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Extra resources for Robust and Insensitive Design of Multivariable Feedback Systems — Multimodel Design —

Example text

K( 1) r-. 29) := was used. 5). 19a). Up to this point, the design represents a generalizati on of the method of procedure applied in [11] to the multivariable case. 5 CANONICAL FORM FOR THE CONTROLLER The following p sets of equations will be obtained after a row-by-row transposition of Eq. • p . 30) It is assumed that each partial controller will be complete observable. m) is arranged in a complete observable canonical form. 31) 0 0 -13 0 -131 [ 1 . k':. m is the m-series unit matrix. 30) will have a solution, independently of B0 , •..

It is not our intention to give a complete description and discussion of the design methods. We confine us on the following two methods: pole placement design, design with quadratic performance criterion. Pole placement One important question in the design of controller is the stabilizability problem. Given a state space model ~(v+1) =~ ~(v) + ~ ~(v) one has to check whether a linear feedback law u(v) = - ! e. J take a negative real parts ? , ~) stabilizable. 48) wherever we like (but for complexe eigenvalues appearing in conjugate pairs) by suitable choice of the state regulator then it will be called the ~e placement problem.

7: Let the open-loop matrix QK(z) proper and det[l+QK(=)] t 0. Then the closed-loop system is exponentially stable if and only if inf ldet[I+QK(z)JI > 0. 45) we introduce the angle function n(a) := Im [log det[l+~K(eja)]J with n(O) = O(n) if det[l+~(1)] > 0 (< 0). 45) is satisfied if and only if (a) (b) det [.!.. + QK("") J ldet[.!. inf etdO,ZxJ + +0 QK(ejet)J I > 0 (if QK(z) has a pole on the unit circle then the contour [0,211] must be indented on the inside) (c) n(2n) - n(O) = 2n n r where nr is the number of poles of QK(z) outside the closed unit disk.

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