- Email: [email protected]
L2-gain and passivity techniques in nonlinear control
1118
Book reviews / Automatica 39 (2003) 1113 – 1124
processes with multiple dead times which are very frequent in industry. The way to incorporate dead times in a space state framework is mentioned in one of the study cases, however I missed an organized section as such, discussing implementation issues and problems related to dead times. (2) Although constraint handling is well treated in the book, I feel that it should have been given more space, especially considering that constraints are mentioned in the title of the book and the importance of this issue for the success of MPC in industry. As an overall conclusion I think this is an excellent textbook and it could be used as a main reference for a course on Model Predictive Control. E.F. Camacho Dept. de Ingenier!9a de Sistemas y Autom!atica; Universidad de Sevilla; Camino de los Descubrimientos S/N; 41092 Seville; Spain E-mail address: [email protected] References Bitmead, R. R., Gevers, M., & Wertz, V. (1990). Adaptive optimal control. The thinking man’s GPC. Upper Saddle River, NJ: Prentice-Hall. Camacho, E. F., & Bordons, C. (1999). Model predictive control. London: Springer.
Martin-Sanchez, J. M., & Rodellar, J. (1996). Adaptive predictive control. From the concepts to plant optimization. Hemel Hempstead, Hertfordshire, UK: Prentice-Hall International (UK). SSoeterboek, R. (1992). Predictive control. A uni
About the reviewer Eduardo F. Camacho has a Ph.D. in electrical engineering from the University of Seville, where he is now Professor and Chairman of the Department of System Engineering and Automatic Control. He has written the books: “Model Predictive Control in the Process Industry” (1995), “Advanced Control of Solar Plants” (1997) and “Model Predictive Control” (1999) published by Springer-Verlag and “Control e Instrumentacin de Procesos Quimicos” published by Ed. Sintesis. He has authored and co-authored more than 150 technical papers in international journals and conference proceedings. He has served on various IFAC technical committees and was a member of the Board of Governors of the IEEE/CSS appointed by the president for 2001. At present he is the chair of the IEEE/CSS International A6airs Committee, vice president of the European Control Association and chair of the IFAC Publication Committee. He has carried out review and editorial work for various conferences and technical journals. At present he is one of the editors of the IFAC journal, Control Engineering Practice, and an associate editor of the European Journal of Control. He was Publication Chair for the IFAC World Congress b’02 and he is the General Chair of the joint Control and Decision Conference (CDC) and European Control Conference (ECC) to be held in 2005. He has acted as evaluator of projects at national and European level. He was appointed for 4 years Manager of the Advanced Production Technology Program of the Spanish National R&D Program. He was one of the Spanish representatives on the Program Committee of the Growth Research program and is now acting as expert for the Program Committee of the NMP research priority of the European Union.
doi:10.1016/S0005-1098(03)00056-6
L2 -gain and passivity techniques in nonlinear control Arjan van der Schaft: Springer, London, 2000, ISBN 1-85233-073-2 The input/output theory of nonlinear systems, initiated by Popov, Zames, and Sandberg, in the 1960s, has been a fruitful area of research, and has paved the way to many of the recent developments in control theory, including robust control (both linear and nonlinear), and numerous nonlinear stabilization techniques that rely on passivity and small-gain principles. Although most textbooks for nonlinear systems cover L2 -gain, passivity, and their implications for stability of interconnected systems, few of them discuss the prominent role played by these techniques as feedback design tools. The book by van der Schaft gives a unique historical perspective, starting with the fundamental input/output theory developed in the 1960s, further enriched in the 1970s by state–space formulations of Willems, Moylan, and Hill; and proceeding with their recent use in selected areas of nonlinear control design, such as Hamiltonian systems and nonlinear H∞ -control. The 7rst three chapters introduce the fundamental input/output concepts, which are crucial for the material studied later in the book. Chapter 1 gives the standard notions
