Characterization of strong structural controllability of uncertain linear time-varying discrete-time systems

Abstract

We extend earlier characterizations of strong structural controllability of linear systems depending on parameters to the time-varying case x(t+1) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ·x(t)+B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ·u(t). Our main result is that the time-varying system is strongly structurally controllable iff the corresponding time-invariant system (whose matrices have the same zero-nonzero structure) is so. We also establish that every strongly structurally controllable time-varying system is controllable on every time interval of the form {t, t + 1, ..., t + n}, where n is the dimension of the state space, a property which is not, in general, valid for linear time-varying systems that are merely controllable. Finally, we present an algorithm for verifying the property of strong structural controllability. Our results cover the single- and multi-input cases and apply without any assumptions on the systems' structure or the time-varying entries of A and B.