Automation and Remote Control, Vol. 63, No. 11, 2002, pp. 1745–1763. Translated from Avtomatika i Telemekhanika, No. 11, 2002, pp. 56–75. Original Russian Text Copyright c  2002 by Polyak, Shcherbakov. DETERMINATE SYSTEMS Superstable Linear Control Systems. II. Design 1 B. T. Polyak and P. S. Shcherbakov Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia Received April 24, 2002 Abstract—The notion of superstability introduced in [1] is applied to the design of stabilizing and optimal controllers. It is shown that a static output feedback controller which ensures superstability of the closed-loop system can be found (provided it exists) by means of linear programming (LP) techniques; finding a superstable matrix in the given affine family is a generalization of this problem. The ideology of superstability is also shown to be fruitful in optimal and robust control. This is exemplified by the problems of rejection of bounded disturbances, optimization of the integral performance index which involves absolute values (rather than squares) of the control and state, and by stabilization of an interval matrix family and simultaneous stabilization. 1. INTRODUCTION In the first part of the paper, [1], the notion of superstability was introduced and the analysis of superstable systems was performed (the behavior of solutions was examined both in the absence and in the presence of exogenous disturbances, the location of the spectrum was analyzed, the robust properties were studied, etc.). However, the most important feature of this notion is its applicability to design. Namely, it turns out that many hard classical problems such as static output stabilizaton, fixed-order controller design, uniform rejection of exogenous disturbances, etc., admit simple solutions provided that the stability requirement for the closed-loop system is replaced with superstability. Thus, all the above-mentioned problems can be solved using LP techniques. Furthermore, some nonstandard problems of optimal control are made tractable in this framework such as the linear-linear (rather than linear-quadratic) regulator problem discussed below. Clearly, this approach has its own drawbacks. For instance, a controllable linear system is not necessarily superstabilizable using state feedback; also, in optimal control problems, estimates of performance indices are used rather than exact values. We start with superstabilization. Said another way, we seek a (state or output) feedback which ensures superstability of the closed-loop system. Similarly to the first part of the paper, the ∞-norm for vectors, ‖x‖ = max 1≤i≤n |x i |, x ∈ R n , and the induced 1-norm for matrices, ‖A‖ = max 1≤i≤n  n ∑ j =1 |a ij |  , A = ((a ij )) ∈ R n×n , are used unless otherwise indicated. 2. SUPERSTABILIZATION OF CONTINUOUS-TIME SYSTEMS Let us consider the following continuous-time system: ˙ x = Ax + Bu, y = Cx, (1) 1 This work was supported by the Russian Foundation for Basic Research, projects nos. 00-15-96018 and 02-01-00127, and carried out within the framework of the complex program of the Presidium of the Russian Academy of Sciences. 0005-1179/02/6311-1745$27.00 c  2002 MAIK “Nauka/Interperiodica”