📻Adaptive and Self-Tuning Control Unit 2 – System ID and Parameter Estimation

Key Concepts and Terminology

• System identification involves estimating mathematical models of dynamic systems based on observed input-output data
• Parameter estimation techniques aim to determine the values of unknown parameters in a given model structure
• Black-box modeling assumes no prior knowledge of the system's internal workings and relies solely on input-output data
• Grey-box modeling incorporates some prior knowledge of the system's physical properties or structure
• Model validation assesses the quality and accuracy of the estimated model by comparing its predictions with actual system behavior
• Bias and variance trade-off refers to the balance between model complexity and generalization ability
• Overfitting occurs when a model fits the noise in the training data, leading to poor generalization on new data
• Underfitting happens when a model is too simple to capture the underlying system dynamics accurately

System Identification Basics

• The goal of system identification is to develop mathematical models that describe the behavior of a dynamic system
• System identification typically involves four main steps: data collection, model structure selection, parameter estimation, and model validation
• The choice of input signals (e.g., step, sinusoidal, or random) affects the quality and informativeness of the collected data
• Sampling time and data length are crucial factors in capturing the relevant system dynamics
• Static and dynamic models differ in their ability to represent time-dependent behavior
• Linearity and time-invariance assumptions simplify the modeling process but may not always hold for real-world systems
• Noise and disturbances can significantly impact the accuracy of the identified model

Parameter Estimation Techniques

• Least Squares (LS) estimation minimizes the sum of squared differences between the model predictions and the observed outputs
  • Ordinary Least Squares (OLS) assumes independent and identically distributed (i.i.d.) noise
  • Weighted Least Squares (WLS) assigns different weights to each data point based on their relative importance or reliability
• Maximum Likelihood (ML) estimation finds the parameter values that maximize the likelihood of observing the given data
  • Assumes a probabilistic model for the noise distribution (e.g., Gaussian)
  • Requires knowledge or assumptions about the noise characteristics
• Bayesian estimation incorporates prior knowledge about the parameters in the form of probability distributions
  • Updates the prior distributions using the observed data to obtain posterior distributions
  • Provides a probabilistic framework for parameter uncertainty quantification
• Gradient-based optimization methods (e.g., gradient descent, conjugate gradient) iteratively update the parameter estimates based on the gradient of the objective function
• Evolutionary algorithms (e.g., genetic algorithms, particle swarm optimization) explore the parameter space using population-based search techniques

Data Collection and Preprocessing

• Proper data collection is essential for accurate system identification
• Input signals should be designed to excite the system dynamics of interest
  • Pseudo-random binary sequences (PRBS) are commonly used for linear systems
  • Chirp signals sweep through a range of frequencies to capture frequency-dependent behavior
• Sampling rate should be chosen based on the system's bandwidth and the desired model accuracy
• Data preprocessing steps include:
  • Outlier detection and removal
  • Filtering to remove high-frequency noise or low-frequency trends
  • Scaling and normalization to improve numerical stability and convergence
• Data partitioning into training, validation, and testing sets helps assess model performance and generalization
• Input-output data can be organized into regression matrices for batch processing

Model Selection and Validation

• Model structure selection involves choosing the appropriate model class (e.g., ARX, ARMAX, state-space) and model order
• Model complexity should balance the trade-off between fitting the data and generalization ability
• Information criteria (e.g., Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC)) provide quantitative measures for model comparison
• Cross-validation techniques (e.g., k-fold, leave-one-out) estimate the model's performance on unseen data
• Residual analysis examines the properties of the model's prediction errors
  • Residuals should be uncorrelated, zero-mean, and have constant variance
  • Autocorrelation and cross-correlation plots help assess the model's adequacy
• Validation metrics (e.g., mean squared error (MSE), root mean squared error (RMSE), coefficient of determination (R^2)) quantify the model's predictive performance

Recursive Estimation Methods

• Recursive estimation updates the parameter estimates as new data becomes available, without storing the entire dataset
• Recursive Least Squares (RLS) is an online version of the least squares estimation
  • Efficiently updates the parameter estimates and the covariance matrix using the matrix inversion lemma
  • Forgetting factor can be introduced to give more weight to recent data and adapt to time-varying systems
• Kalman Filter (KF) is a recursive estimation technique for linear systems with Gaussian noise
  • Consists of a prediction step (time update) and a correction step (measurement update)
  • Provides optimal estimates in the sense of minimizing the mean squared error
• Extended Kalman Filter (EKF) extends the Kalman filter to nonlinear systems by linearizing the system dynamics around the current estimate
• Recursive Maximum Likelihood (RML) updates the parameter estimates by maximizing the likelihood function recursively

Applications in Adaptive Control

• Adaptive control aims to adjust the controller parameters in real-time based on the identified system model
• Model Reference Adaptive Control (MRAC) adjusts the controller parameters to minimize the error between the system output and a reference model output
• Self-Tuning Regulators (STR) estimate the system model parameters and update the controller parameters based on the estimated model
  • Indirect STR first estimates the system model and then computes the controller parameters
  • Direct STR estimates the controller parameters directly from input-output data
• Gain Scheduling adapts the controller parameters based on the operating conditions or system state
• Adaptive Pole Placement assigns the closed-loop poles to desired locations based on the identified system model
• Adaptive control can handle time-varying systems, uncertainties, and disturbances

Challenges and Limitations

• Identifiability issues arise when different parameter values lead to the same input-output behavior
• Persistence of excitation condition ensures that the input signal provides sufficient information for accurate parameter estimation
• Closed-loop identification requires special techniques to handle the correlation between the input and output signals introduced by the feedback loop
• Nonlinear system identification is more challenging due to the increased complexity and diversity of nonlinear model structures
• High-dimensional systems with a large number of parameters may suffer from computational complexity and numerical issues
• Adaptive control may exhibit instability or poor performance if the identified model is inaccurate or the adaptation is too aggressive
• Robustness to unmodeled dynamics, disturbances, and noise is a critical consideration in practical applications
• Balancing the trade-off between adaptation speed and stability is a key challenge in adaptive control design