随机系数和带跳的线性随机微分系统的H_2/H_∞控制
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摘要
本文研究随机系数和带跳的线性随机微分系统在有限时区上的H∞及H2/H∞控制问题.
第一章介绍H2/H∞控制问题以及线性二次最优控制问题研究的历史和现状.利用Nash均衡方法,H2/H∞控制的存在性转化为Riccati方程解的存在性,而与H∞鲁棒性能有关的Riccati方程则为不定型的Riccati方程.
第二章讨论系数矩阵是随机过程且噪声仅依赖于状态的线性随机微分系统的H2/H∞控制问题.利用拟线性化方法我们证明不定型的倒向随机Riccati方程存在唯一解,并由此证明随机系数的界实引理.在此基础上,我们证明H2/H∞控制的存在性与两个耦合的倒向随机Riccati方程解的存在性等价.
第三章讨论系数矩阵是随机过程且噪声可以依赖于控制的线性随机微分系统的界实引理.我们证明存在某个常数,使得当输入输出算子的范数小于该常数时,对应的LQ问题可解,从而我们可以用随机Hamilton系统的解构造不定型的倒向随机Riccati(?)方程的解.作为随机系数界实引理的特殊情形,我们给出了系数含Markov跳的界实引理.最后我们研究一类来源于满足H∞鲁棒性能要求的特殊的倒向随机Riccati方程的可解性.
第四章首先讨论由Brown运动及Poisson随机跳共同驱动且所有的系数是确定性函数的线性随机微分系统的界实引理.我们利用常微分方程解的局部存在性以及实质为拟线性化方法的技巧获得了不定型的Ricccati方程解的全局存在性.我们同时指出随机系数的带Poisson跳的随机微分系统,当跳项和扩散项不依赖于控制时,界实引理仍然成立.我们随后证明H2/H∞控制的存在性等价于四个耦合的矩阵值方程解的存在性.最后我们给出系统由Brown运动及Poisson随机跳共同驱动且系数含Markov跳的界实引理.
The thesis is concerned with H∞control and H2/H∞control of linear stochastic differential systems with random coefficients or with jumps.
Chapter1reviews the theories of H∞control and H2/H∞control, and linear quadratic optimal control. Using a Nash equilibrium approach, the existence of so-lutions to H2/H∞control is converted into that of solutions to the Riccati equation, while the Riccati equation associated with H∞robustness is indefinite.
Chapter2is concerned with the H2/H∞control of linear stochastic differential systems with random coefficients and only state-dependent noise. By using Bell-man's quasilinear principle and a method of monotone convergence, we prove the existence and uniqueness of solutions to indefinite backward stochastic differential equation(BSRE), thus obtaining the bounded real lemma with random coefficients. Consequently, we present sufficient and necessary conditions for the existence of H2/H∞control in terms of a pair of coupled BSREs.
Chapter3is concentrated on a bounded real lemma of linear stochastic dif-ferential systems with random coefficients and state-and control-dependent noise. We prove that there exists some constant such that when the norm of the input-output operator is less than it, the associated LQ problem is solvable. We then use the solution of stochastic Hamiltonian system to construct that of the indefinite BSRE. As a special case of the bounded real lemma with random coefficients, we give the bounded real lemma with Markov jumping parameters. We also consider the existence of solution to a class of special BSRE arising from H∞robustness.
Chapter4is firstly concerned with a bounded real lemma for linear stochastic differential systems driven by a Brownian motion and a Poisson point process with deterministic coefficients. Using the existence of local solutions of ordinary differ-ential equation and the essential technique of quasilinear principle, we obtain the global solution of the indefinite Riccati equation. We point out that in the Ito system driven by a Brownian motion and a Poisson point process with random coefficients, we can prove the bounded real lemma when the martingale part is independent of control. We then prove the equivalence between the existence of H2/H∞control and that of the solutions to four coupled matrix-valued equations. In the end, when the system is driven by a Brownian motion and Poisson point process with Markovian jumping parameters, we prove the associated bounded real lemma.
