- Email: [email protected]
Actuator saturation control V. Kapila, K.M. Grigoriadis; Marcel Dekker, New York, ISBN 0-8247-0751-6
762
Book reviews / Automatica 39 (2003) 757 – 765
Kimura, H. (1989). Conjugation, interpolation and model-matching in H∞ . International Journal of Control, 49, 269–307. Kimura, H. (1997). Chain-scattering approach to H∞ -control. Boston: BirkhRauser. Liu, K., Chen, B. M., & Lin, Z. (2001). On the problem of robust and perfect tracking for linear systems with external disturbances. International Journal of Control, 74(2), 158–174. Maciejowski, J. W. (1989). Multivariable feedback design. Reading, MA: Addison-Wesley Publishing Co. McFarlane, D., & Glover, K. (1992). A loop shaping design procedure using H∞ synthesis. IEEE Transactions on Automatic Control, 37, 759–769. Mustafa, D., & Glover, K. (1990). Minimum entropy H∞ control. In Lecture notes in control and information sciences, Vol. 16. Berlin: Springer. R Nevanlinna, R. (1919). Uber beschrRankte Funktionen, die in gegeben Punkten vorgeschriebenne Werte annehmen. Annales Academiae Scientiarum Fennicaeo, 13, 27–43. R Pick, G. (1916). Uber BeschrRankuungen analytischer Funktionen, welche durch vorgegebene Funktions wert bewirkt sind. Mathematischen Annalen, 77, 7–23. Stoorvogel, A. (1992). The H∞ control problem: A state space approach. Englewood CliBs, NJ: Prentice-Hall. Subrahmanyam, B. (1995). Finite horizon H∞ and related control problems. Boston: BirkhRauser. Van Keulen, B. (1993). H∞ control for distributed parameter systems: A state space approach. Boston: BirkhRauser. Wieland, S., & Willems, J. C. (1989). Almost disturbance decoupling with internal stability. IEEE Transactions on Automatic Control, 34, 277–286. Youla, D. C. (1961). A new theory of cascade synthesis. IRE Transactions on Circuit Theory, 9, 244–260.
Zames, G. (1966). On the input–output stability of time-varying nonlinear feedback systems, Part I and II. IEEE Transactions on Automatic Control, 11, 228–238, 465 – 476. Zames, G. (1981). Feedback and optimal sensitivity: model reference transformations, multiplicity seminorms, and approximation inverses. IEEE Transactions on Automatic Control, AC-26, 301–320. Zhou, K., Doyle, J. C., & Glover, K. (1995). Robust and optimal control. Englewood CliBs, NJ: Prentice-Hall.
About the reviewer Patrizio Colaneri was born in Palmoli, Italy on 12 October 1956. He received the Doctor’s degree (Laurea) in Electrical Engineering in 1981 from the Politecnico di Milano, Italy, and the Ph.D. degree (Dottorato di Ricerca) in Automatic Control in 1987 from the Ministero della Pubblica Istruzione of Italy. From 1982 to 1984, he worked in industry on simulation and control of electrical power plants. From 1984 to 1992, he was with the Centro di Teoria dei Sistemi of the Italian National Research Council (CNR). He spent a period of research at the Systems Research Center of the University of Maryland and held a visiting position at the Johannes Kepler University in Linz. He is currently Professor of Automatica at the Faculty of Engineering of the Politecnico di Milano. Dr. Colaneri was a YAP (Young Author Prize) Cnalist at the 1990 IFAC World Congress, Tallin, USSR. He is a member of the IFAC Technical Committee on Robust Control, the chair the IFAC Technical Committee on Control Design and a senior member of the IEEE. He was a member of the International Program Committee of the 1999 Conference of Decision and Control. Dr. Colaneri serves as Associate Editor of Automatica. His main interests are in the area of periodic systems and control, robust Cltering and control, and digital and multi-rate control. On these subjects, he has authored or co-authored about 120 papers and the book “Control Systems Design: an RH-2 and RH-inCnity viewpoint”, published by Academic Press.
