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## Material Information- Title:
- A performance bound for nonlinear control systems
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- Fanjul, Rafael J., 1963-
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- 1996
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- English
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- viii, 80 leaves : ill. ; 29 cm.
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## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1996.
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- Includes bibliographical references (leaves 77-79).
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- Vita.
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- by Rafael J. Fanjul.
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A PERFORMANCE BOUND FOR NONLINEAR CONTROL SYSTEMS By RAFAEL J. FANJUL JR. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 Copyright 1996 by Rafael J. Fanjul Jr. To my parents, Rafael James and Estela Linares Fanjul, my grandmother, Margaret Stewart Fanjul, my aunt, Sheila Stewart, and my nephews, Frankie and Rafi Montalvo ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Professor Hammer, for his encouragement, guidance and wisdom throughout the course of my studies. Professor Hammer has always found time to discuss my work and for that I am especially thankful. I am grateful to Professor Crisalle for his many hours of assistance in formu- lating my research. I wish to thank Professor Schwartz for her help during the early stages of my research. In addition, I would like to thank Professor Anderson and Professor Principe for serving on my committee. A number of my fellow students have provided both inspiration and advice. In particular, I wish to thank Victor Brennan, Patrick Walker, Aiguo Yan, and Kuo- Huei Yen for their help over the past four-and-a-half years. Finally, I would like to thank Margaret Fanjul Montalvo and Christopher Riffer for editing the manuscript. TABLE OF CONTENTS ACKNOWLEDGEMENTS .............. ABSTRACT ...................... CHAPTERS 1 INTRODUCTION ............... 1.1 Background ................ 1.2 Notation ...... ........... 2 TERMINOLOGY AND BASICS ....... 2.1 Right Fraction Representations 2.2 Generalized Right Inverse . 3 THE PERFORMANCE BOUND.. and Coprimeness 3.1 Causality of System s ......................... 3.2 Systems with the Lipschitz Norm . 3.3 Approximate Right Inverse . 3.3.1 The Existence Theorem of Best Approximate Right Inverse 3.4 The Measure of Right Singularity . 4 CALCULATION OF THE PERFORMANCE BOUND ........ 4.1 The Estimate of the Performance Bound . 4.2 Practical Implementation Techniques . 4.3 Permanent Magnet Stepper Motor Model Example . 4.3.1 Simulation ............................ 4.3.2 The Estimate of the Performance Bound P() . 4.4 Aerodynamic Model Example . 4.4.1 Sim ulation ... . 4.4.2 The Estimate of the Performance Bound P() . 4.5 Multivariable Process Control Model Example . 4.5.1 Simulation ............................ 4.5.2 The Estimate of the Performance Bound P() . 5 CONCLUSION .............................. 5.1 Sum m ary . 5.2 Future Directions .......................... REFERENCES ......................... .......... 77 BIOGRAPHICAL SKETCH .............................. 80 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 A PERFORMANCE BOUND FOR NONLINEAR CONTROL SYSTEMS By Rafael J. Fanjul Jr. December 1996 Chairman: Professor Jacob Hammer Major Department: Electrical and Computer Engineering This research focuses on the control of a nonlinear system whose output sub- ject to an additive disturbance. The main interest is in the investigation of controllers that reduce the effect of the disturbance on the system output. Usually, it is not possi- ble to construct a controller that completely eliminates the effects of the disturbance. It is then of interest to find how well the "best" controller can attenuate the effect of the disturbance. The main result of this dissertation is a performance bound, that provides an estimate of the best disturbance attenuation that can be achieved for a given system, using a causal controller that renders the system internally stable. An approximate right inverse of a nonlinear system E is introduced to fa- cilitate the derivation of the performance bound and the development of nonlinear controllers. An approximate right inverse is constructed to be stable and causal for implementation purposes. The difference between an approximate right inverse and a right inverse is that the right inverse may not be both stable and causal. The role of an approximate right inverse is to approximate a disturbed signal with a signal in the image of the system E. An approximate right inverse can be constructed for any system E. The calculation of the performance bound involves an optimization process of finding a global maximum of a non-convex function. For several cases, a nonlinear programming algorithm is developed to handle the optimization. To demonstrate the application of this performance bound, it is calculated for three practical systems: 1. the voltage control of a permanent magnet stepper motor; 2. the longitudinal control of an aircraft; and 3. the multivariable process control of a regulator that regulates the liquid level in a pressurized tank. CHAPTER 1 INTRODUCTION Over the last 2 decades, there has been considerable interests in the literature in the derivation of optimal controllers that minimizes the effect of the disturbance on the output of a control system. This dissertation addresses this question for the case of nonlinear systems. The main objective is to derive an estimate of the performance of an optimal controller, by deriving a bound on the effect of the disturbance has on the system output when the optimal controller is used. Using this bound, we can then gauge the performance of suboptimal controller, to see how well they compare to the optimal ones. Suboptimal controllers may be much easier to implement than their optimal counterparts. Thus, our performance bound can be use to find simple controllers whose disturbance attenuation properties are close to those of optimal disturbance attenuating controllers. The performance bound derived in this dissertation came from the require- ment to characterize the performance of a controller that reduces the effect of the disturbance on the output to a minimum. The original work provided is the derivation of the performance bound and nonlinear programming algorithm in how to calculate the performance bound. The application of the performance bound for disturbance attenuation is proposed as future directions. The basic design problem to which this performance bound relates is the problem of disturbance attenuation for nonlinear control systems. Specifically, this dissertation discusses the following configuration. In the configuration of Figure 1.1, E is the nonlinear system to be controlled. C represents an equivalent controller that incorporates all the control elements of the loop. The external (or reference) signal is denoted by v; the disturbance signal is denoted by d; and the output signal is denoted by z. The closed loop system is required to be internally stable. Internal stability signifies that a configuration can tolerate small disturbances on its external and internal ports (including ports within the equivalent controller C) without losing stability. The equations that describe Figure 1.1 are z = d +y, y = Eu, u= C(v,z). (1.1) In Figure 1.1, EC represents the appropriate equivalent system. This can be expressed in notation z = c(v, d) (1.2) where the output signal z is determined by the signals v and d and depends on the system E as well as the equivalent controller C. We would like to reduce as much as possible the effects of d on z. Our bound, which is the attainable performance for control of a nonlinear system whose output is subject to an additive disturbance, provides an estimate of the minimal effect of d on z. Using the estimates of the min- imal effect, we can evaluate controllers. The analysis of the dissertation is restricted to the case of discrete-time systems. The desired response, with or without the disturbance signal for the configu- ration in Figure 1.1, is z = Ev. To null out the disturbance signal, the design of the equivalent controller C would be such that y = Eu = Ev d. Substituting for y in d v u , Figure 1.1: The block diagram of Ec. the first equation of (1.1), this yields z = Ev which is our stated goal. For the sake of argument, assume that the system E has a stable and causal inverse system E-1. The input signal u to the system E would then be required to be u = E-I(Ev d). The inverse system E-1 in some cases might not be implementable because it is not sta- ble or it is not causal. For those cases, approximations of the inverse systems would then be used for practical implementation. The approximations will be selected in a manner suited to approximate the signal Ev d with a signal in the image of E. The composition of the system and its approximate inverse system forms a nearly identity system. For points inside the image of the system E, the composition will appear as the identity. For external points, which are points outside the image of the system E, the composition will produce a point in the image of E that is closest to the external point. Our performance bound is a norm that gauges the closeness of the identity system and a nearly identity system formed from the system E and its approximate inverse. The performance where Ec can attain by using our bound is determined by the inherent properties of the system E. The bound is directly related to an approximate inverse. The role of an approximate inverse is to approximate the signal u with a signal v in the image of E. Therefore, the approximate invertibility of the system E provides a measure of the ability to match a set S by a subset of the image of the system E. A measure of singularity is introduced as an indicator of approximate invertibility which is then used to derive the performance bound. To demonstrate the application of this performance bound, it is calculated for three practical systems: 1. the voltage control of a permanent magnet stepper motor; 2. the longitudinal control of an aircraft; and 3. the multivariable process control of a regulator that regulates the liquid level in a pressurized tank. The dissertation is organized as follows. Chapter 2 contains the basic notions of nonlinear systems including the theory of fraction representation. Chapter 3 dis- cusses an approximate right inverse and the performance bound. Chapter 4 discusses the main results of the dissertation, including calculations of the performance bound for applications. Chapter 5 contains a summary and future directions. 1.1 Background This section contains a qualitative survey of some topics in nonlinear control that are important to the dissertation. A more technical discussion of these topics is provided in Chapter 2. The description of Ec in Figure 1.1 has its origins in the theory of fraction rep- resentation of nonlinear systems. It can be stated that a right fraction representation of a nonlinear system E is a factorization of E into a composition form E = PQ-1, where P and Q are stable systems with Q being invertible. In Hammer [16, 18], tools for a compact and perceptive statement of results were developed for the theory of fraction representation of nonlinear systems. The fraction representation E = PQ-1 is said to be coprime when the systems P and Q are right coprime. An attribute of right coprime fraction representation is that every instability of the inverse system Q-1 is also an instability of the system E; i.e., there is no cancellation of instabilities within the composition PQ-1. A causal (respectively, strictly causal) system E is one where the values of the output sequence Eu up to and including index i (respectively, i + 1) depend only on the values of the input sequence u up to index i. A system is bicausal if it is causal and if it possesses a causal inverse. In this dissertation, it is assumed that the system E being controlled can be stabilized, that it is strictly causal, and that it possesses a right coprime fraction representation of the form E = PQ-1, with Q being bicausal. The stabilization 5 assumption on the system E is necessary because the closed loop system is required to be internally stable. Strict causality is placed on the system E as a handy assumption to certify that the closed loop system is well posed. Strict causality is not an essential condition and it can be replaced with plain causality combined with a well-posedeness requirement. This dissertation relies on stabilization theory which applies only to systems possessing right coprime fraction representations. A result from [22] gives a simple parameterization of the set of all system responses that can be obtained through internally stable control of a given system. The control scheme to control E is shown Figure 1.1. The parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system. This result can be summarized as follows. Let E = PQ-1 be a right coprime fraction representation (with a bicausal "denominator" Q) of the system being controlled. Then, 1. For every causal equivalent controller C for which the closed loop system of Figure 1.1 is internally stable, there exists a stable and causal system q(v, d) such that Ec(v, d) = d + Pq(v, d) = [I + PO(v, .)]d (1.3) where P is the "numerator" of the right coprime fraction representation of E and I denotes the identity system. 2. Conversely, for every stable and causal system q(v, d), there is an internally sta- ble control configuration around the system E for which the equivalent system Ec(v, d) satisfies Ec(v, d) = [I + PO(v, .)]d. Thus, equation (1.3) provides a complete parameterization of the class of all responses {Ec} that can be obtained by internally stable control of the system E, with the stable and causal system 0 serving as the sole parameter. In other words, for every q4, there is an equivalent controller C that internally stabilizes E and yields the response (1.3). Conversely, every equivalent controller C that internally stabilizes E generates an explicit 4. On a superficial inspection of (1.3), the attainable performance is prescribed by the "numerator" system P of E because it is the only fixed quantity apart from the identity. For a linear time-invariant system, it was shown in [50] that the at- tainable performance was determined by the location of its right half-plane zeros. For a nonlinear system E, the instabilities of P-1 (the inverse of the "numerator" system P) would be analogous to the right half-plane zeros of a linear time-invariant system. Using this analogy it could be inferred that the instabilities of P-1 would limit performance. The performance bound developed in the dissertation will further confirm this hypothesis. 1.2 Notation The following notational convention will apply unless otherwise stated. ?m S(Rm) Dx f(E) > A lul p(u) IIL | B Lip(), S(RP) II1 IILip M, (U, S2) oe(S1, S2) P(E) The set of m-dimensional real vectors. The set of m-dimensional real vector sequences. The A-step shift operator. The system E has latency of at least A. The -norm of the sequence u. The weighted -norm of the sequence u. The Lipschitz semi-norm of the system E. The set of all stable, causal and recursive systems E: D S(R) where D is a bounded subset of S(3?m). The subset of B with ||EI| < 0c. The Lipschitz norm of the system E. The distance from any u E S1 and the set S2. The maximal distance from any u E Si and the set S2- The right singularity of measure of the system E. CHAPTER 2 TERMINOLOGY AND BASICS This chapter contains a summary of the principal mathematical results which are needed in the dissertation. The presentation is for discrete-time time-invariant nonlinear systems. Almost all the necessary mathematics are contained in [9, 14, 17, 18, 22, 27]. The set of real numbers is denoted by R. The set of m-dimensional real vectors is denoted by Rm. The set of all sequences u is denoted by S(Rm) where u = {uo, 1, U2, ...