Homogeneous polynomial forms for robustness analysis of uncertain systems

Lecture Notes in Control and Information Sciences

Volumn 390, Issue , 2009, Pages

  Chesi, Graziano ^a   Garulli, Andrea ^b   Tesi, Alberto ^c   Vicino, Antonio ^b  

^a UNIVERSITY OF HONG KONG   (Hong Kong)

^b UNIVERSITY OF SIENA   (Italy)

^c UNIVERSITY OF FLORENCE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70450130982     PISSN: 01708643     EISSN: None     Source Type: Book Series    
DOI: None     Document Type: Article

References (150)

1. Almeida, H.L.S., Bhaya, A., Falcao, D.M., Kaszkurewicz, E.: A team algorithm for robust stability analysis and control design of uncertain time-varying linear systems using piecewise quadratics Lyapunov functions. Int. Journal of Robust and Nonlinear Control 11, 357-371 (2001)
2. Altafini, C.: Homogeneous polynomial forms for simultaneous stabilizability of families of linear control systems: a tensor product approach. IEEE Transactions on Automatic Control 51(9), 1566-1571 (2006)
3. Barmish, B.R.: New Tools for Robustness of Linear Systems. Mcmillan Publishing Company, New York (1994)
4. Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York (1974)
5. Bhattacharyya, S.P., Chapellat, H., Keel, L.H.: Robust Control: The Parametric Approach. Prentice Hall, NJ (1995)
6. Blanchini, F.: Nonquadratic Lyapunov functions for robust control. Automatica 31, 451-461 (1995)
7. Blanchini, F., Miani, S.: On the transient estimate for linear systems with time-varying uncertain parameters. IEEE Trans. on Circuits and Systems I 43, 592-596 (1996)
8. Blanchini, F., Miani, S.: A universal class of smooth functions for robust control. IEEE Trans. on Automatic Control 44(3), 641-647 (1999)
9. Blekherman, G.: There are significantly more nonnegative polynomials than sums of squares. Israel Journal of Mathematics 153(1), 355-380 (2006)
10. Bliman, P.-A.: Nonconservative LMI approach to robust stability for systems with uncertain scalar parameters. In: IEEE Conf. on Decision and Control, Las Vegas, Nevada, pp. 305-310 (2002)
11. Bliman, P.-A.: A convex approach to robust stability for linear systems with uncertain scalar parameters. SIAMJournal on Control and Optimization 42(6), 2016-2042 (2004)
12. Bliman, P.-A., Oliveira, R.C.L.F., Montagner, V.F., Peres, P.L.D.: Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex. In: IEEE Conf. on Decision and Control, San Diego, CA, pp. 1486-1491 (2006)
13. Bose, N.K.: Applied Multidimensional Systems Theory. Van Rostrand Reinhold, New York (1982)
14. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)
15. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
16. Boyd, S., Yang, Q.: Structured and simultaneous Lyapunov functions for system stability problems. Int. Journal of Control 49(6), 2215-2240 (1989)
17. Brayton, R.K., Tong, C.H.: Stability of dynamical systems: a constructive approach. IEEE Trans. on Circuits and Systems 26, 224-234 (1979)
18. Brockett, R.W.: Finite dimensional linear systems. John Wiley & Sons, New York (1970)
19. Brockett, R.W.: Lie algebra and Lie groups in control theory. In: Mayne, D.Q., Brockett, R.W. (eds.) Geometric Methods in Systems Theory, pp. 43-82. Reidel, Dordrecht (1973)
20. Chen, Y.H., Chen, J.S.: Robust control of uncertain systems with time-varying uncertainty: a computer-aided setting. In: Skowronski, J.M., Flashner, H., Guttalu, R.S. (eds.) Positive Polynomials in Control. Lecture Notes in Control and Information Sciences, vol. 151, pp. 97-114. Springer, New York (1991)
21. Chesi, G.: Computing output feedback controllers to enlarge the domain of attraction in polynomial systems. IEEE Trans. on Automatic Control 49(10), 1846-1850 (2004)
22. Chesi, G.: Estimating the domain of attraction for uncertain polynomial systems. Automatica 40(11), 1981-1986 (2004)
23. Chesi, G.: Establishing stability and instability of matrix hypercubes. Systems and Control Letters 54(4), 381-388 (2005)
24. Chesi, G.: Establishing tightness in robust H-infinity analysis via homogeneous parameter-dependent Lyapunov functions. Automatica 43(11), 1992-1995 (2007)