of input/output stability and gains. Chapter 2 presents the passivity and small-gain theorems, much in the style of Desoer and Vidyasagar (1975). State–space characterizations of these input/output stability concepts are presented in Chapter 3, within the dissipativity framework of Willems (1972). This chapter is perhaps the most important in the book, because it gives the fundamental state–space tools, such as storage functions, with which input/output properties are veri7ed, or created via feedback design, for physical systems. Indeed, the rest of the book relies heavily on the topics covered in this chapter. Among these topics are the derivation of storage functions for passivity and L2 -gain, which extend to nonlinear systems positive real and bounded real lemmas; stability of dissipative systems, where storage functions are employed as Lyapunov functions; and a connection between dissipativity and optimality, observed 7rst by Kalman (1964), and employed for “inverse-optimal” feedback designs in Freeman and KokotoviFc (1996), and Sepulchre, JankoviFc, and KokotoviFc (1997). The rest of the book addresses several feedback control problems where dissipativity tools play an important role. Chapter 4 studies Hamiltonian systems as an application for passivity theory. This chapter is of great pedagogical value because it gives a physical interpretation of passivity in terms of energy dissipation, and illustrates it on sev-
Book reviews / Automatica 39 (2003) 1113 – 1124
eral examples, such as LC-circuits and rigid-body dynamics. It also presents passivity-based feedback designs for port-controlled Hamiltonian systems, including several new results that appeared since the publication of the 7rst edition of the book in 1996. Another addition to the 7rst edition is Chapter 5, which investigates when a system can be rendered passive by feedback, and how a cascade can be stabilized via feedback passivation, following the seminal papers, Byrnes, Isidori, and Willems (1991), and Sussmann and KokotoviFc (1991), respectively. Although this chapter brie?y mentions backstepping as a recursive passivation design, it does not give a complete account of the rich literature in nonlinear control designs that make use of passivity tools. Instead, the author refers to Sepulchre et al. (1997), and KrstiFc, Kanellakopoulos, and KokotoviFc (1995), for detailed studies of such designs. In Chapters 6 –8, the attention is turned from passivity to L2 -gain and several topics around the central theme of nonlinear H∞ -control theory. Chapter 6 presents factorizations of nonlinear systems which are useful for parametrization of stabilizing controllers and for developing perturbation models of uncertain systems. Chapter 7 uses these perturbation models to derive Hamilton–Jacobi equations for suboptimal H∞ -control design, 7rst by full-state feedback and, next, by output feedback. In particular, it presents Hamilton– Jacobi analogues of the Riccati equations used in linear H∞ -control. Readers familiar with the linear theory will 7nd Chapters 6 and 7 easy to follow. Others, who want to learn H∞ -control theory in a broader nonlinear framework, may 7nd it easier to start with the more pedagogical Chapter 7, which requires no background other than the fundamentals in Chapters 1–3, and refer back to Chapter 6 as needed. The 7nal Chapter 8 explores solvability properties of Hamilton–Jacobi inequalities. The derivations in this chapter rely on invariant manifold techniques for the Hamiltonian corresponding to the Hamilton–Jacobi inequality, and closely parallel the invariant subspace methods for algebraic Riccati equations. The relationship between the local solutions of Hamilton–Jacobi inequalities for nonlinear systems, and those of the Riccati inequalities for their Jacobian linearizations, is studied in detail, and the results are applied to optimal- and H∞ -control problems. Although passivity and small-gain theorems date some 40 years back, their evolution into constructive feedback design tools is still continuing. This book nicely summarizes this evolution, starting with the early phase where fundamental concepts were formulated, and proceeding with recent developments in robust and nonlinear control theory where these concepts have been instrumental. It would be a suitable textbook for a graduate-level course that builds upon a 7rst course in nonlinear systems. In fact, the reviewer was a student in such a course taught by Petar KokotoviFc in Spring 1997, which covered the 7rst edition of this book in doi:10.1016/S0005-1098(03)00057-8
1119
parallel with Sepulchre et al. (1997). Other recent books on the topic of dissipative systems include Lozano, Brogliato, Egeland, and Maschke (2000), which focuses on further details and feedback applications of passivity theory; and Ortega, LorFUa, Sira-RamFUrez, and Nicklasson (1998), which applies passivity-based control techniques to Euler–Lagrange systems. Likewise, nonlinear H∞ -control is further studied in BaVsar and Bernhard (1995), and Helton and James (1999). Murat Arcak Electrical; Computer and Systems Engineering Dept. Rensselaer Polytechnic Institute; 110 8th Street; Troy, New York 12180-3590; USA E-mail address: [email protected] References BaVsar, T., & Bernhard, P. (1995). H∞ optimal control and related minimax design problems (2nd ed.). Boston: Birkhauser. Byrnes, C. I., Isidori, A., & Willems, J. C. (1991). Passivity, feedback equivalence, and global stabilization of minimum phase systems. IEEE Transactions on Automatic Control, 36, 1228–1240. Desoer, C. A., & Vidyasagar, M. (1975). Feedback systems: Input– output properties. New York: Academic Press. Freeman, R. A., & KokotoviFc, P. V. (1996). Robust