第一章介绍H2/H∞控制问题以及线性二次最优控制问题研究的历史和现状.利用Nash均衡方法,H2/H∞控制的存在性转化为Riccati方程解的存在性,而与H∞鲁棒性能有关的Riccati方程则为不定型的Riccati方程.
第二章讨论系数矩阵是随机过程且噪声仅依赖于状态的线性随机微分系统的H2/H∞控制问题.利用拟线性化方法我们证明不定型的倒向随机Riccati方程存在唯一解,并由此证明随机系数的界实引理.在此基础上,我们证明H2/H∞控制的存在性与两个耦合的倒向随机Riccati方程解的存在性等价.
第三章讨论系数矩阵是随机过程且噪声可以依赖于控制的线性随机微分系统的界实引理.我们证明存在某个常数,使得当输入输出算子的范数小于该常数时,对应的LQ问题可解,从而我们可以用随机Hamilton系统的解构造不定型的倒向随机Riccati(?)方程的解.作为随机系数界实引理的特殊情形,我们给出了系数含Markov跳的界实引理.最后我们研究一类来源于满足H∞鲁棒性能要求的特殊的倒向随机Riccati方程的可解性.
第四章首先讨论由Brown运动及Poisson随机跳共同驱动且所有的系数是确定性函数的线性随机微分系统的界实引理.我们利用常微分方程解的局部存在性以及实质为拟线性化方法的技巧获得了不定型的Ricccati方程解的全局存在性.我们同时指出随机系数的带Poisson跳的随机微分系统,当跳项和扩散项不依赖于控制时,界实引理仍然成立.我们随后证明H2/H∞控制的存在性等价于四个耦合的矩阵值方程解的存在性.最后我们给出系统由Brown运动及Poisson随机跳共同驱动且系数含Markov跳的界实引理.
The thesis is concerned with H∞control and H2/H∞control of linear stochastic differential systems with random coefficients or with jumps.
Chapter1reviews the theories of H∞control and H2/H∞control, and linear quadratic optimal control. Using a Nash equilibrium approach, the existence of so-lutions to H2/H∞control is converted into that of solutions to the Riccati equation, while the Riccati equation associated with H∞robustness is indefinite.
Chapter2is concerned with the H2/H∞control of linear stochastic differential systems with random coefficients and only state-dependent noise. By using Bell-man's quasilinear principle and a method of monotone convergence, we prove the existence and uniqueness of solutions to indefinite backward stochastic differential equation(BSRE), thus obtaining the bounded real lemma with random coefficients. Consequently, we present sufficient and necessary conditions for the existence of H2/H∞control in terms of a pair of coupled BSREs.
Chapter3is concentrated on a bounded real lemma of linear stochastic dif-ferential systems with random coefficients and state-and control-dependent noise. We prove that there exists some constant such that when the norm of the input-output operator is less than it, the associated LQ problem is solvable. We then use the solution of stochastic Hamiltonian system to construct that of the indefinite BSRE. As a special case of the bounded real lemma with random coefficients, we give the bounded real lemma with Markov jumping parameters. We also consider the existence of solution to a class of special BSRE arising from H∞robustness.
Chapter4is firstly concerned with a bounded real lemma for linear stochastic differential systems driven by a Brownian motion and a Poisson point process with deterministic coefficients. Using the existence of local solutions of ordinary differ-ential equation and the essential technique of quasilinear principle, we obtain the global solution of the indefinite Riccati equation. We point out that in the Ito system driven by a Brownian motion and a Poisson point process with random coefficients, we can prove the bounded real lemma when the martingale part is independent of control. We then prove the equivalence between the existence of H2/H∞control and that of the solutions to four coupled matrix-valued equations. In the end, when the system is driven by a Brownian motion and Poisson point process with Markovian jumping parameters, we prove the associated bounded real lemma.
