7.1 Parameter estimation and adaptation laws

Adaptive control systems use parameter estimation to determine unknown system values from input-output data. This process updates controller parameters in real-time, improving performance when dealing with uncertain or changing systems.

Adaptation laws guide parameter updates using techniques like the MIT rule, Lyapunov-based methods, and recursive least squares. These algorithms minimize estimation errors while ensuring stability and convergence of the adaptive control system.

Parameter Estimation in Adaptive Control

Concept and Goal of Parameter Estimation

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• Parameter estimation determines unknown parameters in a system model based on measured input-output data
• The goal is to update controller parameters to improve system performance and achieve desired control objectives
• Parameter estimation is used when system parameters are uncertain, time-varying, or subject to external disturbances

Techniques and Applications

• Estimated parameters adjust controller gains or adapt the control law in real-time
• Parameter estimation algorithms minimize an error function between the measured and predicted system outputs
  • Examples: Least squares, gradient descent, Kalman filtering

Adaptation Laws for Parameter Estimation

Types of Adaptation Laws

• Adaptation laws update parameter estimates based on system input-output data and estimation error
• MIT rule updates parameter estimates in the direction of the negative gradient of the error function
• Lyapunov-based adaptation law ensures stability of the parameter estimation process using a Lyapunov function to guide updates
• Recursive least squares (RLS) algorithm minimizes the weighted sum of squared estimation errors
• Extended Kalman filter (EKF) extends the Kalman filter to nonlinear systems and provides optimal parameter estimates in the presence of noise

Convergence and Robustness Analysis

• Analyzing convergence properties and robustness is crucial for reliable parameter estimation under uncertainties and disturbances
  • Convergence rate depends on adaptation gain, initial estimates, and system excitation level
  • Robustness evaluates sensitivity to modeling errors, measurement noise, and external disturbances

Parameter Estimation Applications

Robotics and Aerospace Systems

• Robotics: Identifies dynamic parameters of robot manipulators (mass, inertia, friction coefficients) for accurate motion control
• Aerospace systems: Estimates aerodynamic coefficients, engine parameters, and sensor biases for improved flight control and navigation

Process Control and Automotive Systems

• Process control: Identifies process model parameters (reaction rates, heat transfer coefficients) for optimal control and monitoring
• Automotive systems: Estimates vehicle parameters (tire stiffness, damping coefficients) for enhanced stability control and driver assistance
• Implementing parameter estimation requires careful consideration of system dynamics, available measurements, and computational resources

Stability and Convergence of Parameter Estimation Algorithms

Stability Analysis Techniques

• Stability analysis ensures parameter estimation remains bounded and does not diverge over time
• Lyapunov stability theory analyzes stability by constructing suitable Lyapunov functions
  • Proves boundedness and convergence of estimation errors
• Persistency of excitation (PE) condition is necessary for parameter convergence
  • Input signal must be sufficiently rich and persistent to excite all system modes

Designing Stable Adaptive Control Systems

• Adaptive control systems with parameter estimation should guarantee closed-loop stability
• Maintain desired performance in the presence of parameter uncertainties
  • Robust adaptive control techniques (dead-zone modification, projection operators)
• Careful selection of adaptation gains and update laws based on stability and convergence analysis