doi:10.1016/S0005-1098(02)00282-0
Actuator saturation control V. Kapila, K.M. Grigoriadis; Marcel Dekker, New York, ISBN 0-8247-0751-6 1. Introduction Designing a control system when the actuators are subject to hard constraints is a fundamental problem. As it is well known, actuator limitations are not only one of the main sources of performance degradation, but they may even cause fatal consequences in several situations. The purpose of the present book is to provide an overlook on the current activity and research direction on the topic of constrained system control. The book is divided in chapters written by diBerent authors. This review will describe the contents of each chapter with speciCc comments in the next section. An overall discussion on the book will be provided in the concluding section. 2. Book contents In Chapter 1, the stabilization of exponentially unstable systems in the presence of both magnitude and variation rate constraints is considered. The basic idea to cope with
rate constraints is to include in the model the additional diBerential equation ˙ = R sgn M sat u(t) − (t) ; (t) M where R is the maximum variation rate and M is the maximum amplitude bound, u(t) is the desired (unconstrained) control value and (t) is the actuator control value. The system space is partitioned in the unstable and the stable subspace associated with variables xs and xu , respectively. It is known that the limitations due to the constraints aBect the unstable components xs only. It is shown that if there exists a certain domain U, subset of the (xu ; )-space, and a control that renders such a domain positively invariant for the closed-loop system and asymptotically stable, then there exists a control which tracks an asymptotically constant reference signal as long as such a reference converges to a constraint-admissible point, for every initial state in U. This result is related to that presented in the recent paper (Blanchini & Miani, 2000). The chapter presents also a signiCcant application of the technique to the control design for an unstable aircraft. Chapter 2 faces the problem of determining the most appropriate level of actuator saturation in the presence of gaussian noise. Precisely, for an SISO system the actua-
Book reviews / Automatica 39 (2003) 757 – 765
tor saturation level is deCned as the parameter in us =
sat(u= ). Such parameter is to be chosen as to guarantee that the standard deviation of the output does not exceed a prescribed value. The main result is derived by simplifying the nonlinear term eBects through a stochastic linearization method and leads to a compact formula relating the standard deviation of the output in the presence of actuator saturation and the standard deviation when no saturation is present. The overall derivation is based on this non-rigorous linearization and a set of system norm inequalities, resulting in a simple and ePcient design tool, as evidenced by the examples. In Chapter 3, the problem of determining the maximal domain of attraction for linear unstable systems is considered. Precisely, given the n-dimensional single-input continuous-time system x˙ = Ax + bu with the bounds u ∈ U = [u− ; u+ ], the paper is aimed at determining the null controllable set C. After presenting some introductory results, related to the symmetric case, they consider the problem of determining such set for two-dimensional unstable systems when the bounds on the input are asymmetric. The overall result is that, similar to the symmetric case, the null controllable set can be arbitrarily closely approximated by that of a linear saturated system whose linear gain is computed by solving a Riccati equation. The presentation is very clear and the contribution is almost self-contained. A Cnal interesting point concerns the reviewers’ conjecture of obtaining the presented result by (i) proper symmetrization (if applicable) of the input, (ii) exploiting the results recently reported in Blanchini and Miani (2000) (see also Chapter 1 of book). In Chapter 4, the problem of stabilizing a system with constrained control in order to assure a certain degree of L2 performances is faced. Two methods are exploited: the circle method (which basically deals with saturation as a sector-bounded nonlinearity) and the so-called linear analysis method which consists in determining a region in which a linear controller does not saturate. As an interesting point, it is shown that although the linear method is potentially more conservative for performance analysis, the two methods are “equivalent” in terms of achievable performances if used to synthesize the control. In view of the above result, a linear synthesis technique based on LMI conditions is investigated. To further enhance the control performance the authors consider a switching control strategy pursuing the idea of Wredenhagen and Belanger (1994) (as pointed out in the book) and provide an upper bound for the cost. In Chapter 5, some recent developments in the treatment of gain-scheduled LPV systems subject to position and rate actuator saturation are presented. The main focus is that of providing a stabilizing controller guaranteeing a prescribed L2 -gain margin, based on LMI techniques. Precisely, the problem is tackled by transforming the actuator saturation in a parameter varying term and by Crst using a parameter independent Lyapunov function and subsequently exploiting the less conservative parameter dependent Lyapunov
763
functions (see also Mori and Kokame (2000) for an analysis tool for parameter-dependent uncertain systems). Rate bounds are also analyzed and, under non-restrictive assumptions, it is shown how to recast such problem in one of the already analyzed ones. Finally, when full state measurement is available, the idea of scheduling the controller not only on the plant time-varying parameter, but also on the “distance” of the state from the origin (an idea also pursued in Wredenhagen and Belanger (1994), see also Chapter 5 of this book), is recalled and its advantages over the previously analyzed schemes are evidenced. The author also evidences some problems in the implementation of this latter control scheme, along with an illustrative example showing its effectiveness. In Chapter 6, the stabilization problem in the presence of amplitude and rate nonlinearities is considered. These nonlinearities are sector bounded and are dealt with uncertain time-varying parameters. A suPcient condition is given for quadratic stability which leads to a global stabilizability condition in both state and output feedback case. As observed, these results which are global, can be adapted to achieve local results when dealing with saturation type nonlinearities and exponentially unstable systems. As claimed by the authors the major feature of the paper is to consider discrete-time systems instead of continuous-time systems since the discrete-time case received less attention in the literature. We share this opinion, although we would like to point out some additional references such as Tarbouriech and Garcia (1997, 1998). In Chapter 7, the authors, after an extensive presentation of existing literature on input-saturated systems, face the problem of improving the performances of linear systems with input saturation by means of