25. Chesi, G.: On the gap between positive polynomials and SOS of polynomials. IEEE Trans. on Automatic Control 52(6), 1066-1072 (2007)
26. Chesi, G.: On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems. Automatica 44(11), 2973-2976 (2008)
27. Chesi, G.: Tightness conditions for semidefinite relaxations of forms minimizations. IEEE Trans. on Circuits and Systems II 55(12), 1299-1303 (2008)
28. Chesi, G.: Camera displacement via constrained minimization of the algebraic error. IEEE Trans. on Pattern Analysis and Machine Intelligence 31(2), 370-375 (2009)
29. Chesi, G.: On the minimum stable commutation time for switching nonlinear systems. IEEE Trans. on Automatic Control 54(6), 1284-1289 (2009)
30. Chesi, G.: Visual servoing path-planning via homogeneous forms and LMI optimizations. IEEE Trans. on Robotics 25(2), 281-291 (2009)
31. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: A convex approach to the characterization of the frequency response of ellipsoidal plants. Automatica 38(2), 249-259 (2002)
32. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: LMI-based construction of homogeneous Lyapunov functions for systems with structured uncertainties. In: 41st IEEE Conf. on Decision and Control, Las Vegas, Nevada, pp. 281-286 (2002)
33. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach. Int. Journal of Robust and Nonlinear Control 13(13), 1239-1257 (2003)
34. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Homogeneous Lyapunov functions for systems with structured uncertainties. Automatica 39(6), 1027-1035 (2003)
35. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Robust stability for polytopic systems via polynomially parameter-dependent Lyapunov functions. In: 42nd IEEE Conf. on Decision and Control, Maui, Hawaii, pp. 4670-4675 (2003)
36. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Solving quadratic distance problems: an LMI-based approach. IEEE Trans. on Automatic Control 48(2), 200-212 (2003)
37. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Parameter-dependent homogeneous Lyapunov functions for robust stability of linear time-varying systems. In: 43rd IEEE Conf. on Decision and Control, Paradise Island, Bahamas, pp. 4095-4100 (2004)
38. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions. IEEE Trans. on Automatic Control 49(7), 1211-1216 (2004)
39. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems. Int. Journal of Robust and Nonlinear Control 15(1), 35-49 (2005)
40. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: An LMI-based technique for robust stability analysis of linear systems with polynomial parametric uncertainties. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control. Lecture Notes in Control and Information Sciences, vol. 312. Springer, London (2005)
41. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Polynomially parameter-dependent Lyapunov functions for robust HHHperformance analysis. In: 16th IFAC World Congress on Automatic Control, Prague, Czech Republic (2005)
42. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE Trans. on Automatic Control 50(3), 365-370 (2005)
43. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions. Automatica 43(2), 309-316 (2007)
44. Chesi, G., Garulli, A., Vicino, A., Cipolla, R.: Estimating the fundamental matrix via constrained least-squares: a convex approach. IEEE Trans. on Pattern Analysis and Machine Intelligence 24(3), 397-401 (2002)
45. Chesi, G., Henrion, D. (eds.): Special Issue on Positive Polynomials in Control. IEEE Trans. on Automatic Control, vol. 54 (2009)
46. Chesi, G., Hung, Y.S.: Image noise induced errors in camera positioning. IEEE Trans. on Pattern Analysis and Machine Intelligence 29(8), 1476-1480 (2007)
47. Chesi, G., Hung, Y.S.: Analysis and synthesis of nonlinear systems with uncertain initial conditions. IEEE Trans. on Automatic Control 53(5), 1262-1267 (2008)
