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

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 o f the characteristic polynomial is greater than the sum of absolute values of all other coefficients, while superstable MIMO systems have a state matr ix with l(1) norm less than one. Such systems have many special features. F irst, non asymptotic bounds for the output of such systems with bounded inp ut can be easily obtained. In particular, for small enough initial conditio ns, we get the equalized performance property, recently introduced for the SISO case by Blanchini and Sznaier (36th CDC, San Diego, 1997, pp. 1540-154 5). Second, the same bounds can be obtained for LTV systems, provided all t he frozen LTI systems are super stable. This makes the notion well suited f or adaptive control. These bounds can be used as the performance index for optimal controller de sign, as proposed by Blanchini and Sznaier for the SISO case. Then to obtai n disturbance rejection in SISO or MIMO systems, we design a controller whi ch guarantees super stability of the closed-loop system and minimizes the p roposed performance index (gamma -optimality). This problem happens to be q uasiconvex with respect to the controller coefficients and can be solved vi a parametric linear programming. Compared with the well-known l(1) optimiza tion-based design technique, the approach allows low-order controllers to b e designed (while l(1) optimal controllers may have high order) and can tak e into account non-asymptotic time-domain behaviour of the system with non- zero initial conditions. For unbounded controller orders we prove the exist ence and finite dimensionality of gamma -optimal designs. We also address t he robustness issues for transfer functions with coprime factor uncertainty bounded in l(1) norm. A robust performance problem can be formulated and s imilarly solved via linear programming. Numerous examples are provided to c ompare the proposed design with optimal l(1) and H-infinity controllers. Co pyright (C) 2001 John Wiley & Sons, Ltd.