} of m-dimensional real vectors ui E RM", i = 0, 1,2,.... Given the sequence u E S(Rm), the ith element is denoted by ui. The set of elements {ui, ui+i,..., Uj where j > i > 0 is denoted by ui. A system E, from an input/output perspective, is a map E: S(Rm) --+ S(RP) which transforms input sequences of m- dimensional real vectors into output sequences of p-dimensional real vectors. The image of a subset S C S("m) through the system E is denoted by E[S]. The entire image of system the E is denoted by Im where ImE d- E[S(Rm)]. Given a system E: S(Rm) S(RP) and an input sequence u E S(Rm), we denote by Eu]i = y, the ith element of the output sequence y = Eu, and by y = Eu]j the set of elements {yi, yi+1,... j} where j > i > 0 are integers. There are two kinds of binary operations, addition and composition. For a pair of systems E, E2: S(Rm) -- S(RP), the sum is defined, as usual, by (iE + E2)u = Elu + E2u for all sequences u E S(Rm); the right side of the last formula is the usual elementwise addition of sequences of real vectors. Composition is the usual composition of maps. Definition 2.0.1 A system E: S(Rm) -- S(RP) is causal (respectively, strictly caus- al) if it satisfies the following condition. For every integer i > 0 and for every pair of input sequences u,v E S(Rm) satisfying uo = V, the output sequences satisfy Eu]o = Ev]b (respectively, Eu]o' = Ev]'+). A system M: S(Rm) -+ S(,m) is bicausal if it is causal and if it possesses a causal inverse. A system E: S(Rm) -- S(RP) is called a recursive system if there is a pair of integers r, yi > 0 and a function f: (,),+71 X (Rm)P+1 --+ RP such that, for every input sequence u E S(Rm), the corresponding output sequence y = Eu can be computed recursively in the form Yk+q+l = f(Yk,..., Yk+,, Uk,..., Uk+p) (2.1) for all integers k > 0. The initial conditions yo,..., y, must of course, be specified and fixed. The function f is called a recursion function of E. For causality (strict causality), the condition q +1 > p (q > y_) is placed on system E. The class of strictly causal systems include every system E: S(Rm) -+ S(RP) that can be represented in the form Xk+1 f(xk, Uk) yk = h(xk), k = 1,2,.... (2.2) Here, u E S(Rm) is the input sequence; y E S(RP) is the output sequence; and x E S(Rn) is an intermediate sequence of "states." In the case that the maps f: Rn x Rm -,* and h: R" -+ R are continuous, then the system (2.2) constitutes a continuous realization of the system E. For a real number 0 > 0, the set of all vectors in m with components in the closed interval [-0, ] is denoted by [-0, ]m. The set of all sequences u E S(~m) with elements ui belonging to [-0,0]m for all integers i > 0 is denoted by S(Om). Thus, S(Om) consists of all sequences bounded by 0. It follows then that a system E: S(Rm) -- S(RP) is BIBO (Bounded-Input Bounded-Output)-stable if for every real number 0 > 0, there exists a real number M > 0 such that E[S(0m)] C S(MP). A sequence u E S(R"m) is said to be bounded if there is a real number 0 > 0 such that u e S(Om). The basic notion of stability that is used in this dissertation is related to con- tinuity with respect to a metric. Two norms are particularly useful in this context to derive a metric: the e-norm and the weighted ~-norm. The t"-norm is denoted by - ; for a vector a = (al, a2,... ,am) Rm, it is simply lal d= max{ ai|, la21,. ., *amm}. For a sequence u E S(Rm), the e"-norm is given by Iul df sup IuIl. (2.3) i>O The weighted t"-norm is denoted by p, and is given by p,(u) f sup (1 + e)-i u, > 0 (2.4) i>O for a sequence u e S(Rm). For purposes of this dissertation different values of e will not affect our results; hence, the subscript will be dropped and the weighted e-norm is simply denoted by p. The use of the weighted oo-norm simplifies mathematical arguments over the -norm because the bounded set of sequences S(0m) is compact with respect to p. The norm p induces a metric p on a given S(Rm), for every pair of elements u, v S(Rm), by p(u, v) =f p(u v). Formally, the notion of stability employed in this dissertation is as follows. Definition 2.0.2 A system E: S(Rm) --- S(RP) is stable with respect to the metric p if it is BIBO-stable, and if the restriction E: S(am) -- S(RP) is continuous with respect to p for every real number a > 0. Definition 2.0.2 is usually referred to as input/output stability. The following concept, which describes a weak form of uniform continuity with respect to the "- norm, plays a fundamental role in stabilization theory (see [17]). Definition 2.0.3 A stable system E: S(3m) -- S(RP) is differentially bounded if there is a pair of real numbers e, 0 > 0 such that, for every pair of sequences u E S(Rm) and v E S(em), one has IE(u + v) E(u)Il 8. So far, only stability properties of individual systems have been mentioned. When several individual systems are combined into a composite system, a stronger notion of stability is required, and it is usually referred to as internal stability. Internal stability guarantees desirable stability properties of the composition, and takes into account the effects of various disturbances and noises that may affect the component systems. Consider a composite system E(s) that consists of s individual systems, labeled El,.. ., E where SE: S(Rmm()) -- S(RP()), i = 1,..., s. Individual entries in the list E1,..., ES may represent summers, multipliers, etc. Let u E S(Rm) be the external input sequence of the composite system, and let y E S(Rp) be its output sequence. Let uj E S(Rm(j)) be the input sequence of the system Ej within the configuration, and let y3 E S(Rp(j)) be its output sequence. The interconnections among the subsystems are then characterized by a set of qualities ui = yj(), which determine to which output each input is connected. The external signal u is now augmented by s new input signals qi E S(Rm(i)), i = 1,..., s, and set u'i yj(i) + . For each i, the 7i acts as an additive disturbance on the input port of the system E. The disturbances are all assumed to be bounded by a real number 6 > 0, so that in fact i'i E S(bm(i)), i= 1,...,s. Let E*s*:S(Rm) x S(Rm(i)) x ... x S(GRm()) -_ S(RP) x S(Rp(x)) x ... x S(R(S)): (u, 71, ..., .) -> E*S*(u, 1,..., 9) denote the system induced by the in- terconnected system E(S) and the disturbances, having the input signals u, 1i/,... ,7/ and the output signals y, y1,..., ys, respectively. Definition 2.0.4 The composite system E(s) is internally stable if the system E*`* is stable in the sense of Definition 2.0.2. The composite system E() is strictly internally stable if, besides being stable, the system E*`* is also differentially bounded. Definition 2.0.5 A system E: S(Rm) --+ S(RP) is entirely stabilizable if there is a strictly internally stable control configuration that stabilizes E over the entire input space S(Rm). 2.1 Right Fraction Representations and Coprimeness A right fraction representation of a system E: S(Rm) -+ S(RP) is determined by three quantities: a subset S C S(Rq9), q > 0, called the factorization space; and two stable systems P: S -+ S(RP) and Q: S -- S(Rm), where Q is invertible, such that E = PQ-1. A right fraction representation E = PQ-1 is coprime whenever the stable systems P and Q are right coprime according to the following definition ([16, 18]). (Let G: S1 -- S2 be a map, where S1 C S(Rm) and S2 C S(Rn) are subsets. For a subset S C S(R"), we denote G*[S] the inverse image of S through G, i.e., the set of all sequences u E S1 satisfying Gu E S.) Definition 2.1.1 Let S C S(Rq() be a subset. Two stable systems P: S -* S(RP) and Q: S --+ S(Rm) are right coprime whenever the following conditions hold. 1. For every real number r > 0 there is a real number 0 > 0 such that P*[S(TP)] n Q*[S(Tm)] c S(0q). 2. For every real number 7 > 0 the set S n S(rq) is a closed subset of S(rq) (with respect to the topology induced by p). The concept of a homogeneous system is of key importance to the theory of right coprime fraction representations of nonlinear systems. A homogeneous system has the property of being a continuous map whenever its outputs are bounded. The precise definition is as follows. Definition 2.1.2 A system E: S(Rm) -- S(RP) is a homogeneous system if the fol- lowing holds for every real number a > 0: for every subset S C S(am) for which there exists a real number r > 0 satisfying E[S] C S(TP), the restriction of E to the closure 3S of S in S(am) is a continuous map E: S -- S(7r). The importance of homogeneous systems to the dissertation is stated in the next two theorems. Theorem 2.1.3 [17] An injective system E: S(Rm) -* S(RP) has a right coprime fraction representation if and only if it is a homogeneous system. Theorem 2.1.4 [17] Let E: S(Rm) S(RP) be a recursive system. If E has a recur- sive representation yk+7+1, = f(Yk, *, Yk+n,, Uki,..., k+,) with a continuous recursion function f, then E is a homogeneous system. The following two theorems are important for stabilization theory. Theo- rem 2.1.5 defines the technical details in the stabilizing system E in Figure 2.1 with r = B-1 and
ization, which is the denominator system contains the exact information about the |

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PAGE 1 A PERFORMANCE BOUND FOR NONLINEAR CONTROL SYSTEMS By RAFAEL J. FANJUL JR. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREME TS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 PAGE 2 C op y right 1996 b y Rafael J. Fanjul Jr. PAGE 3 To my parents Rafael James and Estela Linares Fanjul my grandmother Margaret Stewart Fanjul my aunt Sheila Stewart and my nephews Frankie and Rafi Montalvo PAGE 4 ACKNOWLEDGEMENTS I would like to express my gratitude to my advisor, Professor Hammer for his encouragement, guidance and wisdom throughout the course of my studies. Professor Hammer has always found time to discuss my work and for that I am especially thankful. I am grateful to Professor Crisalle for his many hours of assistance in formu lating my research. I wish to thank Professor Schwartz for her help during the early stages of my research. In addition, I would like to thank Professor Anderson and Professor Principe for serving on my committee. A number of my fellow students have provided both inspiration and advice. In particular I wish to thank Victor Brennan, Patrick Walker Aiguo Yan, and Kuo Huei Yen for their help over the past four-and-a-half years. Finally I would like to thank Margaret Fanjul Montalvo and Christopher Riffer for editing the manuscript. IV PAGE 5 ACKNOWLEDGEMENTS ABSTRACT CHAPTERS 1 INTRODUCTION 1.1 Background .. 1. 2 Notation . TABLE OF CONTENTS 2 TERMINOLOGY AND BASICS 2.1 Right Fraction Representations and Coprimeness 2.2 Generalized Right Inverse .. 3 THE PERFORMANCE BOUND .... lV Vll 1 4 7 8 12 15 18 3.1 Causality of Systems . . . . 18 3.2 Systems with the Lipschitz Norm 20 3.3 Approximate Right Inverse . . 22 3.3.l The Existence Theorem of Best Approximate Right Inverse 24 3.4 The Measure of Right Singularity . . . . 27 4 CALCULATION OF THE PERFORMANCE BOUND 4.1 The Estimate of the Performance Bound . . . 4.2 Practical Implementation Techniques ....... 4.3 Permanent Magnet Stepper Motor Model Example 4.3.1 Simulation .................. 4.3.2 The Estimate of the Performance Bound 'P(E) 4.4 Aerodynamic Model Example ............. 4.4.1 Simulation . . . . . . . . . . 4.4.2 The Estimate of the Performance Bound 'P(E) 4.5 Multivariable Process Control Model Example ... 4.5.l Simulation ................... 4.5.2 The Estimate of the Performance Bound 'P(E) 5 CONCLUSION .... 5.1 Summary ..... 5.2 Future Directions 30 30 31 35 37 39 44 46 51 60 63 65 73 73 74 PAGE 6 REFERENCES . . . . . . . . . . . . . . . . . 77 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 80 Vl PAGE 7 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 A PERFORMANCE BOUND FOR NONLINEAR CONTROL SYSTEMS By Rafael J. Fanjul Jr. December 1996 Chairman: Professor Jacob Hammer Major Department: Electrical and Computer Engineering This research focuses on the control of a nonlinear system whose output sub ject to an additive disturbance. The main interest is in the investigation of controllers that reduce the effect of the disturbance on the system output. Usually it is not possi ble to construct a controller that completely eliminates the effects of the disturbance. It is then of interest to find how well the "best" controller can attenuate the effect of the disturbance. The main result of this dissertation is a performance bound that provides an estimate of the best disturbance attenuation that can be achieved for a given system, using a causal controller that renders the system internally stable. An approximate right inverse of a nonlinear system I: is introduced to fa cilitate the derivation of the performance bound and the development of nonlinear controllers. An approximate right inverse is constructed to be stable and causal for implementation purposes. The difference between an approximate right inverse and a right inverse is that the right inverse may not be both stable and causal. The role Vll PAGE 8 of an approximate right inverse is to approximate a disturbed signal with a signal in the image of the system E. An approximate right inverse can be constructed for any system E. The calculation of the performance bound involves an optimization process of finding a global maximum of a non-convex function. For several cases, a nonlinear programming algorithm is developed to handle the optimization. To demonstrat e the application of this performance bound, it is calculated for three practical systems: 1. the voltage control of a permanent magnet stepper motor ; 2. the longitudinal control of an aircraft; and 3. the multivariable process control of a regulator that regulates the liquid level in a pressurized tank. Vlll PAGE 9 CHAPTER 1 INTRODUCTION Over the last 2 decades, there has been considerable interests in the literature in the derivation of optimal controllers that minimizes the effect of the disturbance on the output of a control system. This dissertation addresses this question for the case of nonlinear systems. The main objective is to derive an estimate of the performance of an optimal contro ll er, by deriving a bound on the effect of the disturbance has on the system output when the optimal controller is used. Using this bound, we can then gauge the performance of suboptimal controller, to see how well they compare to the optimal ones. Suboptimal controllers may be much easier to implement than their optimal counterparts. Thus, our performance bound can be use to find simple controllers whose disturbance attenuation properties are close to those of optimal disturbance attenuating contro ll ers. The performance bound derived in this dissertation came from the require ment to characterize the performance of a contro ll er that reduces the effect of the disturbance on the output to a minimum. The original work provided is the derivation of the performance bound and nonlinear programming algorithm in how to calcu l ate the performance bound. The application of the performance bound for disturbance attenuation is proposed as future directions. The basic design problem to which this performance bound relates is the problem of disturbance attenuation for nonlinear contro l systems. Specifically, this dissertation discusses the following configuration. In the configuration of Figure 1.1, E is the non lin ear system to be contro lled C represents an equivalent contro ll er that incorporates all the contro l e l ements of the loop. The externa l ( or reference) signal PAGE 10 2 is denoted by v; the disturbance signal is denoted by d; and the output signal is denoted by z. The closed loop system is required to be internally stable. Internal stability signifies that a configuration can tolerate small disturbances on its external and internal ports (including ports within the equivalent controller C) without losing stability. The equations that describe Figure 1.1 are z=d+y, y=Eu, u=C(v,z). (1.1) In Figure 1.1 1 Ee represents the appropriate equivalent system. This can be expressed in notation z = Ec(v, d) (1.2) where the output signal z is determined by the signals v and d and depends on the system :E as well as the equivalent controller C. We would like to reduce as much as possible the effects of d on z. Our bound, which is the attainable performance for control of a nonlinear system whose output is subject to an additive disturbance, provides an estimate of the minimal effect of d on z. Using the estimates of the min imal effect, we can evaluate controllers. The