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Student Number 953203060 Author Jia-ren Wan(萬嘉仁) Author's Email Address No Public. Statistics This thesis had been viewed 1242 times. Download 295 times. Department Mechanical Engineering Year 2007 Semester 2 Degree Master Type of Document Master's Thesis Language zh-TW.Big5 Chinese Title Relaxed, dissipative fuzzy control - Polya theorem Date of Defense 2008-06-20 Page Count 100 Keyword Dissipative control Linear matrix inequality Polya's theorem Relaxed condition T-S model Abstract In this thesis, we propose a general quadratic dissipative state feedback control method to solve a stabilization problem for fuzzy system with sector-bounded type nonlinearities at the input.

The problem covers the bounded real, positive real and sector-bounded performance as a special case by choosing the corresponding quadratic supply rate.

Moreover, we also prove necessary and sufficient conditions to state feedback controllers ensuring quadratic stability for Takagi-Sugeno fuzzy systems in theory.

But our main objective is to generate a family of linear matrix inequalities based on an extension of Polya theorem (a.k.a. Matrix-valued Polya theorem).

The proposed conditions are stated as progressively less conservative sets of linear matrix inequalities,

allowing us to obtain a solution for the quadratic stabilizability problem whenever a solution exists.

At last, an additional relaxed condition is also provided, relying on the use of slack matrix variables.

All proposed methods will be shown via theoretical analysis and numerical simulations.Table of Content 論文摘要 i

Abstract iv

誌謝 v

圖目 x

第一章 簡介 1

1.1 文獻回顧 1

1.2 研究動機 3

1.3 論文結構 4

1.4 符號標記 5

1.5 預備定理 7

1.6 耗散性之物理意義 8

第一部份:耗散性控制(Dissipativecontrol) 13

第二章 系統架構與耗散性檢測條件 13

2.1 系統架構 13

2.1.1 廣義非線性系統 13

2.1.2 非線性模糊系統 15

2.2 檢測耗散性條件 16

第三章 有界非線性輸入控制器之設計 23

3.1 有界非線性輸入系統 23

3.2 狀態回饋控制器 25

第四章 電腦模擬一 32

4.1 連續系統 32

4.1.1 系統模型 32

4.1.2 求解 34

4.2 離散系統 40

4.2.1 系統模型 40

4.2.2 求解 43

第二部份:波雅定理之代數應用 50

第五章 模糊閉迴路系統之充要條件 50

5.1 波雅定理(P′olya’sTheorem) 50

5.2 矩陣波雅定理(Matrix-valued P′olya's Theorem) 52

5.3 範例及寬鬆性 56

第六章 電腦模擬二 58

6.1 解空間 58

6.2 倒單擺系統 63

6.2.1 系統描述 63

6.2.2 求解 63

第三部份:具鬆弛矩陣變數之寬鬆環境 68

第七章 波雅定理之寬鬆環境 69

7.1 鬆弛矩陣變數 69

7.2 範例 76

第八章 電腦模擬三 82

8.1 解空間比較 82

8.1.1 連續系統 82

8.1.2 離散系統 86

8.2 耗時比較 90

第九章 結論與未來研究方向 91

9.1 總結 91

9.2 未來研究方向 92

參考文獻 94Reference [1] K.Tanaka, T.Ikeda, and H.Wang, “Fuzzy regulators and fuzzy observers:relaxed stability conditions and LMI-based designs,” IEEE Trans.Fuzzy Systems, vol. 6, no.2,pp.250–265,May 1998.

[2] E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Systems, vol. 8, no. 5, pp. 523–534, Oct.2000.

[3] X.Liu and Q.Zhang, “New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,” Automatica, vol. 39, pp. 1571–1582, 2003.

[4] C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, and C.-S. Lee, “A new LMI-based approach to relaxed quadratic stabilzation of T-s fuzzy control systems,” IEEE Trans.Fuzzy Systems, vol. 14, no. 3, pp. 386–397, 2006.

[5] A. Sala and C. Ari˜no, “Asymptotically necessary and suﬃcient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem,” Fuzzy Sets and Systems,2007,doi:10.1016/j,fss.2007.06.016.

[6] J.Willems, “Dissipative dynamical systems-Part1:General theory,” Arch.Ratio-nalMech.Analy., vol. 45, pp. 321–351, 1972.

