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## Material Information- Title:
- Decentralized control of flexible space structures using time varying feedback
- Creator:
- Maben, Egbert N., 1964-
- Publication Date:
- 1996
- Language:
- English
- Physical Description:
- vii, 81 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Automatic control ( jstor )
Damping ( jstor ) Degrees of freedom ( jstor ) Feedback control ( jstor ) Fuel consumption ( jstor ) Governing laws clause ( jstor ) Matrices ( jstor ) Systems design ( jstor ) Vibration ( jstor ) Vibration mode ( jstor ) Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1996.
- Bibliography:
- Includes bibliographical references (leaves 76-80).
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- Also available online.
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- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Egbert N. Maben.
## Record Information- Source Institution:
- University of Florida
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- Copyright Egbert N. Maben. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 35078500 ( OCLC )
023228151 ( ALEPH )
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DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK By EGBERT N. MABEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 @ Copyright 1996 by EGBERT N. MABEN To my parents and teachers ACKNOWLEDGEMENTS I would like to express my sincere gratitude to Dr.David Zimmerman and Dr.Carla Schwartz for helping me in carry out this research work. Without their enormous patience, encouragement and guidance, it would not have been possible to complete the work. I consider myself extremely fortunate to have them as my dissertation advisors. I would like to thank Dr.Norman Fitz-Coy, Dr.Haniph Latchman and Dr.Jacob Hammer for serving in my committee. I would like to acknowledge the help provided by the "Project Care" team, specially Dr.Michael Conlon, Dr.Marilou Behnke, Dr.Fonda Eyler Davis and Kathie Wobie, by providing me with finacial support during my stay at the University of Florida. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS .......... .................. iv ABSTRACT ...... .......... .... .............. vii CHAPTERS 1 INTRODUCTION .. .......... .... ............... 1 1.1 Decentralized Control of Large Systems ................. 1 1.2 Advantage of Using Decentralized Control ............... 2 1.3 Decentralized Stabilization and Pole Placement ........... 7 1.4 An illustration to demonstrate the advantages of decentralized structural control ...................... ..... .... 11 1.5 Objective of the Present Study .................. .. 13 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 16 2.1 Introduction ...... ........... .............. 16 2.2 Controlling Vibrations Using On-Off Controllers Without External Energ ..... . .... . . .. . . . 17 2.2.1 Switching procedure ................... .... 18 2.2.2 Procedure for switching with one controller .......... 20 2.2.3 Example ............................. 20 2.3 Summary .............. ... .............. 22 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING . 25 3.1 Introduction ................... ........... 25 3.2 Designing On-Off Controller for SDOF system ............ 25 3.2.1 Analytical Results ................... ..... 25 3.2.2 Proposition ......... ................... 28 3.2.3 SDOF Example ................... ... ... 28 3.3 Designing On-Off Controll for an MDOF system ..... ....... 29 3.3.1 Modified Independent Modal Control Methods ........ 29 V 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems ................. ........... 32 3.3.3 Four Degree of Freedom example with damping in one mode 36 3.3.4 Four Degree of Freedom example with damping in all modes 37 3.3.5 Kabe's Problem With damping in only two modes ...... 38 3.3.6 Kabe's Problem With damping in all modes ........ 58 3.4 Summary ................................. 63 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL ........ 64 4.1 Introduction ................... ... ......... 64 4.2 Minimum Fuel Control of Mechanical Systems ............. 64 4.3 Effects of changes in window size and sampling period ......... 65 4.3.1 Example 1 ................... ........... 65 4.3.2 Example 2 ................... ........ 69 4.4 Summary ................... .......... ..69 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK ...... 74 5.1 Discussion ........ ... .... ... .. ....... 74 5.2 Future Research ................... .. ...... .. 74 REFERENCES ......... .. ..... . ........ ...... 76 BIOGRAPHICAL SKETCH ................... ........ 81 vi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK By EGBERT N. MABEN May 1996 Chairman: David Zimmerman Co Chairman: Carla Schwartz Major Department: Aerospace Engineering, Mechanics, and Engineering Science New energy-based methods are proposed for improving the performance of flexible structures to external disturbances. The schemes are based on decentralized control methods, which make them appealing for the solutions of large structural control problems. The proposed on-off controller schemes use the work done on the actuators for switching. Energy consumption to damp out the vibrations is minimized by switching the controllers ON only when the work done on them by the structure is positive. Design methodology is proposed to design the controllers to achieve desired damping in vibrating modes of the structure. Simulation results show that the proposed scheme is effective in introducing damping into low or undamped modes of the system. A method similar to the Independent Modal Space Control Scheme is used to get desired damping in modes of vibration using on-off control. vii CHAPTER 1 INTRODUCTION 1.1 Decentralized Control of Large Systems With the space shuttle transportation system now a practical reality, there is considerable interest in large space structures. Two problems which are inherent in the control of large space structures are shape control and attitude control. The latter involves maintaining a given orientation of the spacecraft with regards to an inertial reference system, while the former involves the shape of critical components of the space structures such as a phased array antenna. In both cases, structural flexibility plays important roles since the translational and attitude motions are coupled with the structural vibrations. The motion of a Large Scale Flexible Structure(LFSS) is usually modeled via finite element methods, which can result in very large order models. This means that the order of the system may be extremely high, which can make the tasks of control system design challenging. Model reduction methods that are applied in many cases (Inman, 1989) to control only a subset of the elastic body modes may lead to spillover problems, in which the control effect in stabilizing the subset of modes may cause instability in the uncontrolled modes. The approaches considered to date for investigating the LFSS control have generally been directed towards "centralized control." e.g. model reduction methods (Hughes & Skelton. 1981), modal control methods (Balas, 1978b), output feedback 1 control (Lin. 1981), and adaptive control methods (Behnhabib, Iwens, & Jackson, 1979). Two good survey papers deal with the problem of LFSS control (Gran & Rossi, 1979; Balas, 1982). VWest-vukovich and Davison (1984) have shown how the design of active structural controllers that emulate the real structural elements such as dampers and springs can produce effective control laws. A suboptimal control approach is used in that work to design the structural characteristics (gains) of the controller elements. This allows integration of passive controller, active controller and sensor/actuator locations design. This method exhibits problems due to delays introduced due to digital implementation, and consideration of transfer function of real sensors and actuators. Definition. Fixed modes of the system are modes of the system that remain invariant under any nonzero parametric variation of the system or the controller. The fixed modes of a LFSS result when a sensor and actuator are located at a node of a flexural mode. It was shown that for LFSS with col-located actuators and sensors, the fixed modes of the decentralized system are the same as the centralized fixed modes of the system. A controller that will eliminate the spillover problem is demonstrated. It is also shown that a solution to the decentralized control problem exists only if there is a solution to the centralized system. A further explanation of fixed modes is provided in section 3. 1.2 Advantage of Using Decentralized Control Over the past twenty-five years, engineers and scientists have developed a variety of procedures for analyzing systems and for designing control strategies for controlling LFSS. These procedures are classified into three types: 1. Procedures for modeling dynamical systems (state space formulation, inputoutput transfer function description. etc) 3 2. Procedures for describing the qualitative properties of system behavior (controllability, stability, observability etc.) 3. Procedures for controlling system behavior (stabilizing feedback, optimal control, etc) All of these procedures rest on the common presupposition of centrality; all the information available about the system, and the calculations based upon this information, are centralized. It is useful to distinguish between two kinds of available information: 1. Information about the system model: (off -line or a priori information) 2. Sensor information about the system response: set of all real time measurements made of system response. The common modern estimation and control is based on centralized control theory. When considering large scale systems, the centralized control assumption fails due either to the lack of centralized information or the lack of centralized computing capability. There are many examples of large scale systems that present a great challenge to both analysts and control system designers. Examples include power systems, urban Traffic networks, digital communication networks, flexible manufacturing networks and economic systems. The control of such physical systems are often characterized by spatial separation (such as physical separation between sensors and actuators) so that issues such as the economic cost and reliability of the communication link have to be taken into account in control, thus providing an impetus to decentralized control design. Research in decentralized control has been motivated by the inadequacy of modern control theory to deal with certain issues which are of concern in large scale systems. A key concept in modern control theory is that of state feedback. By using techniques such as Linear Quadratic (LQ) optimal control or pole placement, it is possible to achieve improved system behavior by using state feedback. However, it is often impossible to instrument a system to the extent required for full state feedback, so techniques ranging from linear-quadratic Gaussian (LQG) control, to observer based control to time domain compensator design techniques have been employed to overcome this difficulty. However, a key concept in these types of designs is that every sensor output affects every actuator input. This situation is termed as centralized control. In many large scale systems, it is impossible to implement this in real-time. Thus for economic and possibly reliability reasons, there is a trend for decentralized decision making, distributed computing and hierarchical control. However, these desirable goals of structuring a distributed information and decision framework for large scale systems do not "mesh" with the available centralized methodologies and tools of modern control system design. Reduction of computation and simplification of structure are of particular concern in decentralized control of large scale systems, but are also of concern in almost all areas of control theory and its applications. The basic characteristic of decentralized control is that there are restrictions on the information transfer between certain groups of sensors and actuators. For example, in figure 1, state variables X1 are used to form the control U and state variables X2 are used to form the control U2. This depicts total decentralization. However, intermediate restrictions on the information between controllers are also possible. Partial decentralization takes place when the system is not fully decentralized but the rate of information transfer is constrained so that full centralized control is not possible. It is to be noted that the concept of decentralization refers to the control structure implementation; the control laws may be designed in a completely centralized way (provided there are no other physical limitations preventing this). The advantages of using decentralized control were mentioned in the Ph.D Thesis of Ahmed Tarras and by several other researchers (Tarras, 1987, West-vukovich & 5 SYSTEM CONTROLLER I CONTROLLER 2 Partial Information Figure 1.1. A decentralized system Davison, 1984). The main advantages of using decentralized control laws are to overcome the problems due to the complexity of the system that results from 1. Large Dimensions 2. Uncertainty- deterministic or stochastic 3. Structural Constraints- these make the flow of information between the subsystems difficult. In controlling the complex systems, massive calculations, expensive computer time and high computer costs may be encountered. In addition, there may be large storage requirements, and multiple criteria may be encountered (due to spatial separations, different levels of operation, etc). Definition. A control system is called decentralized, if and only if all local controls are calculated only as an explicit function of the local information (states, output etc). This is a control with one level and no single controller has an overall view of the entire process. In centralized control, one viable approach to design of feedback control laws for time invariant linear systems is by minimization of an infinite horizon quadratic performance index. LQ design method allows asymptotic pole placement by appropriate choice of the performance index (Kwakarnaak & Sivan, 1972) has excellent sensitivity and robustness properties according to various criteria, including the classical gain 6 and phase margins (Sofonov & Athans, 1977), and handles multi-loop problems in exactly the same framework as single loop problems. These results generally extend to the stochastic(LQG) case, in which the disturbances are modeled by passing white Gaussian noise through finite dimensional filters, and in which only partial state information is available. However, the solution of the LQG problem requires that a Kalman filter be implemented to reconstruct the missing state variables. Thus, an additional Riccati equation must be solved. The modes of the filter will appear in the closed-loop response of the system, and the gain and phase margins will be more sensitive and less robust than a corresponding LQ design. In nonclassical stochastic control problems, where the information available to the device controlling a channel input is different from that of the device observing the channel output, the input has a dual purpose: communication through the system dynamics and sensors to the other controllers and direct control of the system. From the control sharing point of view, one can speculate that nonlinear solutions in the nonclassical LQG control are due to actions of the decentralized controllers as they try to signal information to one another using the control system as a communication channel. It is impossible to ascertain what portion of a given system input is applied to purposes of signaling and what portion for control. One can question the desirability of using nonlinear signaling strategies to communicate through the control system dynamics on several grounds. Implementation of such scheme would be extremely complex and performance would be quite sensitive to system parameter variations. In any case, determination of these signaling strategies have been shown to be equivalent to an infinite dimensional, non-convex optimal control problem with neither analytical nor computational solution likely to be forthcoming (Witsenhausen, 1973). Most research in stochastic decentralized control is concerned with determining the optimal parameters of given decentralized control structure (Date & Chow, 1994; Levine & Athans, 1970). 1.3 Decentralized Stabilization and Pole Placement A fundamental result in modern control theory is that the poles of a controllable system can be arbitrarily assigned (subject to complex pole-pairing constraints) by state feedback. This result has been extended to show that the poles of a closed loop system consisting of controllable and observable linear system with a dynamic compensator can be freely assigned. These results are of great theoretical significance and have served as the basis of practical synthesis procedure. A natural generalization of the pole-placement question arises when the restriction to decentralized feedback control is made. Although several authors had looked at this question,(McFadden, 1967; Aoki, 1972; Corfmat & Morse, 1973) the most distinct results are those of Wang and Davison (1973), Davison (1976) and R.P.Corfmat and A.S.Morse (1976). Here, their results are briefly summarized. For a linear system the problem of decentralized pole placement can be formulated as follows. Consider the linear system N k(t) = Az(t) + Biui(t) (1.3.1) i= 1 yi(t) = Cx(t) (1.3.2) where i = 1,. ... N indexes the input and output variables of the various controllers, x E Rn is the state, ui E Rmy and yi E Rri are the input and output, respectively, of the ith local control station. A is the state matrix, Bi is the input matrix and Ci is the output matrix of control station i. The ith controller employs dynamic compensation of the form ui(t) = Mizi(t) + Fivi(t) + Giv (t) (1.3.3) i(t) = Hizi (t) + Liyi(t) + Rivi(t) (1.3.4) where zi E Ri is the state of the ith feedback controller, vi E Rmi is the ith local external input, and Al,. Fi, Gi, Hi, Li, R, are real constant matrices of appropriate sizes. The decentralized pole placement problem is to find matrices Mi, Fi, Gi, Hi, Li, R such that the closed loop system described by (1.3.1)-(1.3.4) has prespecified poles. Of course, if [Ci, A, Bil is controllable and observable from all the stations, the solution to such problem is guaranteed. The interesting case is to assume that (1.3.2) is controllable from all controls ui,......... uN but not from any single control ui, with similar observability assumptions. Consider first the special case Ili = 0 in the above problem. This corresponds to a static decentralized output feedback controller. If F denotes the collection of feedback matrices (FI, F2,..... Fy), then the pole placement problem is to determine F such that the matrix N AF A.4 + BiFC, (1.3.5) i=1 has an arbitrarily specified set of eigenvalues. Clearly, a necessary condition for pole placement in this case is that the polynomials I AI AF I have no common factor, i.e. that ac(A) 1 g.c.d. I AI AF I= 1 (1.3.6) where g.c.d. is greatest common divisor. What is more interesting is that this condition is both necessary and sufficient for pole placement with dynamic compensation(Wang & Davison, 1973). More generally, since the zeros of a (A) (termed as the fixed modes of the system) are invariant under decentralized dynamic compensation, it follows that a necessary and sufficient condition for stabilizability is that the roots of a (A) have strictly negative real parts. Computation of the fixed modes of a system can be accomplished by computing the eigenvalues of AF and A. for randomly selected F and checking for common eigenvalues. A simpler way of checking for the fixed modes of the system is given in 9 Anderson and Moore (1981). Implicit in the pole placement result quoted above is a constructive algorithm. This algorithm requires as a first step the selection of F such that the poles of AF are distinct from that of A. Then, the dynamic feedback is successively employed at the control stations to place the poles that are controllable and observable from a given station. R.P.Corfmat and A.S.Morse (1976) have studied the decentralized feedback control problem from the point of view of determining a more complete characterization of conditions for stabilizablity and pole placement. Their basic approach is to determine conditions under which a system of the form (1.3.1)-(1.3.4) can be made controllable and observable from the input and output variables of a given system by applying static feedback to other controllers. Then dynamic compensation can be employed at this controller in a standard way to place the poles of the system. It is not hard to see that a necessary and sufficient condition to make (1.3.1)(1.3.4) controllable and observable from a single controller is that none of the transfer functions Gij(s) = Ci(sl- A)B i,j ...... 1, ,N (1.3.7) vanish identically. A system satisfying this condition is termed strongly connected. If a system is not strongly connected, it is impossible to make the system controllable and observable from a single controller. In this case, it is necessary to decompose the system into a set of strongly connected subsystems and to make each subsystem controllable and observable from one of its controllers. For a strongly connected system, Corfmat and Morse have given a highly interesting and rather intuitive condition that is necessary and sufficient to make (1.3.1)-(1.3.4) controllable and observable from a single controller. They have shown that if a strongly connected system can be made controllable and observable from a single controller, it can be made controllable and observable from any controller, and a necessary and sufficient 10 condition for this to occur is that the system be complete. Completeness of the system is defined in terms of the transmission polynomials of the system. For example, in case N = 2, a test for completeness (Morse, 1973) is that for all s rank (sI-A)C B1] > 2, (1.3.8) since rank (sI A) > 2 (1.3.9) except when s is an eigenvalue of A. It is only necessary to check the above condition on the spectrum of A. Anderson and Moore (1981) suggested that time varying controllers can be used to eliminate the fixed modes. In their work, they showed that under the assumption of centralized controllability and observability, for a two channel system with feedback of the form: u2(t) = K2(t)y2(t) where K2(t) is periodic and piecewise constant, taking p 1 + max (dim(u2), dim(y2)) values, then strong connectivity, even when a fixed mode is present, is enough to ensure that the system is uniformly controllable from ul and uniformly observable from yl. Most of the work in the decentralized control schemes with time varying controller focuses on making the system controllable and observable and/or on robust control. The concept is to develop an algorithm to place the poles of the systems using time varying controllers. Once this algorithm is available, fixed modes ( the modes of the system which are not affected by TIV controller gains) of the system can be treated the way the non-fixed modes are treated (Wang, 1982). The concept of using time varying periodic compensators with time invariant plants was further investigated by Khargonekar, Poolla, and Tannenbaum (1985). In their work, they showed that for a large class of robustness problems, periodic compensators are superior to time invariant ones. They also gave design techniques which 11 can be implemented easily. They also showed that for weighted sensitivity minimization for linear time invariant plants, time varying controllers have no advantages over time invariant ones. However, the eigenstructure assignment problem was beyond the scope of Poolla's work. Their design was used to design a stabilizing compensator and later, the time varying controller and time invariant plant combinations were used to generate the time response of the system. Identification packages were used to identify this system and separate compensator was designed for pole placement. The order of the system seems to be going higher and higher in this case. Nevertheless, this looks like a first pass solution to the problem of eigenstructure assignment. 1.4 An illustration to demonstrate the advantages of decentralized structural control This example shows the advantages and simplicity of building a decentralized controller for structures. Figure 1.4 shows a 6-bay truss whose model is taken from Young (1990). This truss is modeled as a coupled structure of three substructures. Substructure boundaries are marked in the figure. The truss member mass and stiffness matrices expressed with respect to the local coordinates and used in the assembly process are EA mL[ I 21 Ketem= L -1 6 1 2 The nodal coordinates are defined as vertical and horizontal displacements at the joints. The internal degrees of freedom at which the actuator and displacement sensors are placed are marked in the figure. The finite element model of the individual substructures are obtained using a FORTRAN program and validated using the common FEM package ANSYS (Swanson Analysis Inc, 1992). For convenience, the material properties are assumed to be have unit magnitudes. An Interlocking control design 12 concept is used to place the actuators and sensors at internal degrees of freedom. Minimization of internal coordinate motions of different substructures would localize the dynamic interaction of coupled structure in the components. The component control action is designed to lock up its own boundary condition, that better approximates the one assumed in the component modeling of its adjacent components. 62 6 I 5 9 7 11 u 1.y u2.y2 u3.y3 xb2 xbl Figure 1.2. A Six-Bay Truss A convenient control design technique for this concept is the linear quadratic optimal control regulator approach, in which the aim is to minimize the internal coordinates motion. The component control law is the one that minimizes the performance index, 1 oa J8 = 2j (YsTys + u'TRu') dt where y" and us denote the output and control inputs at the internal degrees of freedom of each substructure. These are marked in the figure by numbers. The optimal component control is a state feedback control law with positions and velocities as inputs. This type of controller is built for individual substructures and the controlled coupled structures state matrix is obtained. The effect of having such a controller on the eigenvalues is studied. Failures in the component controllers by changing the elements of the gain matrices of individual substructures. The gains of some channels 13 were set to zero ( a scenario mimicking a break in communication line) and the effect of this on the overall eigenvalues were studied. With the decentralized control, it was observed that all the poles of the coupled structure have negative real parts. The individual controller failures don't seem to affect the pole locations very much. This of course depends on the type of controller failure that we simulate. Open loop poles of the substructures and coupled structure are plotted in Figure(1.4). In table 1. Values of the coupled loop poles without and with controller failures are listed. 