Solved Problems on Lyapunov Stability for Nonlinear Systems

📚 Lyapunov Stability for Nonlinear Systems: A Comprehensive Guide

Lyapunov stability theory provides a powerful framework for analyzing the stability of nonlinear systems without explicitly solving the differential equations. This guide explores the fundamental concepts and illustrates them with solved problems.

📜 Historical Background

Aleksandr Mikhailovich Lyapunov, a Russian mathematician and physicist, developed the theory in the late 19th century. His work, initially published in 1892, laid the foundation for modern stability analysis, particularly in control systems and dynamical systems.

• 🕰️ Early Development: Lyapunov's original work focused on the stability of equilibrium points of ordinary differential equations.
• ⚙️ Applications in Control: The theory gained prominence with the rise of control engineering in the mid-20th century.
• 🚀 Modern Extensions: Contemporary research extends Lyapunov's methods to time-varying systems, stochastic systems, and systems with delays.

🔑 Key Principles of Lyapunov Stability

Lyapunov stability is assessed by defining a scalar function, known as the Lyapunov function, $V(x)$, which provides information about the system's energy or distance from an equilibrium point. The key principles are:

• 📏 Positive Definiteness: A function $V(x)$ is positive definite if $V(0) = 0$ and $V(x) > 0$ for all $x \neq 0$ in a region around the origin.
• 📉 Negative Definiteness of Derivative: If the derivative of $V(x)$ along the system trajectories, denoted as $\dot{V}(x)$, is negative definite (i.e., $\dot{V}(0) = 0$ and $\dot{V}(x) < 0$ for all $x \neq 0$), then the equilibrium point is asymptotically stable.
• ⚖️ Lyapunov's Stability Theorem: If a positive definite function $V(x)$ can be found such that its derivative $\dot{V}(x)$ is negative semi-definite (i.e., $\dot{V}(x) \leq 0$), then the equilibrium point is stable.

📝 Solved Problems

Problem 1: Stability of a Simple Pendulum

Consider a simple pendulum described by the equation:

$\ddot{\theta} + \sin(\theta) = 0$

where $\theta$ is the angle from the vertical.

Solution:

1. State-space representation: Let $x_1 = \theta$ and $x_2 = \dot{\theta}$. Then the system can be written as:

• $\dot{x_1} = x_2$
• $\dot{x_2} = -\sin(x_1)$

2. Lyapunov function candidate: Choose $V(x) = \frac{1}{2}x_2^2 + (1 - \cos(x_1))$. This represents the total energy of the pendulum.
3. Derivative of Lyapunov function: $\dot{V}(x) = x_2\dot{x_2} + \sin(x_1)\dot{x_1} = x_2(-\sin(x_1)) + \sin(x_1)x_2 = 0$.
4. Conclusion: Since $\dot{V}(x) = 0$, the equilibrium point $(0, 0)$ is stable (but not asymptotically stable).

Problem 2: Stability Analysis of $\dot{x} = -x^3$

Consider the nonlinear system:

$\dot{x} = -x^3$

Solution:

1. Lyapunov function candidate: Choose $V(x) = \frac{1}{2}x^2$.
2. Derivative of Lyapunov function: $\dot{V}(x) = x\dot{x} = x(-x^3) = -x^4$.
3. Conclusion: Since $V(x)$ is positive definite and $\dot{V}(x)$ is negative definite, the equilibrium point $x = 0$ is asymptotically stable.

Problem 3: Stability of a System with Two State Variables

Consider the system:

• $\dot{x_1} = -x_1 + x_2^2$
• $\dot{x_2} = -x_2$

Solution:

1. Lyapunov function candidate: Choose $V(x_1, x_2) = x_1^2 + x_2^2$.
2. Derivative of Lyapunov function: $\dot{V}(x_1, x_2) = 2x_1\dot{x_1} + 2x_2\dot{x_2} = 2x_1(-x_1 + x_2^2) + 2x_2(-x_2) = -2x_1^2 + 2x_1x_2^2 - 2x_2^2$.
3. Analysis: $\dot{V}(x_1, x_2)$ is not immediately negative definite. However, we can rewrite it as $\dot{V} = -2(x_1^2 - x_1x_2^2 + x_2^4/4) - \frac{3}{2}x_2^4 = -2(x_1 - x_2^2/2)^2 - \frac{3}{2}x_2^4$.
4. Conclusion: Since $\dot{V}(x_1, x_2)$ is negative definite, the equilibrium point $(0, 0)$ is asymptotically stable.

💡 Tips for Choosing Lyapunov Functions

• 🔍 Energy Functions: For physical systems, consider energy functions (kinetic + potential energy) as potential Lyapunov function candidates.
• ➕ Quadratic Forms: For linear systems, quadratic forms like $V(x) = x^T P x$ (where $P$ is a positive definite matrix) often work.
• 🧪 Trial and Error: Sometimes, finding a Lyapunov function involves trial and error. Start with simple functions and gradually increase complexity.

🌍 Real-world Applications

• ✈️ Aerospace Engineering: Stability analysis of aircraft and spacecraft control systems.
• 🤖 Robotics: Ensuring the stability of robot movements and control algorithms.
• ⚡ Power Systems: Analyzing the stability of electrical grids and preventing blackouts.

🏁 Conclusion

Lyapunov stability theory is an indispensable tool for analyzing the stability of nonlinear systems. By understanding the key principles and practicing with solved problems, you can effectively apply this theory to various engineering and scientific applications. Keep practicing, and you'll master it!