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Transcription

1 Bibliography [1] F. Albertini and D. D'Alessandro, "Asymptotic stability of continuous-time systems with saturation nonlinearities," Systems 1:'3 GontTOl Letters, Vol. 29, pp , [2] J. Alvarez, R. Suarez and J. Alvarez, "Planar linear systems with single saturated feedback," Systems [1 GontTOI Letters, Vol. 20, pp , [3] B. R. Barmish and W. E. Schmitendorf, "A necessary and sufficient condition for local constrained controllability of a linear system," IEEE Transactions on Automatic GontTOI, Vol. 25, pp , [4] D. S. Bernstein and A. N. Michel, "A chronological bibliography on saturating actuators," International Journal of Robust and NonlineaT Gontml, Vol. 5, pp , [5] G. Bitsoris, "On the positive invariance of polyhedral sets for discrete-time systems," Systems 1 GontTOI Letters, Vol. ll, pp , [6] F. Blanchini, "Feedback control for LTI systems with state and control bounds in the presence of disturbances," IEEE Transactions on Automatic GontTOI, Vol. 35, pp , [7] F. Blanchini, "Ultimate boundedness control for uncertain discretetime systems via set-induced Lyapunov function," IEEE Transactions on Automatic GontTOI, Vol. 39, pp , [8] F. BIanchini, "Set invariance in control - a survey," Automatica, Vol. 35, pp ,

2 376 Bibliography [9] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, [10] P. Bruck, "A new result in time optimal control of discrete systems," IEEE Transactions on Automatic Control, Vol. 19, pp , [11] P. J. Campo and M. Morari, "Robust control of processes subject to saturation nonlinearities," Computers and Chemical Engineering, Vol. 14, pp , [12] P. J. Campo, M. Morari and C. N. Nett, "Multivariable anti-windup and bump less transfer: a general theory," Proceedings of American Control Conference, pp , [13] A. Casavola and E. Mosca, "Global regulation of null-controllable input-saturated systems with arbitrary 12 state disturbance," preprint. [14] H. Choi, "On the stabilization of linear discrete-time systems subject to input saturation," Systems 3 Control Letters, Vol. 36, pp , [15] J. Collado, R. Lozano and A. Ailon, "Semi-global stabilization of discrete-time systems with bounded inputs using periodic controller," Systems 3 Control Letters, Vol. 36, pp , [16] E. J. Davison and K. C. Cowan, "A computational method for determining the stability region of second-order non-linear autonomous system," International Journal of Control, Vol. 9, pp , [17] E. J. Davison and E. M. Kurak, "A computational method for determining quadratic Lyapunov functions for non-linear systems," Automatica, Vol. 7, pp , [18] C. A. Desoer and J. Wing, "An optimal strategy for a saturating sampled data system," IRE Transactions on Automatic Control, Vol. 6, pp. 5-15, [19] M. E. Evans and D. N. P. Murthy, "Controllability of discrete time systems with positive controls," IEEE Transactions on Automatic Contml, Vol. 22, pp , 1977.

3 Bibliography 377 [20] A. Feuer and M. Heymann, "Admissible sets in linear feedback systems with bounded controls," International Journal of Control, Vol. 23, pp , [21] M. E. Fisher and.t. E. Gayek, "Estimating reachable sets for twodimensional linear discrete systems," Journal of Optimization Theory and Applications, Vol. 56, pp , [22] B. A. Francis, "The linear multivariable regulator problem," SIAM Journal on Control and Optimization, Vol. 15, pp , [23] G. F. Franklin,.T. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, third edition, Addison 'Wesley, New York, [24] A. T. Fuller, "In-the-Iarge stability of relay and saturating control systems with linear controller," International Journal of Control, Vol. 10, pp , [25] E. G. Gilbert and K. T. Tan, "Linear systems with state and control constraints: the theory and application of maximal output admissible sets," IEEE Transactions on Automatic Control, Vol. 36, pp , [26] P.-O. Gutman and M. Cwikel, "Admissible sets and feedback control for discrete-time linear dynamical systems with bounded control and dynamics," IEEE Transactions on Automatic Control, Vol. 31, pp , [27] P.-O. Gutman and P. Hagander, "A new design of constrained controllers for linear systems," IEEE Tmnsactions on Automatic Control, Vol. 30, pp , [28] M. Hamza and M. E. Rasmy, "A simple method for determining the reachable set for linear discrete systems," IEEE Tmnsactions on A1J.tomatic Control, Vol. 16, pp , [29] O. Hajek, Control Theory in the Plane, Springer-Verlag, Berlin, [30] R. Hanus, M. Kinnaert and.t. L. Henrotte, "Conditioning technique, a general anti-windup and bumpleslci transfer method," Automatica, Vol. 23, pp , 1987.

