不确定广义系统的保性能控制
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摘要
本文研究了不确定连续广义系统的弹性保性能控制，离散广义系统的H_∞和具有圆盘极点约束的保性能控制，以及性能函数中含有时间乘积因子的连续广义系统保性能控制等问题。实际系统中不确定性的来源是多样的，也往往会具有不同的性质。通过适当的等价变换，这些不确定性可以表示成具有对角结构的形式，即结构化的不确定性，可将这种结构化的不确定性看作是对满足范数匹配条件的不确定性的一种扩展，从而可以利用结构化不确定性的结构特点来改善设计方法，减少保守性。在二次型性能指标中如果含有时间乘积因子，那么这样的性能指标会使得状态响应随着时间的增加更快地趋于零，从而缩短系统达到稳定所需的时间。论文中也研究了一些多目标保性能控制，例如在所要求的性能指标基础上加入其他诸如H_∞、圆盘极点约束等性能要求，能够使被控系统具有更好的动态性能。
本文主要内容如下：
第二章研究了不确定广义系统的弹性保性能控制问题，首先给出了对所有容许的系统不确定性和控制器扰动，连续广义系统存在弹性保性能控制器的一个充分条件，接着证明了该充分条件等价于一组不等式的可解性问题，并利用变量替换法进行处理得到广义弹性动态输出反馈控制器。并且控制器是满足其微分矩阵的秩等于不确定系统微分矩阵秩条件的任意阶形式。最后用实例演示了所提供方法的有效性。
第三章针对一类具有范数有界不确定性的连续广义系统和一种含有时间乘积因子的二次型性能指标，设计状态反馈控制器使得对所有容许的系统不确定性，闭环系统正则，无脉冲，稳定且最小化闭环性能指标的一个上界值。文中给出了保性能控制器存在的充分必要条件并基于双线性矩阵不等式(BMIs)，二次矩阵不等式(QMIs)和线性矩阵不等式(LMIs)给出其设计方法，并利用SDP软件和BMI软件举例说明。
第四章考虑了不确定离散广义系统的H_∞保性能控制问题，基于线性矩阵不等式给出了使得闭环系统具有给定的H_∞性能γ的输出反馈H_∞保性能控制器的存在条件及设计方法，并举例说明所得结论的可行性。
In this paper, we study problems of the resilient guaranteed cost control for uncertain continuous singular systems, the guaranteed cost control with H_∞ and pole placement in a disk of discrete singular systems , and the guaranteed cost control of continuous singular systems with a time-multiplied linear quadratic cost function. For a physical system, the sources of uncertainties are various and the uncertainties always own different characters. With a appropriate equivalent transformation, these uncertainties could be shown as a diagonal construction, namely structured uncertainties. This kind of uncertainties can be regarded as an extension of the norm bounded ones, so the structures can be used to improve the designed method and reduce the constraint of the conclusion. If there is a time-multiplied gene in the quadratic cost function, it could help the state response approach zero more quickly, and shorten the stabilizing time of the systems. We also study several kinds of multi-objective guaranteed cost control in this paper. For example, based on the cost function required, other requests such as H_∞, pole placement in a disk and so on, all of which would make the systems possess better dynamic performances.
The main content of this paper is as follows:
In chapter 2, the resilient guaranteed cost control for singular systems with structured uncertainties is considered. Firstly, the sufficient condition in terms of LMIs for the existence of resilient guaranteed cost controllers is obtained for all the admissible uncertainties of systems and disturbances of controllers. Using the method of variable replacement to deal with the LMIs, we obtain the singular resilient dynamical output-feedback controllers. At last the numerical example is given to illustrate the designed method.
Chapter 3 considers the guaranteed cost control for a class of norm bounded uncertain singular systems with a time-multiplied linear quadratic cost function, and
本文主要内容如下：
第二章研究了不确定广义系统的弹性保性能控制问题，首先给出了对所有容许的系统不确定性和控制器扰动，连续广义系统存在弹性保性能控制器的一个充分条件，接着证明了该充分条件等价于一组不等式的可解性问题，并利用变量替换法进行处理得到广义弹性动态输出反馈控制器。并且控制器是满足其微分矩阵的秩等于不确定系统微分矩阵秩条件的任意阶形式。最后用实例演示了所提供方法的有效性。
第三章针对一类具有范数有界不确定性的连续广义系统和一种含有时间乘积因子的二次型性能指标，设计状态反馈控制器使得对所有容许的系统不确定性，闭环系统正则，无脉冲，稳定且最小化闭环性能指标的一个上界值。文中给出了保性能控制器存在的充分必要条件并基于双线性矩阵不等式(BMIs)，二次矩阵不等式(QMIs)和线性矩阵不等式(LMIs)给出其设计方法，并利用SDP软件和BMI软件举例说明。
第四章考虑了不确定离散广义系统的H_∞保性能控制问题，基于线性矩阵不等式给出了使得闭环系统具有给定的H_∞性能γ的输出反馈H_∞保性能控制器的存在条件及设计方法，并举例说明所得结论的可行性。
In this paper, we study problems of the resilient guaranteed cost control for uncertain continuous singular systems, the guaranteed cost control with H_∞ and pole placement in a disk of discrete singular systems , and the guaranteed cost control of continuous singular systems with a time-multiplied linear quadratic cost function. For a physical system, the sources of uncertainties are various and the uncertainties always own different characters. With a appropriate equivalent transformation, these uncertainties could be shown as a diagonal construction, namely structured uncertainties. This kind of uncertainties can be regarded as an extension of the norm bounded ones, so the structures can be used to improve the designed method and reduce the constraint of the conclusion. If there is a time-multiplied gene in the quadratic cost function, it could help the state response approach zero more quickly, and shorten the stabilizing time of the systems. We also study several kinds of multi-objective guaranteed cost control in this paper. For example, based on the cost function required, other requests such as H_∞, pole placement in a disk and so on, all of which would make the systems possess better dynamic performances.
