On eigenvalue-eigenvector assignment for componentwise ultimate bound minimisation in MIMO LTI discrete-time systems

We consider eigenvalue-eigenvector assignment in order to minimise ultimate bounds on the states of a linear time-invariant (LTI) discrete-time system in the presence of non-vanishing bounded disturbances. As opposed to continuous-time systems, for which eigenstructure assignment with large magnitude stable eigenvalues can yield arbitrarily small ultimate bounds for “matched” perturbations, for discrete-time systems, ultimate bounds cannot be smaller than certain values depending on the disturbance bounds. Moreover, these smallest bounds are not achievable, in general, by assigning the closed-loop eigenvalues to zero (an intuitive conjecture that parallels the continuous-time case). The first contribution of the paper, for single-input systems, are conditions on the zeros of the transfer function between the control input and a state to minimise the ultimate bound corresponding to that state. These conditions generalise a result recently presented by the authors. The second, and main, contribution of the current paper is to characterise, for multiple-input systems, the eigenstructure of the closed-loop system so that some ultimate bounds are minimised to their minimum values. The number of ultimate bound components that can be minimised is constrained by the number of control inputs. For m-input system, the minimisation problem of m - 1 ultimate bound components can be solved without restrictions, while in order to minimise an additional bound, an additional restrictive condition should be satisfied.