4 378 Bibliography [31] M. Hautus, "Linear matrix equations with applications to the regulator problem," in Outils and Modeles Mathematique pour l'automatique, 1. D. Landou, Ed., Paris: C.N.R.S., pp , [32] H. Hindi and S. Boyd, "Analysis of linear systems with saturating using convex optimization," Proceedings of the 37th IEEE Conference on Decision 8 Control, pp , [33] L. M. Hocking, Optimal Control, An Introduction to the Theory and Applications, Oxford University Press, Oxford, [34] L. Hou and A. N. Michel, "Asymptotic stability of systems with saturation constraints," IEEE Transactions on Automatic Control, Vol. 43, pp , [35] T. Hu and Z. Lin, "A complete stability analysis of planar linear systems under saturation," IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, Vol. 47, pp , [36] T. Hu and Z. Lin, "A complete stability analysis of planar linear discrete-time systems under saturation," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, to appear. [37] T. Hu and Z. Lin, "Output regulation for general linear systems with saturating actuators," submitted to Automatica, [38] T. Hu and Z. Lin, "On enlarging the basin of attraction for linear systems under saturated linear feedback," Systems 8 Control Letters, Vol. 40, pp , [39] T. Hu and Z. Lin, "Robust stabilization of exponentially unstable linear systems with saturating actuators," International Journal of Robust and Nonlinear Control, to appear. Also in Proceedings of American Control Conference, pp , [40] T. Hu, Z. Lin and B. M. Chen, "An analysis and design method for linear systems subject to actuator saturation and disturbance," submitted to Automatica, Also in Proceedings of American Control Conference, pp , 2000.

5 Bibliography 379 [41] T. Hu, Z. Lin and L. Qiu, "An explicit description of the null controllable regions of linear systems with saturating actuators," submitted for publication, [42] T. Hu, Z. Lin and L. Qiu, "Stabilization of exponentially unstable linear systems with saturating actuators," IEEE Tmnsaction on Automatic Contml, to appear. [43] T. Hu, Z. Lin and Y. Shamash, "Semi-global stabili,mtion with guaranteed regional performance of linear systems subject to actuator saturation," Systems 8 Contml LetteTs, to appear. Also in Pmceedings of American Contml ConfeTence, pp , [44] T. Hu, D. E. Miller and L. Qiu, "Controllable regions of LTI discretetime systems with input saturation," Pmceedings of the 37th IEEE Con/emcee on Decision and ContTol, pp , [45] T. Hu, D. E. Miller and L. Qiu, "Null controllability and stabilizability of LTI discrete-time systems with input saturation," submitted for publication, [46] T. Hu, A. N. Pitsillides and Z. Lin, "Null controllability and stabilization of linear systems subject to asymmetric actuator saturation," in Actuator Satumtion ContTOl, edited by V. Kapila and K.M. Grigoriadis, Marcel Dekker, New York, to appear. [47] T. Hu, L. Qiu and Z. Lin, "The controllability and stabilization of unstable LTI systems with input saturation," PToceedings of the 36th IEEE Confe'rence on Decision and Contml, pp , [48] A. Isidori and C. I. Byrnes, "Output regulation for nonlinear systems," IEEE Transactions on Automatic ContTol, Vol. 35, pp , [49] L. Jin, P. N. Nikiforuk and M. M. Gupta, "Absolute stability conditions for discrete-time recurrent neural networks," IEEE Transactions on Neural Networks, Vol. 5, pp , [50] S. M. Joshi, "Stability of multiloop LQ regulators with nonlinearities -part I: region of attraction," IEEE Transactions on Automatic ContTOl, Vol. 31, pp , 1986.

