Nonlinear control design for linear differential inclusions via convex hull of quadratics

This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions (LDIs). A recently introduced non-quadratic Lyapunov function, the convex hull of quadratics, will be used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least L"2 gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Numerical examples demonstrate the advantages of using nonlinear feedback control over linear feedback control for LDIs. It is also observed through numerical computation that nonlinear control strategies help to reduce control effort substantially.

[1]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[2]  R. Shorten,et al.  A numerical technique for the stability analysis of linear switched systems , 2004 .

[3]  Tingshu Hu,et al.  Conjugate convex Lyapunov functions for dual linear differential inclusions , 2006, IEEE Transactions on Automatic Control.

[4]  A. Polański.  Lyapunov function construction by linear programming , 1997, IEEE Trans. Autom. Control..

[5]  P. Khargonekar,et al.  H/sub infinity /-optimal control with state-feedback , 1988 .

[6]  Tingshu Hu,et al.  Composite quadratic Lyapunov functions for constrained control systems , 2003, IEEE Trans. Autom. Control..

[7]  Z. Jarvis-Wloszek,et al.  An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[8]  Alexandre Megretski,et al.  Robust state feedback control of LTV systems: Nonlinear is better than linear , 1997 .

[9]  Tingshu Hu,et al.  Conjugate Lyapunov functions for saturated linear systems , 2005, Autom..

[10]  A. Teel,et al.  Stability and performance for saturated systems via quadratic and non-quadratic Lyapunov functions ∗ , 2022 .

[11]  M. Fu,et al.  Piecewise Lyapunov functions for robust stability of linear time-varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[12]  Tingshu Hu,et al.  Properties of the composite quadratic Lyapunov functions , 2004, IEEE Trans. Autom. Control..

[13]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[14]  Tingshu Hu,et al.  Composite quadratic Lyapunov functions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[15]  Tingshu Hu,et al.  Properties of the composite quadratic Lyapunov functions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[16]  Stephen P. Boyd,et al.  A path-following method for solving BMI problems in control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[17]  Andrzej Polanski,et al.  On absolute stability analysis by polyhedral Lyapunov functions , 2000, Autom..

[18]  Graziano Chesi,et al.  Homogeneous Lyapunov functions for systems with structured uncertainties , 2003, Autom..

[19]  Y. Pyatnitskiy,et al.  Criteria of asymptotic stability of differential and difference inclusions encountered in control theory , 1989 .

[20]  Franco Blanchini,et al.  Nonquadratic Lyapunov functions for robust control , 1995, Autom..

[21]  Tingshu Hu,et al.  Dual Matrix Inequalities in Stability and Performance Analysis of Linear Differential/Difference Inclusions , 2006 .

[22]  Luca Zaccarian,et al.  Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions , 2006, IEEE Transactions on Automatic Control.