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Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. The so-called nonlinear X,-control problem in state space is considered with an emphasis on developing machinery with promising computational properties. Both state feedback and output feedback %,-control problems for a class of nonlinear systems are characterized in terms of continuous positive definite solutions of algebraic nonlinear matrix inequalities which are convex feasibility problems. The issue of existence of solutions to these nonlinear matrix inequalities (NLMIs) is justified.

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The standard state space solutions to the H, control problem for linear time invariant systems are generalized to nonlinear timeinvariant systems. A class of nonlinear H,-controllers are parametrized as nonlinear fractional transformations on contractive, stable free nonlinear parameters. As in the linear case, the H, control problem is solved by its reduction to state feedback and output injection problems, together with a separation argument. The sufficient conditionsfor %,-control problem to be solved are also derived with this machinery. The solvability for nonlinear %,-control problem requires positive definite solutions to two parallel decoupled Hamilton-Jacobi inequalities and these two solutions satisfy an additional coupling condition. An illustrative example, which deals with a pwive plant, is given at the end. This paper is a condensed version of [ll). &-Gains of Nonlinear SystemsConsider the following input-affine nonlinear time-invariant (NLTI) system: k = f(I) + g ( z ) w z = h (~) + k ( i ) w G : (Where I E Rn is state vector, w E RP and z E Rq are input and output vectors, respectively. We will assume f, g, h, k E C O , and f(0) = O , h ( O ) = 0. Therefore, 0 E R" is the equilibrium of the system with w = 0. The state transition function 4 : R+ x R" x RP -* Rn is so defined that I = ~( T , I o , w * ) means that system G evolves from initial state IO to state I in time T under the control action w * .