Lyapunov Stability
Motivation¶
For a Linear Time-Invariant (LTI) system, stability is often proven by analyzing the eigenvalues \(\lambda\) of the system matrix. For instance, in continuous-time systems, we require the real parts of all eigenvalues to be strictly negative (\(\text{Re}(\lambda_k) < 0\) for all \(k=1,\ldots,m\)).
However, in the case of a non-normal system matrix (where the eigenvectors are not orthogonal), standard eigenvalue analysis can mask problematic behaviors. Even if all eigenvalues satisfy the stability condition, the lack of orthogonality can lead to transient growth, meaning the state's magnitude may initially spike before eventually decaying. If the system is non-linear, this temporary spike might push the state outside the safe linear operating region. Therefore, a more robust stability proof is required: one that ensures the system trajectories from all possible initial states remain bounded and asymptotically approach an equilibrium state.
Lyapunov Stability¶
Lyapunov stability addresses this by guaranteeing that for any chosen maximum radius \(\epsilon > 0\), there exists an initial radius \(\delta > 0\) such that if the system starts within \(\delta\) of the equilibrium, it will never exceed the distance \(\epsilon\) over time:
This guarantees that for any small perturbation from the equilibrium state, the system's reaction is strictly bounded.
- Uniform Stability: Stability is defined for an initial time instance \(t_0\). If \(\delta\) can be chosen independently of \(t_0\) (meaning the stability bounds do not change over time), then the system is uniformly stable.
- Asymptotic Stability: If, additionally, there exists a \(\delta_0 > 0\) such that starting within this radius guarantees convergence to the equilibrium (\(\|x(t_0)\| < \delta_0 \implies \lim_{t\to \infty}x(t)=0\)), then the system is asymptotically stable.
Lyapunov stability can be proven using Lyapunov Functions