analysis of the dissertation is restricted to the case of discrete-time systems. The desired response, with or without the disturbance signal for the configu ration in Figure 1.1 is z = Ev. To null out the disturbance signal, the design of th e equivalent controller C would be such that y = Eu = Ev d. Substituting for y in d V y +, ~ u + ~ C E Figur 1.1: The blo k diagram of ~ c PAGE 11 3 the first equation of ( 1.1), this yields z = Ev which is our stated goal. For the sake of argument, assume that the system E has a stable and causal inverse system E1 The input signal u to the system E would then be required to be u = E1 (Ev d). The inverse system E1 in some cases might not be implementable because it is not sta ble or it is not causal. For those cases, approximations of the inverse systems would then be used for practical implementation. The approximations will be selected in a manner suited to approximate the signal Ev d with a signal in the image of E. The composition of the system and its approximate inverse system forms a nearly identity system. For points inside the image of the system E the composition will appear as the identity. For external points which are points outside the imag e of the system E the composition will produce a point in the image of E that is closest to the external point. Our performance bound is a norm that gauges the closeness of the identity system and a nearly identity system formed from the system E and its approximate inverse. The performance where E e can attain by using our bound is determined by the inherent properties of the system E. The bound is directly related to an appro xim at e i nv e rs e The role of an approximate inverse is to approximate the signal u with a signal v in the image of E. Therefore the approximate invertibility of the system E provides a measure of the ability to match a set S by a subset of the imag e of the system E. A measure of singularity is introduced as an indicator of approximate invertibility which is then used to derive the performance bound. To demonstrate the application of this performance bound, it is calculated for three practical systems: 1. the voltage control of a permanent magnet stepper motor; 2. the longitudinal control of an aircraft; and 3. the multivariable process control of a regulator that regulates the liquid level in a pressurized tank. PAGE 12 4 The dissertation is organized as follows. Chapter 2 contains the basic notions of nonlinear systems including the theory of fraction representation. Chapter 3 dis cusses an approximate right inverse and the performance bound. Chapter 4 discusses the main results of the dissertation, including calculations of the performance bound for applications. Chapter 5 contains a summary and future directions. 1.1 Background This section contains a qualitative survey of some topics in nonlinear control that are important to the dissertation. A more technical discussion of these topics is provided in Chapter 2. The description of Ee in Figure 1.1 has its origins in the theory of fraction rep resentation of nonlinear systems. It can be stated that a right fraction representation of a nonlinear system E is a factorization of E into a composition form E = PQ1 where P and Q are stable systems with Q being invertible. In Hammer [16, 18], tools for a compact and perceptive statement of results were developed for the theory of fraction representation of nonlinear systems. The fraction representation E = PQ1 is said to be coprime when the systems P and Q are right coprime. An attribute of right coprime fraction representation is that every instability of the inverse system Q1 is also an instability of the system E; i.e., there is no cancellation of instabilities within the composition PQ1 A causal (respectively strictly causal) system E is one where the values of the output sequence Eu up to and including index i (respectively i + 1) depend only on the values of the input sequence u up to index i. A system is bicausal if it is cau al and if it possesses a causal inverse. In this dissertation, it is assumed that the system E being controlled can be stabilized that it is strictly causal and that it possesses a right coprime fraction r presentation of the form E = PQ1 with Q being bicausal. The stabilization PAGE 13 5 assumption on the system Eis necessary because the closed loop system is required to be internally stable. Strict causality is placed on the system E as a handy assumption to cert if y that the closed loop system i s well posed. Strict causa lit y i s not an essentia l cond ition and it can be replaced with plain causa lit y comb in ed with a well-poseden ess requirement. This dissertation relies on stab ilization theory which applies only to systems possessing right copr im e fraction representations. A result from [22] gives a simple parameterization of the set of a ll system responses that can be obtained through internally stable control of a given system. The contro l scheme to contro l E is shown Figure 1.1. The parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system. This result can be summarized as follows. Let E = PQ1 be a right coprime fraction representation ( with a bi causa l denominator Q) of the system being contro ll ed. Then 1. For every causa l equivalent controller C for wh i ch the closed l oop system of Figure 1.1 is internally stab l e, there exists a stab l e and causal system v, d) such that E c(v, d) = d + P(v d) =[I+ P(v )]d (1 3) where P is the numerator" of the right copr im e fraction representation of E and I denotes the identity system. 2. Conversely, for every stable and ca u sa l system( v, d) there is an internally sta ble contro l config uration around the system E for which the equ i valent system Ec(v, d) satisfies Ec(v, d) =[I+ Pcp(v, )]d. Thus equat i on (1.3) provides a complete parameterization of the class of all responses {E e } that can be obtained by int erna ll y stab l e contro l of the system E, with the stable and causa l system serv ing as the so l e parameter. In other words, for every there is an equiva l ent contro ll er C that int erna ll y stab iliz es E and yie ld s PAGE 14 6 the response (1.3). Conversely, every equivalent controller C that internally stabilizes E generates an explicit On a superficial inspection of (1.3), the attainable performance is prescribed by the "numerator" system P of E because it is the only fixed quantity apart from the identity. For a linear time-invariant system, it was shown in [50] that the at tainable performance was determined by the location of its right half-plane zeros. For a nonlinear system E, the instabilities of p-I ( the inverse of the "numerator" system P) would be analogous to the right half-plane zeros of a linear time-invariant system. Using this analogy it could be inferred that the instabilities of p-I would limit performance. The performance bound developed in the dissertation will further confirm this hypothesis. PAGE 15 1 2 Notation The following notational convention will apply unless otherwise stated. S(?Rm) n >. (E) A lul p(u) IIEII The set of m-dimensional real vectors. The set of m-dimensional real vector sequences. The A-step shift operator. The system E has latency of at least A. The 00 -norm of the sequence u. The weighted 00 -norm of the sequence u. The Lipschitz semi-norm of the system E. B The set of all stable, causal and recursive systems E: V --+ S(?R P ) where V is a bounded subset of S(?Rm). Lip(V S(?RP)) : The subset of B with IIEII < oo. IIEIIL ip The Lipschitz norm of the system E. (}M ( u S2) e c (S1 S2) P(E) The distance from any u E S 1 and the set S 2 The maximal distance from any u E S 1 and the set S 2 The right singularity of measure of the system E. 7 PAGE 16 CHAPTER 2 TERMINOLOGY AND BASICS This chapter contains a summary of the principal mathematical results whi c h are needed in the dissertation. The presentation is for discrete-time time-invariant nonlinear systems. Almost all the necessary mathematics are contained in [9 14 17 18 22 27]. The set of real numbers is denoted by ?R. The set of m-dimensional r ea l vectors is denoted by ?Rm. The set of all sequences u is denoted by S(?Rm) wh e re u = { u 0 u 1 u 2 ... } of m-dimensional real vectors U i E ?Rm, i = 0 1 2 .. .. Given the sequence u E S (?Rm) the i th e l ement is denoted by U i The set of elements {u i, u i +I ... Uj} wherej 2: i 2: 0 is denoted by u {. A system E from an input/output perspective is a map E: S(?Rm) ----+ S(?RP) which transforms input sequences of dimensional real vectors into output sequences of p-dimensional real vectors. The image of a subset SC S(?Rm) through the system Eis denoted by E[S]. The entire imag e of system the E is denoted by Im E where Im E d e f E[S(?R m) ]. Given a system E: S(?Rm) ----+ S(?RP) and an input sequence u E S(?Rm ), we denote by Eu ] i def Y i the i th element of the output sequence y = Eu and by y = Eu] i the set of elements {Y i, Y i +l ... Yi} where j 2: i 2: 0 are integers. There are two kinds of binary operations addition and composition. For a pair of systems E 1 E 2 : S(?Rm) ----+ S(?RP) the sum is defined as usual by (E 1 + E2)u ~f E 1 u + E 2 u for all sequences u E S(?Rm ); the right side of th e last formula is th usual e l e mentwise addition of sequences of real vectors. Composition i the usual c omposition of maps. PAGE 17 9 D e finition 2.0.1 A system E: S(?Rm) ----+ S(?RP) is causa l (respectively } strictly ca u al) if it satisfies th e following condition. For ever y integer i 0 and for every pa ir of i nput sequences u v E S(?Rm) satisfying ub = vt th e output sequences s at is fy Eu]b = Ev]b (respectively } Eu]&+ 1 = Ev]& + 1 ). A system M: S(?Rm) ----+ S(?Rm) is bicausal if it is causal and if it possesses a c ausal inv e rse. A system E: S(?Rm) ----+ S(?RP) is called a recursive system if ther e is a pair of integers T/ > 0 and a function f: (?RP)77+1 x (?Rm )+l ----+ ?RP such that for every input sequence u E S(?Rm), the corresponding output sequence y d e f Eu c an be c omput e d recursively in the form (2.1) for all integers k 0. The initial conditions y 0 .. Y TJ must of co ur se, be s p eci fi ed and fix e d. Th e function f is called a recursion function of E. For causality (str i ct ca usality) the condition T/ + 1 ( T/ is placed on system E. The class of strictly causal systems include every system E: S(?Rm) ----+ S(?RP) that c an b e r eprese nt ed in the form Yk k = 1 2, .... (2.2) H e r e, u E S(?Rm) is the input sequence ; y E S(?RP) is the output sequence; and x E S(?Rn) is an intermediate sequence of "s tates. In the c as e that th e maps f: ?R n x ?R m ----+ ?R n and h: ?R n ----+ ?RP are continuous, then the system (2.2) c onstitut es a continu ou s realization of the system E. For a real number 0 > 0 the set of all vectors in ?R m with components in th e closed interval [ -0 0 ] is denoted by [ 0 0]m. The set of a ll sequences u E S(?Rm) with elements U i belonging to [ 0 0]m for a ll integers i 0 is denot e d by S( 0 m). PAGE 18 10 Thus, S( 0m) consists of all sequences bounded by 0. It follows then that a system ~: S(?Rm) -----+ S(?RP) is BIBO (Bounded-Input Bounded-Output )-stable if for every real number 0 > 0 there exists a real number M > 0 such that ~[S(0m)] C S(MP). A sequence u E S(?Rm) is said to be bounded if there is a real number 0 > 0 such that The basic notion of stabi lit y that is used in this dissertation is related to con tinuity with respect to a metric. Two norms are particularly useful in this context to derive a metric: the .t' 00 -norm and the weighted .t' 00 -norm. The .t' 00 -norm is denoted by I I; for a vector a= (a1, a2, ... am) E ?Rm, it is simply lal def max{ la1 I, la2I ... laml}. For a sequence u E S(?Rm), the .t' 00 -norm is given by (2.3) The weighted .t' 00 -norm is denoted by pl and is given by (2.4) for a sequence u E S(?Rm ). For purposes of this dissertation different values of E will not affect our results; hence, the subscript will be dropped and the weighted .t' 00 -norm is simp l y denoted by p. The use of the weighted .t' 00 -norm simplifies mathematical arguments over the .t' 00 -norm because the bounded set of sequences S(0m) is compact with respect top. The norm p induces a metric p on a given S(?Rm), for every pair of e l ements u v E S(?Rm ), by p ( u v) def p( u v ). Formally the notion of stability employed in this dissertation is as follows. Definition 2.0.2 A system ~: S(?Rm) -----+ S(?RP) is stable with respect to the metric p if it is BIBO-stable ) and if the restriction E : S( am) -----+ S(?RP) is continuous with respect to p for every real number a > 0. Definition 2.0.2 is usually referred to as input/output stability. The following concept, which describes a weak form of uniform continuity with respect to the .t' 00 norm, plays a fundamental role in stab ilization theory ( see [17]). PAGE 19 11 D e fini ti o n 2.0.3 A stable system E: S(~m) ----+ S(~P) is differentially bounded if there is a pair of real numbers c, 0 > 0 such that 1 for every pair of sequences u E So far, only stability properties of individual systems have been mentioned. When several individual systems are combined into a composite system a stronger notion of stability is required, and it is usually referred to as internal stability. Internal stability guarantees desirable stability properties of the composition and takes into account the effects of various disturbances and noises that may affect the component systems. Consider a composite system E(s) that consists of s individual systems labeled E 1 ... Es where E i : S(~m(i)) ----+ S(~p( i )) i = 1 ... s. Individual entries in the list E 1 ... Es may represent summers multipliers etc. Let u E S (~ m ) be the external input sequence of the composite system and let y E S(~P) be its output sequence. Let u i E S(~m(j)) be the input sequence of the system E i within the configuration and let yi E S(~PU)) be its output sequence. The interconnections among the subsystems are then characterized by a set of equalities u i = y i ( i ), which determine to which output each input is connected. The external signal u is now augmented by s new input signals r/ E S(~m( i )), i = 1 ... s and set u i def y i ( i ) + r( For each i, the r/ acts as an additive disturbance on the input port of the system E i The disturbances are all assumed to be bounded by a real number 8 > 0, so that in f t i E S ( Cm ( i) ) 1 ac T/ u i ... s. Let E*S*: S(~m) X S(~m(l)) X X S(~m(s)) ----+ S(~P) X S(~p(l)) X X S(~p(s)): ( u rt 1 ... T/s) f---+ E*S*( u, rt 1 ... T/s) denote the system induced by the in terconnected system E(s) and the disturbances, having the input signals u rt 1 ... T/s and the output signals y, y 1 ... ys, respectively. De fin i t io n 2 0.4 The composite system E(s) is internally stable if the system E*s* is stable in the sense of Definition 2. 0.2. The composite system E(s) is strictly internally stable i,l besides being stable 1 the system E*S* is also differentially bounded. PAGE 20 12 Definition 2.0.5 A system E: S(?Rm) ----+ S(?J?P) is entire l y stabilizable if there is a strictly internally sta bl e control configuration that sta bil izes E over the entire input space S(?Rm). 2.1 Right Fraction Representations and Coprimeness A right fraction representation of a system E: S(?Rm) ----+ S(?J?P) is determined by three quantities: a subset S C S(?Rq), q > 0, called the factorization space; and two stab l e systems P: S ----+ S(?RP) and Q: S ----+ S(?Rm ), where Q is invertible, such that E = PQ1 A right fraction representation E = PQ1 is coprime whenever the stable systems P and Q are right coprime according to the following definition ([16, 18)). (Let G: S1 ----+ S2 be a map, where S 1 C S(?Rm) and S 2 C S(?Rn) are subsets. For a subset S C S(?Rn), we denote G* [ S] the inverse image of S through G, i.e., th set of a ll sequences u E S 1 satisfying Gu E S .) Definition 2.1.1 Let S C S(?Rq) be a subset. Two stable systems P: S ----+ S(?RP) and Q: S ----+ S(?Rm) are right coprime whenever the following conditions hold. 1. For every rea l number T > 0 there is a real number 0 > 0 such that 2. For every rea l number T > 0 the set Sn S( Tq) is a closed subset of S( Tq) (with respect to the topology induced by p ). The concept of a homogeneous system is of key importance to the theory of right copr im e fraction representations of nonlinear systems. A homogeneous system has the property of being a continuous map whenever its outputs are bounded. Th pr cise definition i s as follows. Definition 2.1.2 A system E: S(?Rm) ----+ S(?RP) is a homogeneous system if the fol lowing holds for every real number a > 0: for every su b set S C S( am) for which PAGE 21 13 there exists a real number T > 0 satisfying E( S] C S( rP) t h e r es triction of E to t h e closure S of S in S( am) is a continuous map E: S -. S( rP). The importance of homogeneous systems to the dissertation is stat e d in the next two theorems. Theorem 2.1.3 (17] An injective system E: S(~m) -. S(~P) has a right coprime fraction representation if and only if it is a homogeneous system Theorem 2.1.4 [17] Let E: S(~m) -. S(~P) be a recursive system If E has a recur sive representation Yk+ 11 + 1 = f (Yk, ... Yk+ 11 uk, ... Uk+ ) with a continuous recurs i on function f, then E is a homogeneous system. The following two theorems are important for stabilization theory. Theo rem 2 1. 