离散时间参数不确定系统的分析与综合
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摘要
参数不确定型系统是一类重要的不确定系统,对其进行深入研究不仅具有理论价值,而且对工程应用也有指导意义。利用线性矩阵不等式和Lyapunov稳定性理论,本文对参数属于有界凸多面体的不确定系统(简称凸多面体系统)的稳定性及干扰抑制问题进行了较详细地研究。因已有研究成果多针对于连续时间系统,故本文主要论及离散时间凸多面体系统,特别是时变参数情况。
对不确定系统鲁棒稳定性研究具有开创性意义的当属二次稳定性(quadraticstability)思想,其核心是用单一的备选Lyapunov函数去处理整个系统族。既便系统族的每一成员皆是稳定的,也很难求得满足全体成员的公共Lyapunov函数,或者说,基于二次稳定性理念所得到的结果蕴涵着大的保守性。为了减少保守性,特别是涉及时变参数情况,人们想到用参数依赖的备选Lyapunov函数,这样既克服了备选Lyapunov函数的单一性,又能充分利用参数变化率(增量)上界信息(工程中多可估计出),这就是所谓的“参数依赖Lyapunov函数法”,基于此法所获结论的保守性显然大为降低。
基于参数依赖Lyapunov函数法,本文重点研究离散时间凸多面体系统的稳定性和干扰抑制问题。首先针对常参数系统,分别用独立Lyapunov函数和参数依赖Lyapunov函数方法,分析了系统的鲁棒稳定性和干扰至输出的增益,并设计了状态反馈控制器和静态输出反馈控制器使系统鲁棒镇定且具有一定干扰抑制能力。然后针对时变参数系统,也进行了上述相应研究,强调地是在求Lyapunov函数增量时加入了参数增量上界信息,所得结果虽然复杂,但可降低保守性(对参数慢时变系统尤为明显)。本文所获结果具有一定的普遍性,当所有时变参数的增量上界趋于零时,得到定常参数情况相应结果,当所有参数增量上界趋于无穷大时推得二次稳定结果。
对上述结果本文皆进行算例演算,通过计算结果的比较,验证了基于(时变)参数依赖Lyapunov函数方法所得结果的保守性明显降低,且可推出参数极端情况的有关结果。
An important class of uncertain systems is of parameter-uncertain type, deeply studying for them has not only theoretic value, but also engineering significance. analysis and synthesis of stability and robust control of this system is worth for theory, and also has been more and more recognized by engineering. Based on the theory of Linear Matrix Inequality and Lyapunov stability, problems of stability and disturbance attenuation were investigated in detail in this thesis for linear uncertain discrete-time systems whose parameters belong to polytopic domains. This kind of system is simply called as polytopic uncertain systems. Majority of investigations concentrate on continuous-time systems rather than discrete-time systems, so the polytopic uncertain discrete-time systems are mainly investigated in this paper, in particular, time-varying systems.
Quadratic stability has an important meaning on the research of robust stability for uncertain systems, which manages the whole group of systems by one alternative Lyapunov function. Even though every members of the group of systems are stable, the common Lyapunov function is hardly worked out to satisfy every members. In other words, the results based on quadratic stability have great conservativeness. To reduce the conservativeness, we select parameter-dependent Lyapunov functions to overcome the oneness of independent Lyapunov functions and take good use of the information of the upper bounds of the rate of variation or parameter increments, in particular, for time-varying systems. It's so called the approach of parameter-dependent Lyapunov function, which obviously has less conservative results.
