Data-Driven Stochastic Predictive Control with Chance Constraint Satisfaction Using Identified Multi-Step Predictors

Centrala begrepp

A novel data-driven stochastic model predictive control framework is proposed for uncertain linear systems with noisy output measurements. The approach leverages multi-step predictors to efficiently propagate uncertainty and ensure chance constraint satisfaction with minimal conservatism.

Sammanfattning

The paper presents a data-driven stochastic model predictive control (MPC) framework for uncertain linear systems with noisy output measurements. The key aspects of the proposed approach are:

1. Multi-step Predictors:

   • The system is reformulated using multi-step predictors, which directly relate the current output to the past inputs and initial state.
   • This avoids the limitations of sequential propagation in traditional state-space models.

2. Parameter Identification:

   • A method is proposed to identify the multi-step predictors and quantify the associated uncertainty using maximum likelihood estimation (MLE) based on the innovation form of the system obtained through Kalman filter recursions.
   • This allows effective utilization of data from a single input-output trajectory, without requiring precise knowledge of the system parameters.

3. Chance Constraint Satisfaction:

   • The uncertainty in the estimated multi-step predictor parameters is explicitly accounted for in the predictive control problem formulation.
   • A constraint tightening approach is developed to ensure chance constraint satisfaction with minimal conservatism, by directly using the distribution of the parameter estimates.

The numerical example demonstrates that the proposed data-driven approach can significantly reduce conservatism compared to state-of-the-art solutions while reliably satisfying the probabilistic constraints, despite the lack of precise model knowledge.

Statistik

The system has 3 masses, 3 position measurements, and 1 control input.
The disturbance covariance is diag(0, 0, 0, 1e-3, 1e-3, 1e-3).
The measurement noise covariance is 1e-3 * I.

Citat

"The primary contribution of this paper is the development of a data-driven predictive control framework that employs multi-step predictors to ensure chance-constraint satisfaction for linear stochastic systems using only input-output data."
"Ensuring satisfaction of safety-critical constraints when the data is subject to noise remains as a significant challenge. The proposed framework addresses this problem by identifying multi-step predictors to directly enforce probabilistic constraints."

Viktiga insikter från

Stochastic Data-Driven Predictive Control: Chance-Constraint Satisfaction with Identified Multi-step Predictors

by Hald... på arxiv.org 09-17-2024

https://arxiv.org/pdf/2409.10405.pdf

Djupare frågor

How can the proposed framework be extended to handle time-varying or nonlinear systems?

The proposed framework for stochastic data-driven predictive control can be extended to accommodate time-varying or nonlinear systems through several strategies.

Adaptive Multi-step Predictors: For time-varying systems, the multi-step predictors can be adapted to account for changes in system dynamics over time. This can be achieved by implementing a recursive identification approach where the model parameters are updated continuously as new data becomes available. Techniques such as online learning or adaptive filtering can be employed to refine the multi-step predictors dynamically.

Nonlinear State-Space Models: To handle nonlinearities, one could utilize nonlinear state-space models, such as those based on the Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF). These methods allow for the propagation of uncertainty in nonlinear systems by approximating the nonlinear dynamics with linear models around the current state. The multi-step predictors can then be derived from these nonlinear state-space representations.

Data-Driven Nonlinear Control Techniques: Incorporating data-driven approaches such as Gaussian Processes or Neural Networks can enhance the framework's ability to model complex nonlinear relationships. These methods can learn the underlying dynamics from data without requiring explicit model formulations, thus providing flexibility in capturing time-varying behaviors.

Robustness to Nonlinearities: The framework can also integrate robust control techniques that account for model uncertainties and nonlinearities. By formulating the control problem with robust optimization principles, the chance constraints can be satisfied even in the presence of significant nonlinear disturbances.

Hierarchical Control Structures: Implementing a hierarchical control structure can also be beneficial. A high-level controller can manage the overall system behavior while low-level controllers can handle local nonlinearities and time-varying dynamics, ensuring that the overall system remains stable and adheres to the chance constraints.