1.6 1.4 0 c. 0.8 0.6 t 0 5 10 15 20 25 30 Mode number Figure 1.3. Open loop poles of substructures and coupled structure 1.5 Objective of the Present Study The present study investigates the development of new and promising energy efficient ways of controlling vibrations in mechanical structures. Although considerable research has been clone in vibration control of flexible structures, no method uses 14 Without Component With End controlled With Center Controlled failure Component Failure Component Failure -0.0005 + 0.1877i -0.3e-8 + 0.1876i -0.0005 + 0.1877i -0.0005 + 0.1877i -0.0005 + 0.1877i -0.0005 + 0.1877i -0.0037 + 0.2515i -0.0037 + 0.2515i -0.0003 + 0.2514i -0.0007 + 0.2679i -0.0004 + 0.2679i -0.0003 + 0.2679i -0.0101 + 0.6109i -0.0004 + 0.6092i -0.0097 + 0.6106i -0.0093 + 0.6117i -0.0093 + 0.6117i -0.0093 + 0.6117i -0.0054 + 0.6437i -0.0024 + 0.6428i -0.0035 + 0.6441i -0.0059 + 0.6445i -0.0042 + 0.6448i -0.0027 + 0.6441i -0.0044 + 0.6488i -0.0036 + 0.6489i -0.0030 + 0.6485i -0.0029 + 0.6563i -0.0028 + 0.6563i -0.0023 + 0.6562i -0.0059 + 0.6909i -0.0001 + 0.6903i -0.0058 + 0.6907i -0.0055 + 0.6910i -0.0056 + 0.6910i -0.0055 + 0.6910i -0.0196 + 1.0343i -0.0181 + 1.0343i -0.0041 + 1.0309i -0.0062 + 1.0600i -0.0029 + 1.0587i -0.0047 + 1.0596i -0.0045 + 1.2389i 0.0000 + 1.2375i -0.0044 + 1.2388i -0.0043 + 1.2389i -0.0044 + 1.2389i -0.0043 + 1.2389i -0.6676 + 2.3206i -0.6674 + 2.3318i -0.6991 + 2.4474i -0.6813 + 2.3933i -0.6932 + 2.4306i -0.6997 + 2.4557i -0.7125 + 2.4606i -0.7125 + 2.4642i -0.6914 + 2.4778i -0.7252 + 2.4778i -0.0058 + 2.5418i -0.0126 + 2.4818i -0.7066 + 2.4807i -0.7247 + 2.5627i -0.7084 + 2.4821i -0.7325 + 2.5748i -0.0044 + 2.5898i -0.0162 + 2.5758i -1.1823 + 2.8937i -1.1842 + 2.9214i -1.2102 + 2.9729i -1.2108 + 2.9733i -1.2149 + 2.9781i -1.2061 + 2.9779i -1.2238 + 3.0177i -0.0079 + 3.2286i -0.0085 + 3.1672i -2.2575 + 3.7678i -2.2922 + 3.7970i -2.3014 + 3.8008i -2.3022 + 3.8011i -2.3427 + 3.8591i -2.3012 + 3.8040i -2.3828 + 3.8887i -0.0032 + 4.4488i -0.0067 + 4.5129i Table 1.1. Eignevalues of the 6-bay truss example energy transfer from the structure to the controller as a criterion to decide on-off times. The main objective is to demonstrate that such a method is viable and to give a controller design methodology. In Chapter 2. analytical results are presented to demonstrate the use of on-off controllers to damp the vibrations in an energy efficient way. Energy input into the controller is used as a criterion to select the switching times. The controllers are switched on only when work is (one on them. This does not require the use of an 15 external energy source. Since energy supplied to the controller would require that fuel be consumed, the proposed design methodology reduces the fuel consumption for vibration suppression. Additionally, the proposed control laws are decentralized since the on-off switching laws are based on the local displacements and velocities. In Chapter 3, controller design methodologies are given to select the controller parameters to achieve the desired modal damping. Analytical results are given to prove the validity of this. A method which is similar to the Modified Independent Modal Space Control (MIMSC) is used for designing on-off controller for multi degree of freedom systems. Simulation results are presented to show the effectiveness and accuracy of this method. In Chapter 4, the use of optimal control laws with fuel consumption as a cost criterion to design vibration controllers is presented. First, the concept of on-off vibration control is reviewed with an emphasis on fuel consumption minimization. Next, the problems associated with window sizing and selecting the sampling interval are brought to light. A discussion of energy efficient switching control laws future research problems are presented in Chapter 5. CHAPTER 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 2.1 Introduction In this work a minimum energy control method for on-off decentralized control of mechanical systems is introduced. Energy consumption is minimized by turning on the controllers when the structure does work on the controller and turning them off when the controller would impart energy on the structure. Vibrating energy from lightly damped modes is transferred to highly damped modes to introduce damping. In the next chapter, this method is applied to solve pole placement problems for vibrating systems. The work presented here is based on the intuition that using an energy minimization criterion that works with conservation principles will produce energy efficient methods for control of vibrating systems. The methodology presented here is quite feasible, since the switching of the electro-mechanical controller is performed electronically via an on-board processor. The development uses energy as an optimization criterion. The delays introduced by the decision making are not included, but can be very easily incorporated into the analytical results presented here. Several authors used or proposed the use of on-off control for attitude and shape control of large flexible space structures ( Velde and He (1983), Foster and Silverberg (1991), Arbel and Gupta (1981). Masri. Bekey, and Caughey (1981), Rohman and Leipholz (1978), Seywald, Kumar. Deshpande, and Heck (1994), Redmond and 16 17 Silverberg (1992). Neustadt (1960) and Reinhorn. Soong, Riley, Lin, Aizawa, and Higashino (1993)). The approaches using on-off control devices are relatively simple and require less on-line computational effort than other modern control techniques. High energy pulses can be input into the system using on-off devices. In the works of Zimmerman. Inman, and Juang (1991), Zimmerman (1990), Masri et al. (1981), in order to conserve energy, control devices are activated when some specified thresholds of states has been exceeded. The pulse magnitude was determined analytically so as to minimize a non-negative cost function related to the system energy. The degree of control obtained depended on the threshold level considered and the cost function that was minimized. Foster and Silverberg (1991), Arbel and Gupta (1981), Masri et al. (1981), Seywald et al. (1994), Redmond and Silverberg (1992) and Neustadt (1960) used the principle of fuel minimization as a criterion for the implementation of on-off switching control. Flight duration of spacecraft is limited in many cases by the amount of on-board fuel. For satellites, on-board fuel is needed to fire the thrusters and apogee/perigee motors to place them in the final orbit. Thruster firings will be needed for station-keeping and to correct orbital errors. As the size of spacecraft increases, the need to suppress the vibrations introduced into the spacecraft body must also be incorporated into the controller design. The work described in this chapter differs from previous works in that, here, fuel consumption is measured by the amount of external energy needed to damp out the vibrations. More discussions about the proposed scheme and the types of controllers that may be used with the proposed scheme may be found in the next two sections. 2.2 Controlling Vibrations Using On-Off Controllers Without External Energy In general. when subject to vibrational control, a structure does work on its controller and vice-versa. Here it is proposed that the work done on the controllers by the 18 vibrating structure be taken advantage of to arrive at a fuel-optimal solution. This is accomplished by switching the controller "on" only when the structure is doing work on the controller and switching it "off" otherwise. 2.2.1 Switching procedure In this section it is demonstrated how an on-off control scheme using only the energy input rate into the controller can get a better response from the system. Preliminary versions of this work are presented in Schwartz and Maben (1996, 1995). Consider an n-dimensional second order dynamical system of the form M + D + Kx = 0, (2.2.1) where M, D, K are the mass, damping and stiffness matrices respectively. D is assumed to be of proportional type, i.e. D satisfies D = aM + 3K, where a, / are scalar constants. Also, K = KT > 0 and M = MT > 0. When the controller is off, the energy in the system is given by Eo f = 2 [TM + zTKx iJ Dx]. (2.2.2) Since the system is dissipative, the rate of change of energy is given by Eoff = D[M + (K)x] 'TD. (2.2.3) If the controller is of the spring type, the rate of change of energy while the controller is on is given by Eon = 'T[MS + (K + AK)x] ITD, (2.2.4) where AK is the added stiffness due to the controller. It is assumed that AK is symmetric and positive definite. In this study we are not concerned with measuring the energy stored in a controlling device. Rather we are interested in studying the energy input into such a 19 device by observing the rate of change of energy function. Our aim is to increase the energy into such devices (on the assumption that a unidirectional force on a spring type of controller can be converted into some other form of energy and released into an energy dissipating or storage device and not released back into the structure). The controllers are thus switched on only when this energy rate is positive. It should also be noted that x in all these cases represents the displacement over a bias value, measured in reference to an inertial level. The signal Eo, Eoff = Ti'AKx gives a measure of the rate of work done on all controllers if they were on continuously. In the case of one controller, if AKa is the stiffness introduced by the controller, xa is the relative displacement and a is the velocity at the controller location, E~ Eoff is positive (work is done on the controller by the structure), if the product of force acting on the controller AKaz and the velocity at the controller location a is positive. In the case of multiple controllers, if i denotes the location of the ith controller, [AKixi] is monitored and each controller is switched on when the force signal acting on it and the velocity at the controller location are positive. The force acting on the elements of AK matrix can be positive or negative. Assuming that the force in the positive direction (representing compression in spring type of controller) can be released into some other energy dissipating device (for example a resistor network), controllers are switched on only when the forces acting on them are positive. This concept is summarized in the following switching procedure. It should be noted that the location of the controller plays a role in the controllability of the vibrating modes of the structure. If the location of an controller is close to the node of a vibrating mode, that controller will have very little effect on the damping of that particular mode of vibration. 20 2.2.2 Procedure for switching with one controller 1. Between switching times, monitor the signals AKaxa and di where AKa is the nonzero entry on the AK matrix. 2. When both AKaX, and x, are positive, switch the controller on. Switch the controller off at the end of a predetermined time which is related to the system time constant as determined by the lowest period of oscillation in the structure.' (and release the energy absorbed by the spring when it is switched off). 2.2.3 Example Consider a 5 degree of freedom spring mass system shown in Figure 2.2.3. Here we have a second order system, AMf+Ci+Kx = f where the mass and stiffness matrices 1 0 0 0 0 3 -1 0 0 0 0 1 0 0 0 -1 2 -1 0 0 are chosen as, mass= 0 0 1 0 0 stif= 0 -1 2 -1 0 0 0 0 1 0 0 0 -1 2 -1 0 0 0 0 1 0 0 0 -1 3 X r3 m1 2 Figure 2.1. A 5 DOF Spring/Mass System In this example xm is x4. All but the fourth mode have very little damping (equivalent damping factor of 0.002), while the fourth mode is highly damped(equivalent damping factor of 0.4). The simulations of the control procedure of Section 2.2.2 show this method to be very effective in damping out the vibrations of the measured 'In the example in the next subsection, this time is set to be one tenth of lowest period of oscillation. 21 response even when the initial conditions were very close to the mode shapes of a lesser damped mode of the uncontrolled structure. a worst case scenario. Figures 2.2 through 2.6 show the nodal (position) response of the uncontrolled and controlled system when the initial conditions were set to the mode shape of one of the lesser damped mode of the uncontrolled structure. The results are only shown for one set of initial conditions, however the simulation was carried out for each of the mode shapes. From these plots it is clear that the proposed method is effective in increasing the damping. 0.5 0 -0.5 0 50 100 150 200 250 300 O -0.5 -1 0 50 100 !50 200 250 300 Time in seconds Figure 2.2. Response at node 1 22 0.5 -0.5 0 50 100 150 200 250 300 Time in seonds Figure 2.3. Response at node 2 0.5 the structure. 0 50 100 150 200 250 300 Time in seconds 0.5 9-0.5 -1.5 0 50 100 150 200 250 300 Time in 9xond, Figure 2.4. Response at node 3 2.3 Summary A minimum energy on-off vibration control law for structures was introduced. The control procedure takes advantage of the energy which can be imparted to the controllers from switching by turning on the controllers when the structure is doing work on the controller and turning them off when the controllers are doing work on the structure. 23 o0.5 -0.5 0 50 100 150 200 250 300 Time in s.onds 0.5 -1 0 50 100 150 200 250 300 Time in seconds Figure 2.5. Response at node 4 Time in seconds 0.5 -1 0 50 100 150 200 250 300 Time in smcond. Figure 2.6. Response at node 5 The proposed scheme uses the fact that a structure does work on the controllers and vice versa while undergoing vibrations. The controllers are switched on only when they are absorbing energy from the structure. This absorbed energy can be transformed into some other form (to do some useful work like storing energy in a device like battery or gyro). The proposed scheme achieves the desired goal without the use of an external power source (except for decision making and measurements). In one application which gives promise to the proposed scheme, Yang, Mikulas. Park, 24 and Su (1995) have shown that controlled moment gyros can be used for the slewing maneuvers and vibration control of space structures. CHAPTER 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING 3.1 Introduction This chapter provides methods for choosing controller parameters AK in order to achieve specified modal damping for single degree of freedom systems (SDOF) and multi degree of freedom systems (MDOF). First, analytical results are presented for SDOF systems. Second, results are given for MDOF systems. Modified Independent Modal Control Methods (MIMC) are applied to MDOF systems to achieve the desired modal damping. It is also shown that introducing damping in only few modes of the uncontrolled system may cause poor responses in other modes. In view of this, it is recommended that all dominant modes of the structure be damped when using the proposed switching control design. 3.2 Designing On-Off Controller for SDOF system 3.2.1 Analytical Results Consider the scalar single of freedom system described by m + kx + pAkx = 0, (3.2.1) where m is the mass, k is the stiffness, Ak is the added stiffness due to the controller action, x is the displacement of the mass, and 0i is a parameter which takes values 1 or 0, depending, respectively, upon whether or not the controller is on. The switching 25 26 control is expected to introduce damping into the system. The controller is assumed to be located at the node associated with this mass and Ak is positive. The control objective is to achieve a specified average damping rate. The following is a design procedure for specifying Aik so as to achieve this objective. The energy in the system is given by E = 1 (m2 + kx2 + pAkx2) (3.2.2) Let w be the natural frequency of the uncontrolled system, (i.e. w2 = k). The transformation iAx = u gives E = -(i2 22 t2 U 2). (3.2.3) By imposing the control scheme described in the previous chapter, CP is set to 1 when t is such that Aku > 0, or equivalently when u > 0. It is expected that the proposed switching control scheme introduces damping so that u is assumed to be of the form: u = a sin(wt + 3), (3.2.4) where a = a(t) is an exponentially decaying function of time, and w is the uncontrolled natural frequency of oscillation. A functional relationship between the rate of decay function a and the feedback gain Ak will be used to determine a law for introducing a desired average damping into the system. Taking the time derivative of both sides of equation (3.2.4), it = awcos(wt + 3) + &sin(wt + [3). (3.2.5) The decay rate a is assumed to be small compared to aw, so the & term may be neglected in equation (3.2.5). This gives 27 it aw cos(wt + 3). (3.2.6) By conservation of energy, E = 0. Note that this is true at all instances except switching: Thus Sd (iL 2U2) + 22 u) = 0. (3.2.7) Using equations (3.2.6) and (3.2.4) in equation (3.2.7) yields pw2 Ak 2 W2a& + -a w sin 2 (wt + 3) = 0. (3.2.8) 2 k Since the decay rate of the response is assumed to be slow, the method of averaging in (Caughey & O'Kelly, 1965) may be applied to compute &: Ak w rlw a = k a-2 sin 2(wt + 3)dt. (3.2.9) k 20in Since u > 0 holds for only half a cycle of sin 2(wt + 4), p is set to 1 for only half of a cycle of sin 2(wt + /). In that case, the value of the integral in equation (3.2.9) is 1/w and S= a- = Ak (3.2.10) k 27rw k or Ak k a = aoe where k (3.2.11) Thus, for the single degree of freedom case, determines a control law for achieving a desired average damping rate = This result is summarized in the following proposition. 28 3.2.2 Proposition For a single degree of freedom system of the form (3.2.1), on-off switching control may be used to attain a desired overall average damping rate ( using the proposed minimum energy control scheme in conjunction with equation (3.2.11) 3.2.3 SDOF Example The main objective of this example is to show the effectiveness of this method and to show the validity of assumption made earlier, that the decay rate a is assumed to be small compared to aw. Consider the single degree of freedom system with magnitudes of mass and stiffness respectively given by m=l1 and k=100. Simulation of the system with on-off controller was carried out for three values of the damping. For each of these damping, Ak was calculated using equation (3.2.11). The on-off controller was simulated and the response of the system to an initial displacement of 1 and velocity of 0 was observed. The response of the system for values of damping of 0.0015, 0.020 and .6 are given in Figures 3.1, 3.2 and 3.3 respectively. In each of these figures, exponentially decayed curves are plotted to show the exactness of this method for low values of damping. From these time responses, it is concluded that the assumption that a is an exponentially decayed function is valid. Also, it can be seen that when the assumption that 6 << aw is violated, the damping introduced into the system because of switching controller deviates much from the desired value. 29 0.8 exp(-0.015't) 0.6 0 .4 0.2 0 Z -02 -0.4 -O.e 0 5 10 15 20 25 30 35 40 45 50 time in seconds Figure 3.1. Response of SDOF ( = 0.0015 0.8 0.6 0.4 exp(-0.20*t) S0.2 o -0.2 -0.4 -0.6 -0.8 0 5 10 15 20 25 30 35 40 45 50 time in seconds Figure 3.2. Response of SDOF ( = 0.02 3.3 Designing On-Off Controll for an MDOF system 3.3.1 Modified Independent Modal Control Methods Active control of the vibrations of flexible structures based primarily on modal control methods whereby the vibrations are suppressed by controlling the dominant modes of vibration has been the focus of many researchers (Lindberg Jr. & Longman. 30 0.8 0.6 exp(-6.0t) 0.4 o 0 -0.2 -0.4 -0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time in seconds Figure 3.3. Response of SDOF = 0.6 1984; Balas, 1978b, 1978a; Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Canfield & Meirovitch, 1994). Generally these modal control methods belong either to the class of coupled methods or to the class of independent modal space control methods developed by Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Canfield & Meirovitch. 1994). Using coupled methods, the closed loop equations of the system are coupled using feedback control wherein the optimal computation of the feedback gains require the solution of a couple matrix Riccati equation (Balas, 1978b, 1978a). For large flexible structures the solution of the Riccati equation can pose serious difficulties which limit significantly the applicability of the coupled modal control methods. The IMSC method, however, avoids such serious limitations, as the control laws are designed completely in the modal space, using the uncoupled open loop equations of the system as a set of independent second order equations even after including the feedback controllers. Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh. 1983; Canfield & Meirovitch, 1994) showed that 31 under such conditions it is possible to compute the closed form the optimal modal feedback gains. However, the IMSC method requires the use of as many controllers as the number of modes to be controlled. Such a requirement results in a practical limitation of the method when applied to large structures where the number of modeled modes can be very large. Lindberg and Longman (1984) proposed to modify the IMSC by using a small number of controllers to control all the modeled modes through the approximate pseudo-inverse(PI) realization of the modal controller. This modification can result in physical control forces which can be far from desired because the PI is, in effect, a least squares fit of N modal forces to obtain P physical forces. When N equals P the realization is exact and is also the same as IMSC. As N becomes much smaller than P, the accuracy in obtaining P physical forces from N model forces becomes poor. Accordingly, when the realized forces are transformed back to the modal space the resulting modal forces will be very far from the optimal forces and this will result in deterioration in the performance of the controller. For these reasons, the Modified Independent Modal Space Control (MIMSC) method was initiated (Baz & Poh, 1987). The MIMSC modifies the IMSC algorithm to account for the control spillover from the controlled modes into the uncontrolled modes when a small number of controllers is used to control a large number of modes. This method also incorporates an optimal placement procedure for determining the optimal location of the controllers in the structure. Moreover, the MIMSC method relies on an efficient algorithm for "time-sharing" a small number of controllers in the modal space to control a large number of modes. The MIMSC uses N optimally placed controllers to control the N modes that have the highest modal energy at any instant of time and time share these controllers among the other residual modes when the control spillover makes the modal energy higher than the controlled modes. 32 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems In this section., a new design methodology is proposed for using on-off control to achieve desired damping in modes of interest. This method is based on energy transfer (from kinetic to potential), as well as on the dissipative nature of damping forces. Time-sharing of controllers was proposed in (Baz & Poh, 1987, 1991), and these ideas are used here. Consider a second order dynamical system for the n-dimensional displacement vector x given by M.2 + Kx = F. (3.3.12) where l = MT > 0 is the mass matrix of the structure, K = K' T 0 is the stiffness matrix of the structure, and F is n-dimensional vector of forces acting on the structure. Equation (3.3.12) may written in modal coordinates by using the following weighted modal transformation: x = Q
Thus, under the transformation (3.3.13), equation (3.3.12) reduces to the form ii + Au = f, (3.3.14) where A is the diagonal matrix of eigenvalues of M- K matrix and f = 4Q F. The controller being on, yields the dynamics: M.i + Kx = -AK.r. (3.3.15) 33 where (K + AK)T = (K + AK) >_ 0. Under switching control. it is desired that the system behave like a damped dynamical system of the form MYf + D: + Kx = 0, (3.3.16) where D is an n x n desired proportional damping matrix: i.e. D = aM + 3K, where a,3 are scalar constants. Note that the mass and stiffness matrices are the same. The energy stored (or released) in time T due to the added stiffness at the controller is given by EAK j f d (XTAKx) dt. (3.3.17) The energy dissipated due to the damping term D over an interval of length T is given by ED =lTD( dt. (3.3.18) Assuming the controller is only on for T seconds, and using equations (3.3.13) and (3.3.17) the contribution of the ith mode to the energy stored in the controller is 1 Td S -EAKT u [ AK]i ui) dt, (3.3.19) where (i is the ith column of D. The contribution of the ith mode to the energy which would be dissipated in the desired dynamics through the damping over the period Ti is EDi T d(, i4T D[-j iii) dt. (3.3.20) 20 dt Here T, is taken as the period of the ith mode of vibration. Since proportional damping is assumed, fiD()i = 2(iwi, (3.3.21) 34 where (i is the desired modal damping and wi is the natural frequency of the ith mode. If ui is assumed to be of the form ui = e-Citsin (wit), then iti(t) = -(ie-Citsin (wit) + e-Citwicos (wit) (3.3.22) and iii(t) = (ie-C'tsin (wit) 2(ie-(itwicos (wit) e-('tw2sin (wit). (3.3.23) The structure does work on the controller during only half of the cycle of vibration of a given mode and during the other half of the cycle, the controller releases energy into the structure. This logic motivates the scheme for switching the controller on only for half of the period of the mode to be controlled. Using T = T the two 2 integrals (in equations (3.3.18) and (3.3.19)) may be evaluated, and the contribution of mode i to the energy stored in the controller over its period (when the controller is only on for half the period) is equated to the contribution of mode i to the energy dissipated in the damping term over its period. This yields an expression for AK (nonzero elements of AK). Although there is no restriction that AK be diagonal, for simplifying the expressions (and to make them more clear), AK is assumed to be diagonal. Thus ( AKP m = jmAKjjjm. (3.3.24) j such that AKjj34O This depends only on one those entries in the mth column of D, for which the corresponding entries on the diagonal matrix AK are non zero. Take first the case that only one entry in AK is nonzero. Let j mark the index of this entry. Evaluating the derivatives in the integrals of equations (3.3.19) and (3.3.20) (evaluating the first integral for only half the cycle as described earlier) gives EAK. = -j 2 dt (3.3.25) 35 and 2 J (iii2(wi + iii) dt. (3.3.26) Using the expressions given earlier for u, ii. ii. rAK#i, 4TD i, given by equations (3.3.22,3.3.23,3.3.24,3.3.21) define gli, g2i. g3i (the expanded form of the integrand in equation (3.3.25) and terms in the integrand of equation ( 3.3.26)) as gl = (-ie-tsin(wit) + e-Citwicos(wit))e-citsin( it)AKjjfD2 )dt (3.3.27) g2i = j" 2(2 e -isin(wit)+2(- ()e-i cos (w)-e-'wsi(wt)e- tsin(wit)) widt. (3.3.28) g3i = 2(-(i e-('tsin(wit) + e-it'wi cos(wit))2wi Cidt. (3.3.29) Since we are considering the contribution of only one mode at this point, there will be a nonzero contribution from only one diagonal element of AK matrix. Solving gli = g2i +g3i gives an expression for the nonzero diagonal element in the AK matrix as AK C (3.3.30) When there is more than one mode to control, the matrix AK should have only as many non-zero diagonal entries as the number of modes to be controlled. In this case, gli takes the form: f (-Gie-(' sin(wit) + e-it'icos(wit))e-casin(wit)AKfjf 2dt. j such that AKiyO 0 In such cases, the integrand in equation (3.3.25) and equation (3.3.26) will have as many terms as the number of modes to be controlled and as many equations of 36 the form gl = g2i + g32 as are necessary will have to be solved in order to obtain values of different non-zero elements of the AK matrix. With more than one mode to control, there will be m expressions of the form (3.3.25) and m expressions of the form (3.3.26) (one for each of the desired modes to be controlled). This will lead to m equations of the form (3.3.27)-(3.3.29) which can be solved for the m nonzero diagonal entries of AK matrix. The flowchart for the proposed MDOF on-off control scheme is given in Figure 3.4. Note that at any given time, controllers are only set to act on one mode in order to increase its damping. The switching procedure computes the energy in different modes of the structure and checks whether the energy in one of the desired modes is the highest. If it finds that energy in one of the desired modes is taking the highest value, for half the period of vibration of this mode, all the controllers are dedicated to damp the vibrations in this mode (the timing is kept track by the use of a counter T,). During this half cycle of vibration, controllers are individually switched on and off depending upon whether or not work is done on them by the structure. 3.3.3 Four Degree of Freedom example with damping in one mode Consider a four degree of freedom undamped system with mass and stiffness properties F2 0 0 0 Mass = 0 0 (3.3.31) 3 -2 0 0 Stiffness = -2 0 3.3.32) 0 -2 5 -2 0 0 -2 6 37 The controller was placed at the first mass and the damping in the first mode was desired to be 0.2. Unlike the third example (Kabe's problem) in this section, this model had mass normalized eigenvector components of the same order (components of different eigenvectors at nodal locations). Equation 3.3.30 suggests that controller gains are inversely proportional to the eigenvector components. This suggested that placement of controller at any of the masses would have resulted in a value of AK in the same order. The on-off controller scheme proposed in Section 4.3.2 was implemented. Nodal and Modal time responses of the system are given in Figures 3.5 and 3.6 respectively. From the plot of modal responses, it is clear that the control scheme introduces damping in first mode. The value of this damping is found to be 0.19. Figure 3.7 gives the energy in different modes and Figure 3.8 gives the velocity and modal force for the first mode of vibration. From Figure 3.8 it is clear that the control scheme introduces damping into first mode. 3.3.4 Four Degree of Freedom example with damping in all modes In this example, controller parameters used in the simulation are such that all four modes of the four DOF system given in equations (3.3.31) and (3.3.32) exhibit the desired damping. The desired damping values in the four modes are .2,.15,0.20 and 0.17. The natural frequencies of the uncontrolled system are 0.556, 0.9657, 1.2791 and 1.5511 radians. The system was simulated and the algorithm given in the previous section was implemented. It was observed that system behaves like a well damped system with modal damping very close to the desired values. This was simulated with several initial conditions. Figures 3.9-3.12 give the nodal response and Figures 3.133.16 give the modal response of this system. The observed modal damping values are 0.18, 0.14. 0.18 and 0.16. The natural frequencies of the system with control acting upon it are identified as 0.7454, 0.9827, 1.1310 and 1.2454 radians/second respectively, matching exactly with the natural frequencies of the uncontrolled system. 38 3.3.5 Kabe's Problem With damping in only two modes Kabe's eight degree of freedom model is shown in Figure 3.17. The natural frequencies of this uncontrolled system are 30.0280, 31.7163, 31.7557, 32.3020, 33.3267, 35.5763, 38.7887 and 41.8884 radians/second respectively. The mass-normalized eigenvectors of this system are listed in Table 4.1. In order to avoid high values of controller gains, the controllers were placed at 4th and 6th masses. The controller locations were selected based upon the mass normalized eigenvector components. The proposed design procedure shows that the reciprocal of the square of the eigenvector component at the location of the controller decides the gain gain of the controller (i.e. if damping is desired in mode j and controller is placed at location g, reciprocal of the square of the gth component of the jth mass normalized eigenvector decides the controller gain). Note that the fourth and sixth mass normalized eigenvector have fourth and sixth components which are within reasonable limits (leading to reasonable values for the elements of AK matrix. The desired damping in the fourth and sixth mode were set to 0.10 and 0.15, respectively. The initial conditions were set such that the initial energy was concentrated in the fourth and sixth modes. When these conditions were simulated, it was seen that the control algorithm achieves the desired goals within reasonable limits. Figures 3.18-3.25 give the nodal response and Figures 3.26-3.25 give the modal response of this system. In order to show the importance of the controller locations, the same system was simulated with the controllers located at the first and the second mass. The desired damping in fourth mode and sixth mode were set to 0.1 and 0.15. Controller gains in this case were 42 and 38166 compared to 2724 and 3486 in the earlier case. Dampings achieved in this case were .06007 and 0.1076 in mode 4 and 6 compared to the values of 0.08 and 0.14 in the previous case. In order to implement the proposed scheme, the controller gains would have to be such that they do not significantly affect the mass of the system. If the 39 size and weight of the controllers change the mass of the system the effects of this will have to be taken into consideration. