Robust adaptive control for nonlinear uncertain systems

A direct robust adaptive control framework for non linear uncertain systems with constant linearly parame terized uncertainty and nonlinear state dependent uncer tainty is developed The proposed framework is Lyap unov based and guarantees partial asymptotic robust sta bility of the closed loop system that is asymptotic robust stability with respect to part of the closed loop system states associated with the plant Finally a numerical example is provided to demonstrate the e cacy of the proposed approach Introduction In a recent paper a direct nonlinear adaptive con trol framework for adaptive stabilization disturbance re jection and command following was developed In partic ular a Lyapunov based direct adaptive control framework was developed that guarantees partial asymptotic stabil ity of the closed loop system that is asymptotic stabil ity with respect to part of the closed loop system states associated with the plant Furthermore the remainder of the state associated with the adaptive controller gains was shown to be Lyapunov stable In the case where the nonlinear system was represented in normal form with input to state stable zero dynamics the nonlin ear adaptive controller was constructed without requiring knowledge of the system dynamics As is the case in the adaptive control literature the system errors in are captured by a constant linearly parameterized uncertainty model of a known structure but unknown variation This uncertainty characterization al lows the system nonlinearities to be parameterized by a nite linear combination of basis functions within a class of function approximators such as rational functions spline functions radial basis functions sigmoidal functions and wavelets However this linear parameterization of ba sis functions cannot exactly capture the unknown system nonlinearity In this paper we generalize the results of to nonlinear uncertain systems with constant linearly parameterized uncertainty and nonlinear state dependent uncertainty Speci cally we consider a robust adaptive control problem that guarantees asymptotic robust sta bility of the system states in the face of structured un certainty with unknown variation and structured possi bly nonlinear parametric uncertainty with bounded vari ation Hence the overall adaptive control framework cap tures the residual approximation error inherent in linear parameterizations of system uncertainty via basis func tions Robust Adaptive Control for Nonlinear Un certain Systems In this section we introduce an adaptive feedback con trol problem for nonlinear uncertain systems with con This research was supported in part by the National Sci ence Foundation under Grant ECS and the Air Force O ce of Scienti c Research under Grant F 16 stant linearly parameterized uncertainty structure and nonlinear state dependent parametric uncertainty Specif ically consider the controlled nonlinear uncertain system G given by x t f x t f x t G x t u t x x t where x t R t is the state vector u t R t is the control input f R R and satis es f f R R and satis es f and G R R m Here we assume that f and f are uncertain with f belonging to the uncertainty set F given by F f f R R f x G x h x x R g where satis es f R R T y y m y m y y R g and where G R n R m and h R n R satisfy ing h are xed functions denoting the structure of the uncertainty R R is an uncertain func tion and m R R is a given function such that m The special case m y y where is worth noting Speci cally in this case specializes to f R R T y y yy y R g which corresponds to a nonlinear small gain type norm bounded uncertainty characterization The control u in is restricted to the class of ad missible controls consisting of measurable functions such that u t R t Furthermore for the nonlinear un certain system G we assume that the required properties for the existence and uniqueness of solutions are satis ed that is f f G and u satisfy su cient regular ity conditions such that has a unique solution forward in time Theorem Consider the nonlinear uncertain sys tem G given by Assume there exists a matrix Kg R m s and functions G R R m and F R R with F such that the zero solution x t to x t f x t G x t G x t KgF x t fc x t x x t is globally asymptotically stable Furthermore assume there exist functions Vs R n R and R R such that Vs is continuously di erentiable positive de nite and radially unbounded Vs and for all x R n