5 defines the technical details in the stabilizing system E in Figure 2 .1 with 1r = B1 and <.p = A. Theorem 2.1.6 uses a basic property or right coprime factor ization, which is the denominator system contains the exact information about the instabilities of the system of right coprime fraction representation for s tabilization. Theorem 2.1.5 [19] Let E: S ( a m) -. S ( ~ P) b e a s y stem wit h a bo un d e d in p u t space S( am) a > 0 and a ss um e i t ha s a ri gh t cop rime f raction re p resentat i on E = P Q 1 w h ere P: S -. S( ~ P) a n d Q: S -. S( a m) and w h ere SC S( ~ q) f or some inte g er q > s + u E y 7r '-P Figur e 2.1: Th e blo c k diagr a m of E(1r,<,0 ) PAGE 22 14 0. Then, for every stable system M: S-. S with a stable inverse system M1 : S-. S, there exists a pair of stable systems A: S(~P) -. S(~q) and B: S( am) -. S(~q) such that AP + BQ = M. Theorem 2.1.6 [22] Let E: S(~m) -. S(~P) be a system having a right coprime fraction representation E = PQ1 where P: S -. S(~P), Q: S -. S(~m), S C S(~m), and where Q is bicausal. Let D: S(~r) -. S(~m) be any stable and causal system for which the composition ED: S(~r) -. S(~P) is stable. Then, there is a stable and causal system : (~r) -. S such that D = Q. In Figure 1.1, E: S(~m) -. S(~P) is a strictly causal system that needs to be controlled. It will be convenient to regard the external input sequence v as a fixed "parameter" while regarding the disturbance d as an external input, i.e., to consider appropriate partial functions. Since no restriction will be placed on v and d, this will have no effect on the validity of the final result. In the same spirit, we shall use the notation W ( V) Z def C ( V, Z) in which v can be intuitively viewed as a parameter of the system w(v) while z is its input. Control is achieved by a causal nonlinear dynamic controller C: S(~m) x S(~P)-. S(~m): (v,z) f-t C(v,z). For Figure 2.2, w(v) z d~f C(v,z) was defined with the system W ( v): S(~P) -. S(~m): z f--t W ( v )z = u. A result of [22] is now stated d + z + y 1l E w( v) Figure 2.2: The block diagram of EIV(v). PAGE 23 15 formally in Theorem 2.1.7. which furnishes a parameterization of the class of all systems that can be obtained from a given system :E by internally stable control. Theorem 2.1.7 [22] Let :E: S(~m) -+ S(~P) b e a strict ly causal system having a right coprime fraction representation :E = PQ1 with a bicausal denominator Q: S(~m) -+ S(~m). Assume that :E can b e strict ly interna lly sta b ili z ed by a con troller that admits representation Ci ( s, z ) = c1 ( z) [ s + T z ] = u. Then ) ref erring to (1 2) ; the following is true. The class of all inputdisturbance / output r e sponses :E e that can be achieved through internally stab l e control of :E is given by {:E e (v,d) =[I+ P(v)]d ; (): S(~m) x S(~P)-+ S(~m): (v,d) (v)d is a stabl e and causal system} A special case of Theorem 2.1. 7 is when the system :E is stab l e. In that case the right coprime factorization of :E = PQ1 can be taken as P = :E and Q = I and the following is obtained. Corollary 2.1.8 [22] Let E: S(~m) -+ S(~P) b e a strictly causal ) stabl e and differ entially bounded system. Then ) referring to (1.2) ) the following is tru e Th e class of all input&disturbance/ output responses :Ee that can b e achieved through internally stabl e control of :E is given by {:Ee(v,d) =[I+ :E(v)]d ; () : S(~m) x S(~P)-+ S(~m): (v,d) (v)d is a stabl e and causal system}. 2.2 Generalized Right Inverse For the nonlinear recursive system :E: S(~m) -+ S(~P) to possess a right in verse the system :E is required to be surjective. By restricting the range of :E to the image of :E, a surjective system :Er: S(~m) -+ Im :E is obtained with its right inverse system :E*: Im :E -+ S(~m). Let :E 9 : S(ReP) -+ S(~m) be any extens ion of :E* from Im :E to the whole space S(~P). Then, for every element y E Im :E, it is evident that PAGE 24 16 EE 9 y = EE*y = y. The system E 9 is a g e nerali ze d right inverse of the syste m E. Th e gen e ralized right inverse of system E is non-unique when system E is not bijectiv e Th e next theorem and its coro llari es from [14] state that a recursive system ha s a recursiv e generalized right inverse. Theorem 2.2.1 [14] A r ec ursiv e system E: S(?Rm) S(?RP) has a recursive g ener ali ze d right inverse E 9 : S(?RP) S(?Rm). A c ouple of notes from the proof of Th e orem 2.2.1. Let the system E*: Im E S( ?R m) b e the r ec ursiv e system represented by (2.5) In order to extend the domain of E from Im E to all S(?RP), let g e : (?Rm t x (?RPY,++ 2 ?R m b e any extension of the function g. Then the recursive system E 9 : S( R eP) S( ?R m) represented by (2.6) is a genera liz ed right inverse of E. Corollary 2.2.2 [1 4] L e t E: S(?Rm) S(?RP) b e a recursive isomorphism. Th en ) its inverse E1 : S(?RP) S(?Rm) is also a recursive is omorph ism Theorem 2.2.l cannot be generalized to the case of an arbitrary domain D. Nevertheless, Theorem 2.2.l can be generalized to the case when the domain D i s in the following particular form. A subset D C S(?Rm) is recursive if there ex i sts an int eg r ( and a function CJ assigning each point ( a 0 ... a() E ( ?R m)( + 1 a s ub set CJ( a~) C ?Rm such that D consists of exactly all sequences u E S( ?R m) sat isf y ing ukH+I E CJ( utH) for all integers k. Th e function CJ is ca ll e d the g enerating function of D. For instance if II: S(?Rq) S( ?R m) is a recursiv e system, then Im II i s a r ec ur 1v ubset of S( ?R m) PAGE 25 17 Corollary 2.2.3 (14] A recursive system E: V -+ S(~P) 1 where V zs a recurszv su bs et of S(~m) has a recursive right inverse E*: Im E -+ V. PAGE 26 CHAPTER 3 THE PERFORMANCE BOUND This chapter starts by discussing causality of systems and Lipschitz norms which will be useful in establishing the existence theorem of a best approximate right inverse. The performance bound will later be defined as a lower bound for the performance index of the approximate right inverse optimization problem. 3.1 Causality of Systems Recall a causal system E is one for which the values of the output sequence Eu up to and including index i depend only on the values of the input sequence u up to index i. This was precisely defined in Definition 2.0.1. For an integer >., the >.-step shift operator is denoted by D>.. defined, on any sequence u by D )... ] def Uk = Uk->.. (3.1) for all integer k for which Uk->.. exists. Definition 3.1.1 Let E: S 1 S 2 be a system ) where S 1 C S(~m) and S 2 C S(~P). Th e system E has latency of at least ). if there is an integer >. such that ) for every pair of input sequences u, v E S 1 and for every integer k 0 the equality u~ = v~ implies Eurn+>.. = Ev]~+>... We write .C(E) ). if the system E has latency of at least >.. Intuitively the latency represents a 'time delay' incurred in the propagation of changes from th input of E to the output of E. It is a simple consequence of the definition that a system E is causal if and only if .C(E) 0, and it is strictly causal if and only if .C(E) 1. From [17], a few simple properties of latency are listed. 1 Q PAGE 27 19 T h eorem 3.1.2 [17) L e t E 1 :S 1 -+ S 2 and E 2 :S 2 -+ S 31 wh ere S 1 c S( ~ m) 1 S 2 c n >1 [ S(~P)] and S 3 C n>1 + >2 [ S(~q)L b e systems 1 e ach with a well-defined lat enc y 1 respectively 1 ) 2 ,\ 1 and 2 ) 2 ,\ 2 Under this hypoth esis th e c ompo siti on E d e f E 2 E 1 has a well d e fined lat e ncy 1 and (E) 2 ,\ 1 + A2. In part ic ular if A1 + A2 2 0 th en E is a causal system. Theore m 3 1.3 [17) L e t E: S 1 -+ S 2 wh e r e S 1 C S(~m) and S 2 C S(~P) b e a re c urs ive system with a recursive representation Yk + 11 + 1 = f (Yk, .. Yk+r, uk, .. Uk+.) (and some fi xe d initial conditions). Under this hypoth esis E ha s well-defined lat ency and ( E) 2 T/ + 1 T h eo r e m 3 1.4 [17) A system E: S 1 -+ S 2 1 where S 1 C S(~m) and S2 C S( ~ P) ha s well-defined lat ency if and only i f there exis t s an i nt e g er ,\ suc h that the system n>E: S 1 -+ n>[ S 2 ] is a causal system. Th e system E: S 1 -+ S 2 where S 1 C S(~m) and S 2 C S(~P), is a r ec ursive system with a r ec ursive representation Yk+ 11 + 1 = f(Yk ... Yk+ 71 uk, ... Uk+.)B y restricting the range of E to the image of E we obtain the map E r : S 1 -+ E [ S 1 ) whi c h is evident l y surjective and possesses a right inverse E*: E[S 1 ) -+ S 1 such that EE*y = y for y E E[S 1 ) From Theorem 2.2.1 the system E* is r ec ursiv e with a recursiv e repr ese ntation giv e n by (2.5) Examining (2.5), T/ + 1 shifts are required on the output of a right inv e rs e system E* to guarantee ca usality i .e D( 11 + 1 ) E*) 2 0. Le mm a 3 1.5 L e t E: S 1 -+ S 2 1 wh e r e S 1 C S(~m) and S 2 C S(~P) 1 b e a recursive s y stem with a recursive repr ese ntation Yk+ 71 +1 = f (Yk, .. Yk + 11 uk .. Uk+.). L e t E*: E[S1) -+ S 1 b e a right inverse o f E by r es tricting th e rang e of E to th e image of E with a r ec ursiv e r e presentation Uk+. = g(uk ... Uk+.-1, Yk .. Yk+.+ 11 + 1). Un d er thi s hypoth esis th e system E has a latency of at most T/ + 1 1 i.e. 1 (D( 11 + 1 )E*) 2 0. PAGE 28 20 3.2 Systems with the Lipschitz Norm In the previous chapter we discussed two norms: the 00 -norm and the weight ed 00 -norm. We will need to define a new norm for a set of systems that has the convergence property which is that every Cauchy sequence in a compact metric space (X p ) converges to some point of X. The Lipschitz norm for a set of systems is introduced here because it has the necessary convergence property. The Lipschitz norm will be derived from a semi-norm. The semi-norm definition from [41] is given below. D e fini t ion 3.2.1 A semi-norm on a vector space X is a real valued function p on X for all x and y in X and all scalars a such that 1. p(x + y) p(x) + p(y) 2 p(a x ) = lalp(x) 3. p( X) =/Q if X =/Q. In this section, we let (S(~m), p ) and (S(~P) p ) be two metric spaces over th set of all sequences of m-dimensional and p-dimensional real vectors. Let B be the set of all stable causa l and recursive systems E: V ----+ S(~P) where V is a bounded subset of S(~m ). Introduce an operator I I II: B ----+ ~+ defined by IIEII def sup u1 u 2 EV u1::/=u 2 p (E(u1) E(u2)) p ( U1 U2) Th e number II E II has the following properties. Lemma 3 2 2 If E, \ll E B and a E ~ ) th e n 1. IIEII = 0 if and only if E is a constant syst e m on V ; 2 llaEII = lal IIEII; and (3.2 ) PAGE 29 21 The number IIEII is called the Lipschit z semi-norm ([7]) of the system Eon D. A semi-norm has the property that IJEII = 0 does not necessarily imply that E = 0 In fact it can be easily seen that IIEII = 0 if and only if Eis a constant system (ne d not be zero) that maps all sequences from D to the same sequence in S(?RP). The Lipschitz semi-norm is like a derivative bound or gain for all systems in B. Let 's define Lip(D S(?RP)) as the subset of B satisfying IIEIJ < oo. It is clear that an element E of Bis in Lip(D S(?RP)) if and only if there is a number L 0 such that for all u 1 u 2 E D. Moreover JJEII is the "least" such number L. It is also evident that a system with th e Lipschit z semi norm is both bound e d and continuous on its domain. The Lipschitz semi-norm will be used to define the boundary of a set from which an approximate right inverse can b e chosen. The semi-norm II II can be made into a norm as seen in the following theorem. T he o rem 3 .2. 3 L et u 0 b e an e l ement of D and l e t II def EIILip P ( E ( uo)) + 11 E 11 p (E(uo)) + sup u1,u2 EV ,u1 # u2 p (E(u1) E(u2)) p ( U1 U2) then the number 11 EI k ip defin es a norm for all E E Lip(D S(?RP)). (3.3) II E II Lip will be called the Lipschit z norm ([7]) of the system E defined by uo E D. A convenient choice of u 0 is of course u 0 = 0 if O E D wh e re note that E( O ) is not zero in general. To prove the theorem it amounts to showing that IIEIILip = 0 implies E = 0 the zero system. This however is an immediate consequenc of part 1. of Lemma 3.2.2. PAGE 30 22 Theorem 3.2.4 [7] The family Lip(1J, S(~P)) is a complete metric space under the Lipschitz norm II IILip A complete metric space (Lip(1J, S(~P)), II IILip) has the property that every Cauchy sequence of systems {Ek} converges. 3.3 Approximate Right Inverse In [50], the concept of an approximate inverse was formulated for linear sys tems. The problem of disturbance attenuation was addressed with a configuration very similar to Figure 1.1. As a result of [ 50], the approximate in verti bili ty of the system was shown to be a necessary and sufficient condition for disturbance attenu ation. The optimization problem from [50] used the 'h'. 00 -norm II 11 00 and a weighted semi-norm II llw with the property II llw :S II IJ 00 The weighted semi-norm has a weighing filter where W will denote a stable and causal system of unit norm such that lldllw = IIW dll oo The definition from [50] of an approximate inverse is as follows. The set of all stable, causal, and linear systems is denoted by Cs. For any stable and strictly causal system P, an approximate right inverse of P is any stable and causal system W for which III P\llllw < IIIllwThe right singularity measure of P, denoted by (P) is (P) def inf III P\llllwWEC s (3.4) In general P) is a number in the interval 0 :S P) :S 1. For nonlinear systems we cannot define a weighted semi-norm with th with th property II ll w :S II ll oo As a result, we cannot use (3.4) to derive a right singularity measure of E between 0 and 1. Nevertheless we can define an approximate inver e optimization problem. Let's first review some properties of right inverses for nonlinear systems. In Section 2.2 the generalized right inverse was introduced for a class of recursive systems in the form of (2.1). For systems E that are not bijective, PAGE 31 23 the generalized right inverse of systems E are non-unique. For the systems that are bijective, the generalized right inverse is stable but not necessarily causal. By assumption, the system E is strictly causal. This implies .C(E) 2: 1. The design integer ,\, where A 2: 0, is used to characterize the class of approximate right inverse systems that takes in account the latency of E. The design integer A is selected before the optimization process. In the three examples of Chapter 4, the design integer was selected at the minimum latency of E, i.e., when .C(E) 2: Ai, the design integer A = A 1 The approximate right inverse optimization problem of the stable, strictly causal and recursive system E: 1) -t S(~P) with V, a compact subset of S(~m ), is shown in Figure 3.1 withe= [I(u)n->-Ew(u)] where e E S(~P). The performance index is inf sup p(e) lllEA uES(aP) (3.5) where A is the set of all stable and causal systems W: S( aP) -t S(~m ). The system W is a stable approximate right inverse of E while the best approximant \ll* is not guar anteed to be stable. To guarantee stability of the best approximant \ll*, the systems E and Ware chosen from a class of systems Lip(V S(~P)) and Lip(S(aP) S(~m)) respectively. The set A has to be compact subset of Lip(S(aP) S(~m ) ) The next two I u n>Figure 3.1: The approximate right inv rs e optimization problem of E: inf s up p ( ) PAGE 32 24 paragraphs will exp lain why we need the set A to be compact to guarantee stability of the best approximant W Approximation theory is needed to solve the problem of finding a best ap proximant W*. In solving the best approximation problem, additional hypotheses are placed to guarantee the existence and uniqueness of the best approximation. Our practical problem is that we need approximate inverses that are stab le causal, and recursive but for now we will take approximate inverses that are only stable and causal. We sha ll be therefore be interested in a basic existence Theorem 3.3.1 which gives sufficient conditions to guarantee existence of closest point. Theorem 3.3.1 [6] Let K denote a compact set in a metric space. To e ach point p of the space th ere corresponds a point in K of minimum distanc e from p. In the previous section, the class of systems with the Lipschitz norm was introduced so that we can create a compact set of systems in a metric space. Using the hypothesis of Theorem 3.3.l as a starting point we can then prove the existence of best approximate right inverse. 