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 -2.1170 -30.0002 -2.5940 -10.6279 -5.6896 -2.6102 -31.6226 0.0002 -0.0267 -0.3125 -0.0269 -0.1023 -0.0467 -0.0129 0.0030 0.00001 -0.2832 -0.0319 0.0040 0.3769 0.4901 0.3401 -0.0001 0.0001 -0.5873 -0.0020 0.0068 0.2612 0.0014 -0.5282 0.0001 -0.0005 -0.5973 0.0329 -0.0806 -0.6016 -0.0013 0.1988 0.0001 0.0001 -0.2841 -0.0029 0.0827 0.3724 -0.4890 0.3361 0.0001 0.0032 -0.0308 -0.0085 0.6378 -0.1347 0.0598 -0.0172 0.0001 0.0020 -8.0992 -0.2782 16.9175 8.5882 -15.5332 14.9251 -0.0005 -22.3605 Table 3.1. Mass normalized eigenvectors of Kabe's model 40 INPUT STRUCTURAL PARAMETERS, CONTROLLER LOCATION AND LOADING Compute time history of all nodes in physical coordinates Compute time history of all nodes in modal coordinates Compute energy in all modes. Controller F No Find max Energy in desired mode j On? Yes Ys Is energy in No et Controller fla Tc > T desired mode j off. Reset T.sired mode highest? No Yes Switch the controllers on if the Store j. work done on them is positive. Set Controller flag on. Figure 3.4. Flowchart of MDOF damping control algorithm 41 2 2 0 0 S0 0 0 0 o o z-1 z-1 -2 -2 0 10 20 30 40 0 10 20 30 40 time in seconds time in seconds 1.5 1.5 C 1 1 .o 0 0.5 0 0.5 (. 0.. o 0 0 0 z -0.5 z -0.5 -1 -1 0 10 20 30 40 0 10 20 30 40 time in seconds time in seconds Figure 3.5. Nodal Response of a 4-DOF System with (ld,, = 0.2 42 0.5 1 a 2 0 0 -0.5 -1 -1 -2 0 10 20 30 40 0 10 20 30 40 Time in seconds Time in seconds 3 4 ca 1 0 o Mo -2 -1 -2 -4 0 10 20 30 40 0 10 20 30 40 Time in seconds Time in seconds Figure 3.6. Modal Response of a 4-DOF system with (ides = 0.2 43 0.2 0.2 -0.15 -0.15 C w w S0.1C m 0.1 0 0 M 0.05 M 0.05 0 0 0 10 20 30 40 0 10 20 30 40 Time in seconds Time in seconds 0.8 2 >0.6 1.5 S0.4 1 o o 20.2 2 0.5 0 0 0 10 20 30 40 0 10 20 30 40 Time in seconds Time in seconds Figure 3.7. Energy in different modes of a 4-DOF system with (lde, = 0.2 1.5 0 -0.5 -2 0 5 10 15 20 25 30 35 40 Time in Seconds Figure 3.8. Modal force and velocity (for the first mode) 44 0.8 0.6 0.4 ~ 0.2 -0.2 -0.4 0 10 20 30 40 50 60 70 80 90 100 Time in Seconds Figure 3.9. Response at Node 1 of a 4-DOF system (with all modes controlled) 0.8 0.6 0.4 C 0.2 NO 0 S-0.2 -0.4 -0.6 -0.8 0 10 20 30 40 50 60 70 80 90 100 Time in Seconds Figure 3.10. Response at Node 2 of a 4-DOF system (with all modes controlled) 45 0.5 0 0 C -0.5 X -1 -1.5 0 10 20 30 40 50 60 70 80 90 100 Time in Seconds Figure 3.11. Response at Node 3 of a 4-DOF system (with all modes controlled) 1.5 0.5 S-10 -0.5 S-1 X -1.5 -2.5 0 10 20 30 40 50 60 70 80 90 100 Time in Seconds Figure 3.12. Response at Node 4 of a 4-DOF system (with all modes controlled) 46 1.5 1 0.5 0 -0.5 -1 -1.5 0 10 20 30 40 50 60 70 80 90 100 Time in seconds Figure 3.13. Mode 1 Response of a 4-DOF system (with all modes controlled) 3 2 1 -2 -3 -4 0 10 20 30 40 50 60 70 80 90 100 Time in seconds Figure 3.14. Mode 2 Response of a 4-DOF system (with all modes controlled) 47 6 4 2 0 -2 -4 -61 - I 0 10 20 30 40 50 60 70 80 90 100 Time in seconds Figure 3.15. Mode 3 Response of a 4-DOF system (with all modes controlled) 5 4 3 2 1 0 -1 -2 -3 0 10 20 30 40 50 60 70 80 90 100 Time in seconds Figure 3.16. Mode 4 Response of a 4-DOF system (with all modes controlled) 48 k2 k4 k2 k5 m3 m5 m7 k4 kl k3 kl Figure 3.17. Kabe's Model 49 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -t 0 1 2 3 4 5 6 time in seconds Figure 3.18. Controlled response at Node 1 of Kabe's Model (with only two modes controlled) 2 1.5 -1 -1.5 -2 0 1 2 3 4 5 6 time in seconds Figure 3.19. Controlled response at Node 2 of Kabe's Model (with only two modes controlled) 5( 2 1.5 j 1 0.5 0 -0.5 -1 -1.5 -203 4 time in seconds Figure 3.20. Controlled response at Node 3 of Kabe's Model (with only two modes controlled) 1.5 0.5 -1.5 time in seconds Figure 3.21. Controlled response at Node 4 of Kabe's Model (with only two modes controlled) 51 1.5 -1 0 1 2 3 4 5 6 time in seconds Figure 3.22. Controlled response at Node 5 of Kabe's Model (with only two modes controlled) 1.5 0.5 o0 -0.5 -1 -1.5 -2 -0 1 2 3 4 5 6 time in seconds Figure 3.23. Controlled response at Node 6 of Kabe's Model (with only two modes controlled) 52 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 5 6 time in seconds Figure 3.24. Controlled response at Node 7 of Kabe's Model (with only two modes controlled) 53 2 1.5 1 0.5 -0.5 -1 -1.5 -2 0 1 2 3 4 5 6 time in seconds Figure 3.25. Controlled response at Node 8 of Kabe's Model (with only two modes controlled) 1 0.8 0.6 0.4 0.2 E 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 5 6 time in seconds Figure 3.26. Controlled mode 1 Response of Kabe's Model (with only two modes controlled) 54 0.8 0.6 0.4 0.2 E 0 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 time in seconds Figure 3.27. Controlled mode 2 Response of Kabe's Model (with only two modes controlled) 1.5 0.5 0,5 E 0 40 -0.5 -! -1.5 0 1 2 3 4 5 6 time in seconds Figure 3.28. Controlled mode 3 Response of Kabe's Model (with only two modes controlled) 00 0.2 0.15 0.1 0.05 C0 x -0.05 -0.1 -0.15 -0.2 -0.25 0 1 2 3 4 5 6 time in seconds Figure 3.29. Controlled mode 4 Response of Kabe's Model (with only two modes controlled) 56 0.6 0.4 0.2 C 0 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 time in seconds Figure 3.30. Controlled mode 5 Response of Kabe's Model (with only two modes controlled) 57 0.5 0.4 0.3 0.2 0.1 -0.2 -0.3 -04 -0.5 0 1 2 3 4 5 6 time in seconds Figure 3.31. Controlled mode 6 Response of Kabe's Model (with only two modes controlled) 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 time in seconds Figure 3.32. Controlled mode 7 Response of Kabe's Model (with only two modes controlled) 1.5 0.5 0 0 0 -0.5 -1 -1.5 0 1 2 3 4 5 6 time in seconds Figure 3.33. Controlled mode 8 Response of Kabe's Model (with only two modes controlled) 3.3.6 Kabe's Problem With damping in all modes In this section Kabe's eight degree of freedom system was simulated with damping introduced in all modes. The desired damping values in the eight modes are .20,0.18,0.15,0.10.0.16,0.15,0.13 and 0.12. Figures 3.34-3.41 show the modal responses of this system. From these figures it is clear that controlling all the modes of the system gives better performance when compared to the case with only few modes controlled. The achieved damping values in the eight modes are 0.19,0.16,0.14.0.08,0.14.0.14,0.11 and 0.11. The oscillations that are seen in Figure 3.41 are due to the activation of the local modes of vibration. The natural frequencies of the controlled modes were identified as 30.0280, 31.7163, 31.7557, 32.3020, 33.3267, 35.5763. 38.7887 and 41.8884 radians/second respectively, which match the uncontrolled natural frequencies. 59 0.8 0.6 0.4 0.2 0 .-0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 5 6 time in seconds Figure 3.34. Controlled mode 1 Response of Kabe's Model (with all modes controlled) 0.8 0.6 0.4 0.2 E 0 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 time in seconds Figure 3.35. Controlled mode 2 Response of Kabe's Model(with all modes controlled) 60 1.5 1 0.5 -0.5 -1 -1.5 0 1 2 3 4 5 6 time in seconds Figure 3.36. Controlled mode 3 Response of Kabe's Model (with all modes controlled) 0.25 0.2 0.15 0.1 S0.05 -o.o05 -0.1 -0.15 -0.2 -0.25 0 1 2 3 4 5 6 time in seconds Figure 3.37. Controlled mode 4 Response of Kabe's Model (with all modes controlled) 61 0.5 0.4 0.3 0.2 ., 0.1 E 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 1 2 3 4 5 6 Figure 3.38. Controlled mode 5 Response of Kabe's Model (with all modes controlled) 0.5 0.4 0.3 0.2 0.1 E 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 1 2 3 4 5 6 time in seconds Figure 3.39. Controlled mode 6 Response of Kabe's Model (with all modes controlled) 62 0.8 0.6 0.4 0.2 2 0 -0.2 -0.4 -0.6 -0.8 0 1 2 3 4 5 6 time in seconds Figure 3.40. Controlled mode 7 Response of IKabe's Model (with all modes controlled) 1.5 1 -1 -215 0 1 2 3 4 5 6 time in seconds Figure 3.41. Controlled mode 8 Response of Kabe's Model (with all modes controlled) 63 3.4 Summary A design methodology for the energy based on-off switching decentralized control of mechanical structures is presented. Energy consumption is minimized by turning on the controllers when a structure is doing work on the controllers, and turning them off after prescribed on-times. For a SDOF system, closed form solution is given for designing the controller parameter. For a MDOF system the proposed methodology can be used to achieve a desired average damping rate when each mode may be uniquely controlled. The controller design methodology and the switching schemes are presented. Though the proposed scheme suggests the use of no external energy, in practice, such methods would require some energy input for signal monitoring, as well as to drive an electro-mechanically switched controller using a computer or a microprocessor. Since altering the configuration of sensors and controllers does not alter the fact that structures with positive damping dissipate energy, the passivity theorem (Morris & Juang, 1994) can be applied in this case to prove that modal forces from the controllers will not destabilize the system. Knowledge of the system is used in obtaining the controller parameters. Changes or errors in the system or controller parameters, however, will not destabilize the system because it is assumed to be dissipating energy. CHAPTER 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL 4.1 Introduction The work on switching control presented in Chapter 2 and 3 was motivated by a study of the literature which used on-off control of vibrations, such as Athanassiades (1963), Athans (1964b, 1964a), Foster and Silverberg (1991), Silverberg (1986), Redmond and Silverberg (1992) and Medith (1964). The work of Foster and Silverberg (1991) is one such study which uses fuel minimization as an optimization criterion for on-off control of structural vibrations. In that work, the minimum fuel use control law imposed that control pulses be applied every time a zero crossing of the the position (and maximum value of the velocity) would occur. A plot of fuel consumption versus sampling time was also presented in (Foster & Silverberg, 1991). This chapter is devoted to the study of the effects of sampling time and window size on the fuel consumption. 4.2 Minimum Fuel Control of Mechanical Systems Foster and Silverberg (1991) developed a minimum fuel based switching control law for MDOF second order systems of the form: il -t- Kx = F, (4.2.1) where M1 = M' is the n x n mass matrix of the structure, K is the n x n stiffness matrix of the structure, x and I are, respectively, the n-dimensional displacements 64 65 and accelerations at the nodal points of the structure, and F is the n-dimensional vector of forces acting on the structure. Since the control law depends on points (in time) of maximum velocity (or minimum position) these are determined using running averages of velocities and positions: tPf(t) = J a(s, t)f(s)ds, ] a(s, t)ds = 1 (4.2.2) Uv(t) = a(s,t)(f(s) pi(s))2ds, (4.2.3) where a(s, t) is a window function of the form: 1/T, s>t T 0, s < t T. (4.2.4) Here, T is the window size. Equations (4.2.2) and (4.2.3) are used in determining the control weightings for a piecewise constant (vector) control law. In that work, the fuel consumption C(t) in time t is measured using N t C(t)= jI ft ds (4.2.5) i=l where fi(s) is the ith component of the control force at time s, in modal coordinates. 4.3 Effects of changes in window size and sampling period 4.3.1 Example 1 Foster and Silverberg (1991) worked out an example prescribed by the following parameters: M = 1, K = 1 and T = 10. 66 where T is the number of samples in the window. In Figure 4.1 the fuel consumption was plotted as a function of number of samples per period of oscillation (denoted by N., a parameter which determines the sampling period). Notice the sharp increase in fuel consumption for certain values of N. As it turns out, when these increases in fuel consumption occur, so does a triggering of the higher harmonics in the control pulses. The fast fourier analysis of the control pulses shows that the ratio of signal power at the fundamental frequency to the sum of the signal power at the higher harmonic frequencies decreases at the points of positive slope in this plot. 1.05 0.95 .o 0.9 8 0.85 0.65 0.8 0.75 0 10 20 30 40 50 60 Number of samples per period of oscillation Figure 4.1. Effect of Number of Samples per period of oscillation on fuel consumption (example 1) The frequency spectra of the power of control signal are shown in Figures 4.2-4.5 for selected points on the plot shown in Figure 4.1. The fundamental frequency of this system is 0.159 Hz. The ratio of the power in the control signal at the fundamental frequency to that represented by the higher harmonics decreases at the points of 67 positive slope on the plot in Figure 4.1. This is seen in the plots in 4.3 and 4.5. In these plots note that for values of N = 15 and N = 25, the power at higher harmonics take higher values when compared to plots for values of N where the slope of the fuel curve shown in Figure 4.1 is negative. At the points of positive slope of the plot in Figure 4.1, sampling time is such that the window size becomes an integral multiple of one of the higher harmonics. Note that for N = 15 window size is twice the period of third harmonic and for N = 25 window size is twice the period of fifth harmonic. 0.025 0.02 S0.015 0 .CL W 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency in Hz (Number of samples=1 1) Figure 4.2. Frequency spectrum of control pulse in example 1 (N=11) 68 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.2 0.4 0.6 0.8 1 1.2 Frequency in Hz (Number of samples=1 5) Figure 4.3. Frequency spectrum of control pulse in example 1 (N=15) 69 0.025 0.02 0.015 0 S0.01 0.005 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Frequency in Hz (Number of samples=21) Figure 4.4. Frequency spectrum of control pulse in example 1 (N=21) 4.3.2 Example 2 In this example, the system parameters used are: M = 1, K = 2, and T = 10. The system was simulated using the same control scheme and the effects of sampling time on fuel consumption was studied. The plots of fuel consumption and spectra of control signal are given in Figures 4.6-4.10. The natural frequency of vibration is .23 Hz. ,Just like in the previous example, the higher harmonics in the control signal make the fuel consumption to increase at certain values of sampling times. 4.4 Summary The effects of sampling on fuel consumption using Foster and Silverberg (1991)'s minimum fuel on-off control law was studied. It was noted that when the significant 70 0.016 0.014 0.012 0.01 0 S0.008 .) 0.006 0.004 0.002 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency in Hz (Number of samples=25) Figure 4.5. Frequency spectrum of control pulse in example 1 (N=25) harmonics in the control signal impose that window size is integer multiple of the period of higher harmonics, fuel consumption was increased. Ti 171 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0 10 20 30 40 50 60 Number of samples per period of oscillation Figure 4.6. Effect of Number of Samples per period of oscillation on fuel consumption (example 2) 0.012 0.01 0.008 0 o0 0.006 0.004 0.002 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency in Hz (Number of samples=9) Figure 4.7. Frequency spectrum of control pulse in example 2 (N=9) 72 0.015 0.01 0.005 0 0.2 0.4 0.6 0.8 1 1.2 Frequency in Hz (Number of samples=l 1) Figure 4.8. Frequency spectrum of control pulse in example 2 (N=11) 0.014 0.012 0.01 9 0.008 j 0.006 0.004 0.002 0 0 0.5 1 1.5 2 2.5 3 Frequency in Hz (Number of samples=25) Figure 4.9. Frequency spectrum of control pulse in example 2 (N=25) 73 0.018 0.016 0.014 0.012 0.01 C.0.008 0.006 0.004 0.002 0 ' 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Frequency in Hz (Number of samples=38) Figure 4.10. Frequency spectrum of control pulse in example 2 (N=38) CHAPTER 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 5.1 Discussion This study proposed new schemes for structural control design using the energy input into the controller to damp out the vibrations. Through simulations it was shown that such a decentralized control law can be efficient in improving the performance of structures. For SDOF systems, the proposed design methodology introduces desired damping within very close bounds. For MDOF systems, using the techniques of Independent Modal Space Control, a design algorithm was developed to introduce specified damping into desired modes of vibration. The controllers were decentralized in that the decision for switching a given controller was based on feedback from local states only. It was also shown how the uncontrolled modes are affected by controlling only few modes in multi degree of freedom systems. This suggests that the proposed methodology be used to control all of the dominant modes of the vibrating structure. 5.2 Future Research The minimum energy on-off control methods proposed in this work should increase the life span of space structures since very minimal energy is needed from the external power with such a method. To fully demonstrate the practical advantage of these schemes. one should determine the kind of controllers that can be used for such applications. A simple 74 gyroscope, magnetic bearing gyros, or torque wheels provide potentially viable options for such applications. Control momentum gyros have been effectively used for the attitude control of spacecrafts. For instance, single gimbaled gyros have been installed on the Soviet MIR station, while double gimbaled gyros were used in the NASA Skylab. 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Uniform Damping Control of Spacecraft. Journal of Guidance, Control and Dynamics, 9, 221-227. Sofonov, M. G., & Athans, M. (1977). Gain and phase margins for multiloop LQG regulators. IEEE Transactions on Automatic Control, AC-22, 173-177. Souza, M. L. 0. (1988). Exactly solving the weighted time/fuel optimal control of an undamped harmonic oscillator. Journal of Guidance, Control and Dynamics, 11, 488494. Swanson Analysis Systems Inc. Houston, Pennsylvania (1992). ANSYS 5.0 Users Manual -Revision 5.0a Vol 1-4. Tarras, A. (1987). Decentralized Control of Large Scale Systems, Sensitivity and Parameter Robustness. Ph.D. thesis, National Center For Scientific Research- France. Velde, W. E. V., & He, J. (1983). Design of Space structure control systems using on-off thrusters. Journal of Guidance. Control and Dynamics, 6, 53-60. Wang, S. H. (1982). Stabilization of decentralized controller via time varying controllers. IEEE Transactions on Automatic control, AC-27, 741-744. Wang, S. H., & Davison, E. J. (1973). On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control. AC-18, 473-478. West-vukovich, G. S., & Davison. E. J. (1984). The Decentralized control of large flexible space structures. IEEE Transactions on Automatic Control, 29, 866-879. Witsenhausen, H. S. (1973). A standard form for sequential stochastic control. Journal of Mathematical Systems Theory. 1. 80 Yang, L., Mikulas, M. M., Park, K., & Su, R. (1995). Slewing Maneuvers and Vibration Control of Space Structures by Feedforward/Feedback Moment-Gyro Controls. Journal of Dynamic Systems, Measurement, and Control, 117, 343-351. Young, D. (1990). Distributed Finite Element modeling and Control Approach for large flexible Structures. Journal of Guidance, 13. 703-713. Zimmerman. D. C. (1990). Threshold Control for Nonlinear and time varying structures using equivalent transformation. Journal of Intelligent Material Systems and Structures, 1, 76-90. Zimmerman, D. C., Inman, D. J., & Juang, J. N. (1991). Vibration Suppression Using a Constrained Rate Feedback Control Strategy. Journal of Vibration and Acoustics, 113, 345-352. BIOGRAPHICAL SKETCH I was born in Mysore City, India, on September 14, 1964. I spent most of my early childhood at Mangalore with my parents, brother and two sisters. I graduated from St. Aloysius College in Mangalore in 1982. I went on to get my Bachelor of Engineering degree from Karnataka Regional Engineering College in Surathkal. India in 1986. I received my Master of Technology degree from the Indian Institute of Technology in Bombay in 1988. After working in the Indian Institute of Technology, Bombay for a brief period of three months, I joined Hindustan Aeronautics Limited. Bangalore. Within a short time of six months, I was appointed as a Scientist at the Indian Space Research organization. After serving the Indian Space Industry for two years, I attended University of Florida where I have received my Ph.D. in Engineering Mechanics. 81 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David .erman, Chairman Associa e Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Carla Schwartz, Co-Chairm Assistant Professor of Elect rl Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Docto hilosophy. Assistant Professot of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. -"-Pfwph Latchman Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosop y} Jacob Ham Professor of Electrical and Computer Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1996 W ed M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School LD 1780 199 /vl /i UNIVERSITY OF FLORIDA I3 1 i II 11111111111 III I 3 1262 08554 9219 |

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TABLE OF CONTENTS
ACKNOWLEDGEMENTS iv ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 1.1 Decentralized Control of Large Systems 1 1.2 Advantage of Using Decentralized Control 2 1.3 Decentralized Stabilization and Pole Placement 7 1.4 An illustration to demonstrate the advantages of decentralized struc tural control 11 1.5 Objective of the Present Study 13 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 16 2.1 Introduction 16 2.2 Controlling Vibrations Using On-Off Controllers Without External Energy 17 2.2.1 Switching procedure 18 2.2.2 Procedure for switching with one controller 20 2.2.3 Example 20 2.3 Summary 22 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING ... 25 3.1 Introduction 25 3.2 Designing On-Off Controller for SDOF system 25 3.2.1 Analytical Results 25 3.2.2 Proposition 28 3.2.3 SDOF Example 28 3.3 Designing On-Off Controll for an MDOF system 29 3.3.1 Modified Independent Modal Control Methods 29 v Magnitude of imaginary part 13 were set to zero ( a scenario mimicking a break in communication line) and the effect of this on the overall eigenvalues were studied. With the decentralized control, it was observed that all the poles of the coupled structure have negative real parts. The individual controller failures dont seem to affect the pole locations very much. This of course depends on the type of controller failure that we simulate. Open loop poles of the substructures and coupled structure are plotted in Figure(1.4). In table 1, Values of the coupled loop poles without and with controller failures are listed. 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 o coupled structure x end component + centre component 0 10 15 20 Mode number 25 30 Figure 1.3. Open loop poles of substructures and coupled structure 1.5 Objective of the Present Study The present study investigates the development of new and promising energy effi cient ways of controlling vibrations in mechanical structures. Although considerable research has been done in vibration control of flexible structures, no method uses 32 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems In this section, a new design methodology is proposed for using on-off control to achieve desired damping in modes of interest. This method is based on energy transfer (from kinetic to potential), as well as on the dissipative nature of damping forces. Time-sharing of controllers was proposed in (Baz & Poh, 1987, 1991), and these ideas are used here. Consider a second order dynamical system for the n-dimensional displacement vector x given by Mx + Kx = F. (3.3.12) where M = MT > 0 is the mass matrix of the structure, K = KT > 0 is the stiffness matrix of the structure, and F is n-dimensional vector of forces acting on the structure. Equation (3.3.12) may written in modal coordinates by using the following weighted modal transformation: x = 4>u, (3.3.13) where u represents the vector of modal coordinates of the system, and $ is the systems modal shape matrix (i.e. the matrix of eigenvectors). Thus, under the transformation (3.3.13), equation (3.3.12) reduces to the form + \u = f, (3.3.14) where A is the diagonal matrix of eigenvalues of M~lK matrix and / = <&rF. The controller being on, yields the dynamics: Mi + Kx AKx. (3.3.15) 65 and accelerations at the nodal points of the structure, and F is the n-dimensional vector of forces acting on the structure. Since the control law depends on points (in time) of maximum velocity (or mini mum position) these are determined using running averages of velocities and positions: = j a(s,t)f(s)ds, J a(s,t)ds = 1 rf(t) = Jo aisi t)(f{s) nf(s))2ds, (4.2.2) (4.2.3) where a(s, t) is a window function of the form: / .X r 1 /T, s> t-T ( ,f| o, s < t T. (4.2.4) Here, T is the window size. Equations (4.2.2) and (4.2.3) are used in determining the control weightings for a piecewise constant (vector) control law. In that work, the fuel consumption C(t) in time t is measured using CW = f l/.l ds (4.2.5) i= 1 J0 where /(s) is the ilh component of the control force at time s, in modal coordinates. 4.3 Effects of changes in window size and sampling period 4.3.1 Example 1 Foster and Silverberg (1991) worked out an example prescribed by the following parameters: M = 1, K = 1 and T = 10. 53 Figure 3.25. controlled) Figure 3.26. controlled) Controlled response at Node 8 of Kabes Model (with only two modes Controlled mode 1 Response of Kabes Model (with only two modes 79 Reinhorn, A. M., Soong, T. T., Riley, M. A., Lin, R. C., Aizawa, S., &: Higashino, M. (1993). Full Scale Implementation of Active Control. II: Installation and Performance. ASCE Journal Of Structure Engineering, 119, 1935-1960. Rohman, M. A., & Leipholz, H. H. E. (1978). Structural Control By Pole Assignment method. Journal of Engineering mechanics division, ASCE, 104. 1159-1175. R.P.Corfmat, & A.S.Morse (1976). Control of linear system with specified input channels. SIAM Journal of Control, 1\. Schwartz, C. A., & Maben, E. N. (1995). A Minimum Energy Approach to Switching- Control for Mechanical Systems. Proceedings of the first block island workshop on logic and switching control Schwartz, C. A., & Maben, E. N. (1996). On-Off Minimum Energy control of Structures. To Appear in 1st European Structural Control Conference in Barcelona, Seywald, H., Kumar, R. R., Deshpande, S. S., & Heck, M. (1994). Minimum fuel spacecraft reorientation. Journal of Guidance ,Control and Dynamics, 17, 21-29. Silverberg, L. (1986). Uniform Damping Control of Spacecraft. Journal of Guidance, Control and Dynamics, 9, 221-227. Sofonov, M. G., & Athans, M. (1977). Gain and phase margins for multiloop LQG regula tors. IEEE Transactions on Automatic Control, AC-22, 173-177. Souza, M. L. O. (1988). Exactly solving the weighted time/fuel optimal control of an undamped harmonic oscillator. Journal of Guidance, Control and Dynamics, 11, 488- 494. Swanson Analysis Systems Inc, Houston, Pennsylvania (1992). ANSYS 5.0 Users Manual -Revision 5.0a Vol 1-4 Tarras, A. (1987). Decentralized Control of Large Scale Systems, Sensitivity and Parameter Robustness. Ph.D. thesis, National Center For Scientific Research- France. Velde, W. E. V., & He, J. (1983). Design of Space structure control systems using on-off thrusters. Journal of Guidance, Control and Dynamics, 6, 53-60. Wang, S. H. (1982). Stabilization of decentralized controller via time varying controllers. IEEE Transactions on Automatic control, AC-27, 741-744. Wang, S. H., & Davison, E. J. (1973). On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control, AC-18, 473-478. West-vukovich, G. S., &; Davison. E. J. (1984). The Decentralized control of large flexible space structures. IEEE Transactions on Automatic Control, 29, 866-879. Witsenhausen, H. S. (1973). A standard form for sequential stochastic control. Journal of Mathematical Systems Theory, 1. 28 3.2.2 Proposition For a single degree of freedom system of the form (3.2.1), on-off switching control may be used to attain a desired overall average damping rate ( using the proposed minimum energy control scheme in conjunction with equation (3.2.11) 3.2.3 SDOF Example The main objective of this example is to show the effectiveness of this method and to show the validity of assumption made earlier, that the decay rate a is assumed to be small compared to au>. Consider the single degree of freedom system with magnitudes of mass and stiff ness respectively given by m=l and k=100. Simulation of the system with on-off controller was carried out for three values of the damping. For each of these damp ing, Ak was calculated using equation (3.2.11). The on-off controller was simulated and the response of the system to an initial displacement of 1 and velocity of 0 was observed. The response of the system for values of damping of 0.0015, 0.020 and .6 are given in Figures 3.1, 3.2 and 3.3 respectively. In each of these figures, expo nentially decayed curves are plotted to show the exactness of this method for low values of damping. From these time responses, it is concluded that the assumption that a is an exponentially decayed function is valid. Also, it can be seen that when the assumption that A << au is violated, the damping introduced into the system because of switching controller deviates much from the desired value. 