Lyapunov stability
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Lyapunov stability is applicable to only unforced (no control input) dynamical systems. It is used to study the behaviour of dynamical systems under initial perturbations around equilibrium points.
Let us consider that the origin is the equilibrium point (EP) of the system.
A system is said to be stable "in the sense of Lyapunov" (i.s.L.) if for every ε, there is a δ such that:
- <math>
\|x(t_o)\| < \delta \quad => \quad \|x(t)\| < \epsilon \quad \forall t \in R_+<math>
The system is said to be asymptotically stable if as
- <math> t \rightarrow \infty, \quad \|x(t)\| \rightarrow 0 (EP) <math>
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Lyapunov stability theorems
Lyapunov stability theorems give only sufficient condition.
Lyapunov second theorem on stability
Consider a function V(x) : Rn → R such that
- <math>V(x) > 0 : \forall{x} \neq 0<math> (positive definite)
- <math> \dot{V}(x) < 0 <math> (negative definite)
Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov.
It is easier to visualise this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the attractor. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not applicable.
Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints.
Stability for state space models
A state space model <math>\dot{\textbf{x}} = A\textbf{x}<math> is asymptotically stable iff
- <math>A^{T}M + MA + N = 0<math>
has a solution where <math>N = N^{T} > 0<math> and <math>M = M^{T} > 0<math> (positive definite matrices).