There are two steps to investigate polytopic uncertain discrete-time systems in this paper. (1) For constant parameter-dependent systems, respectively, using the approaches based on independent Lyapunov function and parameter-dependent Lyapunov function, robust stability and disturbance attenuation of this kind of systems are analyzed and synthesized; then state feedback controller and output feedback controller are designed. (2) For parameter-varying systems, the same investigation before-mentioned has been done; emphatically, information of the upper bound of parameter increment has been considered, when we compute increment of parameter-dependent Lyapunov function for parameter-varying situation. Conservativeness of the results can be reduced however the results are more complicated, in particular, for slowly-varying parameters. Constant parameters can be regarded as the extremeness of time-varying parameters when the upper bounds of parameter increments tend to zero; Independent Lyapunov function can be regarded as parameter-dependent Lyapunov function when the upper bounds of parameter increments tend to infinity. Therefore, for parameter-varying systems, the results based on the approach of parameter-dependent Lyapunov function have universality.
The above results are all confirmed with examples. Through comparing the results of the examples, the approach of parameter-dependent Lyapunov function could reduce conservativeness of the results, and could educe the extremeness of parameters.
对不确定系统鲁棒稳定性研究具有开创性意义的当属二次稳定性(quadraticstability)思想,其核心是用单一的备选Lyapunov函数去处理整个系统族。既便系统族的每一成员皆是稳定的,也很难求得满足全体成员的公共Lyapunov函数,或者说,基于二次稳定性理念所得到的结果蕴涵着大的保守性。为了减少保守性,特别是涉及时变参数情况,人们想到用参数依赖的备选Lyapunov函数,这样既克服了备选Lyapunov函数的单一性,又能充分利用参数变化率(增量)上界信息(工程中多可估计出),这就是所谓的“参数依赖Lyapunov函数法”,基于此法所获结论的保守性显然大为降低。
基于参数依赖Lyapunov函数法,本文重点研究离散时间凸多面体系统的稳定性和干扰抑制问题。首先针对常参数系统,分别用独立Lyapunov函数和参数依赖Lyapunov函数方法,分析了系统的鲁棒稳定性和干扰至输出的增益,并设计了状态反馈控制器和静态输出反馈控制器使系统鲁棒镇定且具有一定干扰抑制能力。然后针对时变参数系统,也进行了上述相应研究,强调地是在求Lyapunov函数增量时加入了参数增量上界信息,所得结果虽然复杂,但可降低保守性(对参数慢时变系统尤为明显)。本文所获结果具有一定的普遍性,当所有时变参数的增量上界趋于零时,得到定常参数情况相应结果,当所有参数增量上界趋于无穷大时推得二次稳定结果。
对上述结果本文皆进行算例演算,通过计算结果的比较,验证了基于(时变)参数依赖Lyapunov函数方法所得结果的保守性明显降低,且可推出参数极端情况的有关结果。
An important class of uncertain systems is of parameter-uncertain type, deeply studying for them has not only theoretic value, but also engineering significance. analysis and synthesis of stability and robust control of this system is worth for theory, and also has been more and more recognized by engineering. Based on the theory of Linear Matrix Inequality and Lyapunov stability, problems of stability and disturbance attenuation were investigated in detail in this thesis for linear uncertain discrete-time systems whose parameters belong to polytopic domains. This kind of system is simply called as polytopic uncertain systems. Majority of investigations concentrate on continuous-time systems rather than discrete-time systems, so the polytopic uncertain discrete-time systems are mainly investigated in this paper, in particular, time-varying systems.
Quadratic stability has an important meaning on the research of robust stability for uncertain systems, which manages the whole group of systems by one alternative Lyapunov function. Even though every members of the group of systems are stable, the common Lyapunov function is hardly worked out to satisfy every members. In other words, the results based on quadratic stability have great conservativeness. To reduce the conservativeness, we select parameter-dependent Lyapunov functions to overcome the oneness of independent Lyapunov functions and take good use of the information of the upper bounds of the rate of variation or parameter increments, in particular, for time-varying systems. It's so called the approach of parameter-dependent Lyapunov function, which obviously has less conservative results.
