Reachability of Polynomial Matrix Descriptions (PMDs)

We consider the concept reachability for Polynomial Matrix Descriptions (PMDs); i.e., systems of the form ∑: A(ρ)β(t)=B(ρ)u(t),y(t)=C(ρ)β(t), whereρ:=d/dt the differential operator,A(ρ)=A0+A1ρ+...+ Avρv εRr×r[ρ], AiεRr×r,i=0, 1,..., ν ≥ 1 with rankRAv ≤r B(ρ) =B0+B1ρ+...+Bσρσ εRr×m[ρ], Bi εRr×m,i=0,1,...,σ ≥ 0 C(ρ)=C0+C1ρ+...+Cσ1ρσ1 εRm1×r[ρ],Ci εRm1×r,i=0, 1,..., σ1 ≥ 0, β(t): (0−, ∞) →Rr is the pseudostate of (∑),u(t): [0, ∞) →Rm is the control input to (∑), and y(t) is the output of the system (∑). Starting from the fact that generalized state space systems, i.e., systems of the form ∑1: Ex(t)=Ax(t)+ Bu(t), y(t)=Cx(t), whereE εRr×r, rankRE <r, A εRr×r,B εRr×m,C εRm1×r represent a particular case of PMDs, we generalize various known results regarding the smooth and impulsive solutions of the homogeneous and the nonhomogeneous system (∑1) to the more general case of PMDs (∑). Relying on the above generalizations we develop a theory regarding the reachability of PMDs using time-domain analysis, which takes into account finite and infinite zeros of the matrix A(s)=L.[A(ρ)]. The present analysis extends in a general way many results previously known only for regular and generalized state space systems.

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