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Dual approaches to strictly positive real controller synthesis with a math formula performance using linear matrix inequalities


  • James Richard Forbes

    Corresponding author
    • Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, Montreal, Quebec, Canada
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  • This article was published online on March 22, 2012. An error was subsequently identified. This notice is included in the online and print versions to indicate that both have been corrected on January 29, 2013.

Correspondence to: James Richard Forbes, Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0C3.



The synthesis of controllers that minimize a math formula performance index subject to a strictly positive real (SPR) constraint is considered. Two controller synthesis methods are presented that are then combined into an iterative algorithm. Each method synthesizes optimal SPR controllers by posing a convex optimization problem where constraints are enforced via linear matrix inequalities. Additionally, each method fixes the controller state-feedback gain matrix and finds an observer gain matrix such that an upper bound on the closed-loop math formula-norm is minimized and the controller is SPR. The first method retools the standard math formula-optimal control problem by using a common Lyapunov matrix variable to satisfy both the math formula criteria and the SPR constraint. The second method overcomes bilinear matrix inequality issues associated with the math formula performance and the SPR constraint by employing a completing the square method and an overbounding technique. Both synthesis methods are used within an iterative scheme to find optimal SPR controllers in a sequential manner. Comparison of our synthesis methods to existing methods in the literature is presented. Copyright © 2012 John Wiley & Sons, Ltd.