Proceedings of the American Control Conference Chicago, Illinois * June 2000 Analysis and Control of Polytopic Uncertain/Nonlinear Systems in the Presence of Bounded Disturbance Inputs T. PANCAKE, M. CORLESS 1 School of Aeronautics and Astronautics Purdue University West Lafayette,IN 47907 M. BROCKMAN Dynetics Inc. Huntsville, AL 35806 Abstract For a general input-output system subjected t.o disturbance inputs whose magnitudes are uniformly bounded, we intro- duce the notion of Loo stability with level of performance % We present a sufficient condition which guarantees Loo stability of this system. Furthermore, for a specific class of uncertain/nonlinear systems, characterized by a polytopic condition, this condition results in a bunch of matrix in- equalities. An iterative algorithm based on LMIs (Linear Matrix Inequalities) is proposed for minimizing the level of performance. We also investigate the use of linear state feedback controllers to minimize the level of performance and illustrate our results with a nontrivial example. 1 Introduction When analyzing or controlling a system, one is quite often concerned with the peak magnitude of some per- formance output. Clearly, the peak magnitude of im- portant variables is of concern in many engineering sys- tems, from the motion of an aircraft or spacecraft to the path of a robotic arm. Here, we are concerned with uncertain/nonlinear systems which are subject to disturbance inputs whose magnitudes are uniformly bounded. For a given performance output, we wish to either compute an ultimate bound on its magni- tude or design a controller which achieves some desired bound. To this end, we introduce the notion of Loo stability with level of performance '7 for a general un- certain/nonlinear system. For zero initial state, "7 is an upper bound on the Loo gain of the system, that is, the gain of the system when viewed as an operator acting on Loo inputs and producing Loo outputs. This concept is similar to that in the work of Sontag et al. in [6, 7, 8]. Using a Lyapunov based approach, we introduce a re- sult which yields a sufficient condition for our notion of L~ stability. This condition is applied to a class of polytopic uncertain/nonlinear systems to obtain a bunch of matrix inequalities which, if satisfied, guar- antee Lc¢ stability with a level of performance. An 1 Corresponding author 0-7803-5519-9/00 $10.00 © 2000 AACC iterative algorithm based on LMIs (Linear Matrix In- equalities) is proposed to minimize the level of per- formance. These results are based on quadratic Lya- punov functions. When applied to a fixed linear time- invariant system, they recover the corresponding re- sults of Brockman [2] and Abedor, Nagpal, and Poolla [1]. We also investigate the use of linear state feedback con- trollers to minimize the level of performance. Using our analysis results, we obtain conditions involving matrix inequalities which can be used for controller design. We then use these results to obtain a disturbance attenua- tion controller for a single link manipulator with a flex- ible joint which is subject to an unknown but bounded disturbance torque. 2 Analysis Consider an input-output system described by = F(x,w) (la) z = H(x,w) (lb) where x(t) E IRn is the state vector at time t and w(t) E IRl is the exogenous (or, disturbance) input while z(t) E IRP is the performance output. For a vector y E IRk, we define its norm as [lyil = v/~ • The Loo norm of a continuous function v(.) : [0, c<~) --+ IRk is defined as the supremum over all time t of the norm of v(t), that is, IIv(.)lloo = sup IIv(t)ll . t_>0 First, we introduce a concept of stability and perfor- mance that we will use throughout this paper. Definition 1 The input-output system (1) is Loo sta- ble with level of performance '7 if the following conditions are satisfied. 1. The undisturbed system, = F(x,0), (2) 159