Analysis of Non-Monotonic Piecewise Lyapunov Stability and Semi-PID Design for MIMO Piecewise Affine Systems

In this study, we introduce the Non-monotonic Lyapunov (NML) approach, aimed at stability examining and designing stabilizable controller for a specific category of discrete piecewise affine dynamical systems. Traditional Lyapunov techniques are often characterized by their conservative nature, which has spurred the creation of more adaptable methods like NML. A key distinction of the NML approach is its lack of reliance on strict monotonicity when establishing the decrease of a Lyapunov functional. As a result, we derive new stability and stabilization criteria based on the NML framework, expressed as LMIs (linear matrix inequalities) tailored for piecewise affine systems. This NML-based methodology is employed to develop a semi-PID controller which is also able to handle multi-input multi-output (MIMO) piecewise linear systems. Additionally, we present an algorithm for optimal semi-PID controller design, aimed at achieving both optimal trajectories and refined control signals. The effectiveness of this innovative approach is highlighted through various examples and simulation results, demonstrating its significant contributions to PID design within complex systems. The findings underscore the practical applicability and advantages of the NML method in enhancing control strategies.

کلیدواژه ها:

Piecewise Affine Systems ، Lyapunov-based stability ، Linear Matrix Inequality ، Non-Monotonic Lyapunov Theorem ، PID Controller