Glossary

2.2 Recursive least squares (RLS) estimation

2 min read•july 25, 2024

(RLS) estimation is a powerful technique for building mathematical models of dynamic systems. It updates model parameters in real-time as new data comes in, making it ideal for adaptive control and tracking time-varying systems.

RLS minimizes the sum of squared errors between predicted and actual outputs. Its recursive nature allows for efficient processing of large datasets and lower memory requirements, making it well-suited for real-time applications in robotics, finance, and signal processing.

Recursive Least Squares Estimation

Concept of recursive least squares

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  • builds mathematical models of dynamic systems from observed data enables control and prediction (aircraft autopilots, weather forecasting)

  • Recursive Least Squares (RLS) estimation updates model parameters recursively as new data becomes available minimizes sum of squared errors between predicted and actual outputs

  • Purpose of RLS enables efficient adaptation to time-varying systems reduces computational complexity compared to batch methods suits real-time applications and adaptive control (robotics, financial modeling)

  • Advantages over non-recursive methods include lower memory requirements faster processing for large datasets ability to track time-varying parameters (adaptive noise cancellation, channel equalization)

Derivation of RLS algorithm

  • Linear regression model y(t)=ϕT(t)θ+e(t)y(t) = \phi^T(t)\theta + e(t) represents system output regressor vector parameter vector and measurement noise

  • Cost function uses weighted sum of squared errors J(θ)=∑i=1tλt−i(y(i)−ϕT(i)θ)2J(\theta) = \sum_{i=1}^t \lambda^{t-i}(y(i) - \phi^T(i)\theta)^2 with forgetting factor λ\lambda (0 < λ\lambda ≤ 1)

  • RLS update equations:

    1. Parameter update: θ^(t)=θ^(t−1)+K(t)(y(t)−ϕT(t)θ^(t−1))\hat{\theta}(t) = \hat{\theta}(t-1) + K(t)(y(t) - \phi^T(t)\hat{\theta}(t-1))
    2. : K(t)=P(t−1)ϕ(t)(λ+ϕT(t)P(t−1)ϕ(t))−1K(t) = P(t-1)\phi(t)(\lambda + \phi^T(t)P(t-1)\phi(t))^{-1}
    3. update: P(t)=1λ(P(t−1)−K(t)ϕT(t)P(t−1))P(t) = \frac{1}{\lambda}(P(t-1) - K(t)\phi^T(t)P(t-1))

  • Initialization requires setting initial parameter estimate θ^(0)\hat{\theta}(0) and covariance matrix P(0)P(0)

Implementation for online estimation

  • Algorithm steps:

    1. Initialize θ^(0)\hat{\theta}(0) and P(0)P(0)
    2. For each time step:
      • Collect new
      • Compute prediction error
      • Update gain vector
      • Update parameter estimate
      • Update covariance matrix

  • Implementation considerations include choice of forgetting factor numerical stability and conditioning computational efficiency (matrix inversion techniques)

  • Apply to different system structures such as ARX ARMAX and state-space models adapts to various system dynamics (mechanical systems, electrical circuits)

  • Practical issues involve handling missing data dealing with outliers and measurement noise adapting to sudden changes in system dynamics (sensor faults, mode switches)

Convergence and performance analysis

  • Convergence analysis examines asymptotic convergence to true parameters under certain conditions considers influence of on parameter estimates

  • Factors affecting convergence and performance include signal-to-noise ratio forgetting factor selection initial conditions system order and model structure

  • Performance metrics evaluate parameter output prediction error convergence rate quantify algorithm effectiveness

  • Robustness and stability assess sensitivity to measurement noise behavior with unmodeled dynamics numerical stability in finite precision arithmetic

  • Compare with other estimation techniques such as Kalman filter gradient-based methods batch least squares highlight strengths and weaknesses

  • Extensions and variations include exponentially weighted RLS square root RLS algorithms RLS with variable forgetting factor enhance performance in specific scenarios (fast-changing environments, ill-conditioned problems)

Key Terms to Review (16)

