4.3 Lyapunov stability-based adaptation laws

Lyapunov stability theory is a powerful tool for analyzing adaptive control systems without solving complex equations. It uses energy-like functions to determine if a system will converge to its desired state over time. This approach is particularly useful in Model Reference Adaptive Control (MRAC).

In MRAC, Lyapunov theory helps design adaptation laws that ensure stable parameter estimation. By analyzing error dynamics and choosing appropriate Lyapunov functions, engineers can create robust control systems that adapt to changing conditions while maintaining stability.

Lyapunov Stability in Model Reference Adaptive Control (MRAC)

Lyapunov stability in MRAC

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• Lyapunov stability theory analyzes system behavior without solving differential equations
  • Direct method evaluates energy-like function to determine stability (pendulum)
  • Energy-like function approach measures system's "energy" as it evolves over time
• Stability definitions characterize system behavior
  • Asymptotic stability: system converges to equilibrium point as time approaches infinity
  • Exponential stability: system converges at exponential rate (faster than asymptotic)
  • Global stability applies to entire state space, local stability only near equilibrium
• Lyapunov function properties ensure valid stability analysis
  • Positive definite: function value always positive except at origin (quadratic forms)
  • Radially unbounded: function grows without bound as state variables increase
• Application to MRAC involves analyzing system dynamics
  • Error dynamics describe difference between plant and reference model outputs
  • Parameter estimation errors represent discrepancy between true and estimated parameters
  • Composite Lyapunov function combines tracking error and parameter estimation error

Adaptation laws from Lyapunov theory

• Steps for deriving adaptation laws ensure stable parameter estimation
  1. Define error dynamics: express system behavior in terms of tracking error
  2. Choose Lyapunov function candidate: typically quadratic form of errors
  3. Compute time derivative of Lyapunov function: reveals system's energy change
  4. Design adaptation laws: make derivative negative semi-definite for stability
• Gradient method updates parameters proportional to error
  • $\dot{\hat{\theta}} = -\gamma e^T Pb\phi$ where $\gamma$ is adaptation gain, $e$ is error, $P$ solves Lyapunov equation, $b$ and $\phi$ are system matrices
• Least squares method minimizes sum of squared errors
  • $\dot{\hat{\theta}} = -P\phi\phi^T\hat{\theta} + P\phi y$ where $P$ is covariance matrix, $\phi$ is regressor, $y$ is system output
• Normalization techniques improve robustness to input variations
  • $\dot{\hat{\theta}} = -\frac{\gamma e^T Pb\phi}{1 + \phi^T\phi}$ reduces sensitivity to large inputs

Stability Analysis and Performance Improvement

Stability proofs for MRAC systems

• Barbalat's lemma proves asymptotic stability for non-autonomous systems
• LaSalle's invariance principle extends Lyapunov stability to invariant sets
• Persistence of excitation ensures parameter convergence (sinusoidal inputs)
• Convergence properties describe long-term system behavior
  • Parameter convergence: estimated parameters approach true values
  • Tracking error convergence: system output matches reference model
• Stability analysis steps ensure rigorous proof
  1. Show boundedness of signals: all system states remain finite
  2. Prove asymptotic convergence of tracking error: error approaches zero
  3. Analyze parameter estimation convergence: estimated parameters stabilize

Robust adaptation law modifications

• Robustness modifications improve stability in presence of disturbances
  • $\sigma$-modification adds leakage term: $\dot{\hat{\theta}} = -\gamma e^T Pb\phi - \sigma\hat{\theta}$
  • e-modification scales leakage with error magnitude: $\dot{\hat{\theta}} = -\gamma e^T Pb\phi - \sigma|e|\hat{\theta}$
  • Projection algorithm constrains parameter estimates within known bounds
• Transient performance improvements enhance adaptation speed and accuracy
  • Composite adaptation combines multiple error signals
  • Multiple models use parallel estimators for faster convergence
  • Switching and tuning alternate between different adaptation strategies
• Dead-zone modification stops adaptation when error is small to prevent parameter drift
• Adaptive gain methods dynamically adjust adaptation rate
  • Time-varying adaptation gain increases/decreases based on error
  • Kalman filter-based adaptation uses optimal estimation techniques