nonlinear control design, state-space and Lyapunov techniques. Boston: Birkhauser. Helton, J. W., & James, M.R. (1999). Extending H∞ control to nonlinear systems. SIAM Frontiers in Applied Mathematics. Kalman, R. (1964). When is a linear control system optimal?. Transactions of the ASME, Series D, Journal of Basic Engineering, 86, 1–10. KrstiFc, M., Kanellakopoulos, I., & KokotoviFc, P. (1995). Nonlinear and adaptive control design. New York: Wiley. Lozano, R., Brogliato, B., Egeland, O., & Maschke, B. (2000). Dissipative systems analysis and control. London: Springer. Ortega, R., LorFUa, A., Sira-RamFUrez, H., & Nicklasson, P. J. (1998). Passivity-based control of Euler–Lagrange systems: Mechanical, electrical and electromechanical applications. London: Springer. Sepulchre, R., JankoviFc, M., & KokotoviFc, P. (1997). Constructive nonlinear control. New York: Springer. Sussmann, H. J., & KokotoviFc, P. V. (1991). The peaking phenomenon and the global stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 36, 424–439. Willems, J. C. (1972). Dissipative dynamical systems Part I: General theory; Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis, 45, 321–393. About the reviewer Murat Arcak was born in Istanbul, Turkey, in 1973. He received the B.S. degree in Electrical and Electronics Engineering from the BoXgaziVci University, Istanbul, in 1996, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of California, Santa Barbara, in 1997 and 2000. In 2001 he joined the Rensselaer Polytechnic Institute, Troy, New York, as an assistant professor of Electrical, Computer and Systems Engineering. His research interests are in nonlinear control theory and applications. Dr. Arcak is a member of IEEE and SIAM, and an associate editor on the Conference Editorial Board of IEEE Control Systems Society. He was a 7nalist for the student best paper award at the 2000 American Control Conference, and received an NSF CAREER Award in 2003.
Book reviews / Automatica 39 (2003) 1113 – 1124
processes with multiple dead times which are very frequent in industry. The way to incorporate dead times in a space state framework is mentioned in one of the study cases, however I missed an organized section as such, discussing implementation issues and problems related to dead times. (2) Although constraint handling is well treated in the book, I feel that it should have been given more space, especially considering that constraints are mentioned in the title of the book and the importance of this issue for the success of MPC in industry. As an overall conclusion I think this is an excellent textbook and it could be used as a main reference for a course on Model Predictive Control. E.F. Camacho Dept. de Ingenier!9a de Sistemas y Autom!atica; Universidad de Sevilla; Camino de los Descubrimientos S/N; 41092 Seville; Spain E-mail address: [email protected] References Bitmead, R. R., Gevers, M., & Wertz, V. (1990). Adaptive optimal control. The thinking man’s GPC. Upper Saddle River, NJ: Prentice-Hall. Camacho, E. F., & Bordons, C. (1999). Model predictive control. London: Springer.
Martin-Sanchez, J. M., & Rodellar, J. (1996). Adaptive predictive control. From the concepts to plant optimization. Hemel Hempstead, Hertfordshire, UK: Prentice-Hall International (UK). SSoeterboek, R. (1992). Predictive control. A uni
About the reviewer Eduardo F. Camacho has a Ph.D. in electrical engineering from the University of Seville, where he is now Professor and Chairman of the Department of System Engineering and Automatic Control. He has written the books: “Model Predictive Control in the Process Industry” (1995), “Advanced Control of Solar Plants” (1997) and “Model Predictive Control” (1999) published by Springer-Verlag and “Control e Instrumentacin de Procesos Quimicos” published by Ed. Sintesis. He has authored and co-authored more than 150 technical papers in international journals and conference proceedings. He has served on various IFAC technical committees and was a member of the Board of Governors of the IEEE/CSS appointed by the president for 2001. At present he is the chair of the IEEE/CSS International A6airs Committee, vice president of the European Control Association and chair of the IFAC Publication Committee. He has carried out review and editorial work for various conferences and technical journals. At present he is one of the editors of the IFAC journal, Control Engineering Practice, and an associate editor of the European Journal of Control. He was Publication Chair for the IFAC World Congress b’02 and he is the General Chair of the joint Control and Decision Conference (CDC) and European Control Conference (ECC) to be held in 2005. He has acted as evaluator of projects at national and European level. He was appointed for 4 years Manager of the Advanced Production Technology Program of the Spanish National R&D Program. He was one of the Spanish representatives on the Program Committee of the Growth Research program and is now acting as expert for the Program Committee of the NMP research priority of the European Union.