48. Chesi, G., Hung, Y.S.: Establishing convexity of polynomial Lyapunov functions and their sublevel sets. IEEE Trans. on Automatic Control 53(10), 2431-2436 (2008)
49. Chesi, G., Hung, Y.S.: Stability analysis of uncertain genetic SUMregulatory networks. Automatica 44(9), 2298-2305 (2008)
50. Chesi, G., Tesi, A., Vicino, A., Genesio, R.: A convex approach to a class of minimum norm problems. In: Garulli, A., Tesi, A., Vicino, A. (eds.) Robustness in Identification and Control, pp. 359-372. Springer, London (1999)
51. Chesi, G., Tesi, A., Vicino, A., Genesio, R.: On convexification of some minimum distance problems. In: 5th European Control Conf., Karlsruhe, Germany (1999)
52. Chesi, G., Tesi, A., Vicino, A., Genesio, R.: An LMI approach to constrained optimization with homogeneous forms. Systems and Control Letters 42(1), 11-19 (2001)
53. Choi, M., Lam, T., Reznick, B.: Sums of squares of real polynomials. In: Symposia in Pure Mathematics, pp. 103-126 (1995)
54. Cobb, J.D., DeMarco, C.L.: The minimal dimension of stable faces to guarantee stability of a matrix polytope. IEEE Trans. on Automatic Control 34(9), 990-992 (1989)
55. Curto, R.E., Fialkow, L.A.: The truncated complex K-moment problem. Transactions of the American Mathematical Society 352(6), 2825-2855 (2000)
56. Dasgupta, S., Chockalingam, G., Anderson, B.D.O., Fu, M.: Lyapunov functions for uncertain systems with applications to the stability of time varying systems. IEEE Trans. on Automatic Control 41, 93-106 (1994)
57. de Klerk, E., Laurent, M., Parrilo, P.: On the equivalence of algebraic approaches to the minimization of forms on the simplex. In: Henrion, D., Garulli, A. (eds.) Positive polynomials in Control. Lecture Notes in Control and Information Sciences, pp. 121-132. Springer, Heidelberg (2005)
58. de Oliveira, M.C., Bernussou, J., Geromel, J.C.: A new discrete-time robust stability condition. Systems and Control Letters 37, 261-265 (1999)
59. de Oliveira, P.J., Oliveira, R.C.L.F., Leite, V.J.S., Montagner, V.F., Peres, P.L.D.: HH guaranteed cost computation by means of parameter-dependent Lyapunov functions. Automatica 40(6), 1053-1061 (2004)
60. Ebihara, Y., Onishi, Y., Hagiwara, T.: Robust performance analysis of uncertain LTI systems: Dual LMI approach and verifications for exactness. In: Chesi, G., Henrion, D. (eds.) Special Issue on Positive Polynomials in Control. IEEE Trans. on Automatic Control, vol. 54, pp. 938-951 (2009)
61. El-Samad, H., Prajna, S., Papachristodoulou, A., Doyle, J., Khammash, M.: Advanced methods and algorithms for biological networks analysis. Proceedings of the IEEE 94(4), 832-853 (2006)
62. Fujisaki, Y., Sakuwa, R.: Estimation of asymptotic stability regions via homogeneous polynomial Lyapunov functions. International Journal of Control 79(6), 617-623 (2006)
63. Fukumoto, H., Fujisaki, Y.: Exact robust H2 performance analysis for linear singleparameter dependent systems. In: IEEE Conf. on Decision and Control, New Orleans, Louisiana, pp. 2743-2748 (2007)
64. Fuller, A.T.: Conditions for a matrix to have only characteristic roots with negative real parts. Journal of Mathematical Analysis and Applications 23, 71-98 (1968)
65. Gahinet, P., Apkarian, P., Chilali, M.: Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Trans. on Automatic Control 41(3), 436-442 (1996)
66. Garulli, A., Masi, A., Vicino, A.: Convex relaxations for quadratic distance problems. In: IEEE Conf. on Decision and Control, Cancun (Mexico), December 2008, pp. 5444-5449 (2008)
67. Genesio, R., Tartaglia, M., Vicino, A.: On the estimation of asymptotic stability regions: State of the art and new proposals. IEEE Trans. on Automatic Control 30, 747-755 (1985)
68. Geromel, J.C., Colaneri, P.: Robust stability of time varying polytopic systems. Systems and Control Letters 55(1), 81-85 (2006)
69. Goebel, R., Teel, A.R., Hu, T., Lin, Z.: Conjugate convex Lyapunov functions for dual linear differential inclusions. IEEE Transactions on Automatic Control 51(4), 661-666 (2006)
70. Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988)
