Data-Driven Control for Unknown Nonlinear Systems Using Koopman Operator and Sum-of-Squares Optimization

Core Concepts

This paper proposes a novel data-driven control approach for unknown nonlinear systems using the Koopman operator and sum-of-squares (SOS) optimization, achieving improved stability guarantees and data efficiency compared to existing methods.

Abstract

Bibliographic Information:

Strässer, R., Berberich, J., & Allgöwer, F. (2024). Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency. arXiv preprint arXiv:2411.03875.

Research Objective:

This paper aims to develop a data-driven control method for unknown nonlinear systems that leverages the Koopman operator and sum-of-squares optimization to achieve improved stability guarantees and data efficiency.

Methodology:

The authors propose a novel approach that combines the stability-and-certificate-oriented extended dynamic mode decomposition (SafEDMD) architecture with sum-of-squares (SOS) optimization. First, SafEDMD is used to generate a data-driven bilinear surrogate model with certified error bounds for the unknown nonlinear system. Then, a rational controller is designed by formulating an SOS program that explicitly accounts for the bilinearity of the surrogate model and the associated error bounds. This controller is guaranteed to stabilize the error-affected bilinear surrogate model and, consequently, the underlying nonlinear system.

Key Findings:

• The proposed SOS-based controller design guarantees global exponential stability for uncertain bilinear systems, outperforming existing methods that offer only local stability guarantees.
• The approach significantly reduces conservatism compared to existing Koopman-based control methods that over-approximate the bilinearity of the surrogate model.
• This leads to a larger region of attraction and improved data efficiency, requiring fewer data samples for feasible controller design.

Main Conclusions:

The paper demonstrates the effectiveness of combining Koopman operator theory with SOS optimization for data-driven control of unknown nonlinear systems. The proposed method provides strong stability guarantees, reduces conservatism, and improves data efficiency compared to existing approaches.

Significance:

This research contributes to the growing field of data-driven control by providing a novel and effective method for designing controllers for complex nonlinear systems with limited model knowledge. The improved stability guarantees and data efficiency offered by the proposed approach have significant implications for practical applications in various domains, including robotics, aerospace, and process control.

Limitations and Future Research:

• The paper primarily focuses on state-feedback control and does not explicitly address output feedback scenarios.
• The impact of noise and disturbances on the performance of the proposed controller is not extensively studied.
• Future research could explore extensions of the approach to incorporate performance specifications and handle more complex system dynamics, such as those involving time delays or hybrid behavior.

Stats

For the bilinear building control example, the authors used a zone volume (Vz) of 2, a supply air temperature (T0) of -1, and a sampling time (Ts) of 1.
In the inverted pendulum example, the parameters were set as mass (m) = 1, length (l) = 1, damping coefficient (b) = 0.5, and gravitational acceleration (g) = 9.81.
The data length (d) for each constant input in the inverted pendulum example was 200 data pairs.
The proportional error bound constants used in the inverted pendulum example were cx = 1 × 10^-2 and cu = 1 × 10^-3 for the SOS-based controller, and cx = 2 × 10^-3 and cu = 2 × 10^-4 for the LMI-based controller.

Quotes

"In this paper, we propose a novel approach to design controllers for unknown nonlinear systems based on the Koopman operator and sum-of-squares optimization (SOS)."
"Our approach significantly reduces conservatism by establishing a larger region of attraction and improved data efficiency."
"Compared to existing Koopman-based control methods with stability guarantees, which over-approximate the bilinearity of the surrogate model, our approach explicitly accounts for the bilinearity."

Key Insights Distilled From

Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency

by Robi... at arxiv.org 11-07-2024

https://arxiv.org/pdf/2411.03875.pdf

Deeper Inquiries

How does the computational complexity of the proposed SOS-based controller design scale with the system dimension and complexity compared to other data-driven control methods?