21 response even when the initial conditions were very close to the mode shapes of a lesser damped mode of the uncontrolled structure, a worst case scenario. Figures 2.2 through 2.6 show the nodal (position) response of the uncontrolled and controlled system when the initial conditions were set to the mode shape of one of the lesser damped mode of the uncontrolled structure. The results are only shown for one set of initial conditions, however the simulation was carried out for each of the mode shapes. From these plots it is clear that the proposed method is effective in increasing the damping. Time in seconds Figure 2.2. Response at node 1 I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David Zjmpaerman, Chairman Associate Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Carla Schwartz, Co-Chairm Assistant Professor of Elect Engineering Computer I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoj^of-Philosophy. KbimarfFitz-Coy Assistant Professo: of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. a OU\A Ffarrfph Latchman Associate Professor of Electrical and Computer Engineering 10 condition for this to occur is that the system be complete. Completeness of the sys tem is defined in terms of the transmission polynomials of the system. For example, in case N = 2. a test for completeness (Morse, 1973) is that for all s rank (si-A) Bi C2 0 >2, (1.3.8) since rank (si A) >2 (1.3.9) except when s is an eigenvalue of A. It is only necessary to check the above condition on the spectrum of A. Anderson and Moore (1981) suggested that time varying controllers can be used to eliminate the fixed modes. In their work, they showed that under the assumption of centralized controllability and observability, for a two channel system with feedback of the form: u2(t) = K2(t)y2(t) where K2(t) is periodic and piecewise constant, taking p > 1 + max (dim(u2), dim(y2)) values, then strong connectivity, even when a fixed mode is present, is enough to ensure that the system is uniformly controllable from iq and uniformly observable from y^. Most of the work in the decentralized control schemes with time varying controller focuses on making the system controllable and observable and/or on robust control. The concept is to develop an algorithm to place the poles of the systems using time varying controllers. Once this algorithm is available, fixed modes ( the modes of the system which are not affected by TIV controller gains) of the system can be treated the way the non-fixed modes are treated (Wang, 1982). The concept of using time varying periodic compensators with time invariant plants was further investigated by Khargonekar, Poolla, and Tannenbaum (1985). In their work, they showed that for a large class of robustness problems, periodic com pensators are superior to time invariant ones. They also gave design techniques which 46 Figure 3.13. Mode 1 Response of a 4-DOF system (with all modes controlled) Figure 3.14. Mode 2 Response of a 4-DOF system (with all modes controlled) Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK By EGBERT N. MABEN May 1996 Chairman: David Zimmerman Co Chairman: Carla Schwartz Major Department: Aerospace Engineering, Mechanics, and Engineering Science New energy-based methods are proposed for improving the performance of flexible structures to external disturbances. The schemes are based on decentralized control methods, which make them appealing for the solutions of large structural control problems. The proposed on-off controller schemes use the work done on the actuators for switching. Energy consumption to damp out the vibrations is minimized by switching the controllers ON only when the work done on them by the structure is positive. Design methodology is proposed to design the controllers to achieve desired damping in vibrating modes of the structure. Simulation results show that the proposed scheme is effective in introducing damping into low or undamped modes of the system. A method similar to the Independent Modal Space Control Scheme is used to get desired damping in modes of vibration using on-off control. vn 22 Time in seconds Figure 2.3. Response at node 2 Figure 2.4. Response at node 3 2.3 Summary A minimum energy on-off vibration control law for structures was introduced. The control procedure takes advantage of the energy which can be imparted to the controllers from switching by turning on the controllers when the structure is doing work on the controller and turning them off when the controllers are doing work on the structure. 59 Figure 3.34. Controlled mode 1 Response of Kabes Model (with all modes controlled) Figure 3.35. Controlled mode 2 Response of Kabe's Model(with all modes controlled) 6 and phase margins (Sofonov & Athans, 1977), and handles multi-loop problems in exactly the same framework as single loop problems. These results generally extend to the stochastic(LQG) case, in which the disturbances are modeled by passing white Gaussian noise through finite dimensional filters, and in which only partial state in formation is available. However, the solution of the LQG problem requires that a Kalman filter be implemented to reconstruct the missing state variables. Thus, an additional Riccati equation must be solved. The modes of the filter will appear in the closed-loop response of the system, and the gain and phase margins will be more sensitive and less robust than a corresponding LQ design. In nonclassical stochastic control problems, where the information available to the device controlling a channel input is different from that of the device observing the channel output, the input has a dual purpose: communication through the system dynamics and sensors to the other controllers and direct control of the system. From the control sharing point of view, one can speculate that nonlinear solutions in the nonclassical LQG control are due to actions of the decentralized controllers as they try to signal information to one another using the control system as a commu nication channel. It is impossible to ascertain what portion of a given system input is applied to purposes of signaling and what portion for control. One can question the desirability of using nonlinear signaling strategies to communicate through the control system dynamics on several grounds. Implementation of such scheme would be extremely complex and performance would be quite sensitive to system parameter variations. In any case, determination of these signaling strategies have been shown to be equivalent to an infinite dimensional, non-convex optimal control problem with neither analytical nor computational solution likely to be forthcoming (Witsenhausen, 1973). Most research in stochastic decentralized control is concerned with determin ing the optimal parameters of given decentralized control structure (Date & Chow, 1994; Levine & Athans, 1970). DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK Bv EGBERT N. MABEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 Copyright 1996 by EGBERT N. MABEN To my parents and teachers ACKNOWLEDGEMENTS I would like to express my sincere gratitude to Dr.David Zimmerman and Dr.Carla Schwartz for helping me in carry out this research work. Without their enormous patience, encouragement and guidance, it would not have been possible to complete the work. I consider myself extremely fortunate to have them as my dissertation advisors. I would like to thank Dr.Norman Fitz-Coy, Dr.Haniph Latchman and Dr.Jacob Hammer for serving in my committee. I would like to acknowledge the help provided by the 'Project Care team, specially Dr.Michael Conlon, Dr.Marilou Behnke, Dr.Fonda Eyler Davis and Kathie Wobie, by providing me with finacial support during my stay at the University of Florida. IV TABLE OF CONTENTS ACKNOWLEDGEMENTS iv ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 1.1 Decentralized Control of Large Systems 1 1.2 Advantage of Using Decentralized Control 2 1.3 Decentralized Stabilization and Pole Placement 7 1.4 An illustration to demonstrate the advantages of decentralized struc tural control 11 1.5 Objective of the Present Study 13 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 16 2.1 Introduction 16 2.2 Controlling Vibrations Using On-Off Controllers Without External Energy 17 2.2.1 Switching procedure 18 2.2.2 Procedure for switching with one controller 20 2.2.3 Example 20 2.3 Summary 22 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING ... 25 3.1 Introduction 25 3.2 Designing On-Off Controller for SDOF system 25 3.2.1 Analytical Results 25 3.2.2 Proposition 28 3.2.3 SDOF Example 28 3.3 Designing On-Off Controll for an MDOF system 29 3.3.1 Modified Independent Modal Control Methods 29 v 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems 32 3.3.3 Four Degree of Freedom example with damping in one mode 36 3.3.4 Four Degree of Freedom example with damping in all modes 37 3.3.5 Kabes Problem With damping in only two modes 38 3.3.6 Kabes Problem With damping in all modes 58 3.4 Summary 63 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL 64 4.1 Introduction 64 4.2 Minimum Fuel Control of Mechanical Systems 64 4.3 Effects of changes in window size and sampling period 65 4.3.1 Example 1 65 4.3.2 Example 2 69 4.4 Summary 69 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 74 5.1 Discussion 74 5.2 Future Research 74 REFERENCES 76 BIOGRAPHICAL SKETCH 81 vi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK By EGBERT N. MABEN May 1996 Chairman: David Zimmerman Co Chairman: Carla Schwartz Major Department: Aerospace Engineering, Mechanics, and Engineering Science New energy-based methods are proposed for improving the performance of flexible structures to external disturbances. The schemes are based on decentralized control methods, which make them appealing for the solutions of large structural control problems. The proposed on-off controller schemes use the work done on the actuators for switching. Energy consumption to damp out the vibrations is minimized by switching the controllers ON only when the work done on them by the structure is positive. Design methodology is proposed to design the controllers to achieve desired damping in vibrating modes of the structure. Simulation results show that the proposed scheme is effective in introducing damping into low or undamped modes of the system. A method similar to the Independent Modal Space Control Scheme is used to get desired damping in modes of vibration using on-off control. vn CHAPTER 1 INTRODUCTION 1.1 Decentralized Control of Large Systems With the space shuttle transportation system now a practical reality, there is considerable interest in large space structures. Two problems which are inherent in the control of large space structures are shape control and attitude control. The latter involves maintaining a given orientation of the spacecraft with regards to an inertial reference system, while the former involves the shape of critical components of the space structures such as a phased array antenna. In both cases, structural flexibility plays important roles since the translational and attitude motions are coupled with the structural vibrations. The motion of a Large Scale Flexible Structure(LFSS) is usually modeled via finite element methods, which can result in very large order models. This means that the order of the system may be extremely high, which can make the tasks of control system design challenging. Model reduction methods that are applied in many cases (Inman, 1989) to control only a subset of the elastic body modes may lead to spillover problems, in which the control effect in stabilizing the subset of modes may cause instability in the uncontrolled modes. The approaches considered to date for investigating the LFSS control have gen erally been directed towards centralized control. e.g. model reduction methods (Hughes & Skelton. 1981), modal control methods (Balas, 1978b), output feedback 1 2 control (Lin. 1981), and adaptive control methods (Behnhabib, Iwens, & Jackson, 1979). Two good survey papers deal with the problem of LFSS control (Gran & Rossi, 1979; Balas, 1982). West-vukovich and Davison (1984) have shown how the design of active structural controllers that emulate the real structural elements such as dampers and springs can produce effective control laws. A suboptimal control approach is used in that work to design the structural characteristics (gains) of the controller elements. This allows integration of passive controller, active controller and sensor/actuator locations design. This method exhibits problems due to delays introduced due to digital implementation, and consideration of transfer function of real sensors and actuators. Definition. Fixed modes of the system are modes of the system that remain in variant under any nonzero parametric variation of the system or the controller. The fixed modes of a LFSS result when a sensor and actuator are located at a node of a flexural mode. It was shown that for LFSS with col-located actuators and sensors, the fixed modes of the decentralized system are the same as the centralized fixed modes of the system. A controller that will eliminate the spillover problem is demonstrated. It is also shown that a solution to the decentralized control problem exists only if there is a solution to the centralized system. A further explanation of fixed modes is provided in section 3. 1.2 Advantage of Using Decentralized Control Over the past twenty-five years, engineers and scientists have developed a variety of procedures for analyzing systems and for designing control strategies for controlling LFSS. These procedures are classified into three types: 1. Procedures for modeling dynamical systems (state space formulation, input- output transfer function description, etc) 3 2. Procedures for describing the qualitative properties of system behavior (controllability, stability, observability etc.) 3. Procedures for controlling system behavior (stabilizing feedback, optimal control, etc) All of these procedures rest on the common presupposition of centrality; all the infor mation available about the system, and the calculations based upon this information, are centralized. It is useful to distinguish between two kinds of available information: 1. Information about the system model: (off -line or a priori information) 2. Sensor information about the system response: set of all real time mea surements made of system response. The common modern estimation and control is based on centralized control the ory. When considering large scale systems, the centralized control assumption fails due either to the lack of centralized information or the lack of centralized comput ing capability. There are many examples of large scale systems that present a great challenge to both analysts and control system designers. Examples include power systems, urban Traffic networks, digital communication networks, flexible manufac turing networks and economic systems. The control of such physical systems are often characterized by spatial separation (such as physical separation between sensors and actuators) so that issues such as the economic cost and reliability of the communi cation link have to be taken into account in control, thus providing an impetus to decentralized control design. Research in decentralized control has been motivated by the inadequacy of modern control theory to deal with certain issues which are of concern in large scale systems. A key concept in modern control theory is that of state feedback. By using techniques such as Linear Quadratic (LQ) optimal control or pole placement, it is possible to 4 achieve improved system behavior by using state feedback. However, it is often im possible to instrument a system to the extent required for full state feedback, so techniques ranging from linear-quadratic Gaussian (LQG) control, to observer based control to time domain compensator design techniques have been employed to over come this difficulty. However, a key concept in these types of designs is that every sensor output affects every actuator input. This situation is termed as centralized control. In many large scale systems, it is impossible to implement this in real-time. Thus for economic and possibly reliability reasons, there is a trend for decentral ized decision making, distributed computing and hierarchical control. However, these desirable goals of structuring a distributed information and decision framework for large scale systems do not mesh with the available centralized methodologies and tools of modern control system design. Reduction of computation and simplification of structure are of particular concern in decentralized control of large scale systems, but are also of concern in almost all areas of control theory and its applications. The basic characteristic of decentralized control is that there are restrictions on the information transfer between certain groups of sensors and actuators. For example, in figure 1, state variables AT are used to form the control U\ and state variables A2 are used to form the control t/2. This depicts total decentralization. However, inter mediate restrictions on the information between controllers are also possible. Partial decentralization takes place when the system is not fully decentralized but the rate of information transfer is constrained so that full centralized control is not possible. It is to be noted that the concept of decentralization refers to the control structure implementation; the control laws may be designed in a completely centralized way (provided there are no other physical limitations preventing this). The advantages of using decentralized control were mentioned in the Ph.D Thesis of Ahmed Tarras and by several other researchers (Tarras, 1987; West-vukovich & 5 Information Figure 1.1. A decentralized system Davison, 1984). The main advantages of using decentralized control laws are to overcome the problems due to the complexity of the system that results from 1. Large Dimensions 2. Uncertainty- deterministic or stochastic 3. Structural Constraints- these make the flow of information between the subsystems difficult. In controlling the complex systems, massive calculations, expensive computer time and high computer costs may be encountered. In addition, there may be large storage requirements, and multiple criteria may be encountered (due to spatial separations, different levels of operation, etc). Definition. A control system is called decentralized, if and only if all local controls are calculated only as an explicit function of the local information (states, output etc). This is a control with one level and no single controller has an overall view of the entire process. In centralized control, one viable approach to design of feedback control laws for time invariant linear systems is by minimization of an infinite horizon quadratic per formance index. LQ design method allows asymptotic pole placement by appropriate choice of the performance index (Kwakarnaak & Sivan, 1972) has excellent sensitivity and robustness properties according to various criteria, including the classical gain 6 and phase margins (Sofonov & Athans, 1977), and handles multi-loop problems in exactly the same framework as single loop problems. These results generally extend to the stochastic(LQG) case, in which the disturbances are modeled by passing white Gaussian noise through finite dimensional filters, and in which only partial state in formation is available. However, the solution of the LQG problem requires that a Kalman filter be implemented to reconstruct the missing state variables. Thus, an additional Riccati equation must be solved. The modes of the filter will appear in the closed-loop response of the system, and the gain and phase margins will be more sensitive and less robust than a corresponding LQ design. In nonclassical stochastic control problems, where the information available to the device controlling a channel input is different from that of the device observing the channel output, the input has a dual purpose: communication through the system dynamics and sensors to the other controllers and direct control of the system. From the control sharing point of view, one can speculate that nonlinear solutions in the nonclassical LQG control are due to actions of the decentralized controllers as they try to signal information to one another using the control system as a commu nication channel. It is impossible to ascertain what portion of a given system input is applied to purposes of signaling and what portion for control. One can question the desirability of using nonlinear signaling strategies to communicate through the control system dynamics on several grounds. Implementation of such scheme would be extremely complex and performance would be quite sensitive to system parameter variations. In any case, determination of these signaling strategies have been shown to be equivalent to an infinite dimensional, non-convex optimal control problem with neither analytical nor computational solution likely to be forthcoming (Witsenhausen, 1973). Most research in stochastic decentralized control is concerned with determin ing the optimal parameters of given decentralized control structure (Date & Chow, 1994; Levine & Athans, 1970). I 1.3 Decentralized Stabilization and Pole Placement A fundamental result in modern control theory is that the poles of a controllable system can be arbitrarily assigned (subject to complex pole-pairing constraints) by state feedback. This result has been extended to show that the poles of a closed loop system consisting of controllable and observable linear system with a dynamic compensator can be freely assigned. These results are of great theoretical significance and have served as the basis of practical synthesis procedure. A natural generalization of the pole-placement question arises when the restriction to decentralized feedback control is made. Although several authors had looked at this question,(McFadden, 1967; Aoki, 1972; Corfmat & Morse. 1973) the most distinct results are those of Wang and Davison (1973), Davison (1976) and R.P.Corfmat and A.S.Morse (1976). Here, their results are briefly summarized. For a linear system the problem of decentralized pole placement can be formulated as follows. Consider the linear system N x(t) = Ax(t) + E BMt) (1.3.1) 1=1 yi(t) = CiX(t) (1.3.2) where i = 1, N indexes the input and output variables of the various controllers. x Rn is the state, Rmi and G RTi are the input and output, respectively, of the ith local control station. .4 is the state matrix, BÂ¡ is the input matrix and C, is the output matrix of control station i. The ith controller employs dynamic compensation of the form Ui{t) = MtZi(t) + Fiyiit) + GiVi(t) (1.3.3) i{t) = HiZi(t) + Liyi{t) + RiVi{t) (1.3.4) 8 where z, 6 RVi is the state of the ith feedback controller, v{ e Rm' is the ith local external input, and Mt, F, G, Hi, Li, RÂ¡ are real constant matrices of appropriate sizes. The decentralized pole placement problem is to find matrices Mi, Fi,Gi, Hi, Li, Ri such that the closed loop system described by (1.3.1)-(1.3.4) has prespecified poles. Of course, if [G,, ,4, B,] is controllable and observable from all the stations, the solution to such problem is guaranteed. The interesting case is to assume that (1.3.2) is controllable from all controls U(, U/v but not from any single control with similar observability assumptions. Consider first the special case j\/t- = 0 in the above problem. This corresponds to a static decentralized output feedback controller. If F denotes the collection of feedback matrices (Flt F2, F.v), then the pole placement problem is to determine F such that the matrix AFA + 'Â£iBiFiCi (1.3.5) =i has an arbitrarily specified set of eigenvalues. Clearly, a necessary condition for pole placement in this case is that the polynomials | XI A? \ have no common factor, i.e. that cv( A) = g.c.d. | AI Ap |= 1 (1.3.6) where g.c.d. is greatest common divisor. What is more interesting is that this con dition is both necessary and sufficient for pole placement with dynamic compensa tion Wang Davison, 1973). More generally, since the zeros of a (A) (termed as the fixed modes of the system) are invariant under decentralized dynamic compensation, it follows that a necessary and sufficient condition for stabilizability is that the roots of a (A) have strictly negative real parts. Computation of the fixed modes of a system can be accomplished by computing the eigenvalues of Ap and A for randomly selected F and checking for common eigenvalues. A simpler wav of checking for the fixed modes of the system is given in 9 Anderson and Moore (1981). Implicit in the pole placement result quoted above is a constructive algorithm. This algorithm requires as a first step the selection of F such that the poles of Ap are distinct from that of A. Then, the dynamic feedback is successively employed at the control stations to place the poles that are controllable and observable from a given station. R.P.Corfmat and A.S.Morse (1976) have studied the decentralized feedback con trol problem from the point of view of determining a more complete characterization of conditions for stabilizablity and pole placement. Their basic approach is to de termine conditions under which a system of the form (1.3.1)-(1.3.4) can be made controllable and observable from the input and output variables of a given system by applying static feedback to other controllers. Then dynamic compensation can be employed at this controller in a standard way to place the poles of the system. It is not hard to see that a necessary and sufficient condition to make (1.3.1)- (1.3.4) controllable and observable from a single controller is that none of the transfer functions Gij(s) = Ci(sI-A)-lBj i,j = 1, ,N (1.3.7) vanish identically. A system satisfying this condition is termed strongly connected. If a system is not strongly connected, it is impossible to make the system con trollable and observable from a single controller. In this case, it is necessary to decompose the system into a set of strongly connected subsystems and to make each subsystem controllable and observable from one of its controllers. For a strongly connected system, Corfmat and Morse have given a highly interesting and rather in tuitive condition that is necessary and sufficient to make (1.3.1)-(1.3.4) controllable and observable from a single controller. They have shown that if a strongly connected system can be made controllable and observable from a single controller, it can be made controllable and observable from any controller, and a necessary and sufficient 10 condition for this to occur is that the system be complete. Completeness of the sys tem is defined in terms of the transmission polynomials of the system. For example, in case N = 2. a test for completeness (Morse, 1973) is that for all s rank (si-A) Bi C2 0 >2, (1.3.8) since rank (si A) >2 (1.3.9) except when s is an eigenvalue of A. It is only necessary to check the above condition on the spectrum of A. Anderson and Moore (1981) suggested that time varying controllers can be used to eliminate the fixed modes. In their work, they showed that under the assumption of centralized controllability and observability, for a two channel system with feedback of the form: u2(t) = K2(t)y2(t) where K2(t) is periodic and piecewise constant, taking p > 1 + max (dim(u2), dim(y2)) values, then strong connectivity, even when a fixed mode is present, is enough to ensure that the system is uniformly controllable from iq and uniformly observable from y^. Most of the work in the decentralized control schemes with time varying controller focuses on making the system controllable and observable and/or on robust control. The concept is to develop an algorithm to place the poles of the systems using time varying controllers. Once this algorithm is available, fixed modes ( the modes of the system which are not affected by TIV controller gains) of the system can be treated the way the non-fixed modes are treated (Wang, 1982). The concept of using time varying periodic compensators with time invariant plants was further investigated by Khargonekar, Poolla, and Tannenbaum (1985). In their work, they showed that for a large class of robustness problems, periodic com pensators are superior to time invariant ones. They also gave design techniques which 11 can be implemented easily. They also showed that for weighted sensitivity minimiza tion for linear time invariant plants, time varying controllers have no advantages over time invariant ones. However, the eigenstructure assignment problem was beyond the scope of Poollas work. Their design was used to design a stabilizing compensator and later, the time varying controller and time invariant plant combinations were used to generate the time response of the system. Identification packages were used to identify this system and separate compensator was designed for pole placement. The order of the system seems to be going higher and higher in this case. Nevertheless, this looks like a first pass solution to the problem of eigenstructure assignment. 1.4 An illustration to demonstrate the advantages of decentralized structural control This example shows the advantages and simplicity of building a decentralized controller for structures. Figure 1.4 shows a 6-bay truss whose model is taken from Young (1990). This truss is modeled as a coupled structure of three substructures. Substructure boundaries are marked in the figure. The truss member mass and stiffness matrices expressed with respect to the local coordinates and used in the assembly process are EA 1 -1 ' ,, mL ' 2 1 L -1 1 Meiem ~ c b 1 2 The nodal coordinates are defined as vertical and horizontal displacements at the joints. The internal degrees of freedom at which the actuator and displacement sensors are placed are marked in the figure. The finite element model of the individual sub structures are obtained using a FORTRAN program and validated using the common FEM package ANSYS (Swanson Analysis Inc, 1992). For convenience, the material properties are assumed to be have unit magnitudes. An Interlocking control design 12 concept is used to place the actuators and sensors at internal degrees of freedom. Min imization of internal coordinate motions of different substructures would localize the dynamic interaction of coupled structure in the components. The component control action is designed to lock up its own boundary condition, that better approximates Figure 1.2. A Six-Bav Truss A convenient control design technique for this concept is the linear quadratic op timal control regulator approach, in which the aim is to minimize the internal coordi nates motion. The component control law is the one that minimizes the performance index, JSC = \ /o (ysTys + usTRus) dt where ys and us denote the output and control inputs at the internal degrees of freedom of each substructure. These are marked in the figure by numbers. The opti mal component control is a state feedback control law with positions and velocities as inputs. This type of controller is built for individual substructures and the controlled coupled structures state matrix is obtained. The effect of having such a controller on the eigenvalues is studied. Failures in the component controllers by changing the elements of the gain matrices of individual substructures. The gains of some channels Magnitude of imaginary part 13 were set to zero ( a scenario mimicking a break in communication line) and the effect of this on the overall eigenvalues were studied. With the decentralized control, it was observed that all the poles of the coupled structure have negative real parts. The individual controller failures dont seem to affect the pole locations very much. This of course depends on the type of controller failure that we simulate. Open loop poles of the substructures and coupled structure are plotted in Figure(1.4). In table 1, Values of the coupled loop poles without and with controller failures are listed. 