There are two steps to investigate polytopic uncertain discrete-time systems in this paper. (1) For constant parameter-dependent systems, respectively, using the approaches based on independent Lyapunov function and parameter-dependent Lyapunov function, robust stability and disturbance attenuation of this kind of systems are analyzed and synthesized; then state feedback controller and output feedback controller are designed. (2) For parameter-varying systems, the same investigation before-mentioned has been done; emphatically, information of the upper bound of parameter increment has been considered, when we compute increment of parameter-dependent Lyapunov function for parameter-varying situation. Conservativeness of the results can be reduced however the results are more complicated, in particular, for slowly-varying parameters. Constant parameters can be regarded as the extremeness of time-varying parameters when the upper bounds of parameter increments tend to zero; Independent Lyapunov function can be regarded as parameter-dependent Lyapunov function when the upper bounds of parameter increments tend to infinity. Therefore, for parameter-varying systems, the results based on the approach of parameter-dependent Lyapunov function have universality.
The above results are all confirmed with examples. Through comparing the results of the examples, the approach of parameter-dependent Lyapunov function could reduce conservativeness of the results, and could educe the extremeness of parameters.
引文
1郑大中.线性系统理论.北京:清华大学出版社,1990
2俞立.鲁棒控制线性矩阵不等式处理方法.北京,清华大学出版社,2002
3S.P.Boyd,V.Balakrishnan,E.Feron,et al.Control System Analysis and Synthesis via Linear Matrix Inequalties.in Proceedings of the American Control Conference.1993.San Francisco,California.
4S.P.Boyd,L.E.Ghaoui,E.Feron,et al.Linear Matrix Inequality in System and Control Theory.Philadelphia:SIAM,1994
5B.R.Barmish.Stabilization of Uncertain System via Linear Control.IEEE Transactions on automatic control.1983,28:848-850.
6B.R.Barmish,M.Corless,G.Leitmann.A New Class of Stabilizing Controllers of an Uncertain System.SIAM Journal on Control and Optimization.1983,21:246-252.
7B.R.Barmish.Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain system.Journal of Optimization Theory and Applications.1985,46(4):399-408.
8J.C.Geromel,J.Bernussou,P.L.D.Peres.Stabilizability of Uncertain Linear Systems Via Linear Programming.in Proceedings of the 27th Conference on Decision and Control.1988.Austin,Texas,USA:1771-1775.
9J.C.Geromel,P L.D.Peres,J.Bernussou.On a Convex Parameter Space Method for Linear Control Design of Uncertain Systems.SIAM Journal on Control And Optimization.1991,29(2):381-402.
10V.Nesterov,A.Nemirovskii.Interior-Point Polynomial Algorithms in Convex Programming.Philadelphia:SIAM,1994
11P.Gahinet,A.Nemirovskii,A.J.Laub,et al.The LMI Control Toolbox.in Proceedings of the 33rd Conference on Decision and Control.1994.Lake Buena,Vista:2038-2041.
12J.C.Geromel,J.Bernussou.M.C.de Oliveira.H_2 Norm Optimization with Constrained Dynamic Output Feedback Controllers:decentralized and reliable Control.in Proceedings of the 36th Conference on Decision and Control.1997.San Diego,California USA:1307-1312.
13K.Takaba.Robust Preview Tracking Control for Polytopic Uncertain Systems.in Proceedings of the 37th IEEE Conference on Decision and Control.1998.Tampa,Florida USA.
14W.M.Haddad,D.S.Bernstein.Parameter-Dependent Lyapunov Functions,Constant Real Parameter Uncertainty,and the Popov Criterion in Robust Analysis and Synthesis Part 1.in Proceedings of the 30th Conference on Decision and Control.1991.Brighton,England:2274-2279.
15W.M.Haddad,D.S.Bernstein.Parameter-Dependent Lyapunov Functions,Constant Real Parameter Uncertainty,and the Popov Criterion in Robust Analysis and Synthesis Part 2. in Proceedings of the 30th Conference on Decision and Control. 1991. Brighton, England: 2274-2279.