71. Henrion, D., Garulli, A. (eds.): Positive Polynomials in Control. Lecture Notes in Control and Information Sciences, vol. 312. Springer, London (2005)
72. Henrion, D., Lasserre, J.-B.: GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi. ACM Trans. on Mathematical Software 29(2), 165-194 (2003)
73. Henrion, D., Lasserre, J.-B.: Detecting global optimality and extracting solutions in GloptiPoly. In: Henrion, D., Garulli, A. (eds.) Positive Polynomials in Control. Lecture Notes in Control and Information Sciences, vol. 312. Springer, London (2005)
74. Henrion, D., Lasserre, J.-B.: Convergent relaxations of polynomial matrix inequalities and static output feedback. IEEE Trans. on Automatic Control 51(2), 192-202 (2006)
75. Henrion, D., Sebek, M., Kucera, V.: Positive polynomials and robust stabilization with fixed-order controllers. IEEE Trans. on Automatic Control 48, 1178-1186 (2003)
76. Hershkowitz, D.: Recent directions in matrix stability. Linear Alg. Appl. 171, 161-186 (1992)
77. Hol, C.W.J., Scherer, C.W.: Computing optimal fixed order HH-synthesis values by matrix sum of squares relaxations. In: IEEE Conf. on Decision and Control, Paradise Island, Bahamas (2004)
78. Hol, C.W.J., Scherer, C.W.: A sum-of-squares approach to fixed order HH-synthesis. In: Henrion, D., Garulli, A. (eds.) Positive polynomials in Control. Lecture Notes in Control and Information Sciences, pp. 45-71. Springer, Heidelberg (2005)
79. Horisberger, H.P., Belanger, P.R.: Regulators for linear time invariant plants with uncertain parameters. IEEE Trans. on Automatic Control 21(5), 705-708 (1976)
80. Hu, T., Lin, Z.: Composite quadratic Lyapunov functions for constrained control systems. IEEE Transactions on Automatic Control 48(3), 440-450 (2003)
81. Jarvis-Wloszek, Z., Feeley, R., Tan, W., Sun, K., Packard, A.: Control applications of sum of squares programming. In: Henrion, D., Garulli, A. (eds.) Positive polynomials in Control. Lecture Notes in Control and Information Sciences, pp. 3-22. Springer, Heidelberg (2005)
82. Jarvis-Wloszek, Z., Packard, A.K.: An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions. In: IEEE Conf. on Decision and Control, Las Vegas, Nevada, pp. 287-292 (2002)
83. Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. on Automatic Control 43, 555-559 (1998)
84. Khalil, H.K.: Nonlinear Systems. McMillan Publishing Company, New York (1992)
85. Lasserre, J.-B.: Global optimization with polynomials and the problem of moments. SIAM Journal of Optimization 11(3), 796-817 (2001)
86. Lasserre, J.-B., Netzer, T.: Sos approximations of nonnegative polynomials via simple high degree perturbations. Math. Zeitschrift 256, 99-112 (2006)
87. Lasserre, J.-B., Netzer, T.: Sufficient conditions for a polynomial to be a sum of squares. Arch. Math. 89, 390-398 (2007)
88. Lavaei, J., Aghdam, A.G.: Robust stability of LTI systems over semi-algebraic sets using sum-of-squares matrix polynomials. IEEE Trans. on Automatic Control 53(1), 417-423 (2008)
89. Lavaei, J., Sojoudi, S., Aghdam, A.: Rational Optimization using Sum-of-Squares Techniques. In: IEEE Conf. on Decision and Control, New Orleans (December 2007)
90. Leite, V.J.S., Peres, P.L.D.: An improved LMI condition for robust D-stability of uncertain polytopic systems. IEEE Trans. on Automatic Control 48(3), 500-504 (2003)
91. Lofberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: CACSD Conf., Taipei, Taiwan (2004)
92. Margaliot, M., Langholz, G.: Necessary and sufficient conditions for absolute stability: the case of second-order systems. IEEE Trans. on Circuits and Systems I 50(2), 227-234 (2003)
93. Montagner, V.F., Peres, P.L.D.: A new LMI condition for the robust stability of linear time-varying systems. In: IEEE Conf. on Decision and Control, Maui, Hawaii, pp. 6133-6138 (2003)
94. Montagner, V.F., Peres, P.L.D.: Robust stability and HHHperformance of linear timevarying systems in polytopic domains. International Journal of Control 77(15), 1343-1352 (2004)