3.3.1 The Existence Theorem of Best Approximate Right Inverse For clarity the following sets are defined for this subsection only. Let a 0 be a real number. Denote by Lip( S( aP), S(~m)) the set of all stable and casual systems W: S( aP) -+ S(~m) satisfying II '1111 < oo. Let D be a compact subset of S(~m) and let Lip(D S(~P)) be the set of all stab l e and causa l systems E: D-+ S(~P) satisfying IIEI\ < oo Let G > 0 be a real number. We denote by Ac the compact subset Lip(S(a P), S(~m)) given by A c= {w\w E Lip(S(aP) S(~m)) and \\w\\ :::; G}. (3.6) Theorem 3.3.2 Let EE Lip(D, S(~P)) be a strictly causal and r cursive system for a g iv n real number G > 0 and for a d esi gn integer ,\ 0. There is a tabl and PAGE 33 25 causal system w* E A c such that sup p (I(u) n-,\Ew*(u)) ::; sup p (I(u) n-,\Ew(u)) (3.7) uES( a, P) uES( aP) for all W E Ac. Proof :The right hand side of (3. 7) is a continuous function of W. If we take W = 0 then supuES( aP) p(I(u) n-,\Ew(u)) = a. There exists an infimum of th right hand side of (3. 7) since it is continuous with respect to W and it is bounded between O and a. Let and suppose that r= inf sup p(I(u)-D-,\Ew(u)), WEAG uES( aP) sup p (I(u) n-,\Ewk(u)) = rk k = 0 1 2 ... uES( aP) with rk monotonically decreasing tor. Then, for k sufficiently large sup p (I(u) n-,\Ewk(u)) ::; r + 1::; M. uES( aP) (3.8) (3.9) (3.10) Since the sequences of systems { I n-,\ Ew k} are bounded and monotonic it follows that the sequence of systems { I n-,\ Ew k} converges. This implies that for every c > 0 there exists an integer N such that l n 2'. N implies or sup p (n-,\Ewn(u) n-,\Ew 1 (u)) ::; c u ES( aP) (3.12) The Lip(S(aP) S(~P)) systems set is a subset of the stable and causal sys tems f: S( a P ) -t S(~P) satisfying llfll < oo. The composition of two systems of bounded Lipschitz norm is a l so a system of bounded Lipschitz norm hence for all W E Lip(S(aP), S(~m)) and a given E E Lip( D S(~P)), the composite system PAGE 34 26 Ew E Lip(S(aP), S(?R.P)). Now we define fk E Lip(S(aP), S(5R.P)) such that fk def Ewk and rewriting (3.12) we have (3.13) This establishes limkoo n->. fk = n->. J* where the system n->. f* is the limit of the sequence of systems { n->. Jk}. To translate n->. J* into a Lip( S( aP), S(~P)) system we get n>-(n->. J*) = J*] f The system J*] f has its first A outputs y 0 Yi ... Y >.1 set to zero because they were not used in the convergence of the sequence {n->. fk}It follows by the comp l eteness of Lip(S(aP), S(~P)) that the system J*] f E Lip(S(aP) The sequence of systems { n->. f k} converges to the limit system n->. f*. There fore, the sequence of systems {n->-Ewk} converges a l so to the limit system n->. J*. In other words, for every c > 0 and for all u E S( am), there exists an integer N such that l n N, this implies sup p (n >-Ewn(u)n->-Ew1(u)) Sc:. uES(aP) From the propert i es of a system of bounded Lipschitz norm we can we write sup p (n->-Ewn(u) n->-Ew 1 (u)) uES(aP) < 11n->.EIIL ip sup p (wn(u) W1(u)). uES( aP ) (3.14) (3.15) We will assume II D->.EIILip =/0, which implies that E is not the zero system. The sequence of systems { W k} is a continuous mapping from a compact metric space (S(aP) p) into a compact metric space (D,p) Now the sequence of systems {wk} is a sequence in a compact metric space ( A c, II IILip)Therefore some subsequence of {Wk} converges to a system in A c. Thus we have limi= W k 1 = \ll* where \ll* E A c for the subsequence of systems { W k 1 } Now we can define a subsequence of {n->. Jd by {n->. fkJ such that n>. f k1 def n->. Ew kl Every subsequence of { n->. fd converges to n->. J* so PAGE 35 27 limj-+oo n->. !k) = n->. J*; therefore (3.16) The system \ll* E Ac is best approximate right inverse of the system E with optimal sequence of systems { \l1 k 1 } E Ac that converges to \ll*. 3.4 The Measure of Right Singularity As has been previously noted, for every pair of elem en ts u, v E S ( ~m), the metric pf is defined (3.17) The distance from any u E S(~m) and the set S 2 C S(~m) is defined by the number (3.18) We can now define the maximal distance from any u E S 1 C S(~m) and the set S 2 by the number (3.19) From Theorem 3.3.2 there exists a best approximate right inverse system \ll* E Ac for a design integer ,,\ 0 such that the inequality (3.20) I u E n>Figure 3.2: The performance bound optimization problem of E: supuES(aP) p ,(e). PAGE 36 28 holds for all '11 E Ao and for all E > 0. Using '11* for '11 withe= [I(u)nAE '1f*(u)] in Figure 3.1 Figure 3.2 represents the performance bound optimization problem of the causal and recursive system EE Lip(V, S(~P)) where the performance index i s sup P c(e) u ES( aP) (3 .21 ) wh e r e e E S(~P). Now we want to put a lower bound on the term supuES( aP) P c(e) = supuES(a-P) P c (I(u ) nAE '1f *(u)) Using (3.18) and (3.19) we ca n write sup eM (u DAE 'll*[ S(an ]) u ES( a-P) e c (S(a ?), DAE 'll*[S(a ?) ]) > e c (S(a ?), DA imE). (3.22) Not e that even if Im E is unbounded e c ( S( aP), nA Im E) is bounded since the first operand of e c , S(aP), is bounded. Definition 3.4.1 For any s y stem E E L ip( V S( ~ P)) that is a stric tly causal recur sive of the system (2.1 )) th e right singularity measure ) d e not e d by P(E) ) for a rea l a > 0 and a d esi gn in t e g er A 0 is (3 .23) The p er formanc e bound is finally defined as the right singularity m eas ur e P (E). It provides a measure of the ability to match a set by subset of the image of a system as see n in (3.23). The performance bound came from the requir eme nt to provide an est imat e of the minimal effect of the disturbance signal d on the output s ignal z of Figur e 1.1. By assumption th e system E: V S(~P) is stable where Vis a compact s ub se t of S( ~ m) In that case, the right c oprime factorization of E = PQ1 can be taken as P = E and Q = I where J: V V the identity system, and P: V E[ V]. When the system E: S( am) S(~P) is unstable the right coprime factoriza tion of E = PQ1 is taken as P: S E[S(am)] and Q: S S(am) where SC S( ~ m) PAGE 37 29 and Q is bicausal. From Theorem 2.1.5, any stable and bicausal system M: S ----+ S with a stable inverse system M1 : S ----+ Scan be selected so ther exists a pair of ta ble systems A: S(~P) ----+ S(~m) and B: S(o:m) ----+ S(~m) such that AP+ BQ = M. The stabi lization of E can be seen in Figure 2.1 with 1r = B1 and c.p = A. The closed loop response of the stab ilization is P M1 Now M can be se l ected as the identity system J: S----+ S. The factorization space Scan be taken as S(o:m) and [21] describes the construct ion of the stabilizing contro ll ers that y i e ld the factorization space of S (om) As a result the closed loop response is P: S (om) ----+ E [ S (om) ]. The preceding paragraphs illustrate that the numerator system P directly drives the performance bound by the image of its system E. The image of E for discrete-time nonlinear systems is analogous to t h e right half-plane zeros of li near continuous-t im e systems Both of these quantities are fixed and cannot be a l tered by feedback. Now we can use the performance bound for systems of practical origin. PAGE 38 CHAPTER 4 CALCULATION OF THE PERFORMANCE BOUND 4.1 The Estimate of the Performance Bound It turns out that the calculation of the right singularity measure P(E) for nonlinear systems is rather laborious. We develop in this section an estimate 'P(E) of P(E) which is easier to calculate. This estimate satisfies P(E) < 'P(E). ( 4.1) From Theorem 3.3.2, there exists a best approximate right inverse system \JI* E Ac such that the inequality (3.20) holds for all 'II E Ac and for a design integer >. 2'.: 0. Theorem 3.3.2 demonstrates the existence of a best approximate right inverse system but does not provide the construct ion of a best approximate right inverse system. I u n-).. Figure 4 1: The est imate of the performance bound optimization problem of the system E with an approximate right inverse system \JI: 'P(E) = supuES(aP) p((e). 30 PAGE 39 31 Using a causal approximate right inverse system WE Lip(S(o :P ) S(?R m )) for a given design integer,,\ 2 0 from Figure 3.1, Figure 4.1 with e = [I(u)n>Ew(u)], represents the estimate of the performance bound optimization problem of the causal and recursive system EE Lip(V, S(?RP)), where the performance index is (4.2) where e E S(?R P ). ow we want to express th e term sup u E S ( a-P) p f. ( e ) = s up u E S(a-P ) p f. (I(u) n->-Ew(u)) using the operator e f. ). Using (3.18) and (3.19) we can writ e sup p f. (I(u) n>Ew(u)) u E S ( a P) sup eM ( u n>Ew[ S (o :P) ]) u ES( aP) e f. (S(a ? ) n>Ew(S(a ? )]). ( 4 .3) Definition 4.1.1 For any system E E Lip(V S(?RP)) that is strictly causal and r e c ursi ve an estimate of the performance bound d e noted by r>(E) giv e n r e al a > 0 and d e sign integ e r ,,\ 2 0 is given by ( 4.4) wh e r e WE Lip(S(aP) S(?R m )) is a causal appro x imat e right inv e rs e syst e m. 4.2 Practical Implementation Techniques For the applications presented here we will be interested in the response of the systems over a finite interval of time say for a discrete set (0, T], where T > 0 is a fixed integer. Let Cr(?Rn) be the set of all functions h: (0 T] -+ ?R n Denote by I hi def sup i E(O T] lh( i) I the 00 -norm on Cr(?Rn) and by pf.( h) def sup i E[O T](l + E t i lh( i) I for E > 0 the norm p f. on Cr(?Rn). Let h E Cr(?Rn) be any function. Then sup (1 + et i lh(i)I :s; sup lh(i)I iE[0,T] i E[0 T] and (1 + el sup (1 + et i lh(i)I = sup (1 + e?i lh(i)I 2 sup lh(i)I i E[0 T] i E[0,T] i E[0,T] PAGE 40 32 so that on Cr(~n). These inequalities imply the equivalence of the norms I I and Pi(-) on Cr(~n ). We will use the 00 -norm for its simplicity in the calculation of the estimate of the performance bound P(E) (so the norms I I and Pi() are interchangeable over finite intervals of time). For the mathematical analysis presented in Chapter 2 and Chapter 3 the signals of interest are always bounded. For implementation purposes, the maxi mum peak deviation of the input signal, maxi~o ID-uil where D.ui = Ui+t u i is limited. To limit the maximum peak deviation the signals are transformed by using a discrete-time lowpass filter. A new family of sequences, denoted by SnLP(a), is formed by using a second order discrete-time lowpass filter with a cutoff frequency nLP in rad/sec on S(a). The transformation between S(a) and SnLP(a) is bijective since the lowpass filter is one-to-one and the image of the lowpass filter is Sn LP (a). Thus the transformed space Sn LP (a) is isomorphic to the input space S( a). The alternate definitions of P(E) are given in ( 4.3) and using the f 00 -norm ( for t = 0) we g e t P ( E ) = sup jI ( u ) n,\ Ew(u)I = sup eMo (u D,\ Ew [S( a P) J). (4 .6 ) u E S ( aP ) uE S(a P ) ,...-----------------------------I u L P F -I I e n,\ f ~----------------------------~ F i g ur e 4 2 : Th e Fun c tion f: S(aP) --+ ~+: u f-+ f (u) e E ~ + PAGE 41 33 Define a function f: S( a?) --+ ~+ that incorporates the lowpass filter as displayed in Figure 4.2, such that P(LJ) = sup f(u), (4.7) uES( a P) where the function f is defined as follows f ( u) def I I ( L p F ( u)) n,\ L, '11 ( L p F ( u)) I (4.8) The parameters of the function f are the system LJ an approximate inverse s y st e m '11 the design integer \ the optimization set S( aP), and the lowpass filter cutoff frequency nLP The function f must also satisfy f(u) = eM 0 (u n->.LJ'11[SoLP(a P )]) where eM 0 (-, is defined in (3.18). It then follows that f() is a convex function when its input or optimization set S( aP) and the image set LJ'P [ So LP ( aP)] are convex since eM a is a distance function from a point in the optimization set to the image set. The distance function eM o by its construction is a convex function. In practice the imag e set is not convex; however studying the optimization problem of finding a global maximum of a convex function will give us insight to the non-convex optimization problem. Unlike the minimization of a convex function once a local maximum has been found there is more or less by definition no local information to tell you how to proceed to a higher local maximum. In particular there is no local criterion for deciding whether a given local maximum is really the global maximum. Hence maximizing a convex function is usually a much harder task than minimizing a convex function. Theorem 4.2.1 shows that a convex function achieves a maximum over a c ompact polyhedral set at an extreme point. Theorem 4.2.1 [3] L e t g: S(~P) --+ b e a convex function and l e t S b e a non e mpty compact polyhedral s e t in S(~ P ). For the problem of maximi z ing g( u) subj e ct to u E S ) th e r e i s an optimal solution ) ii, where ii, is an extreme point of S such that g( u) 2: g( u) for all u ES. PAGE 42 34 Using Theorem 4.2.1, he global maximum of f(u) maybe found by evaluating all the extreme points of S( aP) and finding the maximum on this set. For example if we have a one-dimensional set of sequences which is a finite length of 250 elements the number of extreme points and hence the number of evaluations off would be 2 250 = 1.8903 x 10 75 This procedure is laborious as far as computation is concerned. Although the image set S is non-convex the property that the optimization set is convex is enough to suspect that global maximum would be found on an extreme point of the optimization set. If we take an interior point Ui from the optimization set which is a nonempty compact polyhedral set and form a neighborhood Nr, ( u i) of radius r i, then we can find a Ui+1 E Nr,(u i ) such that lJM 0 (u i +I S) > lJM 0 (ui, S) and U i +I is closer to the boundary than U i The radius r i of a neighborhood Nr,(u i) is se l ecte d so that we do not accumulate in an interior local maximum. Since the optimization set is convex and lJMo is a distance function the sequence { ui} will converge to a point on the boundary. Therefore the global maximum will b e on th e boundary. For simulation purposes, the global maximum is assumed to be an extreme point. The three systems of practical origin that hav e been selected for in vestigat ion ar e all fourth order nonlinear systems. Each of the systems has a distinct memoryle ss nonlinear subsystem. Thes e subsystems represent respectively dead-zon es, position c onstraint s, and rate constraints for the problems: permanent magnet stepper motor mod e l exa mple multi variable process control model example and aerodynamic model examp l e Nonlinear programming is used to calculate the estimate of the performan ce bound. The algorithms that are developed take advantage of the traits of eac h of the nonlinear systems and of the finite bandwidth of the input signal space. A us e ful trait of a system is a property that would help reduce the region of optimization in the search for a global maximum. It will be shown in th e n ext sections that the comp utation is not as laborious as pr ev iousl y conjectured. PAGE 43 35 4.3 Permanent Magnet Stepper Motor Model Example This section is concerned with voltage control of a permanent magnet (PM) stepper motor. This example comes from [5], wherein the authors developed a model based contro l law using exact linearization implemented on an industrial setup. The PM stepper motor model consists of a s lott ed stator with two phases and a permanent magnet rotor. On e side of the rotor is a north pole, and the other side is a south pole. The teeth on each side of the rotor are out of alignment by a tooth-width. The equations of motion for the PM stepper motor model are given by: dia 1 [va Ria+ I PAGE 44 36 For the PM stepper motor model, there is an appropriate