6 380 Bibliography [51] J. Junkins, J. Valasek and D. Ward, Report of ONR UCAV Modeling Effort, Department of Aerospace Engineering, Texas A&M University, [52] R. E. Kalman, "Optimal nonlinear control of saturating systems by intermittent action," IRE Wescon Convention Record, Pt. 4, pp , [53] P. Kapasouris, M. Athans and G. Stein, "Design of feedback control systems for stable plants with saturating actuators," Proceedings of the 21th IEEE Conference on Decision and Control, pp , [54] S. S. Keerthi and E. G. Gilbert, "Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints," IEEE Transactions on Automatic Control, Vol. 32, pp , [55] H. K. Khalil, Nonlinear Systsems, second edition, Prentice Hall, Upper Saddle River, [56] P. P. Khargonekar, K. Poolla and A. Tannenbaum, "Robust control of linear time invariant plants using periodic compensation," IEEE Transactions on Automatic Control, Vol. 30, pp , [57] R. B. Koplon, M. L. J. Hautus and E. D. Sontag, "Observabilityof linear systems with saturated outputs," Linear Algebra and Applications, Vols , pp , [58] R. L. Kosut, "Design of linear systems with saturating linear control and bounded states," IEEE Transactions on Automatic Control, Vol. 28, pp [59] G. Kreisselmeier, "Stabilization of linear systems in the presence of output measurement saturation," Systems &J Control Letters, Vol. 29, pp , [60] N. J. Krikelis and S. K. Barkas, "Design of tracking systems subject to actuator saturation and integrator windup," International Journal of Control, Vol. 39, pp , 1984.

7 Bibliography 381 [61] J. B. Lasserre, "On reachable and controllable sets for twodimensional linear discrete-time systems," Journal of Optimization Theory and Appl'lcations, Vol. 70, pp , [62] J. B. Lasserre, "Reachable, controllable sets and stabilizing control of constrained linear systems," Automatica, Vol. 29, pp , [63] E. B. Lee and L. Markus, Foundations of Optimal Contml, John Wiley and Sons Inc., New York, [64] J. N. Lin, "Determination of reachable set for a linear discrete system," IEEE Transactions on Automatatic Contml, Vol. 15, pp , [65] Z. Lin, Low Gain Feedback, Lecture Notes in Control and Information Sciences, Vol. 240, Springer-Verlag, London, 1998, [66] Z. Lin and T. Hu, "Semi-global stabilization of linear system subject to output saturation," Systems 8 Contml Letters, to appear. [67] Z. Lin and A. Saberi, "Semi-global exponential stabilization of linear systems subject to 'input saturation' via linear feedbacks," Systems 8 Control Letters, Vol. 21, pp , [68] Z, Lin and A. Saberi, "Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedbacks," Systems 8 Control Letters, Vol. 24, pp , [69] Z, Lin and A. Saberi, "A semi-globallow-and-high design for linear systems with input saturation - stabilization and disturbance rejection," International Journal of Robust and Nonlinear Control, Vol. 5, pp , [70] Z. Lin and A. Saberi, "Semi-global exponential stabilization of linear discrete-time systems subject to 'input saturation' via linear feedbacks," Systems 8 Control Letters, Vol. 24, pp , [71] Z. Lin, A. A. Stoorvogel and A. Saberi, "Output regulation for linear systems subject to input saturation," Automatica, Vol. 32, pp , 1996.

8 382 Bibliography [72] D. Liu and A.N. Michel, "Asymptotic stability of systems operating on a closed hypercube," Systems 1!3 Control Letters, vol. 19, pp , [73] D. Liu and A.N. Michel, "Sparsely interconnected neural networks for associative memories with applications to cellular neural networks," IEEE Transactions on Circuits and Systems - II: Analog and Digital Signal Processing, Vol. 41, pp , [74] D. Liu and A.N. Michel, Dynamical Systems with Saturation Nonlinearities, Lecture Notes in Control and Information Sciences, Vol. 195, Springer-Verlag, London, [75] K. A. Loparo and G. L. Blankenship, "Estimating the domain of attraction of of nonlinear feedback systems," IEEE Transactions on Automatic Control, Vol. 23, pp , [76] J. Macki and M. Strauss, Introduction to Optimal Control, Springer Verlag, Berlin, [77] R. Mantri, A. Saberi and V. Venkatasubramanian, "Stability analysis of continuous time planar systems with state saturation nonlinearity," IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, Vol. 45, pp , [78] A. Megretski, "Lz BIBO output feedback stabilization with saturated control," 13th IFAC World Congress, Vol. D, pp , [79] A. Nagata, S. Kodama and S. Kumagai, "Time optimal discrete control system with bounded state variable," IEEE Transactions on Automatic Control, Vol. 10, pp , [80] L. Pandolfi, "Linear control systems: controllability with constrained controls," Journal of Optimization Theory and Applications, Vol. 19, pp , [81] C. Pittet, S. Tarbouriech and C. Burgat, "Stability regions for linear systems with saturaing controls via circle and Popov criteria," Proceedings of the 36th IEEE Conference on Decision and Control, pp , 1997.

