UCLA

This paper presents an adaptive robust control strategy for a class of uncertain linear systems with disturbance input. The uncertain linear system contains unknown parameters in both the system and the input matrices. The system makes perfect partial measurement of the state. A disturbance attenuation function is transformed into a minimax differential game where the disturbance and the unknown parameters act as the cooperative players that maximize the cost function, whereas the controller is the opposing player that minimizes the cost function. Having shown the existence of a saddle point in the cost function, the optimization yields a minimax controller coupled with a stable nonlinear modified gain observer, which estimates the state and the unknown parameters. By maximizing the cost function with respect to the uncertain parameters, the minimax controller takes the parameter estimation confidence level into account to generate the worst case input for a given instance of time. This leads to two sets of Riccati equations, one for the controller and one for the observer.

In this class of parameter estimation problem, the measurement function and the augmented system, which is composed of the state and the uncertain parameters, are modifiable functions or can be transformed into modifiable functions using an appropriate change of coordinate system, such as the observable canonical form. The essence of a modifiable function is that although the observer dynamics are nonlinear, the error in the observer's estimation error is linear. The existence of the saddle point in the performance index is presented. Under certain conditions, such as the observability and controllability of the system, the existence of the solution to the two Riccati equation and finiteness of the value function, the close loop system is stable and the estimation error is bounded, which demonstrates the disturbance attenuation properties of the observer.

We present an unstable single-input single-output (SISO) four state vector case example with four unknown system parameters, subject to worst case disturbance. The simulation demonstrates the applicability of the disturbance attenuation controller coupled with the nonlinear modified gain observer, and assess its performance against the linear quadratic regulator (LQR) coupled with a modified gain extended Kalman observer (MGEKO).

Furthermore, the derived disturbance attenuation controller is applied to an air breathig hypersonic flight vehicle with full state measurement. The nonlinear longitudinal dynamics of the aircraft, which is subject to large aerodynamic uncertainty, is linearized around a nominal trim condition to derive a nominal linearized perturbation model. The nonlinear modified gain observer estimates four system parameters to yield an estimated linearized perturbation model and the corresponding estimation error weightings for the worst case controller. The simulation result demonstrates the applicability of the worst case controller coupled with the nonlinear modified gain observer and its superior performance when compared against the Sum-of-Squares method shown in the previous literature.