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- Permanent Link:
- https://ufdc.ufl.edu/UF00097391/00001
## Material Information- Title:
- Limit cycles for systems with nonlinear friction
- Creator:
- James, Michael R. (
*Dissertant*) Bullock, Thomas E. (*Thesis advisor*) Latchman, Haniph A. (*Reviewer*) Couch, Leon W. (*Reviewer*) Popov, Vasile M. (*Reviewer*) Wilson, David C. (*Reviewer*) Phillips, Winfred M. (*Degree grantor*) Lockhart, Madelyn M. (*Degree grantor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1989
- Copyright Date:
- 1989
- Language:
- English
- Physical Description:
- vii, 140 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Eigenvalues ( jstor )
Limit cycles ( jstor ) Mathematical variables ( jstor ) Matrices ( jstor ) Nonlinearity ( jstor ) Sliding ( jstor ) Symmetry ( jstor ) Torque ( jstor ) Trajectories ( jstor ) Velocity ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Limit cycles ( lcsh ) Nonlinear mechanics ( lcsh ) Piecewise linear topology ( lcsh ) Servomechanisms ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- Exact conditions for the existence of limit cycles due to friction in a servomechanism have been determined for the general (n-dimensional) case. These conditions are in the form of nonlinear algebraic equations, for which a solution consistent with certain assumptions must be found. A two-input model for friction is required for accurate results, including both dry friction and static friction components. Therefore, standard nonlinear analysis methods such as absolute stability and describing functions have limited applicability. A piecewise linear method is used, which constructs the limit cycle trajectory with pieces from each region of the state space. The system model is first converted to a control canonical form, which simplifies the analysis. Assumptions made in the formation of the model are specified exactly; the model represents a rotational servomechanism, but is shown to be applicable to other systems. Next, the piecewise linear method is applied to an assumed limit cycle. The resulting algebraic equations can be solved for the exact limit cycle period, amplitude, and trajectory, if any exist. The limit cycles found are of simple type; that is, complex orbits such as those which traverse a region more than once would not be found. Necessity and sufficiency of the existence conditions is proved. An analysis of the local orbital stability of limit cycles found by this method is also presented. Eigenvalues of a given matrix determine the asymptotic stability of the periodic orbit. The property of symmetry of limit cycles for odd systems in general and friction in particular is examined. Special cases are presented in which symmetry holds, and counter-examples for the general case of odd systems are given. The question of symmetry for friction limit cycles remains open. Examples are included to illustrate the method. One example is a 3D system that exhibits two limit cycles in its phase portrait, each a different type. Listings of useful analysis computer programs are given, in addition to suggestions for further research in this area.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1989.
- Bibliography:
- Includes bibliographical references (leaves 137-138)
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Michael R. James.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
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- Copyright [name of dissertation author]. 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.
- Resource Identifier:
- 001530507 ( AlephBibNum )
22331252 ( OCLC ) AHE3883 ( NOTIS )
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LIMIT CYCLES FOR SYSTEMS WITH NONLINEAR FRICTION By MICHAEL R. JAMES 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 1989 ACKNOWLEDGEMENTS I am indebted to many people for their cooperation and assistance over the long period of my doctoral studies. First, my adviser, Dr. Thomas Bullock, and the other members of my graduate committee, for their suggestions, help and cooperation. Dr. V. M. Popov provided a very helpful discussion on symmetry. Dean Eugene Chenette and Art Zirger were very helpful during my coursework and residence, as were many other UF personnel too numerous to mention. I received help from many fellow employees of Harris Corporation: Pete Pitard, for his hard work in obtaining financial assistance, Lee Almond, for his support, Kevin Arter, for his review and suggestions, and the Training/FEEDS and Human Resources personnel. Special thanks go to Rich and Donna Phelan, whose enthusiastic support solved many problems during the first years. Most of all, I want to thank my wife, parents, and family for their understanding and support during these six long years. Without their sacrifices I could not have succeeded, nor would it be meaningful without them. All of these people share in this achievement, and I deeply thank them for their support. TABLE OF CONTENTS ACKNOWLEDGMENTS............................ .. ....... ii LIST OF FIGURES ................................. ... v ABSTRACT............................................ vi CHAPTERS I INTRODUCTION............................... 1 Problem Statement .......................... 1 The Piecewise Linear Method................ 4 Illustrative Examples...................... 8 Organization of Dissertation............... 21 II REVIEW OF EXISTING RESULTS................. 23 An Unpopular Area of Study.................. 23 Phase Plane (2D) Analysis of Friction...... 24 Literature on Piecewise Linear Method...... 25 Extensions to Multi-Dimensional Case (Approximate Analyses by Describing Function) ................................ 26 General Results on Periodic Orbits.......... 27 Summary of Literature Review............... 28 III DEVELOPMENT OF STANDARD MODEL.............. 30 System Model with Physical State Variables. 31 Modelling Friction By a Nonlinear, Two-Input Function .................... 32 The Physical State Model in Piecewise Linear Regions..... .................... 34 Conversion to Control Canonical Form........ 38 Simplification of DE Solution Via Coordinate Translation................ 41 Summary of Assumptions and Discussion of Generality of Model ................... 43 iii IV EXISTENCE CONDITIONS FOR LIMIT CYCLES...... 49 Necessary Conditions for a Simple Limit Cycle ................................. 49 Illustrative Examples...................... 59 Exact (Necessary and Sufficient) Conditions 68 V LIMIT CYCLE SYMMETRY AND STABILITY......... 71 Stability of Predicted Limit Cycles......... 71 Stability Calculations for Example Problems 91 Symmetry of Limit Cycles for Odd Systems... 100 VI CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH................................. 116 Results and Conclusions of Current Research 116 Recommendations for Further Research........ 120 APPENDIX ANALYSIS COMPUTER PROGRAMS ............. 123 LIST OF REFERENCES ................................. 137 BIOGRAPHICAL SKETCH ................................ 139 LIST OF FIGURES Figure pa I-i BLOCK DIAGRAM OF SYSTEM UNDER STUDY... 3 I-2 TWO-INPUT FRICTION MODEL.............. 3 I-3 PIECEWISE-LINEAR REGIONS FOR 2D CASE.. 6 I-4 LIMIT CYCLE AMPLITUDES FOR 2D CASE, (B=-l, J=l, LC=1)................... 10 I-5 2D SLIDING LIMIT CYCLE, PHASE PLANE PLOT (B=-l, K=l). .................... 12 I-6 BLOCK DIAGRAM OF 3D SYSTEM (EXAMPLE 2) 13 I-7 3D STICKING LIMIT CYCLE, VIEWED ALONG VELOCITY AXIS (B=l).......... 15 I-8 3D STICKING LIMIT CYCLE, VIEWED ALONG AXIS OF INTEGRATOR STATE (B=1)...... ......................... 16 I-9 3D STICKING LIMIT CYCLE, VIEWED ALONG POSITION AXIS (B=l).......... 17 I-10 3D SLIDING LIMIT CYCLE, VIEWED ALONG AXIS OF INTEGRATOR STATE (B=.9)..... .......................... 19 I-11 3D SLIDING LIMIT CYCLE, VIEWED ALONG VELOCITY AXIS (B=.9)......... 20 IV-1 3D LIMIT CYCLE EXAMPLE STICKY CASE, DETERMINATION OF PERIOD Tl (ZEROS OF F(T1))............................ 65 V-1 3D LIMIT CYCLE EXAMPLE SLIDING CASE, LOCAL STABILITY NEAR XO............ 94 V-2 3D LIMIT CYCLE EXAMPLE STICKY CASE, LOCAL STABILITY NEAR X3............ 96 V-3 3D EXAMPLE SYSTEM PHASE PORTRAIT, VIEWED ALONG AXIS OF INTEGRATOR STATE ............................. 97 V-4 3D EXAMPLE SYSTEM PHASE PORTRAIT, VIEWED ALONG AXIS OF INTEGRATOR STATE (EXPANDED) ....... ............ 98 V-5 3D EXAMPLE SYSTEM PHASE PORTRAIT, VIEWED ALONG AXIS OF INTEGRATOR STATE (EXPANDED) ........ ........... 99 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 LIMIT CYCLES FOR SYSTEMS WITH NONLINEAR FRICTION By MICHAEL R. JAMES December, 1989 Chairman: Dr. Thomas E. Bullock Major Department: Electrical Engineering Exact conditions for the existence of limit cycles due to friction in a servomechanism have been determined for the general (n-dimensional) case. These conditions are in the form of nonlinear algebraic equations, for which a solution consistent with certain assumptions must be found. A two-input model for friction is required for accurate results, including both dry friction and static friction components. Therefore, standard nonlinear analysis methods such as absolute stability and describing functions have limited applicability. A piecewise linear method is used, which constructs the limit cycle trajectory with pieces from each region of the state space. The system model is first converted to a control canonical form, which simplifies the analysis. Assumptions made in the formation of the model are specified exactly; the model represents a rotational servomechanism, but is shown to be applicable to other systems. vi Next, the piecewise linear method is applied to an assumed limit cycle. The resulting algebraic equations can be solved for the exact limit cycle period, amplitude, and trajectory, if any exist. The limit cycles found are of simple type; that is, complex orbits such as those which traverse a region more than once would not be found. Necessity and sufficiency of the existence conditions is proved. An analysis of the local orbital stability of limit cycles found by this method is also presented. Eigenvalues of a given matrix determine the asymptotic stability of the periodic orbit. The property of symmetry of limit cycles for odd systems in general and friction in particular is examined. Special cases are presented in which symmetry holds, and counter-examples for the general case of odd systems are given. The question of symmetry for friction limit cycles remains open. Examples are included to illustrate the method. One example is a 3D system that exhibits two limit cycles in its phase portrait, each a different type. Listings of useful analysis computer programs are given, in addition to suggestions for further research in this area. vii CHAPTER I INTRODUCTION Problem Statement The objective of this research is to define exact conditions on the existence and behavior of limit cycles due to friction. The analysis method to be used is a piecewise linear approach which, unlike approximate methods such as describing functions or absolute stability, has the potential for providing exact (i.e., necessary and sufficient) conditions for existence. This problem is of interest in the design of servomechanisms, which must deal with the nonlinearities in typical electromechanical devices, such as friction. Experience has shown that the effect of friction is generally benign, since it tends to provide increased damping to the system. However, this cannot be guaranteed for all systems. Furthermore, it is a standard practice to precompensate for this effect by deliberately tuning the system to be underdamped without friction, so that it is not excessively damped in actual operation with friction. It is postulated that this technique, when pushed to extremes, results in large-signal instability, since the damping effect of friction varies inversely with amplitude. It is, therefore, a significant research problem to determine the point where limit cycles can exist. 1 An approach has been developed during this research that uses the piecewise linear method, that provides an exact condition for the existence of a limit cycle in terms of a nonlinear algebraic system of equations. When a solution to the system is found (and thus a limit cycle exists), the method automatically gives exact frequency, amplitude, and state trajectories for the limit cycle. On the other hand, exact conditions under which the nonlinear system of equations is solvable are more difficult because of the nonlinear nature of the system. The specific problem under study is shown in block diagram form in Figure I-1. The plant in the standard system is represented by a Laplace-transform model of a motor/load combination, which includes the effect of load inertia and damping. The model does not include compliance in the load, although, as described in Chapter III, this may be included. The inertia (J) and damping (B) represent the lumped quantities for the load and motor, with any gear ratio taken into account. The servo controller is represented as a linear system with motor drive torque as the output quantity. Note that despite the specific description of the problem model, it will be shown in Chapter III (Development of Standard Model) that the model is actually quite general. The model used for nonlinear friction includes both static and sliding coulomb friction components (see Figure 1-2). Therefore, it is a two-input nonlinearity and is FIGURE I-1. BLOCK DIAGRAM OF SYSTEM UNDER STUDY L F(TORQUE) L C SLIDING MODE Y (FOR NONZERO VELOCITY, Yn O) VELO F_ n(VELOCITY) -L LF= LCsgn[Yn] c L (TORQUE) STATIC OR STICTION MODE (FOR ZERO VELOCITY, Y =0) L n S LF= LS sat[LIN/LS ] L (INPUT L TORQUE) /-- S FIGURE 1-2. TWO-INPUT FRICTION MODEL thus difficult to handle by standard techniques (such as describing functions or absolute stability criteria). The extra complexity caused by the use of the two-input model is necessary; examples are known where modelling sliding friction only indicates a limit cycle exists, when, in fact, the "sticky" effect of the static friction prevents it. On the other hand, the sliding friction can provide much of the damping of many servomechanisms which have little or no viscous friction damping in the load. These comments indicate that a model including both effects is necessary. Note that the friction model used is odd, and therefore the system derivative vector will be an odd function of the state. The Piecewise Linear