随机非线性系统的控制器设计与分析
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摘要
本文主要研究了一类高阶次随机非线性系统的逆最优镇定问题和一类高阶次随机非线性系统的输出反馈镇定问题．全文分为以下两部分：
一、一类高阶次随机非线性系统的逆最优镇定问题．
考虑如下的高阶次随机非线性系统：dx_i=(x_(i+1)~(pi)+f_i(x_i))dt+g_i~T(x_i)dω,i=1，…，n-1，dx_n=(u~(pn)+f_n(x_n))dt+g_n~T(x_n)dω，其中z=[x_1…，x_n]~T∈R~n和u∈R分别是系统的可测状态和控制输入；ω∈R~r是独立标准Wiener过程向量；p_i≥1，i=1，…，n，是奇数；f_i：R~i→R和g_i：R~i→R~r是满足如下假设1的光滑函数且f_i(0)=0，g_i(0)=0．
假设1：对f_i(x_i)∈R和g_i(x_i)∈R~r，i=1，…，n，x_i=[x_1，…，x_i]~T∈R~i(i=1，…，n)，存在已知的非负光滑函数f_(il)(·)和g_(i1)(·)，i=1，…，n，使得|f_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))f_(i1)(x_i)，|g_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))g_(i1)(x_i)，i=1，…，n．p_i≥1，i=1，…，n，是满足假设2的奇数．
假设2：p_1≥…≥p_n≥1．
该部分的控制目标是在假设1，2的条件下，设计一个光滑的状态反馈控制器，使得闭环系统在概率意义下是全局渐近稳定的，并且是逆最优稳定的．
二、一类高阶次随机非线性系统的输出反馈镇定问题．
考虑如下的随机非线性系统：dζ_i=ζ_(i+1)~p dt+φ_i~T(ζ_i)dω，i=1，…，n-1，dζ_n=v~pdt+φ_n~T(ζ_n)dω，y=ζ_1，其中ζ=[ζ_1，…，ζ_n]~T∈R~n，v∈R和y∈R分别是系统的不可量测的状态向量，控制输入和可测的输出；ω∈R~m是独立标准Wiener过程向量；p≥1是奇数．φ_i((?)_i)∈R~r是满足如下假设3的光滑函数且φ_i(0)=0．
假设3：对φ_i((?)_i)，i=1，…，n，存在常数k>0使得[φ_i((?)_i)|≤k|y|~(p+1/2)．
控制目标是在假设3的条件下，设计一个观测器和一个输出反馈控制器：(?)=(?)(x)，u=μ(x)，保证闭环系统在概率意义下的全局渐近稳定性．
Inverse optimal stabilization for a class of high-order stochastic nonlinear systems, and output-feedback stabilization for high-order stochastic nonlinear systems are considered in the paper, which is composed of the following two parts.
1. The problem of Inverse optimal stabilization for a class of high-order stochastic nonlinear systems.
Consider the following high-order stochastic nonlinear systems described by:dx_i=(x_(i+1)~(pi)+f_i(x_i))dt+g_i~T(x_i)dω,i=1,…,n-1,dx_n=(u~(pn)+f_n(x_n))dt+g_n~T(x_n)dω,where x = [x_1,…,x_n]~T∈R~n and u∈R are the measurable state and control input, respectively;ω∈R~r is independent standard Wiener process vector; P_i≥1, i = 1,…,n, are odd integers. The functions f_i : R~i→R and g_i : R~i→R~r, i = 0,1,…, n, are assumed to be smooth which satisfies the following Assumption 1, vanished at the origin.
Assumption 1: There are nonnegative smooth functions f_(i1)(x_i) and g_(i1)(x_i),i=1,…,n, such that |f_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))f_(i1)(x_i)，|g_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))g_(i1)(x_i), i=1,…,n．p_i≥1,i=1,…,n,are odd integers satisfying Assumption 2.
Assumption 2:p_1≥…≥p_n≥1．
The objective of this part is to design a smooth state-feedback controller under Assumption 1 and 2, such that the closed-loop system is globally asymptotically stable in probability and inverse optimal stabilization in probability.
2. The problem of output-feedback stabilization for high-order stochastic nonlinear systems.
Consider the following stochastic nonlinear systems:dζ_i=ζ_(i+1)~p dt+φ_i~T(ζ_i)dω,i=1,…,n-1,dζ_n=v~pdt+φ_n~T(ζ_n)dω,y=ζ_1,where x = [x_1,…,x_n]~T∈R~n, u∈R and y∈E are unmeasurable state vector, control input and measurable output, respectively;ω∈R~m is independent standard Wiener process vector; p > 0 is an odd integer,φ_i : R~i→R~r, i = 1,…, n, are C~1 functions which satisfy the following Assumption 3 andφ_i(0) = 0.
