6.4 Lyapunov functions

Lyapunov functions are powerful tools for analyzing stability in nonlinear systems. They provide a way to assess system behavior without solving complex equations, using energy-like scalar functions to determine if a system will settle to equilibrium.

These functions have specific properties that make them useful for stability analysis. By examining how Lyapunov functions change over time, we can draw conclusions about a system's stability and even design controllers to ensure desired behavior.

Definition of Lyapunov functions

• Lyapunov functions are scalar functions used to analyze the stability of nonlinear dynamical systems in control theory
• Named after the Russian mathematician Aleksandr Lyapunov who introduced the concept in the late 19th century
• Provide a generalization of the concept of energy functions used in the stability analysis of physical systems

Properties of Lyapunov functions

Positive definite functions

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• A function $V(x)$ is positive definite if $V(x) > 0$ for all $x \neq 0$ and $V(0) = 0$
• Positive definite functions are used to construct Lyapunov functions
• Examples of positive definite functions include $V(x) = x^2$ and $V(x) = x_1^2 + x_2^2$

Continuously differentiable functions

• Lyapunov functions are required to be continuously differentiable
• Continuous differentiability ensures that the derivative of the Lyapunov function exists and is continuous
• Enables the use of calculus techniques in the stability analysis

Radially unbounded functions

• A function $V(x)$ is radially unbounded if $V(x) \rightarrow \infty$ as $\|x\| \rightarrow \infty$
• Radially unbounded functions are used to ensure global stability properties
• Examples of radially unbounded functions include $V(x) = x^2 + y^2$ and $V(x) = \sqrt{x^2 + y^2}$

Lyapunov stability theory

Lyapunov stability vs asymptotic stability

• Lyapunov stability means that the system trajectories remain bounded in a neighborhood of the equilibrium point
• Asymptotic stability implies that the system trajectories converge to the equilibrium point as time approaches infinity
• Asymptotic stability is a stronger notion than Lyapunov stability

Lyapunov stability theorems

• Lyapunov's direct method provides sufficient conditions for stability using Lyapunov functions
• If a positive definite function $V(x)$ has a negative semi-definite derivative $\dot{V}(x) \leq 0$, then the equilibrium point is Lyapunov stable
• If a positive definite function $V(x)$ has a negative definite derivative $\dot{V}(x) < 0$, then the equilibrium point is asymptotically stable

Lyapunov instability theorem

• Lyapunov's instability theorem provides a sufficient condition for instability
• If there exists a continuously differentiable function $V(x)$ such that $V(0) = 0$, $V(x) > 0$ for $x \neq 0$ in a neighborhood of the origin, and $\dot{V}(x) > 0$ for $x \neq 0$ in that neighborhood, then the equilibrium point is unstable

Construction of Lyapunov functions

Quadratic Lyapunov functions

• Quadratic Lyapunov functions have the form $V(x) = x^T P x$, where $P$ is a positive definite matrix
• Commonly used for linear systems and systems with quadratic nonlinearities
• Can be constructed using the solution of the Lyapunov equation $A^T P + P A = -Q$, where $A$ is the system matrix and $Q$ is a positive definite matrix

Non-quadratic Lyapunov functions

• Non-quadratic Lyapunov functions are used for systems with more complex nonlinearities
• Examples include polynomial Lyapunov functions, logarithmic Lyapunov functions, and sum-of-squares Lyapunov functions
• Construction of non-quadratic Lyapunov functions often requires problem-specific insights and techniques

Lyapunov functions for linear systems

• For linear systems $\dot{x} = Ax$, quadratic Lyapunov functions are commonly used
• The stability of a linear system can be determined by the eigenvalues of the matrix $A$
• If all eigenvalues of $A$ have negative real parts, then the system is asymptotically stable and a quadratic Lyapunov function can be constructed

Lyapunov functions for nonlinear systems

• Constructing Lyapunov functions for nonlinear systems is more challenging than for linear systems
• Common approaches include linearization, energy-based methods, and sum-of-squares techniques
• The choice of Lyapunov function depends on the specific structure and properties of the nonlinear system

Applications of Lyapunov functions

Stability analysis of equilibrium points

• Lyapunov functions are used to determine the stability of equilibrium points in nonlinear systems
• The stability of an equilibrium point can be established by constructing a suitable Lyapunov function and analyzing its derivative
• Examples include the stability analysis of the inverted pendulum and the Van der Pol oscillator

Stability analysis of periodic orbits

• Lyapunov functions can be used to analyze the stability of periodic orbits in nonlinear systems
• The stability of a periodic orbit can be determined by constructing a Lyapunov function in the vicinity of the orbit
• Examples include the stability analysis of limit cycles in the Fitzhugh-Nagumo model and the Lorenz system

Controller design using Lyapunov functions

• Lyapunov functions can be used as a tool for designing stabilizing controllers for nonlinear systems
• The control law is chosen such that the derivative of the Lyapunov function is negative definite, ensuring stability
• Examples include the design of feedback linearization controllers and backstepping controllers

Adaptive control using Lyapunov functions

• Lyapunov functions are used in the design of adaptive control systems, where the controller parameters are adjusted online
• The adaptation law is derived by ensuring that the derivative of the Lyapunov function is negative definite
• Examples include the design of model reference adaptive controllers and adaptive observers

Limitations of Lyapunov functions

Conservativeness of Lyapunov stability criteria

• Lyapunov stability criteria provide sufficient conditions for stability, but not necessary conditions
• There may exist stable systems for which a Lyapunov function cannot be easily constructed
• The conservativeness of Lyapunov stability criteria can lead to overdesign and suboptimal performance

Difficulty in finding Lyapunov functions

• Constructing Lyapunov functions for complex nonlinear systems can be challenging
• There is no systematic method for finding Lyapunov functions, and the process often relies on intuition and trial-and-error
• The difficulty in finding Lyapunov functions can limit the applicability of Lyapunov-based methods in practice

Extensions of Lyapunov stability theory

Barbalat's lemma

• Barbalat's lemma is an extension of Lyapunov stability theory that deals with the asymptotic behavior of functions
• It states that if a function $f(t)$ is uniformly continuous and has a finite limit as $t \rightarrow \infty$, then its derivative $\dot{f}(t)$ converges to zero as $t \rightarrow \infty$
• Barbalat's lemma is useful for analyzing the convergence of adaptive control systems and observer-based controllers

LaSalle's invariance principle

• LaSalle's invariance principle is an extension of Lyapunov stability theory that relaxes the requirement of a negative definite derivative
• It states that if a positive definite function $V(x)$ has a negative semi-definite derivative $\dot{V}(x) \leq 0$, then the system trajectories converge to the largest invariant set within the set where $\dot{V}(x) = 0$
• LaSalle's invariance principle is useful for analyzing the asymptotic behavior of systems with non-strict Lyapunov functions

Lyapunov-like functions

• Lyapunov-like functions are generalizations of Lyapunov functions that do not necessarily satisfy all the properties of classical Lyapunov functions
• Examples include semi-definite Lyapunov functions, vector Lyapunov functions, and integral Lyapunov functions
• Lyapunov-like functions can be used to analyze the stability of systems that do not admit classical Lyapunov functions, such as systems with non-smooth dynamics or time-varying systems