By Zijad Aganovic, Zoran Gajic
This ebook is designed to be a complete remedy of linear how you can optimum keep watch over of bilinear structures. The unified topic of this publication is using dynamic programming to be able to simplify and decompose required computations for the optimum keep an eye on of bilinear-quadratic structures. there are various examples of bilinear keep an eye on structures that offer nice demanding situations to engineers, mathematicians and computing device scientists: those comprise nuclear reactors, missile intercept difficulties and mechanical brake structures. The e-book additionally examines distinct periods of bilinear-quadratic keep an eye on difficulties: particularly singularly perturbed and weakly coupled bilinear regulate structures. The usefulness of the awarded the right way to those forms of keep an eye on challenge is validated via numerous actual regulate process examples.
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Extra info for Linear Optimal Control of Bilinear Systems: with Applications to Singular Perturbations and Weak Coupling
Sample text
In general it has no analytical solution. 44) i = 1, 2, 3, ... 47) with u~(t) ~ u*oo(t) = - R - 1 B P ~ x o o ( t ) - R-1B{xoo(t)M}Pooxoo(t) This expression indicates the quadratic nature of the optimal "closed-loop" control for bilinear systems. 47) is uniform in t, and is guaranteed under the following assumption. 1 The pair (A, B) is controllable and x stays in the controllability domain X~ = {x E RnI(A, B + {xM}) controllable}. A Note that the algebraic Riccati equation whose coefficient matrices are functions of a parameter t is studied in (Ran and Rodman, 1988) for the case t E (to,tf).
16), respectively. It is assumed that ~(k)(t, to, e) and @(k)(t, to, e) are known for every t. The initial conditions ~(k+D(to) and ~(k+D(to) have to be determined. This can be done as follows. 25) 44 k"/ Since both matrices + ~ ) ( t + , t o , 0 ) - F,+~k)(t+,to, O) and +~k)(t+,to,O)F3)~)(t+, to, o) are nonsingul~rfor all choice, of F1 and F3 (Kalman, 1960; see also Kirk, 1970, page 211), so does A(k+D(e) for 0 < e < O and el sufficiently small. Thus, in summary, we have established the following theorem.
The near-optimal "closed-loop" control is obtained in the form of an approximate linear "feedback" law, with the feedback gains calculated from two reduced-order independent time varying linear-quadratic optimal control problems. , 1990) and the recursive scheme for the optimal control of general bilinear systems with quadratic performance criteria (Cebuhar and Constanza, 1984). An algorithm which produces an arbitrary degree of accuracy for the "closed-loop" control is derived. The results are demonstrated on a real physical bilinear model of a paper making machine.