doi:10.1016/S0005-1098(03)00056-6
L2 -gain and passivity techniques in nonlinear control Arjan van der Schaft: Springer, London, 2000, ISBN 1-85233-073-2 The input/output theory of nonlinear systems, initiated by Popov, Zames, and Sandberg, in the 1960s, has been a fruitful area of research, and has paved the way to many of the recent developments in control theory, including robust control (both linear and nonlinear), and numerous nonlinear stabilization techniques that rely on passivity and small-gain principles. Although most textbooks for nonlinear systems cover L2 -gain, passivity, and their implications for stability of interconnected systems, few of them discuss the prominent role played by these techniques as feedback design tools. The book by van der Schaft gives a unique historical perspective, starting with the fundamental input/output theory developed in the 1960s, further enriched in the 1970s by state–space formulations of Willems, Moylan, and Hill; and proceeding with their recent use in selected areas of nonlinear control design, such as Hamiltonian systems and nonlinear H∞ -control. The 7rst three chapters introduce the fundamental input/output concepts, which are crucial for the material studied later in the book. Chapter 1 gives the standard notions
of input/output stability and gains. Chapter 2 presents the passivity and small-gain theorems, much in the style of Desoer and Vidyasagar (1975). State–space characterizations of these input/output stability concepts are presented in Chapter 3, within the dissipativity framework of Willems (1972). This chapter is perhaps the most important in the book, because it gives the fundamental state–space tools, such as storage functions, with which input/output properties are veri7ed, or created via feedback design, for physical systems. Indeed, the rest of the book relies heavily on the topics covered in this chapter. Among these topics are the derivation of storage functions for passivity and L2 -gain, which extend to nonlinear systems positive real and bounded real lemmas; stability of dissipative systems, where storage functions are employed as Lyapunov functions; and a connection between dissipativity and optimality, observed 7rst by Kalman (1964), and employed for “inverse-optimal” feedback designs in Freeman and KokotoviFc (1996), and Sepulchre, JankoviFc, and KokotoviFc (1997). The rest of the book addresses several feedback control problems where dissipativity tools play an important role. Chapter 4 studies Hamiltonian systems as an application for passivity theory. This chapter is of great pedagogical value because it gives a physical interpretation of passivity in terms of energy dissipation, and illustrates it on sev-
Book reviews / Automatica 39 (2003) 1113 – 1124
eral examples, such as LC-circuits and rigid-body dynamics. It also presents passivity-based feedback designs for port-controlled Hamiltonian systems, including several new results that appeared since the publication of the 7rst edition of the book in 1996. Another addition to the 7rst edition is Chapter 5, which investigates when a system can be rendered passive by feedback, and how a cascade can be stabilized via feedback passivation, following the seminal papers, Byrnes, Isidori, and Willems (1991), and Sussmann and KokotoviFc (1991), respectively. Although this chapter brie?y mentions backstepping as a recursive passivation design, it does not give a complete account of the rich literature in nonlinear control designs that make use of passivity tools. Instead, the author refers to Sepulchre et al. (1997), and KrstiFc, Kanellakopoulos, and KokotoviFc (1995), for detailed studies of such designs. In Chapters 6 –8, the attention is turned from passivity to L2 -gain and several topics around the central theme of nonlinear H∞ -control theory. Chapter 6 presents factorizations of nonlinear systems which are useful for parametrization of stabilizing controllers and for developing perturbation models of uncertain systems. Chapter 7 uses these perturbation models to derive Hamilton–Jacobi equations for suboptimal H∞ -control design, 7rst by full-state feedback and, next, by output feedback. In particular, it presents Hamilton– Jacobi analogues of the Riccati equations used in linear H∞ -control. Readers familiar with the linear theory will 7nd Chapters 6 and 7 easy to follow. Others, who want to learn H∞ -control theory in a broader nonlinear framework, may 7nd it easier to start with the more