State feedback synthesis is considered for linear discrete-time systems to
minimize an upper bound of the closed-loop induced Il-norm. A sufficient co
ndition, called Generalized Bounded Real Inequality (GBRI), is presented to
establish such a bound. An algorithm similar to the Invariant Kernel Algor
ithm (Shamma, IEEE Trans. Automat. Control 41 (1996) 533-544) and the contr
active 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 exa
mple is given for which the optimal induced il-norm achieved by a linear co
ntroller is achieved by our synthesis approach. (C) 1999 Elsevier Science L
td. All rights reserved.