引文
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[3]D. S. Berstain and W. M. Haddad. LQG control with an H∞ performance bound:A Riccati equation approach. IEEE Trans. Automat. Contr.,34:293-305,1989.
[4]A. Bensoussan. Lectures on Stochastic Control. Proc. Cortona, Lecture Notes in Mathematics, Springer, New York,972:1-62,1982.
[5]J. M. Bismut. Linear quadratic optimal stochastic control with random coeffi-cients, SIAM J. Control Optim.,14:419-444,1976.
[6]R. K. Boel and D. Varaiya. Optimal control of jump processes. SIAM Journal on Control and Optim.,15:92-119,1977.
[7]A.EL Bouhtouri, D. Hinrichsen and A. J. Pritchard. H∞ type control for discrete-time stochastic systems. Int.J. Robust Nonlinear Control 9:923-948, 1999.
[8]R. Boel and P. Varaiya. Optimal control of jump processes. SIAM Journal on Control and Optim.,15(1):92-119,1977.
[9]S. Chen and J. Yong. Stochastic linear quadratic optimal control problems with random coefficients, Chinese Ann. Math. Ser. B,21:323-338,2000.
[10]S. Chen and J. Yong. Stochastic linear quadratic optimal control problems. Appl. Math. Optim.,43:21-45,2001.
[11]S. Chen, X. Li and X. Zhou. Stochastic linear quadratic regulators with indefi-nite control weight costs. SIAM J. Control Optim.36:1685-1702,1998.
[12]S. Chen and X. Zhou. Stochastic linear quadratic regulators with indefinite control weight costs, Ⅱ, SIAM J. Control Optim.,39:1065-1081,2000.
[13]B. Chen and W. Zhang. Stochastic H2/H∞ control with state-dependent noise. IEEE Trans. Automat. Control 49:45-57,2004.
[14]X. Chen and K. Zhou. Multiobjective H2/H∞ control design. SIAM J. Control Optim.,40:628-660,2001
[15]O. L. V. Costa and R. P. Marques. Mixed H2/H∞ control of discrete-time Markovian jump linear systems. IEEE Trans. Automat. Control 43:95-100, 1998.
[16]M. C. De Oliviera, J. C. Geromel and J. Bernussou. An LMI optimization ap-proach to multiobjective controller design for discrete-time systems. Proc.38th IEEE conf. decision and control 3611-3616,1999.
[17]F. Delbaen and S. Tang. Harmonic Analysis of Stochastic Equations and Back-ward Stochastic Differential Equations. arXiv:0801.3505.2008.
[18]J. C. Doyle, K. Glover, P. P. Khargonekar and B. Francis. State-space solutions to standard H2 and H∞ problems. IEEE Trans. Automat. Contr.,34:831-847, 1989.
[19]V. Dragan and T. Morozan. Stability and robust stabilization to linear stochas-tic systems described by differential equations with markovian jumping and multiplicative white noise. Stochastic Analysis and Applications,20(1),33-92, 2002.
[20]V. Dragan and T. Morozan. The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping. IEEE Transactions on Automatic Control,49:665-675,2004.
[21]L. I. Gal'chuk. Existence and uniqueness of a solution for stochastic equations with respect to semimartingales. Theory Probab.Appl.,23:751-763,1978.
[22]E. Gershon, U. Shaked and I. Yaesh. Estimation of state-multiplicative linear systems. Springer, London,2005.
[23]G. Guatteri and F. Masiero. Infinite horizon and ergodic optimal quadratic control for an affinc equation with stochastic coefficients. SIAM J. Control Optim.,48(3):1600-1631,2009.
[24]D. Hinrichsen and A. J. Pritchard. Stochastic H∞. SIAM J. Control Optim., 36:1504-1538,1998.
[25]Y. Hu and X. Zhou. Indefinite stochastic Riccati equations. SIAM J. Control Optim.,42:123-137,2003.
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