pedagogical Chapter 7, which requires no background other than the fundamentals in Chapters 1–3, and refer back to Chapter 6 as needed. The 7nal Chapter 8 explores solvability properties of Hamilton–Jacobi inequalities. The derivations in this chapter rely on invariant manifold techniques for the Hamiltonian corresponding to the Hamilton–Jacobi inequality, and closely parallel the invariant subspace methods for algebraic Riccati equations. The relationship between the local solutions of Hamilton–Jacobi inequalities for nonlinear systems, and those of the Riccati inequalities for their Jacobian linearizations, is studied in detail, and the results are applied to optimal- and H∞ -control problems. Although passivity and small-gain theorems date some 40 years back, their evolution into constructive feedback design tools is still continuing. This book nicely summarizes this evolution, starting with the early phase where fundamental concepts were formulated, and proceeding with recent developments in robust and nonlinear control theory where these concepts have been instrumental. It would be a suitable textbook for a graduate-level course that builds upon a 7rst course in nonlinear systems. In fact, the reviewer was a student in such a course taught by Petar KokotoviFc in Spring 1997, which covered the 7rst edition of this book in doi:10.1016/S0005-1098(03)00057-8
1119
parallel with Sepulchre et al. (1997). Other recent books on the topic of dissipative systems include Lozano, Brogliato, Egeland, and Maschke (2000), which focuses on further details and feedback applications of passivity theory; and Ortega, LorFUa, Sira-RamFUrez, and Nicklasson (1998), which applies passivity-based control techniques to Euler–Lagrange systems. Likewise, nonlinear H∞ -control is further studied in BaVsar and Bernhard (1995), and Helton and James (1999). Murat Arcak Electrical; Computer and Systems Engineering Dept. Rensselaer Polytechnic Institute; 110 8th Street; Troy, New York 12180-3590; USA E-mail address: [email protected] References BaVsar, T., & Bernhard, P. (1995). H∞ optimal control and related minimax design problems (2nd ed.). Boston: Birkhauser. Byrnes, C. I., Isidori, A., & Willems, J. C. (1991). Passivity, feedback equivalence, and global stabilization of minimum phase systems. IEEE Transactions on Automatic Control, 36, 1228–1240. Desoer, C. A., & Vidyasagar, M. (1975). Feedback systems: Input– output properties. New York: Academic Press. Freeman, R. A., & KokotoviFc, P. V. (1996). Robust nonlinear control design, state-space and Lyapunov techniques. Boston: Birkhauser. Helton, J. W., & James, M.R. (1999). Extending H∞ control to nonlinear systems. SIAM Frontiers in Applied Mathematics. Kalman, R. (1964). When is a linear control system optimal?. Transactions of the ASME, Series D, Journal of Basic Engineering, 86, 1–10. KrstiFc, M., Kanellakopoulos, I., & KokotoviFc, P. (1995). Nonlinear and adaptive control design. New York: Wiley. Lozano, R., Brogliato, B., Egeland, O., & Maschke, B. (2000). Dissipative systems analysis and control. London: Springer. Ortega, R., LorFUa, A., Sira-RamFUrez, H., & Nicklasson, P. J. (1998). Passivity-based control of Euler–Lagrange systems: Mechanical, electrical and electromechanical applications. London: Springer. Sepulchre, R., JankoviFc, M., & KokotoviFc, P. (1997). Constructive nonlinear control. New York: Springer. Sussmann, H. J., & KokotoviFc, P. V. (1991). The peaking phenomenon and the global stabilization of nonlinear systems. IEEE Transactions on Automatic Control, 36, 424–439. Willems, J. C. (1972). Dissipative dynamical systems Part I: General theory; Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis, 45, 321–393. About the reviewer Murat Arcak was born in Istanbul, Turkey, in 1973. He received the B.S. degree in Electrical and Electronics Engineering from the BoXgaziVci University, Istanbul, in 1996, and the M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of California, Santa Barbara, in 1997 and 2000. In 2001 he joined the Rensselaer Polytechnic Institute, Troy, New York, as an assistant professor of Electrical, Computer and Systems Engineering. His research interests are in nonlinear control theory and applications. Dr. Arcak is a member of IEEE and SIAM, and an associate editor on the Conference Editorial Board of IEEE Control Systems Society. He was a 7nalist for the student best paper award at the 2000 American Control Conference, and received an NSF CAREER Award in 2003.