95. Narendra, K.S., Taylor, J.H.: Frequency domain criteria for absolute stability. Academic Press Inc., New York (1973)
96. Narendra, K.S., Tripathi, S.S.: Identification and optimization of aircraft dynamics. Journal of Aircraft 10, 193-199 (1973)
97. Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., Roos, K., Terlaky, T., Shang, S. (eds.) High Performance Optimization. Kluwer, Norwell (2000)
98. Nesterov, Y., Nemirovsky, A.: Interior-Point Polynomial Methods in Convex Programming. SIAM, Philadelphia (1994)
99. Neto, A.T.: Parameter dependent Lyapunov functions for a class of uncertain linear systems: an LMI approach. In: IEEE Conf. on Decision and Control, Phoenix, Arizona, pp. 2341-2346 (1999)
100. Oishi, Y.: An asymptotically exact approach to robust semidefinite programming problems with function variables. In: Chesi, G., Henrion, D. (eds.) Special Issue on Positive Polynomials in Control. IEEE Trans. on Automatic Control, vol. 54, pp. 1000-1006 (2009)
101. Olas, A.: Construction of optimal Lyapunov function for systems with structured uncertainties. IEEE Trans. on Automatic Control 39, 167-171 (1994)
102. Oliveira, R.C.L.F., de Oliveira, M.C., Peres, P.L.D.: Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions. Systems and Control Letters 57(8), 680-689 (2008)
103. Oliveira, R.C.L.F., Peres, P.L.D.: Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations. IEEE Trans. on Automatic Control 52(7), 1334-1340 (2007)
104. Papachristodoulou, A.: Analysis of nonlinear time delay systems using the sum of squares decomposition. In: American Control Conf., Boston, MA (2004)
105. Papachristodoulou, A., Prajna, S.: A tutorial on sum of squares techniques for system analysis. In: American Control Conf., Portland, Oregon, pp. 2686-2700 (2005)
106. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology (2000)
107. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Programming Ser. B 96(2), 293-320 (2003)
108. Peaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J.: A new robust D-stability condition for real convex polytopic uncertainty. Systems and Control Letters 40, 21-30 (2000)
109. Peaucelle, D., Ebihara, Y., Arzelier, D., Hagiwara, T.: General polynomial parameterdependent Lyapunov functions for polytopic uncertain systems. In: International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, pp. 2238-2242 (2006)
110. Peaucelle, D., Sato, M.: LMI tests for positive definite polynomials: Slack variable approach. IEEE Trans. on Automatic Control 54(4), 886-891 (2009)
111. Peet, M., Papachristodoulou, A., Lall, S.: Positive forms and stability of linear timedelay systems. In: IEEE Conf. on Decision and Control, San Diego, CA, pp. 187-192 (2006)
112. Peres, P.L.D., Souza, S.R., Geromel, J.C.: Optimal H2 control for uncertain systems. In: American Control Conf. (1992)
113. Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Linear Algebra 127, 99-104 (1998)
114. Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477-492. Springer, Heidelberg (2004)
115. Prajna, S., Papachristodoulou, A.: Analysis of switched and hybrid systems-beyond piecewise quadratic methods. In: American Control Conf., Denver, CO (2003)
116. Prajna, S., Papachristodoulou, A., Parrilo, P.A.: SOSTOOLS: a general purpose sum of squares programming solver. In: IEEE Conf. on Decision and Control, Las Vegas, Nevada (2002)
117. Prajna, S., Parrilo, P.A., Rantzer, A.: Nonlinear control synthesis by convex optimization. IEEE Trans. on Automatic Control 49(2), 310-314 (2004)
118. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. 42, 969-984 (1993)
119. Radziszewski, B.: O najlepszej funkcji lapunowa. Technical report, IFTR Reports (1977)
120. Ramos, D.C.W., Peres, P.L.D.: A less conservative LMI condition for the robust stability of discrete-time uncertain systems. Systems and Control Letters 43, 371-378 (2001)
121. Reznick, B.: Extremal psd forms with few terms. Duke Mathematical Journal 45(2), 363-374 (1978)
122. Reznick, B.: Some concrete aspects of Hilbert's 17th problem. Contemporary Mathematics 253, 251-272 (2000)