Assumption 2:φ_i(x_i),i= 1,…,n, there exists a real number k > 0 such that for all i= 1,…n,[φ_i((?)_i)|≤k|y|~(p+1/2)
The objective is to design an observer and an output-feedback controller:(?)=(?)(x), u=μ(x),under Assumption 3, such that the closed-loop system is globally asymptotically stable in probability.
一、一类高阶次随机非线性系统的逆最优镇定问题．
考虑如下的高阶次随机非线性系统：dx_i=(x_(i+1)~(pi)+f_i(x_i))dt+g_i~T(x_i)dω,i=1，…，n-1，dx_n=(u~(pn)+f_n(x_n))dt+g_n~T(x_n)dω，其中z=[x_1…，x_n]~T∈R~n和u∈R分别是系统的可测状态和控制输入；ω∈R~r是独立标准Wiener过程向量；p_i≥1，i=1，…，n，是奇数；f_i：R~i→R和g_i：R~i→R~r是满足如下假设1的光滑函数且f_i(0)=0，g_i(0)=0．
假设1：对f_i(x_i)∈R和g_i(x_i)∈R~r，i=1，…，n，x_i=[x_1，…，x_i]~T∈R~i(i=1，…，n)，存在已知的非负光滑函数f_(il)(·)和g_(i1)(·)，i=1，…，n，使得|f_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))f_(i1)(x_i)，|g_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))g_(i1)(x_i)，i=1，…，n．p_i≥1，i=1，…，n，是满足假设2的奇数．
假设2：p_1≥…≥p_n≥1．
该部分的控制目标是在假设1，2的条件下，设计一个光滑的状态反馈控制器，使得闭环系统在概率意义下是全局渐近稳定的，并且是逆最优稳定的．
二、一类高阶次随机非线性系统的输出反馈镇定问题．
考虑如下的随机非线性系统：dζ_i=ζ_(i+1)~p dt+φ_i~T(ζ_i)dω，i=1，…，n-1，dζ_n=v~pdt+φ_n~T(ζ_n)dω，y=ζ_1，其中ζ=[ζ_1，…，ζ_n]~T∈R~n，v∈R和y∈R分别是系统的不可量测的状态向量，控制输入和可测的输出；ω∈R~m是独立标准Wiener过程向量；p≥1是奇数．φ_i((?)_i)∈R~r是满足如下假设3的光滑函数且φ_i(0)=0．
假设3：对φ_i((?)_i)，i=1，…，n，存在常数k>0使得[φ_i((?)_i)|≤k|y|~(p+1/2)．
控制目标是在假设3的条件下，设计一个观测器和一个输出反馈控制器：(?)=(?)(x)，u=μ(x)，保证闭环系统在概率意义下的全局渐近稳定性．
Inverse optimal stabilization for a class of high-order stochastic nonlinear systems, and output-feedback stabilization for high-order stochastic nonlinear systems are considered in the paper, which is composed of the following two parts.
1. The problem of Inverse optimal stabilization for a class of high-order stochastic nonlinear systems.
Consider the following high-order stochastic nonlinear systems described by:dx_i=(x_(i+1)~(pi)+f_i(x_i))dt+g_i~T(x_i)dω,i=1,…,n-1,dx_n=(u~(pn)+f_n(x_n))dt+g_n~T(x_n)dω,where x = [x_1,…,x_n]~T∈R~n and u∈R are the measurable state and control input, respectively;ω∈R~r is independent standard Wiener process vector; P_i≥1, i = 1,…,n, are odd integers. The functions f_i : R~i→R and g_i : R~i→R~r, i = 0,1,…, n, are assumed to be smooth which satisfies the following Assumption 1, vanished at the origin.
Assumption 1: There are nonnegative smooth functions f_(i1)(x_i) and g_(i1)(x_i),i=1,…,n, such that |f_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))f_(i1)(x_i)，|g_i(x_i)|≤(|x_1|~(pi)+…+|x_i|~(pi))g_(i1)(x_i), i=1,…,n．p_i≥1,i=1,…,n,are odd integers satisfying Assumption 2.
Assumption 2:p_1≥…≥p_n≥1．
The objective of this part is to design a smooth state-feedback controller under Assumption 1 and 2, such that the closed-loop system is globally asymptotically stable in probability and inverse optimal stabilization in probability.