the Popov criteria. After considering the classical problem of determining an output feedback controller that maximizes the (ellipsoidal) set of states which can be asymptotically driven to the origin when a saturation is present (i.e., no disturbance aBects the system) they consider mixed L2 =L∞ optimization problems. They show how to determine good suboptimal approximations of the maximal set of initial conditions and of the suboptimal mixed controller, exploiting coupled BMI and LMI equations, whose underlying common denominator is the Popov stability criteria. In Chapter 8, some results concerning the problem of output regulation with constraints are proposed. This work is along the line proposed by the authors in a previous book (Saberi, Stoorvogel, & Sannuti, 2000). The regulation problem consists in zeroing the system output, in the presence of disturbances which are generated by a Cctitious linear system named exosystem, whose model is known but with unknown initial conditions. The novelty consists in the problem setup which incorporates a constrained output which takes into account both input and state variable constraints. The basic problem considered is the global (respectively, semiglobal) stabilization under constraints. The solvability of the
764
Book reviews / Automatica 39 (2003) 757 – 765
problem is stated in some theorems, which involve the so-called weak non-minimum-phase condition, namely, the non-existence of exponentially unstable zeros. Some critical assumptions such as the so-called right invertibility of the constraints are analyzed. The case of constraints which are non-minimum phase for which the global or semiglobal problems are not solvable is Cnally considered. The chapter presentation is clear and mathematicall sound. In Chapter 9, the problem of the directionality compensation for input constrained system is considered. The work starts by observing that in the presence of a control which is componentwise saturated there are diBerent ways to associate a feasible input to the desired input (clearly the problem does not hold for single-input systems). For instance, the feasible input achieved by clipping each saturated component of the “desired control vector” does not preserve its direction. Therefore, the question how we can associate a feasible input to the one computed by a controller arises (whenever the latter is infeasible). After presenting some possible directionality compensation strategies, the authors analyze the optimal directionality compensation previously investigated in a paper of their own. They also propose a diBerent scheme for windup compensation based on input–output linearization, under the assumption of minimum-phase (possibly nonlinear) system. It is claimed that the systems with the proposed control satisCes some optimality conditions and it is locally stable (point (c) Theorem 9.2). However, the details on this last claim are omitted, and the reader is referred to a diBerent source. In Chapter 10, position and rate constraints are considered, and the main focus is the robust local stabilization of norm-bounded (somehow structured) uncertain systems via an output feedback controller having the same order as that of the plant under consideration. The nonlinear terms are treated as sector-bounded nonlinearities, leading to a polytopic representation of the overall uncertain system to be controlled. After the necessary mathematical preliminaries, a non-straightforward result based on the solution of two coupled Riccati equation is presented, a result which furnishes a suPcient condition for the robust stabilization problem to have a solution. The conservativity deriving from using a parameter-independent Lyapunov function is subsequently enhanced resorting to matrix inequalities techniques, whose computational issues are also analyzed. The resulting technique explicitly provides a guaranteed ellipsoidal domain of attraction to the origin. In Chapter 11, control saturation is modeled as a time-varying parameter and the control synthesis problem is embedded in a gain-scheduling design. Known LMI conditions, are then invoked in order to provide a compensator assuring that the closed-loop system satisCes a degree of performance speciCed by the designer. The technique is applied to an interesting ?ight control problem, in which actuator saturation is well known to be a crucial problem in which an inaccurate design may have severe consequences.
A potential weakness of the approach is that a priori assumptions on the saturation level (namely the bounds for the saturation indicator parameter) have to be made. These assumptions must be veriCed a posteriori, by simulation (see Fig. 6 p. 291 and the comments below). This leaves open the necessity of a trial-and-error procedure, in the case in which the foreseen level of saturation is lower than the actual one. 3. Discussion As in the spirit of the Series, these books are contributed by several authors who normally present their own research results and, therefore, their personal point of view. Clearly, the overall quality of these books strictly depends upon the choice of the contributors. From this point of view, the Editors did a good job, since all the authors are well qualiCed. There are several nice connections established in the book between concepts derived from the robust control, such as the circle criterion or the idea of modeling input saturations via time-varying parameters, and the problem of input saturation. Also, in most chapters, the signiCcant problem of simultaneous presence of amplitude and rate constraints is analyzed. The perspective oBered to the reader in terms of variety of approaches is quite wide, although the reviewers encountered a certain prevalence of methods based on LMIs, which is quite reasonable in view of the popularity of these techniques. As a result, most of the proposed approaches lead to algorithms which are practically implementable with ePcient tools. The derived compensators are of limited complexity and suitable for industrial applications. In some way the dominance of sections based on LMI and Riccati equation tools distracted the attention from other approaches which are, from the reviewers’ point of view, equally interesting for the problem. Clearly a certain level of arbitrariness in the choice of the material is welcome but the reviewers would have appreciated mentioning also other synthesis techniques, in particular the so-called model predictive control (Mayne, 2001), the L1 approach (Dahleh & Pearson, 1987), and the wider area based on dynamic programming techniques (Bertsekas, 2000; Glover & Schweppe, 1971). Although some of these approaches are older and well established in the literature (besides requiring a heavy computational load), we think that they still provide some conceptually important properties. Also the approach based on the explicit determination of the (possibly largest) null controllable set (also known as domain of attraction), basically the invariant set approach (Blanchini, 1999), is essentially limited to a pair of chapters although the concept of domain of attraction is often implicit in the quadratic Lyapunov functions computed via LMIs. This consideration might allow to face some problems from another perspective.
