H/sub ∞/ control of nonlinear systems via output feedback: a class of controllers

Abstract

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 * .

A family of adaptive H ∞ controllers with full information for dissipative hamiltonian systems

“…From the proof of Theorem 1, we know that the Hamiltonian function H(x) can be used to build a corresponding Lyapunov function of the system. As a result, unlike the parameterization method presented in [12][13][14][15], the one obtained in this paper avoids solving Hamilton-Jacobi-Issacs equations.…”

CONTROLLER REDUCTION FOR NONLINEAR PLANTS—ANL2APPROACH

“…Consider the nonlinear input affine plant P, (1), and the HJE: (8). Then an NRCF of the scaled plant P M " P, (13), with "(1!…”

Nonlinear H/sub ∞/ control: a J-dissipative approach

“…A parameterization of controllers solving locally the -control problem via state feedback was obtained in [10] for a restricted class of input affine nonlinear plants, assuming simplified output structure. Under the same simplifying assumptions, a class of controllers was provided in [15], which solves this problem locally by output feedback in terms of two uncoupled Hamilton-Jacobi inequalities. This was done using the approach in [5] and [6] for linear plants characterized by their scattering description.…”

“…Therefore, an important problem which we take up here is the parameterization of controllers which solve the nonlinear -control problem. Our work continues a series of investigations on the -control problem for nonlinear systems [1]- [4], [9]- [11], [15], [20]- [23]. A parameterization of controllers solving locally the -control problem via state feedback was obtained in [10] for a restricted class of input affine nonlinear plants, assuming simplified output structure.…”