${\cal H}$-Representation and Applications to Generalized Lyapunov Equations and Linear Stochastic Systems

This paper introduces an <i>H</i>-representation method to express an <i>n</i><sup>2</sup> × 1 vector <i>X</i><sup>→</sup> as <i>X</i><sup>→</sup>=<i>H</i>[(<i>X</i>)\tilde]. Based on the introduced <i>H</i>-representation approach, several topics are extensively discussed, including the generalized Lyapunov equations (GLEs) arising from stochastic control, stochastic observability, generalized <i>D</i>-stability and <i>D</i>-stabilization, weak stability, and stabilization. A necessary and sufficient condition for the existence and uniqueness of the symmetric and skew-symmetric solutions of GLEs is presented, respectively. Moreover, the solution structure of GLEs is also clarified. Through the <i>H</i>-representation method, several necessary and sufficient conditions are also obtained for stochastic observability, generalized <i>D</i>-stability and <i>D</i>-stabilization, weak stability, and stabilization.

[1]  Vasile Dragan,et al.  Stochastic observability and applications , 2004, IMA J. Math. Control. Inf..

[2]  Yulin Huang,et al.  Infinite horizon stochastic H2/Hinfinity control for discrete-time systems with state and disturbance dependent noise , 2008, Autom..

[3]  A. G. Mazko.  Lyapunov matrix Equations for a Certain Class of Regions Bounded by Algebraic Curves , 1980 .

[4]  Pierre Apkarian,et al.  Robust pole placement in LMI regions , 1999, IEEE Trans. Autom. Control..

[5]  Jan C. Willems,et al.  Feedback stabilizability for stochastic systems with state and control dependent noise , 1976, Autom..

[6]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[7]  Oswaldo Luiz do Valle Costa,et al.  Mean square stability conditions for discrete stochastic bilinear systems , 1985 .

[8]  Shengyuan Xu,et al.  H∞ Output Feedback Control with Spectrum Constraints for Uncertain Stochastic Systems , 2007 .

[9]  J. D. Val,et al.  On the observability and detectability of continuous-time Markov jump linear systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[10]  Lihua Xie,et al.  Interval Stability and Stabilization of Linear Stochastic Systems , 2009, IEEE Transactions on Automatic Control.

[11]  P. Gahinet,et al.  H∞ design with pole placement constraints: an LMI approach , 1996, IEEE Trans. Autom. Control..

[12]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[13]  D. Hinrichsen,et al.  Stochastic H∞ , 1998 .

[14]  S. Gutman,et al.  A general theory for matrix root-clustering in subregions of the complex plane , 1981 .

[15]  Xun Yu Zhou,et al.  Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls , 2000, IEEE Trans. Autom. Control..

[16]  D. Hinrichsen,et al.  H∞-type control for discrete-time stochastic systems , 1999 .

[17]  Guang-Ren Duan,et al.  On unified concepts of detectability and observability for continuous-time stochastic systems , 2010, Appl. Math. Comput..

[18]  Weihai Zhang,et al.  Infinite horizon stochastic H2/Hºº control for discrete-time systems with state and disturbance dependent noise. , 2008 .

[19]  Pascal Gahinet,et al.  H/sub /spl infin// design with pole placement constraints: an LMI approach , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[20]  R. Penrose.  On best approximate solutions of linear matrix equations , 1956, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  Tobias Damm.  On detectability of stochastic systems , 2007, Autom..

[22]  William L. Brogan,et al.  Modern control theory (3rd ed.) , 1991 .

[23]  Zidong Wang,et al.  Mixed H/sub 2//H/sub /spl infin// filtering for uncertain systems with regional pole assignment , 2005, IEEE Transactions on Aerospace and Electronic Systems.

[24]  Bor-Sen Chen,et al.  On stabilizability and exact observability of stochastic systems with their applications , 2023, Autom..

[25]  Ian R. Petersen,et al.  Notions of observability for uncertain linear systems with structured uncertainty , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[26]  R. Kálmán.  On the general theory of control systems , 1959 .

[27]  M. Fragoso,et al.  A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control , 1998 .

[28]  Brian D. O. Anderson,et al.  An iterative algorithm to solve state-perturbed stochastic algebraic Riccati equations in LQ zero-sum games , 2010, Syst. Control. Lett..

[29]  Valery A. Ugrinovskii.  Observability of linear stochastic uncertain systems , 2003, IEEE Trans. Autom. Control..

[30]  Zidong Wang.  Robust state estimation for perturbed systems with error variance and circular pole constraints: The discrete-time case , 2000 .

[31]  D. Hinrichsen,et al.  Stochastic $H^\infty$ , 1998 .

[32]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[33]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[34]  P. Shi.  Backward Stochastic Differential Equation and Exact Controllability of Stochastic Control Systems , 1994 .

[35]  Venkataramanan Balakrishnan,et al.  Semidefinite programming duality and linear time-invariant systems , 2003, IEEE Trans. Autom. Control..

[36]  Bor-Sen Chen,et al.  Generalized Lyapunov Equation Approach to State-Dependent Stochastic Stabilization/Detectability Criterion , 2008, IEEE Transactions on Automatic Control.

[37]  Hiroaki Mukaidani,et al.  Soft-constrained stochastic Nash games for weakly coupled large-scale systems , 2009, Autom..