The main content of this paper is as follows:
In chapter 2, the resilient guaranteed cost control for singular systems with structured uncertainties is considered. Firstly, the sufficient condition in terms of LMIs for the existence of resilient guaranteed cost controllers is obtained for all the admissible uncertainties of systems and disturbances of controllers. Using the method of variable replacement to deal with the LMIs, we obtain the singular resilient dynamical output-feedback controllers. At last the numerical example is given to illustrate the designed method.
Chapter 3 considers the guaranteed cost control for a class of norm bounded uncertain singular systems with a time-multiplied linear quadratic cost function, and
引文
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40．张建林，费树岷．不确定广义时滞系统的保性能控制[C]，Proceedings of 2004Chinese Control and Decision Conference, 2004:43-47
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42．熊军林．不确定广义系统的最优保性能控制[J]，自动化学报，2004，30(4)：588-591
43．董增福．矩阵分析教程[M]，哈尔滨：哈尔滨工业大学出版社，2003
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2. Francis B A. A course in H_∞ control theory [M]. New York: Springer- Verlag, 1987
3. Garcia G, Bernussou J, Arzelier D. Robust stabilization of discrete- time linear systems with norm-bounded time-varying uncertainty[J]. Systems & Control Letters, 1994, 22:327-339
4．俞立．不确定离散线性系统的鲁棒镇定[J]，控制与决策，1999，4(2)：169-172
5. Chang S L, and Peng T K C. Adaptive guaranteed cost control systems with uncertain parameters[J]. IEEE Transactions on Automatic Control, 1972, 17: 474-483
6. Petersen I R, McFarlane D C. Optimal guaranteed cost control of uncertain linear systems[A]. In Proceedings of the 1992 American Control Conference[C], Chicago, IL: 2929-2930
7. Moheimani S O R, Petersen I R. Optimal guaranteed cost control of uncertain systems via static and dynamic output feedback[J]. Automatic, 1996, 32(4): 575-579
8. Fishman A M, Dion J, Duggard L, Troffino A. A linear matrix inequality approach for guaranteed cost control[C]. Proceedings of the 13th World Congress. IFAC, San Francisco, CA, 1996, 1:197-202
9. Boukas E K, Liu Z K, and Al-Sunni F. Guaranteed cost control of a markov jump linear uncertain system using a time-multiplied cost function[J]. Journal of Optimization Theory and Applications, 2003, 116(1): 183-204
10．薛安克，孙优贤．不确定线性系统保代价控制的鲁棒性分析[J]，自动化学报，2001，27(1)：346-352
11. Petersen I R, Mcfarlane D C. Optimal guaranteed cost control and filtering for uncertain linear systems[J]. IEEE Transactions on Automatic Control, 1994, AC-37: 1971-1977
12. Savkin A V, Petersen I R. Minimax optimal control of uncertain systems withstructured uncertainty[J]. Int J Robust and Nonlinear control to appear, 1995
13. Savkin A V, Petersen I R. Output feedback guaranteed cost control of uncertain systems on an infinite time interval[J]. Submitted for publication, 1995
14. Valer Y A, Ugrinovskii, Petersen I R. Guaranteed cost control of uncertain systems via Lur'e-Postnikov Lyapunov functions[J]. Automatic, 2000, 36:279-285
15. Yu L, Wang J C, and Chu J. Guaranteed cost control of uncertain linear discrete time systems. Proceedings of American control conference[J]. Albuquerque New Mexico, 1997,5:3181-3184
16．俞立，不确定离散系统的最优保性能控制[J]，控制理论与应用，1999，16(5)：639-642
17. Bersteid S, Haddadw M. The optimal projection equations with Petersen-Hollot bounds: robust stability and performance via fixed order dynamic compensation[J]. IEEE Transactions on Automatic Control, 1988, AC-33:578-582
18. Bersteid S, Haddadw M, Robust stability and performance via fixed-order dynamic compensation with guaranteed cost bounds[J]. Math Control sig Syst, 1990,(3):139-163
19．王德进．线性不确定系统多目标鲁棒保代价控制[J]，控制与决策，2004，19(3)：327-330
20．俞立，陈国定，潘海天．不确定离散时间系统的H_2／H_∞最优保性能控制[J]，控制与决策，2001，16(2)：151-154
21．张瑞金，杨成梧．反馈控制系统Delta算子理论的研究与发展[J]，控制理论与应用，1998，15(2)：153-160
22．胡刚，任俊超，谢湘生．Delta算子不确定系统的最优保性能控制[J]，电机与控制学报，2003，7(2)：139-142
