8.2 Robustness issues in adaptive control

Adaptive control systems face challenges in maintaining stability and performance under uncertainties. Robustness is crucial for handling modeling errors, disturbances, and parameter variations. This topic explores the sources of uncertainty and their impact on system behavior.

Stability analysis techniques help evaluate the system's robustness. The trade-off between adaptation speed and robustness is a key consideration, with various modification techniques available to enhance system performance while maintaining stability.

Robustness in Adaptive Control Systems

Robustness in adaptive control

• Robustness maintains stability and performance under uncertainties tolerates modeling errors and external disturbances
• Ensures consistent performance across various operating conditions maintains system stability despite parameter variations
• Challenges include interaction between adaptation mechanism and system dynamics potential for instability due to rapid parameter changes
• Evaluated using stability margins (gain margin, phase margin) sensitivity functions and $H_{\infty}$ norm

Sources of control uncertainty

• Parameter uncertainties arise from manufacturing tolerances aging and wear of components environmental factors (temperature, humidity)
• Unmodeled dynamics include high-frequency modes nonlinearities and time delays
• External disturbances encompass sensor noise actuator limitations and load variations
• System changes involve sudden parameter shifts and structural modifications
• Measurement errors result from sensor inaccuracies and quantization effects

Stability and Performance Analysis

Impact of unmodeled dynamics

• Excites high-frequency modes potentially causing instability due to spillover degrades tracking performance
• Parameter variations lead to temporary loss of stability during adaptation oscillatory behavior in control signals slow convergence or divergence of parameter estimates
• Analyzed using Lyapunov stability theory small-gain theorem and passivity-based analysis
• Convergence issues include parameter drift bursting phenomena and persistent excitation requirements

Adaptation speed vs robustness

• Fast adaptation quickly responds to changes improves transient performance but increases sensitivity to noise and potential instability
• Slow adaptation improves robustness provides smoother control action but results in sluggish response to changes and prolonged settling time
• Design considerations involve selecting adaptation gains using normalization in adaptation laws implementing projection algorithms
• Robustness enhancement techniques include $\sigma$-modification $e$-modification and dead-zone modification
• Trade-off balances tracking accuracy with stability margins considers specific application requirements