Parametric robust positively invariant sets for linear systems with scaled disturbances

This paper presents novel results on robust positively invariant (RPI) sets for linear discrete-time systems with additive disturbances. In particular, we determine how RPI sets change with scaling of the disturbance set. In this context, we analyze families of RPI sets for a given interval of scaling factors. Such parametric RPI sets are useful to study the sensitivity of the system to changes in the disturbance strength.

I. Introduction

Robust positively invariant (RPI) sets are important for performance analysis and synthesis of controllers for uncertain systems (see, e.g., [1, Sects. 6.4 and 6.5]). In particular, RPI sets are used for the design of robust model predictive control (MPC) schemes with guaranteed stability (see, e.g., [2]–[5]). In this paper, we address RPI sets for linear disturbed systems of the form $$\begin{equation*} x(k+1)=Ax(k)+w(k) \tag{1} \end{equation*}$$ with state and disturbance constraints $$\begin{equation*} x(k)\in \mathcal{X}\ \text{and}\ w(k)\in \mathcal{W}^{\alpha}\ \text{for every}\ k\in \mathbb{N}. \tag{2} \end{equation*}$$