Book reviews / Automatica 39 (2003) 757 – 765
To conclude, we think the material covered in this book gives a quite extensive overview of currently used synthesis techniques for actuator saturated systems, although, as a pure question of taste, the reviewers would have preferred to see a little bit more emphasis on other, equally important, existing synthesis techniques for input saturated systems. Franco Blanchini E-mail address: [email protected] Stefano Miani Dipartimento di Matematica e Informatica University of Udine Via Delle Scienze 208 33100 Udine; Italy E-mail address: [email protected] References Bertsekas, D. P. (2000). Dynamic programming and optimal control. Belmont, MA: Athena ScientiCc. Blanchini, F. (1999). Set invariance in control—a survey. Automatica, 35(11), 1747–1767. Blanchini, F., & Miani, S. (2000). Any domain of attraction for a linear constrained system is a tracking domain of attraction. SIAM Journal on Control and Optimization, 38(3), 971–994. Dahleh, M. A., & Pearson, J. B. (1987). l1 -Optimal feedback controllers for MIMO discrete-time systems. IEEE Transactions on Automatic Control, 32(4), 314–322. Glover, D., & Schweppe, F. (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Transactions on Automatic Control, 16(5), 411–423. Mayne, D. Q. (2001). Control of constrained dynamic systems. European Journal of Control, 7, 87–99. Mori, T., & Kokame, H. (2000). A parameter-dependent Lyapunov function for a polytope of matrices. IEEE Transactions on Automatic Control, 45(8), 1516–1519. Saberi, A., Stoorvogel, A., & Sannuti, P. (2000). Control of linear systems with regulation and input constraints. London: Springer. doi:10.1016/S0005-1098(02)00281-9
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Tarbouriech, S., & Garcia, G. (1998). Local stabilization for linear discrete-time systems with bounded controls and norm-bounded time-varying uncertainty. International Journal of Robotics Nonlinear Control, 8(10), 831–844. Tarbouriech, S., & Garcia, G. (Eds.) (1997). Control of uncertain systems with bounded inputs, Lecture Notes in Control and Information Sciences, Vol. 227. London: Springer. Wredenhagen, G. F., & Belanger, P. R. (1994). Piecewise-linear LQ control for systems with input constraints. Automatica, 30(3), 403–416. About the reviewers Franco Blanchini was born in Legnano (MI) on 29 December 1959. He is Full Professor at the Engineering Faculty of the University of Udine where he teaches Dynamic System Theory and Automatic Control. He is a member of the Department of Mathematics and Computer Science of the same university and is also the Director of the System Dynamics Laboratory. He has been Associate Editor of Automatica since 1996. In 1997, he was member of the Program Committee of the 36th IEEE Conference on Decision and Control, San Diego, CA. In 1999, he was a member of the Program Committee of the 38th IEEE Conference on Decision and Control, Phoenix, AR. In 2001, he was a member of the Program Committee of the 40th IEEE Conference on Decision and Control, Orlando, FL. He was Chairman of the 2002 IFAC workshop on Robust Control, Cascais, Portugal. He is the recipient of 2001 ASME Oil & Gas Application Committee Best Paper Award as a co-author of the article “Experimental evaluation of a High-Gain Control for Compressor Surge Instability.” He is the recipient of the 2002 IFAC prize survey paper award as author or the article “Set Invariance in Control—a survey,” Automatica, November, 1999. Stefano Miani was born in Parma, Italy, in 1967. He received the Electrical Engineering Laurea degree summa cum laude from the University of Padova, Italy, in 1993 and the Ph.D. degree in Control Engineering from the same university in 1996. From 1996 to 1997, he was a Lecturer at the University of Udine, in 1997 he joined the Department of Electronics and Computer Science (DEI), University of Padova as a Researcher. Since November 1998, he has been a Researcher with the Department of Electrical, Mechanical and Management Engineering (DIEGM), at the University of Udine, Italy, where he is teaching Automatic Control courses. His research interests include the areas of constrained control, l∞ disturbance attenuation problems, gain scheduling control and uncertain and production–distribution systems via set-valued techniques.
