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.