This article provides a detailed explanation of robust invariant set estimation for discrete-time switched linear systems with peak-bounded disturbances. Related articles, related papers, and MATLAB links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

Okajima and K. Fujinami, Estimation of Robust Invariant Set for Switched Linear Systems Using Recursive State Updating and Robust Invariant Ellipsoid, SICE Journal of Control, Measurement, and System Integration, Vol. 14, No. 1, pp. 97–106 (2021) (Open Access)

This paper is co-work with Kakeru Fujinami (Master's student, Kumamoto University at the time of publication).

Why Robust Invariant Set Estimation Matters

In practical control systems, disturbances are unavoidable, and understanding how these disturbances affect the system state is critical for guaranteeing safe and reliable operation. A robust invariant set is a region in state space such that, once the state enters this region, it will remain there for any admissible disturbance. Accurately estimating the size of this set provides essential guarantees for system safety and performance.

Applications include:

Power systems, where bounding the state ensures reliable operation under load disturbances.
Constrained control systems, where the state must remain within physical limits.
Quantized control systems, where finite-precision signals introduce bounded errors that must be accounted for.

Switched linear systems — systems that switch among multiple linear subsystems — arise frequently in practice (e.g., automobiles, aircraft, and power converters). However, estimating the robust invariant set for switched systems is significantly more challenging than for standard linear time-invariant (LTI) systems, because the switching introduces combinatorial complexity.

This paper proposes a novel method that combines two complementary approaches — recursive state updating (inner approximation) and robust invariant ellipsoid (outer approximation) — to achieve a less conservative estimation of the robust invariant set for switched linear systems.

Robust Invariant Set and State Reachable Set for LTI Systems

Definitions

Consider the discrete-time LTI system:

$\displaystyle x_{k+1} = Ax_k + Bw_k$

where $x_{k} \in \mathbb{R}^{n}$ is the state and $w_{k} \in \mathcal{W}$ is a bounded disturbance. The matrix $A$ is assumed to be Schur stable.

A set $\mathcal{X}$ is called a robust invariant set if, whenever $x_{k} \in \mathcal{X}$ , the next state $Ax_{k} + Bw_{k}$ also belongs to $\mathcal{X}$ for all admissible disturbances.

The state reachable set $\mathcal{R}_{\infty}$ is the set of all states reachable from the origin under all admissible disturbance sequences. It can be shown that $\mathcal{R}_{\infty}$ is the smallest robust invariant set, and any robust invariant set $\mathcal{X}$ satisfies $\mathcal{R}_{\infty} \subset \mathcal{X}$ .

Therefore, to precisely evaluate the impact of disturbances on the system state, it is important to estimate the robust invariant set as small as possible, approaching $\mathcal{R}_{\infty}$ .

Estimation by Recursive State Updating (Inside Approximation)

One approach is to compute the set of reachable states step by step. Starting from $\mathcal{L}_{0} = {0}$ , the reachable set at each time step is computed by applying the state equation:

$\displaystyle \mathcal{L}_k = \{x \in \mathbb{R}^n \mid x = Az + Bw,\; z \in \mathcal{L}_{k-1},\; w \in \mathcal{W}_p\}$

This produces a nested sequence of convex polyhedral sets:

$\displaystyle \mathcal{L}_0 \subset \cdots \subset \mathcal{L}_k \subset \mathcal{L}_{k+1} \subset \mathcal{R}_\infty$

As $k \to \infty$ , these sets converge to $\mathcal{R}_{\infty}$ . However, in practice, computation must stop at a finite $k$ , so $\mathcal{L}_{k}$ provides an inner approximation of $\mathcal{R}_{\infty}$ . This means it does not guarantee that the actual state stays within $\mathcal{L}_{k}$ .

Estimation by Robust Invariant Ellipsoid (Outside Approximation)

An alternative is to estimate the robust invariant set as an ellipsoid using a Lyapunov function. Given a positive-definite matrix $P$ , the ellipsoid is defined as:

$\displaystyle \mathcal{E}(P) = \{x \in \mathbb{R}^n \mid x^T P x \leq 1\}$

Theorem 2.1 (from the paper): For an LTI system with bounded disturbance $w_{k}^{T} w_{k} \leq 1$ , the ellipsoid $\mathcal{E}(P)$ is a robust invariant set if and only if there exists $\alpha \in \lbrack 0, 1 - \rho(A)^{2} \rbrack$ satisfying the matrix inequality involving $A$ , $B$ , $P$ , and $\alpha$ . For the explicit form, see Eq. (12) in the paper.