Book reviews / Automatica 39 (2003) 757 – 765
Kimura, H. (1989). Conjugation, interpolation and model-matching in H∞ . International Journal of Control, 49, 269–307. Kimura, H. (1997). Chain-scattering approach to H∞ -control. Boston: BirkhRauser. Liu, K., Chen, B. M., & Lin, Z. (2001). On the problem of robust and perfect tracking for linear systems with external disturbances. International Journal of Control, 74(2), 158–174. Maciejowski, J. W. (1989). Multivariable feedback design. Reading, MA: Addison-Wesley Publishing Co. McFarlane, D., & Glover, K. (1992). A loop shaping design procedure using H∞ synthesis. IEEE Transactions on Automatic Control, 37, 759–769. Mustafa, D., & Glover, K. (1990). Minimum entropy H∞ control. In Lecture notes in control and information sciences, Vol. 16. Berlin: Springer. R Nevanlinna, R. (1919). Uber beschrRankte Funktionen, die in gegeben Punkten vorgeschriebenne Werte annehmen. Annales Academiae Scientiarum Fennicaeo, 13, 27–43. R Pick, G. (1916). Uber BeschrRankuungen analytischer Funktionen, welche durch vorgegebene Funktions wert bewirkt sind. Mathematischen Annalen, 77, 7–23. Stoorvogel, A. (1992). The H∞ control problem: A state space approach. Englewood CliBs, NJ: Prentice-Hall. Subrahmanyam, B. (1995). Finite horizon H∞ and related control problems. Boston: BirkhRauser. Van Keulen, B. (1993). H∞ control for distributed parameter systems: A state space approach. Boston: BirkhRauser. Wieland, S., & Willems, J. C. (1989). Almost disturbance decoupling with internal stability. IEEE Transactions on Automatic Control, 34, 277–286. Youla, D. C. (1961). A new theory of cascade synthesis. IRE Transactions on Circuit Theory, 9, 244–260.
Zames, G. (1966). On the input–output stability of time-varying nonlinear feedback systems, Part I and II. IEEE Transactions on Automatic Control, 11, 228–238, 465 – 476. Zames, G. (1981). Feedback and optimal sensitivity: model reference transformations, multiplicity seminorms, and approximation inverses. IEEE Transactions on Automatic Control, AC-26, 301–320. Zhou, K., Doyle, J. C., & Glover, K. (1995). Robust and optimal control. Englewood CliBs, NJ: Prentice-Hall.
About the reviewer Patrizio Colaneri was born in Palmoli, Italy on 12 October 1956. He received the Doctor’s degree (Laurea) in Electrical Engineering in 1981 from the Politecnico di Milano, Italy, and the Ph.D. degree (Dottorato di Ricerca) in Automatic Control in 1987 from the Ministero della Pubblica Istruzione of Italy. From 1982 to 1984, he worked in industry on simulation and control of electrical power plants. From 1984 to 1992, he was with the Centro di Teoria dei Sistemi of the Italian National Research Council (CNR). He spent a period of research at the Systems Research Center of the University of Maryland and held a visiting position at the Johannes Kepler University in Linz. He is currently Professor of Automatica at the Faculty of Engineering of the Politecnico di Milano. Dr. Colaneri was a YAP (Young Author Prize) Cnalist at the 1990 IFAC World Congress, Tallin, USSR. He is a member of the IFAC Technical Committee on Robust Control, the chair the IFAC Technical Committee on Control Design and a senior member of the IEEE. He was a member of the International Program Committee of the 1999 Conference of Decision and Control. Dr. Colaneri serves as Associate Editor of Automatica. His main interests are in the area of periodic systems and control, robust Cltering and control, and digital and multi-rate control. On these subjects, he has authored or co-authored about 120 papers and the book “Control Systems Design: an RH-2 and RH-inCnity viewpoint”, published by Academic Press.
