System Identification for MPC

The Prediction Error Method (PEM) is a fundamental framework where model parameters are estimated by minimizing the discrepancy between the model's predicted output and the actual measured output. It provides a statistically sound foundation for many identification techniques.

• Core Principle: Find parameters that make the model's one-step-ahead predictions as accurate as possible.
• Flexibility: Can be applied to a wide range of model structures, including ARX, ARMAX, OE, and BJ models.
• Optimization: Typically involves solving a nonlinear optimization problem, which can be computationally intensive but yields consistent estimates under general conditions.

Subspace Identification is a state-space oriented technique that estimates models directly from input-output data without requiring nonlinear optimization. It is particularly effective for identifying multi-input, multi-output (MIMO) systems.

• Direct State Estimation: Operates by projecting data into subspaces to reveal the system's extended observability matrix, from which the state-space matrices (A, B, C, D) are derived.
• Numerical Robustness: Relies on reliable linear algebra operations like QR decomposition and Singular Value Decomposition (SVD).
• Advantage: Avoids local minima issues common in PEM for high-order systems, providing a good initial model for refinement.

The choice between linear and nonlinear model structures is a critical trade-off in system identification for MPC, balancing complexity with predictive accuracy.

• Linear Models (e.g., ARX, State-Space): Result in a Quadratic Programming (QP) problem for Linear MPC. They are simpler to identify and validate but may lack fidelity for highly nonlinear dynamics.
• Nonlinear Models (e.g., NARX, Hammerstein-Wiener, Neural Networks): Enable Nonlinear MPC (NMPC) for higher accuracy but introduce a Nonlinear Programming (NLP) problem. Identification is more complex and requires careful regularization to prevent overfitting.
• Grey-Box Modeling: Combines prior physical knowledge (white-box) with data-driven estimation (black-box) to create a partially structured model, often improving generalization with less data.

The quality of the identified model is fundamentally limited by the information content of the experimental data. Persistently exciting input signals are required to properly stimulate all relevant system dynamics.

• Signal Types: Common choices include Pseudo-Random Binary Sequences (PRBS), multi-sine signals, or chirp signals. Each has specific spectral properties.
• Design Criteria: The experiment must cover the expected operating range and frequency bandwidth relevant for MPC control. The signal amplitude must be large enough to overcome noise but remain within safe operating limits.
• Practical Consideration: For online identification or adaptive MPC, the controller's own operation must provide sufficient excitation, which can conflict with regulation objectives.

Model validation is the process of assessing the predictive capability of an identified model using a fresh dataset not used for estimation. It is essential to avoid overfitting and ensure the model is suitable for MPC.

• Key Metrics: Analyze the simulation error (model running free) and prediction error (one-step-ahead) on validation data. Statistical tests like the Auto-Correlation Function (ACF) of residuals check for unmodeled dynamics.
• Cross-Validation: Techniques like k-fold validation help assess model robustness, especially for nonlinear or neural network models where overfitting is a significant risk.
• MPC-Specific Test: The ultimate validation is often performed in simulation or Hardware-in-the-Loop (HIL) testing, where the closed-loop performance of the MPC using the identified model is evaluated.

Recursive identification algorithms update model parameters in real-time as new data arrives. This enables adaptive MPC, where the controller's internal model is refined during operation to handle slowly changing system dynamics or degradation.

• Common Algorithms: Recursive Least Squares (RLS) and its variants (e.g., with forgetting factors) are standard for linear models. For nonlinear models, approaches like Extended Kalman Filtering can be used for parameter estimation.
• Forgetting Factor: A tuning parameter that discounts older data, allowing the model to track time-varying parameters.
• Dual Control Challenge: There is an inherent conflict between control (regulating the system) and identification (exciting it to learn); adaptive schemes must carefully manage this exploration-exploitation trade-off.

To accurately identify all dynamic modes of a system, the input signal used for data collection must be persistently exciting. This means it must contain enough frequency content and amplitude variation to probe the system's full behavior. Insufficient excitation leads to an underdetermined problem and poor model generalization.

• Practical Implication: Simple step tests are often inadequate for complex, multi-variable systems. Engineers must design specialized test signals, such as pseudo-random binary sequences (PRBS) or multi-sine waves, to ensure all relevant dynamics are captured in the data.

Model complexity must be carefully balanced. A model that is too simple (high bias) cannot capture the true system dynamics, leading to structural errors. A model that is too complex (high variance) will overfit the specific noise in the training data, resulting in poor performance on new data.

• Model Structures: Choosing between linear ARX, nonlinear NARX, state-space, or neural network models is a direct application of this trade-off. Cross-validation on a separate dataset is the primary method for navigating this challenge and selecting the appropriate model order.