[7] ——, “Dissipative dynamical systems-Part2: Linear systems with quadratic supply rates,” Arch.Rational Mech.Analy., vol. 45, pp. 352–393, 1972.

[8] D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans.Automatic Control, vol. 21, pp. 708–711, 1976.

[9] ——, “Dissipative dynamical systems: Basic input-output and state properties,”

J.franklin Inst., vol. 309, pp. 327–357, 1980.

[10] L. Xie, “Robust output feedback dissipative control for uncertain nonlinear systems,” in Intelligent Control and Automation, 2004. WCICA 2004. Fifth World Congress on, vol.1,Hangzhou,China,Jun.2004,pp.809–813.

[11] S.Yuliar and M.James, “General dissipative output feedback control for nonlinear systems,” in Proceedings of the 34th IEEE Conference on Decision and Control, vol.3,New Orleans,LA,Dec.1995,pp.2221–2226.

[12] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM.

[13] P.Gahinet, A.Nemirovskii, A.Laub, and M.Chilali, “The LMI control toolbox,” in Proceedings of the 33rd IEEE Conferenceon Decision and Control, LakeBuena Vista,FL,USA,Dec.1994,pp.2038–2041.

[14] Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms. Philadelphia, PA: SIAM, 1994.

[15] S. Yuliar and M. James, “Stabilization of linear systems with sector bounded nonlinearities at the input and output,” in Proc. of the 36th Conf. on Deci. & Contr., vol. 3, Kobe, JP, Dec. 1996, pp. 4759–4764.

[16] S. Gupta, “ Robust Stabilization of Uncertain Systems Based on Energy Dissipation Concepts,” NASAContractorReport4713,1996.

[17] V. Chellaboina, W. Haddad, and A. Kamath, “A dissipative dynamical systems approach to stability analysis of time delay systems,” in American Control Conference, 2003. Proceedings of the 2003, vol. 1, Denver,Colorado, Jun. 2003, pp. 363–368.

[18] L.Xie, “Robust Dissipative Control for Uncertain Descriptor Linear Systems with TimeDelay,”in The Sixth World Congresson Intelligent Control and Automation, vol.1,Dalian,China,Jun.2006,pp.2327–2333.

[19] S. Xie, L. Xie, and C. de Souza, “Robust dissipative control for linear systems with dissipative uncertainty,” Int.J.Contr., vol. 70, no. 2, pp. 169–191, 1998.

[20] Z. Tan, Y. Soh, and L.Xie, “Dissipative control for linear discrete-time systems,” Automatica, vol. 35, pp. 1557–1564, 1999.

[21] S. Yuliar, M. James, and J. Helton, “Dissipative Control Systems Synthesis with Full State Feedback,” Mathematics of Control, Signals, and Systems,vol.11,no.4, 1998.

[22] T. Takagi and M. Sugeno, “Fuzzy identiﬁcation of systems and its applications to modelling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, no. 1, pp. 116–132,Jan.1985.

[23] K. Tanaka and H. Wang, Fuzzy Control Systems Design: A Linear Matrix Inequality Approach. New York, NY: John Wiley & Sons, Inc., 2001.

[24] H. Uang, “On the dissipativity of nonlinear systems: fuzzy control approach,” Fuzzy Sets and Systems, vol. 156, pp. 185–207, 2005.

[25] J. Lo and J. Wan, “Dissipative control to fuzzy systems with nonlinearity at the input,” in The 2007 CACS International Automatic Control Conference, Taichung,Tw,Nov.2007,pp.329–334.

[26] J. Lo and D. Wu, “Dissipative ﬁltering for nonlinear fuzzy systems,” in The2007 CACSInternationalAutomaticControlConference,Taichung,Tw,Nov.2007,pp. 623–627.

[27] Y.Li,Y.fu,and G.Duan, “Robust dissipative control for T-S fuzzy systems with time-delays,” in IEEE ISIE,Montreal,Ca,Jul.2006,pp.97–101.

[28] T. Hu, Z. Lin, and B. Chen, “An anlysis and design method for linear systems subject toactuatorsaturationanddisturbance,” Automatica,vol.38,pp.351–359, 2002.

[29] ——, “ Analysis and design for discrete-time linear systems subject to actuator saturation,” Syst. & Contr. Lett., vol. 45, pp. 97–112, 2002.

[30] C. Pittet, S. Tarbouriech, and C. Burgat, “Stability regions for linear systems with saturating controls via circle and Popov criteria,” in Proc. of the 36th IEEE Conf. on Decision & Control,SanDiego,CA.,Dec.1997,pp.4518–4521.