doi:10.1016/S0005-1098(02)00282-0
Actuator saturation control V. Kapila, K.M. Grigoriadis; Marcel Dekker, New York, ISBN 0-8247-0751-6 1. Introduction Designing a control system when the actuators are subject to hard constraints is a fundamental problem. As it is well known, actuator limitations are not only one of the main sources of performance degradation, but they may even cause fatal consequences in several situations. The purpose of the present book is to provide an overlook on the current activity and research direction on the topic of constrained system control. The book is divided in chapters written by diBerent authors. This review will describe the contents of each chapter with speciCc comments in the next section. An overall discussion on the book will be provided in the concluding section. 2. Book contents In Chapter 1, the stabilization of exponentially unstable systems in the presence of both magnitude and variation rate constraints is considered. The basic idea to cope with
rate constraints is to include in the model the additional diBerential equation ˙ = R sgn M sat u(t) − (t) ; (t) M where R is the maximum variation rate and M is the maximum amplitude bound, u(t) is the desired (unconstrained) control value and (t) is the actuator control value. The system space is partitioned in the unstable and the stable subspace associated with variables xs and xu , respectively. It is known that the limitations due to the constraints aBect the unstable components xs only. It is shown that if there exists a certain domain U, subset of the (xu ; )-space, and a control that renders such a domain positively invariant for the closed-loop system and asymptotically stable, then there exists a control which tracks an asymptotically constant reference signal as long as such a reference converges to a constraint-admissible point, for every initial state in U. This result is related to that presented in the recent paper (Blanchini & Miani, 2000). The chapter presents also a signiCcant application of the technique to the control design for an unstable aircraft. Chapter 2 faces the problem of determining the most appropriate level of actuator saturation in the presence of gaussian noise. Precisely, for an SISO system the actua-
Book reviews / Automatica 39 (2003) 757 – 765
tor saturation level is deCned as the parameter in us =
sat(u= ). Such parameter is to be chosen as to guarantee that the standard deviation of the output does not exceed a prescribed value. The main result is derived by simplifying the nonlinear term eBects through a stochastic linearization method and leads to a compact formula relating the standard deviation of the output in the presence of actuator saturation and the standard deviation when no saturation is present. The overall derivation is based on this non-rigorous linearization and a set of system norm inequalities, resulting in a simple and ePcient design tool, as evidenced by the examples. In Chapter 3, the problem of determining the maximal domain of attraction for linear unstable systems is considered. Precisely, given the n-dimensional single-input continuous-time system x˙ = Ax + bu with the bounds u ∈ U = [u− ; u+ ], the paper is aimed at determining the null controllable set C. After presenting some introductory results, related to the symmetric case, they consider the problem of determining such set for two-dimensional unstable systems when the bounds on the input are asymmetric. The overall result is that, similar to the symmetric case, the null controllable set can be arbitrarily closely approximated by that of a linear saturated system whose linear gain is computed by solving a Riccati equation. The presentation is very clear and the contribution is almost self-contained. A Cnal interesting point concerns the reviewers’ conjecture of obtaining the presented result by (i) proper symmetrization (if applicable) of the input, (ii) exploiting the results recently reported in Blanchini and Miani (2000) (see also Chapter 1 of book). In Chapter 4, the problem of stabilizing a system with constrained control in order to assure a certain degree of L2 performances is faced. Two methods are exploited: the circle method (which basically deals with saturation as a sector-bounded nonlinearity) and the so-called linear analysis method which consists in determining a region in which a linear controller does not saturate. As an interesting point, it is shown that although the linear method is potentially more conservative for performance analysis, the two methods are “equivalent” in terms of achievable performances if used to synthesize the control. In view of the above result, a linear synthesis technique based on LMI conditions is investigated. To further enhance the control performance the authors consider a switching control strategy pursuing the idea of Wredenhagen and Belanger (1994) (as pointed out in the book) and provide an upper bound for the cost. In Chapter 5, some recent developments in the treatment of gain-scheduled LPV systems subject to position and rate actuator saturation are presented. The main focus is that of providing a stabilizing controller guaranteeing a prescribed L2 -gain margin, based on LMI techniques. Precisely, the problem is tackled by transforming the actuator saturation in a parameter varying term and by Crst using a parameter independent Lyapunov function and subsequently exploiting the less conservative parameter dependent Lyapunov
763
functions (see also Mori and Kokame (2000) for an analysis tool for parameter-dependent uncertain systems). Rate bounds are also analyzed and, under non-restrictive assumptions, it is shown how to recast such problem in one of the already analyzed ones. Finally, when full state measurement is available, the idea of scheduling the controller not only on the plant time-varying parameter, but also on the “distance” of the state from the origin (an idea also pursued in Wredenhagen and Belanger (1994), see also Chapter 5 of this book), is recalled and its advantages over the previously analyzed schemes are evidenced. The author also evidences some problems in the implementation of this latter control scheme, along with an illustrative example showing its effectiveness. In Chapter 6, the stabilization problem in the presence of amplitude and rate nonlinearities is considered. These nonlinearities are sector bounded and are dealt with uncertain time-varying parameters. A suPcient condition is given for quadratic stability which leads to a global stabilizability condition in both state and output feedback case. As observed, these results which are global, can be adapted to achieve local results when dealing with saturation type nonlinearities and exponentially unstable systems. As claimed by the authors the major feature of the paper is to consider discrete-time systems instead of continuous-time systems since the discrete-time case received less attention in the literature. We share this opinion, although we would like to point out some additional references such as Tarbouriech and Garcia (1997, 1998). In Chapter 7, the authors, after an extensive presentation of existing literature on input-saturated systems, face the problem of improving the performances of linear systems with input saturation by means of the Popov criteria. After