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Structure at infinity, model matching and disturbance rejection for linear systems with delays. (English) Zbl 0805.93008
The structur at infinity is studied for rational transfer matrices with two unknows \(T(s,v)\), where \(s\) is the classical Laplace variable, and \(v= e^ s\) is the unitary shift operator; \(v^{-1}\) is interpreted as the unit delay. Denote by \(D\) the set of all such matrices. Using the Smith-Macmillan form at infinity for \(T\in D\), the following Model Matching Problem is solved: Given proper matrices \(T\in D\), \(T_ m\in D\), determine if there is a proper \(T_ c\in D\) that solves the equation \(T\cdot T_ c= T_ m\). Consider now the class of systems with one delay and disturbance vector \(d(t)\): \[ x'(t)= A_ 0 x(t)+ A_ 1 x(t-1)+ Bu(t)+ Ed(t);\quad y(t)= Cx(t).\tag{1} \] The subscripts “\(m\)” and “\(c\)” stand for “model” and “compensator”, respectively. The Disturbance Rejection Problem with Dynamic Precompensator for systems (1) reads as follows: Does there exists a proper (or strictly proper) precompensator \(u= T_ c d\), where \(T_ c\in D\), that rejects the effect of \(d(t)\) on the output \(y(t)\)? The solution is given in terms of the supremal frequency invariant subspaces. This notion is well-known for the linear time invariant systems, and it is suitably generalized in the reviewed paper for the delay systems (1). The structure at infinity is also used to obtain a criterion for the partial disturbance rejection property of the system (1); namely, there exists a nonanticipative state feedback law \(u(t)\) that depends linearly on \[ x(t),x(t- 1),\dots, x(t-k),\;d(t),\dots,d(t- k) \] and is such that the output \(y(t)\) of the compensated system is not affected by \(d(t)\) over the interval \([0,k+ 1]\).

93B17 Transformations
93C05 Linear systems in control theory
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