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Adaptive Control based on Disturbance Attenuation and the Modified Gain Observer for Parameter Uncertain Linear Systems

This paper examines an adaptive control strategy for a class of uncertain linear systems with disturbance input. The uncertain linear system contains unknown parameters in both the system coefficient matrix and the control coefficient matrix. The system makes perfect partial measurement of the system states. The goal is to find the worst case control strategy assuming the system is perturbed by worst case disturbances and unknown parameters. A disturbance attenuation function is defined and is transformed into a minimax differential game, where the disturbance and the unknown parameters act as cooperative players to maximize the cost function, while the controller is the opposing player and attempts to minimize the cost function. Applying dynamic programming to the cost function naturally divides into two sequential operations. As a result, the optimization yields a minimax controller coupled with a stable nonlinear observer called a modified gain observer, which estimates the state and the parameters. Under certain conditions, the observer and controller is shown to be stable. It is proved that the minimax controller is a saddle point, so that the decomposition is valid. A vector case example using the disturbance attenuation controller versus the LQR is presented to demonstrate applicability as well as assessment of performance.