MATLAB Simulation of Switched Systems

Switched systems represent an important class of hybrid dynamic systems characterized by their dynamic behavior switching among different subsystems. MATLAB provides powerful tools for modeling, analyzing, and simulating switched systems. This article introduces simulation methods for switched systems in MATLAB, including LMI (Linear Matrix Inequality) solving and simulation plot generation.

### 1. Application of LMI Solving in Switched Systems LMI (Linear Matrix Inequality) serves as a crucial tool for stability analysis of switched systems. In MATLAB, the Robust Control Toolbox's LMI solver can handle stability problems in switched systems. LMI solving is typically used to find common Lyapunov functions or design switching rules to ensure global system stability. By defining appropriate matrix variables and inequality constraints, users can construct LMI problems and solve them using MATLAB solvers like `feasp` (feasibility problem solver) or `mincx` (minimization of linear objectives under LMI constraints). The implementation involves creating LMI variables using `lmivar`, defining constraints with `lmiterm`, and solving with the appropriate solver function.

### 2. Simulation Plot Generation for Switched Systems In MATLAB, switched system simulation can be implemented either through Simulink or by writing direct scripts. Simulation plots typically include system state trajectories, switching signals, and performance indicators (such as settling time, overshoot). For systems containing multiple subsystems, switching logic (time-based or state-based switching) can be designed to simulate actual operation processes. Simulation results can be visualized using the `plot` function, and system performance can be optimized by parameter adjustment. Key MATLAB functions include `ode45` for solving differential equations and `switch` or `if-else` statements for implementing switching logic in code-based simulations.

### 3. Extended Applications In practical applications, switched system simulations extend beyond stability analysis to include optimal control, robustness, and disturbance rejection studies. Furthermore, by integrating machine learning and reinforcement learning methods, more intelligent switching strategies can be designed to enhance system adaptability. Implementation approaches may involve using MATLAB's Reinforcement Learning Toolbox or developing custom algorithms using function approximation and policy optimization techniques.