A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

Technical Summary: A Piecewise Ellipsoidal Reachable Set Estimation Method for Continuous Bimodal Piecewise Affine Systems

Problem Statement
The paper addresses the problem of estimating the reachable sets for continuous bimodal piecewise affine (PWA) systems subject to exogenous disturbances. While PWA systems are widely used to model physical phenomena (e.g., relays, hysteresis, saturation) and approximate nonlinear dynamics, calculating their reachable sets is generally an undecidable problem. Existing methods for general PWA hybrid systems often suffer from high computational complexity or are ill-suited for the specific dynamical structure of bimodal systems. Furthermore, while ellipsoidal bounding techniques exist for linear and switched systems, they often rely on common Lyapunov functions or require specific matrix decompositions that may lead to conservative estimates or fail to handle state-dependent switching with non-vanishing disturbances effectively.

Methodology
The authors propose a new approach within the framework of ellipsoidal bounding, utilizing piecewise quadratic Lyapunov functions tailored to the bimodal nature of the system. The system is defined by two affine dynamics switching based on a hyperplane $c^T x + f = 0$. The disturbances are constrained within an ellipsoid.

The methodology proceeds through the following technical steps:

1. Half-Space Positivity Conditions: The authors first derive conditions (Lemma 3.1) to ensure that a quadratic function is positive on a specific half-space. This is achieved by characterizing the half-space using a specific matrix transformation involving the switching surface parameters.
2. Continuity Constraints: To ensure the piecewise Lyapunov function is continuous across the switching hyperplane, the authors establish conditions (Lemma 3.2) relating the parameters of the quadratic functions in the two modes.
3. Homogenization and S-Procedure: The authors employ the concept of homogenization for sets and a generalized S-procedure (Lemma 3.5) to handle the constraints imposed by the disturbance ellipsoid and the switching surface simultaneously.
4. LMI Formulation: By combining these lemmas with a Lyapunov characterization of the reachable set (Lemma 2.3), the authors derive Linear Matrix Inequality (LMI) conditions (Theorem 3.6 and Corollary 3.7). These conditions guarantee that the derivative of the piecewise Lyapunov function is negative definite outside a specific region, thereby bounding the reachable set within a piecewise ellipsoidal set.

Key Contributions

• Novel LMI Characterizations: The paper introduces new LMI conditions (Lemmas 3.1, 3.2, 3.4, and Theorem 3.5) specifically designed to exploit the bimodal dynamics. Unlike previous methods that assume a common Lyapunov matrix or require special decompositions (e.g., $P_i = F_i^T T F_i$), this approach constructs the piecewise ellipsoid without such restrictive assumptions.
• Handling Non-Vanishing Disturbances: The method accommodates bounded, non-vanishing disturbances, a feature where some existing literature (e.g., [28]) requires disturbances to vanish at infinity.
• State-Dependent Switching: The approach is formulated for state-dependent switching, distinguishing it from methods that rely on time-dependent switching.
• Computational Efficiency: The proposed method results in a convex optimization problem that can be solved efficiently, offering a significant reduction in computational time compared to zonotope-based approaches for higher-order systems.

Results
The proposed method is validated through two numerical examples:

1. Example 4.1 (2D System): A bimodal system with "orthogonal" phase planes is analyzed. The results demonstrate that the piecewise quadratic Lyapunov function provides a tighter (less conservative) bound on the reachable set compared to a common quadratic Lyapunov function. The piecewise method successfully captures the system's geometry that the common method misses.
2. Example 4.2 (4D Mechanical System): A mechanical system with a one-sided spring is used to compare the proposed method against the CORA toolbox (which uses zonotopes).
   • Accuracy: The piecewise ellipsoidal estimate is shown to be smaller (tighter) than the common ellipsoidal estimate.
   • Performance: The proposed method solves the 4th-order system in approximately 0.1 seconds. In contrast, the CORA toolbox required over 200 seconds for a 40th-order zonotope and over 215 seconds for a 50th-order zonotope. Notably, increasing the zonotope order in CORA increased computation time without significantly improving the estimate, whereas the proposed method achieved high precision with minimal computational cost.

Significance and Claims
The paper claims that its primary significance lies in exploiting the specific dynamical structure of continuous bimodal PWA systems to propose a simpler and less conservative approach than general hybrid system methods. The authors emphasize that their conditions are novel because they avoid the need for special decompositions of Lyapunov matrices, which are often difficult to derive optimally in existing literature.

The authors modestly note that while the method is effective for continuous bimodal systems, extending it to general multimodal (trimodal or higher) PWA systems is non-trivial and requires future investigation, potentially involving special representations like hinging hyperplanes. The work positions itself as a specialized, efficient tool for a class of systems that includes important engineering applications like linear relay systems.