Higher-order continuous-time implicit systems: Consistency and weak consistency, impulse controllability, geometric concepts, and invertibility properties Original Research Article

The paper provides a simple, and fully algebraic, distributional setup for a higher-order linear implicit system, with arbitrary constant coefficients, on the continuous, nonnegative time axis. The distributional system equation exhibits impulses, depending on arbitrary points in the state space, and for these points several concepts of weak consistency, of increasing degree, are introduced in such a way that weak consistency of degree 0 coincides with the standard concept of consistency, whereas weak consistency of the highest degree is related to impulse controllability. All weakly consistent subspaces are expressed in the consistent subspace, and for the latter space a numerically stable algorithm is presented. Also, we derive for every concept of weak consistency a condition, in terms of the system coefficients only, for the statement that the associated subspace is as large as possible. In particular, we get a condition for impulse controllability that generalizes the celebrated condition for impulse controllability of a first-order system. Further, we state two conditions for control solvability, and we specify when distributional state and/or input trajectories are unique. Finally, we define and characterize various subspaces for a higher-order system, in combination with an output equation, and link these spaces to two of our four invertibility concepts, namely those in the strong sense, and we establish that the above composite system is left- (right-)invertible in the strong sense if and only if the corresponding system matrix is left- (right-)invertible as a rational matrix, even if the transfer matrix does not exist.