REACHABILITY OF POLYNOMIAL MATRIX DESCRIPTIONS (PMDS)

We consider the concept reachability for Polynomial Matrix Description s (PMDs); i.e., systems of the form Sigma : A(rho)beta(t) = B(rho)u(t) , y(t) = C(rho)beta(t), where rho : = d/dt the differential operator, A (rho) = A(0) + A(1) rho +...+A(upsilon)rho(upsilon) is an element of R(rxr) [rho], A(i) is an element of R(rxr) i = 0, 1,...,upsilon great er than or equal to 1 with rank(R) A(upsilon) less than or equal to r, B(rho) = B-0 + B-1 rho +...+ B(sigma)rho(sigma) is an element of R(rx m) [rho] B-i is an element of R(rxm), i = 0, 1,...,sigma greater than or equal to 0, C(rho) = C-0+C-1 rho+...+C(sigma 1)rho(sigma 1) is an e lement of R(m1xr) [rho], C-i is an element of R(m1xr), i = 0, 1,..., s igma(1) greater than or equal to 0, beta(t) : (0(-), infinity) --> R(r ) is the pseudostate of (Sigma), u(t) : [0, infinity) --> R(m) is the control input to (Sigma), and y(t) is the output of the system (Sigma) . Starting from the fact that generalized state space systems, i.e., s ystems of the form Sigma(1) : Ex(t) = Ax(t) + Bu(t), y(t) = Cx(t), whe re E is an element of R(rxr), rank(R) E < r, A is an element of R(rxr) , B is an element of R(rxm), C is an element of R(m1xr) represent a pa rticular case of PMDs, we generalize various known results regarding t he smooth and impulsive solutions of the homogeneous and the nonhomoge neous system (Sigma(1)) to the more general case of PMDs (Sigma). Rely ing on the above generalizations we develop a theory regarding the rea chability of PMDs using time-domain analysis, which takes into account finite and infinite zeros of the matrix A(s) = L_[A(rho)]. The presen t analysis extends in a general way many results previously known only for regular and generalized state space systems.