Method The piecewise linear method is an approach where systems with piecewise linear nonlinearities can be analyzed using the tools of linear system theory. Since the nonlinearity is linear over portions of its domain, the state space can be divided into regions in which the system behaves in a purely linear fashion. The linear system theory then applies, although there will be a different linear system in each region. Solution trajectories must then be pasted together from different regions through the use of boundary conditions (called commutation equations in relay servo or on-off system analysis such as in Weischedel, 1973). The nonlinear friction to be analyzed in this study is piecewise linear, even though it requires two inputs. Five regions of the state space are required (Figure I-3 shows the case for 2-dimensional state space): (I) velocity < 0, (II) velocity > 0, (III) velocity = 0 and magnitude of accelerating input torque < LS, (IV) velocity = 0 and torque > LS, and (V) velocity = 0 and torque < -Ls, where LS is the maximum static friction torque (breakaway torque). The first two regions are always open half-spaces, while the last three divide the (n-l)-dimensional sub-space where velocity equals zero into three regions. For example, for the 3D case (n=3), the last three regions are portions of the plane y3 = velocity = 0. As can be seen by examination of the friction model, the nonlinearity either has a constant output throughout each region (I, II, IV, and V), or its output is proportional to a linear combination of states (region III). Even a simple limit cycle must traverse at least three of these regions, so the linear systems of each must be considered in this limit cycle analysis. There are several advantages in the use of the piecewise linear technique. Once a limit cycle is found (i.e., solutions from four regions that can be pasted together to form a closed trajectory), the exact trajectory VELOCITY Y2 OR DY1/DT LINEOF 0F ZERO N INPUTTORQUE \ Y2 = -(K/B) Y1 (FOR 2D CASE) ' REGION IV: VELOCITY = 0 AND TORQUE > LS REGION II: VELOCITY > 0 SAMPLE PERIODIC ORBIT REGION III: UNESEGMENT OF EQUIL. POINTS -LS / K Y1 REGION V: \ / VELOCITY= 0 AND TORQUE <-LS 1 \ I ----- *~~~~~~^ REGION I: VELOCITY < 0 FIGURE 1-3. PIECEWISE-LINEAR REGIONS FOR 2D CASE POSITION ~ __ is known. Therefore, the limit cycle amplitude, period, and characteristics are known exactly, rather than approximately as in describing functions. Furthermore, the method leads to a more exact analysis, since the commutation conditions require the solution trajectories to match exactly at the region boundaries. Finally, the technique has a geometric approach that is intuitively appealing and lends itself to graphical solution. The method is applicable to the majority of servo nonlinearities. Backlash, friction, and most other nonlinearities found in practice are piecewise linear. The disadvantage of the method is in the difficulty of solving the commutation conditions for a solution--finding the trajectories that will "paste" together to form a cycle. One way to surmount this problem applies the method as a secondary step in the analysis in order to refine an approximate solution found by other methods. For example, a limit cycle period and amplitude are determined approximately by describing functions; then the piecewise linear method is used (with the approximate solution as an initial solution trajectory) in an iterative way to converge to the true solution. A composite computer analysis program could integrate these functions for the convenience of the servo designer. Therefore, the difficulty of solution does not really limit the applicability of the method. Illustrative Examples Before going into the detailed development of the method and other results, it is useful to present simple example systems exhibiting friction limit cycles. The limit cycles were analyzed by the methods to be presented later, so we can return to these examples at the appropriate time to illustrate the method; here only the results are shown. The second example system is particularly interesting since it is only 3D (three state variables describe the system), yet it exhibits two distinct limit cycles of differing types (when the appropriate parameter values are selected). The fact that such a simple system can have such complex behavior indicates the richness of the study of piecewise linear systems. Example I-1: Two-Dimensional (2D) System with Sliding Friction Limit Cycle The question of friction limit cycles for the case n=2 can be completely solved; in fact, the piecewise linear method is the same as standard phase plane methods for this case. For n=2, the controller can be a constant gain only (since there are two states in the plant); the state variable model of the system is then (1.1) dyl/dt = y2 dy2/dt = -(K/J) yl (B/J) y2 LF/J where LF is the friction torque from the model in Figure 1-2, and any feedback gain on y2 is lumped in with B. It can be shown (see Thaler and Pastel, 1962, pp. 96-104) that for B>0 and for K>0, the system is globally asymptotically stable (i.e., no limit cycles). Thus a stable system remains stable (in fact it is more stable) with friction (2D case only). The case B=0 is also stable; the linear system has imaginary eigenvalues, the friction gives enough damping to give asymptotic stability (trajectories spiral to an equilibrium point that is not necessarily the origin). Interestingly, a limit cycle exists for every case with negative damping (B<0). If the position feedback gain K is sufficiently large to give the linear system a complex pair of poles, an unstable, unique limit cycle exists, whose amplitude depends on K, B and the inertia J and sliding friction level LC. This "sliding" type of friction limit cycle is one in which the motor or other object being moved never "sticks"; the reversing torque is sufficient when it comes to rest to immediately breakaway in the opposite direction. Therefore, the orbit consists of two parts, each representing motion, with two switching per cycle. The initial conditions can be expressed in terms of the system parameters as (1.2) X10 = (Lc/K) [(L+1)/(X-l)], x20 = 0 where (1.3) p = exp[-Br/(2JP)], 3 = [4K/J B2/J2]k Figure I-4 shows some values of amplitude (which equals x10) as a function of gain K for the case J = LC = 1, B = -1; it 0 0 0 0 co 0 S-- Q N o oo II II u- E- II HO p4 U H __________________g 0 * O a appears that a limit cycle of any amplitude can be found by adjusting gain properly. The unstable limit cycle bounds the region of asymptotic stability: trajectories inside spiral in to equilibrium near the origin, while those outside go to infinity (Figure I-5 shows phase plane plot). This example shows why it is dangerous to design underdamped systems, and depend on the sliding friction to stabilize the system; the friction damping is amplitude-dependent, and at some critical amplitude, is insufficient for stability. Example 1-2: Three-Dimensional (3D) System with Both Sliding and Sticking Limit Cycles The second example system to be considered is defined by the differential equations (1.4) dy/dt = A + b LF where (1.5) A = 0 1 0 0 0 1 -K1 -K2 -B with b' = [ 0 0 -1 ], LF = friction torque, and where the state variables are y2 = position, y3 = velocity, and yl = compensator integrator. Figure I-6 shows a block diagram of this system. Case I: One Stable Pole and One Imaginary Pole Pair (Only Sticking Limit Cycle Exists) For specific results, let us first set B = K1 = K2 = 1, and set the sliding friction torque to 1.0 and sticky o o 0 0 0 S0 C > O OU u C4 0 3 >4l U II "-" I-I H c I II 1- H oi a 3 &< c?;~ 1- FIGURE 1-6. BLOCK DIAGRAM OF 3D SYSTEM (EXAMPLE 2) friction (breakaway torque) to 1.2. Simulation of this case demonstrates a sticking limit cycle with an initial condition of y0' = [ 1.0 0.2 0.0 ]. This type of friction limit cycle repeats a four-part cycle of sticking (until torque integrates up to breakaway), sliding to a new position, sticking again, and sliding back to the start position. A visual display of this behavior can be seen in Figures I-7 through 1-9, which show projected views of the closely similar limit cycle found using the Case II parameters. The solution of the piecewise-linear equations results in a valid limit cycle solution where y = [ 1 0.2 0 ], T1 (the sliding period) = r, and the sticking period T2 = 10, yielding an overall limit cycle period of approx. 26.3 seconds, matching the simulation results. Case II: One Stable Pole and One Unstable Pole Pair (Both Sticking and Sliding Limit Cycles Exist) If the same example system is used with B = 0.9, and K = K2 = 1 still, the closed-loop eigenvalues (of the linear portion of the system) move into the right-half plane, to .026 j 1.024, while the other pole is at -.9524. As might be expected, there is still a sticking limit cycle solution close to that of Case I (Figures I-7 through I-9). Numerical methods give a solution at approximately O = [0.98 0.22 0 ], and approximately the same period (confirmed by simulation). 1- + + OM E-4OZ I - I o 0 oo o U< E-i o o I 0D U- U II E-4 H lX]X HE i< H UU O aH 0> ,H Ct< i ll-- O 0 0 O 0 0 0 0 ______ _________ __________ _______ ---- 0 0 0 0 0 0 I 01 > c OU-OU E II U C E- 1 0 uO U E HZ U) EH co z o0 Z H HO 3M CC 3 ___ __ __ _ _ _ _ _ _ _4_____ I__ _ _ t t + + 0 CD o o o E-4 CD o I 0 -4 The interesting point about this case, however, is the similarity to Example I-1, the 2D case; that is, an unstable complex pole pair exists. Since the nonlinear damping effect of the sliding friction is largest at small amplitudes, we might expect trajectories to converge for small amplitudes and diverge at large amplitudes where the unstable linear poles overcome the nonlinear damping (as in the 2D case). Therefore, this case would have both a sticking and a pure sliding friction limit cycle! Simulation having shown a limit cycle with a half-period T1 approximately equal to 3.1 seconds, numerical solution of the equations for the symmetric, sliding case was performed, yielding a solution at T1 = 3.1455 seconds, and y0 = [-0.283 12.4 0.]. Figures 1-10 and I-11 show some views of the two 3D limit cycles for this case. It is expected from physical intuition and simulation results that the sticking orbit would be stable, while the sliding orbit would be unstable. This is in fact the case, with the sliding orbit exhibiting saddle point behavior--one stable and one unstable mode (real eigenvalue / eigenvector pair), while the sticking orbit has stable node behavior-- two real stable modes. The analysis of the orbital stability will be examined in Chapter V, where Example 1-2, case 2 will be used to demonstrate the method used for limit cycle orbital stability calculations. O O o r-( S -1 -- O - o -- 1 I -- ---- -- \ I -- -- I -- I I I ---- -- -- I \ ---- - > OU)HE-q > (4 o o -4 0 - 4 0 o ..I -- 0 0 0 0 0 -4 -4 ( ) U. 0 U* II H' H *Z rt> L M .x U 1-1 Q &> "Z=E-HuC < E-,OW OE-4 Organization of Dissertation The main body of this dissertation is organized as follows. After a review of existing results on this problem and related areas, the standard model introduced above is developed more fully. The objective in the model development (in Chapter III) is to provide more convenient forms for analysis (a control canonical form and a normal form), while maintaining the generality and traceability to actual servomechanisms of the original model. Once the model is in the proper form, Chapter IV develops the nonlinear system of equations that represent the necessary and sufficient conditions for the existence of a friction limit cycle. Several special cases are examined that are somewhat simplified from the general problem and easier to solve. For example, if it can be assumed that the limit cycle is symmetric about the origin (that is, half-wave symmetric), the equation system can be simplified considerably. The benefits of symmetry in the solution of the limit cycle conditions led to an examination of the conditions under which this symmetry existed. Chapter V examines this problem and describes special cases for which symmetry exists. Unfortunately, it is not known if friction limit cycles are symmetric in general, although a search for counterexamples was unsuccessful. Using the results of the previous chapters, a complete stability analysis of friction limit cycles is also included in Chapter V. Using the eigenvalues of the local stability matrix developed in this analysis, the stability of limit cycles previously found is determined. This allows the global phase portrait of the system to be pieced together from a definition of all the equilibrium points, limit cycles, and their stability characteristics. The results presented here are considered only a start in an essentially new area: use of piecewise linear techniques for limit cycle analysis. Many possibilities for further research exist, both in the completion of the solution for friction, and in application of the method to other piecewise linear nonlinearities. As pointed out above, the majority of the plant nonlinearities encountered in servo design are piecewise linear, including gear backlash, saturation, quantization, and dead zone. The last chapters provide a summary of results and suggestions for further research in these areas. CHAPTER II REVIEW OF EXISTING RESULTS An Unpopular Area of Study Surprisingly little work has been done in this area, for various reasons. First, the nonlinearity is not "nice," i.e., continuous or smooth, which prevents the application of many mathematical tools. Second, when modelled with two inputs it really becomes a system with multiple nonlinearities, for which standard approaches cannot be used, and extremely few results are available. Finally, and, I believe, most importantly, it is perceived as generally benign in effect. It can be shown fairly easily (see Thaler and Pastel, 1962, for example) in the 2D case (i.e., two state variables), that if the linear portion of the system is already stable, the system with pure sliding friction added is also stable. .It seems probable that this result could be extended to arbitrary- dimensioned systems, based on the following energy argument: the linear system is stable (hence contracting), and because the (sliding) friction opposes motion, it damps the system further, so the map is more contracting than without it. Therefore, friction is generally held to be benign in effect (it increases the stability