Adaptive filtering: Adaptive filtering is a technique used in signal processing where the filter parameters automatically adjust based on the input signal characteristics. This allows for improved performance in environments where conditions may change over time, such as noise reduction or echo cancellation. It plays a crucial role in various applications, enabling systems to learn from data and optimize their responses accordingly.
Bounded Input: Bounded input refers to a condition in which the input signal to a system is restricted within a certain range, ensuring that it does not exceed predefined limits. This concept is crucial in adaptive control systems as it ensures stability and reliability, allowing algorithms, like Recursive Least Squares (RLS) estimation, to effectively update and optimize system parameters without becoming unstable due to excessive input variability.
Covariance matrix: A covariance matrix is a square matrix that provides a measure of how much each of the variables in a dataset varies from the mean with respect to each other. It summarizes the relationships between multiple variables, showing the degree to which they change together, which is particularly useful in multivariate statistics and estimation methods. In the context of Recursive Least Squares (RLS) estimation, the covariance matrix plays a crucial role in updating the estimates of parameters over time as new data is received.
Estimation Error: Estimation error refers to the difference between the actual value of a parameter and the estimated value obtained through a specific estimation technique. In the context of recursive least squares (RLS) estimation, it represents how accurately the model can predict or estimate the true parameters of a system. A smaller estimation error indicates a more accurate model, while larger errors suggest that the model may need adjustments or improvements.
Gain Vector: A gain vector is a set of coefficients that determine the contribution of each parameter to the output in a recursive least squares (RLS) estimation framework. It plays a crucial role in adapting the model to incoming data by adjusting these weights based on the error between the estimated and actual outputs. This adaptability allows for improved accuracy over time, making the gain vector an essential component of adaptive control systems.
Input Signal: An input signal is a variable that carries information or data into a system for processing, typically representing the system's response to external influences. In adaptive and self-tuning control, the input signal serves as a crucial element that helps adjust and optimize the control parameters in real-time, based on the system's performance and characteristics. Understanding the nature and behavior of input signals is essential for effective estimation and control strategies.
Input-output data: Input-output data refers to the information collected regarding the inputs applied to a system and the resulting outputs produced. This data is essential in understanding the relationship between what goes into a system and what comes out, enabling engineers and researchers to analyze system performance and make informed decisions about control strategies, especially in adaptive and self-tuning control scenarios.
Least squares estimation: Least squares estimation is a mathematical method used to find the best-fitting curve or line by minimizing the sum of the squares of the differences between observed and predicted values. This technique is widely used in various fields for estimating parameters of models, helping to adjust and improve predictions based on data. Its applications are significant in areas like system identification, adaptive control, and data fitting, where accuracy and adaptability are crucial.
Minimum Mean Square Error: Minimum Mean Square Error (MMSE) is a criterion used to evaluate the performance of estimators by measuring the average of the squares of the errors—that is, the difference between the estimated values and the actual values. In the context of estimation techniques, achieving MMSE means that the estimation minimizes the expected squared error, leading to more accurate predictions. This concept is crucial in adaptive filtering and Recursive Least Squares (RLS) estimation, where the goal is to continuously update estimates to achieve this optimal performance.
Observational Data: Observational data refers to information collected through direct observation of subjects in their natural environment without any manipulation or intervention. This type of data is crucial in fields like adaptive and self-tuning control because it provides insights into the actual behavior and characteristics of systems, enabling better modeling and estimation processes.
Output Signal: An output signal is the response generated by a system or a controller based on its input and internal state, often used to convey information or control actions in various applications. In adaptive and self-tuning control systems, the output signal is critical as it reflects how well the system is performing relative to desired objectives, providing feedback that can be analyzed for further adjustments. The dynamics of the output signal can reveal insights about system stability, response characteristics, and the effectiveness of parameter estimation techniques such as Recursive Least Squares (RLS).
Parameter Convergence: Parameter convergence refers to the process through which the estimated parameters of an adaptive control system approach their true values over time. This concept is essential for ensuring that adaptive control techniques effectively adjust to changing conditions and system dynamics, leading to improved performance. Understanding parameter convergence is crucial for various adaptive strategies, as it helps establish the stability and reliability of control systems under different operating scenarios.
Persistent Excitation: Persistent excitation refers to the condition in which the input signals to a system provide sufficient information over time to allow accurate estimation of the system parameters. This concept is crucial because, without persistent excitation, adaptive control algorithms may not converge to the correct parameter values, leading to instability or poor performance.
Recursive Least Squares: Recursive least squares (RLS) is an adaptive filtering algorithm that recursively minimizes the least squares cost function to estimate the parameters of a system in real-time. It allows for the continuous update of parameter estimates as new data becomes available, making it highly effective for dynamic systems where conditions change over time.
Stochastic Process: A stochastic process is a collection of random variables that represent the evolution of a system over time, where each random variable is dependent on some probabilistic behavior. These processes are crucial in modeling systems that exhibit uncertainty and variability, enabling predictions about future states based on past information. Understanding stochastic processes helps in analyzing dynamic systems, especially in fields where data is collected over time.
System Identification: System identification is the process of building mathematical models of dynamic systems based on measured input-output data. This process allows for understanding, predicting, and controlling system behavior in various applications, making it crucial for effective control design and analysis.
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