2. The problem of output-feedback stabilization for high-order stochastic nonlinear systems.
Consider the following stochastic nonlinear systems:dζ_i=ζ_(i+1)~p dt+φ_i~T(ζ_i)dω,i=1,…,n-1,dζ_n=v~pdt+φ_n~T(ζ_n)dω,y=ζ_1,where x = [x_1,…,x_n]~T∈R~n, u∈R and y∈E are unmeasurable state vector, control input and measurable output, respectively;ω∈R~m is independent standard Wiener process vector; p > 0 is an odd integer,φ_i : R~i→R~r, i = 1,…, n, are C~1 functions which satisfy the following Assumption 3 andφ_i(0) = 0.
Assumption 2:φ_i(x_i),i= 1,…,n, there exists a real number k > 0 such that for all i= 1,…n,[φ_i((?)_i)|≤k|y|~(p+1/2)
The objective is to design an observer and an output-feedback controller:(?)=(?)(x), u=μ(x),under Assumption 3, such that the closed-loop system is globally asymptotically stable in probability.
引文
[1] E.D. Sontag. A Lyapunov-like characterization of asymptotic controllability [J]. SIAM Journal of Control and Optimization, 1983, 21: 462-471.
[2] Z. Artstein. Stabilizations with relaxed controls [J]. Nonlinear Analysis, 1998, 7(11): 1163-1173.
[3] H.K. Khalil. Nonlinear Systems [M]. Third Edition, New York: Prentice Hall, 2002.
[4] R. Marino, P. Tomei. Nonlinear Control Design-Geometric, Adaptive and Robust [M]. London: Prentice-Hall, 1995.
[5] A. Isidori. Nonlinear Control Systems [M]. Third Edition, London: Springer-Verlag, 1995.
[6] I. Kanellakopoulos, P.V. Kokotovic, A.S. More. Systematic design of adaptive controllers for feedback linearizable systems [J]. IEEE Transactions on Automatic Control, 1991, 36: 1241-1253.
[7] M. Krstic, I. Kanellakopouls, P.V. Kokotovic. Nonlinear and Adaptive Control Design [M]. New York: John Wiley, 1995.
[8] R. Freeman, P.V. Kokotovic. Robust Nonlinear Control Design [M]. Boston: Birkhauser, 1995.
[9] P.V. Kokotovic, M. Arcak. Constructive nonlinear control: a historical perspective [J]. Automatica, 2001, 37: 637-662.
[10] J.T. Spooner, M. Maggiore, R. Ordonez, K.M. Passino. Stable Adaptive Control and Estimation for Nonlinear Systems [M]. New York: John Wiley & Sons, 2002.
[11] W. Lin. http://nonlinear.caseedu/ -linwei/.
[12] Z.P. Jiang. http://ctrl.poly.edu/faculty/ProLJiang.html.
[13]郭雷．时变随机系统-稳定性、估计与控制[M]．长春：吉林科学技术出 版社，1993．
[14] R.Z. Has'minskii. Stochastic Stability of Differential Equations [M]. Rockville, Maryland: S & N International publisher, 1980.
[15] H.J. Kushner. Stochastic Stability and Control [M]. New York: Academic Press, 1967.
[16] R.A. Gihman, A.V. Skorohod. Stochastic Differential Equations [M]. New York: Springer-Verlag, 1972.
[17]胡宣达．随机微分方程稳定性理论[M]．南京：南京大学出版社，1986．
[18] Z.G. Pan, T. Bar. Adaptive controller design for tracking and disturbanceattenuation in parametric-feedback nonlinear systems [J]. IEEE Transactions on Automatic Control, 1998, 43: 1066-1083.
[19] Z.G. Pan, T. Basar. Backstepping controller design for nonlinear stochasticsystems under a risk-sensitive cost criterion [J]. SIAM Journal of Control and Optimization, 1999, 37: 957-995.
[20] Z.G. Pan, Y.G. Liu, S. Shi. Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics [J]. Science in China (Series F), 2001, 44: 292-308.
[21] Z.G. Pan, K. Ezal, A. Krener, P.V. Kokotovic. Backstepping design with local optimality matching [J]. IEEE Transactions on Automatic Control, 2001, 46: 1014-1027.
[22] Y.G. Liu, Z.G. Pan, S. Shi. Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitivie cost [J]. IEEE Transactions on Automatic Control, 2003, 48(3): 509-513.
