Optimal design for discrete‐time linear systems via new performance index

Abstract

Abstract We introduce a class of linear discrete‐time systems called ‘superstable’. For the SISO case this means that the absolute value of the constant term of the characteristic polynomial is greater than the sum of absolute values of all other coefficients, while superstable MIMO systems have a state matrix with l 1 norm less than one. Such systems have many special features. First, non‐asymptotic bounds for the output of such systems with bounded input can be easily obtained. In particular, for small enough initial conditions, we get the equalized performance property, recently introduced for the SISO case by Blanchini and Sznaier (36th CDC, San Diego, 1997, pp. 1540–1545). Second, the same bounds can be obtained for LTV systems, provided all the frozen LTI systems are super stable. This makes the notion well suited for adaptive control. These bounds can be used as the performance index for optimal controller design, as proposed by Blanchini and Sznaier for the SISO case. Then to obtain disturbance rejection in SISO or MIMO systems, we design a controller which guarantees super stability of the closed‐loop system and minimizes the proposed performance index ( γ ‐optimality). This problem happens to be quasiconvex with respect to the controller coefficients and can be solved via parametric linear programming. Compared with the well‐known l 1 optimization‐based design technique, the approach allows low‐order controllers to be designed (while l 1 optimal controllers may have high order) and can take into account non‐asymptotic time‐domain behaviour of the system with non‐zero initial conditions. For unbounded controller orders we prove the existence and finite dimensionality of γ ‐optimal designs. We also address the robustness issues for transfer functions with coprime factor uncertainty bounded in l 1 norm. A robust performance problem can be formulated and similarly solved via linear programming. Numerous examples are provided to compare the proposed design with optimal l 1 and H ∞ controllers. Copyright © 2001 John Wiley & Sons, Ltd.