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 o coupled structure x end component + centre component 0 10 15 20 Mode number 25 30 Figure 1.3. Open loop poles of substructures and coupled structure 1.5 Objective of the Present Study The present study investigates the development of new and promising energy effi cient ways of controlling vibrations in mechanical structures. Although considerable research has been done in vibration control of flexible structures, no method uses 14 Without Component failure With End controlled Component Failure With Center Controlled Component Failure -0.0005 + 0.1877 -0.0005 + 0.1877 -0.0037 + 0.2515 -0.0007 + 0.2679 -0.0101 + 0.6109 -0.0093 + 0.6117 -0.0054 + 0.6437 -0.0059 4- 0.6445 -0.0044 4- 0.6488 -0.0029 + 0.6563 -0.0059 + 0.6909 -0.0055 4- 0.6910 -0.0196 + 1.0343 -0.0062 4- 1.0600 -0.0045 4- 1.2389 -0.0043 + 1.2389 -0.6676 4- 2.3206 -0.6813 + 2.3933 -0.7125 4- 2.4606 -0.7252 + 2.4778 -0.7066 + 2.4807 -0.7325 4- 2.5748 -1.1823 + 2.8937 -1.2108 + 2.9733 -1.2238 4- 3.0177 -2.2575 + 3.7678 -2.3022 + 3.801 li -2.3828 + 3.8887 -0.3e-8 + 0.1876 -0.0005 + 0.1877 -0.0037 4- 0.2515 -0.0004 + 0.2679 -0.0004 + 0.6092 -0.0093 4- 0.6117 -0.0024 + 0.6428 -0.0042 + 0.6448 -0.0036 + 0.6489 -0.0028 4- 0.6563 -0.0001 + 0.6903 -0.0056 + 0.6910 -0.0181 + 1.0343 -0.0029 4- 1.0587 0.0000 + 1.2375 -0.0044 4- 1.2389 -0.6674 4- 2.3318 -0.6932 4- 2.4306 -0.7125 4- 2.4642 -0.0058 4- 2.5418 -0.7247 + 2.5627 -0.0044 + 2.5898 -1.1842 4- 2.9214 -1.2149 4- 2.9781 -0.0079 + 3.2286 -2.2922 4- 3.7970 -2.3427 4- 3.8591 -0.0032 + 4.4488 -0.0005 4- 0.1877 -0.0005 + 0.1877 -0.0003 4- 0.2514 -0.0003 4- 0.2679 -0.0097 + 0.6106 -0.0093 4- 0.6117 -0.0035 4- 0.6441 -0.0027 + 0.6441 -0.0030 + 0.6485 -0.0023 + 0.6562 -0.0058 4- 0.6907 -0.0055 + 0.6910 -0.0041 4- 1.0309 -0.0047 4- 1.0596 -0.0044 4- 1.2388 -0.0043 4- 1.2389 -0.6991 + 2.4474 -0.6997 + 2.4557 -0.6914 4- 2.4778 -0.0126 4- 2.4818 -0.7084 + 2.4821 -0.0162 4- 2.5758 -1.2102 4- 2.9729 -1.2061 + 2.9779 -0.0085 + 3.1672 -2.3014 4- 3.8008 -2.3012 4- 3.8040 -0.0067 + 4.5129 Table 1.1. Eignevalues of the 6-bay truss example energy transfer from the structure to the controller as a criterion to decide on-off times. The main objective is to demonstrate that such a method is viable and to give a controller design methodology. In Chapter 2. analytical results are presented to demonstrate the use of on-off controllers to damp the vibrations in an energy efficient way. Energy input into the controller is used as a criterion to select the switching times. The controllers are switched on only when work is done on them. This does not require the use of an 15 external energy source. Since energy supplied to the controller would require that fuel be consumed, the proposed design methodology reduces the fuel consumption for vibration suppression. Additionally, the proposed control laws are decentralized since the on-off switching laws are based on the local displacements and velocities. In Chapter 3, controller design methodologies are given to select the controller parameters to achieve the desired modal damping. Analytical results are given to prove the validity of this. A method which is similar to the Modified Independent Modal Space Control (MIMSC) is used for designing on-off controller for multi degree of freedom systems. Simulation results are presented to show the effectiveness and accuracy of this method. In Chapter 4, the use of optimal control laws with fuel consumption as a cost criterion to design vibration controllers is presented. First, the concept of on-off vibration control is reviewed with an emphasis on fuel consumption minimization. Next, the problems associated with window sizing and selecting the sampling interval are brought to light. A discussion of energy efficient switching control laws future research problems are presented in Chapter 5. CHAPTER 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 2.1 Introduction In this work a minimum energy control method for on-off decentralized control of mechanical systems is introduced. Energy consumption is minimized by turning on the controllers when the structure does work on the controller and turning them off when the controller would impart energy on the structure. Vibrating energy from lightly damped modes is transferred to highly damped modes to introduce damping. In the next chapter, this method is applied to solve pole placement problems for vibrating systems. The work presented here is based on the intuition that using an energy minimiza tion criterion that works with conservation principles will produce energy efficient methods for control of vibrating systems. The methodology presented here is quite feasible, since the switching of the electro-mechanical controller is performed electron ically via an on-board processor. The development uses energy as an optimization criterion. The delays introduced by the decision making are not included, but can be very easily incorporated into the analytical results presented here. Several authors used or proposed the use of on-off' control for attitude and shape control of large flexible space structures ( Velde and He (1983), Foster and Silver- berg (1991), Arbel and Gupta (1981), Masri. Bekev, and Caughev (1981), Rohman and Leipholz (1978), Sevwald, Kumar, Deshpande, and Heck (1994). Redmond and 16 17 Silverberg (1992), Neustadt (1960) and Reinhorn. Soong, Riley, Lin. Aizawa, and Higashino (1993)). The approaches using on-off control devices are relatively simple and require less on-line computational effort than other modern control techniques. High energy pulses can be input into the system using on-off devices. In the works of Zimmerman. Inman, and Juang (1991), Zimmerman (1990), Masri et al. (1981), in order to conserve energy, control devices are activated when some specified thresholds of states has been exceeded. The pulse magnitude was determined analytically so as to minimize a non-negative cost function related to the system energy. The degree of control obtained depended on the threshold level considered and the cost function that was minimized. Foster and Silverberg (1991), Arbel and Gupta (1981), Masri et al. (1981), Sey- wald et al. (1994), Redmond and Silverberg (1992) and Neustadt (1960) used the principle of fuel minimization as a criterion for the implementation of on-off switch ing control. Flight duration of spacecraft is limited in many cases by the amount of on-board fuel. For satellites, on-board fuel is needed to fire the thrusters and apogee/perigee motors to place them in the final orbit. Thruster firings will be needed for station-keeping and to correct orbital errors. As the size of spacecraft increases, the need to suppress the vibrations introduced into the spacecraft body must also be incorporated into the controller design. The work described in this chapter differs from previous works in that, here, fuel consumption is measured by the amount of external energy needed to damp out the vibrations. More discussions about the proposed scheme and the types of controllers that may be used with the proposed scheme may be found in the next two sections. 2.2 Controlling Vibrations Using On-Off Controllers Without External Energy In general, when sub ject to vibrational control, a structure does work on its con troller and vice-versa. Here it is proposed that the work done on the controllers by the 18 vibrating structure be taken advantage of to arrive at a fuel-optimal solution. This is accomplished by switching the controller on only when the structure is doing work on the controller and switching it off otherwise. 2.2.1 Switching procedure In this section it is demonstrated how an on-off control scheme using only the energy input rate into the controller can get a better response from the system. Preliminary versions of this work are presented in Schwartz and Maben (1996. 1995). Consider an n-dimensional second order dynamical system of the form Mx + Dx + Kx = 0, (2.2.1) where M, D, K are the mass, damping and stiffness matrices respectively. D is assumed to be of proportional type, i.e. D satisfies D = aM + 0K, where a, p are scalar constants. Also, K = KT > 0 and M = MT > 0. When the controller is off. the energy in the system is given by Eaff = -[xr Mx + xr Kx xT Dx], (2.2.2) Since the system is dissipative, the rate of change of energy is given by ff = xt[Mx + {K)x] xT Dx. (2.2.3) If the controller is of the spring type, the rate of change of energy while the controller is on is given by = xt[Mx + (K + A K)x] xT Dx, (2.2.4) where AK is the added stiffness due to the controller. It is assumed that AK is symmetric and positive definite. In this study we are not concerned with measuring the energy stored in a con trolling device. Rather we are interested in studying the energy input into such a 19 device by observing the rate of change of energy function. Our aim is to increase the energy into such devices (on the assumption that a unidirectional force on a spring type of controller can be converted into some other form of energy and released into an energy dissipating or storage device and not released back into the structure). The controllers are thus switched on only when this energy rate is positive. It should also be noted that x in all these cases represents the displacement over a bias value, measured in reference to an inertial level. The signal Eon E0Â¡Â¡ xT AKx gives a measure of the rate of work done on all controllers if they were on continuously. In the case of one controller, if AKa is the stiffness introduced by the controller, xa is the relative displacement and xa is the velocity at the controller location, 0fÂ¡ is positive (work is done on the controller by the structure), if the product of force acting on the controller AKaxa and the velocity at the controller location xa is positive. In the case of multiple controllers, if i denotes the location of the ittl controller, [AKiXi] is monitored and each controller is switched on when the force signal acting on it and the velocity at the controller location are positive. The force acting on the elements of AK matrix can be positive or negative. Assuming that the force in the positive direction (representing compression in spring type of controller) can be released into some other energy dissipating device (for example a resistor network), controllers are switched on only when the forces acting on them are positive. This concept is summarized in the following switching procedure. It should be noted that the location of the controller plays a role in the controlla bility of the vibrating modes of the structure. If the location of an controller is close to the node of a vibrating mode, that controller will have very little effect on the damping of that particular mode of vibration. 20 2.2.2 Procedure for switching with one controller 1. Between switching times, monitor the signals AKaxa and xa where AKa is the nonzero entry on the AA' matrix. 2. When both AKaxa and xa are positive, switch the controller on. Switch the controller off at the end of a predetermined time which is related to the system time constant as determined by the lowest period of oscillation in the structure.1 (and release the energy absorbed by the spring when it is switched off). 2.2.3 Example Consider a 5 degree of freedom spring mass system shown in Figure 2.2.3. Here we have a second order system, Mx+Cx+Kx = f where the mass and stiffness matrices are chosen as, mass = 1 0 0 0 1 o 1 CO -1 0 0 O < 0 1 0 0 0 -1 2 -1 0 0 0 0 1 0 0 stif = 0 -1 2 -1 0 0 0 0 1 0 0 0 -1 2 -1 o i 0 0 0 1 0 0 0 -1 3 \-VWW~ p p p |4i p5 ml ]\/VWV m2 1WWV~j^ WWV m4 -WWv m5 |_WW\^ \ Figure 2.1. A 5 DOF Spring/Mass System In this example xa is X\. All but the fourth mode have very little damping (equiv alent damping factor of 0.002), while the fourth mode is highly damped(equivalent damping factor of 0.4). The simulations of the control procedure of Section 2.2.2 show this method to be very effective in damping out the vibrations of the measured lIn the example in the next subsection, this time is set to be one tenth of lowest period of oscillation. 21 response even when the initial conditions were very close to the mode shapes of a lesser damped mode of the uncontrolled structure, a worst case scenario. Figures 2.2 through 2.6 show the nodal (position) response of the uncontrolled and controlled system when the initial conditions were set to the mode shape of one of the lesser damped mode of the uncontrolled structure. The results are only shown for one set of initial conditions, however the simulation was carried out for each of the mode shapes. From these plots it is clear that the proposed method is effective in increasing the damping. Time in seconds Figure 2.2. Response at node 1 22 Time in seconds Figure 2.3. Response at node 2 Figure 2.4. Response at node 3 2.3 Summary A minimum energy on-off vibration control law for structures was introduced. The control procedure takes advantage of the energy which can be imparted to the controllers from switching by turning on the controllers when the structure is doing work on the controller and turning them off when the controllers are doing work on the structure. 23 Time in seconds Figure 2.5. Response at node 4 Time in seconds Figure 2.6. Response at node 5 The proposed scheme uses the fact that a structure does work on the controllers and vice versa while undergoing vibrations. The controllers are switched on only when they are absorbing energy from the structure. This absorbed energy can be transformed into some other form (to do some useful work like storing energy in a device like battery or gyro). The proposed scheme achieves the desired goal without the use of an external power source (except for decision making and measurements). In one application which gives promise to the proposed scheme, Yang, Mikulas. Park, 24 and Su (1995) have shown that controlled moment gyros can be used for the slewing maneuvers and vibration control of space structures. CHAPTER 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING 3.1 Introduction This chapter provides methods for choosing controller parameters AK in order to achieve specified modal damping for single degree of freedom systems (SDOF) and multi degree of freedom systems (MDOF). First, analytical results are presented for SDOF systems. Second, results are given for MDOF systems. Modified Independent Modal Control Methods (MIMC) are applied to MDOF systems to achieve the desired modal damping. It is also shown that introducing damping in only few modes of the uncontrolled system may cause poor responses in other modes. In view of this, it is recommended that all dominant modes of the structure be damped when using the proposed switching control design. 3.2 Designing On-Off Controller for SDOF system 3.2.1 Analytical Results Consider the scalar single of freedom system described by mx + kx + iiAkx = 0, (3.2.1) where m is the mass, k is the stiffness, Ak is the added stiffness due to the controller action, x is the displacement of the mass, and n is a parameter which takes values 1 or 0, depending, respectively, upon whether or not the controller is on. The switching 25 26 control is expected to introduce damping into the system. The controller is assumed to be located at the node associated with this mass and Ak is positive. The control objective is to achieve a specified average damping rate. The following is a design procedure for specifying Ak so as to achieve this objective. The energy in the system is given by E (mx2 + kx2 + fiAkx2') (3.2.2) Let ui be the natural frequency of the uncontrolled system, (i.e. u/2 = . The transformation sfmx = u gives By imposing the control scheme described in the previous chapter, // is set to 1 when t is such that Aku > 0, or equivalently when u > 0. It is expected that the proposed switching control scheme introduces damping so that u is assumed to be of the form: u = a sin(uit + P), (3.2.4) where a = a(t) is an exponentially decaying function of time, and uj is the uncon trolled natural frequency of oscillation. A functional relationship between the rate of decay function a and the feedback gain Ak will be used to determine a law for introducing a desired average damping into the system. Taking the time derivative of both sides of equation (3.2.4), = ctu)cos(ujt + Â¡3) + sin(uit 4- 0). (3.2.5) The decay rate a is assumed to be small compared to aw, so the term may be neglected in equation (3.2.5). This gives U2 + 27 aujcos(u)t + (3). (3.2.6) By conservation of energy, E = 0. Note that this is true at all instances except switching: Thus = i-(ii2 + uj2u2) + imjj2vi\ =0. (3.2.7) y 2 at k ) Using equations (3.2.6) and (3.2.4) in equation (3.2.7) yields i2a + ^-^-a2iosm2 (ujt +/3) = 0. (3.2.8) Z K Since the decay rate of the response is assumed to be slow, the method of averaging- in (Caughev & OKelly, 1965) may be applied to compute : A A: l =a / ij, sin 2(cut + d)dt. (3.2.9) k 2 Jo Since u > 0 holds for only half a cycle of sin 2(out + /3), /z is set to 1 for only half of a cycle of sin2(af + /3). In that case, the value of the integral in equation (3.2.9) is 1 /ui and or Ak uj 27TUI (3.2.10) a = a0e Thus, for the single degree of freedom case, determines a control law for achieving a desired average damping rate Â£ = This result is summarized in the following proposition. 28 3.2.2 Proposition For a single degree of freedom system of the form (3.2.1), on-off switching control may be used to attain a desired overall average damping rate ( using the proposed minimum energy control scheme in conjunction with equation (3.2.11) 3.2.3 SDOF Example The main objective of this example is to show the effectiveness of this method and to show the validity of assumption made earlier, that the decay rate a is assumed to be small compared to au>. Consider the single degree of freedom system with magnitudes of mass and stiff ness respectively given by m=l and k=100. Simulation of the system with on-off controller was carried out for three values of the damping. For each of these damp ing, Ak was calculated using equation (3.2.11). The on-off controller was simulated and the response of the system to an initial displacement of 1 and velocity of 0 was observed. The response of the system for values of damping of 0.0015, 0.020 and .6 are given in Figures 3.1, 3.2 and 3.3 respectively. In each of these figures, expo nentially decayed curves are plotted to show the exactness of this method for low values of damping. From these time responses, it is concluded that the assumption that a is an exponentially decayed function is valid. Also, it can be seen that when the assumption that A << au is violated, the damping introduced into the system because of switching controller deviates much from the desired value. 29 Figure 3.1. Response of SDOF Â£ = 0.0015 Figure 3.2. Response of SDOF Â£ = 0.02 3.3 Designing On-Off Controll for an MDOF system 3.3.1 Modified Independent Modal Control Methods Active control of the vibrations of flexible structures based primarily on modal control methods whereby the vibrations are suppressed by controlling the dominant modes of vibration has been the focus of many researchers (Lindberg Jr. & Longman, 30 Figure 3.3. Response of SDOF Â£ = 0.6 1984; Balas, 1978b, 1978a; Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Can- field & Meirovitch, 1994). Generally these modal control methods belong either to the class of coupled methods or to the class of independent modal space control meth ods developed by Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Canfield & Meirovitch, 1994). Using coupled methods, the closed loop equations of the system are coupled using feedback control wherein the optimal com putation of the feedback gains require the solution of a couple matrix Riccati equation (Balas, 1978b, 1978a). For large flexible structures the solution of the Riccati equation can pose serious difficulties which limit significantly the applicability of the coupled modal control methods. The IMSC method, however, avoids such serious limitations, as the control laws are designed completely in the modal space, using the uncoupled open loop equations of the system as a set of independent second order equations even after including the feedback controllers. Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh. 1983; Canfield & Meirovitch, 1994) showed that 31 under such conditions it is possible to compute the closed form the optimal modal feedback gains. However, the IMSC method requires the use of as many controllers as the number of modes to be controlled. Such a requirement results in a practical limitation of the method when applied to large structures where the number of modeled modes can be very large. Lindberg and Longman (1984) proposed to modify the IMSC by using a small number of controllers to control all the modeled modes through the approximate pseudo-inverse(PI) realization of the modal controller. This modification can result in physical control forces which can be far from desired because the PI is, in effect, a least squares fit of N modal forces to obtain P physical forces. When N equals P the realization is exact and is also the same as IMSC. As N becomes much smaller than P, the accuracy in obtaining P physical forces from N model forces becomes poor. Accordingly, when the realized forces are transformed back to the modal space the resulting modal forces will be very far from the optimal forces and this will result in deterioration in the performance of the controller. For these reasons, the Modified Independent Modal Space Control (MIMSC) method was initiated (Baz & Poh, 1987). The MIMSC modifies the IMSC algorithm to account for the control spillover from the controlled modes into the uncontrolled modes when a small number of controllers is used to control a large number of modes. This method also incorporates an optimal placement procedure for determining the optimal location of the controllers in the structure. Moreover, the MIMSC method relies on an efficient algorithm for time-sharing a small number of controllers in the modal space to control a large number of modes. The MIMSC uses N optimally placed controllers to control the N modes that have the highest modal energy at any instant of time and time share these controllers among the other residual modes when the control spillover makes the modal energy higher than the controlled modes. 32 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems In this section, a new design methodology is proposed for using on-off control to achieve desired damping in modes of interest. This method is based on energy transfer (from kinetic to potential), as well as on the dissipative nature of damping forces. Time-sharing of controllers was proposed in (Baz & Poh, 1987, 1991), and these ideas are used here. Consider a second order dynamical system for the n-dimensional displacement vector x given by Mx + Kx = F. (3.3.12) where M = MT > 0 is the mass matrix of the structure, K = KT > 0 is the stiffness matrix of the structure, and F is n-dimensional vector of forces acting on the structure. Equation (3.3.12) may written in modal coordinates by using the following weighted modal transformation: x = 4>u, (3.3.13) where u represents the vector of modal coordinates of the system, and $ is the systems modal shape matrix (i.e. the matrix of eigenvectors). Thus, under the transformation (3.3.13), equation (3.3.12) reduces to the form + \u = f, (3.3.14) where A is the diagonal matrix of eigenvalues of M~lK matrix and / = <&rF. The controller being on, yields the dynamics: Mi + Kx AKx. (3.3.15) 33 where (K + AA')r = (K + A A') > 0. Under switching control, it is desired that the system behave like a damped dy namical system of the form Mx + Dx + Kx 0, (3.3.16) where D is an n x n desired proportional damping matrix: i.e. D = alVI + 0K. where a,P are scalar constants. Note that the mass and stiffness matrices are the same. The energy stored (or released) in time T due to the added stiffness at the con troller is given by eak = \ [ ~ (xtAKx) dt. (3.3.17) The energy dissipated due to the damping term D over an interval of length T is given by If! (*'")* (3-3-l8) Assuming the controller is only on for T seconds, and using equations (3.3.13) and (3.3.17) the contribution of the ith mode to the energy stored in the controller is Eaio = \[ Jf (ui KAK*i] ui) dt (3.3.1.9) where is the ith column of would be dissipated in the desired dynamics through the damping over the period T, is E, = \[' Jt (i ui) dt. (3.3.20) Here Tt is taken as the period of the ith mode of vibration. Since proportional damping is assumed, D$i = 2CiUi, (3.3.21) 34 where Q is the desired modal damping and u>i is the natural frequency of the ith mode. If Ui is assumed to be of the form -u, = e-iiisin (w), then i(t) = Qe-^sin (ujit) + e_,o;cos (urf) (3.3.22) and i(t) = Q2e_,tsm (ujit) 2Qe~^'tLicos (Ujt) e^uPsin (a;). (3.3.23) The structure does work on the controller during only half of the cycle of vibration of a given mode and during the other half of the cycle, the controller releases energy into the structure. This logic motivates the scheme for switching the controller on only for half of the period of the mode to be controlled. Using T = E- the two integrals (in equations (3.3.18) and (3.3.19)) may be evaluated, and the contribution of mode i to the energy stored in the controller over its period (when the controller is only on for half the period) is equated to the contribution of mode i to the energy dissipated in the damping term over its period. This yields an expression for AK (nonzero elements of AK). Although there is no restriction that AK be diagonal, for simplifying the expressions (and to make them more clear), AK is assumed to be diagonal. Thus j such that AKjj^O This depends only on one those entries in the mlh column of $, for which the corre sponding entries on the diagonal matrix AA' are non zero. Take first the case that only one entry in AA' is nonzero. Let j mark the index of this entry. Evaluating the derivatives in the integrals of equations (3.3.19) and (3.3.20) (eval uating the first integral for only half the cycle as described earlier) gives 1 rT'/2 \ Eak> = 9 yo 2 (y-i&jiAKjjUi) dt (3.3.25) 35 and Ei = \ Jo {i^iUJiUi + ^2C*^t) dt. (3.3.26) Using the expressions given earlier for u, . AA'$j, &[D$i, given by equa tions (3.3.22.3.3.23.3.3.24,3.3.21) define gli, ,g2. g3i (the expanded form of the inte grand in equation (3.3.25) and terms in the integrand of equation ( 3.3.26)) as gli [' ( sin(uiit) + e~C'ituiicos(ijJit))e~citsin{uit)AKjj^i2ij)dt (3.3.27) J o /2s. g2i = / 2(Cf e~('itsin(uJit)+2{Q)e~^tu}iCOs(u!lt)e~^itLi>fsin(it))e~citsin(uJit)) Q uiidt. J o (3.3.28) /at g3i = I 2(C e sin(ujit) + e ^Ui cos{ojit))2uji Qdtt. (3.3.29) Jo Since we are considering the contribution of only one mode at this point, there will be a nonzero contribution from only one diagonal element of AK matrix. Solving gli = g2i+g3i gives an expression for the nonzero diagonal element in the AK matrix as AK = (3.3.30) When there is more than one mode to control, the matrix AA^ should have only as many non-zero diagonal entries as the number of modes to be controlled. In this case, gli takes the form: / (sin(u>it) + e~citu>iCOs(uJit))e~(itsin(u)it)AKjj$2jdt. j such that AKjj^O In such cases, the integrand in equation (3.3.25) and equation (3.3.26) will have as many terms as the number of modes to be controlled and as many equations of 36 the form c/1, = g'2i + #3, as are necessary will have to be solved in order to obtain values of different non-zero elements of the AA' matrix. With more than one mode to control, there will be m expressions of the form (3.3.25) and m expressions of the form (3.3.26) (one for each of the desired modes to be controlled). This will lead to m equations of the form (3.3.27)-(3.3.29) which can be solved for the m nonzero diagonal entries of AA' matrix. The flowchart for the proposed MDOF on-off control scheme is given in Figure 3.4. Note that at any given time, controllers are only set to act on one mode in order to increase its damping. The switching procedure computes the energy in different modes of the structure and checks whether the energy in one of the desired modes is the highest. If it finds that energy in one of the desired modes is taking the highest value, for half the period of vibration of this mode, all the controllers are dedicated to damp the vibrations in this mode (the timing is kept track by the use of a counter Tc). During this half cycle of vibration, controllers are individually switched on and off depending upon whether or not work is done on them by the structure. 3.3.3 Four Degree of Freedom example with damping in one mode Consider a four degree of freedom undamped system with mass and stiffness prop erties Mass = ' 2 0 0 0 0 0 3 0 0 4 0 0 0 ' 0 0 5 (3.3.31) 3-20 O' -24-20 0-25-2 0 0-26 Stiffness (3.3.32) 37 The controller was placed at the first mass and the damping in the first mode was desired to be 0.2. Unlike the third example (Kabes problem) in this section, this model had mass normalized eigenvector components of the same order (components of different eigenvectors at nodal locations). Equation 3.3.30 suggests that controller gains are inversely proportional to the eigenvector components. This suggested that placement of controller at any of the masses would have resulted in a value of AA' in the same order. The on-off controller scheme proposed in Section 4.3.2 was im plemented. Nodal and Modal time responses of the system are given in Figures 3.5 and 3.6 respectively. From the plot of modal responses, it is clear that the control scheme introduces damping in first mode. The value of this damping is found to be 0.19. Figure 3.7 gives the energy in different modes and Figure 3.8 gives the velocity and modal force for the first mode of vibration. From Figure 3.8 it is clear that the control scheme introduces damping into first mode. 3.3.4 Four Degree of Freedom example with damping in all modes In this example, controller parameters used in the simulation are such that all four modes of the four DOF system given in equations (3.3.31) and (3.3.32) exhibit the desired damping. The desired damping values in the four modes are .2,.15,0.20 and 0.17. The natural frequencies of the uncontrolled system are 0.556. 