Reachability of Polynomial Matrix Descriptions (PMDs)

Abstract

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+A₁ ρ+...+ A_v ρ ^v εR ^r×r[ρ], A_iεR ^r×r,i=0, 1,..., ν ≥ 1 with rank _R A _v ≤r B(ρ) =B ₀+B ₁ρ+...+B _σρ^σ εR ^r×m[ρ], B_i εR ^r×m,i=0,1,...,σ ≥ 0 C(ρ)=C₀+C₁ ρ+...+C_σ1 ρ ^σ1 εR ^m1×r[ρ],C _i εR ^m1×r,i=0, 1,..., σ₁ ≥ 0, β(t): (0⁻, ∞) →R ^r is the pseudostate of (∑),u(t): [0, ∞) →R ^m 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 εR ^r×r, rank _R E <r, A εR ^r×r,B εR ^r×m,C εR ^m1×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.