considering the classical problem of determining an output feedback controller that maximizes the (ellipsoidal) set of states which can be asymptotically driven to the origin when a saturation is present (i.e., no disturbance aBects the system) they consider mixed L2 =L∞ optimization problems. They show how to determine good suboptimal approximations of the maximal set of initial conditions and of the suboptimal mixed controller, exploiting coupled BMI and LMI equations, whose underlying common denominator is the Popov stability criteria. In Chapter 8, some results concerning the problem of output regulation with constraints are proposed. This work is along the line proposed by the authors in a previous book (Saberi, Stoorvogel, & Sannuti, 2000). The regulation problem consists in zeroing the system output, in the presence of disturbances which are generated by a Cctitious linear system named exosystem, whose model is known but with unknown initial conditions. The novelty consists in the problem setup which incorporates a constrained output which takes into account both input and state variable constraints. The basic problem considered is the global (respectively, semiglobal) stabilization under constraints. The solvability of the
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Book reviews / Automatica 39 (2003) 757 – 765
problem is stated in some theorems, which involve the so-called weak non-minimum-phase condition, namely, the non-existence of exponentially unstable zeros. Some critical assumptions such as the so-called right invertibility of the constraints are analyzed. The case of constraints which are non-minimum phase for which the global or semiglobal problems are not solvable is Cnally considered. The chapter presentation is clear and mathematicall sound. In Chapter 9, the problem of the directionality compensation for input constrained system is considered. The work starts by observing that in the presence of a control which is componentwise saturated there are diBerent ways to associate a feasible input to the desired input (clearly the problem does not hold for single-input systems). For instance, the feasible input achieved by clipping each saturated component of the “desired control vector” does not preserve its direction. Therefore, the question how we can associate a feasible input to the one computed by a controller arises (whenever the latter is infeasible). After presenting some possible directionality compensation strategies, the authors analyze the optimal directionality compensation previously investigated in a paper of their own. They also propose a diBerent scheme for windup compensation based on input–output linearization, under the assumption of minimum-phase (possibly nonlinear) system. It is claimed that the systems with the proposed control satisCes some optimality conditions and it is locally stable (point (c) Theorem 9.2). However, the details on this last claim are omitted, and the reader is referred to a diBerent source. In Chapter 10, position and rate constraints are considered, and the main focus is the robust local stabilization of norm-bounded (somehow structured) uncertain systems via an output feedback controller having the same order as that of the plant under consideration. The nonlinear terms are treated as sector-bounded nonlinearities, leading to a polytopic representation of the overall uncertain system to be controlled. After the necessary mathematical preliminaries, a non-straightforward result based on the solution of two coupled Riccati equation is presented, a result which furnishes a suPcient condition for the robust stabilization problem to have a solution. The conservativity deriving from using a parameter-independent Lyapunov function is subsequently enhanced resorting to matrix inequalities techniques, whose computational issues are also analyzed. The resulting technique explicitly provides a guaranteed ellipsoidal domain of attraction to the origin. In Chapter 11, control saturation is modeled as a time-varying parameter and the control synthesis problem is embedded in a gain-scheduling design. Known LMI conditions, are then invoked in order to provide a compensator assuring that the closed-loop system satisCes a degree of performance speciCed by the designer. The technique is applied to an interesting ?ight control problem, in which actuator saturation is well known to be a crucial problem in which an inaccurate design may have severe consequences.
A potential weakness of the approach is that a priori assumptions on the saturation level (namely the bounds for the saturation indicator parameter) have to be made. These assumptions must be veriCed a posteriori, by simulation (see Fig. 6 p. 291 and the comments below). This leaves open the necessity of a trial-and-error procedure, in the case in which the foreseen level of saturation is lower than the actual one. 3. Discussion As in the spirit of the Series, these books are contributed by several authors who normally present their own research results and, therefore, their personal point of view. Clearly, the overall quality of these books strictly depends upon the choice of the contributors. From this point of view, the Editors did a good job, since all the authors are well qualiCed. There are several nice connections established in the book between concepts derived from the robust control, such as the circle criterion or the idea of modeling input saturations via time-varying parameters, and the problem of input saturation. Also, in most chapters, the signiCcant problem of simultaneous presence of amplitude and rate constraints is analyzed. The perspective oBered to the reader in terms of variety of approaches is quite wide, although the reviewers encountered a certain prevalence of methods based on LMIs, which is quite reasonable in view of the popularity of these techniques. As a result, most of the proposed approaches lead to algorithms which are practically implementable with ePcient tools. The derived compensators are of limited complexity and suitable for industrial applications. In some way the dominance of sections based on LMI and Riccati equation tools distracted the attention from other approaches which are, from the reviewers’ point of view, equally interesting for the problem. Clearly a certain level of arbitrariness in the choice of the material is welcome but the reviewers would have appreciated mentioning also other synthesis techniques, in particular the so-called model predictive control (Mayne, 2001), the L1 approach (Dahleh & Pearson, 1987), and the wider area based on dynamic programming techniques (Bertsekas, 2000; Glover & Schweppe, 1971). Although some of these approaches are older and well established in the literature (besides requiring a heavy computational load), we think that they still provide some conceptually important properties. Also the approach based on the explicit determination of the (possibly largest) null controllable set (also known as domain of attraction), basically the invariant set approach (Blanchini, 1999), is essentially limited to a pair of chapters although the concept of domain of attraction is often implicit in the quadratic Lyapunov functions computed via LMIs. This consideration might allow to face some problems from another perspective.