This ellipsoid provides an outer approximation: $\mathcal{R}_{\infty} \subset \mathcal{E}(P)$ . The advantage is that it guarantees the state remains within the estimated region. The disadvantage is that a single ellipsoid may be conservative — the gap between $\mathcal{E}(P)$ and $\mathcal{R}_{\infty}$ can be large.

Problem: Robust Invariant Set for Switched Linear Systems

The paper considers the discrete-time switched linear system:

$\displaystyle x_{k+1} = A_{\sigma(k)} x_k + B_{\sigma(k)} w_k$

where $\sigma(k) \in {1, \ldots, N}$ is the switching signal and $w_{k} \in \mathcal{W}_{p}$ is a peak-bounded disturbance. Each subsystem $(A_{i}, B_{i})$ is Schur stable.

The robust invariant set $\mathcal{Y}$ must satisfy the invariance condition for all switching signals and all admissible disturbances. The state reachable set $\mathcal{R}_{\infty}$ is defined analogously.

Challenges for Switched Systems

Recursive state updating becomes much more expensive, because at each step, all $N$ subsystem matrices must be considered. The number of polyhedral sets grows as $N^{k}$ , making computation infeasible for large $k$ .

Robust invariant ellipsoid requires a common Lyapunov matrix $P$ for all subsystems. This introduces additional conservativeness compared to the LTI case.

The paper demonstrates these issues with a numerical example using $N = 2$ subsystems. The recursive state updating provides a good inner approximation $\mathcal{V}_{8}$ , but cannot guarantee performance. The robust invariant ellipsoid $\mathcal{E}(P)$ guarantees performance, but has a large gap from the true reachable set.

Proposed Method: Combining Recursive State Updating and Robust Invariant Ellipsoid

Basic Idea

The key insight is to decompose the state into two components: the zero-state response (driven by disturbance from the origin) and the zero-input response (starting from an invariant set with no further disturbance). This corresponds to computing the direct sum (Minkowski sum) of two sets.

Starting from the robust invariant ellipsoid $\mathcal{E}(P)$ as the initial set, after applying zero disturbance for $k$ steps, the ellipsoid shrinks:

$\displaystyle \{0\} \subset \cdots \subset \mathcal{X}_{k+1} \subset \mathcal{X}_k \subset \cdots \subset \mathcal{X}_0 = \mathcal{E}(P)$

Combining this with the growing polyhedral sets $\mathcal{V}_{k}$ from recursive state updating:

$\displaystyle \mathcal{P}_k = \bigcup_{j_k=1}^{N^k} (\mathcal{V}_{k,j_k} \oplus \mathcal{E}(P)_{k,j_k})$

where $\oplus$ denotes the Minkowski sum.

Key Result (Theorem 4.1)

Theorem 4.1: For any positive integer $\ell$ , $\mathcal{X}_{\ell} \oplus \mathcal{L}_{\ell}$ is a robust invariant set, and the relation $\mathcal{X}_{\ell+1} \oplus \mathcal{L}_{\ell+1} \subset \mathcal{X}_{\ell} \oplus \mathcal{L}_{\ell}$ holds.

This produces a nested sequence of robust invariant sets:

$\displaystyle \mathcal{P}_0 = \mathcal{E}(P) \supset \cdots \supset \mathcal{P}_{k-1} \supset \mathcal{P}_k \supset \mathcal{P}_{k+1} \supset \mathcal{R}_\infty$

As $k \to \infty$ , the shrinking ellipsoid converges to the origin while the polyhedral set converges to $\mathcal{R}_{\infty}$ , so theoretically $\lim_{k \to \infty} \mathcal{P}_{k} = \mathcal{R}_{\infty}$ . Furthermore, for any finite $k \geq 0$ :

$\displaystyle \mathcal{V}_k \subset \mathcal{R}_\infty \subset \mathcal{P}_k$

This means $\mathcal{P}_{k}$ always provides a guaranteed outer approximation of $\mathcal{R}_{\infty}$ , and the approximation becomes tighter as $k$ increases.