9 Bibliography 383 [82] X. Qian and J. Song, Engineering Cybernetics, in Chinese, Science Academics, Beijing, [83] J. H. F. Ritzerfeld, "A condition for the overflow stability of secondorder digital filters that is satisfied by all scaled state-space structures using saturation," IEEE Transactions on Circuits and Systems, Vol. 36, pp , [84] B. C. Romanchuk, "Computing regions for attraction with polytopes: planar case," Automatica, Vol. 32, pp , [85] A. Saberi, Z. Lin and A. R. Teel, "Control of linear systems with saturating actuators," IEEE Transactions on Automatic Control, Vol. 41, pp , [86] A. Saberi, P. Sannuti and B. M. Chen, H 2 -Optimal Control, Prentice Hall, London, [87] P. Sannuti and A. Saberi, "A special coordinate basis of multivariable linear systems-finite and infinite zero structure, squaring down and decoupling," International Journal of Control, Vol. 45, pp , [88] R. De Santis and A. Isidori, "Output regulation for linear systems with anti-stable eigenvalues in the presence of input saturation," Proceedings of the 38th IEEE Conference on Decision and Control, pp , [89] W. E. Schmitendorf and B. R. Barmish, "Null controllability of linear systems with constrained controls," SIAM Journal on Control and Optimization, Vol. 18, pp , [90] E. D. Sontag, "An algebraic approach to bounded controllability of linear systems," International Journal of Control, Vol. 39, pp , [91 ] E. D. Sontag, Mathematical Control TheoTY, second edition, Springer, New York, [92] E. D. Sontag, and H. J. Sussmann, "Complete controllability of continuous-time recurrent neural networks," Systems f3 Control Letters, Vol. 30, pp , 1997.

10 384 Bibliography [93] E. D. Sontag and H.J. Sussmann, "Nonlinear output feedback design for linear systems with saturating controls," Proceedings of the 29th IEEE Conference Decision and Control, pp , [94] R. Suarez, J. Alvarez-Ramirez and J. Solis-Daun, "Linear systems with bounded inputs: global stabilization with eigenvalue placement," International Journal of Robust and Nonlinear Control, Vol. 7, pp , [95] H. J. Sussmann, E. D. Sontag and Y. Yang, "A general result on the stabilization of linear systems using bounded controls," IEEE Transactions on Automatic Control, Vol. 39, pp , [96] H. J. Sussmann and Y. Yang, "On the stabilizability of multiple integrators by means of bounded feedback controls," Proceedings of the 30th IEEE Conference on Decision and Control, pp , [97] M. Sznaier, "A set induced norm approach to the robust control of constrained systems," SIAM Journal on Control and Optimization, Vol. 31, pp , [98] A. R. Teel, "Global stabilization and restricted tracking for multiple integrators with bounded controls," Systems fj Control Letters, Vol. 18, pp , [99] A. R. Teel, "Linear systems with input nonlinearities: global stabilization by scheduling a family of Hoo-type controllers," International Journal of Robust and Nonlinear Control, Vol. 5, pp , [100] A. R. Teel, "Semi-global stabilization of linear null controllable systems with input nonlinearities," IEEE Transactions on Automatic Control, Vol. 40, pp , [101] A. R. Teel, Feedback Stabilization: Nonlinear Solutions to Inherently Nonlinear Problems, Ph.D dissertation, University of California, Berkeley, [102] A. R. Teel, "A nonlinear small gain theorem for the analysis of control systems," IEEE Transactions on Automatic Control, Vol. 42, pp , 1996

11 Bibliography 385 [103] R. P. V. Til and W. E. Schmitendorf, "Constrained controllability of discrete-time systems," International Journal of Control, Vol. 43, pp , [104] A. Vanelli and M. Vidyasagar, "Maximal Lyapunov functions and domain of attraction for autonomous nonlinear systems," Automatica, Vol. 21, pp , [105] S. Weissenberger, "Application of results from the absolute stability to the computation of finite stability domains," IEEE Transactions on Automatic Control, Vol. 13, pp , [106] J. Wing and C. A. Desoer, "The multiple-input minimal time regulator problem (general theory)," IEEE Transactions on Automatic Control, Vol. 8, pp , [107] G. F. \Vredenhagen and P. R. Belanger, "Piecewise-linear LQ control for systems with input constraints," Automatica, Vol. 30, pp. 403~ 416, [108] J. C. Willems, "Least squares stationary optimal control and the algebraic Riccati equations," IEEE Transactions on Automatic Contml, Vol. 16, pp , [109] W. M. Wonham, Linear Multivariable Contml: A Geometric Approach, Springer-Verlag, New York, [110] Y. Yang, E. D. Sontag and H. J. Sussmann, "Global stabilization of linear discrete-time systems with bounded feedback," Systems CJ Contml Letters, Vol. 30, pp , 1997.