16 P. Gahinet, P. Apkarian, M. Chilali. Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty. in Proceedings of the 33rd Conference on Decision and Control December. 1994. Lake Buena, Vista, FL: 2026-2031.
17 P. Gahinet, P Apkarian, M. Chilali. Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty. IEEE Transactions on Automatic Control. 1996, 41(3): 436-442.
18 A.G.Sparks. Analysis of Affinely Parameter-Varying Systems using Parameter-Dependent Lyapunov Functions. in Proceedings of the 36th Conference on Decision and Control. 1997. San Diego, California USA: 990-991.
19 E. Feron, P. Apkarian,P Gahinet. Analysis and Synthesis of Robust Control Systems via Parameter-Dependent Lyapunov Functions. IEEE Transactions on Automatic Control. 1996, 41(7): 1041-1046.
20 J. C. Geromel, M. C. De Oliveira, L. Hsu. LMI Characterization of Structural and Robust Stability. Linear Algebra and Its Applications. 1998, 285(1-3): 69-80.
21 M. C. de Oliveira, J. C. Geromel, L. Hsu. LMI Characterization of Structural and Robust Stability: the Discrete-Time Case. Linear Algebra and Its Applications. 1999, 296(1-3): 27-38.
22 G. Chesi, A. Garulli, A. Tesi, et al. Robust Stability of Polytopic Systems via Polynomially Parameter-Dependent Lyapunov Functions. in Proceedings of the 42nd IEEE Conference on Decision and Control. 2003. Maui, Hsnaii USA: 4670-4675.
23 G. Chesi, A. Garulli, A. Tesi, et al. Polynomially Parameter-Dependent Lyapunov Functions for Robust Stability of Polytopic Systems: An LMI Approach. IEEE Transactions on Automatic Control. 2005, 50(3): 365-370.
24 G. Chesi, A. Garulli, A. Tesi, et al. Parameter-Dependent Homogeneous Lyapunov Functions for Robust Stability of Linear Time-Varying Systems. In 43rd IEEE Conference an Decision and Control. 2004. Atlantis, Paradise Island, Bahamas: 14-17.
25 M. C. de Oliveira, J. Bernussou, J. C. geromel. A New Discrete-Time Robust Stability Condition. Sysrems & Control Letters. 1999, 37: 261-265.
26 M. C. de Oliveira, R. E. Skelton. Stability Tests for Constrained Linear Systems. In: Perspectives in Robust Control 2001:241-257.
27 O. Bachelier, J. Bernussou, M. C. de Oliveira, et al. Parameter Dependant Lyapunov Control Design: Numerical Evaluation. in Proceedings of the 38th Conference on Decision & Control. 1999. Phoenix, Arizona USA: 293-297.
28 M. C. de Oliveira, J. C. Geromel, L.Hsu. A New Absolute Stability Test for Systems with State-Dependent Perturbations. International Journal of Robust and Nonlinear Control. 2002, 12(14): 1209-1226.
29 J. Daafouza, J. Bernussou. Parameter Dependent Lyapunov Functions for Discrete Time Systems with Time Varying Parametric Uncertainties.Sysrems & Control Letters.2001,43:355-359.
30J.Daafouz,J.Bernussou.Poly-Quadratic Stability and H∞ Performance for Discrete Systems with Time Vary Uncertainties.in Proceedings of 40th IEEE Conference on Decision and Control.2001.Orlando,Florida USA:267-272.
31M.C.de Oliveira,J.C.Gerome],J.Bernussou.Extended H_2 and H_∞ Norm Characterizations and Controller Parametrizations for Discrete-Time Systems.International Journal of Control.2002,75(9):666-679.
32D.Rosinova,V.Vesely.Robust Static Output Feedback for Discrete-Time Systems-LMI Approach.Periodica Polytechnica Series Electronic Engineering.2004,48(3-4):151-163.
33V Suplin,U.Shaked.Robust H_∞ Output-Feedback Control of Linear Discrete-Time Systems.Sysrems & Control Letters.2005,54:799-808.