0.9657, 1.2791 and 1.5511 radians. The system was simulated and the algorithm given in the previous section was implemented. It was observed that system behaves like a well damped system with modal damping very close to the desired values. This was simulated with several initial conditions. Figures 3.9-3.12 give the nodal response and Figures 3.13- 3.16 give the modal response of this system. The observed modal damping values are 0.18, 0.14. 0.18 and 0.16. The natural frequencies of the system with control acting upon it are identified as 0.7454, 0.9827, 1.1310 and 1.2454 radians/second respectively, matching exactly with the natural frequencies of the uncontrolled system. 38 3.3.5 Kabes Problem With damping in only two inodes Kabes eight degree of freedom model is shown in Figure 3.17. The natural fre quencies of this uncontrolled system are 30.0280. 31.7163, 31.7557, 32.3020, 33.3267, 35.5763, 38.7887 and 41.8884 radians/second respectively. The mass-normalized eigenvectors of this system are listed in Table 4.1. In order to avoid high values of controller gains, the controllers were placed at 4i/l and 6th masses. The controller locations were selected based upon the mass normalized eigenvector components. The proposed design procedure shows that the reciprocal of the square of the eigenvector component at the location of the controller decides the gain gain of the controller (i.e. if damping is desired in mode j and controller is placed at location g, reciprocal of the square of the gth component of the jth mass normalized eigenvector decides the controller gain). Note that the fourth and sixth mass normalized eigenvector have fourth and sixth components which are within reasonable limits (leading to reason able values for the elements of AK matrix. The desired damping in the fourth and sixth mode were set to 0.10 and 0.15, respectively. The initial conditions were set such that the initial energy was concentrated in the fourth and sixth modes. When these conditions were simulated, it was seen that the control algorithm achieves the desired goals within reasonable limits. Figures 3.18-3.25 give the nodal response and Figures 3.26-3.25 give the modal response of this system. In order to show the impor tance of the controller locations, the same system was simulated with the controllers located at the first and the second mass. The desired damping in fourth mode and sixth mode were set to 0.1 and 0.15. Controller gains in this case were 42 and 38166 compared to 2724 and 3486 in the earlier case. Dampings achieved in this case were .06007 and 0.1076 in mode 4 and 6 compared to the values of 0.08 and 0.14 in the previous case. In order to implement the proposed scheme, the controller gains would have to be such that they do not significantly affect the mass of the system. If the 39 size and weight of the controllers change the mass of the system the effects of this will have to be taken into consideration. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 -2.1170 -30.0002 -2.5940 -10.6279 -5.6896 -2.6102 -31.6226 0.0002 -0.0267 -0.3125 -0.0269 -0.1023 -0.0467 -0.0129 0.0030 0.00001 -0.2832 -0.0319 0.0040 0.3769 0.4901 0.3401 -0.0001 0.0001 -0.5873 -0.0020 0.0068 0.2612 0.0014 -0.5282 0.0001 -0.0005 -0.5973 0.0329 -0.0806 -0.6016 -0.0013 0.1988 0.0001 0.0001 -0.2841 -0.0029 0.0827 0.3724 -0.4890 0.3361 0.0001 0.0032 -0.0308 -0.0085 0.6378 -0.1347 0.0598 -0.0172 0.0001 0.0020 -8.0992 -0.2782 16.9175 8.5882 -15.5332 14.9251 -0.0005 -22.3605 Table 3.1. Mass normalized eigenvectors of Kabes model 40 INPUT STRUCTURAL PARAMETERS. CONTROLLER LOCATION? AND LOADING Set Controller flag off. Reset To Compute time history of all nodes in physical coordinates Compute time history of all nodes in modal coordinates No Compute energy in all modes. Find Energy in desired mode j Switch the controllers on if the Store j. work done on them is positive. Set Controller flag on. Figure 3.4. Flowchart of MDOF damping control algorithm 41 Figure 3.5. Nodal Response of a 4-DOF System with Cides = 0.2 Mode 4 Figure 3.6. Modal Response of a 4-DOF system with Cides = 0.2 Mode 4 Energy 43 Figure 3.7. Energy in different modes of a 4-DOF system with Â£ldes = 0.2 Figure 3.8. Modal force and velocity (for the first mode) 44 Figure 3.9. Response at Node 1 of a 4-DOF system (with all modes controlled) Figure 3.10. Response at Node 2 of a 4-DOF system (with all modes controlled) 45 Figure 3.11. Response at Node 3 of a 4-DOF system (with all modes controlled) Figure 3.12. Response at Node 4 of a 4-DOF system (with all modes controlled) 46 Figure 3.13. Mode 1 Response of a 4-DOF system (with all modes controlled) Figure 3.14. Mode 2 Response of a 4-DOF system (with all modes controlled) 47 Figure 3.15. Mode 3 Response of a 4-DOF system (with all modes controlled) Figure 3.16. Mode 4 Response of a 4-DOF system (with all modes controlled) 48 Figure 3.17. Kabes Model 49 Figure 3.18. Controlled response at Node 1 of Kabes Model (with only two modes controlled) Figure 3.19. Controlled response at Node 2 of Kabes Model (with only two modes controlled) 50 Figure 3.20. Controlled response at Node 3 of Kabes Model (with only two modes controlled) Figure 3.21. Controlled response at Node 4 of Kabes Model (with only two modes controlled) 51 Figure 3.22. Controlled response at Node 5 of Kabes Model (with only two modes controlled) Figure 3.23. Controlled response at Node 6 of Kabes Model (with only two modes controlled) 52 Figure 3.24. Controlled response at Node 7 of Kabe's Model (with only two modes controlled) 53 Figure 3.25. controlled) Figure 3.26. controlled) Controlled response at Node 8 of Kabes Model (with only two modes Controlled mode 1 Response of Kabes Model (with only two modes 54 Figure 3.27. Controlled mode 2 Response of Kabes Model (with only two modes controlled) Figure 3.28. Controlled mode 3 Response of Kabes Model (with only two modes controlled) 55 Figure 3.29. Controlled mode 4 Response of Kabes Model (with only two modes controlled) 56 Figure 3.30. Controlled mode 5 Response of Kabes Model (with only two modes controlled) 57 Figure 3.31. controlled) Figure 3.32. controlled) Controlled mode 6 Response of Kabes Model (with only two modes Controlled mode 7 Response of Kabes Model (with only two modes 58 Figure 3.33. Controlled mode 8 Response of Kabes Model (with only two modes controlled) 3.3.6 Kabes Problem With damping in all modes In this section Kabes eight degree of freedom system was simulated with damp ing introduced in all modes. The desired damping values in the eight modes are .20.0.18,0.15.0.10,0.16.0.15,0.13 and 0.12. Figures 3.34-3.41 show the modal responses of this system. From these figures it is clear that controlling all the modes of the system gives better performance when compared to the case with only few modes con trolled. The achieved damping values in the eight modes are 0.19,0.16,0.14.0.08,0.14.0.14,0.11 and 0.11. The oscillations that are seen in Figure 3.41 are due to the activation of the local modes of vibration. The natural frequencies of the controlled modes were iden tified as 30.0280. 31.7163, 31.7557, 32.3020, 33.3267, 35.5763. 38.7887 and 41.8884 radians/second respectively, which match the uncontrolled natural frequencies. 59 Figure 3.34. Controlled mode 1 Response of Kabes Model (with all modes controlled) Figure 3.35. Controlled mode 2 Response of Kabe's Model(with all modes controlled) 60 Figure 3.36. Controlled mode 3 Response of Kabes Model (with all modes controlled) Figure 3.37. Controlled mode 4 Response of Kabes Model (with all modes controlled) 61 Figure 3.38. Controlled mode 5 Response of Kabes Model (with all modes controlled) Figure 3.39. Controlled mode 6 Response of Kabes Model (with all modes controlled) 62 Figure 3.40. Controlled mode 7 Response of Kabe's Model (with all modes controlled) Figure 3.41. Controlled mode 8 Response of Kabes Model (with all modes controlled) 63 3.4 Summary A design methodology for the energy based on-off switching decentralized control of mechanical structures is presented. Energy consumption is minimized by turning on the controllers when a structure is doing work on the controllers, and turning them off after prescribed on-times. For a SDOF system, closed form solution is given for designing the controller parameter. For a MDOF system the proposed methodology can be used to achieve a desired average damping rate when each mode may be uniquely controlled. The controller design methodology and the switching schemes are presented. Though the proposed scheme suggests the use of no external energy, in practice, such methods would require some energy input for signal monitoring, as well as to drive an electro-mechanically switched controller using a computer or a microprocessor. Since altering the configuration of sensors and controllers does not alter the fact that structures with positive damping dissipate energy, the passivity theorem (Morris & Juang, 1994) can be applied in this case to prove that modal forces from the con trollers will not destabilize the system. Knowledge of the system is used in obtaining the controller parameters. Changes or errors in the system or controller parameters, however, will not destabilize the system because it is assumed to be dissipating energy. CHAPTER 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL 4.1 Introduction The work on switching control presented in Chapter 2 and 3 was motivated by a study of the literature which used on-off control of vibrations, such as Athanassiades (1963), Athans (1964b, 1964a), Foster and Silverberg (1991), Silverberg (1986), Red mond and Silverberg (1992) and Medith (1964). The work of Foster and Silverberg (1991) is one such study which uses fuel minimization as an optimization criterion for on-off control of structural vibrations. In that work, the minimum fuel use control law imposed that control pulses be applied every time a zero crossing of the the po sition (and maximum value of the velocity) would occur. A plot of fuel consumption versus sampling time was also presented in (Foster & Silverberg, 1991). This chapter is devoted to the study of the effects of sampling time and window size on the fuel consumption. 4.2 Minimum Fuel Control of Mechanical Systems Foster and Silverberg (1991) developed a minimum fuel based switching control law for MDOF second order systems of the form: Mx 4- Kx = F, (4.2.1) where M = Ml is the n x n mass matrix of the structure, K is the n x n stiffness matrix of the structure, x and x are, respectively, the n-dimensional displacements 64 65 and accelerations at the nodal points of the structure, and F is the n-dimensional vector of forces acting on the structure. Since the control law depends on points (in time) of maximum velocity (or mini mum position) these are determined using running averages of velocities and positions: = j a(s,t)f(s)ds, J a(s,t)ds = 1 rf(t) = Jo aisi t)(f{s) nf(s))2ds, (4.2.2) (4.2.3) where a(s, t) is a window function of the form: / .X r 1 /T, s> t-T ( ,f| o, s < t T. (4.2.4) Here, T is the window size. Equations (4.2.2) and (4.2.3) are used in determining the control weightings for a piecewise constant (vector) control law. In that work, the fuel consumption C(t) in time t is measured using CW = f l/.l ds (4.2.5) i= 1 J0 where /(s) is the ilh component of the control force at time s, in modal coordinates. 4.3 Effects of changes in window size and sampling period 4.3.1 Example 1 Foster and Silverberg (1991) worked out an example prescribed by the following parameters: M = 1, K = 1 and T = 10. 66 where T is the number of samples in the window. In Figure 4.1 the fuel consump tion was plotted as a function of number of samples per period of oscillation (denoted by N, a parameter which determines the sampling period). Notice the sharp increase in fuel consumption for certain values of N. As it turns out, when these increases in fuel consumption occur, so does a trig gering of the higher harmonics in the control pulses. The fast fourier analysis of the control pulses shows that the ratio of signal power at the fundamental frequency to the sum of the signal power at the higher harmonic frequencies decreases at the points of positive slope in this plot. ; 1 i 1 1 1 0 10 20 30 40 50 60 Number of samples per period of oscillation Figure 4.1. Effect of Number of Samples per period of oscillation on fuel consumption (example 1) The frequency spectra of the power of control signal are shown in Figures 4.2-4.5 for selected points on the plot shown in Figure 4.1. The fundamental frequency of this system is 0.159 Hz. The ratio of the power in the control signal at the fundamental frequency to that represented by the higher- harmonics decreases at the points of 67 positive slope on the plot in Figure 4.1. This is seen in the plots in 4.3 and 4.5. In these plots note that for values of N = 15 and N = 25, the power at higher harmonics take higher values when compared to plots for values of N where the slope of the fuel curve shown in Figure 4.1 is negative. At the points of positive slope of the plot in Figure 4.1, sampling time is such that the window size becomes an integral multiple of one of the higher harmonics. Note that for N = 15 window size is twice the period of third harmonic and for N = 25 window size is twice the period of fifth harmonic. Figure 4.2. Frequency spectrum of control pulse in example 1 (N=ll) Signal power 68 Figure 4.3. Frequency spectrum of control pulse in example 1 (N=15) 69 Figure 4.4. Frequency spectrum of control pulse in example 1 (N=21) 4.3.2 Example 2 In this example, the system parameters used are: M = 1, K = 2, and T 10. The system was simulated using the same control scheme and the effects of sam pling time on fuel consumption was studied. The plots of fuel consumption and spectra of control signal are given in Figures 4.6-4.10. The natural frequency of vi bration is .23 Hz. Just like in the previous example, the higher harmonics in the control signal make the fuel consumption to increase at certain values of sampling times. 4.4 Summary The effects of sampling on fuel consumption using Foster and Silverberg (1991)s minimum fuel on-off control law was studied. It was noted that when the significant 70 Figure 4.5. Frequency spectrum of control pulse in example 1 (N=25) harmonics in the control signal impose that window size is integer multiple of the period of higher harmonics, fuel consumption was increased. 71 Figure 4.6. Effect of Number of Samples per period of oscillation on fuel consumption (example 2) Figure 4.7. Frequency spectrum of control pulse in example 2 (N=9) 72 Figure 4.8. Frequency spectrum of control pulse in example 2 (N=ll) Figure 4.9. Frequency spectrum of control pulse in example 2 (N=25) Signal power 73 CHAPTER 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 5.1 Discussion This study proposed new schemes for structural control design using the energy input into the controller to damp out the vibrations. Through simulations it was shown that such a decentralized control law can be efficient in improving the perfor mance of structures. For SDOF systems, the proposed design methodology introduces desired damping within very close bounds. For MDOF systems, using the techniques of Independent Modal Space Control, a design algorithm was developed to introduce specified damping into desired modes of vibration. The controllers were decentralized in that the decision for switching a given controller was based on feedback from local states only. It was also shown how the uncontrolled modes are affected by controlling only few modes in multi degree of freedom systems. This suggests that the proposed methodology be used to control all of the dominant modes of the vibrating structure. 5.2 Future Research The minimum energy on-off control methods proposed in this work should increase the life span of space structures since very minimal energy is needed from the external power with such a method. To fully demonstrate the practical advantage of these schemes, one should de termine the kind of controllers that can be used for such applications. A simple 74 10 gyroscope, magnetic bearing gyros, or torque wheels provide potentially viable op tions for such applications. Control momentum gyros have been effectively used for the attitude control of spacecrafts. For instance, single gimbaled gyros have been installed on the Soviet MIR station, while double gimbaled gyros were used in the NASA Skylab. Momentum gyro-based control designs have been tried in the recent past for slewing maneuvers and vibration control of space structures. 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ASCE Journal Of Structure Engineering, 119, 1935-1960. Rohman, M. A., & Leipholz, H. H. E. (1978). Structural Control By Pole Assignment method. Journal of Engineering mechanics division, ASCE, 104. 1159-1175. R.P.Corfmat, & A.S.Morse (1976). Control of linear system with specified input channels. SIAM Journal of Control, 1\. Schwartz, C. A., & Maben, E. N. (1995). A Minimum Energy Approach to Switching- Control for Mechanical Systems. Proceedings of the first block island workshop on logic and switching control Schwartz, C. A., & Maben, E. N. (1996). On-Off Minimum Energy control of Structures. To Appear in 1st European Structural Control Conference in Barcelona, Seywald, H., Kumar, R. R., Deshpande, S. S., & Heck, M. (1994). Minimum fuel spacecraft reorientation. Journal of Guidance ,Control and Dynamics, 17, 21-29. Silverberg, L. (1986). Uniform Damping Control of Spacecraft. Journal of Guidance, Control and Dynamics, 9, 221-227. Sofonov, M. G., & Athans, M. (1977). Gain and phase margins for multiloop LQG regula tors. IEEE Transactions on Automatic Control, AC-22, 173-177. Souza, M. L. O. (1988). Exactly solving the weighted time/fuel optimal control of an undamped harmonic oscillator. Journal of Guidance, Control and Dynamics, 11, 488- 494. Swanson Analysis Systems Inc, Houston, Pennsylvania (1992). ANSYS 5.0 Users Manual -Revision 5.0a Vol 1-4 Tarras, A. (1987). Decentralized Control of Large Scale Systems, Sensitivity and Parameter Robustness. Ph.D. thesis, National Center For Scientific Research- France. Velde, W. E. V., & He, J. (1983). Design of Space structure control systems using on-off thrusters. Journal of Guidance, Control and Dynamics, 6, 53-60. Wang, S. H. (1982). Stabilization of decentralized controller via time varying controllers. IEEE Transactions on Automatic control, AC-27, 741-744. Wang, S. H., & Davison, E. J. (1973). On the stabilization of decentralized control systems. IEEE Transactions on Automatic Control, AC-18, 473-478. West-vukovich, G. S., &; Davison. E. J. (1984). The Decentralized control of large flexible space structures. IEEE Transactions on Automatic Control, 29, 866-879. Witsenhausen, H. S. (1973). A standard form for sequential stochastic control. Journal of Mathematical Systems Theory, 1. 80 Yang, L., Mikulas, M. M., Park, K., k Su, R. (1995). Slewing Maneuvers and Vibration Control of Space Structures by Feedforward/Feedback Moment-Gyro Controls. Journal of Dynamic Systems, Measurement, and Control, 117, 343-351. Young, D. (1990). Distributed Finite Element modeling and Control Approach for large flexible Structures. Journal of Guidance, 13, 703-713. Zimmerman. D. C. (1990). Threshold Control for Nonlinear and time varying structures using equivalent transformation. Journal of Intelligent Material Systems and Structures, 1, 76-90. Zimmerman. D. C., Inman, D. J., & Juang, J. N. (1991). Vibration Suppression Using a Constrained Rate Feedback Control Strategy. Journal of Vibration and Acoustics. 113, 345-352. BIOGRAPHICAL SKETCH I was born in Mysore City, India, on September 14, 1964. I spent most of my early childhood at Mangalore with my parents, brother and two sisters. I graduated from St. Aloysius College in Mangalore in 1982. I went on to get my Bachelor of Engineering degree from Karnataka Regional Engineering College in Surathkal. India in 1986. I received my Master of Technology degree from the Indian Institute of Technology in Bombay in 1988. After working in the Indian Institute of Technology, Bombay for a brief period of three months, I joined Hindustan Aeronautics Limited. Bangalore. Within a short time of six months, I was appointed as a Scientist at the Indian Space Research organization. After serving the Indian Space Industry for two years, I attended University of Florida where I have received my Ph.D. in Engineering Mechanics. 81 I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David Zjmpaerman, Chairman Associate Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Carla Schwartz, Co-Chairm Assistant Professor of Elect Engineering Computer I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctoj^of-Philosophy. KbimarfFitz-Coy Assistant Professo: of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. a OU\A Ffarrfph Latchman Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy/ Jacob Hamper Professor of Electrical and Computer Engineering This dissertation was submitted to the Graduate Faculty of the College of En gineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1996 Karen A. Holbrook Dean, Graduate School LD 1780 19% M IIX* UNIVERSITY OF FLORIDA 3 1262 08554 9219 69 Figure 4.4. Frequency spectrum of control pulse in example 1 (N=21) 4.3.2 Example 2 In this example, the system parameters used are: M = 1, K = 2, and T 10. The system was simulated using the same control scheme and the effects of sam pling time on fuel consumption was studied. The plots of fuel consumption and spectra of control signal are given in Figures 4.6-4.10. The natural frequency of vi bration is .23 Hz. Just like in the previous example, the higher harmonics in the control signal make the fuel consumption to increase at certain values of sampling times. 4.4 Summary The effects of sampling on fuel consumption using Foster and Silverberg (1991)s minimum fuel on-off control law was studied. It was noted that when the significant To my parents and teachers 55 Figure 3.29. Controlled mode 4 Response of Kabes Model (with only two modes controlled) 63 3.4 Summary A design methodology for the energy based on-off switching decentralized control of mechanical structures is presented. Energy consumption is minimized by turning on the controllers when a structure is doing work on the controllers, and turning them off after prescribed on-times. For a SDOF system, closed form solution is given for designing the controller parameter. For a MDOF system the proposed methodology can be used to achieve a desired average damping rate when each mode may be uniquely controlled. The controller design methodology and the switching schemes are presented. Though the proposed scheme suggests the use of no external energy, in practice, such methods would require some energy input for signal monitoring, as well as to drive an electro-mechanically switched controller using a computer or a microprocessor. Since altering the configuration of sensors and controllers does not alter the fact that structures with positive damping dissipate energy, the passivity theorem (Morris & Juang, 1994) can be applied in this case to prove that modal forces from the con trollers will not destabilize the system. Knowledge of the system is used in obtaining the controller parameters. Changes or errors in the system or controller parameters, however, will not destabilize the system because it is assumed to be dissipating energy. 3.3.2 Analytical Results For Designing On-Off Controller for MDOF systems 32 3.3.3 Four Degree of Freedom example with damping in one mode 36 3.3.4 Four Degree of Freedom example with damping in all modes 37 3.3.5 Kabes Problem With damping in only two modes 38 3.3.6 Kabes Problem With damping in all modes 58 3.4 Summary 63 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL 64 4.1 Introduction 64 4.2 Minimum Fuel Control of Mechanical Systems 64 4.3 Effects of changes in window size and sampling period 65 4.3.1 Example 1 65 4.3.2 Example 2 69 4.4 Summary 69 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 74 5.1 Discussion 74 5.2 Future Research 74 REFERENCES 76 BIOGRAPHICAL SKETCH 81 vi 60 Figure 3.36. Controlled mode 3 Response of Kabes Model (with all modes controlled) Figure 3.37. Controlled mode 4 Response of Kabes Model (with all modes controlled) Mode 4 Figure 3.6. Modal Response of a 4-DOF system with Cides = 0.2 31 under such conditions it is possible to compute the closed form the optimal modal feedback gains. However, the IMSC method requires the use of as many controllers as the number of modes to be controlled. Such a requirement results in a practical limitation of the method when applied to large structures where the number of modeled modes can be very large. Lindberg and Longman (1984) proposed to modify the IMSC by using a small number of controllers to control all the modeled modes through the approximate pseudo-inverse(PI) realization of the modal controller. This modification can result in physical control forces which can be far from desired because the PI is, in effect, a least squares fit of N modal forces to obtain P physical forces. When N equals P the realization is exact and is also the same as IMSC. As N becomes much smaller than P, the accuracy in obtaining P physical forces from N model forces becomes poor. Accordingly, when the realized forces are transformed back to the modal space the resulting modal forces will be very far from the optimal forces and this will result in deterioration in the performance of the controller. For these reasons, the Modified Independent Modal Space Control (MIMSC) method was initiated (Baz & Poh, 1987). The MIMSC modifies the IMSC algorithm to account for the control spillover from the controlled modes into the uncontrolled modes when a small number of controllers is used to control a large number of modes. This method also incorporates an optimal placement procedure for determining the optimal location of the controllers in the structure. Moreover, the MIMSC method relies on an efficient algorithm for time-sharing a small number of controllers in the modal space to control a large number of modes. The MIMSC uses N optimally placed controllers to control the N modes that have the highest modal energy at any instant of time and time share these controllers among the other residual modes when the control spillover makes the modal energy higher than the controlled modes. 48 Figure 3.17. Kabes Model 36 the form c/1, = g'2i + #3, as are necessary will have to be solved in order to obtain values of different non-zero elements of the AA' matrix. With more than one mode to control, there will be m expressions of the form (3.3.25) and m expressions of the form (3.3.26) (one for each of the desired modes to be controlled). This will lead to m equations of the form (3.3.27)-(3.3.29) which can be solved for the m nonzero diagonal entries of AA' matrix. The flowchart for the proposed MDOF on-off control scheme is given in Figure 3.4. Note that at any given time, controllers are only set to act on one mode in order to increase its damping. The switching procedure computes the energy in different modes of the structure and checks whether the energy in one of the desired modes is the highest. If it finds that energy in one of the desired modes is taking the highest value, for half the period of vibration of this mode, all the controllers are dedicated to damp the vibrations in this mode (the timing is kept track by the use of a counter Tc). During this half cycle of vibration, controllers are individually switched on and off depending upon whether or not work is done on them by the structure. 3.3.3 Four Degree of Freedom example with damping in one mode Consider a four degree of freedom undamped system with mass and stiffness prop erties Mass = ' 2 0 0 0 0 0 3 0 0 4 0 0 0 ' 0 0 5 (3.3.31) 3-20 O' -24-20 0-25-2 0 0-26 Stiffness (3.3.32) 26 control is expected to introduce damping into the system. The controller is assumed to be located at the node associated with this mass and Ak is positive. The control objective is to achieve a specified average damping rate. The following is a design procedure for specifying Ak so as to achieve this objective. The energy in the system is given by E (mx2 + kx2 + fiAkx2') (3.2.2) Let ui be the natural frequency of the uncontrolled system, (i.e. u/2 = . The transformation sfmx = u gives By imposing the control scheme described in the previous chapter, // is set to 1 when t is such that Aku > 0, or equivalently when u > 0. It is expected that the proposed switching control scheme introduces damping so that u is assumed to be of the form: u = a sin(uit + P), (3.2.4) where a = a(t) is an exponentially decaying function of time, and uj is the uncon trolled natural frequency of oscillation. A functional relationship between the rate of decay function a and the feedback gain Ak will be used to determine a law for introducing a desired average damping into the system. Taking the time derivative of both sides of equation (3.2.4), = ctu)cos(ujt + Â¡3) + sin(uit 4- 0). (3.2.5) The decay rate a is assumed to be small compared to aw, so the term may be neglected in equation (3.2.5). This gives U2 + 35 and Ei = \ Jo {i^iUJiUi + ^2C*^t) dt. (3.3.26) Using the expressions given earlier for u, . AA'$j, &[D$i, given by equa tions (3.3.22.3.3.23.3.3.24,3.3.21) define gli, ,g2. g3i (the expanded form of the inte grand in equation (3.3.25) and terms in the integrand of equation ( 3.3.26)) as gli [' ( sin(uiit) + e~C'ituiicos(ijJit))e~citsin{uit)AKjj^i2ij)dt (3.3.27) J o /2s. g2i = / 2(Cf e~('itsin(uJit)+2{Q)e~^tu}iCOs(u!lt)e~^itLi>fsin(it))e~citsin(uJit)) Q uiidt. J o (3.3.28) /at g3i = I 2(C e sin(ujit) + e ^Ui cos{ojit))2uji Qdtt. (3.3.29) Jo Since we are considering the contribution of only one mode at this point, there will be a nonzero contribution from only one diagonal element of AK matrix. Solving gli = g2i+g3i gives an expression for the nonzero diagonal element in the AK matrix as AK = (3.3.30) When there is more than one mode to control, the matrix AA^ should have only as many non-zero diagonal entries as the number of modes to be controlled. In this case, gli takes the form: / (sin(u>it) + e~citu>iCOs(uJit))e~(itsin(u)it)AKjj$2jdt. j such that AKjj^O In such cases, the integrand in equation (3.3.25) and equation (3.3.26) will have as many terms as the number of modes to be controlled and as many equations of 8 where z, 6 RVi is the state of the ith feedback controller, v{ e Rm' is the ith local external input, and Mt, F, G, Hi, Li, RÂ¡ are real constant matrices of appropriate sizes. The decentralized pole placement problem is to find matrices Mi, Fi,Gi, Hi, Li, Ri such that the closed loop system described by (1.3.1)-(1.3.4) has prespecified poles. Of course, if [G,, ,4, B,] is controllable and observable from all the stations, the solution to such problem is guaranteed. The interesting case is to assume that (1.3.2) is controllable from all controls U(, U/v but not from any single control with similar observability assumptions. Consider first the special case j\/t- = 0 in the above problem. This corresponds to a static decentralized output feedback controller. If F denotes the collection of feedback matrices (Flt F2, F.v), then the pole placement problem is to determine F such that the matrix AFA + 'Â£iBiFiCi (1.3.5) =i has an arbitrarily specified set of eigenvalues. Clearly, a necessary condition for pole placement in this case is that the polynomials | XI A? \ have no common factor, i.e. that cv( A) = g.c.d. | AI Ap |= 1 (1.3.6) where g.c.d. is greatest common divisor. What is more interesting is that this con dition is both necessary and sufficient for pole placement with dynamic compensa tion Wang Davison, 1973). More generally, since the zeros of a (A) (termed as the fixed modes of the system) are invariant under decentralized dynamic compensation, it follows that a necessary and sufficient condition for stabilizability is that the roots of a (A) have strictly negative real parts. Computation of the fixed modes of a system can be accomplished by computing the eigenvalues of Ap and A for randomly selected F and checking for common eigenvalues. A simpler wav of checking for the fixed modes of the system is given in 18 vibrating structure be taken advantage of to arrive at a fuel-optimal solution. This is accomplished by switching the controller on only when the structure is doing work on the controller and switching it off otherwise. 2.2.1 Switching procedure In this section it is demonstrated how an on-off control scheme using only the energy input rate into the controller can get a better response from the system. Preliminary versions of this work are presented in Schwartz and Maben (1996. 1995). Consider an n-dimensional second order dynamical system of the form Mx + Dx + Kx = 0, (2.2.1) where M, D, K are the mass, damping and stiffness matrices respectively. D is assumed to be of proportional type, i.e. D satisfies D = aM + 0K, where a, p are scalar constants. Also, K = KT > 0 and M = MT > 0. When the controller is off. the energy in the system is given by Eaff = -[xr Mx + xr Kx xT Dx], (2.2.2) Since the system is dissipative, the rate of change of energy is given by ff = xt[Mx + {K)x] xT Dx. (2.2.3) If the controller is of the spring type, the rate of change of energy while the controller is on is given by = xt[Mx + (K + A K)x] xT Dx, (2.2.4) where AK is the added stiffness due to the controller. It is assumed that AK is symmetric and positive definite. In this study we are not concerned with measuring the energy stored in a con trolling device. Rather we are interested in studying the energy input into such a 17 Silverberg (1992), Neustadt (1960) and Reinhorn. Soong, Riley, Lin. Aizawa, and Higashino (1993)). The approaches using on-off control devices are relatively simple and require less on-line computational effort than other modern control techniques. High energy pulses can be input into the system using on-off devices. In the works of Zimmerman. Inman, and Juang (1991), Zimmerman (1990), Masri et al. (1981), in order to conserve energy, control devices are activated when some specified thresholds of states has been exceeded. The pulse magnitude was determined analytically so as to minimize a non-negative cost function related to the system energy. The degree of control obtained depended on the threshold level considered and the cost function that was minimized. Foster and Silverberg (1991), Arbel and Gupta (1981), Masri et al. (1981), Sey- wald et al. (1994), Redmond and Silverberg (1992) and Neustadt (1960) used the principle of fuel minimization as a criterion for the implementation of on-off switch ing control. Flight duration of spacecraft is limited in many cases by the amount of on-board fuel. For satellites, on-board fuel is needed to fire the thrusters and apogee/perigee motors to place them in the final orbit. Thruster firings will be needed for station-keeping and to correct orbital errors. As the size of spacecraft increases, the need to suppress the vibrations introduced into the spacecraft body must also be incorporated into the controller design. The work described in this chapter differs from previous works in that, here, fuel consumption is measured by the amount of external energy needed to damp out the vibrations. More discussions about the proposed scheme and the types of controllers that may be used with the proposed scheme may be found in the next two sections. 2.2 Controlling Vibrations Using On-Off Controllers Without External Energy In general, when sub ject to vibrational control, a structure does work on its con troller and vice-versa. Here it is proposed that the work done on the controllers by the 23 Time in seconds Figure 2.5. Response at node 4 Time in seconds Figure 2.6. Response at node 5 The proposed scheme uses the fact that a structure does work on the controllers and vice versa while undergoing vibrations. The controllers are switched on only when they are absorbing energy from the structure. This absorbed energy can be transformed into some other form (to do some useful work like storing energy in a device like battery or gyro). The proposed scheme achieves the desired goal without the use of an external power source (except for decision making and measurements). In one application which gives promise to the proposed scheme, Yang, Mikulas. Park, 37 The controller was placed at the first mass and the damping in the first mode was desired to be 0.2. Unlike the third example (Kabes problem) in this section, this model had mass normalized eigenvector components of the same order (components of different eigenvectors at nodal locations). Equation 3.3.30 suggests that controller gains are inversely proportional to the eigenvector components. This suggested that placement of controller at any of the masses would have resulted in a value of AA' in the same order. The on-off controller scheme proposed in Section 4.3.2 was im plemented. Nodal and Modal time responses of the system are given in Figures 3.5 and 3.6 respectively. From the plot of modal responses, it is clear that the control scheme introduces damping in first mode. The value of this damping is found to be 0.19. Figure 3.7 gives the energy in different modes and Figure 3.8 gives the velocity and modal force for the first mode of vibration. From Figure 3.8 it is clear that the control scheme introduces damping into first mode. 3.3.4 Four Degree of Freedom example with damping in all modes In this example, controller parameters used in the simulation are such that all four modes of the four DOF system given in equations (3.3.31) and (3.3.32) exhibit the desired damping. The desired damping values in the four modes are .2,.15,0.20 and 0.17. The natural frequencies of the uncontrolled system are 0.556. 0.9657, 1.2791 and 1.5511 radians. The system was simulated and the algorithm given in the previous section was implemented. It was observed that system behaves like a well damped system with modal damping very close to the desired values. This was simulated with several initial conditions. Figures 3.9-3.12 give the nodal response and Figures 3.13- 3.16 give the modal response of this system. The observed modal damping values are 0.18, 0.14. 0.18 and 0.16. The natural frequencies of the system with control acting upon it are identified as 0.7454, 0.9827, 1.1310 and 1.2454 radians/second respectively, matching exactly with the natural frequencies of the uncontrolled system. 2 control (Lin. 1981), and adaptive control methods (Behnhabib, Iwens, & Jackson, 1979). Two good survey papers deal with the problem of LFSS control (Gran & Rossi, 1979; Balas, 1982). West-vukovich and Davison (1984) have shown how the design of active structural controllers that emulate the real structural elements such as dampers and springs can produce effective control laws. A suboptimal control approach is used in that work to design the structural characteristics (gains) of the controller elements. This allows integration of passive controller, active controller and sensor/actuator locations design. This method exhibits problems due to delays introduced due to digital implementation, and consideration of transfer function of real sensors and actuators. Definition. Fixed modes of the system are modes of the system that remain in variant under any nonzero parametric variation of the system or the controller. The fixed modes of a LFSS result when a sensor and actuator are located at a node of a flexural mode. It was shown that for LFSS with col-located actuators and sensors, the fixed modes of the decentralized system are the same as the centralized fixed modes of the system. A controller that will eliminate the spillover problem is demonstrated. It is also shown that a solution to the decentralized control problem exists only if there is a solution to the centralized system. A further explanation of fixed modes is provided in section 3. 1.2 Advantage of Using Decentralized Control Over the past twenty-five years, engineers and scientists have developed a variety of procedures for analyzing systems and for designing control strategies for controlling LFSS. These procedures are classified into three types: 1. Procedures for modeling dynamical systems (state space formulation, input- output transfer function description, etc) 4 achieve improved system behavior by using state feedback. However, it is often im possible to instrument a system to the extent required for full state feedback, so techniques ranging from linear-quadratic Gaussian (LQG) control, to observer based control to time domain compensator design techniques have been employed to over come this difficulty. However, a key concept in these types of designs is that every sensor output affects every actuator input. This situation is termed as centralized control. In many large scale systems, it is impossible to implement this in real-time. Thus for economic and possibly reliability reasons, there is a trend for decentral ized decision making, distributed computing and hierarchical control. However, these desirable goals of structuring a distributed information and decision framework for large scale systems do not mesh with the available centralized methodologies and tools of modern control system design. Reduction of computation and simplification of structure are of particular concern in decentralized control of large scale systems, but are also of concern in almost all areas of control theory and its applications. The basic characteristic of decentralized control is that there are restrictions on the information transfer between certain groups of sensors and actuators. For example, in figure 1, state variables AT are used to form the control U\ and state variables A2 are used to form the control t/2. This depicts total decentralization. However, inter mediate restrictions on the information between controllers are also possible. Partial decentralization takes place when the system is not fully decentralized but the rate of information transfer is constrained so that full centralized control is not possible. It is to be noted that the concept of decentralization refers to the control structure implementation; the control laws may be designed in a completely centralized way (provided there are no other physical limitations preventing this). The advantages of using decentralized control were mentioned in the Ph.D Thesis of Ahmed Tarras and by several other researchers (Tarras, 1987; West-vukovich & I 1.3 Decentralized Stabilization and Pole Placement A fundamental result in modern control theory is that the poles of a controllable system can be arbitrarily assigned (subject to complex pole-pairing constraints) by state feedback. This result has been extended to show that the poles of a closed loop system consisting of controllable and observable linear system with a dynamic compensator can be freely assigned. These results are of great theoretical significance and have served as the basis of practical synthesis procedure. A natural generalization of the pole-placement question arises when the restriction to decentralized feedback control is made. Although several authors had looked at this question,(McFadden, 1967; Aoki, 1972; Corfmat & Morse. 1973) the most distinct results are those of Wang and Davison (1973), Davison (1976) and R.P.Corfmat and A.S.Morse (1976). Here, their results are briefly summarized. For a linear system the problem of decentralized pole placement can be formulated as follows. Consider the linear system N x(t) = Ax(t) + E BMt) (1.3.1) 1=1 yi(t) = CiX(t) (1.3.2) where i = 1, N indexes the input and output variables of the various controllers. x Rn is the state, Rmi and G RTi are the input and output, respectively, of the ith local control station. .4 is the state matrix, BÂ¡ is the input matrix and C, is the output matrix of control station i. The ith controller employs dynamic compensation of the form Ui{t) = MtZi(t) + Fiyiit) + GiVi(t) (1.3.3) i{t) = HiZi(t) + Liyi{t) + RiVi{t) (1.3.4) 58 Figure 3.33. Controlled mode 8 Response of Kabes Model (with only two modes controlled) 3.3.6 Kabes Problem With damping in all modes In this section Kabes eight degree of freedom system was simulated with damp ing introduced in all modes. The desired damping values in the eight modes are .20.0.18,0.15.0.10,0.16.0.15,0.13 and 0.12. Figures 3.34-3.41 show the modal responses of this system. From these figures it is clear that controlling all the modes of the system gives better performance when compared to the case with only few modes con trolled. The achieved damping values in the eight modes are 0.19,0.16,0.14.0.08,0.14.0.14,0.11 and 0.11. The oscillations that are seen in Figure 3.41 are due to the activation of the local modes of vibration. The natural frequencies of the controlled modes were iden tified as 30.0280. 31.7163, 31.7557, 32.3020, 33.3267, 35.5763. 38.7887 and 41.8884 radians/second respectively, which match the uncontrolled natural frequencies. 80 Yang, L., Mikulas, M. M., Park, K., k Su, R. (1995). Slewing Maneuvers and Vibration Control of Space Structures by Feedforward/Feedback Moment-Gyro Controls. Journal of Dynamic Systems, Measurement, and Control, 117, 343-351. Young, D. (1990). Distributed Finite Element modeling and Control Approach for large flexible Structures. Journal of Guidance, 13, 703-713. Zimmerman. D. C. (1990). Threshold Control for Nonlinear and time varying structures using equivalent transformation. Journal of Intelligent Material Systems and Structures, 1, 76-90. Zimmerman. D. C., Inman, D. J., & Juang, J. N. (1991). Vibration Suppression Using a Constrained Rate Feedback Control Strategy. Journal of Vibration and Acoustics. 113, 345-352. Mode 4 Energy 43 Figure 3.7. Energy in different modes of a 4-DOF system with Â£ldes = 0.2 Figure 3.8. Modal force and velocity (for the first mode) 29 Figure 3.1. Response of SDOF Â£ = 0.0015 Figure 3.2. Response of SDOF Â£ = 0.02 3.3 Designing On-Off Controll for an MDOF system 3.3.1 Modified Independent Modal Control Methods Active control of the vibrations of flexible structures based primarily on modal control methods whereby the vibrations are suppressed by controlling the dominant modes of vibration has been the focus of many researchers (Lindberg Jr. & Longman, 5 Information Figure 1.1. A decentralized system Davison, 1984). The main advantages of using decentralized control laws are to overcome the problems due to the complexity of the system that results from 1. Large Dimensions 2. Uncertainty- deterministic or stochastic 3. Structural Constraints- these make the flow of information between the subsystems difficult. In controlling the complex systems, massive calculations, expensive computer time and high computer costs may be encountered. In addition, there may be large storage requirements, and multiple criteria may be encountered (due to spatial separations, different levels of operation, etc). Definition. A control system is called decentralized, if and only if all local controls are calculated only as an explicit function of the local information (states, output etc). This is a control with one level and no single controller has an overall view of the entire process. In centralized control, one viable approach to design of feedback control laws for time invariant linear systems is by minimization of an infinite horizon quadratic per formance index. LQ design method allows asymptotic pole placement by appropriate choice of the performance index (Kwakarnaak & Sivan, 1972) has excellent sensitivity and robustness properties according to various criteria, including the classical gain 14 Without Component failure With End controlled Component Failure With Center Controlled Component Failure -0.0005 + 0.1877 -0.0005 + 0.1877 -0.0037 + 0.2515 -0.0007 + 0.2679 -0.0101 + 0.6109 -0.0093 + 0.6117 -0.0054 + 0.6437 -0.0059 4- 0.6445 -0.0044 4- 0.6488 -0.0029 + 0.6563 -0.0059 + 0.6909 -0.0055 4- 0.6910 -0.0196 + 1.0343 -0.0062 4- 1.0600 -0.0045 4- 1.2389 -0.0043 + 1.2389 -0.6676 4- 2.3206 -0.6813 + 2.3933 -0.7125 4- 2.4606 -0.7252 + 2.4778 -0.7066 + 2.4807 -0.7325 4- 2.5748 -1.1823 + 2.8937 -1.2108 + 2.9733 -1.2238 4- 3.0177 -2.2575 + 3.7678 -2.3022 + 3.801 li -2.3828 + 3.8887 -0.3e-8 + 0.1876 -0.0005 + 0.1877 -0.0037 4- 0.2515 -0.0004 + 0.2679 -0.0004 + 0.6092 -0.0093 4- 0.6117 -0.0024 + 0.6428 -0.0042 + 0.6448 -0.0036 + 0.6489 -0.0028 4- 0.6563 -0.0001 + 0.6903 -0.0056 + 0.6910 -0.0181 + 1.0343 -0.0029 4- 1.0587 0.0000 + 1.2375 -0.0044 4- 1.2389 -0.6674 4- 2.3318 -0.6932 4- 2.4306 -0.7125 4- 2.4642 -0.0058 4- 2.5418 -0.7247 + 2.5627 -0.0044 + 2.5898 -1.1842 4- 2.9214 -1.2149 4- 2.9781 -0.0079 + 3.2286 -2.2922 4- 3.7970 -2.3427 4- 3.8591 -0.0032 + 4.4488 -0.0005 4- 0.1877 -0.0005 + 0.1877 -0.0003 4- 0.2514 -0.0003 4- 0.2679 -0.0097 + 0.6106 -0.0093 4- 0.6117 -0.0035 4- 0.6441 -0.0027 + 0.6441 -0.0030 + 0.6485 -0.0023 + 0.6562 -0.0058 4- 0.6907 -0.0055 + 0.6910 -0.0041 4- 1.0309 -0.0047 4- 1.0596 -0.0044 4- 1.2388 -0.0043 4- 1.2389 -0.6991 + 2.4474 -0.6997 + 2.4557 -0.6914 4- 2.4778 -0.0126 4- 2.4818 -0.7084 + 2.4821 -0.0162 4- 2.5758 -1.2102 4- 2.9729 -1.2061 + 2.9779 -0.0085 + 3.1672 -2.3014 4- 3.8008 -2.3012 4- 3.8040 -0.0067 + 4.5129 Table 1.1. Eignevalues of the 6-bay truss example energy transfer from the structure to the controller as a criterion to decide on-off times. The main objective is to demonstrate that such a method is viable and to give a controller design methodology. In Chapter 2. analytical results are presented to demonstrate the use of on-off controllers to damp the vibrations in an energy efficient way. Energy input into the controller is used as a criterion to select the switching times. The controllers are switched on only when work is done on them. This does not require the use of an 52 Figure 3.24. Controlled response at Node 7 of Kabe's Model (with only two modes controlled) 19 device by observing the rate of change of energy function. Our aim is to increase the energy into such devices (on the assumption that a unidirectional force on a spring type of controller can be converted into some other form of energy and released into an energy dissipating or storage device and not released back into the structure). The controllers are thus switched on only when this energy rate is positive. It should also be noted that x in all these cases represents the displacement over a bias value, measured in reference to an inertial level. The signal Eon E0Â¡Â¡ xT AKx gives a measure of the rate of work done on all controllers if they were on continuously. In the case of one controller, if AKa is the stiffness introduced by the controller, xa is the relative displacement and xa is the velocity at the controller location, 0fÂ¡ is positive (work is done on the controller by the structure), if the product of force acting on the controller AKaxa and the velocity at the controller location xa is positive. In the case of multiple controllers, if i denotes the location of the ittl controller, [AKiXi] is monitored and each controller is switched on when the force signal acting on it and the velocity at the controller location are positive. The force acting on the elements of AK matrix can be positive or negative. Assuming that the force in the positive direction (representing compression in spring type of controller) can be released into some other energy dissipating device (for example a resistor network), controllers are switched on only when the forces acting on them are positive. This concept is summarized in the following switching procedure. It should be noted that the location of the controller plays a role in the controlla bility of the vibrating modes of the structure. If the location of an controller is close to the node of a vibrating mode, that controller will have very little effect on the damping of that particular mode of vibration. 56 Figure 3.30. Controlled mode 5 Response of Kabes Model (with only two modes controlled) I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy/ Jacob Hamper Professor of Electrical and Computer Engineering This dissertation was submitted to the Graduate Faculty of the College of En gineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1996 Karen A. Holbrook Dean, Graduate School 77 Baz, A., k Poh, S. (Eds.). (1991). Modified Independent Modal Space Control With Positive Position Feedback and a Neural Observer. Eight VPI and SU Symposium on Dynamics and Control of Large Structures, Blacksburg, Va. Baz, A., Poh, S.. & Fedor, J. (1992). Independent Modal Space Control With Positive Position Feedback. Transactions of the ASME. 114, 96-103. Baz, A., Poh, S., k Studer, P. (1989). Modified Independent Modal Space Control Method For Active Control of Flexible Systems. Proceedings of Institution of Mechanical Engi neers, 203. 103-112. Behnhabib. R. J.. Ivvens, R. P., k Jackson, R. L. (1979). Adaptive control for large space structures. In Proceedings of IEEE Conference on Decision and Control, pp. 214-217. Brown, M. E. (Ed.). (1983). Rapid slewing maneuvers of a flexible spacecraft using on-off thrusters. Vol. CSDL-T-825. Charles Stark Draper Laboratory Inc, Cambridge,Ma. Canfield. R., k Meirovitch, L. (1994). Integrated structural design and vibration suppres sion using independent modal space control. AIAA Journal of Guidance, Control and Dynamics, 32, 2053-2060. Caughey, T. K., k OKelly, M. E. J. (1965). Classical Normal Modes in Damped Linear Dynamic Systems. ASME Journal of Applied mechanics, 32, 583-588. Corfmat, R., k Morse, A. (1973). Stabilization with decentralized feedback control. IEEE Transactions on Automatic Control, AC-18, 679-682. Date, R. A., k Chow, J. H. (1994). Decentralized Stable Factors and a Parameterization of Decentralized Controllers. IEEE Transactions on Automatic Control, AC-39, 347-351. Davison, E. J. (1975). The robust decentralized control of a general servomechanism prob lem. IEEE Transactions on Automatic Control, 21, 14-24. Davison, E. J. (1976). The robust decentralized control of a general servomechanism prob lem. IEEE Transactions on Automatic control, AC-21, 14-24. Foster, L. A., k Silverberg, L. (1991). On-Off Decentralized Control of Flexible structures. ASME Journal of Dynamic Systems Measurement and Control, 113, 41-47. Gran, R.. k Rossi. M. (1979). A survey of large space structures control problem. In IEEE Conference on Decision and Control, pp. 1002-1007. Hughes, P. C., k Skelton, R. E. (1981). Modal truncation for flexible spacecraft. Journal of Guidance Control and Dynamics, \, 291-297. Inman, D. J. (1989). Vibrations, with Control, Measurement and Stability (first edition). Prentice Hall, New Jersey. Katti, S. (1981). Comments on Decentralized control of Linear Multivariable Systems. Automtica, 17. 665. 44 Figure 3.9. Response at Node 1 of a 4-DOF system (with all modes controlled) Figure 3.10. Response at Node 2 of a 4-DOF system (with all modes controlled) 10 gyroscope, magnetic bearing gyros, or torque wheels provide potentially viable op tions for such applications. Control momentum gyros have been effectively used for the attitude control of spacecrafts. For instance, single gimbaled gyros have been installed on the Soviet MIR station, while double gimbaled gyros were used in the NASA Skylab. Momentum gyro-based control designs have been tried in the recent past for slewing maneuvers and vibration control of space structures. When used in conjunction with the proposed on-off control scheme, momentum gyros might pose additional problems associated with the controller dynamics due to the delays in troduced by them. The proposed methodology is based on the assumption that the controllers that are used do not release the absorbed energy back into the structure. The energy absorbed by the controller can be either stored or dissipated into some sink (like a heating element). Peizo-electric crystal based controllers can convert the force into electricity which may be stored. Gyro based controllers can convert the forces into angular momentum which may be used to drive small generators to gener ate electrical energy. The effects of such controllers on the mass of the structure will have to be investigated. 72 Figure 4.8. Frequency spectrum of control pulse in example 2 (N=ll) Figure 4.9. Frequency spectrum of control pulse in example 2 (N=25) CHAPTER 3 DESIGNING CONTROLLER TO ACHIEVE DESIRED DAMPING 3.1 Introduction This chapter provides methods for choosing controller parameters AK in order to achieve specified modal damping for single degree of freedom systems (SDOF) and multi degree of freedom systems (MDOF). First, analytical results are presented for SDOF systems. Second, results are given for MDOF systems. Modified Independent Modal Control Methods (MIMC) are applied to MDOF systems to achieve the desired modal damping. It is also shown that introducing damping in only few modes of the uncontrolled system may cause poor responses in other modes. In view of this, it is recommended that all dominant modes of the structure be damped when using the proposed switching control design. 3.2 Designing On-Off Controller for SDOF system 3.2.1 Analytical Results Consider the scalar single of freedom system described by mx + kx + iiAkx = 0, (3.2.1) where m is the mass, k is the stiffness, Ak is the added stiffness due to the controller action, x is the displacement of the mass, and n is a parameter which takes values 1 or 0, depending, respectively, upon whether or not the controller is on. The switching 25 51 Figure 3.22. Controlled response at Node 5 of Kabes Model (with only two modes controlled) Figure 3.23. Controlled response at Node 6 of Kabes Model (with only two modes controlled) Signal power 73 54 Figure 3.27. Controlled mode 2 Response of Kabes Model (with only two modes controlled) Figure 3.28. Controlled mode 3 Response of Kabes Model (with only two modes controlled) 24 and Su (1995) have shown that controlled moment gyros can be used for the slewing maneuvers and vibration control of space structures. LD 1780 19% M IIX* UNIVERSITY OF FLORIDA 3 1262 08554 9219 47 Figure 3.15. Mode 3 Response of a 4-DOF system (with all modes controlled) Figure 3.16. Mode 4 Response of a 4-DOF system (with all modes controlled) 20 2.2.2 Procedure for switching with one controller 1. Between switching times, monitor the signals AKaxa and xa where AKa is the nonzero entry on the AA' matrix. 2. When both AKaxa and xa are positive, switch the controller on. Switch the controller off at the end of a predetermined time which is related to the system time constant as determined by the lowest period of oscillation in the structure.1 (and release the energy absorbed by the spring when it is switched off). 2.2.3 Example Consider a 5 degree of freedom spring mass system shown in Figure 2.2.3. Here we have a second order system, Mx+Cx+Kx = f where the mass and stiffness matrices are chosen as, mass = 1 0 0 0 1 o 1 CO -1 0 0 O < 0 1 0 0 0 -1 2 -1 0 0 0 0 1 0 0 stif = 0 -1 2 -1 0 0 0 0 1 0 0 0 -1 2 -1 o i 0 0 0 1 0 0 0 -1 3 \-VWW~ p p p |4i p5 ml ]\/VWV m2 1WWV~j^ WWV m4 -WWv m5 |_WW\^ \ Figure 2.1. A 5 DOF Spring/Mass System In this example xa is X\. All but the fourth mode have very little damping (equiv alent damping factor of 0.002), while the fourth mode is highly damped(equivalent damping factor of 0.4). The simulations of the control procedure of Section 2.2.2 show this method to be very effective in damping out the vibrations of the measured lIn the example in the next subsection, this time is set to be one tenth of lowest period of oscillation. 