12 Index actuator saturation, 1 admissible control, 6, 13, 38, 57, 270, 308 algebraic Riccati equation, 6, 67, 75, 127 ANCBC, 6, 56, 79, 111, 129 anti-stable, 7 anti-windup, 2 ARE, 6, 67, 75, 127, 141 associate memory, 327 asymptotically null controllable definition, 34, 51 asymptotically null controllable region, 34, 51, 75 definition, 34, 52 description, 34, 52 asymptotically null controllable with bounded controls (see: ANCBC), 6 asymptotically stable, 7 auxiliary feedback matrix, 165 bang-bang control, 21, 24, 34, 43-45, 232, 234, 242, 247, 254, 257 bounded global attractor, 328 chattering, 231, 242, 285 circle criterion, 164 closed trajectory, 331, 334, 338 contractive invariance definition, 164, 174 example, 168 under saturated high gain feedback, 244 contractive invariant example, 177 convergence rate, 229, 231 maximal, 233, 258 maximal, with disturbance, 255 of a linear system, 230 of a nonlinear system, 231 overall, 247 sub-optimal, 243 convex hull, 9, 160 convex set, 8 strictly, 8 DARE, 6,108 detect ability, 268, 307, 363 digital filter, 328, 344 disturbance rejection, 211 example, 219 LMI formulation, 216, 225 problem, 214, 223 with guaranteed domain of attraction, 214, 216, 223,

13 388 Index with guaranteed domain of attraction, LMI formulation, 218,227 domain of attraction, 57, 73, 74, 164 continuity, 124 convexity, 65 definition, 57, 163, 174 enlargement, 183, 185 enlargement, example, 189, 190 enlargement, LMI formulation, 184, 185, 187 estimation, 157, 164, 169, 179 estimation, example, 173, 177 expansion, 197 expansion, example, 207 expansion, theorem, 197 monotonicity, 120, 127 of a set, definition, 129, 213, 222 ellipsoid, 6 volume of, 159 equilibrium, 58 equivalent transformation, 329 exogenous system (or exosystem), 265, 267, 306 exponentially unstable, 7 extremal control, 44, 45 definition, 19 description, 23, 28, 29, 32, 46 equivalence, 19 in finite steps, definition, 41 minimal representative, 20 property, 21 extremal point, 8 extremal trajectory, 15 definition, 19 global stabilization at set of equilibria, theorem, 89 example, 104 Green's Theorem, 123, 334, 337 Hausdorff distance, 8, 67, 250 high gain feedback, 72, 85 Hurwitz, 7 index theory, 62 integrator windup, 1 internal stability, 267, 307 invariant ellipsoid, 234 contractive, 234 contractive, condition, , 175, 176, 259 contractive, definition, 164, 174 contractive, example, 168, 177 contractive, necessary and sufficient condition, 167, 177, 238, 260 contractive, under saturated high gain feedback, 244 maximal, 235 with disturbance, 255, 256 with disturbance, strictly, 255 invariant set, 196 definition, 164, 174 with disturbance, 136 Jordan block, 7

14 Index 389 Jordan canonical form, 23 Lagrange multiplier method, 170, 238, 261 level set, 7 lifted system, 186 lifting technique, 186, 199 limit cycle, 59 limit set, 348 limit trajectory, 347, 348 properties, 348, 349 linear matrix equation, 274, 280, 311,316 regulator equations, 269, 272, 307, 309 linear region of saturation, 9, 163 LMI, 6, 158, 166, 216, 219, 225, 228, 252 LQR, 6, 67, 208 maximal convergence control, 233, 258, 263 example, 246 with disturbance, 255 maximal invariant ellipsoid, 235 minimum energy regulation, 67 negative definite, 6 negative semi-definite, 6 neural network, 328, 344 neutrally stable, 7, 230, 271, 308 norm of matrix, 4 of signal, 4 of vector, 4 null controllability, 11, 13, 37, 39 null controllable in a finite time, 13, 39 null controllable region, 11, 13,39, 219,220 asymptotically, 34, 51, 212 at a finite time, 13, 33, 39 general description, 15, 41 separation result, 13, 39 null controllable region, boundary, 15 example, 26, 29, 30, 32 formula, 25, 28 general description, 16 trajectory description, 20, 24, 29, 32 null controllable region, extremes, 43 example, 50 formula, 47, 48 trajectory description, 48, 49 observability, 290, 324 observer, 290, 324 estimation error, 290 fast observer, 290 output regulation, 265, 305 definition, 267, 307 output regulation, linear theory, 267, 306 assumptions, 268, 307 error feedback, 267, 306 error feedback, problem, 268 solutions, 269, 307 state feedback, 267, 306 state feedback, problem, 268 output regulation, with saturating actuator, 270, 307 assumptions, 268, 271, 272, 307, 308

