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lec7_6245_2004

# lec7_6245_2004 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Algorithms for H-Infinity Optimization 1 This lecture covers the algorithmic side of H-Infinity optimization. It presents classical necessary and suﬃcient conditions of existence of a β -suboptimal controller, stated in terms of stabilizing solutions of Riccati equations, and provides explicit formulae for the so-called “central” β -suboptimal controller. 7.1 Problem Formulation and Algorithm Objectives This section examines the specifics of the way in which the H-Infinity optimal LTI feedback design problem is formulated (and, eventually, solved). 7.1.1 Suboptimal Feedback Design H-infinity optimization problem appears to be formulated as the task of designing a sta- bilizing controller K ( s ), which minimizes the H-infinity norm of the closed-loop transfer matrix T zw from w to z for a given open loop plant P ( s ) (see Figure 7.1), defined by state space equations ˙ x = Ax + B 1 w + B 2 u z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u A set of standard well posedness constraints is imposed on the setup: 1 Version of March 29, 2004

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2 w z P ( s ) y u ± K ( s ) Figure 7.1: General LTI design setup for output feedback stabilizability, the pairs ( A, B 2 ) and ( C 2 , A ) must be respectively stabilizable and detectable for non-singularity, D 21 must be right invertible (full measurement noise), D 12 must be left invertible (full control penalty), and matrices A sI B 2 A sI B 1 C 1 D 12 , C 2 D 21 must be respectively left and right invertible for all s j R However, in contrast with the case of H2 optimization, basic H-Infinity algorithms solve a suboptimal controller design problem, formulated as that of finding whether, for a given β > 0, a controller achieving the closed loop L2 gain T zw < β exists, and, in case the answer is aﬃrmative, calculating one such controller. 7.1.2 Why Suboptimal Controllers? There are several reasons to prefer suboptimal controllers over the optimal one in H- Infinity optimization. One of the most compelling reasons is that the optimal closed loop transfer matrix T zw can be shown to have a constant largest singular number over the complete frequency range. In particular, this means that the optimal controller is not strictly proper, and the optimal frequency response to the cost output does not roll off at high frequencies. Example 7.1 Consider the standard LTI feedback optimization setup with 1 1 s +1 P ( s ) = 1 s 0 , 1+ s
3 (a case of state estimation). Since there is no feedback loop to close here, the controller transfer function K ( s ) must be a stable, and the cost is the H-Infinity norm 1 s 1 T zw min , T zw = K ( s ) .

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