123. Reznick, B.: On hilberts construction of positive polynomials (submitted for publication) (2007), http://www.math.uiuc.edu/-reznick
124. Sakuwa, R., Fujisaki, Y.: Robust stability analysis of single-parameter dependent descriptor systems. In: IEEE Conf. on Decision and Control and European Control Conf., Seville, Spain, pp. 2933-2938 (2005)
125. Sato, M., Peaucelle, D.: Robust stability/performance analysis for linear time-invariant polynomially parameter dependent systems using polynomially parameter-dependent Lyapunov functions. In: IEEE Conf. on Decision and Control, San Diego, CA, pp. 5807-5813 (2006)
126. Scherer, C.W.: Relaxations for robust linear matrix inequality problems with verifications for exactness. SIAM J. Matrix Anal. Appl. 27(1), 365-395 (2005)
127. Scherer, C.W.: LMI relaxations in robust control. European Journal of Control 12(1), 3-29 (2006)
128. Scherer, C.W., Hol, C.W.J.: Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Programming Ser. B 107(1-2), 189-211 (2006)
129. Schmuedgen, K.: The K-moment problem for compact semi-algebraic sets. Mathematische Annalen 289, 203-206 (1991)
130. Seiler, P.: Stability region estimates for SDRE controlled systems using sum of squares optimization. In: American Control Conf., vol. 3, pp. 1867-1872 (2003)
131. Shor, N.Z.: Class of global minimum bounds of polynomial functions (in Russian). Cybernetics 11-12, 731-734 (1987)
132. Shor, N.Z.: Nondifferentiable optimization and polynomial problems. Kluwer, Dordrecht (1998)
133. Stengle, G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87-97 (1974)
134. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11-12, 625-653 (1999)
135. Tan, W., Packard, A.: Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Trans. on Automatic Control 53(2), 565-571 (2008)
136. Tesi, A., Vicino, A.: Robust stability of state space models with structured uncertainties. IEEE Trans. on Automatic Control 35, 191-195 (1990)
137. Tibken, B.: Estimation of the domain of attraction for polynomial systems via LMI's. In: IEEE Conf. on Decision and Control, Sydney, Australia, pp. 3860-3864 (2000)
138. Topcu, U., Packard, A., Seiler, P.: Local stability analysis using simulations and sumof-squares programming. Automatica 44(10), 2669-2675 (2008)
139. Šiljak, D.D.: Nonlinear Systems: Parametric Analysis and Design. John Wiley & Sons, New York (1969)
140. Šiljak, D.D.: Parameter space methods for robust control design: a guided tour. IEEE Trans. on Automatic Control 34(7), 674-688 (1989)
141. van der Schaft, A.J.: L2-gain analysis of nonlinear state feedback HH-control. IEEE Trans. on Automatic Control 37(6), 770-784 (1992)
142. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38, 49-95 (1996)
143. Vicino, A., Tesi, A., Milanese, M.: Computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties. IEEE Trans. on Automatic Control 35, 835-841 (1990)
144. Wang, F., Balakrishnan, V.: Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems. IEEE Trans. on Automatic Control 47(5), 720-734 (2002)
145. Wu, F., Prajna, S.: Sos-based solution approach to polynomial lpv system analysis and synthesis problems. Int. Journal of Control 78(8), 600-611 (2005)
146. Xie, L., Shishkin, S., Fu, M.: Piecewise Lyapunov functions for robust stability of linear time-varying systems. Systems and Control Letters 31, 165-171 (1997)
147. Xu, J., Xie, L.: Homogeneous polynomial Lyapunov functions for piecewise affine systems. In: American Control Conf., pp. 581-586 (2005)
148. Zelentsovsky, A.L.: Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems. IEEE Trans. on Automatic Control 39(1), 135-138 (1994)
149. Zhang, X., Tsiotras, P., Iwasaki, T.: Parameter-dependent Lyapunov functions for exact stability analysis of single parameter dependent LTI systems. In: IEEE Conf. on Decision and Control, Maui, Hawaii, pp. 5168-5173 (2003)
150. Zhou, K., Khargonekar, P.P.: Stability robustness bounds for linear state-space models with structured uncertainties. IEEE Trans. on Automatic Control 32, 621-623 (1987)