Numerical Examples

The paper demonstrates the method with a 2-state switched linear system with $N = 2$ subsystems:

$\displaystyle A_1 = \begin{pmatrix} 0.375 & -0.9 \cr 0.3 & 0.45 \end{pmatrix}, \quad B_1 = \begin{pmatrix} 1.2 \cr 0.4 \end{pmatrix}$

$\displaystyle A\_{2} = \begin{pmatrix} 0.225 & -0.75 \cr 0.45 & 0.225 \end{pmatrix}, \quad B\_{2} = \begin{pmatrix} 0.1 \cr 0.4 \end{pmatrix}$

The disturbance satisfies $-1 \leq w_{k} \leq 1$ .

Computation Time of Recursive State Updating

The computation time for the polyhedral set $\mathcal{V}_{k}$ increases rapidly:

$\mathcal{V}_{k}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$	$\mathcal{V}_{6}$	$\mathcal{V}_{7}$	$\mathcal{V}_{8}$
Time (sec.)	3.24	9.89	25.73	65.98	168.63	425.8

Results

The robust invariant ellipsoid alone gives $\gamma = 10.61$ (for the evaluation based on the major axis of the ellipsoid), which is significantly conservative. The proposed method produces progressively tighter approximations:

$\mathcal{P}_{2}$ : Clearly tighter than $\mathcal{E}(P)$ , but still has noticeable gap from $\mathcal{V}_{8}$
$\mathcal{P}_{6}$ : Much closer to $\mathcal{V}_{8}$
$\mathcal{P}_{8}$ : Very close to $\mathcal{V}_{8}$ , with $\mathcal{P}_{8} \subset \mathcal{P}_{6} \subset \mathcal{P}_{2}$

The paper also compares with the method of Chen, Lam, and Zhang (2016), which uses multiple Lyapunov strategies with a genetic algorithm. The proposed method $\mathcal{P}_{8}$ yields a smaller (less conservative) robust invariant set than this existing method.

This work connects to a broader research program:

Dynamic Quantizer Design — H. Okajima, K. Sawada and N. Matsunaga, Dynamic Quantizer Design under Communication Rate Constraints, IEEE Transactions on Automatic Control (2016). This paper uses robust invariant set analysis as part of the dynamic quantizer design for networked control systems.

Model Error Compensator (MEC) — The robust invariant set estimation can be combined with the Model Error Compensator to guarantee performance bounds in the presence of model uncertainties.

Multi-Rate Observer — H. Okajima, Y. Hosoe and T. Hagiwara, State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm, IEEE Access (2023). LMI-based state estimation for systems with multiple sensors operating at different rates shares the common Lyapunov function analysis framework.

Conference Paper — K. Fujinami and H. Okajima, Less-conservative state reachable set estimation using feed forward state updating and robust invariant ellipsoid, Proceedings of the SICE Annual Conference 2020, pp. 1156–1158. This is the preliminary version for LTI systems; the journal paper extends the method to switched linear systems.

MATLAB Code

GitHub (Dynamic Quantizer with Invariant Set Analysis): MATLAB_Dynamic_Quantizer01
GitHub (LMI basics): MATLAB Fundamental Control LMI

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Dynamic Quantizer / Publications / LMI / MEC

Video

YouTube: Control Engineering Channel / Portal

Paper Information

Okajima and K. Fujinami, "Estimation of Robust Invariant Set for Switched Linear Systems Using Recursive State Updating and Robust Invariant Ellipsoid", SICE Journal of Control, Measurement, and System Integration, Vol. 14, No. 1, pp. 97–106, 2021. DOI: 10.1080/18824889.2021.1918372 (Open Access)

Co-author: Kakeru Fujinami (Department of Computer Science and Electrical Engineering, Kumamoto University)

Conference paper: K. Fujinami and H. Okajima, "Less-conservative state reachable set estimation using feed forward state updating and robust invariant ellipsoid," Proceedings of the SICE Annual Conference 2020, pp. 1156–1158.

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.