System Theory

Download Adaptive approximation based control: unifying neural, fuzzy by Jay A. Farrell, Marios M. Polycarpou PDF

By Jay A. Farrell, Marios M. Polycarpou

A hugely available and unified method of the layout and research of clever regulate platforms Adaptive Approximation established regulate is a device each regulate dressmaker must have in his or her keep an eye on toolbox. blending approximation thought, parameter estimation, and suggestions regulate, this booklet provides a unified procedure designed to let readers to use adaptive approximation established keep an eye on to latest structures, and, extra importantly, to realize adequate instinct and knowing to control and mix it with different regulate instruments for purposes that experience now not been encountered prior to. The authors supply readers with a thought-provoking framework for conscientiously contemplating such questions as: * What homes may still the functionality approximator have? * Are sure households of approximators enhanced to others? * Can the soundness and the convergence of the approximator parameters be assured? * Can keep an eye on platforms be designed to be strong within the face of noise, disturbances, and unmodeled results? * Can this technique deal with major alterations within the dynamics because of such disruptions as procedure failure? * What kinds of nonlinear dynamic platforms are amenable to this process? * What are the constraints of adaptive approximation established keep watch over? Combining theoretical formula and layout suggestions with wide use of simulation examples, this ebook is a stimulating textual content for researchers and graduate scholars and a worthy source for training engineers.

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Extra info for Adaptive approximation based control: unifying neural, fuzzy and traditional adaptive approximation approaches

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R)y(r)dTand P - l ( t ) = (b(7)4(T)'d.. 27) Note by the defini- = #(t)y(t) Since P ( t ) P - l ( t ) = I , differentiation and rearrangement shows that in general the time derivative of a matrix and its inverse must satisfy P = -P$ [P-'(t)]P; therefore, in least squares estimation P = -P(t)$(t)q(t)TP(t). 28) Finally, to show that the continuous-time least squares estimate of 0 satisfies eqn. 25) we differentiate both sides of eqn. At)) = d(t) = W d t ) ( Y ( 4 - B ( t ) ) . 29). Typically, the initial value of the matrix P is selected to be large.

This simulation uses the actual plant dynamics. Initially, the tracking error is large, but as the online data is used to estimate the approximator parameters, the tracking performance improves significantly. It is important that the designer understands the relationship between the tracking error and the function approximation error. It is possible for the tracking error to approach zero without the approximation error approaching zero. 22). If the last three terms sum to zero, then ij will converge to zero.

Test this in simulation. Analyze the performance of this gain scheduled controller using the actual dynamics. 2. Derive the error dynamics of eqns. 1 1)-( I . 14) for the linear adaptive control law. (Hint: add -tu EIJ to eqn. 5) and substitute eqn. ) + Show that the correct values for the model of eqn. 5) to match eqn. 2. First, duplicate the results of the example. Do the estimated parameters converge to the same values each time TI is commanded to the same operating point? Using the Lyapunov function 71 Yz 73 show that the time derivative of V evaluated along the error dynamics of the adaptive control system is negative semidefinite.

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