Discrete System Robust Model Reference Adaptive Control

This paper mainly deals with two theoretical problem of robust adaptive control: Robust direct model reference adaptive control for discrete-time systems and model reference adaptive control for linear discrete-time systems under bounded disturbances, which is composed of the following two parts.1. Discrete-time robust direct model reference adaptive controlLet us consider the discrete-time linear plantwhere G(z) is the transfer function of the plant,△(z) represent multiplicative plant uncertainties.d₁ and d₂ is bounded input disturbances.Control objective is to design an output feedback control u such that all the signals in the closed-loop system are all uniformly bounded and the output y can track the following reference model output y_m as close as possiblefor any reference signal r(t), where r(t) are uniformly bounded. To meet the control objective, the assumptions of the system and the reference model are needed in the second chapter.In this part, by using the discrete-time conclusions on the (?)_p and (?)_2δ relationship properties between the input and the output, and the swapping lemmasⅠandⅡ, establishing the properties of discrete-time adaptive law, defining the normalizing signal, and relating the signal with all the signals in the closed-loop system, the stability and robustness properties of the discrete-time MRAC scheme are analyzed rigorously in systematic fashion as in the continuous-time case.2. Model reference adaptive control for linear discrete-time systems under bounded disturbancesConsider a SISO linear discrete-time system described by the following modelwhere u and y are the input and the output of the system respectively, 5 is the disturbance, A and B are coprime polynomials in backward shift operator q^-1 given below with d=n-m being time delay and b₀ known.This paper considers robustness analysis of linear discrete-time adaptive control using a Lyapunov function. For a new adaptive law with a dead zone, we develop a method for stability analysis based on Lyapunov functions. With the chosen Lypunov function, the boundedness of all the variables in the system is established and furthermore, the tracking error is shown to asymptotically converge to the size of dead zone.