The computational complexity of the SOS-based controller design proposed in the paper is primarily influenced by the size of the resulting semidefinite program (SDP). Here's a breakdown of the factors involved:

Size of the SDP: The number of decision variables and constraints in the SDP grows polynomially with the system dimension (state and input dimensions) and the degree of the polynomials used in the SOS program (parameter α). Higher-degree polynomials offer more flexibility in the controller design but lead to larger SDPs.
Data-driven aspect: While the data-driven nature of the approach (using SafEDMD) impacts the accuracy of the bilinear surrogate model, it doesn't directly affect the complexity of the SOS optimization itself. The size of the data used influences the dimensions of matrices A, B0, and ˜B, but their effect on the overall complexity is less significant compared to the polynomial degrees.
Comparison to other data-driven control methods:

Linear methods: Methods based on linear approximations (e.g., those using linear EDMD or those over-approximating the bilinearity in the paper's context) generally lead to smaller SDPs and are computationally less demanding.
Nonlinear methods: Other nonlinear data-driven control methods, such as those based on neural networks, often involve non-convex optimization problems. These can be significantly harder to solve than SDPs, especially for guaranteeing global stability.
In summary: The SOS-based approach offers a trade-off between controller flexibility and computational complexity. While it can handle more complex systems and provide stronger guarantees compared to linear methods, it comes at the cost of increased computational burden. However, it can be more tractable than other nonlinear methods relying on non-convex optimization.

Could the reliance on a data-driven bilinear surrogate model limit the applicability of this approach to highly nonlinear systems where accurate bilinear approximations are difficult to obtain?

Yes, the reliance on a data-driven bilinear surrogate model could potentially limit the applicability of this approach to highly nonlinear systems where achieving accurate bilinear approximations proves challenging. Here's why:

Bilinear approximation limitations: While the Koopman operator framework allows representing nonlinear dynamics in a lifted space where they become bilinear, the accuracy of a finite-dimensional bilinear approximation depends on the system's inherent nonlinearity and the choice of observable functions. For highly complex nonlinearities, capturing the dynamics accurately with a tractable number of observable functions might be difficult.
Data requirements: Obtaining an accurate bilinear surrogate model for highly nonlinear systems might necessitate a large amount of data, potentially making the approach infeasible for data-scarce applications.
Region of attraction: Even if a reasonably accurate bilinear model is obtained, the guaranteed region of attraction (RoA) for the controller might be limited if the system exhibits strong nonlinearities outside the region covered by the training data.
Potential mitigation strategies:

Enriching the dictionary of observable functions: Using a richer set of observable functions (e.g., including nonlinear functions of the original state variables) can improve the approximation capability. However, this increases the dimensionality of the lifted space and, consequently, the computational complexity.
Local approximations: Instead of seeking a global bilinear approximation, focusing on local approximations around specific operating points could be more effective for highly nonlinear systems. This might require designing multiple controllers for different operating regions.
Hybrid approaches: Combining the Koopman-based approach with other techniques, such as learning a residual model to capture the unmodeled dynamics, could improve accuracy and broaden applicability.
In conclusion: While the reliance on bilinear approximations poses a potential limitation, ongoing research on Koopman operator theory and its applications in control aims to address these challenges and extend its applicability to a wider range of nonlinear systems.

How can the insights from this research on Koopman operator theory and control be leveraged to develop new methods for system identification and model reduction of complex dynamical systems?

The insights from this research, particularly the use of SafEDMD and SOS optimization within the Koopman operator framework, open up promising avenues for developing novel methods in system identification and model reduction of complex dynamical systems:
System Identification:

Guaranteed error bounds: SafEDMD provides a systematic way to learn bilinear surrogate models with certified error bounds. This can be valuable for system identification by providing confidence in the learned model's accuracy and enabling robust control design.
Handling nonlinearities: The Koopman operator approach offers a powerful framework for identifying nonlinear systems from data. By leveraging richer dictionaries of observable functions and exploring different lifting techniques, more accurate nonlinear system identification methods can be developed.
Data efficiency: The combination of SafEDMD and SOS optimization allows for data-efficient controller design. This insight can be extended to system identification by developing methods that can learn accurate models from limited data, which is crucial for many practical applications.
Model Reduction:

Preserving stability: The use of SOS optimization for controller design with stability guarantees can be adapted to develop model reduction techniques that preserve stability properties of the original system. This is crucial for ensuring that the reduced-order model is suitable for control design and analysis.
Structure-preserving reduction: The Koopman operator framework can guide the development of model reduction techniques that preserve specific structures inherent in the original system, such as bilinearity or other dominant nonlinearities. This can lead to more accurate and physically interpretable reduced-order models.
Data-driven reduction: The data-driven nature of SafEDMD can be exploited to develop model reduction methods that tailor the reduced-order model to the specific operating conditions or regions of interest based on the available data.
Overall, the research highlights the potential of combining Koopman operator theory, data-driven methods like SafEDMD, and optimization techniques like SOS programming to advance the field of system identification and model reduction. By addressing the limitations of existing methods and leveraging the strengths of each technique, new opportunities arise for analyzing and controlling complex dynamical systems.