By employing these strategies, the proposed framework can be effectively adapted to manage the complexities associated with time-varying and nonlinear systems, thereby enhancing its applicability in real-world scenarios.

What are the potential limitations of the assumption that the disturbance and measurement noise covariances are known or can be accurately estimated from data?

The assumption that disturbance and measurement noise covariances are known or can be accurately estimated from data presents several potential limitations:

Modeling Errors: If the true noise characteristics deviate from the assumed models, the performance of the predictive control framework may be compromised. For instance, if the noise is non-Gaussian or exhibits time-varying properties, the static covariance estimates may lead to incorrect predictions and suboptimal control actions.

Data Limitations: Accurate estimation of noise covariances relies heavily on the availability of sufficient and representative data. In scenarios where data is scarce or unrepresentative, the estimated covariances may not reflect the true system dynamics, leading to increased conservatism in the control strategy and potential violations of chance constraints.

Overconfidence in Estimates: Relying on estimated covariances can lead to overconfidence in the model's predictions. If the estimated covariances are underestimated, the control strategy may not adequately account for the actual uncertainty, resulting in a higher likelihood of constraint violations.

Computational Complexity: The process of estimating noise covariances can introduce additional computational complexity, especially in high-dimensional systems. This can lead to increased computational time and resource requirements, which may not be feasible in real-time applications.

Assumption of Stationarity: The assumption that noise covariances are constant over time may not hold in practice, particularly in dynamic environments. Non-stationary noise can lead to inaccurate predictions and necessitate frequent re-estimation of covariances, complicating the control design.

Robustness to Model Mismatch: The framework may lack robustness to model mismatches if the noise characteristics are not accurately captured. This can result in poor performance in the presence of unexpected disturbances or changes in the system dynamics.

To mitigate these limitations, it is essential to incorporate robust estimation techniques, adaptive filtering methods, and uncertainty quantification strategies that can dynamically adjust to changes in the system and provide more reliable estimates of noise covariances.

Can the multi-step predictor identification and uncertainty quantification be further improved to enhance the computational efficiency and scalability of the overall approach?

Yes, the multi-step predictor identification and uncertainty quantification can be further improved to enhance computational efficiency and scalability through several approaches:

Sparse Identification Techniques: Utilizing sparse identification methods can significantly reduce the computational burden associated with estimating multi-step predictors. By focusing on the most relevant features and parameters, these techniques can streamline the identification process, making it more efficient and scalable for high-dimensional systems.

Parallel Computing: Implementing parallel computing strategies can enhance the efficiency of the identification process. By distributing the computational load across multiple processors or cores, the time required for estimating multi-step predictors and quantifying uncertainties can be significantly reduced.

Dimensionality Reduction: Applying dimensionality reduction techniques, such as Principal Component Analysis (PCA) or Autoencoders, can simplify the data used for identification. This can lead to faster convergence in the estimation process and reduce the complexity of the resulting models, making them more manageable.

Incremental Learning: Incorporating incremental learning algorithms allows for the continuous updating of multi-step predictors as new data becomes available. This approach reduces the need for complete re-estimation, thus saving computational resources and time.

Efficient Sampling Methods: Enhancing uncertainty quantification through more efficient sampling methods, such as Latin Hypercube Sampling or Quasi-Monte Carlo methods, can provide better estimates of the uncertainty bounds with fewer samples. This can lead to faster computations while maintaining accuracy.

Model Order Reduction: Implementing model order reduction techniques can simplify the multi-step predictors by reducing the number of states or parameters involved. This can lead to faster computations and easier implementation in real-time control applications.

Adaptive Algorithms: Developing adaptive algorithms that can adjust the complexity of the multi-step predictors based on the current state of the system can improve scalability. For instance, simpler models can be used in less complex scenarios, while more detailed models can be employed when necessary.

By integrating these improvements, the overall approach can achieve greater computational efficiency and scalability, making it more suitable for real-time applications in complex and high-dimensional systems.