34Vinicius F.Montagnerand Pedro L.D.Peres.Robust Stability and H_∞ Performance of Linear Time-Varying Systems in Polytopic Domains.INT.J.CONTROL,2004,vol.77.No.15,1343-1352.
35段广仁.线性系统理论.第二版.哈尔滨:哈尔滨工业大学出版社.2004.
36胡寿松.自动控制原理(下册).修订版.国防工业出版社.1987.
2俞立.鲁棒控制线性矩阵不等式处理方法.北京,清华大学出版社,2002
3S.P.Boyd,V.Balakrishnan,E.Feron,et al.Control System Analysis and Synthesis via Linear Matrix Inequalties.in Proceedings of the American Control Conference.1993.San Francisco,California.
4S.P.Boyd,L.E.Ghaoui,E.Feron,et al.Linear Matrix Inequality in System and Control Theory.Philadelphia:SIAM,1994
5B.R.Barmish.Stabilization of Uncertain System via Linear Control.IEEE Transactions on automatic control.1983,28:848-850.
6B.R.Barmish,M.Corless,G.Leitmann.A New Class of Stabilizing Controllers of an Uncertain System.SIAM Journal on Control and Optimization.1983,21:246-252.
7B.R.Barmish.Necessary and Sufficient Conditions for Quadratic Stabilizability of an Uncertain system.Journal of Optimization Theory and Applications.1985,46(4):399-408.
8J.C.Geromel,J.Bernussou,P.L.D.Peres.Stabilizability of Uncertain Linear Systems Via Linear Programming.in Proceedings of the 27th Conference on Decision and Control.1988.Austin,Texas,USA:1771-1775.
9J.C.Geromel,P L.D.Peres,J.Bernussou.On a Convex Parameter Space Method for Linear Control Design of Uncertain Systems.SIAM Journal on Control And Optimization.1991,29(2):381-402.
10V.Nesterov,A.Nemirovskii.Interior-Point Polynomial Algorithms in Convex Programming.Philadelphia:SIAM,1994
11P.Gahinet,A.Nemirovskii,A.J.Laub,et al.The LMI Control Toolbox.in Proceedings of the 33rd Conference on Decision and Control.1994.Lake Buena,Vista:2038-2041.
12J.C.Geromel,J.Bernussou.M.C.de Oliveira.H_2 Norm Optimization with Constrained Dynamic Output Feedback Controllers:decentralized and reliable Control.in Proceedings of the 36th Conference on Decision and Control.1997.San Diego,California USA:1307-1312.
13K.Takaba.Robust Preview Tracking Control for Polytopic Uncertain Systems.in Proceedings of the 37th IEEE Conference on Decision and Control.1998.Tampa,Florida USA.
14W.M.Haddad,D.S.Bernstein.Parameter-Dependent Lyapunov Functions,Constant Real Parameter Uncertainty,and the Popov Criterion in Robust Analysis and Synthesis Part 1.in Proceedings of the 30th Conference on Decision and Control.1991.Brighton,England:2274-2279.
15W.M.Haddad,D.S.Bernstein.Parameter-Dependent Lyapunov Functions,Constant Real Parameter Uncertainty,and the Popov Criterion in Robust Analysis and Synthesis Part 2. in Proceedings of the 30th Conference on Decision and Control. 1991. Brighton, England: 2274-2279.
16 P. Gahinet, P. Apkarian, M. Chilali. Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty. in Proceedings of the 33rd Conference on Decision and Control December. 1994. Lake Buena, Vista, FL: 2026-2031.
17 P. Gahinet, P Apkarian, M. Chilali. Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty. IEEE Transactions on Automatic Control. 1996, 41(3): 436-442.
18 A.G.Sparks. Analysis of Affinely Parameter-Varying Systems using Parameter-Dependent Lyapunov Functions. in Proceedings of the 36th Conference on Decision and Control. 1997. San Diego, California USA: 990-991.
