Core Concepts

For bilinear systems with unknown parameters, a data-driven controller can be designed to asymptotically stabilize a desired state setpoint and provide a guaranteed basin of attraction, even in the presence of noisy data.

Abstract

The paper considers the problem of designing a controller for an unknown bilinear system using only noisy input-state data points generated by it. The goal is to achieve regulation to a given state setpoint and provide a guaranteed basin of attraction, without knowledge of the system parameters.
The authors propose two design scenarios:
When the equilibrium input corresponding to the desired setpoint is known:
A linear matrix inequality (LMI) based design is provided, which ensures local asymptotic stability of the setpoint with a guaranteed basin of attraction.
This design improves upon the authors' previous work by not requiring any knowledge of upper bounds on the norms of the system matrices.
When the equilibrium input is unknown:
The authors show that it is impossible to find an input that makes the desired setpoint an equilibrium for all possible systems consistent with the data.
Instead, the authors design a controller that stabilizes a small neighborhood of the desired setpoint, with the size of the basin of attraction as a design parameter.
The designs are validated numerically on a Cuk converter example.

Stats

The bilinear system dynamics are described by the following equation:
x◦= A⋆x + B⋆u + C⋆(u ⊗x) + d⋆
The noisy data points collected from the system satisfy:
x◦(ti) = A⋆x(ti) + B⋆u(ti) + C⋆(u(ti) ⊗x(ti)) + d⋆+ e(ti)
The noise sequence e(t0), e(t1), ..., e(tT −1) is assumed to be bounded in energy, i.e., EE⊤⪯ΞΞ⊤.

Quotes

"Bilinear systems are a halfway house between linear systems, which are endowed with strong structure and properties, and nonlinear systems, which are less tractable but capture more complex phenomena."
"Approaches such as Carleman linearization [21, p. 110] or Koopman operator [12] lead to bilinear systems after truncation and, in this sense, bilinear system approximate more complex nonlinear ones."

Key Insights Distilled From

by Andrea Bisof... at **arxiv.org** 04-05-2024

Deeper Inquiries

The proposed data-driven control design for bilinear systems can be extended to handle more general nonlinear system models by incorporating techniques from nonlinear control theory. One approach is to use data-driven identification methods to estimate the parameters of a nonlinear system model, such as neural networks or fuzzy models, from the collected noisy data points. Once the model is identified, the control design can be formulated using the estimated model to achieve regulation to a desired setpoint. Additionally, techniques like adaptive control or reinforcement learning can be employed to adapt the controller online based on the evolving system dynamics captured by the data.

When applying this data-driven control approach to real-world systems, several limitations and challenges may arise. One major limitation is the assumption of bounded noise in the data, which may not always hold in practical scenarios. Dealing with unbounded or time-varying noise can lead to inaccuracies in the identified system model and affect the performance of the controller. Moreover, the complexity of the system dynamics and the high dimensionality of the data can pose challenges in terms of computational efficiency and real-time implementation. Practical constraints such as actuator saturation, sensor noise, and communication delays also need to be considered in the control design to ensure stability and robustness of the system.

The insights gained from the setpoint control of bilinear systems can indeed be leveraged to develop data-driven control strategies for other classes of nonlinear systems with unknown parameters. By generalizing the control design framework to accommodate more complex nonlinear dynamics, such as polynomial systems or Hammerstein-Wiener models, the data-driven approach can be extended to achieve stabilization and tracking objectives for a wider range of systems. Techniques like system identification, adaptive control, and reinforcement learning can be tailored to suit the specific characteristics of different nonlinear system classes, enabling the design of effective data-driven controllers in the presence of uncertainties and unknown parameters.

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