Robust Static Output Feedback Design with Deterministic and Probabilistic Certificates

Static output feedback design for linear plants is well known to be a challenging non-convex problem. The presence of plant uncertainty makes this challenge even harder. In this chapter, we propose a new BMI formulation with S-variables which includes an interesting link between state feedback, output injection, state injection, and static output feedback gains in a unified framework. Based on this formulation, the robust design problem is suitably addressed by iterative optimization procedures with either deterministic or probabilistic viewpoints exploiting the fact that Lyapunov certificates are separated from the control gain design variables. The deterministic approach is for affine polytopic systems. The probabilistic approach requires no assumption on the uncertain system, and is based on the Scenario with Certificates (SwC) method which was recently proposed to address certain static anti-windup design problems. Numerical results illustrate the effectiveness of the approach in both deterministic and stochastic cases.

[1]  Chaouki T. Abdallah,et al.  Static output feedback: a survey , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[2]  Giuseppe Carlo Calafiore,et al.  Random Convex Programs , 2010, SIAM J. Optim..

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

[4]  Roberto Tempo,et al.  Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms , 2013, Autom..

[5]  Patrizio Colaneri,et al.  RH 2 Control , 1997 .

[6]  Mario Sznaier,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems with Applications, Second Edition, Roberto Tempo, Giuseppe Calafiore, Fabrizio Dabbene (Eds.). Springer-Verlag, London (2013), 357, ISBN: 978-1-4471-4609-4 , 2014, Autom..

[7]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[8]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[9]  O. Toker,et al.  On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[10]  Giuseppe Carlo Calafiore,et al.  Research on probabilistic methods for control system design , 2011, Autom..

[11]  Dimitri Peaucelle,et al.  From static output feedback to structured robust static output feedback: A survey , 2016, Annu. Rev. Control..

[12]  Denis Arzelier,et al.  H2 for HIFOO , 2010, 1010.1442.

[13]  Dimitri Peaucelle,et al.  S-Variable Approach to LMI-Based Robust Control , 2014 .

[14]  Michael Stingl,et al.  PENLAB: A MATLAB solver for nonlinear semidefinite optimization , 2013 .

[15]  D. Henrion,et al.  Solving polynomial static output feedback problems with PENBMI , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[16]  Marco C. Campi,et al.  The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs , 2008, SIAM J. Optim..

[17]  Guang-Hong Yang,et al.  Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties , 2013, Autom..

[18]  Michael L. Overton,et al.  Multiobjective robust control with HIFOO 2.0 , 2009, 0905.3229.

[19]  Arturo Locatelli,et al.  Control Theory and Design: An Rh2 and Rh Viewpoint , 1997 .

[20]  Luca Zaccarian,et al.  Robust Linear Static Anti-Windup With Probabilistic Certificates , 2015, IEEE Transactions on Automatic Control.

[21]  Marco C. Campi,et al.  The exact feasibility of randomized solutions of robust convex programs , 2008 .

[22]  Yasuaki Oishi,et al.  Probabilistic Design of a Robust Controller Using a Parameter-Dependent Lyapunov Function , 2006 .