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Module 10 p2 - Notes on Model Reference Adaptive Control...

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Notes on Model Reference Adaptive Control C.T. Abdallah-Based on Narendra’s book and S. Onori’s Thesis May 2, 2005 1 Introduction The aim of these notes is to present in a compact fashion the design of model reference adaptive controllers. 2 Model-Reference Adaptive Control Model-reference adaptive control (MRAC) may be regarded as an adaptive tracking system in which the desired performance is expressed in terms of a reference model, which gives the desired response to a command signal. This turns out to be a convenient way to give specifications for a tracking problem. A block diagram of the system is shown in Fig. 1, where the system has a feedback loop composed of the process and the controller with another feedback loop that adapts the controller parameters. The parameters are changed on the basis of the feedback from the error, which is the difference between the output of the system and the output of the reference model. The mechanism for adjusting the parameters will be obtained using the Lyapunov stability theory. In this chapter, MRAC is investigated following the results of Narendra, ([4]). In all that follows, the plant is assumed to be single input, single output, linear and time- invariant with relative degree one. The proposed controller is differentiator free, and the objectives of the control system are to make the output of the unknown plant follow the predefined reference model such that all parameters in the system are bounded. Even though the goal in model-reference adaptive systems is to drive the error y p y m to zero, the transient behavior of the closed- loop system is not specified. This is an important issue, because the output of the controlled plant y p ( t ) may grow away from y m ( t ) before converging to it as time goes to infinity. Lemma 1 [Barbalat] ([3]) If f : R + R is a uniformly continuous function and if the limit of the integral lim t integraldisplay t 0 | f ( τ ) | d τ exists and is finite, then lim t f ( t ) = 0 (1) 1
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Adaptive Law Controller Plant Model r u y p y m Controller parameters Figure 1: Block diagram of a MRAS. Proof. The proof can be found in [3]. The definitions of Positive Real (PR) and Strictly Positive Real (SPR) transfer func- tions, ([2]) will be given next. Definition 2 A rational transfer function G ( s ) with real coefficients is positive real (PR) if, (i) poles of all elements of G ( s ) are in Re [ s ] 0 , (ii) for all real ϖ for which j ϖ is not a pole of any element of G ( s ) , the matrix G ( j ϖ )+ G ( j ϖ ) is positive semidefinite, and (iii) any pure imaginary pole j ϖ of any element of G ( s ) is a pole and the residue matrix lim s j ϖ ( s j ϖ ) G ( s ) is positive semidefinite Hermitian. A transfer function G ( s ) is strictly positive real (SPR) if G ( s ε ) is positive real for some ε > 0 .
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