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Mixed H2/H∞ Control via Nonsmooth Optimization

Abstract : We present a new approach to mixed H 2 /H∞ output feedback control synthesis. Our method uses non-smooth mathematical programming techniques to compute locally optimal H 2 /H∞-controllers, which may have a pre-defined structure. We prove global convergence of our method and present numerical tests to validate it numerically. Key words. Mixed H 2 /H∞ output feedback control, multi-objective control, robustness and performance, non-smooth optimization, trust region technique. AMS subject classifications. 93B36, 93B50, 90C29, 49J52, 90C26, 90C34, 49J35 1. Introduction. Mixed H 2 /H ∞ output feedback control is a prominent example of a multi-objective design problem, where the feedback controller has to respond favorably to several performance specifications. Typically in H 2 /H ∞ synthesis, the H ∞ channel is used to enhance the robustness of the design, whereas the H 2 channel guarantees good performance of the system. Due to its importance in practice, mixed H 2 /H ∞ control has been addressed in various ways over the years, and we briefly review the main trends. The interest in H 2 /H ∞ synthesis was originally risen by three publications [22, 23, 27] in the late 1980s and early 1990s. The numerical methods proposed by these authors are based on coupled Riccati equations in tandem with homotopy methods, but the numerical success of these strategies remains to be established. With the rise of LMIs in the later 1990s, different strategies which convexify the problem became increasingly popular. The price to pay for convexifying is either a considerable conservatism, or that controllers have large state dimension [29, 25]. In [45, 47, 48] Scherer developed LMI formations for H 2 /H ∞ synthesis for full-order controllers [48], and reduced the problem to solving LMIs in tandem with non-linear algebraic equalities [48, 45]. In this form, H 2 /H ∞ problems could in principle be solved via nonlinear semidefinite programming techniques like specSDP [24, 39, 49] or Pennon [31, 32, 36], if only these techniques were suited for medium or large size problems. Alas, one of the disappointing lessons learned in recent years from investigating BMI and LMI problems is that this is just not the case. Due to the presence of Lyapunov variables, whose number grows quadratically with the system size, [13, p. 20ff], BMI and LMI programs quickly lead to problem sizes where existing numerical methods fail. Following [3, 4, 5, 6, 7], we address H 2 /H ∞-synthesis by a new strategy which avoids the use of Lyapunov variables. This leads to a non-smooth and semi-infinite optimization program, which we solve with a spectral bundle method, inspired by the non-convex spectral bundle method of [37, 38] and [3, 5]. Important forerunners [19, 40, 28] are based on convexity and optimize functions of the form λ 1 • A with affine A. We have developed our method further to deal with typical control applications like multi-disk [7] and multi frequency band synthesis [6], design under integral quadratic constraints (IQCs) [4, 9, 8], and to loop-shaping techniques [2, 1].
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P. Apkarian, Dominikus Noll, Aude Rondepierre. Mixed H2/H∞ Control via Nonsmooth Optimization. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2008, 47 (3), pp.1516-1546. ⟨10.1137/070685026⟩. ⟨hal-00634512⟩

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