nonlinear coordinate transformation which is known as the direct-quadrature (DQ) transformation. The DQ transformation for the phase voltages and currents is defined as follows: [ ~: l [ :: l def def [ cos(Nr0) sin(Nr0) [ cos(Nr0) sin(Nr0) (4 .13 ) (4.14) The direct current id corresponds to the component of the stator magnetic field along the axis of the rotor magnetic field, while the quadrature current iq corresponds to the orthogonal component. The application of the DQ transformation to the original system equations ( 4.9) through ( 4.12) yields: did 1 [vd Rid + NrLwiq] ( 4.15) dt diq 1 [vq Riq NrLwid I PAGE 45 take the following form: Xk+l j ( Xk) + g( Xk) Vqk Wk [ 0 0 1 0 ] Xk, where x k E ~4, and f, g are vector fields on ~ 4 The analytical expression for the input dead-zone is if VCqk 1 if -1 < VCqk < 1 if VCqk -1, 37 ( 4.19) ( 4.20) ( 4.21) The system ~: 1) -+ S(~) : vcq w, where 1) C S(~), is the composition of the equations of motion with the dead-zone for which vcq is the quadrature voltage command. Figure 4.3 shows the block diagram of the system ~4.3.1 Simulation The signal of interest is the angular velocity signal w. It is assumed that all signals of interest start at rest (i.e., Wn = 0 for all n n 0 ). In one of the applications presented in [5], the desired speed trajectory required the motor to go from to zero to 1350 RPM in 10 ms with a sample rate of 10 kHz The bandwidth of w was set to 50 Hz. The rise time was about 10.6 ms and the absolute bound of lwl 1350 RPM was imposed. I I I Dead Zone rtL I vqk I I I xk+l = f(x k) + g(x k) vq k ... .... Wk = [O O 1 OJ x k ~, I-------------------------------------------------------' Wk+l .... ... Figure 4.3: The block diagram of the system I:: 1) -+ S(~) : vcq w for the PM Stepper Motor Model Example. PAGE 46 38 The block diagram of an approximate right inverse system W: S 2 1r so( a) -t SOR) : we f--+ Vqinv is shown in Figure 4.4. The first block calculates the raw quadra ture voltage v~1c for a given angular velocity command we1c. The analytical expression for the continuous dead-zone inverse ( C DI) is given by { v' + 1 Qlc Vqinv1c = CDJ(v;1c) = G V~k v' 1 Qlc if if if v' > _1_ Q1c G-1 l 1 1 -G-1 < Vqk < G-1 v' < __ 1_ Qk G-1 ( 4.22) Here G = JJCD/(,)11 represents the Lipschitz semi-norm of the CDI system. The range of G is 1 < G < oo. In the C DI block, the voltage quadrature command Vqinv1c is determined by using ( 4.22) on the raw quadrature voltage v~1c. The voltage quadrature command Vqinv1c is then used to update the states Xinv1c of the system W. Two integrations of Vq are required to reach w. This results in the discretized model of at least a two step delay. Therefore, .C(E) 2:: 2. From Figure 4.4 it follows that .C(w) 2:: 0. Theorem 3.1.2 applied to the composite system E\ll: S2,r. 50 (a) -t SOR) : we f--+ w yields .C(Ew) 2:: 2. For the simulation, g(xk) = [0 t~t 0 0]' #0 in (4.19) for all Veq E D and a given x 0 The design integer is taken as ,\ = 2 for an approximate right inverse system W. For the simulation of 40 milli-seconds a sampling period of 100 micro-second s y i e lds a sequence of 400 points. In the previous section, the global maximum wa s ,,,. -------\I Wk ( X k = f(x k-1) + g(x k 1 )vq k I ---~ -1 I vqk I I Wk= [O O I OJ x k I _________ .,,, k 1 L0--xk = f(x k -1) + g ( x k I ) vq k Continuous Dead-Zone Inverse Dead-Zon e Fi g ur e 4.4: Th e blo c k diagram of an approximat e right inv e r s sys t m W : 2 rr-s o ( a) -t S O R) : w e f--+ Vq inv for th e PM St e pp e r Motor Mod e l Exampl PAGE 47 39 shown to be on the boundary. For simulation purposes, the global maximum is assumed to be an extreme point. The pre-filtered binary random sequences have only the values RPM, the extreme points of S(1350). Table 4.2 shows the Monte Carlo results for 4 different system gains G of the C DI system: 2, 10 100 and 1000 where the error lwc n2 E\J!(w c )I is evaluated for each of the sequences. As can be seen from Table 4.2, as G -+ oo, the error lw c n2 E\J!(w c )I -+ 0 in a linear fashion. For the limiting case as G -+ oo, an approximate right inverse system \JI is not continuous. However the performance bound P(E) appears to be directly proportional to the system gain G. This observation will be proven true in the following subsection. From Table 4.2 the maximum value of lw c -D2 E\J!(w c ) I for G = 10 rad/sec is 0.02686. This corresponds to the 18 th sequence in the Monte Carlo run. In the following subsection we will calculate the estimate of the performance bound P(E) over the optimization set S21r 50 (1350). A first estimate is f>(E) 0.02686. In Figure 4.5 the 18 th sequence for w e the filtered square wave is plotted with the response sequence E\J!(w c ) without the n2 shift since the errors are less than 0.01 % of the magnitude of the signal. The absolute angular velocity error sequen c e is plotted in Figure 4.6. The quadrature voltage Vq and the absolute quadrature voltage error l v ~k vq k I are plotted in Figure 4. 7 and Figure 4.8 respectively. ote that Figure 4.6 and Figure 4.8 appear to be the same figure but of different scales. 4.3.2 The Estimate of the Performance Bound P(E) In this section, we will find an estimate of the performance bound P(E) for the voltage control problem of a PM stepper motor by using the optimization definition as follows: P(E) = sup lw c n2 E\J!(w c )I we E S 2 1r s o (1350) ( 4.23) PAGE 48 40 Table 4.2: Family of Sequences, S 2 1r.so(1350), Comparison Approximate Right Inverse Error lw c n2 ~\ll(w c )I Sequence Gain G = 2 I Gain G = 10 I Gain G = 100 I Gain G = 1000 1 0.1339 0.02628 0.001151 0.0002244 2 0.1341 0.02588 0.001021 0.0000000 3 0.1342 0.02510 0.001979 0.0001157 4 0.1343 0.02546 0.001346 0.0000000 5 0.1332 0.02623 0.001882 0.0000000 6 0.1336 0.02631 0.000332 0.0000000 7 0.1343 0.02652 0.002509 0.0000000 8 0.1340 0.02608 0.000279 0.0000000 9 0.1341 0.02659 0.001415 0.0000000 10 0.1326 0.02628 0.001764 0.0000000 11 0.1343 0.02641 0.001312 0.0002664 12 0.1339 0.02650 0.002210 0.0000549 13 0.1344 0.02617 0.000710 0.0000000 14 0.1332 0.02537 0.002150 0.0000000 15 0.1334 0.02670 0.002611 0.0000000 16 0.1328 0.02683 0.002629 0.0000000 17 0.1334 0.02662 0.002360 0.0001910 18 0.1336 0.02686 0.001415 0.0000000 19 0.1341 0.02619 0.002322 0.0000000 20 0.1341 0.02636 0.002085 0.000022 4 21 0.1337 0.02680 0.002059 0.0000000 22 0.1342 0.02615 0.002300 0.0002322 23 0.1337 0.02464 0.000000 0.0000000 24 0.1343 0.02566 0.001393 0.0000000 25 0.1342 0.02677 0.002103 0.0002603 max 0.1344 0.02686 0.002629 0.000266 4 PAGE 49 a.. i ..!... "O C ('tl E E 0 {) "O C ('tl 150 100 50 0 -50 a, -100 (J') C 0 g-150 Q) a: E'-200 0 ai > co -250 "S Cl C -300 : -omega : ..:. oinega _c ... \ .... \ -350'-----'------'-----'-------'--------'----..,___--_._--~ 0 5 10 15 20 25 30 35 40 Time (milli-seconds) 4 1 Figure 4.5: The angular velocity response w = E\ll(w c ) and the command w e E S21r 50 (1350) with system gain G = 10 are shown for the 18 th sequence in the Mont e Carlo run. 0 03 ~--~,---~,---~,--~ ---~ ---~, ---~ -~ 0 025 ,... 0 005 0 0 5 I --~ --~-~~ ~!I __ 10 15 20 25 30 35 40 Time (milli-seconds) [dt=0 1 msec] Figure 4 .6: The absolute angular ve lo city error lw c k -D2 E w (w ck )I is shown for th e 1 8 th sequence in the Monte Carlo run of q c E S21r. 50 (1350) with system gain G = 10 where lw c n2 Ew(w c )I = 0.02686. PAGE 50 6 ,------.-:---.---,---., ----,---,-, -------. ---~ -~ 4 ::, -0 0 a Q) 0) .s 0 >-2 -4 I\ .. .. ,.. . ;A J v. -6'------'----'---'--....1...I __ ___Jl.__ __ ...,__ ; __ ____.1 ___ .J....I __ ___, 0 5 10 15 20 25 30 35 40 Time (milli-seconds) 42 Figure 4.7: The quadrature voltage vq is shown for the 18 th sequence in the Monte Carlo run of w e E S2-rr. 50 (1350) with system gain G = 10. 0 1 ,----.,---,----,,,----....------r----,, -------. ---~ -~ 0 09,_o oa~ c. 0 ::: 0 07 -w 0 06 '"" a o o5 Q) 0) <1l 0.04 > Q) 0.03 Cl) .0 0 02 0 01 . ............... ............ ......... . 0 ......._ __ ...J... 1 ------ILL...a.-''------'-..L...-aL..l.....__......,_. __ __.J....I __ __..__,1 ____.__._,_,LL..II --~ 0 5 10 15 20 25 30 35 40 Time (milli-seconds) [dt=0 1 msec] Figure 4.8: The absolute quadrature voltage error, l v;k Vqk I is shown for the 1 8 t h sequence in th e Monte Carlo run of w e E S2-rr 5 0 (1350) with system gain G = 10 where I v; Vq I = 0.0993. PAGE 51 4 3 where the system :E and an approximate right inverse system '11 are illustrated in Figure 4.3 and Figure 4.4, respectively. The first block of Figure 4.4 calculates the raw quadrature voltage v~k for a given angular velocity command wck. In the CDI block, the voltage quadrature command Vqinvk is determined by using ( 4.22) on the raw quadrature voltage v~k. In Figure 4.3 ( vcqk = Vqinvk ), the quadrature voltage Vqk is the output of the DZ block by using ( 4.21 ). The composition of ( 4.22) and ( 4.21) yields Vqk = DZ (CD I ( v~J) = It follows then that for v~k = b v' qk G v' 1 qk 0 G v~k + 1 v' qk if if if if if I > 1 Vqk G-1 1 / 1 c < vqk < c-1 _l. < v' < l. G qk G __ 1_ < v' < _l. G-1 qk G I < 1 vqk G-1 ( 4.24) ( 4.25) For G = 10 b = 0.1 and this is consistent with Figure 4.8 where Iv~ v ql = 0.09993 b The maximum quadrature voltage error occurs when the magnitude of the raw quadrature voltage lv~k I = b Since there is a feedback loop inside an approximate inverse system '11, the quadrature voltage errors are not accumulative with respect to the angular velocity errors. Evaluating the right hand side of ( 4.23) with lv~k I = b will give us the estimate of the performance bound which requires two integrations of the errors as follows: or 'P(:E) at lv~kl = 1 --~t GL 60 Km A 2 ----D,.t 21r G L J 60 Km A 2 D.t 21r G L J ( 4.26) PAGE 52 44 Using the optimization set we E S21r. 50 (1350), and for G = 10, this yields r>(E) 0.02688. The estimate is 0.00002 higher than the value from Table 4.2 of 0.02686. The estimate of the performance bound r>(E) is directly proportional to the system gain G. Figure 4.6 is a scaling of Figure 4.8 by a factor of [.j 6.t 2 In this example, we have found a closed form solution for an estimate of the performance bound given by r>(E) which was verified by simulation. In the next 2 examples nonlinear programming algorithms will be used to calculate r>(E). 4.4 Aerodynamic Model Example This section will provide an example of the performance bound calculation for the longitudinal control of an aircraft. Control of pitch attitude of an aircraft can be achieved by deflecting all or portion of either a forward or aft tail surface. The derivation of the equations of motion of the aircraft with the assumption that the roll rate, yaw rate and y-velocity component in the body-axes frame are all zero is found in [4]. These equations are: u V q 0 2_ [X (u,w,q 8) + Tmgsin(0)] wq m 1 [Z (u, w, q, 8) + mg cos(0)] + uq m lvf ( u, w, q, 0 8) /lyy q ( 4.27) ( 4.28) (4 29) ( 4.30) where u and w are the body-axis x-velocity component and the body-axis z -velocity component, respectively. The body-axis pitch rate and the Euler pitch angle are given by q and 0, respectively The longitudinal control input or elevator deflection is given by 8. X and Z are the aerodynamic forces and M is aerodynamic moment. The following are constants: T thrust; m mass g gravity ; and ly y, moment of inertia. The aerodynamic forces and moments are represented by the means of the PAGE 53 45 aerodynamic stability coefficients as seen below: a arctan( w/u) a -; (cos(a))2 (wu wu) u Q tp(u2+w2) L (CL+ CL 0 a + CLqq + CL 0 8) QS D (Cn + Cn 0 a) QS X L sin (a) D cos (a) ( 4.31) z -Lcos(a)Dsin(a) ( 4.32) M (C m + Cm 0 a + Cm 0 a + Cmqq + Cm 0 8 ) QSc. ( 4.33) Here a and a are the angle of attack and the angle of attack rate respectivel y. Q is the dynamic pressure. The aerodynamic moment, M, and the aerodynamic forces X and Z, are directly proportional to t he dynamic pressure. L and D are the aerodynamic forces in the wind axes which are known as lift and drag The following are constants: S, the wing reference area and c, the wing mean aerodynamic cord. Table 4.3: The Aerodynamic Model Parameters Parameter Value Parameter Value m 1247.390 kg Cn 0.05 T 1600 Cna 0.33 g 9.81 m-s2 CL 0.41 lyy 4066 kgm 2 CL 0 4.44 p 1.225 kgm3 CLq 3.8c/(2uo) s 17.094 m 2 CLO 0.355 C 1.737 m Cma -0.683 Uo 53.8931 m-s1 Cm Q -4.36 c/(2uo) Vo -0.0931634 m-s1 Cmq -9.96 c/(2uo) qo 0 rads1 Cmo -0.923 0o 0 rad 8_1 0.00128510 rad The reference geometry, mass characteristics, and aerodynamic characteris tics of the general aviation airplane: NAVION were taken from Appendix B of the PAGE 54 46 reference (33). The values of the parameters are shown in Table 4.3. The equa tions of motion are discretized using second order Runge Kutta integration. Let Xk = [uk Vk qk Ok]'. Equations ( 4.27) through ( 4.30) take the following form: Xk+l f ( Xk) + g( Xk) Ok qk [ 0 0 1 0 ] Xk where x k E ~ 4 and J, g are vector fields on ~ 4 ( 4.34) ( 4.35) The actuator has a first-order response modelled by G(s) = 10/(s + 10) which has rate constraints of 15 deg/sec and position constraints of 15 deg. The system E: 1) --+ S(~) : 8c 1-+ q, where 1) C S(~), is the composition of the equations of motion with the actuator dynamics for which 8c is the elevator deflection command. Figure 4.9 shows the block diagram of the system E. 4.4.1 Simulation The signal of interest is the pitch rate signal q. It is assumed that all signals of interest start at rest (i.e., qn = 0 for all n n 0 ). The absolute bound of lql 1.15 deg/sec was imposed so that after 10 seconds of flight, the aircraft will always be within its parameters of good flight conditions. Typical pitch rate commands are gen e rated by the longitudinal autopilot. The altitude hold mode and the mach hold mode have time constants of approximately 100 milli-seconds ([4]). This corresponds to a r e striction on bandwidth of q to be approximately 10 rad/sec. ------------------------------------------------------. ' : Rate Position : .----~ : Constraints Constraints : ~ ~ ~\ x..-1 = f ( x k ) + g(x k) Ok qk+I b ac t --7I q k = [O O I 0] x k E ------------------ --------------------. --------I Figur e 4.9: The block diagram of the system E: 1) --+ S O R) : 8 c 1-+ q for th e erody nami c Mod e l Example. PAGE 55 47 The block diagram of an approximate right inverse system W: SoLP(a) SO R) : qc r-+ Dinv is shown in Figure 4.10. The first block calculates the raw elevator deflection 8k for a given pitch rate command qc1c. In the limiting logic block, the elevator deflection command 8inv is determined by first using the constraints on the raw elevator deflection 8k and then using the inverse actuator dynamics. The elevator deflection command 8inv is then used to update the states Xinv of the system W. One step delay of 8c is required to reach q. Therefore, (~) 1. From Fig ure 4.10 it follows that ('11) 0. Theorem 3.1.2 applied to the composite system Ew: So LP (a) SOR) : qc r-+ q yields (~'11) 1. For the considered flight condi tions, g( xk) i0 in ( 4 .34) for all 8c E 1J and a given x 0 The design integer is taken as A = l for an approximate right inverse system W. For a simulation of 10 seconds, a sampling period of 4 0 milli-seconds yields a sequence of 250 points. The pre-filtered binary random sequences have only the values .15 deg/sec because we are only interested in the extreme points of S(l.15). Table 4.4 shows the Monte Carlo results for 3 different cutoff frequencies D.Lp: 10, 15, and 20 rad/sec, where the error lqc n1 ~\ll(qc)I is evaluated for each of the sequences. For practical purposes, as can be seen from Table 4.4, the system W: S 10 (1.15) S O R) is a right inverse of the system E. ,------------1 ( f ( ) ( ) ~ \ I X k X k I + g X k 1 0 k I __ _,_ _,I I \ qCk = [0 0 I OJ x I ..._ ________ / x. = f ( x k l ) + g ( x k l ) 8, o k Limiting Logic Position Constraints Rate Constraints 1I I \ aa.. .t ---------------------------Figur e 4 .10: The block diagram of an approximate right inverse system W: nLP ( a) S O R) : q c r-+ 8inv for th Aerodynamic Model Example. PAGE 56 48 Table 4.4: Family of Sequences, SoLP(l.15), Comparison Approximate Right Inverse Error lqc n1 ~\ll(q c) I Sequence nLP = 10 rad/sec I nLP = 15 rad/sec I nLP = 20 rad/sec 1 4.996E-16 0.9075 1.353 2 4.163E-16 1.7650 2.265 3 3.331E-16 0.4226 1.553 4 2.776E-16 0.3620 1.121 5 3.608E-16 0.5654 1.592 6 3.331E-16 0.4446 1.356 7 3.331E-16 1.4340 2.153 8 2.498E-16 0.9444 1.788 9 3.331E-16 0.6539 1.896 10 4.441E-16 1.2530 1.931 11 2.220E 16 1.2270 2.121 12 5.551E-16 0.9 812 1.371 13 4.996E-16 0.7 821 1.471 14 5.551E-16 0.7177 1.488 15 4.441E-16 0.6 425 1.351 16 4.441E -16 0.7448 1.602 17 5.551E-16 0.4841 1.454 18 4.996E-16 0. 