of a system by increasing the damping). For these and possibly other reasons, the references available in this area were few, and no analysis of the general multi-dimensional friction limit cycle problem was found. In spite of these comments, the study of friction limit cycles has practical value for control system design. Although the effect of sliding friction is to increase damping, it has become fairly standard to precompensate for this effect in a design. The system is deliberately designed to be underdamped, so that it is not too well- damped when friction is added. It is conjectured that this technique, when pushed to extremes, results in large-signal instability, since the damping effect varies inversely with amplitude. In addition, the stiction component can cause difficulties also. An example was given in Chapter I of a stable (although not asymptotically stable) third-order linear system which has a limit cycle when nonlinear friction (including stiction effects) is added. A slight variation in parameters results in a stable linear system which destabilizes with the addition of nonlinear friction! Phase Plane (2D) Analysis of Friction As mentioned above, Thaler and Pastel (1962), in their classic text on nonlinear systems, completely solved the second-order case for friction, including both sliding and static components. They also give an exact criterion for the existence of limit cycles when an input ramp is present. They showed that the system had no limit cycles (for a system with positive damping) for the zero-input case. Earlier work by the same authors, (Pastel and Thaler, 1960) actually shows a stability boundary for backlash (as a function of system damping), then demonstrates the stabilizing effect of coulomb friction. Analyses of the 2D friction problem were presented in several other references from the period when phase plane analysis was an active area of research. Literature on Piecewise Linear Method The approach used in this dissertation to give exact solutions for limit cycles (if they exist) has been known and used for many years. In the relay servomechanism literature, it is known as the piecewise linear method. The method was used primarily for relay systems, though it is generally applicable to any piecewise linear nonlinearity. Note that the piecewise linear method is essentially the same as phase plane analysis for 2D systems, so there is overlap between the phase plane references, such as those cited previously, and those for the piecewise linear method. Unfortunately, the graphical methods of phase plane analysis cannot be easily extended to higher dimensions (see comment below on the work of Ku). The piecewise linear method was used at least as early as 1963 (by Kovatch); additional studies can be found in references such as Kovatch (1964) for two nonlinearities, O'Donnell (1964) for time-optimal switching, Marstrander (1969) for backlash, Negoescu and Sebastion (1971), Langill (1965), and Urabe (1967). Weischedel (1973) applied the method to on-off control systems. Ku (1958) made significant and early contributions, including attempts to extend the phase plane method to higher dimensions by various graphical projections. The Russian controls literature calls this the Andronov point transformation method; a reference was found in a translation of material developed in 1956 by E.P. Popov (translated 1962). Although significant development of this area has been performed, especially in the Russian journal Automatika, an application to the friction problem has not been found. Extensions to Multi-Dimensional Case (Approximate Analyses by Describing Function) Analysis using approximate techniques such as the describing function go back at least as far as the piecewise linear method, or even farther in the Russian journals (Popov, 1956). Four papers are listed by Gibson (1963), that he refers to as independent developments of the describing function method; all are from the late 1940s! However, the two input difficulty that limits the applicability of describing functions to friction is usually dealt with by ignoring the static friction. No references have been found that discuss the general case for an exact friction model. No other references were found that attempted the multi-dimensional friction problem with other methods. General Results on Periodic Orbits In the mathematics literature on the theory of periodic solutions of ordinary differential equations, the Poincare map (Hirsch and Smale, 1974) is frequently used; this map represents the change in system state after one cycle or period. By searching for fixed points of the map (which represent a return to the original state after one cycle), a periodic solution can be found. This theory was used in generating the stability results to be stated in Chapter V, where the orbital stability of friction limit cycles is examined. The Chapter V results are, therefore, original only in their application to friction limit cycles, not in the theory of orbital stability. There are many results in this literature (although the Poincare theory itself is restricted to planar systems). Pliss (1966) stated some results on the existence and uniqueness of these fixed points, although they could not apply to friction as they were restricted to continuous functions, or non-autonomous systems. Hale (1969) discussed fixed-point theorems for 2D systems and n-dimensional systems on a torus; in addition he stated the orbital stability result (that all eigenvalues of the map except one must be less than one in magnitude). Hayashi (1964) refers to this approach as the topological approach to periodic solutions, since the theorist is concerned with the existence and topological properties of integral curves in state-space. Finally, Hirsch and Smale (1974) provide an excellent discussion on an introductory level of the theory of flow maps; this reference was invaluable in the stability analysis of Chapter V. My survey of this literature was thorough, but not exhaustive; no applications to this nonlinearity were encountered. Summary of Literature Review In the 20 years since the heyday of phase plane and piecewise linear methods, it appears that almost no new results have been discovered in this area. However, the study of friction limit cycles has significant practical value for designers of servomechanisms, since a servo designed using standard techniques may exhibit instabilities in the presence of friction. The literature survey was extensive in the area of controls, and moderate in the mathematics field. In the former, IEEE indices were searched, including 25 years of the Transactions on Automatic Control. Twenty-five years of the Russian journal Automation and Remote Control were examined. The journal Automatica was also reviewed. Although no journals in the field of pure mathematics were checked, many texts in the areas of stability, limit cycles, and dynamical systems were examined. The main contribution of this research is the application of known (although relatively obscure in controls literature) techniques such as the piecewise linear method and Poincare flow maps to the problem of nonlinear friction. No reference was found that considers the multi- dimensional friction limit cycle problem using an exact method such as the piecewise linear technique. All the references found either analyze the 2D case for friction, or derive approximate results by using a simpler friction model. Practical experience with servos indicates that it is unacceptable to ignore either friction component. The development presented here, therefore, appears to be the first to attack the multi-dimensional friction limit cycle problem with sufficient accuracy for practical applications. In addition, the results on limit cycle symmetry for piecewise linear systems, and the derivation of limit cycle orbital stability by the same method, are original, to the author's knowledge. CHAPTER III DEVELOPMENT OF STANDARD MODEL This chapter is concerned with setting up the model of the system to be studied, and converting it to a convenient form for analysis. The original system model is in the form of physical state variables, where the states represent actual physical quantities in the servomechanism. The model form to be used for most results is a control canonical representation, which is developed from the physical states by a linear transformation. This form has advantages in clarifying subsequent developments of the theory, including the role of system zeroes and characterizing limit cycle symmetry. This chapter also demonstrates how the piecewise linear friction torque input can be removed by a translation, which leads to a simplification of the existence formulas of Chapter IV. In addition to the derivation of the state forms, this chapter provides a description of the various assumptions made in the analysis. In other words, the generality of the analysis and the solutions are defined by examining the assumptions made. For example, although the original motor/load model included no compliance or resonance effects, this restriction can be removed, allowing the results to be more generally applicable. System Model with Physical State Variables The two primary physical variables are angular velocity (required as input to the sliding mode friction) and its integral, angular position. The source of this model is a rotating servomechanism. The theory is not restricted to rotational friction problems, however, since angular variables (torque, angular velocity, etc.) could be replaced by linear ones (force, velocity, etc.) without affecting the nature of the problem (this point is discussed further at the end of this chapter). The model was illustrated in block diagram form in Figure I-1; the controller transfer function G(s) is replaced by state equations in the model below. The remaining n-2 state variables, where n is the order of the system, are controller states. That is, they form the model of the servo loop compensation, amplifiers, motor armature effects, and other dynamic portions of the system. Thus we may set up a model of the following form: (3.1) = Ac y + a y + a c2yn c c -cl n-l -c2 n Yn-1 = Yn J Yn = -Kyn-1 Byn + b'yc + f(y) where the yi variables are the physical states and make up the vector y, the dot (') indicates the time derivative d/dt, prime (') indicates transpose, f(.) is the friction torque, J is the lumped system inertia, and B is the damping. The n-2 dimensional linear subsystem in the n-2 states in vector yc is formed by the system matrix Ac c c (n-2 x n-2), the input vectors a cand a (n-2 x 1), and the output vector b (n-2 x 1) that feeds the output of the controller into the torque equation. The only assumptions made in the construction of this model are that yn-1 and yn are angular position and velocity, respectively. These assumptions completely define the first equation, and require that the nonlinear friction torque term f/J appears in the y differential equation (representing angular acceleration due to the sum of torques applied). The rest of the system is linear; the friction term does not appear in any other derivative. Note that there is an assumption made here that restricts the generality of the model: that the friction appears in only one equation. Although examples can be constructed that do not obey this restriction (such as those involving differentiators, full state feedback, torque measurement, or a foundation model), this model is felt to be quite general, and covers essentially all servo problems where friction limit cycle information is of.interest. This point will be discussed in more detail later in this chapter. Modelling Friction By a Nonlinear, Two-Input Function The nonlinearity investigated in this analysis is a model for mechanical friction that depends on both the relative velocity of the surfaces and the driving torque of the moving element. The level of friction opposing the motion depends on velocity (or more precisely, the direction of the velocity) when the element is moving. In the case where the element is stationary, on the other hand, the static friction torque holding the system motionless must be set equal to the input driving torque that is attempting to move the element. Each of these cases is based on the models of Coulomb friction described in any basic statics textbook. Figure I-2 in Chapter I illustrates the friction model. Although an analysis can be performed with only one element modelled, thereby using a single-input nonlinearity and simplifying the analysis, this would lead to results that would not apply in any physical system, and could be misleading or erroneous. Cases can be constructed where the modelling of sliding friction alone predicts a limit cycle, due to its highly nonlinear, discontinuous characteristic. In a practical system, however, no limit cycle would exist. For example, the two-dimensional friction limit cycle is completely solved in Thaler and Pastel (1962), as described in Chapter I. Neglecting the static friction component in this analysis leads to the conclusion that a limit cycle of very small amplitude can be obtained. However, in reality, such a small initial condition will stick at the initial point due to the static friction torque. On the other hand, the sliding component of friction cannot be ignored either. As a qualitative example of a case demonstrating this, consider a system with no damping, which would limit cycle if static friction alone were present. The system starts at an initial position with nonzero error, integrates up in the controller until it breaks free from the static friction, and overshoots the desired position to start the cycle all over again. The inclusion of sliding friction in the model, however, might provide enough damping to the system to cause it to spiral in to the equilibrium point at the origin. Another possibility indicated by the behavior of typical friction limit cycles found is that a system that is actually stable for small signals (bounded region of stability) might appear unstable if analyzed without sliding friction. These cases indicate that accurately predicting limit cycles for physical servomechanisms requires a complete model of friction, as is used in the analysis contained in this paper. The Physical State Model in Piecewise Linear Regions As described in the introduction, application of the piecewise linear method to the system model results in five