15 390 Index error feedback design, 290, 324 error feedback law, 295 error feedback, problem, 271 example, 297 state feedback design, 279, 315 state feedback law, 284, 320 state feedback, problem, 270, 308 performance degradation, 1 periodic trajectory, 62, 346 persistent disturbance, 211 piecewise-linear control(plc), 80 PLC,6 Poincare-Bendixon Theorem, 62, 336 positive definite, 6 positive limit set, 59 positive semi-definite, 6 practical stabilization, 113, 114 semi-global, 113, 114 semi-global, theorem, 115 quadratic Lyapunov function, 164 reachable, 15, 41 in a finite time, 15, 40 reachable region, 15, 16, 41 at a finite time, 15, 40 recurrent neural network, 327 reference signal, 265, 267, 306 regulatable region, 271, 309 asymptotic, definition, 273, 310 definition, 272, 309 description, theorem, 274, 311 reset windup, 2 saturated linear feedback, 57 high gain, 67, 72, 242 saturation of actuator, 1, 57 of sensor, 327, 362 of state, , 344 saturation function, 3 coupled, 263 scalar, 3 vector, 3 Schur complements, 171 Schur stable, 7 semi-global practical stabilization example, 144 problem, 114 theorem, 115 semi-global stabilization, 67, 74, 195, 199, 205 example, 73, 79, 107, 207 theorem, 67, 77, 110,204, 207 with regional performance, 195 with regional performance, example, 207 semi-globalization, 199, 205 semi-stable, 7 sensor saturation, 327, 362 example, 370 semi-global stabilizability, theorem, 363 semi-global stabilization, example, 370

16 Inde_x 3_9_1 set semi-global stabilization, feedback laws, 365 shape reference, 159, 214, 223 size of, 159 set invariance, 234 contractive, 234 contractive, condition, , 175, 176, 259, 260 contractive, definition, 164, 174 contractive, necessary and sufficient condition, 167, 177, 238 definition, 164, 174 set invariance, with disturbance, 213, 255 analysis, 222 analysis, problem, 214 condition, 215, 223 definition, 222 enlargement, LM1 formulation, 215, 224 enlargement, problem, 214, 222 problem, 222 property, 256 strictly, 213, 255 strictly, definition, 222 shape reference set, 159, 184, 214, 223 stability anti-stable, 7 asymptotically stable, 7 exponentially unstable, 7 globally asymptotically stable, 328, 344 locally asymptotically stable, 329, 345 neutrally stable, 7, 230, 271, 308 of a matrix, 7 of a system, 7 semi-stable, 7 stable, 7 stabilizability, 8, 268, 307, 363 stabilization, 55 global, at set of equilibria, 86 global, theorem, 89 local, 55 semi-global, 55, 105, 108, 199, 205 semi-global, example, 207 semi-global, theorem, 67, 77, 110, 204, 207 state saturation, , 344 bounded global attractor, 334 global asymptotic stability, theorem, 330 global boundedness, condition, 334 globally asymptotic stability, theorem, 345 periodic trajectories, example, 347 with one saturated state, 347, 352 state transition map, 57, 59 strictly convex, 8, 16, 33 supremum, 184

17 392 Index switching control law, 81, 110, 143, 197 switching surface, 231, 242 time optimal control, 1, 12, 229 time-reversed system, 15, 40, 58, 86 tracking error, 267, 306, 324 TRANS3 aircraft, 144, 297 longitudinal dynamics, 144, 297 transmission polynomial, 270 vector field state saturation, horizontal shifting property, 333 state saturation, vertical shifting property, 332 with disturbance, 132 with disturbance, properties, 132, 133 with state saturation, 331 with state saturation, partition, 331 vertex, 41