Real sensor data is contaminated with measurement noise and occasional outliers. System identification algorithms must be robust to these disturbances to prevent learning spurious dynamics.

• Pre-processing: Techniques like low-pass filtering and outlier rejection (e.g., using median filters) are essential first steps.
• Algorithm Choice: Prediction Error Methods (PEM) inherently handle measurement noise in the output. For severe outliers, robust estimation techniques that minimize the L1-norm of errors, rather than the standard L2-norm (least squares), may be required.

Identifying nonlinear models (e.g., for Nonlinear MPC) is computationally intensive. The parameter search space is vast, and cost functions are non-convex, leading to potential convergence to local minima.

• Training Time: Methods like nonlinear least squares or training a recurrent neural network can require significant data and compute time.
• Global vs. Local Search: Engineers often use a combination of global optimization (e.g., genetic algorithms) to find a good region and local gradient-based methods (e.g., Levenberg-Marquardt) for fine-tuning.

No identified model is perfect. The residual difference between the model and the true plant (model-plant mismatch) is a primary source of performance degradation in MPC. This mismatch arises from unmodeled dynamics, parameter drift, and operating condition changes.

• MPC's Role: A key strength of MPC is its ability to reject some disturbance via feedback. However, significant mismatch can lead to constraint violation or instability.
• Mitigation: Strategies include designing robust MPC controllers, implementing adaptive MPC that updates the model online, or using Moving Horizon Estimation (MHE) to provide better state estimates in the face of model errors.

Collecting informative data from operational systems, especially safety-critical ones, can be expensive or dangerous. Furthermore, identifying a model from data collected under closed-loop control is challenging because the controller suppresses excitations.

• Direct vs. Indirect Methods: Direct identification ignores the controller and can be biased. Indirect identification first identifies the closed-loop dynamics and then calculates the open-loop plant model, requiring knowledge of the controller.
• Practical Approach: A common strategy is to perform initial open-loop tests in a safe regime to get a baseline model, then refine it using specialized closed-loop identification techniques during normal operation.

These are two fundamental approaches to system identification. Black-box modeling treats the system as an unknown entity, using flexible function approximators (like neural networks or NARX models) to map inputs to outputs without incorporating prior physical knowledge. Grey-box modeling combines a priori physical laws (e.g., Newtonian mechanics) with data-driven estimation to identify unknown parameters within a structured model, often leading to more interpretable and physically consistent results for MPC.

The Prediction Error Method (PEM) is a cornerstone framework for parametric system identification. It works by:

• Postulating a model structure with adjustable parameters.
• Using the model to predict system outputs over a dataset.
• Iteratively adjusting the parameters to minimize a cost function (typically the sum of squared prediction errors).
• Common model structures estimated via PEM include ARX, ARMAX, OE, and BJ models. PEM provides statistically efficient estimates and is the basis for many modern identification toolboxes.

Subspace identification is a state-space oriented technique that directly estimates a linear state-space model from input-output data without requiring nonlinear optimization. Key steps involve:

• Forming large block Hankel matrices from the data.
• Using numerical linear algebra tools (like QR decomposition and Singular Value Decomposition (SVD)) to extract the system's observability matrix and state sequence.
• This method is particularly powerful for identifying multi-input, multi-output (MIMO) systems and provides a balanced state-space realization suitable for MPC.

Persistency of excitation is a critical condition on the input signal used for identification. An input signal is persistently exciting of order n if it contains enough frequency content and variation to uniquely identify all parameters of a model of that order. Without it, the estimation problem becomes ill-conditioned. For MPC model identification, inputs like pseudo-random binary sequences (PRBS) or multisine signals are designed to be persistently exciting, ensuring the collected data is informative enough to reveal the full system dynamics.

After identification, a model must be rigorously validated before deployment in MPC. This involves testing the model on a fresh validation dataset not used for estimation. Key validation metrics include:

• Fit percentage (e.g., Normalized Root Mean Square Error).
• Residual analysis (checking if prediction errors are white and uncorrelated with inputs).
• Cross-validation, where data is partitioned into multiple sets, with the model trained on some and tested on others, guards against overfitting and assesses generalizability to new operating conditions.

Recursive Least Squares (RLS) is an online parameter estimation algorithm. Unlike batch methods, RLS updates model parameters with each new data point, making it suitable for adaptive MPC or systems with slowly varying dynamics. The algorithm maintains an estimate of the parameter covariance matrix and uses it to compute a gain for updating parameters. A forgetting factor can be introduced to discount older data, allowing the model to track time-varying parameters. RLS is computationally efficient for real-time model refinement.