5.3 Certainty equivalence principle in STR design

The certainty equivalence principle simplifies self-tuning regulator design by treating estimated parameters as true values. This approach allows for separate estimation and control, making adaptive control more manageable, but it comes with limitations like potential instability and suboptimal performance.

Self-tuning regulators use this principle in various control laws, including minimum variance and pole placement. While it can work well with rapid parameter convergence, it may struggle with unmodeled dynamics and parameter drift, requiring careful performance evaluation and stability analysis.

Certainty Equivalence Principle in Self-Tuning Regulator Design

Certainty equivalence principle in regulators

• Certainty equivalence principle treats estimated parameters as true values ignoring parameter estimation uncertainty in control design
• Application in self-tuning regulator (STR) design follows two-step process: parameter estimation and control law computation using estimated parameters directly
• Separation of estimation and control allows independent design of estimator and controller simplifying adaptive control problem (LQG control)

Control laws for self-tuning schemes

• Indirect self-tuning regulators employ parameter estimation using recursive least squares (RLS) and derive control law based on estimated model
• Direct self-tuning regulators adapt controller parameters directly implicitly using certainty equivalence
• Minimum variance control derives control law using estimated process parameters optimizing one-step-ahead prediction ($J = E[(y_{k+1} - r_{k+1})^2]$)
• Pole placement control assigns closed-loop poles using estimated model computing controller gains ($A(q^{-1})R(q^{-1}) + B(q^{-1})S(q^{-1}) = T(q^{-1})$)

Limitations of certainty equivalence

• Assumes rapid convergence of parameter estimates negligible estimation errors and persistence of excitation
• Potential instability during transient phase and suboptimal performance due to ignored uncertainty
• Lack of robustness to unmodeled dynamics (high-frequency resonances)
• Parameter drift in absence of excitation and bursting phenomena in certain conditions (insufficient excitation)

Performance of certainty-based regulators

• Performance metrics include tracking error control effort and convergence rate of parameter estimates
• Stability analysis applies Lyapunov stability theory and small gain theorem for robustness assessment
• Comparative evaluation between certainty equivalent and cautious control approaches reveals trade-offs between performance and robustness
• Monte Carlo simulations provide statistical analysis assessing sensitivity to initial conditions and disturbances (measurement noise)