Book reviews / Automatica 39 (2003) 757 – 765
To conclude, we think the material covered in this book gives a quite extensive overview of currently used synthesis techniques for actuator saturated systems, although, as a pure question of taste, the reviewers would have preferred to see a little bit more emphasis on other, equally important, existing synthesis techniques for input saturated systems. Franco Blanchini E-mail address: [email protected] Stefano Miani Dipartimento di Matematica e Informatica University of Udine Via Delle Scienze 208 33100 Udine; Italy E-mail address: [email protected] References Bertsekas, D. P. (2000). Dynamic programming and optimal control. Belmont, MA: Athena ScientiCc. Blanchini, F. (1999). Set invariance in control—a survey. Automatica, 35(11), 1747–1767. Blanchini, F., & Miani, S. (2000). Any domain of attraction for a linear constrained system is a tracking domain of attraction. SIAM Journal on Control and Optimization, 38(3), 971–994. Dahleh, M. A., & Pearson, J. B. (1987). l1 -Optimal feedback controllers for MIMO discrete-time systems. IEEE Transactions on Automatic Control, 32(4), 314–322. Glover, D., & Schweppe, F. (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Transactions on Automatic Control, 16(5), 411–423. Mayne, D. Q. (2001). Control of constrained dynamic systems. European Journal of Control, 7, 87–99. Mori, T., & Kokame, H. (2000). A parameter-dependent Lyapunov function for a polytope of matrices. IEEE Transactions on Automatic Control, 45(8), 1516–1519. Saberi, A., Stoorvogel, A., & Sannuti, P. (2000). Control of linear systems with regulation and input constraints. London: Springer. doi:10.1016/S0005-1098(02)00281-9
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Tarbouriech, S., & Garcia, G. (1998). Local stabilization for linear discrete-time systems with bounded controls and norm-bounded time-varying uncertainty. International Journal of Robotics Nonlinear Control, 8(10), 831–844. Tarbouriech, S., & Garcia, G. (Eds.) (1997). Control of uncertain systems with bounded inputs, Lecture Notes in Control and Information Sciences, Vol. 227. London: Springer. Wredenhagen, G. F., & Belanger, P. R. (1994). Piecewise-linear LQ control for systems with input constraints. Automatica, 30(3), 403–416. About the reviewers Franco Blanchini was born in Legnano (MI) on 29 December 1959. He is Full Professor at the Engineering Faculty of the University of Udine where he teaches Dynamic System Theory and Automatic Control. He is a member of the Department of Mathematics and Computer Science of the same university and is also the Director of the System Dynamics Laboratory. He has been Associate Editor of Automatica since 1996. In 1997, he was member of the Program Committee of the 36th IEEE Conference on Decision and Control, San Diego, CA. In 1999, he was a member of the Program Committee of the 38th IEEE Conference on Decision and Control, Phoenix, AR. In 2001, he was a member of the Program Committee of the 40th IEEE Conference on Decision and Control, Orlando, FL. He was Chairman of the 2002 IFAC workshop on Robust Control, Cascais, Portugal. He is the recipient of 2001 ASME Oil & Gas Application Committee Best Paper Award as a co-author of the article “Experimental evaluation of a High-Gain Control for Compressor Surge Instability.” He is the recipient of the 2002 IFAC prize survey paper award as author or the article “Set Invariance in Control—a survey,” Automatica, November, 1999. Stefano Miani was born in Parma, Italy, in 1967. He received the Electrical Engineering Laurea degree summa cum laude from the University of Padova, Italy, in 1993 and the Ph.D. degree in Control Engineering from the same university in 1996. From 1996 to 1997, he was a Lecturer at the University of Udine, in 1997 he joined the Department of Electronics and Computer Science (DEI), University of Padova as a Researcher. Since November 1998, he has been a Researcher with the Department of Electrical, Mechanical and Management Engineering (DIEGM), at the University of Udine, Italy, where he is teaching Automatic Control courses. His research interests include the areas of constrained control, l∞ disturbance attenuation problems, gain scheduling control and uncertain and production–distribution systems via set-valued techniques.