4835 1.586 19 4.441E-16 0.6658 1.585 20 3.331E-16 0.7861 1.675 21 3.608E-16 0.5702 1.544 22 3.331E-16 0.6071 1.809 23 3.886E-16 0.6047 1.635 24 3.053E-16 0.4456 1.797 25 4.441E-16 0.9205 1.413 max 5.551E-16 1.7650 2.265 PAGE 57 4 9 2 U) 1 5 .. .... .. -(/) Q) 0) I Q) .. 1 ... .... I I ..!_, "C I I C: 0 5 .. I I t <1l E I I I E I 0 I (.) 0 I: "C I I I I C: I I / (1j Q) I I I -0.5 .. I '1 0 I I CL en Q) II Q) -1 cii II ..c: (.) -1 5 -2 0 2 3 4 5 6 7 8 9 10 Time (seconds) Figure 4.11: The shifted pitch rate response q = n1 ~\J! ( q c ) and th e command q c E S 1 5 (1.15) are shown for the 2 nd sequence in the Monte Carlo run. 1.8 ,----~-----.-----r---.-----,----,.----,----------.--~----, 1 6 1 4 u Q) t 1 2 .... e w Q) cii II .c 0 8 ..... .B a: Q) -S 0.6 .. ... 0 (/) .0 <( 0 4 ..... ... ... ,. ..... 0.2 0 L..____JJLo......J...__JL_J.L..Ll...JLIIULJLILJ_llLLJJL.._...illLJ--LJLILIIL....L...IUL-1llLo""1--_L....L.1L.....1..JL____J,..ILJ...--1.W.liUW-Lll--1.......JU 0 2 3 4 5 6 7 8 9 10 Time (seconds) [dt=0.04 sec] Figure 4.12: The absolute pitch rate error lq c k n1 ~\J!(q c k)I is shown for th e 2 nd sequenc e in the Monte Carlo run of q c E S 15 (1.15) where lq c n1 ~\J!(q c )I = 1.76 5 PAGE 58 5 0 2 I I I I I I ' I 1 5 .... . ...... ... 1 .. ci 1 a, 0 5 C J 0 1 u Jlr a, o'"" 0 :u 0 iii .... : .. -v ai -0.5 ... LU i -1 f..... -1 5 -2 I I I I I I I I I 0 2 3 4 5 6 7 8 9 10 Time (seconds) F i gure 4 .1 3 : Th e e l eva tor d e fl ect ion 8 i s s ho wn fo r t h e 2 n d seq u ence m t h e Monte Ca rl o run o f q c E S 1 5 ( 1.1 5) 15 I I I I I , I I 10'"" .. .. .. u a, Cl) --Ol 5 a, "' a, \ (\ iii a: C 0 0 'B a, 0 0 -5 iii > a, LU -10 . -15~-~'_...__._.___..__.._.__.__.........._,'-"-'--_.__..__,,_,_.____._._.L......L.,_...___ ............. ..w._ _.__.__~~_._._~~ LJ 0 2 3 4 5 6 7 8 9 10 Time (seconds) F i gure 4 .1 4: T h e e l evator d e fl ect ion r ate 8 i s s hown for t h e t h e 2 nd seq u ence i n the Monte Ca rlo r u n o f q c E S i s( l.1 5) PAGE 59 51 From Table 4.4, the maximum value of lqc n1 E'll(qc)I for nLP = 15 rad/sec is 1. 765. This corresponds to the 2 nd sequence in the Monte Carlo run. In the following subsection we will calculate the estimate of the performance bound P(E) over the optimization set S 15 (1.15). A first estimate is P(E) 2: 1.765. In Figure 4.11 the 2 nd sequence of qc, the filtered square wave, is plotted with the corresponding shifted sequence E'll(qc) which is shifted one step to the left by 40 milli-seconds. The absolute pitch rate error sequence is plotted in Figure 4.12. The elevator deflection 8 and elevator deflection rate 8 are plotted in Figure 4 .13 and Figure 4.14, respectively. As can be seen from Figure 4.12, the maximum error occurs at 8.0 seconds. Examining the plots between 7 and 8 seconds reveals that the pitch rate response is not keeping up with the pitch rate command. The elevator deflection is never position constrained but is rate constrained from 7.44 to 8.12 seconds for a total of 680 milli-seconds. As seen from this example, when we increase the bandwidth of the pitch rate command from 10 to 15 rad/sec, the rate constraints become effective and thus the measure of right singularity is increased. 4.4.2 The Estimate of the Performance Bound P(E) In this section we will find an estimate of the performance bound P(E) for the longitudinal contro l problem of an aircraft by using the optimization definition as follows: P(E) = sup lqc n1 E'll( qc) I q c ES1 s (l.lS) ( 4.36) where the system E and an approximate right inverse system '11 are illustrated in Figure 4.9 and Figure 4.10, respectively. The discussion following Theorem 4.2 .1 in Section 4.2 suggests that the maximum of the right hand side of ( 4.36) is found by evaluating all extreme points and then finding the maximum. In Section 4.4.1 25 random extreme points out of possible 2 250 (1.8903 x 10 75 ) extreme points were evaluated with the conclusion P(E) 2: 1. 765. PAGE 60 qc = 1. 1 5 qc = -1.15 0 I i+-N; -----. I I I A r ''v-----1 I I I I I .. Figure 4.15: Signal Construction's Nomenclature 52 It is time to take advantage of the traits of the system. The first block of Figure 4 10 calculates the raw elevator deflection 8~ for a given pitch rate command qck. In mathematical terms it is so l ving for 8~ in the following equations: M' k M~ (Cm+ Cm 0 0'.k-l + Cmqqk-1 + Cm o a (Uk-I, Wk-1, 8~) + Cm68~) Q k-1 Sc. ( 4.37) ( 4.38) As can be seen from ( 4 38), the raw aerodynamic moment M~ is affine with respect to the dynamic pressure Qk-l It turns out that it is also affine with respect to the raw e l evator deflection 8~. Thus, 8~ is inverse l y proportional to Qk-I and decreasing dynamic pressure causes the elevator deflection to increase. This increases the likeli hood for the elevator deflection to be rate constrained. Therefore, the optimization region of ( 4.36) can be reduced to a subset of S 15 (1.15) which contains a lower than initial dynamic pressure after a period of flight. A pitch-up command say a com mand q c = 1.15 deg/sec for at l east 2 seconds after initialization, will decrease the dynamic pressure. PAGE 61 53 Two problems of maximization of convex functions are: 1. Once a local maximum has been found. There is no local information to tell you how to proceed to a higher local maximum. 2. There is no local criterion for deciding whether a given local maximum is really the global maximum. These features are also applicable to the maximization of non-convex function since its maximum for simulation purposes will be an extreme point. The signal construction and the grid search method will be techniques used to find a higher local maximum to get around the first problems. These two methods are explained in the following paragraphs. After an extensive search for the signal that maximizes ( 4 36) and an appropriate exiting condition we can say that we have qualitatively found the global maximum. This alleviates the second problem. The signal construction will always be done in the pre-filtered space S(l.15) with out any loss of generality since the pre-filtered space S(l.15) is isomorphic to the :filtered space S 15 (1.15). The baseline signal will be q c = 1.15 deg/sec for the entire simulation of 10 seconds. The construction of the signals begins with a pulse q c = -1.15 for N p 1 steps and starting at N 1 steps. An optimization takes place over N 1 and Np 1 The signal construction s nomenclature is shown in Figure 4.15. A second pulse is constructed with Nu 2 steps at q c = 1.15 and Np 2 steps at q c = -1.15. The optimization takes place over N 2 Nu 2 and Np 2 with Np 1 fixed. This procedures continues until the (i + 1)-th pulse optimization does not produce higher maxima. A signal with i pulses has 2i parameters (N i, Np 1 , NPi Nu 2 .. Nu i )In the search grid method the starting parameter N i (see Figure 4.15) of the pulse train is still optimized over its range. The other 2i 1 parameters ( N p 1 ... N P i, N u 2 ... N U i ) are varied over a selected range. The selected range is based on the available PAGE 62 2 5.-------r---~----r---~----r-----~----~-2 Q) Q) c, 1.5 Q) 1 1 -..!... -0 C 0.5 E 0 (.) 0 -0 C 3l -0.5 C 0 a. gi -1 a: Q) i a: -1.5 .c .B a: -2 hl -2 5~-~--~-~-~~-~-~--~-~--~-~ 0 2 3 4 5 6 7 8 9 10 Time (seconds) 54 Figure 4.16 : The shifted pitch rate response q = n1 E\Jf ( q c) and the co mmand q c E S 15 (1.15) are shown for the first pulse optimization (N 1 = 242 Np 1 = 4). 3 ,------r---,------r---,------r-----,,---~----~-... e w Q) 2.5 i 1 5 a: .c .B a: Q) :i 1 0 rJ) .c 0 5 0 '-------'---'--------'---'--------'---'--------'---'--------'---'-'L.J 0 2 3 4 5 6 7 8 9 10 Time (seconds) [dt=0 04 sec] Figur 4.17: The absolute pitch rate error, lq ck n1 E\Jf(q ck) I i s shown for the first puls e optimization (N 1 = 242, Np 1 = 4) of q c E S 15 (1.15) wher e lq c -D1 E\Jf(q c) I = 1.6 46 PAGE 63 (J) Q) Q) o, Q) 7 ._.!., -0 C ra E E 0 (.) -0 C ra 2 5 2 1 5 0 5 0 5l 0 .5 C 0 C. gi -1 a: Q) iii a: -1 .5 .c a: 2 .. .. .. ... ... .. ... I I I I I I I ..... I I I I I I I I J I f I I I : 1 I I I .I I I I I I I ..... I 1 I 2 _5~-~--~-~ -~ -~--~-~--~-~ -~ 0 2 3 4 5 6 7 8 9 1 0 Ti me ( seconds ) 55 Figur e 4. 1 8 : Th e s hift e d pit c h rat e r e spons e q = n1 ~W ( q c ) and t h e c ommand q c E S 1 5 ( 1.15 ) ar e s hown for th e s ec ond puls e optimization ( N 2 = 222 Nu 2 = 7 N p 2 = 7 ). 3 ~ -~,--~, --~, -~ -~ ,---~ -~ --~ ~-~ -~ 2 5 ~ .... u Q) Q) 2,.. ...... .... e w Q) iii 1 5 a: .c a: Q) :5 1 ,.. 0 (J) .0 <( 0 5~ .. o ~ -~ '-~ '--~ --~-~--~; I I I o 2 3 4 5 6 Time (seconds) (dt=0 04 sec] I I r 7 8 9 10 Figur e 4 .19: Th e absolute pitch rate error l q c k n 1 ~W ( q ck ) I is shown for th e sec ond pulse optim i zation (N 2 = 222 Nu 2 = 7 Np 2 = 7) of q c E S 1 5 (1.15) wh e r e lq c n1 ~w( q c ) I = 2. 713 PAGE 64 56 2 5 I I I I I I I 2 ...... IJ) Q) ,t 1 5 Cl Q) I i , 11 I I -0 I I I I C PAGE 65 2 5 I I I I I 2fen Q) 1 5 f0) Q) 1J -..!.., "C C Ill 0 5 E E 0 (.) 0 "C C Ill 5l -0.5 fC 0 a. GJ en -1 fQ) a: Q) PAGE 66 1.5 ,------,----r--------,---~-~--~--~----,---~-~ 0 5 0 en Q) "O -0 5 0 t5 -1 Q) 0 0 cij -1.5 > Q) [iJ -2 -2 5 -3 -3 5 '-------'-----'------'----'--------''---_.._ __ .,____ _______._ __ _.__ ___, 0 2 3 4 5 6 7 8 9 10 Time (seconds) 58 Figure 4.24: The e l evator deflection 8 is shown for the search grid method optimiza tion (N3 = 215 Np1 = 2 Nu2 = 7, Np2 = 5 Nu3 = 7 Np3 = 8) of q c E Sis(l.15). 15.------,, --~,--~,--~,--~,.----~,----,-,--~,-~~,~---,-~ 10 u Q) -!!! g> 5 >. Q) PAGE 67 59 computing power. All points in the selected range are evaluated and then the max imum is found. The maximum of the search grid method is qualitatively the global maximum provided that the selected range is suitable. The results of the nonlinear programming algorithm, which is the integration of the signal construction and the grid search method, are shown in Table 4.5 and Figures 4.16 4.25. Figure 4.16 and Figure 4.17 represent the first pulse optimization with plots of the equences q c, n1 Ew(q c) and lq c k n1 Ew(q ck) 1. Figure 4.18 and Figure 4.19 represent the second pulse optimization with plots of th e sequences q c, n1 Ew(q c) and lq ck n1 Ew(q ck) 1. Figure 4.20 and Figure 4.21 represent the third pulse optimization with plots of the sequences q 0 n1 Ew(q c) and lq ck n1 Ew(q ck )1. The third pulse optimization parameters are (Np 1 Nu 2 Np 2 Nu 3 Np 3 ) = ( 4, 7 7 8 7). The search grid was taken to be (2: 6 5: 9 5: 9 6: 10 5: 9) which yields 3125 grid points that have to be optimized over the variable N 3 Th e r es ult of the search grid method and the estimate of the performance bound are: (N3, Np1, Nu2, Np2, Nu3, Np3) P(E) (215 2 7 5, 7 8) 2.908. The search grid sequence of q c E S 15 (1.15) is examined. In Figure 4.22, the se quence q c the filtered square wave is plotted with the corresponding shifted se quence Ew(q c) which is shifted one step to the left by 40 milli-seconds. The absolute pit c h rate error sequence is plotted in Figure 4.23. The elevator deflection 8 and elevator deflection rate 8 are plotted in Figure 4.24 and Figure 4.25 respectively. As ca n be see n from Figure 4 23, the maximum error occurs at 9.88 seconds. Examin ing the plots between 8.6 and 10 seconds reveals that the pitch rate r es ponse do es not keep up with the pitch rate command. The elevator deflection is never position PAGE 68 60 constrained but is rate constrained for a total 1.2 seconds out of the last 1.4 seconds of simulation. Table 4.5: The Estimate of the Performance Bound r>(LJ) for the Aerodynamic Model Example Optimization Method Parameters lqc n1 LJ\ll(q c) I (Ni, Np1, Nu2, Np2, Nu3, Np3) for q c E Sis(l.15) First Pulse (242,4) 1.646 Second Pulse (222,4, 7, 7) 2.713 Third Pulse (195,4 7, 7,8, 7) 2.856 Search Grid (215,2, 7,5 7,8) 2.908 i>(LJ) = 2.908 4.5 Multivariable Process Control Model Example This section is concerned with multivariable control of a plant consisting of a cylindrical tank containing a water column pressurized by air. This example comes from [10 11] wherein the authors developed a multi variable predictive control strategy with a variable receding horizon. The process control model consists of a cylindrical tank which contains a water column pressurized by air. Figure 4.26 is a simplified process diagram. A centrifugal pump feeds water into the tank through the pneumatic valve 1 and a compressor feeds air through the pneumatic valve 2. Liquid can drain from the tank through the manual valve 3 while air can exit through the relief valve 4. Measurements of the process output the height of the liquid column y 1 and the gauge pressur of the enclosed air y 2 are obtained by means of pressure transducers. Valves 1 and 2 are manipulated through electro-pneumatic transducers by voltage signals u 1 and u2. The height of the tank is 105 cm and the diameter is 7.5 cm. The dynamics of the process control model can be described in terms of four states. Let x 1 denote the liquid-level height [m], x 2 the air pressure inside the column PAGE 69 61 valve 2 Air DPT Y 1 valve 1 Figure 4.26: The Multivariable Process Control Model with inputs: valve signals u 1 and u 2 ; with outputs: liquid level y 1 and the air pressure y 2 [bar gauge], x 3 the opening of water valve 1 (lift) [dimensionless], and x 4 the opening of air valve 2 (lift) [dimensionless]. The equations of motion of the process control model are given by: 1 ( 4.39) X1 S (q1J q1d) 1 + X2 ( 4.40) X2 S(L xi) [(q11 q1d) + (qgf qgd)] 1 ( 4.41) X 3 (g1U1 X3) T/ 1 ( 4.42) X4 (g U2 X4) T g g where L is the height of the column and S is the cross-sectional area of the column. The volumetric flow rates of the liquid feed and liquid discharge streams are repr e sented by the variables q11 and q1d respectively; and the gas feed and gas discharge rates by the variables q 9 J and q 9 d, respectively. The liquid and air flow rates are modeled using a steady state relationship appropriate for the square-root and equal-percentage characteristics of the valve PAGE 70 used in the process control model: x3K11JPtJ x2 p1G(x1 + ~L') X/dKtd X2 + p1G( X1 + ~L") K ( ) 0.65 X4 gf PgJ X2 62 ( 4.43) ( 4.44) ( 4.45) ( 4.46) where the constants Kij represent the valve coefficients, G is the acceleration of gravity, Pt is density of the liquid p 9 J is the inlet pressure of the air PtJ is the inlet pressure of the liquid, ~L' is the height difference between valve 1 and the bottom of the column and ~L" is the height difference between valve 3 and the bottom of the column. Specific values and definitions of all the multivariable process control mod e l parameters are given in Table 4.6. The states and inputs must satisfy the constraints ( 4.4 7) where u 1 = u 2 = lO[V] (the maximum signal voltage) and x 1 and x 2 are bounds established by the user. The bounds of the output vector are identical to thos e of the states x 1 and x2 because y 1 = x1 and Y2 = x2. Table 4.6: The Multivariable Process Control Model Parameters Parameter Value Parameter Value s 0.0043 m 2 Ktd 4.0 X 104 m 3 s1 -~ L 1.05 m Kif 6.9 X 104 m 3 s1 -~ ~L' 0.16 m Kgd 9.8 X 104 m 3 s1 -~ ~L" 0.40 m Kgf 3.1 X 104 m 3 s1 -~ Ptf 0.5 bar Tt 1.5 s Pt 1.0 x 10 3 kgm3 91 0.1 v1 Pgf 0.4 bar Tg 0.3 s gg 0.1 v1 PAGE 71 63 The equations of motion are discretized using first order Euler integration. Let Xk = [x 1 1c x 2 1c x 3 1c x 4 1c]'. Equations ( 4.39 ) through ( 4.42) take the following form: ( 4.48) ( 4.49) where Xk E ~4, and J, g 1 g 2 are vector fields on ~ 4 The system ~: 1) -t S(~ 2 ) : uc f-+ y of interest, where 1) C S(~ 2 ), is the composition of the equations of motion with the input/states constraints for which uc is the voltage command signals u 1 and u 2 Figure 4.27 shows the block diagram of the system ~. iLc1c I I I I I I I I I I I I I I I I I Input Constraints _E .... .... Uk Xk+I State Constraints _E xLk+I ... rz'Xk+I = f(xL k) + g(xL k) Uk C L-----------------------------------------------------------I :Y1c+1 1 I ... .... I I Figure 4.27: The block diagram of the system ~: 1) -t S(~ 2 ) uc f-+ y for the Multivariable Process Control Model Example. 4 .5 .1 Simulation The signals of interest are the liquid-level height y 1 and the gauge pressure Y2 of the enclosed air. The bounds of ( 4.4 7) have been set to x 1 = 0. 