23. Furuta K, Kim S B. Pole assignment in a specified disc[J]. IEEE Transactions on Automatic Control, 1987, AC-32:423-427
24. Mawasaki N, Shimemura E. Pole placement in aspecified region based on a linear quadratic regulator[J]. Int. J. Control, 1988, 48:225-240
25. Haddad W M, Bernstein D S. Controller design with regional pole constraints[J]. IEEE Transactions on Automatic Control, 1992, AC-37:54-69
26. Wang Z D, Tang G Q, Chen X M. Robust controller design for uncertain linearsystems with circular pole constraints[J]. Int. J. Control, 1996, 65:1045-1054
27. Garcia G, Bernussou J. Pole assignment for uncertain systems in a specified disk by state feedback[J]. IEEE Transactions on Automatic Control, 1995, AC-40:184-190
28. Chilali M, Gahinet P. H_∞ design with pole placement constraints: an LMI approach[J]. IEEE Transactions on Automatic Control, 1996, AC-41: 358-367
29. Garcia G. Quadratic guaranteed cost and disc pole location control for discrete-time uncertain systems[J], IEE Proceedings: Control Theory and Applications, 1997, 144(6): 545-548
30. Yang G H, Wang J L, Yeng C S. Guaranteed cost control for discrete-time linear systems under controller gain perturbations[J]. Linear Algebra and its Applications, 2000, 312:161-180
31. Astrom K J, Wittenmark B. Computer controlled systems: theory and design[M]. Englewood Cliffs, N J: Prentice-Hall Inc., 1993
32．俞力，冯浩．不确定离散时滞系统的保性能控制[J]．自动化学报，2001，27(3)：392-396
33. Rosenbrock H H. Structural properties of linear dynamical systems[J]. Int. J. Control, 1974, 20(2): 191-202
34. Zhang Q L, James L. Robust elimination of impulse of descriptor systems[J]. Dynamics of Continuous, Discrete and Impulsive Systems, 2002, 9(1): 13-28
35. Zhang Q L, Liu W Q, David H. A Lyapunov approach to analysis of discrete singular systems[J]. Systems & Control Letters, 2002, 45(3): 237-247
36. Morihira N, Takaba K and Katayama T. On the H_∞ control for descriptor systems-a J-spectral factorization approach[J]. In Proc. 16th SICE Symp. On Dynamical System Theory, Kobe, Japan, 1993:261-266
37. Dai L. Singular control systems: lecture notes in control and information sciences [M]. New York: Springer-Verlag, 1989
38. Keel L H, Bhattacharyya S P. "Robust, fragile or optimal"[J]. IEEE Transactions on Automatic Control, 1997,42(8): 1098-1105
39．刘岑枫，胡刚．具有容许扰动的不确定广义系统的弹性保成本控制[C]，Proceedings of 2004 Chinese Control and Decision Conference．2004：87-90
40．张建林，费树岷．不确定广义时滞系统的保性能控制[C]，Proceedings of 2004Chinese Control and Decision Conference, 2004:43-47
41. Feng J E, Chen Z L. Guaranteed cost control of linear uncertain singular systems with time-delay[J]. Control Theory and Application, 2002, 19(suppl): 711-714
42．熊军林．不确定广义系统的最优保性能控制[J]，自动化学报，2004，30(4)：588-591
43．董增福．矩阵分析教程[M]，哈尔滨：哈尔滨工业大学出版社，2003
44. Jadbabaie, Ali, Jamshidi, Mohammad. Guaranteed-cost design of continuous-time Takagi-Sugeno fuzzy controllers via linear matrix inequalities[A]. IEEE international Conference on Fuzzy Systems, IEEE World Congress on Computational Intelligence, 1998,v1: 268-273
45. Petersen I R, McFarlane D C. and Rotea M A. Optimal guaranteed cost control of discrete-time uncertain linear systems[A]. In Proceedings of the 12th IFAC World Congress[C], Sydney, Australia, 1993, vol. Ⅰ: 407-410
46. Feng J, Zhu S Q, Cheng Z L. Guaranteed cost control of linear uncertain singular time-delay systems[A]. Proceedings of the IEEE Conference on Decision and Control[C], 2002, 2, 1802-1807
47. Rear S O, Moheiman, Petersen I R. Guaranteed cost control of uncertain systems with a time-multiplied quadratic cost function: an approach based on linear matrix inequalities[J]. Automatica, 1998, 34(5): 651-654
48. Masubuchi I, Kamitane Y, Ohara A, et al. H_∞ Control for Descriptor Systems: A Matrix Inequalities Approach[J]. Automatica, 1997, 33(4): 669-673
49. Petersen I R. A stabilization algorithm for a class of uncertain linear systems[J], Syst. Control Lett., 1987, 8:351-357
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