[23] Y.G. Liu, J.F. Zhang. Reduced-order observer-based control design for nonlinear stochastic systems [J]. System and Control Letters, 2004, 52: 123-135.
[24] Y.G Liu, J.F. Zhang. Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems [J]. Science in China (Ser. F), 2004, 47: 527-544.
[25] Y.G Liu, J.F. Zhang. Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics [J]. SI AM Journal of Control and Optimization, 2006, 45: 885-926.
[26] Y.G. Liu, J.F. Zhang, Z.G. Pan. Design of satisfaction output feedback controls for stochastic nonlinear systems under quadratic tracking risk-sensitive index [J]. Science in China (Ser.F), 2003, 46: 126-145.
[27] H. Deng, M. Krstic. Stochastic nonlinear stabilization part I: a back-stepping design [J]. System and Control Letters, 1997, 32: 143-150.
[28] H. Deng, M. Krstic. Stochastic nonlinear stabilization part II: inverse Optimality [J]. System and Control Letters, 1997, 32: 151-159.
[29] H. Deng, M. Krstic. Output-feedback stochastic nonlinear stabilization [J]. IEEE Transactions on Automatic Control, 1999, 44: 328-333.
[30] H. Deng, M. Krstic. Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J]. System and Control Letters, 2000, 39: 173-182.
[31] H. Deng, M. Krstic, R.Williiams. Stabilization of stochastic nonlinear driven by noise of unknown covariance [J]. IEEE Transaction on Automatic Control, 2001, 46: 1237-1253.
[32] M. Krstic, H. Deng. Stability of Nonlinear Uncertain Systems [M]. New York: Springer-Verlag, 1998.
[33] Z.J. Wu, X.J. Xie, S.Y. Zhang. Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function [J]. International. Journal of Control, 2006, 79(12):1635-1646.
[34] Z.J. Wu, X.J. Xie. Adaptive backstepping controller design using stochastic small-gain theorem [J]. Automatica, 2007, 43(4): 608-620.
[35] S.J. Liu, J.F. Zhang, Z.P. Jiang. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems [J]. Automatica, 2007, 43(2): 238-251.
[36] M. Kristic, Z.H. Li. Inverse optimal design of input to state stabilizing nonlinear controllers [J]. IEEE Transactions on Automatic Control, 1998, 43: 336-350.
[37] R.A. Freeman, P.V. Kokotovic. Robust nonlinear control design: state-space and Lyapunov techniques [J]. IEEE Transactions on Automatic Control, 2003, 11: 73-83.
[38] W.C. Luo, Y.C. Chu, K.V. Ling. Inverse optimal adaptive control for attitude tracking of spacecraft [J]. IEEE Transactions on Automatic Control, 2005, 50: 1639-1654.
[39] X.J. Xie, J. Tian. State-feedback stabilization for high-order stochastic nonlinear systems with stochastuic inverse dynamics [J]. International Journal of Robust and Nonlinear Control, 2007, 17: 1343-1362.
[2] Z. Artstein. Stabilizations with relaxed controls [J]. Nonlinear Analysis, 1998, 7(11): 1163-1173.
[3] H.K. Khalil. Nonlinear Systems [M]. Third Edition, New York: Prentice Hall, 2002.
[4] R. Marino, P. Tomei. Nonlinear Control Design-Geometric, Adaptive and Robust [M]. London: Prentice-Hall, 1995.
[5] A. Isidori. Nonlinear Control Systems [M]. Third Edition, London: Springer-Verlag, 1995.
[6] I. Kanellakopoulos, P.V. Kokotovic, A.S. More. Systematic design of adaptive controllers for feedback linearizable systems [J]. IEEE Transactions on Automatic Control, 1991, 36: 1241-1253.
[7] M. Krstic, I. Kanellakopouls, P.V. Kokotovic. Nonlinear and Adaptive Control Design [M]. New York: John Wiley, 1995.
[8] R. Freeman, P.V. Kokotovic. Robust Nonlinear Control Design [M]. Boston: Birkhauser, 1995.
[9] P.V. Kokotovic, M. Arcak. Constructive nonlinear control: a historical perspective [J]. Automatica, 2001, 37: 637-662.
[10] J.T. Spooner, M. Maggiore, R. Ordonez, K.M. Passino. Stable Adaptive Control and Estimation for Nonlinear Systems [M]. New York: John Wiley & Sons, 2002.
[11] W. Lin. http://nonlinear.caseedu/ -linwei/.
[12] Z.P. Jiang. http://ctrl.poly.edu/faculty/ProLJiang.html.