75 m and i:2 = 0.30 bar. It is assumed that all signals of interest start at their midpoints ( i. e. Y1n = 0.375 m and Y2n = 0.15 bar for all n no)From [11], the time constant of the liquid-level y 1 ( under zero gauge-pressure of air) varies from 50 70 s and the time constant of the air pressure y 2 is 1.3 s. The bandwidth of y 1 was set at 0.3 rad/s c to stress an approximate right inverse system W while the bandwidth of y 2 was set at 1/1.3 rad/sec. PAGE 72 64 The set Vs C S(~ 2 ) is defined with initial conditions y 1 = 0.375 and y 2 = 0.15 for all i ::; 0, and with IY1i 0.3751 ::; 0.357 and IY2i 0.151 ::; 0.1428, for all i > 0. The domain V11, C S(~ 2 ) of an approximate right inverse system \JI takes the sequences of Vs 's components 1 and 2 that are passed through a second order discrete time low pass filter with cutoff frequencies 0.3 and 1/1.3 rad/sec, respectively. The outputs of the filters 1 and 2 are then clipped by [0.0, 0. 75) and [0.0, 0.30), respectively. Ye1c ,--------------------------------------------------------------------------------! Input \JI I i Constraints : x,_, OptiO:~ation ~u, + I 1--~ _____ State Constraints z' x, = f(xL ,_,) + g(xL ,_,) u,. x, j f I ,___ ___. '-------------xLk-t : I ----------------------------------------------------------------------------------------1 Figure 4.28: The block diagram of an approximate right inverse system \JI: V'11 _. S(~ 2 ) : Ye 1--+ Uinv with design integer A = 2 for the Multivariable Process Control Model Example. --------------------------------------------------t u 1 & u 2 Optimization Input \JI i I Constraints State Constraints I z xk-1 = f(xL k 2) + g(xL k-2) u Ik 1 xk 1 .__ ___, '------------~xLk -2 ~ ---' : I --------------------------------------------------------------Figur e 4 .29 : The block diagram of an approximate right inverse system W: Vw _. S( ~ 2 ) : Y e 1--+ Uinv with design integer A = 3 for the Multivariabl e Process C on t rol Mod e l Example. The block diagrams of approximat e right inverse syst e ms W: Vw _. S( ~ 2 ) Y e r-+ Uinv with d e sign integers A = 2 and A = 3 are shown in Figur 4. 2 8 a nd PAGE 73 65 Figure 4.29, respective l y Two integrations of u 1 are required to reach y 1 and sim ilarly for u 2 and y 2 Two integrations of u 1 are also required to reach y 2 but three integrations of u 2 are required to reach y 1 For the design integer A = 2 case, the system W is causal. The u 1 optimization block in Figure 4.28 optimizes u 1 k to give a minimum value of IY c k -D2 Ew(y c k)I each step. For the design integer A = 3 case, the system W is strictly causal. The u 1 and u 2 optimization block in Figure 4.29 optimizes u 1 k and u 2 k to give a minimum value of IY c k -D3 E\ll(ycJ I each step. The procedure involves undertaking a u 2 k optimization if the u 1 k optimization fai l s so as to reduce the error IY c k n3 E\ll(y c k)I. The signal U2k is optimized to give a minimum value of IY c k+l n3 Ew(y c k+JI provided the result does not augment the error IY c k n3 Ew(y c k)I. For the simulation of 60 seconds a sampling period of 150 milli-seconds yields a two-dimensional sequence of 400 points. The pre-filtered binary random sequences {yick} only have values of 0.018 0.0732 m for k 2:: 1 and the pre-filtered binary random sequences {y 2 c k} only have values of 0.0072, 0.2928 bar for k 2:: 1 because we are only interested in extreme points of 1J 5 Table 4. 7 shows the Monte Carlo results for 2 different design integers A of the system W: A = 2 and A = 3 where the error I Y e n->. Ew (Ye) I is evaluated for each of the sequences. Also included in Table 4. 7 is the average number of floating point operations (flops) per Monte Carlo sequence run for the u 1 optimization block and for the u 1 and u 2 optimization block. On the average the u 1 and u 2 optimization subroutine requires 3 x greater the number of flops as the u 1 optimization subroutine, but a single execution of the u 1 and u 2 subroutine can require 4 + x greater the number of flops than the u 1 subroutine. 4.5.2 The Estimate of the Performance Bound 'P(E) In this section, we will find an estimate of the performance bound 'P(E) for the multi variable process control problem of a regulator that regulates the liquid level PAGE 74 66 Table 4.7: Family of Sequences, Dw, Comparison Error IYc n->-I:w(yc)I Average Number of Flops Sequence A=2 I A=3 A=2IA=3 1 0.02481 0.02190 137.0 367.8 2 0.03742 0.03138 140.3 449.3 3 0.01584 0.01506 142.5 396.0 4 0.05440 0.04699 150.0 432.7 5 0.02408 0.02255 135.5 447.5 6 0.02064 0.01732 137.3 441.0 7 0.02147 0.01917 136.1 379.4 8 0.02613 0.02389 140.9 426.5 9 0.01185 0.01124 135.5 401.5 10 0.00601 0.00589 139.7 374.2 11 0.02080 0.02089 144.9 459.0 12 0.03080 0.02840 137.0 439.1 13 0.01270 0.01113 139.1 404.9 14 0.02746 0.02446 135.5 456.5 15 0.03962 0.03634 144.0 513.7 16 0.02189 0.01942 140.3 413.9 17 0.01852 0.01713 134.6 454.2 18 0.01745 0.01819 136.1 467.9 19 0.01757 0.01527 138.5 441.9 20 0.01637 0.01393 143.1 399.8 21 0.03920 0.03406 144.6 435.2 22 0.03529 0.02838 138.2 418.2 23 0.01935 0.01764 140.9 464.1 24 0.01943 0.01733 138.5 420.2 25 0.01594 0.01466 139.4 393.5 max 0.0544 0.04699 150.0 513.7 mm 134.6 367.8 mean 139.6 427.9 st dev 3.703 33.90 PAGE 75 0 8 .------,------,,---......--------,----~---~ 0 7 E -g 0 6 <1l E E 8 0 5 "O C <1l Q) 0.4 0 a. !J) Q) ':: 0 3 .s= Cl ci> I :'2 0 2 :, O" :::i 0 1 I I I I I / I I I I : I .. I : I . \ : ...... 1 \ ............. l ..... : I I i I I o~---~----'----~~---~'----~---~ 0 10 20 30 40 50 60 Time (seconds) 67 Figure 4.30: For ,\ = 2 the liquid height response y 1 = n2 ~ \II (Yic) and the com mand Y1 c E Vw are s hown for the square wave optimization (N 11 = 104 N 1 2 = 103). 0 8 ,------,-----,------,------,,-----......--------, 0 7 E -g 0 6 <1l E E 8 0 5 "O C <1l Q) 0.4 0 a. !J) Q) ':: 0 3 .s= Cl ci> I :'2 0.2 :, O" :::i I / I I I I 0.1 .. / \ I .. \ ..... I I .... I I \ .. I \ : \ 0 .__ ___ __._ ____ ..___ ,:__ __ .___,:__ __ _._ ___ ____,.___ ___ _, 0 10 20 30 40 50 60 Time (seconds) Figure 4.31: For ,\ = 3, the liquid h e ight response y 1 = n3 ~\IJ(y 1 0 ) and the com mand y 1 c E 1Jw are shown for the sq uar e wave optimization (N 11 = 78 N 1 2 = 73). PAGE 76 0 3 ,-----,-=,-~-:.--r-_-_-~~------------r--------h27 -;:-0 25 Ill e "O C Ill E E 0 2 0 (.) "O C Ill 5l 0 15 C 0 a. (/) Q) a: 0.1 (/) (/) Q) a: 0 05 0 '--------'------'--"'----.,___------'-----'------' 0 10 20 30 40 50 60 Time (seconds) 68 Figure 4.32: For ,\ = 2 the air pressure response y 2 = n2 E\JJ (y 2 c) and the command Y2 c E 'Dw are shown for the sq uar e wave optimization (N 11 = 104 N 12 = 103). 0 3 ,---~~--r-----------~-------~ -;:-0 25 Ill e "O C Ill E E 0 2 0 (.) "O C Ill 5l 0 15 C 0 a. (/) Q) a: 0 1 (/) (/) Q) a: 0 05 I l h27 ~ 0 L..._------'-----'---L-L----.,___------'-----'------' 0 10 20 30 40 50 60 Time (seconds) Figure 4 .33: For,\ = 3 the air pressure response y 2 = n3 E\JJ(y 2 c) and the command Y2 c E 'Dw are shown for the square wave optimization (N 11 = 78 N1 2 = 73). PAGE 77 0.5 N ~0.4 C Ill 15 e o 3 w Q) :, 0 1l 0 2 <( 0 1 I I : I I I I i I I / \ I I ;._ \ ... ... I I I I : \ \ \ ' ' ' ~ ' I . . . ,, I I l I I I '\ I \ I ' ' ' 0 L........ PAGE 78 10 9 8 2 7 0 e6 Cl) "O C <1l E 5 E 0 (.) 4 Q) > PAGE 79 71 in a pressurized tank by using the optimization definition as follows: P(E) = sup jye n->E'll (ye) I Y c EVq, ( 4.50) where the system E and an approximate right inverse system '11 with design integers A: 2 and 3 are illustrated in Figure 4.27 Figure 4.28, and Figure 4.29, respectively. In Section 4.4.2, a nonlinear programming algorithm was developed to calcu late the estimate of the performance bound 'P(E). It exploits the traits of the system. Due to the sluggishness of the liquid height response, only one transition was needed from high to low in both components of the signal ye. The transition is at N 11 steps for y 1 e and N 12 steps for y 2 e The results of the nonlinear programming algorithm are shown in Table 4.8 and Figures 4.30 4.37. Table 4.8: The Estimate of the Performance Bound 'P(E) for the Multivariable Pro cess Control Model Example Parameters IY e n-AE'll(y e )I Design Integer (N1 1 N1 2 ) for Y e E Vw A=2 (104,103) P(E) = 0.5983 A=3 (78, 73) 'P(E) = 0.4544 Figure 4.30 and Figure 4.31 represent the optimal liquid height response and command for A = 2 and A = 3, respectively. The u 1 and u 2 subroutine improves the liquid height response for A = 3, at the expense of the air pressure response. Figure 4.32 and Figure 4.33 represent the optimal air pressure response and command for A = 2 and A = 3 respectively. The u 1 and u 2 subroutine increases the air pressure to help drain the tank while the u 1 subroutine is operating with near zero gauge pressure on y 1 which has a time constant between 50 70 s. The optimal absolute error sequences are plotted in Figure 4.34 and Fig ure 4.35 for A = 2 and A = 3, respectively. In the u 1 & u 2 subroutine, the signal u2k PAGE 80 72 is optimized to give a minimum value of IY ck +i n3 1:;w(y ck +JI provided the error IY ck -D3 I:\ll(yck)I does not augment. In the process the error IY2 ck -D3 I: \ll(y2 cJ I is augmented but never above the value of the error IYick n3 I;\lf(yick)I which i s e vident in Figure 4.35. The optimal valve commands u 1 and u 2 are plotted in Fig ur e 4.36 and Figure 4.37 for A = 2 and A = 3 respectively. For A = 2 the signal u 1 is constrained throughout most of the simulation while for A = 3 the signal u 2 which influenc es y 1 in three steps, is optimally used to reduce the error I Yi ck -D3 I: \ll(y ick) I PAGE 81 CHAPTER 5 CONCLUSION 5.1 Summary This dissertation presents new results for the problem of disturbance atten uation for nonlinear control systems. Of primary significance is the derivation of a performance bound that provides a measure of the ability to match a set by subset of the image of a system. Our bound arose from the desire of providing an estimate of the minimal effect of the disturbance signal don the output signal z of Figure 1.1. It turns out that the calcu lation of the performance bound is rather labori ous because its definition (Definition 3.4.1) uses Im E. In practice we will find an estimate of the performance bound that uses an approximate right inverse system in its calculation instead of Im E. The estimate of the performance bound involves an optimization problem which relates to finding a global maximum of a non-convex function. Nonlinear programming is used to calculate the estimate of the performance bound. Three systems of practical origin were se l ected for the analysis of the per formance bound. In the PM stepper motor examp l e, a closed form estimate of th e performance bound was found which was verified by simulation. In the aerodynamic model example, a nonlinear programming algorithm was determined to compute the estimate of the performance bound. The multivariable process model examp l e pro vided us insight into how different design integ ers can influen ce the estimate of the performance bound. 73 PAGE 82 74 5.2 Future Directions The results of this dissertation serve to enhance many of the tools used to obtain the optimal controller for nonlinear control systems. In the process of com pleting one particular objective, other promising areas frequently appear as directions for additional research. One of the more intriguing of such areas emerging from this work is the derivation of a bound that handles constant bias disturbances. Referring to Figure 1.1, and using the parameterization (1.3) with the assump tion that the system E is stable (i.e. P = E and Q = I), we get Ec(v, d) = d + E(v, d). (5.1) The stable and causal system is selected such that ( v, d) = w (Ev d) (5.2) where W is an approximate right inverse system. All that is needed to define a performance bound is an optimization set. In this case, the optimization set is the image of [Ev d] where the domain of [Ev d] is a bounded set. The performance bound is given by P(E) = (!l (Im [Ev d], n->-Im E) (5.3 ) The bound of (5.3) is a theoretical estimate of the minimal effect of the disturbance don the output z Let s use,\ = 0 initially in the bound optimization problem. Note that there is another ,\ associated with the approximate inverse optimization problem. Both are equal for most applications. In this application the design integer ,\ will be associat e d with an approximate inverse and ,\ 2: 0. Now we need to modify our definition of th norm p f. for the bound. For a sequence u E S(?Rm) and for a j 2: 0 the norm P f.,j is defined (5.4) PAGE 83 75 The class of disturbances is a constant bias; i.e., the magnitude of the disturbance is unknown, but it is constant (i.e., a step disturbance). Let us define our performance bound for ,\ = 0, denoted by R(E), such that R(E) def inf sup Pc ,A ( d + E( v, d) Ev) (5.5) where Ev is our desired response. The supremum is over the input signal space and the infimum is over the stable and causal systems . The stable and causal system can be chosen as (5.6) where the design integer A 2: 0 of an approximate right inverse system W is the least latency of the system E. It will be shown that the bound R(E) is well suited for a constant bias dis turbance d. If the system W were a right inverse of the system E, then E\ll = D A I. Let us substitute (5.6) into (5.5), we get R(E) sup Pc,A ( d + E\Il ( D-AEv d) Ev) sup Pc ,A ( d +DAI (D-AEv d) Ev) sup Pc,A ( d + (Ev DAd) Ev) sup P c,A ( d DAd) R(E) 0 where the supremum is taken over the input signal space. The bound R(E) can be calculated using techniques of Chapter 4. The op timization set is the image of [D-AEv d] where the domain of [D-AEv d] is a bounded set. 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Hunt Right coprime factorizations and stabilization for nonlinear systems" IEEE Transactions on Automatic Control, Vol. 38 No. 2 pp. 222 231 1993. [47) M. Vidyasagar Nonlinear Systems Analysis Second Edition. Englewood Cliffs NJ: Prentice Hall 1993. [48) G. Zames "On the input-output stability of time-varying nonlinear feedback systems, Part I ", IEEE Transactions on Automatic Control Vol. AC-11 No. 2 pp. 228 238 1966. [49) G. Zames On the input-output stability of time-varying nonlinear feedback systems, Part II ", IEEE Transactions on Automatic Control, Vol. AC-11 No 3 pp. 465 476, 1966. [50) G. Zames, "Feedback and optimal sensitivity: Model reference transformations multiplicative seminorms, and approximate inverses ", IEEE Transact i ons on Automatic Control, Vol. AC-26, No. 2, pp. 301 320, 1981. PAGE 88 BIOGRAPHICAL SKETCH Rafael J. Fanjul Jr. was born in West Palm Beach, Florida, on May 28, 1963. He received his bachelor's degree in mechanical engineering from Georgia Institute of Technology in 1985. He then entered graduate school at Georgia Institute of Technology and received a master's degree in e l ectrica l engineering in 1986. From 1986 to 1992 he was employed by Martin Marietta Orlando Florida as a guidance and contro l engineer. During his tenure at Martin Marietta he attended night chool at the University of Centra l Florida and received a master's degree in mathematical science in 1992. He has been a Professional Engineer in the State of Florida since September 1990. In August 1992, he was granted an educationa l l eave of absence from Martin Marietta to pursue a doctoral degree. Since August 1992, he has been a Ph.D. student and pre-doctoral fellow at the University of Florida under the supervision of Professor J. Hammer. 80 PAGE 89 f Cr t i fy L ha L I ha r a cl t, hi St LI cl y a 11 I t hat i 11 Ill~ 0 p 1l1 I O ll i ( CO 11 for Jl7 S t 0 ace ptable tandards of sc holarly pr e entation and i fully a PAGE 90 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy . -;;;e Z' ~;;y Carla A. Schwartz Assistant Professor of Electrical and Computer Engineering This dissertation was submitted to the Graduate Faculty of the Department of Electrical and Computer Engineering in the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1996 Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School PAGE 92 LD 1 8 990 I F / UNIVERSITY OF FLORIDA II I II IIIIII Ill Ill lllll lllll II llllll llll llll llll 11111111111111111 3 1262 08554 4434 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EIVXU6IDY_NARWT7 INGEST_TIME 2013-02-14T14:31:39Z PACKAGE AA00013540_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |