MINIMIZATION OF THE L-INFINITY-INDUCED NORM FOR SAMPLED-DATA SYSTEMS

The following problem is addressed: Given a continuous-time plant, wit h continuous-time performance objectives, expressed in terms of the L( infinity)-induced norm, design a digital controller that delivers or o ptimizes this performance. This problem differs from the standard disc rete-time methods in that it takes into consideration the inter-sample behavior of the closed-loop system. The resulting closed-loop system dynamics consist of both continuous-time and discrete-time dynamics an d thus such systems are known as hybrid systems. It is shown that give n any degree of accuracy, there exists a standard discrete-time l1 pro blem, which can be determined a priori, whose solution yields a contro ller that is almost optimal in terms of the hybrid L(infinity)-induced norm. This is accomplished by first converting the hybrid system into an equivalent infinite-dimensional discrete-time system using the lif ting technique in continuous time, then the infinite-dimensional parts of the system which model the inter-sample dynamics are approximated. We present a thorough analysis of the approximation procedure, and sh ow that it is convergent at the rate of (1/n). Explicit bounds that ar e independent of the controller are obtained to characterize the appro ximation. Finally, it is shown that the geometry of the induced norm f or the sampled-data problem is different than that of the standard l1 norm, and hence there might not exist a linear isometry that maps the sampled-data problem exactly to a standard discrete-time problem.