Eigenvalue assignment for componentwise ultimate bound minimisation in LTI discrete-time systems

We address optimal eigenvalue assignment in order to obtain minimum ultimate bounds on every component of the state of a linear time-invariant (LTI) discrete-time system in the presence of non-vanishing disturbances with known constant bounds. As opposed to some continuous-time cases where ultimate bounds can be made arbitrarily small by applying feedback with sufficiently high gain so that the closed-loop eigenvalues are sufficiently fast, the ultimate bound of a discrete-time system with an additive bounded disturbance can never be made smaller than some set that depends on the disturbance bound, even if all closed-loop eigenvalues are set at zero (the fastest possible in discrete-time). In this context, our contribution is twofold: (a) we single out cases where feedback that may not assign all closed-loop eigenvalues at zero achieves the minimum possible ultimate bound for some component of the system state, and (b) by employing an existing componentwise ultimate bound computation formula, we find a class of systems for which assigning all closed-loop eigenvalues at zero indeed yields minimum ultimate bounds. An intermediate result-and our third contribution-in the derivation of (b) is the obtention of the Jordan decomposition that minimises the componentwise ultimate bound formula employed.

I. Introduction

In the presence of bounded disturbances that do not vanish as the state approaches an equilibrium point, asymptotic stability is not possible but, under certain conditions, the ultimate boundedness of the system's trajectories can be guaranteed [6]. A guaranteed ultimate bound on the system's trajectories can be effectively interpreted as a measure of “attenuation” of the effect of disturbances. Thus, the assignment of a prespecified ultimate bound by feedback control and/or the estimation of tight ultimate bounds are problems of interest in control system design that have many applications e.g., in systems involving quantisation [12], unknown disturbance signals [10], etc.