regions of state space, with a linear system for each region. The form of the (now linear) model in each region is described, so that the trajectories in each region can be described. Region I: Velocity < 0 (v <0). This n-dimensional region is described by the same set of differential equations as in (3.1), except that the nonlinear friction term f(y) is replaced by a constant input Lc. This follows from the fact that the friction in sliding mode is constant (independent of velocity) as long as the system is in motion in one direction. From basic mechanics, this constant friction is a function of the normal force, with the proportionality constant being the coefficient of sliding friction. The torque Le is positive, since physical friction always acts to oppose the direction of motion; velocity is negative, so the acceleration torque due to friction is positive. Therefore, the model in region I is a linear system driven by a constant input. Region II: Velocity > 0 (y >0). This region model is exactly the same as in region I, except that f(y) is replaced by -Lc Region III: Velocity = 0 (y =0) and Acceleration Torque 5 Breakaway Torque. In this case, the system is in a static condition, and the model for the static component of friction applies (also known as stiction). The friction opposes the applied acceleration torque with an equal torque so that the net acceleration torque is zero,-up to a maximum amount of static friction. The maximum amount of friction (the breakaway torque) is determined by the coefficient of static friction and the normal force. Once the acceleration torque becomes greater than the maximum stiction, the net torque becomes nonzero, and the system begins to move (enters region I or II). Note that the acceleration torque used to define this region, and which is matched by an opposing friction force, consists of every term in the torque equation (the derivative of y ) except the friction f. Therefore, the linear terms in the torque equation define the boundaries of this region. The region is contained in the (n-l)-dimensional hyperplane y n=0. Since the net torque is zero and the velocity is zero, the first two differential equations in (3.1) drop out (the state yn-i remains constant and y remains zero over trajectories in this region), leaving the n-2 dimensional controller system as the only equations required to define the trajectory in region III. Of course, input vector ac2 drops out (since yn=O), while acl and b are used to model the effect of the constant angular position on the controller and the controller output torque (to determine the breakaway condition). Region IV: Velocity = 0 (y =0) and Acceleration Torque > (+)Breakaway Torque. This region is also contained in the (n-l)-dimensional hyperplane Yn=0, but the driving acceleration is sufficient at any point to overcome the maximum static friction opposing the impending motion. Thus a system that comes to rest with sufficient torque (trajectory enters region IV rather than III) immediately breaks into motion again, entering region II (since acceleration torque is positive, velocity increases from zero and trajectory must enter region II). For this reason, a trajectory can be said to cross this region (in the sense of crossing a plane, for example), but is only in the region for an instant. Since the integral of a finite quantity over a set of measure zero is zero, the applied torque and the differential equations are irrelevant to the system behavior. The trajectory leaves the region at the exact point it entered. It is therefore unnecessary to examine the form of the system equations in this region. Region V: Velocity = 0 (y =0) and Acceleration Torque < (-) Breakaway Torque. The region V model is identical to the region IV model, except for the breakaway conditions. The five regions cover the state space, including the equilibrium point (actually, line segment) at the origin. The trajectories in each of the individual regions can be described by the solution to the linear system of differential equations: (3.2) y(t) = F(t,t0) y(t0) + G(t,t0) L F(t,t0) = exp[A(t-t0)] where L is the (constant) input in that region, A is the system matrix for that region, and the F and G matrices vary from region to region. The initial conditions y(t0 ) are determined by the commutation conditions at the boundary of the region at which the trajectory enters. An algebraic trick, to be presented below, can be used with this form to remove the input term (GL) in equation (3.2). The system then behaves as an autonomous system, with considerable simplification of the defining equations. Conversion to Control Canonical Form The most convenient form for analysis is the control canonical form, because the role of system poles and zeroes in the existence conditions in Chapter IV will be clarified through the use of this representation. In addition, a result in Chapter V on symmetry is proved using this form. Although this section constitutes a digression from the main line of development, it is justified by the usefulness of the control form. The conversion to control form is accomplished by representing the controller (transfer function G(s) in Figure I-1) in control form and combining this (n-2)-dimensional system with the equations for position and velocity. When completed, the system model will have the following form: (3.3) x1 = x2 X2 = x3 Xn-1 = xn xn = -a0X1 alx2 .. axn + f() The coefficients in the differential equation for x are the n coefficients of the closed loop polynomial of the linear system. The procedure used to convert the system to control canonical form is as follows. The physical state model is set up so that the controller (the (n-2)-dimensional subsystem represented by state variables yc) is in control canonical form already. This can be done for any controller representation that is controllable (if the original representation for the controller was a transfer function, a controllable representation can always be found). Next, the model is transformed so the velocity feedback into the controller subsystem is zero (i.e., vector a in -c2 (3.1) is zero). This can be done without loss of generality in the model, as shown by the following argument: Suppose ac2 is not zero in the original system. We can apply a similarity transformation (3.4) y = Ta (1) where y(1) is the new state variable vector, and T = 1 0 0 . 0 a 0 0 1 0 Oa 0 c2 n-2 O0 0 O l a 2 0 c2 0 0 0 1 0 0 0 1 Note this is equivalent to replacing all (n-2) controller states yc by (Yc(1) + ac2 Yn-l)- A straight forward calculation (plug (3.4) into (3.1)) shows that the resulting system has no feedback of velocity into the controller. Since a similarity transformation (or the equivalent change of variables) does not affect the basic system behavior, but merely its representational model, this alteration of the model is valid. Physically, this operation has removed a redundant parameter in the original model, since the velocity feedback in the original model can be achieved by adjusting the damping parameter, B, or adjusting the position feedback gain, K1. Since the models are equivalent, the superscript on the y state variables can then be dropped in subsequent equations. With the controller in canonical form, and the velocity feedback to the controller eliminated, the following intermediate form is obtained: (3.5) yc = Ac + a yn- Yn-1 = Yn J n = -K1n-1 By + b'c + f() In order to complete the conversion of the model to control form, two similarity transformations are then applied to this intermediate model in succession. As previously explained, similarity tranformations do not change the basic behavior, hence are valid alterations of the model equation. The transformations are: -1 (3.6a) A, = T1 A T1 -1 (3.6b) A2 = T2-1 A T2 where the nxn matrix A is obtained from the model in (3.5), and (3.7a) T1 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 n-3 1 0 0 .0 0 1 (3.7b) T2 = 1 0 0 0 0 1 0 0 0 0 1 0 0 60 1 n-3 1 and where the 's are the coefficients of the characteristic polynomial of the controller alone (i.e., the characteristic polynomial of matrix Ac in equation 3.5). The transformed system is then in control canonical form. Note that the friction input is unchanged by these transformations, and is still applied in the yn differential equation. The composite transformation is required for later derivations, as it defines the relationship between the physical states and the control form states: (3.8) y = T1T2x where T = TT = 1 0. 0 0 0 1 0 0 0 1 0 0 P0 16 n-3 1 0 0 60 91 n-3 1 Simplification of DE Solution Via Coordinate Translation Once the system is in control form, the following translation of the state coordinates can be used to convert it to an autonomous system (within each region). Since the friction is a constant in region I or II (L or -L ), and C C since none of the differential equations except that for x n involves xl, the xl coordinate can be translated by (3.9) xla = x1 f/a0 where the subscript "a" indicates autonomous coordinates, and f is the (constant) friction level in that region (I or II). Substitution of this translation shows that the x n equation still gives the same result, the system matrix is unchanged, and the input has disappeared. The autonomous system model is then (3.10) xla = x2 k2 = 3 n-1 = n x = -ax x nx n 0 la 1 2 n n Therefore, by performing a translation on the initial condition and on the final state of the trajectory in the region, the trajectory can now be described by the simpler form (3.11) x (t) = F(t,t0) a (t0) F(t,t0) = exp[A(t-t0)] instead of equation (3.2), where the matrix A is in control form (i.e., the matrix A2 produced by transformation 3.6). Of course, the control form representation is not required to perform this translation. The equivalent translation in terms of the physical state variables can be defined using the transformation matrix as (3.12) ly = T2 1 Ya = Y ly where 1' = [ f/a0 0 . 0 ] is the translation of the control form states (equation (3.9)) and Ya is the autonomous physical state variables. The resulting differential equation for y would be of the form (3.13) ya = A Ya (with A unchanged). A slightly different form of the existence conditions of Chapter IV would then result. The control form has certain advantages, however, as will be seen. Summary of Assumptions and Discussion of Generality of Model The following assumptions and conditions on the system are made during the analysis of friction limit cycles in this dissertation: (Al) The system can be modelled by a model of the form in Figure I-1, and equations (3.1). (A2) The velocity feedback into the controller is zero (without loss of generality, as shown by argument given previously in this chapter). (A3) The controller portion has a controllable representation (this is assured if the controller is representable by a strictly proper transfer function; if transfer function is proper the direct feed-through term can be divided out and lumped with the direct position feedback, so this case is also allowed). (A4) Friction appears only in the differential equation for velocity (the acceleration torque equation). (A5) The closed-loop, linear portion of the system should not have a pole at the origin. (A6) The limit cycles to be analyzed are all simple, that is, they traverse regions I and II once before returning to the initial point, and therefore have only four switching per cycle at most. Note that many of the assumptions were made for convenience in the subsequent development, so that a specific model could be used for concreteness. Most can be removed by a transformation of the original problem as discussed below, or by minor modifications of the theory, and so involve no loss of generality. In fact, the only restriction to the system model that cannot be removed is that the limit cycle is simple (assumption A6). The analysis could be extended, as discussed below, to remove even this restriction. Thus the theory is quite general. Several of these assumptions require further discussion. Assumption (A4) is required to perform the analysis using the piecewise linear method in Chapter IV, since the analysis uses the control canonical form heavily. If the friction input appears in more than one differential equation, the trick used there to eliminate the friction by a coordinate translation is invalid. Examples could be generated that violate this assumption. If acceleration torque was directly measured, the friction torque could be inferred and used as an input to the controller. The same comment holds for systems with differentiators, since the derivative of the velocity involves the friction. Friction also causes a reaction torque on the mounting of the motor or other physical device, so if a foundation or mount model is included, the assumption would be violated. This condition does not restrict the generality of the method, however. Systems which violate this assumption can still be transformed to control form, the method applied, and existence conditions checked. A minor modification of the Chapter IV conditions (involving the P vector only) would be required. Assumption (A5) is not necessary, except for steps involving taking the inverse of certain matrix quantities in Chapter IV. However, the assumption results in no loss of generality, for the following reason: a system with a zero eigenvalue has a free integrator so that, with the proper selection of state variables, one state does not feed back into the system. This can be seen by examining the control form of the system model; when an eigenvalue is zero, the coefficient a0 of the characteristic polynomial is zero, and the first column of the system matrix is zero (see equation 3.3). Therefore the magnitude of the state xl has no effect on the behavior of the system. Any limit cycle existing in the system will thus also exist in the (n-l)-dimensional system formed by deleting the state xl (although not necessarily vice versa). Therefore, any system that fails this assumption can be analyzed by deleting free integrator states, then looking for limit cycles; any limit cycles existing in the original system would be found by this method. Note, however, that solutions may be introduced that do not exist in the original. For example, an asymmetric limit cycle in the reduced system might not exist in the original since the free integrator output might not be periodic (see Chapter V). Assumption (A6) is also necessary to allow the derivation of Chapter IV to proceed. The piecewise linear method requires the overall shape of the limit cycle to be known, including which regions are traversed and the order of traversal. This could be considered a limitation of the method, since the analysis cannot be performed without knowing the shape of limit cycles that are to be found by the analysis! Note, however, that an even more restrictive assumption is made in the classical describing function analysis, a tool that has proven quite useful for decades. That analysis assumes a sinusoidal input to the nonlinearity, therefore implicitly assuming a half-wave symmetric type of solution with two switching per cycle. It is therefore felt that the assumption is reasonable. Future research could remove this assumption by listing all possible solution shapes (and order and number of traversals of regions during each cycle), and applying the method to each possibility. The difficulty lies in the large number of potential solutions to be checked. If it could be proved that all limit cycles are simple, the difficulty would disappear. Without such a proof, the results of this thesis apply only to simple limit cycles. For example, the necessary conditions for the existence of a limit cycle found in the next chapter are necessary only for simple limit cycles. Therefore, the lack of a solution for these conditions for a particular system only guarantees that a simple limit cycle cannot exist for that system. More complicated behaviors are not ruled out. Finally, some comments on the generality of the system model assumed are in order (assumption (Al)). The model was inspired by the case of a rotational servomechanism, such as a DC motor. Although this is an important application, the theory developed here can be applied to many other systems, as long as they can be put in the form specified. For example, a linear drive system has the same characteristics of inertia, damping, friction, and acceleration force (instead of torque), so the theory can be directly applied. Systems with gear-trains and loads can be modelled by lumping the load and gear inertia into the motor, as long as the system is rigid and closely coupled (gear backlash can be ignored). Although the model is for a continuous-time system, the use of discrete-time (i.e., digital) controllers can be accomodated by using their continous-time equivalents. Many systems of practical interest have mechanical flexure, modelled by resonances. If they can be lumped into the model of the controller, the theory can still be applied as is. However, the case of two masses, separated by a spring, where both masses experience nonlinear friction, does not fit into the present model. Additional piecewise linear regions must be specified, with a great increase in the complexity of the analysis, unless the limit cycle is known to be low in frequency so the system could be considered to behave as a rigid body. In summary, certain assumptions are made to allow a specific, concrete model to be used in the development. Examination of the assumptions shows that most are for convenience only, and the theory applies to a quite general class of problems. Given these restrictions and assumptions on the problem to be studied, and the development of a convenient representation, the piecewise linear method can be applied to the problem of friction limit cycles. The next chapter derives exact conditions for the existence of these limit cycles, and, once a solution is found, provides exact information on the limit cycle trajectory. CHAPTER IV EXISTENCE CONDITIONS FOR LIMIT CYCLES This chapter applies the piecewise linear method to the problem of friction limit cycles, using the model developed in the previous chapter. The method provides a set of nonlinear algebraic equations for which a solution must exist in order to have a limit cycle. These represent necessary conditions for limit cycle existence, since every (simple) friction limit cycle for this system model meets these conditions. The issues of sufficient conditions and minimal sets of necessary and sufficient conditions are discussed. Finally, some examples applying these conditions are presented. Necessary Conditions for a Simple Limit Cycle In order to simplify the analysis, the assumption is made that the limit cycle which the system exhibits is simple, in the sense that it traverses each of the four state-space regions exactly once during each limit cycle period. This is assumption (A6) discussed in Chapter III. This assumption makes sense from a physical viewpoint, since it is hard to imagine a system where the velocity changes sign more than twice before returning to the initial state. It is clear that this assumption holds in the 2D case, since the trajectory must go to the left in region I and to the right in region II (see Figure I-3 and I-5) and the 49 trajectory cannot cross over itself. However, no claim can be made that the following analysis is completely general, due to this assumption. Therefore, although this is a restriction of the generality of the results, the solution of the equations requires the limit cycle meet the following definition: Simple Limit Cycle: A periodic orbit which traverses each piecewise linear region once at most (at most twice for region III). As a consequence of this assumption, there will be exactly four distinct switching instants during the period. The system will be in a particular region, and, therefore, its behavior will be governed by that region's system model equations, for T. seconds, for i = 1, 2, 3, 4. As will be seen below, some of these switching periods may be zero. Thus the total limit cycle period T is the sum of the four T.'s: 1 (4.1) T = T1 + T2 + T3 + T4 Note that the subscript on the periods indicate order, and not the region being traversed during period i. Since the limit cycle is periodic, any initial point may be chosen to start the analysis. Define x to be the system state at t=0. Assuming a limit cycle exists, this initial state can be chosen to be the point in region III (or V, see Figure 1-3, p. 6) where motion is impending. In other words, initial velocity is zero, and the initial driving torque is negative, and sufficient to overcome static friction and begin motion. These requirements on the initial point can be expressed as (recalling that y represents the physical state variables, and x the control canonical states): (4.2a) Yn(0) = [ 0 60 n-3 1 ] x = 1' 0X = 0 (4.2b) yn(0) = H' A x" < -LS (or = -LS ) where the notation A' is used to represent the nth row of the x to y transformation in (3.8) (used to calculate the velocity yn from the state x), the parameter LS is the maximum static friction before breakaway, and the matrix A is the control canonical form of the state matrix, in region I. Note that the A matrix from region I (velocity < 0) is used since the rate of change of yn is identically zero in region III, by design. Having characterized the requirements on the initial point, the limit cycle trajectory can be followed in region I (which the system transits after breaking free) using the region I model. The autonomous state variables can be formed as discussed in Chapter III, by making the translation: (4.3) xla = x (-L/an) or x = x - -a with 1' = [ (-Lc/an) 0 0 . 0 ] where a is the value in the nth row and first column of A, and LC > 0 is the sliding coulomb friction torque. This translation is selected so that the friction torque is positive, and opposes the motion in region I in the negative direction. Note that the parameter anl cannot be zero, since this would give a zero column in A, which signals the presence of a free integrator. The trajectory in region I is then At At (4.4) x (t) = e x = e ( 1) -a aO ( so that the trajectory exits region I at the point where y (velocity) again goes to zero (4.5) x(T1) = xa(TI) + 1 AT = e (x 1) +1 with the requirement that n (T ) = 0: (4.6) n(T1) = l' X(T1) = 0 At this point, the limit cycle will display two types of behavior, depending on the driving torque at t=T If the trajectory transits from region I into region V, this torque is already sufficient to break loose the system and drive back in the other direction, then T2 = 0 and the system spends no time in region V (in other words, system does not stick, but is only momentarily motionless). As discussed in Chapter III, the region V equations do not have to be applied, since the entry and exit points for the trajectory in this region are the same. If this also occurs at the other switch point (i.e., T4 = 0), the limit cycle is driven purely by sliding friction. The other case is more complex to analyze, since the trajectory must be calculated through all four switching periods. If the driving torque is insufficient to overcome the static friction at t = T1, the mechanical load stops for a period T2 until the controller can integrate up and break it loose. The system has come to rest in region III after leaving region I, so system behavior is governed by the model for region III during this period. Both static and sliding friction effects will affect this limit cycle. For this case, the system state evolves as (4.7) x(t) = es (t-T1 (T) T1 < t < (T1 + T2) from the initial point when the velocity went to zero at t = T Note that no translation is necessary, since the friction term is absorbed into the system model in the static condition, represented by state matrix A In either the pure sliding friction case (T2 = 0) or the sticky case, the requirements on the system state at t = T1 + T2 is that the driving torque is sufficient to break loose: (4.8) Yn(T1 + T2) = V'A x(T1 + T2) > LS (or = LS) Since the sticky case involved the system evolving until sufficient torque was developed to break away, the system will move as soon as torque equals the static friction, so (4.8) becomes an equality; note that the same comment applies to condition (4.2b). There is also the implicit requirement that the velocity is zero when motion is impending (as in condition (4.2)), but this requirement is not placed explicitly, since it is taken care of by the model equations in region III (which force acceleration to be zero) plus the zero velocity at t = T1. The system goes through the same behavior as above, in the opposite sense, while traversing region II and returning to III (or IV). The equations for the trajectory in these regions, along with the conditions on the state at the switching instants, are found by first performing the translation to autonomous coordinates for region II, as in (4.3): (4.9) xla = x + (-Lc/an) or x = x + 1 -a - with i' = [ (-Lc/a ) 0 0 . 0 ] as before. Note the friction torque is negative in this region for LC > 0, since the velocity is positive. The trajectory in region II is then (4.10) x (t) = eA(t-T1-T2) x (T1 + T2) = eA(t-T1T2 (x(T1 + T2) + i), for T1 + T2 < t < T1 + T2 + T3 and the trajectory exits region II at the point where y (velocity) again goes to zero (4.11) x(T1 + T2 + T3) = x (T1 + T2 + T3) 1 AT = e 3 ( x(T1 + T2) + 1) 1 with the requirement that y (T + T2 + T) = 0: (4.12) Yn(T1 + T2 + T3) = x(T + T2 + T3) = 0 At this point, the system trajectory either immediately breaks free (trajectory in region IV) and leaves the region, or it sticks (trajectory in region III) and follows the trajectory defined by (4.13) x(t) = e (t-T1-T2-T3) x(T1 + T2 + T3), for T1 + T2 + T < t < T + T2 + T3 + T from the point when the velocity went to zero. Note that no translation is necessary, since the friction term is absorbed into the system model in the static condition as in the case of region III. Collecting the conditions (4.5, 4.7, 4.11, 4.13) for the trajectory through all four regions, and placing the requirement that the end-point match the initial point, we have the following nonlinear algebraic matrix equation for the initial point: (4.14) AT AT AT AT (4.14) I es 4 e 3 eAs 2 e 1) x = 1 with AT AT AT AT 1 = es 4 (e 3 [e s 2 (I e A} 1 + 1] 1) AT AT AT AT AT AT AT = -es4 e 3 eAsT2 e 1 1 + e s 4 e 3 e s 2 1 AT AT AT + e s 4 e 3 1 es 4 1 In addition, there are four scalar nonlinear equations from the four conditions on switching points (4.2a/b, 4.6, 4.8, and 4.12, but the last one is redundant). Altogether, there are n+4 (nonlinear) equations in the n+4 unknowns x0' T1, T2, T3, and T4. In the pure sliding friction case, of course, T2 and T4 are both zero, but conditions (4.2b) and (4.8) become inequalities and thus drop out, and (4.14) simplifies to AT AT (4.15) {I e 3 e 1) 0 = 11 with AT AT 1 = e 3 [ (I e 1 1 + 1 ] 1 -1 = -eA3 eAT 1 1 + 2 eAT3 1 1 so there are n+2 equations in the n+2 unknowns xg, T1, and T3 plus the two inequality conditions on the switch points. The results presented so far in this chapter can be summarized as Theorem 4-1: A simple limit cycle exists for the system model under study only if the following algebraic conditions hold on the variables xO, T1, T2, T3, and T4: Sticking Limit Cycle Case AT AT AT AT (4.14) (I e s 4 e 3 e s 2 e 1) 0 = 1 where 1 = es 4 (e AT3 [es 2 (I e AT1) 1+ 1] 1) AT AT AT AT A.T AT AT = -e s 4 e 3 e s 2 e 1 1 + e s 4 e 3 e s 2 1 AT AT AT + esT4 eA3 1 es4 1 (4.12) Yn(T1 + T2 + T3) =[ 0 0 Pi n-3 1 ] x(T1 + T2 + T3) = (T1 + T + T) = 0 (4.2b) Yn(0) = x' A x0 = -LS (4.6) Yn(T1) = X(T1) = 0 (4.8) Yn(T1 + T2) = 2 'A x(T1 + T2) = LS Pure Sliding Limit Cycle Case (T2 = T3 = 0) (4.15) {I e 3 e 1) x0 = 1 -0 1 AT AT where 1 = e 3 [ {I e 1) 1 + 1 ] 1 1-l = -eA3 eAT 1 1 + 2 e AT3 1 1 (4.2a) y (0) = [ 0 P . n-3 1 ] X = X0 = 0 (4.6) Yn(T1) = x(T) = 0 In other words, the conditions above are necessary for the existence of a simple limit cycle. Condition (4.12) was included in the sticking case instead of the equivalent condition (4.2a) to obtain a more symmetric form of the conditions. Note that the conditions requiring velocity = 0 could be applied at either t = T1 or t = T1 + T2 (equation (4.6)), and at either t = 0 or t = T1 + T2 + T3 (equation (4.2a)). This is a consequence of the fact that velocity remains zero between these times. Proof: The derivation presented above is a step-by-step solution, through each region, for the limit'cycle trajectory that was assumed to exist. Since the conditions stated in the theorem were derived directly from the trajectory solutions as seen above, and since a simple limit cycle by definition must traverse those regions as assumed, the conditions are necessary, and the proof is complete. Note that a non-simple limit cycle, should any exist, would not have to meet the necessary conditions. Thus, if a solution to the necessary conditions exists for a system, a limit cycle may exist; if no solution exists, no simple limit cycle exists, but a non-simple one is not ruled out! A complete theory requires an examination of the possibility of more complex limit cycles. It is interesting to note that the same development could be performed using physical state variables. This would result in replacing the control canonical A matrix in the equations by the original system matrix, eliminating the need for the vector P, and replacing 1 by the translation in equation (3.12). The form used here clarifies the role of system poles and zeroes, however, as seen by the following discussion. The definition of the initial state X0, equation (4.14), is completely determined by the poles (for fixed switching periods), since the A matrix contains the characteristic polynomial information only. Now, observing that the transfer function from the friction input to the velocity as an output is Yn(s) / l(s) = [0 0 . 0 1] T12 (sI-A)-1 b (this holds since y(s)=T12x(s), and x(s)=(sI-A)-lbl(s) ), with b' = [0 0 . 1], it is clear that the transfer function zeroes are defined by the bottom row of T12, that is, the vector P. Thus, varying the zeroes only would change the P vector only (in such a way that poles of the closed loop system are unaffected), X0 would be unchanged, and the four (or two) switching conditions alone would be affected. The control form representation clarifies this relationship, which suggests an interesting line for further research. Illustrative Examples Example IV-1: Two-Dimensional (2D) System with Sliding Friction Limit Cycle This is the same as Example I-1 of Chapter I; the piecewise linear method is the same as standard phase plane methods for this 2D case. The state variable model of the system is (4.16) dyl/dt = y2 dy2/dt = -(K/J) yl (B/J) Y2 Lf/J where Lf is the friction torque, yl is position, and y2 is velocity. The analysis performed below assumes K, J > 0, although B can be negative (the other cases can be solved by similar methods, but are left out for brevity). Chapter I presents more information about the system, including results of a previous solution by phase plane methods. A second-order system cannot possibly have a limit cycle with sticking, since once the system sticks, it is at an equilibrium point (there is insufficient torque to break free, and no controller integrator to ramp up). Therefore, we need only examine the necessary conditions for the sliding case. In addition, it will be assumed for this example that the limit cycle is symmetric (this property is proved for 2D systems in Chapter V). Applying the equations developed in this chapter, the initial condition can be found from equations (4.5) (which (4.15) simplifies to using the symmetry assumption) and (4.6): AT (4.17) x(T1) = eAT1 (x 1) + 1 = - (4.18) y2(T1) = 1' x(T1) = x2(T1) = 0 where use is made of the fact that 0' = [ 0 1 ] (physical state model already in canonical form), and x(T1) = -X0 (by symmetry). Making the assumption that K is large enough (or B is small enough) that the system poles are complex, the matrix exponential for this 2D system can be evaluated as (4.19) eA = et cospt+(a/p)sinpt (1/p)sinpt -(K/JP)sinpt cos3t-(a/p)sin3t where a = B/(2J), 3 = (1/2) { (4K/J) (B/J)2 ). Letting 0 = [a 0]' (since x2=0), and noting 1 = [LC/K 0]', equations (4.17) and (4.18) become: -aYT (4.20) x2(T1) = 0 = -[a (LC/K)] eT 1 (K/JP)sin3T1 which implies T1 = 7/3 (note a > (LC/K) since otherwise x0 is an equilibrium point). Theoretically, T1 can also be any integer multiple of -at 7/3. However, noting that x2(t) = velocity = C e sin/t, any multiple greater than one would entail sign changes in the velocity for t This is an example of a solution to the necessary conditions that is not a valid limit cycle; see discussion on sufficient conditions below. The other component of x0 can be evaluated from (4.17) as (4.21) x1(T1) = -a -arT = [a (Lc/K)] eT l(cospT1+(a/P)sinPT1) + (LC/K) = [a (LC/K)] e-a7/8 cosr + (LC/K) Therefore (4.22) a + (LC/K) = [a (LC/K)] e-ar/ The initial condition can now be evaluated as (4.23) a = (Lc/K) ((/+1)/(4-1)), P = e-ar/ which is the result presented in Chapter I. Since x0 is in region IV, a must be greater than zero. This implies j>1, requiring a<0, or B<0. As the solution from Chapter I predicted, a sliding type of limit cycle exists for every case with negative damping (B<0) (if the position feedback gain K is sufficiently large to give the linear system a complex pair of poles). The initial conditions can be expressed in terms of the system parameters by the equations from Chapter I: (1.2) X10 = (LC/K) [(x+1)/(g-1)], x20 = 0 where (1.3) p = exp[-Br/(2JP) ], = P [4K/J B2/J2] Refer to Figure I-4 for a plot of amplitude (which equals x10) as a function of gain K for the case J=LC=1, B=-l. Example IV-2: Three-Dimensional (3D) System with Both Sliding and Sticking Limit Cycles The second example system to be considered (same as Example I-2) is defined by the differential equations (1.4) dy/dt = A y + b L where (1.5) A = 0 1 0 0 0 1 -K1 -K2 -B with b' = [0 0 -1], Lf = friction torque, and where the state variables are y2 = position, y3 = velocity, and yl = compensator integrator. Figure I-6 shows a block diagram of this system. The results presented in Chapter I are now to be derived, based on the equations developed in this chapter. Case I: One Stable Pole and One Imaginary Pole Pair (Only Sticking Limit Cycle Exists) As in the example in Chapter I, let us first set B = K = K = 1, and set the sliding friction torque to 1.0 and sticky friction (breakaway torque) to 1.2. The equations (4.2a/b, 4.6, 4.8) defining the switching periods and initial state (4.14) can be set up and solved for this specific case, in order to demonstrate the analytical calculation of the limit cycle trajectory. The eigenvalues of this system are -1 and jl. The matrix exponential of A is required for the equations, and can be found (by, for example, diagonalizing A) as AT -T -T (4.24) 2eAT e +sinT+cosT 2sinT e +sinT-cosT -T -T -e -sinT+cosT 2cosT -e +sinT+cosT -T s-T e -sinT-cosT -2sinT e -sinT+cosT The equations to be applied are AT (4.25) x = e AT1 ( 1) + 1 AT (4.26) x2 = es 2 x -x0 where a symmetric limit cycle was assumed (breakaway point at t = T1 + T2 is at -x0), and therefore the simpler equations were used in place of (4.14). Note that A is the s A matrix with the bottom row set to zero, hence the matrix exponential in the sticking region is (4.27) eAsT2 = 1 T2 T22/2 0 1 T2 0 0 1 In addition, the breakaway condition that the acceleration torque equal the static friction is required: (4.28) 1'Ax0 = dx3(0)/dt = -LS Noting that the translation vector 1' = [ 1 0 0 ], letting 0 = [ a b 0 ] (since x30 must be zero), and letting xI = [ c d 0 ], we obtain (4.29) c 1 = (1/2)(a-l)(e-T + sinT + cosT) + bsinT d = (1/2)(a-1)(-e-T sinT + cosT) + bcosT 0 = (1/2)(a-1)(e- sinT cosT) -bsinT from application of equation (4.25) and the definition of AT the matrix exponential e, (4.30) c + d T2 = -a d = -b from application of equation (4.26) and the definition of A T e s 2, and (4.31) a + b = L S from application of equation (4.28). By eliminating c and d from these equations and performing additional algebra, they simplify to (4.32) b = LS a -T T2 = [(a-l)e-T1 + (a+l)] / (LS a) -T a [e 1 + sinT cosT ] -T = eT1 + (21sf l)sinT1 cosT1 -T a [-e 1 sinT1 cosT1 -2] = -e T1 (2 LS l)cosT1 sinT1 2LS The first two equations define b and T2 in terms of a and TI, while b and T2 were eliminated from the last two, which can be used to simultaneously solve for a and T1. Eliminating a and after some algebra, the last two equations form a nonlinear algebraic equation, whose zeroes are potential limit cycle solutions: (4.33) f(T1) = (e-T 1 l)(l+cosT1) (e-T1 + l)sinT1 = 0 The plot of this function in Figure (IV-1) and analysis shows that there are zeroes at odd multiples of r, and also near 37/2 (+ 2n7, n = 0, 1, ...). The solution at r results in a valid limit cycle solution where x = [ 1 0.2 0 ], -0 T1 = r, and T2 = 10, yielding an overall limit cycle period of approx. 26.3 seconds, matching the simulation results. The second potential solution is at T1 = 4.73 (approx. 37/2, obtained by numerical solution of f(T1)=0), and is invalid, since it results in a negative value for T2 (of -12.214). Additional solutions are also invalid, for the same reason as in Example IV-1. Examination of the matrix '-- __ _-------- ------- ,^. o i I oo 000 O rr m > 0 o 0 0 u I0 U HYA o-4 o 41 , - I M U *-1 Cn C 11C ti exponential and the resulting formula for the velocity state shows that the velocity changes sign approximately every 7 seconds. Therefore, the trajectory using these longer values for T1 leaves region II before the switching time, so it is invalid (see next section). Therefore, the solution of the piecewise-linear equations results in a valid limit cycle solution that matches simulation results. Chapter I contains plots of the limit cycle trajectory. Case II: One Stable Pole and One Unstable Pole Pair (Both Sticking and Sliding Limit Cycles Exist) If the same example system is used with B = 0.9, and K1 = K2 = 1 still (case II in Chapter I), the closed-loop eigenvalues (of the linear portion of the system) move into the right-half plane, to .026 j 1.024, while the other pole is at -.9524. As stated in Chapter I, there is still a sticking limit cycle solution close to that of Case I (Figures I-7 through 1-9, confirmed by simulation). A similar analysis to that presented above can'be used for this case, so the details are omitted. Note, however, that numerical methods may be required to evaluate the matrix exponential (due to the tediousness of the calculations; case I was a particularly simple form); this forces the use of iterative methods to calculate the limit cycle parameters (see example of this method below). The second limit cycle solution, as discussed in Chapter I, is of the sliding type. Simulation having shown one with a half-period T1 approximately equal to 3.1 seconds, an iterative numerical solution of the equations for the symmetric, sliding case was performed, as follows. The equation for this case is obtained from (4.5), where again the assumption is made that the limit cycle is symmetric: AT (4.34) xl = eAT ( 0 1) + 1 = -xO which can be solved as (4.35) xO = (I + eAT1)-1 (I eAT) 1 A computer program calculated values for the initial condition xg, given trial values of T1, and the process was repeated until the velocity initial condition XO(3) = 0. This was not difficult, since an approximate starting condition was available from the simulation results. Given a T1 that resulted in a zero initial velocity, the entire initial condition could be defined from (4.35). This yielded a solution at T1 = 3.1455 seconds, and 20 = yO = [-0.283 12.4 0.]. The solution is valid since the breakaway condition is also satisfied (torque at zero is greater than LS). Figures 1-10 and I-11 show some views of the limit cycle for this case. An appendix contains computer code that was used to perform this iterative calculation. The code applies to the sliding case only, and would have to be modified for the general case. Exact (Necessary and Sufficient) Conditions Theorem 4-1 presented some conditions that were necessary for the existence of a simple limit cycle. That they are not sufficient is seen by the results in the Examples, where solutions were found that met all of the necessary conditions, yet were not valid limit cycles (the switching period T2 was less than zero, or velocity changed sign during a switching period). In order to expand the previous conditions into a set that is also sufficient, the concept of a consistent solution is useful: Definition: A consistent solution is a solution of the equations in Theorem 4-1 that meets the assumptions about the region containing the trajectory during each switching period. These assumptions are (Sticking Case) (1) Trajectory in region I, 0 < (2) Trajectory in region III, T (3) Trajectory in region II, T1 + T2 < t < T1 + T2 (4) Trajectory in region III, T1 + T2 + T3 5 t T1 (Pure Sliding Case) (1) Trajectory in region I, 0 < (2) Trajectory in region V, t = (3) Trajectory in region II, T, t < T1 1 t 2 T1 + T2 + T3 + T2 + T3 + T4 < t < T1 + T3 (4) Trajectory in region IV, t = T1 + T3 A check of the consistency of a solution is straightforward, based on the definitions of the various regions presented in previous chapters: the sign of the velocity is examined for assumptions (1) and (3), and the magnitude of the torque for (2) and (4). Given this definition, a set of necessary and sufficient existence conditions can now be presented: Theorem 4-2: A simple limit cycle exists in the system under study if and only if a consistent solution of the Theorem 4-1 equations exists. In addition, this limit cycle has the exact properties (switching periods, initial condition, and trajectory) defined by the corresponding solution to the equations. Proof: (Necessity) Suppose a simple limit cycle exists with the given properties. Theorem 4-1 (more exactly, the derivation in the first part of this chapter) demonstrated by analysis of this limit cycle that the listed equations in fact apply to the trajectory. In addition, this derivation, along with the assumption that the limit cycle is simple, shows that the solution must meet the consistency conditions, i.e., it must be in the appropriate region of the state space during each switching period. Therefore this condition is also necessary. (Sufficiency) Now suppose a consistent solution of the equations exists. A limit cycle can be generated, with the same properties as defined in the solution, by starting at x0 and following around to the initial point again. At each step of the process, the consistency conditions require the trajectory to be within a given region, so that the system equations for that region apply. The equations listed in Theorem 4-1, derived from the application of those system equations, then show that the trajectory reaches the next switching point at the appropriate time and state. Therefore, a limit cycle must exist with those exact properties, and the proof is complete. Although this theorem defines a set of exact conditions, i.e., they are equivalent (necessary and sufficient) to the existence of the limit cycle, it is appropriate to ask if these conditions are minimal. In other words, are there redundancies in the set of conditions? Is there a simpler set of conditions that are still exact? For example, it may be sufficient to merely require switch periods greater than zero, thus eliminating cases such as found in Example IV-2. These questions are open, and may be subjects of further research. CHAPTER V LIMIT CYCLE SYMMETRY AND STABILITY This chapter uses the results of chapter IV and other known facts about the friction nonlinearity to examine the behavior and characteristics of friction limit cycles. The model of the system developed previously is used to analyze the stability of limit cycles predicted by the equations in chapter IV. In addition, the relationships between the oddness of the system differential equations, symmetry, and uniqueness of limit cycles is explored. Stability of Predicted Limit Cycles Once a limit cycle has been predicted by the solution of the equations derived in Chapter IV, it is natural to ask about its relationship to the global phase portrait. Since the piecewise linear model used here is exact, the exact limit cycle period, amplitude, and trajectory can be found immediately once an initial point on the limit cycle is known. It is more difficult to determine stability and other information about the phase portrait, however, since it can require extensive simulation. However, local stability of the limit cycle can be determined by the standard method of linearization of the periodic flow map at the initial point previously found. In other words, an approximate model of the trajectories near the limit cycle initial point can be determined by 71 linearization. This analysis provides information about the asymptotic stability of a closed orbit solution as discussed in Hirsch and Smale (1974), chapter 13 (if more information on orbital stability analysis is of interest, this reference provides a good description, and was quite useful in the development of the results in this chapter). Note that the model provided by this analysis is not a continuous-time representation of trajectories near the cycle; instead, this discrete-time model shows the deviation from the initial point and how it evolves after each cycle period. Specifically, an initial sufficiently small deviation from the initial point x0 is assumed (so the trajectory starts at x +6x ) and the trajectory traced once around the limit cycle. The new deviation from the initial point when the trajectory completes the cycle is related to the initial deviation by the equation (5.1) 6xT = Y 6Sx + higher order terms where 6xT is the new deviation from the initial point (0+6xT is the point on the trajectory after one cycle), and Y is the transition matrix, representing the first-order evolution of the deviations from the initial point over time. Eigenvalues of the transition matrix determine stability, while eigenvectors determine the phase portrait near the limit cycle. This analysis is attempted below; however, because of the discontinuities in the system nonlinearity the analysis is difficult. Therefore, the analysis is accomplished, as in the derivation of Chapter IV, by solving one region at a time, and then pasting the trajectories together. The first case analyzed is that where the predicted limit cycle has no sticking (sliding friction only). The case with sticking during the cycle is then analyzed by building on the first case. Case 1: Limit Cycle with no Sticking Assume an initial point x0 that meets the equations in Chapter IV with no sticking. In this case, the first switching period is equal to the constant T1 in chapter IV, the second switching period is T3, and the constants T2 and T4, representing the periods of stickiness, are zero. A deviation 6x0 is assumed from the initial point of the limit cycle. We wish to limit deviations, however, so that the perturbed initial point is still contained in region IV. This is necessary in order to apply the Chapter IV equations, since the initial point to which the equations apply is assumed to be in region IV. Limiting the deviation in this fashion results in no loss of information about the limit cycle stability, however; (n-l) of the eigenvalues of the periodic flow map can still be determined, while the nt eigenvalue (that applies to deviations out of region IV, and along the limit cycle) would be zero, as will be seen in the examples below. Note that the derivative of the flow after one orbital period has an eigenvalue of unity (Hirsch and Smale, 1974, p. 277), but the Poincare map has a zero eigenvalue (Hirsch and Smale, chapter 13, section 3). In order to limit the deviation 6x such that the initial point remains in region IV, 5x can be constructed as -1 ' (5.2) 6x = T12 Ir 6y0r where T12 is the composite linear transformation from Chapter III used to convert between canonical and physical state variables (y = T12 x), I is an n x (n-l) matrix (Ir is (n-1) x n) used to force the nth element of the deviation vector Sy0 to be zero: (5.3) I = 1 0 0 ... 0 0 1 0 ... 0 0 ... 0 1 0 0 ... 0 0 1 0 ... 0 0 0 and, finally, Sy0r is the (n-l)-vector of deviations in physical state variables. This construction forces the 6x0 vector to remain in region IV, since the change in initial velocity (6Sy(n), the nth element of the physical state vector) is zero. Although the derivation below will be done in canonical variables, this construction (5.2) will be used at the end to complete the calculation for physical variables, and force the deviation to have zero initial velocity. Starting at the initial point x +6x the trajectory is determined by the piecewise linear methods as in chapter IV. The trajectory in region I is found using autonomous state variables: (5.4) x (t) = x(t) 1 --a so the trajectory is At At (5.5) x (t) = e x (0) = e [x0 + x- -a -a At x(t) = e [X0 + SX0 1] + 1 For a sufficiently small deviation 6x0, the trajectory reaches region V (velocity goes to zero preparatory to reversing direction), but not necessarily in exactly T1 seconds. Denoting the change in this period is ST, the velocity goes to zero when the trajectory reaches (5.6) x(T1 + 5T) = eA(T1+6T) [x + 5x 1 1 The quantity of interest is the deviation of this point from the point where the original limit cycle reaches region V AT (5.7) x e 1 [X 1] + 1 (5.8) Sx = x(T1+6T) x- AST AT A(T +aT) =(e I} e 1 [ ] + e(T1T) 6x Note that the first term shows the dependence on ST, since it is zero if ST is zero, while the second term shows the dependence on S6x0 -Q Now that the exact value for Sx is determined, this nonlinear formula must be linearized to determine local behavior. One factor that causes difficulties is that 6T depends on 6x0, which complicates the calculation for (5.9) Y = (d(Sx )/d(5x )}|Ix= where Y is the desired transition matrix defined above, and the derivative is evaluated for 6x = 0. The dependence of Sx on both ST and 6x0 explicitly is handled using the chain rule. Note that since x is not a function of 6x or ST, (5.10) [d/d(6x0)] (6x ) = [d/d(6x )] {x(T1+6T) x) = [d/d(Sx0)] (x(T1+6T)) = [d/d(6x0)] (eA(T1+6T) [X0 + S 1]) where the term in x(T1+6T) equal to 1 was also dropped since it does not depend on x 0. Using the chain rule, this derivative is (5.11) d(6x1)/d(6x0) = D(6X1)/D(6x0) + [D(6X1)/D(6T)] [d(6T)/d(6x0)] where the symbol "D" indicates a partial derivative. The first term is (5.12) D(6x )/D(6x0) = eA(T1+6T) AT = eAT1 when evaluated at 6x0=0 (hence 6T=0). The partial derivative in the second term can be evaluated as follows. First define AT m = e 1 [X0 + &x 1] Then, by using (5.10), the desired partial derivative is (5.13) D(6x1)/D(6T) = [D/D(6T)] {eAST M) = AST A e m AT = Ae 1 [ 1] when the equation for m is substituted back in, and the derivative is evaluated at 6x0=0 (6T=0). The differentiation step can be verified by expanding the exponential into a power series, and evaluating term by term. The evaluation of the [d(6T)/d(6x0)] factor in the second term requires the definition of a functional relationship between 6T and 6x0. We know from the derivation of x(T1 + 6T) above that the velocity goes to zero at this point (trajectory reaches region V). As in the analysis in Chapter IV, this can be stated as (5.14) P' x(T1 + ST) = 0 (eA(T +6T) = {e 1 [x + 6x0 1]) since P'1 = 0. This formula implicitly defines the relationship between 6T and 6x in that we have a function of the form f(6T,6x0) = 0. The implicit function theorem (Rudin (1976), pp. 223-8) can be used to evaluate the desired derivative as -i (5.15) d(6T)/d(6x0) = [Df/D(6T)]-1 [Df/D(6x0)] Evaluating these factors, (5.16) Df/D(ST) = [D/D(6T)] {('(eA(T1+T) [0+6x0-11)) A6T = [D/D(6T)] {(' eAST m) A6T = A e A with m defined as above (to see this, again expand exponential into series and evaluate term by term). The second factor is (5.17) Df/D(6x0) = [D/D(6x0)] {('(eA(T1+6T) [x0+60-])) = [D/D(6x0)] {(' eA(T1+6T) 6x0) = eA(T1+6T) The derivative from the implicit function theorem, (5.15), can now be evaluated as AT -1 AT (5.18) d(6T)/d(6x0) = -({'A eAT1 [O-I]} ) ('eAT 1 where the factors are again evaluated at 6x =0 (6T=0). Finally, combining (5.12,13,18) into (5.11), we get the total derivative representing the linearized periodic map (5.19) Y = (d(6xl)/d(6x0)) 16x=o AT AT AT -1 AT = eA1 (A eAT1 ['X-1]) ('A eAT1 [-O i-{e A1) Note that the factor to be inverted is a scalar. This expression can be simplified by noting that AT (5.20) i = eAT1 [x 1] + 1 so substituting into (5.19) gives AT -1 AT (5.21) Y = eAT1 (A [X1-1]) (P'A [X1- i)-1 ('eAT1) = eAT1 (A [xl-1] 'e AT1) / ({'A [x-1]) The linearized flow map for the entire cycle must now be constructed, using this equation (5.19) for the linearized flow map for a half-cycle. Note, first of all, that the linear transformation representing the linearized flow map for the cycle is the product of the'linear transformations of the two halves. This is a consequence of the chain rule, since (5.22) 6xT = f (6x ) = (f2(6x0)) d(6xT)/d(6xo) = (d(6)/d(6x1)) (d(6x1)/d(6x )) The second of these two linear transformations has already been derived; the first must be found and the linear map for the entire cycle found by multiplication. Assume first of all that the limit cycle is symmetric, so that the point at which the limit cycle velocity goes to zero is exactly one-half period after the initial point, at -X0. In this case, the half-period is equal to the constant T1 and T3 equals T l Therefore, the linearization in which we are interested is completely determined by the first half of the limit cycle trajectory, because of the odd symmetry. To show this, assume that the same deviation from the point xl is taken, except with a change of sign, and trace the trajectory from this point x -6x Noting that this initial point equals (-x0-6x0), and using the odd symmetry of the phase portrait, it is clear that the trajectory returns to region IV at the point (-x -6x ) (since the trajectory from x0+6x0 goes to x1+6x ). Thus the resulting deviation on the second half of the cycle is (5.23) 6x = (-x2-6x ) X = -6X1 since xl = -x0 Linearizing this second leg, using the initial deviation of -6x , (5.24) 6x = Z (-6x ) + higher order terms -2 = -6x = -Y 6x + h.o.t. by the analysis of the first half-cycle. Since this holds true for any deviation 6x0, we must have Y = Z. Therefore, the transition matrix for the whole cycle is the square of the transition matrix for the first half-cycle, Y2 Now remove the assumption of symmetry. Assume that the limit cycle is not symmetric (but still involves only sliding, i.e. case 1, no sticking); in this case the linearized flow map must be examined on the second half of the cycle also. The same derivation as above (equations (5.4) through (5.8)) can be performed, but starting at the initial point x1+6x1, in order to determine SxT in terms of 6x1. The trajectory in region II is found using autonomous state variables: (5.25) x (t) = x(t) + 1 -a so the trajectory is (5.26) x(t) = eA(t-T) [ + x1 + 1] -1 For a sufficiently small deviation 6x1, the velocity goes to zero when the trajectory reaches (5.27) x(T + 6T3) = eA(T3+6T3) [x + 6x + 1 - The deviation of this point from the point where the original limit cycle reaches region IV is (5.28) 6xT = x(T+ST3) xT = (eAT3 I) eA3 [x + 1] + eA(T3+T3x Note that by comparing this formula to (5.8), and stating (5.8) as a function with parameters (5.29) 6x= (eAST I) eAT1 [x 1] + eA(T1 T) = f(6x0, 6T; T1, x0-1) then we see (5.30) 6x = f(6xl, ST3; T, x +1) The linearization for this function has already been derived, so the linear map for the second half of the cycle is found by substituting the correct parameters from (5.30) into (5.19), so that (5.31) 6ST = Y(T3, x1+1) 6x, + h. o. t. (5.32) Y(T3, l+1) = d(6XT)/d(6X1) 6x=0 = eAT3 (A eAT3 [+]) (~'A eAT3 [+)-1 ('eAT3) By the argument given previously in (5.22), the total linearized flow map is the product (5.33) Ycompos = Y(T3' +1) Y(T1' 0-) 6T = Y cx + h. o. t. -T compos -0 Note that this reduces to the symmetric case when T3 = T1 and x = 0: (5.34) Ycompos Y(T -x+1) Y(T, xO-1) = Y2(T1, -1) since Y is an even function of its second variable. The final step in the derivation for this case is to convert back to physical variables, while at the same time forcing the initial deviation to the desired region. By following the reasoning given at the start of the analysis for this case (equation (5.2)), the desired matrix for the linearized flow map of the entire cycle is (5.35) Y -compos = Y(T' +1) Y -) r-compos r 3' l1 O ( 0 -1 Yr(T, x -1) = Ir T 2 Y(T x0-1) T Ir (5.36) 6YTr = Yr-compos 60r + higher order terms where 6yTr is the (n-l)-vector representing the deviation in physical variables after going around one cycle (with velocity = 6y left out since it is zero), 6y0r is the initial deviation in the same variables (also has (n-l) elements, no deviation in initial velocity allowed), and Yr' Y are the matrices representing the linearized flow r-compos maps derived above, but in physical variables and reduced to order (n-1). Note that the latter matrix is (n-l)x(n-1), so its (n-1) eigenvalues are those desired for the local stability analysis (as discussed above, the nth eigenvalue is always 0). Case 2: Limit Cycle with Sticking To analyze this case, it is necessary to derive the linearized flow map for the portions of the limit cycle in the sticky region, i.e., region III. These linear maps can then be combined with the maps previously derived for the sliding regions to form the composite map for the cycle. Assume an initial point x3 that meets the equations in Chapter IV, so that this is the point at which the system comes to rest in region III (this is the fourth switch point in the cycle). In this case, the period the system sticks in region III is equal to the constant T4 in Chapter IV. This initial point must be used, instead of the point x at the point of impending motion, in order to derive the behavior within region III. A deviation 6x3 is assumed from the initial point of the limit cycle. This deviation is of course constrained as before, so that the resulting initial point remains inside region III (velocity = 0). Starting at the initial point x3+6x3, the trajectory is determined by the piecewise linear methods as in chapter IV. The trajectory in region III is (5.37) x(t) = eAt x(0) = eAt [X3 + 6x 3 For a sufficiently small deviation 6x3, the trajectory reaches the condition of impending motion (acceleration torque sufficient to break loose), but not necessarily in exactly T4 seconds. Denoting the change in this period is ST, impending motion is reached when (5.38) x(T4 + ST) = eAs(T4+6T) [x3 + 6x3] The quantity of interest is the deviation of this point from the point where the original limit cycle breaks free AT (5.39) x es 4 x (5.40) 6x = x(T4+6T) 4 A 6T AT A (T +6T) = {es I) es 4 x + esT4T) The differentiation of this function to determine the first-order dependence of 6x on 6x3 is performed as before; many of the steps are similar and thus omitted. The desired derivative is (5.41) Ys = d(6x4)/d(6x3) 1=0 where Y is the desired linearized map in the stiction region. Note that since x4 is not a function of 6x3 or 6T, (5.42) [d/d(6x3)] (6x4} = [d/d(6x3)] {x(T4+6T) x4) = [d/d(6x3)] (x(T4+6T)} = [d/d(6x3)] {eAs(T4+6T) [x3 + 6X3]) The dependence of 6x on both ST and 6x3 explicitly is handled using the chain rule, as before: (5.43) d(6x4)/d(6x3) = D(6x4)/D(6x3) + [D(6x4)/D(6T)] [d(6T)/d(6x3)] where "D" is again the symbol for partial derivative. The first term is (5.44) D(6x4)/D(6x3) = e s when evaluated at 6x = 6T = 0. The partial derivative in the second term is A 6T (5.45) D(6x4)/D(6T) = [D/D(6T)] {e s m) A T = A e s 4 [x3] where m is a temporary vector independent of 6T, as in the previous derivation. The evaluation of the [d(6T)/d(6x3)] factor in the second term is handled as before by relating the two variables with an implicit function, and using the implicit function theorem. The implicit relation in this case is determined by the breakaway condition (i.e., impending motion); as in the analysis in Chapter IV, this can be stated as (5.46) P' A x(T4+6T) = -LS or, equivalently, 0' A eAs(T4+6T) [x3 + 6x3] + LS = 0 Thus we have a function of the form f(6T, 6x ) = 0. The derivative is then (5.47) d(6T)/d(63) = [Df/D(6T)]-1 [Df/D(6X3)] Evaluating these factors, (5.48) Df/D(6T) = [D/D(6T)] ({' A e s6T m) A 6T = e' AA es m s The second factor is (5.49) Df/D(6x3) = [D/D(6x3)] ({' A eAs(T4+6T) 6 x = A eAs(T4+6T) so that AT -1 AT (5.50) d(6T)/d(6x ) = -{('AA eAs 4 x3) {('A eAs 4) where the factors are again evaluated at 6x =6T=0. Finally, combining (5.44,45,50) into (5.43), we get the total derivative representing the linearized periodic map (5.51) Ys(T4, x3) = d(6x4)/d(6x3) 6x=0 AT AT AT -1 AT = e s 4 (As e s 4 x3) ('AAs e s 4 x ) ('A e s 4) Note that the factor to be inverted is a scalar. This expression can be simplified by noting that AT (5.52) x4 = e s x3 so substituting into (5.51) gives (5.53) Ys(T4x3) = eAsT4 (As x4)('AAs 4)('A e s4) AT AT = e s 4 (As x4 'A e s 4) / {('A A x4) In order to obtain the linear map for the entire cycle for the case with sticking during the cycle, we can use the chain rule as before: (5.54) Ycompos = d( )d(/d(6 2)) (d(6x2)/d(6x )) x (d(6x )/d(6x0 ) (d(6x4)/d(x3 ) where the last factor was just derived, the first and third are obtained from the sliding case previously analyzed, and the second factor can be found from the fourth by symmetry, as shown below. Note that 6x0 = 6x, since both are the deviations at the point of impending motion in region III; this equivalence justifies the juxtaposition of the last two factors when the chain rule is applied. Note also that in (5.54) the order of the factors is different from that used in the sliding friction case; the initial deviation for the case with sticking is taken as 6x3 at the point x where the system comes to rest, rather than at the point x where motion is impending. This order is required since any (small) deviations Sx3 are allowed that keep the initial velocity zero (hence 6x3 can be constructed as in (5.2)). On the other hand, certain initial perturbations would not be allowed at the point x ; any perturbations at this point of impending motion that altered the forcing torque would either cause motion (so initial point no longer in region III), or would mean motion is no longer impending. This problem does not arise in the sliding friction case, but in the sticking case there are two constraints on x0: zero velocity and forcing torque at breakaway. It is therefore much more convenient to take the initial point to be x3 for the stability analysis; all constraints are then taken care of in the derivation automatically. The second factor in (5.54) can be found by symmetry, as follows: taking a deviation from the point x~ at which the system comes to rest in region III of Sxl, we can derive by the same steps as above that (5.55) 6x2 2=(T2+6T) x =(es I) eAsT2 x + eAs(T2+T)65 1 1 The similarity of this equation to (5.40) can be used to show that the desired linearization is (5.56) Ys(T2, 1) = d(6x2)/d(6 ) 6X= AT AT AT -1 AT = es 2 (As es 2 xl} ('AAs e s 2 xl)1 'A eAsT2} Of course, as in the sliding case, this matrix must be modified to transform to physical variables, and reduced to order (n-l). The linear transformation for the entire, four-part cycle is then (5.57) Yr-compos = Y(T3' 2+1) Y (T x ) r-compos r 2 rs 2'1 x Yr (T1,X-i) Yrs(T4'x3) -1 Y (T1' 0x-1) = I T2 Y(T -1) T2 I 12 r Y1' -0 12 I -1- Y(T2' l) =Ir T 2 Ys(T2' l) T12 I (5.58) STr = Yr-compos XOr + higher order terms where, as before, the Syr vectors are the (n-l) elements of the initial and final deviation, respectively, and Y r-compos is the desired linear transformation that determines local stability of the limit cycle. The results of the stability analysis of this chapter can be summarized as follows: Theorem 5-1: Given the existence of a friction limit cycle, as predicted by the solution of the nonlinear algebraic equations derived in Chapter IV (Theorem 4-1), the local asymptotic stability of the periodic orbit is determined by the eigenvalues of the matrix: (Case 1: Sliding only) (5.35) Y compos = Y (T +) Yr(T -) r-compos r 3' -1 r(1' -0 (Case 2: Sliding plus sticking) (5.57) Yr-compos = Y(T3' x2+1) Y rs(T2, 1) r-compos r 3' 2 rs2' x1 x Y (T1, xO-1) Yrs(T, x3) where -1 Yr(T XO-1) = I1 T12 Y(TI, X--1) T12 I1 -1-i Yrs(T x) = I T2 Y (T, x ) T2- Ir the x 's are the switching points, the T 's are the corresponding switching periods, Ir is the (n-l) x n matrix defined in (5.3), T12 is the transformation from physical to canonical variables defined in Chapter III, and the matrix functions Y and Y are s (5.19) Y(T1, x0-1) = d(6Xl)/d(6x0) 6x=0 SeAT1 (A eAT1 [X0-i]) ('A eAT1 [x -1])-{('eAT1) (5.56) Ys(T2, X1) = d(6x2)/d(6x1) 6x=0 AT AT AT -1 AT = es 2 (As e s 2 X1) ({'AA eAs 2 X}) ({'A es 2 Before this theorem is proved, some terms must be defined more rigorously. Definition: The flow of the differential equation system is the map (t,x0):'(t,x0), where 9(t,x0) = x(t)'when the initial condition x(0) = x . --0' Definition: The Poincare map (P-map) is the map 6x :g(6x ), where g(6x0) = x(t) x0, 6x0 = x(0) x, and tl is the first time at which the trajectory again crosses the local section around the periodic orbit at x . -0 Proof: The proof follows from the theory in Chapter 13 of Hirsch and Smale (1974) and the calculations above, except for one difficulty to be overcome. The reference theory is stated for the case where f(.) is continuously differentiable (so that the flow < is continuous). However, as will be shown below, all that is required to obtain the desired result is that the flow be a continuous function of the deviation about the initial point, x + Sx so that certain limits can be taken, and, in addition, that the P-map is differentiable, so that the derivative may be used to check asymptotic stability. As a preliminary step, then, these properties must be shown. The flow is the integral of a real-valued function of time (see equation 5.69 below), f(x(.)), so it is a continuous function of time (Rudin, (1976), Theorem 6.20). It is also clear that the flow is continuous in x within each piecewise linear region, since the function f(.) is at least C1 there (and Hirsch and Smale, Chapter 15, shows this is sufficient for continuity of the flow). Now, taking the initial condition as a sufficiently small deviation from the initial point x0' the flow at any finite time is the composition of a finite number of continuous functions, formed from the pieces of the trajectory in each region it traverses. This composition can be formed since the trajectory is continuous across the discontinuities in f(.); that is, x(t+) = x(t-), where t is a switching time. Therefore the flow is a continuous function of this initial condition (Rudin, (1976), Theorem 4.7, the composition of continuous functions is continuous), for sufficiently small deviations. The same approach is used to demonstrate the differentiability of the P-map. This map is simply the composition of the flows in each of the regions, so is continuous by the previous argument. But, in fact, the flows in each region are continuously differentiable, so their composition is also differentiable. Using these properties, the rest of the proof proceeds as in Hirsch and Smale (1974). In chapter 13, section 3, it is shown that if the P-map has x0 as a sink, then the orbit is asymptotically stable (using the property of continuity of the flow to show that the flow converges uniformly to the periodic orbit if the deviations after each cycle also converge). But the P-map is a discrete-time dynamical system, so it is a sink if the derivative of this map has all eigenvalues less than one in magnitude (which requires the property of differentiability of the P-map). The calculations of this derivative have been presented above, and show that the matrix Y is in fact the derivative r-compos of the P-map at x Therefore the (n-l) eigenvalues of this matrix determine if the orbit is asymptotically stable, and the proof is complete. To summarize the above stability results, it has been shown that the perturbations from the periodic orbit evolve according to the equation (5.58) yTr = Yr-compos 65Or + higher order terms where 6YTr is the (n-l)-vector representing the deviation in physical variables after going around one cycle (with velocity = Syn left out since it is zero), and Sy0r is the initial deviation in the same variables (also has (n-l) elements, no deviation in initial velocity allowed). The eigenvalues of the matrix Yr-compos' therefore, indicate the stability of the periodic orbit (i.e., if they are all less than one in magnitude, the orbit is asymptotically stable, and in fact a periodic attractor). In addition, it has been shown that the results on orbital stability in Hirsch and Smale (1974), chapter 13, can be extended to the case where f(.) is only piecewise linear, rather than C1. Stability Calculations for Example Problems The limit cycle orbital stability equations derived above can be applied to the example system discussed in Chapters I and IV (example 1-2, case 2, a 3D system with both a sliding and stiction limit cycle). The example system is defined, as in the example, by (5.59) dy/dt = A y + b Lf where A = 0 1 0 0 0 1 -K1 -K2 -B with b' = [0 0 -1], Lf = friction torque, and where the state variables are y2 = position, y3 = velocity, and y = compensator integrator. The transformation matrices T1 and T2 are identity matrices, and the row vector 9' equals [ 0 1]. The particular case has B = 0.9, K = K2 = 1, with closed-loop poles at .026 j 1.024 and -.9524. The stiction limit cycle solution is at approximately x = [0.98 0.22 0], with T1 at approximately 7. There is also a solution of the equations for the symmetric, sliding case, at T1 = 3.1455 seconds, and xO = [-0.283 12.4 0]. Example V-l, Part 1: Stability of Sliding Limit Cycle For this case, equations (5.35, 5.19) from Theorem 5-1 apply. Using the parameters defined above, we find that AT (5.60) Ae 1 [x 1] = [-12.41 .001 13.13]' so that P' AeAT1 [x0 The matrix exponential algorithm was used) - 1] = 13.13 is (approximately, AT K (5.61) eAT = -.5314 -.5227 .5830 Therefore, the matrix Y required flow map is -.1126 -1.054 .0603 to define a numerical .5227 -.5830 -.5294 the linearized (5.62) Y = .0197 -.0556 .0223 -.5227 -1.054 -.583 .0002 .0000 -.0002 Although there are unavoidable numerical errors in this calculation, the true Y matrix apparently does have a zero eigenvalue, as expected. The upper left block sub-matrix is equal to Yr' and the composite stability matrix is (5.63) Y = .02944 .05749 .5407 1.14 This matrix has eigenvalues of .002125 (with corresponding eigenvector at [-2.104 1.]' ), and 1.168 (with eigenvector [.05 1.]' ). Thus it has one stable and one unstable mode (note this is a discrete-time system), and is a saddle point. A simulation of this system was used to verify this, where the xl and x2 states were sampled whenever the velocity x3 went through zero (near x0). The plot in Figure V-1 shows the point x0, the eigenvectors of the stable and unstable modes, and typical trajectories near x The stable mode is very fast so that simulated trajectories quickly jump onto the unstable eigenvector, then move along it. The simulation experimentally verifies the stability predictions. Example V-l, Part 2: Stability of Sticking Limit Cycle Similar calculations can be performed for the other limit cycle that exists for this system, except that equation (5.57) from Theorem 5-1 governs this case. Using the initial condition defined above, with T1 = v and T 9 (as approximate values), and xl = x(T1) = [1. -0.22 0.]', and recalling that the symmetry defines the other switching points and periods, we find after substitution in (5.56) (5.64) Y = 0 -1 -9.9 0 1 9 0 0 1 with Y being the upper left block sub-matrix. The stability matrix for the sliding portions of the limit cycle is also required, and the upper left block of Y is |