Robust control via sequential semidefinite programming

This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under Linear Matrix Inequality (LMI) in tandem with nonlinear matrix equality constraints. The latter type of constraints renders the design numerically and algorithmically difficult. We solve the optimization problem via sequential semidefinite programming (SSDP), a technique which expands on sequential quadratic programming (SQP) known in nonlinear optimization. Global and fast local convergence properties of SSDP are similar to those of SQP, and SSDP is conveniently implemented with available semidefinite programming (SDP) solvers.

[1]  R.T. Reichert,et al.  Robust autopilot design using μ-synthesis , 1990, 1990 American Control Conference.

[2]  M. Safonov,et al.  Real/complex multivariable stability margin computation via generalized Popov multiplier-LMI approach , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[3]  Friedemann Leibfritz,et al.  An Interior Point Constrained Trust Region Method for a Special Class of Nonlinear Semidefinite Programming Problems , 2002, SIAM J. Optim..

[4]  B. Fares,et al.  An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory , 2001 .

[5]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[6]  Pierre Apkarian,et al.  Robust control via concave minimization local and global algorithms , 2000, IEEE Trans. Autom. Control..

[7]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[8]  P. Gahinet,et al.  A convex characterization of gain-scheduled H∞ controllers , 1995, IEEE Trans. Autom. Control..

[9]  Pierre Apkarian,et al.  Concave Programming in Control Theory , 1999, J. Glob. Optim..

[10]  S. M. Robinson.  Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[11]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[12]  L. Ghaoui,et al.  A cone complementarity linearization algorithm for static output-feedback and related problems , 1997, IEEE Trans. Autom. Control..

[13]  J. F. Bonnans,et al.  Local analysis of Newton-type methods for variational inequalities and nonlinear programming , 1994 .

[14]  G. Becker.  Parameter-dependent control of an under-actuated mechanical system , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[15]  Friedemann Leibfritz,et al.  Trust region methods for solving the optimal output feedback design problem , 2003, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[16]  A. Packard.  Gain scheduling via linear fractional transformations , 1994 .

[17]  P. Gahinet,et al.  Affine parameter-dependent Lyapunov functions and real parametric uncertainty , 1996, IEEE Trans. Autom. Control..

[18]  Nicholas J. Higham,et al.  Modifying the inertia of matrices arising in optimization , 1998 .

[19]  Carsten W. Scherer,et al.  Robust mixed control and linear parameter-varying control with full block scalings , 1999 .

[20]  L. Ghaoui,et al.  Measurement‐scheduled control for the RTAC problem: an LMI approach , 1998 .

[21]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[22]  Pierre Apkarian,et al.  Self-scheduled H∞ control of linear parameter-varying systems: a design example , 1995, Autom..

[23]  E. Feron,et al.  Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions , 1996, IEEE Trans. Autom. Control..

[24]  C. Lemaréchal,et al.  Nonsmooth Algorithms to Solve Semidefinite Programs , 1999 .

[25]  J. W. Helton,et al.  Coordinate optimization for bi-convex matrix inequalities , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[26]  Paul T. Boggs,et al.  Sequential Quadratic Programming , 1995, Acta Numerica.

[27]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[28]  K. Goh,et al.  Control system synthesis via bilinear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[29]  C. Scherer.  A full block S-procedure with applications , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[30]  Pierre Apkarian,et al.  Robust Control via Sequential Semidefinite Programming , 2002, SIAM J. Control. Optim..

[31]  Anders Helmersson,et al.  Methods for robust gain scheduling , 1995 .