[13]郭雷．时变随机系统-稳定性、估计与控制[M]．长春：吉林科学技术出 版社，1993．
[14] R.Z. Has'minskii. Stochastic Stability of Differential Equations [M]. Rockville, Maryland: S & N International publisher, 1980.
[15] H.J. Kushner. Stochastic Stability and Control [M]. New York: Academic Press, 1967.
[16] R.A. Gihman, A.V. Skorohod. Stochastic Differential Equations [M]. New York: Springer-Verlag, 1972.
[17]胡宣达．随机微分方程稳定性理论[M]．南京：南京大学出版社，1986．
[18] Z.G. Pan, T. Bar. Adaptive controller design for tracking and disturbanceattenuation in parametric-feedback nonlinear systems [J]. IEEE Transactions on Automatic Control, 1998, 43: 1066-1083.
[19] Z.G. Pan, T. Basar. Backstepping controller design for nonlinear stochasticsystems under a risk-sensitive cost criterion [J]. SIAM Journal of Control and Optimization, 1999, 37: 957-995.
[20] Z.G. Pan, Y.G. Liu, S. Shi. Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics [J]. Science in China (Series F), 2001, 44: 292-308.
[21] Z.G. Pan, K. Ezal, A. Krener, P.V. Kokotovic. Backstepping design with local optimality matching [J]. IEEE Transactions on Automatic Control, 2001, 46: 1014-1027.
[22] Y.G. Liu, Z.G. Pan, S. Shi. Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitivie cost [J]. IEEE Transactions on Automatic Control, 2003, 48(3): 509-513.
[23] Y.G. Liu, J.F. Zhang. Reduced-order observer-based control design for nonlinear stochastic systems [J]. System and Control Letters, 2004, 52: 123-135.
[24] Y.G Liu, J.F. Zhang. Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems [J]. Science in China (Ser. F), 2004, 47: 527-544.
[25] Y.G Liu, J.F. Zhang. Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics [J]. SI AM Journal of Control and Optimization, 2006, 45: 885-926.
[26] Y.G. Liu, J.F. Zhang, Z.G. Pan. Design of satisfaction output feedback controls for stochastic nonlinear systems under quadratic tracking risk-sensitive index [J]. Science in China (Ser.F), 2003, 46: 126-145.
[27] H. Deng, M. Krstic. Stochastic nonlinear stabilization part I: a back-stepping design [J]. System and Control Letters, 1997, 32: 143-150.
[28] H. Deng, M. Krstic. Stochastic nonlinear stabilization part II: inverse Optimality [J]. System and Control Letters, 1997, 32: 151-159.
[29] H. Deng, M. Krstic. Output-feedback stochastic nonlinear stabilization [J]. IEEE Transactions on Automatic Control, 1999, 44: 328-333.
[30] H. Deng, M. Krstic. Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance [J]. System and Control Letters, 2000, 39: 173-182.
[31] H. Deng, M. Krstic, R.Williiams. Stabilization of stochastic nonlinear driven by noise of unknown covariance [J]. IEEE Transaction on Automatic Control, 2001, 46: 1237-1253.
[32] M. Krstic, H. Deng. Stability of Nonlinear Uncertain Systems [M]. New York: Springer-Verlag, 1998.
[33] Z.J. Wu, X.J. Xie, S.Y. Zhang. Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function [J]. International. Journal of Control, 2006, 79(12):1635-1646.
[34] Z.J. Wu, X.J. Xie. Adaptive backstepping controller design using stochastic small-gain theorem [J]. Automatica, 2007, 43(4): 608-620.
[35] S.J. Liu, J.F. Zhang, Z.P. Jiang. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems [J]. Automatica, 2007, 43(2): 238-251.
[36] M. Kristic, Z.H. Li. Inverse optimal design of input to state stabilizing nonlinear controllers [J]. IEEE Transactions on Automatic Control, 1998, 43: 336-350.
[37] R.A. Freeman, P.V. Kokotovic. Robust nonlinear control design: state-space and Lyapunov techniques [J]. IEEE Transactions on Automatic Control, 2003, 11: 73-83.
[38] W.C. Luo, Y.C. Chu, K.V. Ling. Inverse optimal adaptive control for attitude tracking of spacecraft [J]. IEEE Transactions on Automatic Control, 2005, 50: 1639-1654.
[39] X.J. Xie, J. Tian. State-feedback stabilization for high-order stochastic nonlinear systems with stochastuic inverse dynamics [J]. International Journal of Robust and Nonlinear Control, 2007, 17: 1343-1362.