45 Figure 3.11. Response at Node 3 of a 4-DOF system (with all modes controlled) Figure 3.12. Response at Node 4 of a 4-DOF system (with all modes controlled) 34 where Q is the desired modal damping and u>i is the natural frequency of the ith mode. If Ui is assumed to be of the form -u, = e-iiisin (w), then i(t) = Qe-^sin (ujit) + e_,o;cos (urf) (3.3.22) and i(t) = Q2e_,tsm (ujit) 2Qe~^'tLicos (Ujt) e^uPsin (a;). (3.3.23) The structure does work on the controller during only half of the cycle of vibration of a given mode and during the other half of the cycle, the controller releases energy into the structure. This logic motivates the scheme for switching the controller on only for half of the period of the mode to be controlled. Using T = E- the two integrals (in equations (3.3.18) and (3.3.19)) may be evaluated, and the contribution of mode i to the energy stored in the controller over its period (when the controller is only on for half the period) is equated to the contribution of mode i to the energy dissipated in the damping term over its period. This yields an expression for AK (nonzero elements of AK). Although there is no restriction that AK be diagonal, for simplifying the expressions (and to make them more clear), AK is assumed to be diagonal. Thus j such that AKjj^O This depends only on one those entries in the mlh column of $, for which the corre sponding entries on the diagonal matrix AA' are non zero. Take first the case that only one entry in AA' is nonzero. Let j mark the index of this entry. Evaluating the derivatives in the integrals of equations (3.3.19) and (3.3.20) (eval uating the first integral for only half the cycle as described earlier) gives 1 rT'/2 \ Eak> = 9 yo 2 (y-i&jiAKjjUi) dt (3.3.25) 71 Figure 4.6. Effect of Number of Samples per period of oscillation on fuel consumption (example 2) Figure 4.7. Frequency spectrum of control pulse in example 2 (N=9) 41 Figure 3.5. Nodal Response of a 4-DOF System with Cides = 0.2 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to Dr.David Zimmerman and Dr.Carla Schwartz for helping me in carry out this research work. Without their enormous patience, encouragement and guidance, it would not have been possible to complete the work. I consider myself extremely fortunate to have them as my dissertation advisors. I would like to thank Dr.Norman Fitz-Coy, Dr.Haniph Latchman and Dr.Jacob Hammer for serving in my committee. I would like to acknowledge the help provided by the 'Project Care team, specially Dr.Michael Conlon, Dr.Marilou Behnke, Dr.Fonda Eyler Davis and Kathie Wobie, by providing me with finacial support during my stay at the University of Florida. IV REFERENCES Anderson, B. D. O., & Moore, J. B. (1981). Time varying feedback laws for decentralized control. IEEE Transactions on Automatic control, AC-26, 1133-1139. Aoki, M. (1972). On feedback stabilizibility of decentralized dynamic systems. Automtica, 8, 163-173. Arbel, A., & Gupta, N. K. (1981). Robust Collocated control for large flexible space struc tures. Journal of Guidance and Control, f, 480-486. Athanassiades, M. (1963). Optimal control for linear time invariant plants with time, fuel and energy constraints. Transactions of the American Institute of Electrical Engineers, 81, 321-325. Athans, M. (1963). Minimum fuel feedback control systems: Second order case. IEEE Transactions on Applications and Industry, 82, 8-17. Athans, M. (1964a). Fuel optimal control of a double integral plant. IEEE Transactions on Applications and Industry, 83, 240-245. Athans, M. (1964b). Minimum fuel control of second order plants with real poles. IEEE Transactions on Applications and Industry, 83, 148-153. Balas, M. J. (1978a). Feedback Control of Flexible Systems. IEEE Transactions on Auto matic Control. 23, 673-679. Balas, M. J. (1978b). Modal control of certain flexible dynamic systems. SIAM Journal of Control Optimization, 16, 450-462. Balas, M. J. (1980). Enhanced Modal Control of Flexible Structures via Innovations Feedthrough. International Journal of Control, 32, 983-1003. Balas, M. J. (1982). Trends in Large Space Structure Control Theory: Fondest hopes, Wildest dreams. IEEE Transactions on Automatic control, 27. 522-535. Barbieri, E., & Ozguner, U. (1993). A New Minimal Time Control Law For a One Mode Model of a Flexible Slewing Structure. IEEE Transactions on Automatic Control, 38. 142-146. Baz, A., & Poli, S. (1987). A comparison between IMSC. PI and MIMSC Methods in controlling the vibration of flexible systems. Tech. rep. CR-181156, NASA. 76 Copyright 1996 by EGBERT N. MABEN 12 concept is used to place the actuators and sensors at internal degrees of freedom. Min imization of internal coordinate motions of different substructures would localize the dynamic interaction of coupled structure in the components. The component control action is designed to lock up its own boundary condition, that better approximates Figure 1.2. A Six-Bav Truss A convenient control design technique for this concept is the linear quadratic op timal control regulator approach, in which the aim is to minimize the internal coordi nates motion. The component control law is the one that minimizes the performance index, JSC = \ /o (ysTys + usTRus) dt where ys and us denote the output and control inputs at the internal degrees of freedom of each substructure. These are marked in the figure by numbers. The opti mal component control is a state feedback control law with positions and velocities as inputs. This type of controller is built for individual substructures and the controlled coupled structures state matrix is obtained. The effect of having such a controller on the eigenvalues is studied. Failures in the component controllers by changing the elements of the gain matrices of individual substructures. The gains of some channels DECENTRALIZED CONTROL OF FLEXIBLE SPACE STRUCTURES USING TIME VARYING FEEDBACK Bv EGBERT N. MABEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 70 Figure 4.5. Frequency spectrum of control pulse in example 1 (N=25) harmonics in the control signal impose that window size is integer multiple of the period of higher harmonics, fuel consumption was increased. 66 where T is the number of samples in the window. In Figure 4.1 the fuel consump tion was plotted as a function of number of samples per period of oscillation (denoted by N, a parameter which determines the sampling period). Notice the sharp increase in fuel consumption for certain values of N. As it turns out, when these increases in fuel consumption occur, so does a trig gering of the higher harmonics in the control pulses. The fast fourier analysis of the control pulses shows that the ratio of signal power at the fundamental frequency to the sum of the signal power at the higher harmonic frequencies decreases at the points of positive slope in this plot. ; 1 i 1 1 1 0 10 20 30 40 50 60 Number of samples per period of oscillation Figure 4.1. Effect of Number of Samples per period of oscillation on fuel consumption (example 1) The frequency spectra of the power of control signal are shown in Figures 4.2-4.5 for selected points on the plot shown in Figure 4.1. The fundamental frequency of this system is 0.159 Hz. The ratio of the power in the control signal at the fundamental frequency to that represented by the higher- harmonics decreases at the points of xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E7VACTKVO_IHEJ6T INGEST_TIME 2015-04-03T19:19:26Z PACKAGE AA00029742_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 3 2. Procedures for describing the qualitative properties of system behavior (controllability, stability, observability etc.) 3. Procedures for controlling system behavior (stabilizing feedback, optimal control, etc) All of these procedures rest on the common presupposition of centrality; all the infor mation available about the system, and the calculations based upon this information, are centralized. It is useful to distinguish between two kinds of available information: 1. Information about the system model: (off -line or a priori information) 2. Sensor information about the system response: set of all real time mea surements made of system response. The common modern estimation and control is based on centralized control the ory. When considering large scale systems, the centralized control assumption fails due either to the lack of centralized information or the lack of centralized comput ing capability. There are many examples of large scale systems that present a great challenge to both analysts and control system designers. Examples include power systems, urban Traffic networks, digital communication networks, flexible manufac turing networks and economic systems. The control of such physical systems are often characterized by spatial separation (such as physical separation between sensors and actuators) so that issues such as the economic cost and reliability of the communi cation link have to be taken into account in control, thus providing an impetus to decentralized control design. Research in decentralized control has been motivated by the inadequacy of modern control theory to deal with certain issues which are of concern in large scale systems. A key concept in modern control theory is that of state feedback. By using techniques such as Linear Quadratic (LQ) optimal control or pole placement, it is possible to 39 size and weight of the controllers change the mass of the system the effects of this will have to be taken into consideration. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 -2.1170 -30.0002 -2.5940 -10.6279 -5.6896 -2.6102 -31.6226 0.0002 -0.0267 -0.3125 -0.0269 -0.1023 -0.0467 -0.0129 0.0030 0.00001 -0.2832 -0.0319 0.0040 0.3769 0.4901 0.3401 -0.0001 0.0001 -0.5873 -0.0020 0.0068 0.2612 0.0014 -0.5282 0.0001 -0.0005 -0.5973 0.0329 -0.0806 -0.6016 -0.0013 0.1988 0.0001 0.0001 -0.2841 -0.0029 0.0827 0.3724 -0.4890 0.3361 0.0001 0.0032 -0.0308 -0.0085 0.6378 -0.1347 0.0598 -0.0172 0.0001 0.0020 -8.0992 -0.2782 16.9175 8.5882 -15.5332 14.9251 -0.0005 -22.3605 Table 3.1. Mass normalized eigenvectors of Kabes model CHAPTER 2 MINIMUM ENERGY ON-OFF CONTROL FOR MECHANICAL SYSTEMS 2.1 Introduction In this work a minimum energy control method for on-off decentralized control of mechanical systems is introduced. Energy consumption is minimized by turning on the controllers when the structure does work on the controller and turning them off when the controller would impart energy on the structure. Vibrating energy from lightly damped modes is transferred to highly damped modes to introduce damping. In the next chapter, this method is applied to solve pole placement problems for vibrating systems. The work presented here is based on the intuition that using an energy minimiza tion criterion that works with conservation principles will produce energy efficient methods for control of vibrating systems. The methodology presented here is quite feasible, since the switching of the electro-mechanical controller is performed electron ically via an on-board processor. The development uses energy as an optimization criterion. The delays introduced by the decision making are not included, but can be very easily incorporated into the analytical results presented here. Several authors used or proposed the use of on-off' control for attitude and shape control of large flexible space structures ( Velde and He (1983), Foster and Silver- berg (1991), Arbel and Gupta (1981), Masri. Bekev, and Caughev (1981), Rohman and Leipholz (1978), Sevwald, Kumar, Deshpande, and Heck (1994). Redmond and 16 62 Figure 3.40. Controlled mode 7 Response of Kabe's Model (with all modes controlled) Figure 3.41. Controlled mode 8 Response of Kabes Model (with all modes controlled) 61 Figure 3.38. Controlled mode 5 Response of Kabes Model (with all modes controlled) Figure 3.39. Controlled mode 6 Response of Kabes Model (with all modes controlled) 33 where (K + AA')r = (K + A A') > 0. Under switching control, it is desired that the system behave like a damped dy namical system of the form Mx + Dx + Kx 0, (3.3.16) where D is an n x n desired proportional damping matrix: i.e. D = alVI + 0K. where a,P are scalar constants. Note that the mass and stiffness matrices are the same. The energy stored (or released) in time T due to the added stiffness at the con troller is given by eak = \ [ ~ (xtAKx) dt. (3.3.17) The energy dissipated due to the damping term D over an interval of length T is given by If! (*'")* (3-3-l8) Assuming the controller is only on for T seconds, and using equations (3.3.13) and (3.3.17) the contribution of the ith mode to the energy stored in the controller is Eaio = \[ Jf (ui KAK*i] ui) dt (3.3.1.9) where is the ith column of would be dissipated in the desired dynamics through the damping over the period T, is E, = \[' Jt (i ui) dt. (3.3.20) Here Tt is taken as the period of the ith mode of vibration. Since proportional damping is assumed, D$i = 2CiUi, (3.3.21) 30 Figure 3.3. Response of SDOF Â£ = 0.6 1984; Balas, 1978b, 1978a; Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Can- field & Meirovitch, 1994). Generally these modal control methods belong either to the class of coupled methods or to the class of independent modal space control meth ods developed by Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh, 1983; Canfield & Meirovitch, 1994). Using coupled methods, the closed loop equations of the system are coupled using feedback control wherein the optimal com putation of the feedback gains require the solution of a couple matrix Riccati equation (Balas, 1978b, 1978a). For large flexible structures the solution of the Riccati equation can pose serious difficulties which limit significantly the applicability of the coupled modal control methods. The IMSC method, however, avoids such serious limitations, as the control laws are designed completely in the modal space, using the uncoupled open loop equations of the system as a set of independent second order equations even after including the feedback controllers. Meirovitch and co-workers (Meirovitch & Oz, 1978; Meirovitch & Baruh. 1983; Canfield & Meirovitch, 1994) showed that 78 Khargonekar. P., Poolla, K., k Tannenbaum, A. (1985). Robust control of linear time invariant plants using periodic compensation. IEEE Transactions on Automatic control, AC-30. Kwakarnaak, H., k Sivan, R. (1972). The maximally achievable accuracy of linear optimal regulators and linear optimal filters. IEEE Transactions on Automatic Control, AC-17, 79-86. Levine. W. S., k Athans, M. (1970). On the Determination of the Optimal Constant Output Feedback Gains for Linear Multivariable Systems. IEEE Transactions on Automatic Control. AC-15, 41-48. Lin. J. C. (1981). Closed loop asymptotic stability and robustness for large space systems with reduced order controllers. In IEEE Conference on Decision and Control, San Diego, CA., pp. 1497-1502. Lindberg Jr., R., k Longman, R. (1984). On the Number and Placement of Actuators for Independent Modal Space Control. Journal of Guidance and Control, 7, 215-221. Masri. S. F., Bekey, G., k Caughey, T. K. (1981). Optimum Pulse Control of Flexible Structures. Journal of Applied Mechanics, f8, 619-626. McFadden, D. (1967). Mathematical Systems Theory and Applications, chap. On the Con trollability of Decentralized Microeconomic systems: the assignment problem. Springer- Verlag. Medith, J. S. (1964). On minimum fuel satellite attitude control. Transactions of the American Institute of Electrical Engineers, 83, 120-128. Meirovitch, L., k Baruh, H. (1983). Robustness of the Independent Modal-Space Control Method. Journal of Guidance, 6, 20-25. Meirovitch. L., Baruh, H., Montgomery, A. C., k Williams, J. P. (1984). Uniform damping control of spacecraft. Journal of Guidance, Control and Dynamics, 7, 437-442. Meirovitch. L., k Oz, H. (1978). Modal space control of distributed gyroscopic systems. AIAA Journal of Guidance, Control and Dynamics, 3, 140-150. Morris, K. A., k Juang, J. N. (1994). Dissipative Controller Designs for Second Order Dynamic Systems. IEEE Transactions on Automatic Control, 39. 1056-1063. Morse, A. S. (1973). Structural invariants of linear multivariable systems. SIAM Journal of Control, 11, 446-465. Neustadt. L. (1960). Synthesizing time optimal control system. Journal of mathematical analysis and applications, 1, 484-493. Poh, S.. k Baz, A. (1990). Active Control of Flexible Structures Using a Modal Positive Position Feedback Controller. Tech. rep. CR-186336, NASA. Redmond, J., k Silverberg, L. (1992). Fuel consumption in optimal control. Journal of Guidance Control and Dynamics, 15, 424-430. CHAPTER 1 INTRODUCTION 1.1 Decentralized Control of Large Systems With the space shuttle transportation system now a practical reality, there is considerable interest in large space structures. Two problems which are inherent in the control of large space structures are shape control and attitude control. The latter involves maintaining a given orientation of the spacecraft with regards to an inertial reference system, while the former involves the shape of critical components of the space structures such as a phased array antenna. In both cases, structural flexibility plays important roles since the translational and attitude motions are coupled with the structural vibrations. The motion of a Large Scale Flexible Structure(LFSS) is usually modeled via finite element methods, which can result in very large order models. This means that the order of the system may be extremely high, which can make the tasks of control system design challenging. Model reduction methods that are applied in many cases (Inman, 1989) to control only a subset of the elastic body modes may lead to spillover problems, in which the control effect in stabilizing the subset of modes may cause instability in the uncontrolled modes. The approaches considered to date for investigating the LFSS control have gen erally been directed towards centralized control. e.g. model reduction methods (Hughes & Skelton. 1981), modal control methods (Balas, 1978b), output feedback 1 50 Figure 3.20. Controlled response at Node 3 of Kabes Model (with only two modes controlled) Figure 3.21. Controlled response at Node 4 of Kabes Model (with only two modes controlled) 11 can be implemented easily. They also showed that for weighted sensitivity minimiza tion for linear time invariant plants, time varying controllers have no advantages over time invariant ones. However, the eigenstructure assignment problem was beyond the scope of Poollas work. Their design was used to design a stabilizing compensator and later, the time varying controller and time invariant plant combinations were used to generate the time response of the system. Identification packages were used to identify this system and separate compensator was designed for pole placement. The order of the system seems to be going higher and higher in this case. Nevertheless, this looks like a first pass solution to the problem of eigenstructure assignment. 1.4 An illustration to demonstrate the advantages of decentralized structural control This example shows the advantages and simplicity of building a decentralized controller for structures. Figure 1.4 shows a 6-bay truss whose model is taken from Young (1990). This truss is modeled as a coupled structure of three substructures. Substructure boundaries are marked in the figure. The truss member mass and stiffness matrices expressed with respect to the local coordinates and used in the assembly process are EA 1 -1 ' ,, mL ' 2 1 L -1 1 Meiem ~ c b 1 2 The nodal coordinates are defined as vertical and horizontal displacements at the joints. The internal degrees of freedom at which the actuator and displacement sensors are placed are marked in the figure. The finite element model of the individual sub structures are obtained using a FORTRAN program and validated using the common FEM package ANSYS (Swanson Analysis Inc, 1992). For convenience, the material properties are assumed to be have unit magnitudes. An Interlocking control design 40 INPUT STRUCTURAL PARAMETERS. CONTROLLER LOCATION? AND LOADING Set Controller flag off. Reset To Compute time history of all nodes in physical coordinates Compute time history of all nodes in modal coordinates No Compute energy in all modes. Find Energy in desired mode j Switch the controllers on if the Store j. work done on them is positive. Set Controller flag on. Figure 3.4. Flowchart of MDOF damping control algorithm CHAPTER 5 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 5.1 Discussion This study proposed new schemes for structural control design using the energy input into the controller to damp out the vibrations. Through simulations it was shown that such a decentralized control law can be efficient in improving the perfor mance of structures. For SDOF systems, the proposed design methodology introduces desired damping within very close bounds. For MDOF systems, using the techniques of Independent Modal Space Control, a design algorithm was developed to introduce specified damping into desired modes of vibration. The controllers were decentralized in that the decision for switching a given controller was based on feedback from local states only. It was also shown how the uncontrolled modes are affected by controlling only few modes in multi degree of freedom systems. This suggests that the proposed methodology be used to control all of the dominant modes of the vibrating structure. 5.2 Future Research The minimum energy on-off control methods proposed in this work should increase the life span of space structures since very minimal energy is needed from the external power with such a method. To fully demonstrate the practical advantage of these schemes, one should de termine the kind of controllers that can be used for such applications. A simple 74 27 aujcos(u)t + (3). (3.2.6) By conservation of energy, E = 0. Note that this is true at all instances except switching: Thus = i-(ii2 + uj2u2) + imjj2vi\ =0. (3.2.7) y 2 at k ) Using equations (3.2.6) and (3.2.4) in equation (3.2.7) yields i2a + ^-^-a2iosm2 (ujt +/3) = 0. (3.2.8) Z K Since the decay rate of the response is assumed to be slow, the method of averaging- in (Caughev & OKelly, 1965) may be applied to compute : A A: l =a / ij, sin 2(cut + d)dt. (3.2.9) k 2 Jo Since u > 0 holds for only half a cycle of sin 2(out + /3), /z is set to 1 for only half of a cycle of sin2(af + /3). In that case, the value of the integral in equation (3.2.9) is 1 /ui and or Ak uj 27TUI (3.2.10) a = a0e Thus, for the single degree of freedom case, determines a control law for achieving a desired average damping rate Â£ = This result is summarized in the following proposition. CHAPTER 4 FUEL-EFFICIENT WAYS OF VIBRATION CONTROL 4.1 Introduction The work on switching control presented in Chapter 2 and 3 was motivated by a study of the literature which used on-off control of vibrations, such as Athanassiades (1963), Athans (1964b, 1964a), Foster and Silverberg (1991), Silverberg (1986), Red mond and Silverberg (1992) and Medith (1964). The work of Foster and Silverberg (1991) is one such study which uses fuel minimization as an optimization criterion for on-off control of structural vibrations. In that work, the minimum fuel use control law imposed that control pulses be applied every time a zero crossing of the the po sition (and maximum value of the velocity) would occur. A plot of fuel consumption versus sampling time was also presented in (Foster & Silverberg, 1991). This chapter is devoted to the study of the effects of sampling time and window size on the fuel consumption. 4.2 Minimum Fuel Control of Mechanical Systems Foster and Silverberg (1991) developed a minimum fuel based switching control law for MDOF second order systems of the form: Mx 4- Kx = F, (4.2.1) where M = Ml is the n x n mass matrix of the structure, K is the n x n stiffness matrix of the structure, x and x are, respectively, the n-dimensional displacements 64 49 Figure 3.18. Controlled response at Node 1 of Kabes Model (with only two modes controlled) Figure 3.19. Controlled response at Node 2 of Kabes Model (with only two modes controlled) 38 3.3.5 Kabes Problem With damping in only two inodes Kabes eight degree of freedom model is shown in Figure 3.17. The natural fre quencies of this uncontrolled system are 30.0280. 31.7163, 31.7557, 32.3020, 33.3267, 35.5763, 38.7887 and 41.8884 radians/second respectively. The mass-normalized eigenvectors of this system are listed in Table 4.1. In order to avoid high values of controller gains, the controllers were placed at 4i/l and 6th masses. The controller locations were selected based upon the mass normalized eigenvector components. The proposed design procedure shows that the reciprocal of the square of the eigenvector component at the location of the controller decides the gain gain of the controller (i.e. if damping is desired in mode j and controller is placed at location g, reciprocal of the square of the gth component of the jth mass normalized eigenvector decides the controller gain). Note that the fourth and sixth mass normalized eigenvector have fourth and sixth components which are within reasonable limits (leading to reason able values for the elements of AK matrix. The desired damping in the fourth and sixth mode were set to 0.10 and 0.15, respectively. The initial conditions were set such that the initial energy was concentrated in the fourth and sixth modes. When these conditions were simulated, it was seen that the control algorithm achieves the desired goals within reasonable limits. Figures 3.18-3.25 give the nodal response and Figures 3.26-3.25 give the modal response of this system. In order to show the impor tance of the controller locations, the same system was simulated with the controllers located at the first and the second mass. The desired damping in fourth mode and sixth mode were set to 0.1 and 0.15. Controller gains in this case were 42 and 38166 compared to 2724 and 3486 in the earlier case. Dampings achieved in this case were .06007 and 0.1076 in mode 4 and 6 compared to the values of 0.08 and 0.14 in the previous case. In order to implement the proposed scheme, the controller gains would have to be such that they do not significantly affect the mass of the system. If the 57 Figure 3.31. controlled) Figure 3.32. controlled) Controlled mode 6 Response of Kabes Model (with only two modes Controlled mode 7 Response of Kabes Model (with only two modes 15 external energy source. Since energy supplied to the controller would require that fuel be consumed, the proposed design methodology reduces the fuel consumption for vibration suppression. Additionally, the proposed control laws are decentralized since the on-off switching laws are based on the local displacements and velocities. In Chapter 3, controller design methodologies are given to select the controller parameters to achieve the desired modal damping. Analytical results are given to prove the validity of this. A method which is similar to the Modified Independent Modal Space Control (MIMSC) is used for designing on-off controller for multi degree of freedom systems. Simulation results are presented to show the effectiveness and accuracy of this method. In Chapter 4, the use of optimal control laws with fuel consumption as a cost criterion to design vibration controllers is presented. First, the concept of on-off vibration control is reviewed with an emphasis on fuel consumption minimization. Next, the problems associated with window sizing and selecting the sampling interval are brought to light. A discussion of energy efficient switching control laws future research problems are presented in Chapter 5. 67 positive slope on the plot in Figure 4.1. This is seen in the plots in 4.3 and 4.5. In these plots note that for values of N = 15 and N = 25, the power at higher harmonics take higher values when compared to plots for values of N where the slope of the fuel curve shown in Figure 4.1 is negative. At the points of positive slope of the plot in Figure 4.1, sampling time is such that the window size becomes an integral multiple of one of the higher harmonics. Note that for N = 15 window size is twice the period of third harmonic and for N = 25 window size is twice the period of fifth harmonic. Figure 4.2. Frequency spectrum of control pulse in example 1 (N=ll) Signal power 68 Figure 4.3. Frequency spectrum of control pulse in example 1 (N=15) 9 Anderson and Moore (1981). Implicit in the pole placement result quoted above is a constructive algorithm. This algorithm requires as a first step the selection of F such that the poles of Ap are distinct from that of A. Then, the dynamic feedback is successively employed at the control stations to place the poles that are controllable and observable from a given station. R.P.Corfmat and A.S.Morse (1976) have studied the decentralized feedback con trol problem from the point of view of determining a more complete characterization of conditions for stabilizablity and pole placement. Their basic approach is to de termine conditions under which a system of the form (1.3.1)-(1.3.4) can be made controllable and observable from the input and output variables of a given system by applying static feedback to other controllers. Then dynamic compensation can be employed at this controller in a standard way to place the poles of the system. It is not hard to see that a necessary and sufficient condition to make (1.3.1)- (1.3.4) controllable and observable from a single controller is that none of the transfer functions Gij(s) = Ci(sI-A)-lBj i,j = 1, ,N (1.3.7) vanish identically. A system satisfying this condition is termed strongly connected. If a system is not strongly connected, it is impossible to make the system con trollable and observable from a single controller. In this case, it is necessary to decompose the system into a set of strongly connected subsystems and to make each subsystem controllable and observable from one of its controllers. For a strongly connected system, Corfmat and Morse have given a highly interesting and rather in tuitive condition that is necessary and sufficient to make (1.3.1)-(1.3.4) controllable and observable from a single controller. They have shown that if a strongly connected system can be made controllable and observable from a single controller, it can be made controllable and observable from any controller, and a necessary and sufficient BIOGRAPHICAL SKETCH I was born in Mysore City, India, on September 14, 1964. I spent most of my early childhood at Mangalore with my parents, brother and two sisters. I graduated from St. Aloysius College in Mangalore in 1982. I went on to get my Bachelor of Engineering degree from Karnataka Regional Engineering College in Surathkal. India in 1986. I received my Master of Technology degree from the Indian Institute of Technology in Bombay in 1988. After working in the Indian Institute of Technology, Bombay for a brief period of three months, I joined Hindustan Aeronautics Limited. Bangalore. Within a short time of six months, I was appointed as a Scientist at the Indian Space Research organization. After serving the Indian Space Industry for two years, I attended University of Florida where I have received my Ph.D. in Engineering Mechanics. 81 |