Novel Conditions for the Finite-Region Stability of 2D-Systems with Application to Iterative Learning Control

TL;DR

This work addresses finite-region stability (FRS) for discrete-time 2D Roesser systems and links it to iterative learning control (ILC) over finite horizons. It introduces a time-varying Lyapunov-based FRS condition leveraging the forward $+$ operator, and provides LMI/DLMI-based analysis and synthesis methods that are less conservative than prior 2D-FRS approaches. The paper demonstrates how to design ILC laws that guarantee the tracking error bound within a finite number of iterations and derives a finite-region stabilization procedure using state-feedback gains, all solvable with standard optimization tools. The results have practical impact for applications where state trajectories are spatial (2D) or involve coupled space-time coordinates, enabling reliable finite-interval performance guarantees.

Abstract

Some recent papers have extended the concept of finite-time stability (FTS) to the context of 2D linear systems, where it has been referred to as finite-region stability (FRS). FRS methodologies make even more sense than the classical FTS approach developed for 1D-systems, since, typically, at least one of the state variables of 2D-systems is a space coordinate, rather than a time variable. Since space coordinates clearly belong to finite intervals, FRS techniques are much more effective than the classical Lyapunov approach, which looks to the asymptotic behavior of the system over an infinite interval. To this regard, the novel contribution of this paper goes in several directions. First, we provide a novel sufficient condition for the FRS of linear time-varying (LTV) discrete-time 2D-systems, which turns out to be less conservative than those ones provided in the existing literature. Then, an interesting application of FRS to the context of iterative learning control (ILC) is investigated, by exploiting the previously developed theory. In particular, a new procedure is proposed so that the tracking errors of the ILC law converges within the desired bound in a finite number of iterations. Finally, a sufficient condition to solve the finite-region stabilization problem is proposed. All the results provided in the paper lead to optimization problems constrained by linear matrix inequalities (LMIs), that can be solved via widely available software. Numerical examples illustrate and validate the effectiveness of the proposed technique.

Figures (11)

• Figure 1: The region $\mathcal{N}_1\times\mathcal{N}_2$, with $N_1< N_2$. Case a) $0<M\le N_1$; case b): $N_1 < M \le N_2$; case c): $N_2<M\le N_1+N_2$.
• Figure 2: The region $0\le h+k=M \le N_1$.
• Figure 3: The region $N_1\le h+k=M < N_2$.
• Figure 4: The region $N_2<M\le N_1+N_2$.
• Figure 5: Example 1: the initial condition bound, defined by $R_1(k),R_2(h)$, must be surrounded by the trajectory bound, defined by $\Gamma_1(\cdot,0), \Gamma_2(0,\cdot)$. Using a variable weighting matrix $\Gamma(h,k)$ it is possible to shrink or enlarge the bound over $\mathcal{N}_1\times \mathcal{N}_2$. Using a constant $\Gamma$ implies a fixed bound for all $(h,k)$. To graphically render the above concepts, only the diagonal point ($h=k$) have been considerd in the figure.

Theorems & Definitions (16)

• Definition 1
• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• proof
• Theorem 1
• proof
• Remark 4
• Remark 5