The paper "Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems" by Mikael Johansson and Anders Rantzer, published in 1996, addresses the problem of constructing piecewise quadratic Lyapunov functions for nonlinear and hybrid systems. The authors formulate this problem as a convex optimization task involving linear matrix inequalities (LMIs). They demonstrate the flexibility and power of their approach through several examples. The paper begins by introducing the concept of Lyapunov functions and their importance in systems theory, particularly in stability analysis. It highlights the limitations of classical methods, such as the circle criterion, and introduces the use of LMIs for more general and accurate stability analysis. The authors then describe the system dynamics using a cell partition approach, where the state space is divided into a finite number of closed sets (cells). Each cell is defined by a set of affine inequalities, and the dynamics within each cell are piecewise affine. For hybrid systems, which have both continuous and discrete states, the Lyapunov function is quadratic in the continuous state and non-increasing at the jumps of the discrete state. The paper presents a simple example to illustrate the method, showing that a globally quadratic Lyapunov function does not exist for a specific nonlinear system but a piecewise quadratic Lyapunov function can be found. It also discusses the application of classical frequency domain methods and the Popov criterion to the same system, highlighting the limitations of these methods compared to the LMI approach. The authors derive stability theorems for piecewise linear and piecewise affine systems, providing conditions under which a continuous piecewise quadratic Lyapunov function exists. These theorems are then applied to specific examples, demonstrating the effectiveness of the LMI formulation. Finally, the paper extends the LMI formulation to hybrid systems, showing how to formulate stability conditions for systems with switching states. It concludes by discussing the potential applications of the method in performance analysis, global linearization, controller optimization, and model approximation. The authors acknowledge the support of M. Branicky and K.J. Åström and express gratitude to the Institute of Applied Mathematics, Sweden, for their financial support.The paper "Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems" by Mikael Johansson and Anders Rantzer, published in 1996, addresses the problem of constructing piecewise quadratic Lyapunov functions for nonlinear and hybrid systems. The authors formulate this problem as a convex optimization task involving linear matrix inequalities (LMIs). They demonstrate the flexibility and power of their approach through several examples. The paper begins by introducing the concept of Lyapunov functions and their importance in systems theory, particularly in stability analysis. It highlights the limitations of classical methods, such as the circle criterion, and introduces the use of LMIs for more general and accurate stability analysis. The authors then describe the system dynamics using a cell partition approach, where the state space is divided into a finite number of closed sets (cells). Each cell is defined by a set of affine inequalities, and the dynamics within each cell are piecewise affine. For hybrid systems, which have both continuous and discrete states, the Lyapunov function is quadratic in the continuous state and non-increasing at the jumps of the discrete state. The paper presents a simple example to illustrate the method, showing that a globally quadratic Lyapunov function does not exist for a specific nonlinear system but a piecewise quadratic Lyapunov function can be found. It also discusses the application of classical frequency domain methods and the Popov criterion to the same system, highlighting the limitations of these methods compared to the LMI approach. The authors derive stability theorems for piecewise linear and piecewise affine systems, providing conditions under which a continuous piecewise quadratic Lyapunov function exists. These theorems are then applied to specific examples, demonstrating the effectiveness of the LMI formulation. Finally, the paper extends the LMI formulation to hybrid systems, showing how to formulate stability conditions for systems with switching states. It concludes by discussing the potential applications of the method in performance analysis, global linearization, controller optimization, and model approximation. The authors acknowledge the support of M. Branicky and K.J. Åström and express gratitude to the Institute of Applied Mathematics, Sweden, for their financial support.