19 E. Feron, P. Apkarian,P Gahinet. Analysis and Synthesis of Robust Control Systems via Parameter-Dependent Lyapunov Functions. IEEE Transactions on Automatic Control. 1996, 41(7): 1041-1046.
20 J. C. Geromel, M. C. De Oliveira, L. Hsu. LMI Characterization of Structural and Robust Stability. Linear Algebra and Its Applications. 1998, 285(1-3): 69-80.
21 M. C. de Oliveira, J. C. Geromel, L. Hsu. LMI Characterization of Structural and Robust Stability: the Discrete-Time Case. Linear Algebra and Its Applications. 1999, 296(1-3): 27-38.
22 G. Chesi, A. Garulli, A. Tesi, et al. Robust Stability of Polytopic Systems via Polynomially Parameter-Dependent Lyapunov Functions. in Proceedings of the 42nd IEEE Conference on Decision and Control. 2003. Maui, Hsnaii USA: 4670-4675.
23 G. Chesi, A. Garulli, A. Tesi, et al. Polynomially Parameter-Dependent Lyapunov Functions for Robust Stability of Polytopic Systems: An LMI Approach. IEEE Transactions on Automatic Control. 2005, 50(3): 365-370.
24 G. Chesi, A. Garulli, A. Tesi, et al. Parameter-Dependent Homogeneous Lyapunov Functions for Robust Stability of Linear Time-Varying Systems. In 43rd IEEE Conference an Decision and Control. 2004. Atlantis, Paradise Island, Bahamas: 14-17.
25 M. C. de Oliveira, J. Bernussou, J. C. geromel. A New Discrete-Time Robust Stability Condition. Sysrems & Control Letters. 1999, 37: 261-265.
26 M. C. de Oliveira, R. E. Skelton. Stability Tests for Constrained Linear Systems. In: Perspectives in Robust Control 2001:241-257.
27 O. Bachelier, J. Bernussou, M. C. de Oliveira, et al. Parameter Dependant Lyapunov Control Design: Numerical Evaluation. in Proceedings of the 38th Conference on Decision & Control. 1999. Phoenix, Arizona USA: 293-297.
28 M. C. de Oliveira, J. C. Geromel, L.Hsu. A New Absolute Stability Test for Systems with State-Dependent Perturbations. International Journal of Robust and Nonlinear Control. 2002, 12(14): 1209-1226.
29 J. Daafouza, J. Bernussou. Parameter Dependent Lyapunov Functions for Discrete Time Systems with Time Varying Parametric Uncertainties.Sysrems & Control Letters.2001,43:355-359.
30J.Daafouz,J.Bernussou.Poly-Quadratic Stability and H∞ Performance for Discrete Systems with Time Vary Uncertainties.in Proceedings of 40th IEEE Conference on Decision and Control.2001.Orlando,Florida USA:267-272.
31M.C.de Oliveira,J.C.Gerome],J.Bernussou.Extended H_2 and H_∞ Norm Characterizations and Controller Parametrizations for Discrete-Time Systems.International Journal of Control.2002,75(9):666-679.
32D.Rosinova,V.Vesely.Robust Static Output Feedback for Discrete-Time Systems-LMI Approach.Periodica Polytechnica Series Electronic Engineering.2004,48(3-4):151-163.
33V Suplin,U.Shaked.Robust H_∞ Output-Feedback Control of Linear Discrete-Time Systems.Sysrems & Control Letters.2005,54:799-808.
34Vinicius F.Montagnerand Pedro L.D.Peres.Robust Stability and H_∞ Performance of Linear Time-Varying Systems in Polytopic Domains.INT.J.CONTROL,2004,vol.77.No.15,1343-1352.
35段广仁.线性系统理论.第二版.哈尔滨:哈尔滨工业大学出版社.2004.
36胡寿松.自动控制原理(下册).修订版.国防工业出版社.1987.