[31] J. da Silva and S. Jr. Tarbouriech, “ Antiwindup design with guaranteed regions of stability: an LMI-based approach,” IEEE Trans. Automatic Control, vol. 50, pp.106–111,2005.

[32] Y. Cao, Z. Lin, and B. Chen, “An output feedback H∞controller design for linear systems subject to sensor nonlinearities,” IEEE Trans. Circuits and Syst.

I: Fundamental Theory and Applications, vol. 50, no. 7, pp. 914–921, Jul. 2003.

[33] S. Lee, E. Kim, H. Kim, and M. Park, “Robust Analysis and Design for Discrete-Time Nonlinear Systems Subject to Actuator Saturation via Fuzzy Control,” IE-ICE Transactions on Fundamentals of Electronics, vol. E88VA, 2005.

[34] Y. Cao and Z. Lin, “Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation,” IEEE Trans. Fuzzy Systems, vol. 11, no. 1, pp. 57–67, Feb. 2003.

[35] J. Lo and M. Lin, “Feedback control via Popov for Fuzzy Systems with input saturations,” in Proc. the 13th IEEE Conf. Fuzzy Systems, Budapest, Hu, Jul. 2004,pp.1221–1226.

[36] J. Lo and E. Chen, “State feedback control via circle criterion for uncertain fuzzy systems,” in Proc.2004Conf.Auto.Contr., vol. 1, Taipei, TW, Mar. 2004.

[37] G.Feng, “A survey on analysis and design of model-based fuzzy control systems.” IEEE Trans. Fuzzy Systems, vol. 14, no. 5, pp. 676–697, Oct. 2006.

[38] A. Sala, T. Guerra, and R. Babuska, “Perspectives of fuzzy systems and control,” Fuzzy Sets and Systems, vol. 156, pp. 432–444, Jun. 2005.

[39] T. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form,” Automatica, vol. 40, pp.823–829,2004.

[40] S. Zhou, G. Feng, J. Lam, and S. Xu, “Robust H∞ control for discrete-time fuzzy systems via basis-dependent Lyapunov functions,” InformationSciences,vol.174, pp.197–217,2004.

[41] S. Zhou, J. Lam, and W. Zheng, “Control Design for Fuzzy Systems Based on Relaxed Nonquadratic Stability and H∞ Performance Conditions,” IEEE Trans. Fuzzy Systems, vol. 15, pp. 188–199, 2007.

[42] K. Tanaka, T. Hori, and H. Wang, “A multiple Lyapunov Function Approach to Stabilization of Fuzzy Control Systems,” IEEE Trans. Fuzzy Systems, vol. 11, no.4,pp.582–589,Aug.2003.

[43] K. Tanaka, H. Ohtake, and H. Wang, “A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions,” IEEE Trans. Fuzzy Systems, vol. 15, pp. 333–341, Jun. 2007.

[44] M. de Oliveira, J. Geromel, and J. Bernussou, “Extended H2and H∞norm characterizations and controller parameterizations for discrete-time systems,” Int.J. Contr., vol. 75, no. 9, pp. 666–679, 2002.

[45] S.-W.Kau, H.-J.Lee, C.-M.Yang, C.-H.Lee, L.Honga, andC.-H.Fang, “Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties,” Fuzzy Sets and Systems, vol. 158, pp. 135–146, 2007.

[46] M.Teixeira, E.Assuncao, and R.Avellar, “On relaxed LMI-based design for fuzzy regulators and fuzzy observers,” IEEE Trans. Fuzzy Systems, vol. 11, no. 5, pp. 613–623,2003.

[47] B.Ding,H.Sun,andP.Yang, “Further studies onLMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno’s form,” Automatica, vol. 42, pp.503–508,2006.

[48] D. Ramos and P. Peres, “An LMI condition for the robust stability of uncertain continuous-time linear systems,” IEEE Trans. Automatic Control, vol. 47, no. 4, pp.675–678,Apr.2002.

[49] M.deOliveira and J.Geromel, “A class of robust stability conditions where linear parameter dependence of the Lyapunov function is a necessary condition for arbitrary parameter dependencestar,” Syst. & Contr. Lett., vol. 54, pp. 1131–1134, Nov.2005.

