Optimal induced l1-norm state feedback control

State feedback synthesis is considered for linear discrete-time systems to minimize an upper bound of the closed-loop induced Il-norm. A sufficient condition, called Generalized Bounded Real Inequality (GBRI), is presented to establish such a bound. An algorithm similar to the Invariant Kernel Algorithm (Shamma, IEEE Trans. Automat. Control 41 (1996) 533-544) and the contractive set algorithm (Blanchini, IEEE Trans. Automat. Control 39 (1994) 428-433) reduces the analysis and synthesis problems to linear programming. If the problem is feasible, our algorithm gives a polyhedral set that induces a closed-loop Lyapunov function and leads to feasible control laws. An example is given for which the optimal induced il-norm achieved by a linear controller is achieved by our synthesis approach. (C) 1999 Elsevier Science Ltd. All rights reserved.