[50] R. Oliveira and P. Peres, “LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability,” in Proc. of 44th IEEE Conf. on Deci and Contr, Seville,Spain,Dec.2005,pp.1660–1665.

[51] R.C.Oliveira and P.L.Peres, “LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions,” Syst. & Contr. Lett., vol.55,pp.52–61,Jan.2006.

[52] M.deOliveira,J.Bernussou,andJ.Geromel,“A new discrete-time robust stability condition,” Syst. & Contr. Lett., vol. 37, pp. 261–265, 1999.

[53] J. Daafouz and J. Bernussou, “Parameter dependent Lyapunov functions for dis-crete time systems with time varying parametric uncertainties,” Syst. & Contr. Lett., vol. 43, pp. 355–359, Aug. 2001.

[54] C. Arino and A. Sala, “Design of Multiple-Parameterisation PDC Controllers via Relaxed Conditions for Multi-Dimensional Fuzzy Summations,” in IEEE International Conference on Fuzzy Systems,London,UK,July 2007,pp.1–6.

[55] G. Hardy, J. Littlewood, and G. P′olya, Inequalities, second edition. Cambridge, UK.:Cambridge University Press,1952.

[56] V.Power and B.Reznick, “A new bound for P′olya’s Theorem with applications to polynominals positiveon polyhedra,”J.Pure Appl. Algebra,vol.164,pp.221–229, 2001.

[57] J. de Loera and F. Santos, “An eﬀective version of Polya’s theorem on positive deﬁniteforms,” Journal of Pure and Applied Algebra,vol.108,pp.231–240,1996.

[58] C. Scherer, “Higher-order relaxations for robust LMI problems with veriﬁcations for exactness,” in Proc. of 42th IEEE Conf. on Deci and Contr, Maui,Hawaii, USA,Dec.2003,pp.4652–4657.

[59] ——, “Relaxations for robust linear matrix inequality problems with veriﬁcations for exactness,” SIAM Journal on Matrix Analysis and Applications, vol. 27, pp. 365–395,2005.

[60] R.Oliveira and P.Peres, “Stability of polytopes of matrices via aﬃne parameter-dependent Lyapunov functions: Asymptotically exact LMI conditions,” Linear Algebra and its Applications, vol. 405, pp. 209–228, 2005.

[61] V. Montagner, R. Oliveira, P. Peres, and P.-A. Bliman, “Linear matrix inequality characterisation for H∞ and H2guaranteed cost gain-scheduling quadratic stabilisation of linear time-varying polytopic systems,” IET Control Theory and Applications, vol. 1, pp. 1726–1735, 2007.

[62] R.Oliveira and P.Peres, “Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. Automatic Control, vol. 52, no. 7, pp. 1334–1340, Jul. 2007.

[63] V.F.Montagner, R.C.L.F. Oliveira, and P.L.D.Peres,“Necessary and suﬃcient LMI conditions to compute quadratically stabilizing state feedback controllers for Takagi-Sugeno systems,” in American Control Conference, 2007. ACC ’07,New YorkCity,USA,2007,pp.4059–4064.

[64] H. K. Khalil, Nonlinear Systems. New York, NY.: Macmillian Publishing Co., 1992.

[65] P.Gahinet, A.Nemirovskii, A.Laub, and M.Chilali, “LMI control Toolbox User’s Guide,” in TheMathWorksInc., Natick,MA.

[66] B.Ding,H.Sun,andP.Yang, “Further studies onLMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno’s form,” Automatica, vol. 43, pp.503–508,2006.

[67] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans. Automatic Control, vol. 42, no. 7, pp. 896– 911,Jul.1997.

[68] V.Montagner, P.Peres, S.Tarbouriech, and I.Que innec, “Improved Estimation of Stability Regions for Uncertain Linear Systems with Saturating Actuators: an LMI-based Approach,” in Decision and Control, 2006 45th IEEE Conference on, 2006,SanDiego,CA,USA,Dec.2006,pp.5429–5434.

[69] J.LoandM.Lin, “Robust H∞ nonlinear control via fuzzy static output feedback,” IEEE Trans. Circuits and Syst. I: Fundamental Theory and Applications, vol. 50, no.11,pp.1494–1502,Nov.2003.

[70] K. R. GV Reklaitis, ARavindran, Engineering Optimization: Methods and Applications 2th. Cambridge, UK.: Wiley, 2006.Advisor Ji-chang Lo(羅吉昌)

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