Optimal Control of Discrete-Time Systems with Bounded Disturbances

Conceitos essenciais

This paper introduces an optimal control approach for linear discrete-time systems subject to bounded disturbances, based on a novel duality between ellipsoidal approximations of reachable and hardly observable sets.

Resumo

The paper presents a novel approach to the optimal control of linear discrete-time systems subject to bounded disturbances. The key contributions are:

1. Establishment of a duality between ellipsoidal approximations of reachable and hardly observable sets, leading to the definition of the ε-norm.
2. Derivation of exact equations for determining optimal state-feedback and observer gains, which are computationally more efficient than previous optimization-based methods.
3. Introduction of an optimal solution to the output-feedback control problem, which significantly outperforms prior sub-optimal results.

The authors first introduce the concept of the ε-norm, which quantifies the size of the ellipsoidal approximation of the reachable set. They then present the optimal state-feedback and filtering solutions with respect to the ε-norm. The main contribution is the optimal solution to the output-feedback control problem, which was previously unknown.

Numerical simulations demonstrate the superiority of the proposed output-feedback controller over previous sub-optimal methods in terms of the achieved ε-norm.

Estatísticas

The paper provides the following key figures and metrics:

The ε(α)-norm of the closed-loop system as a function of the parameter α, comparing the proposed optimal method with previous sub-optimal methods.
Visualization of the invariant ellipsoids and sample state trajectories for the closed-loop system, showing the smaller ellipsoid obtained with the optimal method.
Numerical values of the controller parameters and the corresponding ε-norm for the sub-optimal and optimal methods.

Citações

"The primary contributions of this paper are as follows:

We establish a duality between the ellipsoidal approximations of reachable and hardly observable sets for discrete-time systems, which leads to a definition of the ε-norm.
For the first time, we present exact equations for determining optimal state-feedback and observer gains in discrete-time systems with respect to the ε-norm. These equations yield the same results as the optimization procedures proposed in [3]–[5], but are computationally more efficient due to a reduced number of variables.
We introduce an optimal solution to the output-feedback control problem for discrete-time systems under bounded disturbances. This solution shows a significant improvement over prior sub-optimal results found in [4], [5]."

Principais Insights Extraídos De

Optimal Control for Discrete-Time Systems under Bounded Disturbances

by Egor Dogadin... às arxiv.org 09-20-2024

https://arxiv.org/pdf/2409.12252.pdf

Perguntas Mais Profundas

How can the proposed optimal control approach be extended to handle more complex system dynamics, such as nonlinear or time-varying systems?

The proposed optimal control approach, which leverages the duality between ellipsoidal approximations of reachable and hardly observable sets, can be extended to handle more complex system dynamics, including nonlinear and time-varying systems, through several strategies.

Nonlinear Systems: For nonlinear systems, one could employ techniques such as feedback linearization or the use of Lyapunov-based methods. By transforming the nonlinear dynamics into an equivalent linear form around an operating point, the existing framework can be applied. Additionally, the concept of invariant ellipsoids can be generalized to encompass nonlinear dynamics by considering local linearizations and ensuring that the ellipsoidal approximations remain valid in the vicinity of the operating point.

Time-Varying Systems: To address time-varying systems, the control approach can be adapted by incorporating time-dependent parameters into the Riccati equations. This could involve formulating a time-varying Lyapunov equation that accounts for the changes in system dynamics over time. Furthermore, adaptive control strategies could be integrated, allowing the controller to adjust its parameters in real-time based on observed system behavior, thus maintaining optimal performance despite variations in system dynamics.

Robustness and Stability: The extension to more complex dynamics would also necessitate a focus on robustness and stability. Techniques such as robust control design, which ensures performance under model uncertainties and external disturbances, can be integrated into the existing framework. This would involve modifying the ε-norm to account for worst-case scenarios, thereby enhancing the system's resilience to disturbances.

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

What are the potential limitations or drawbacks of the ε-norm as a performance metric, and how could it be further improved or combined with other measures?

The ε-norm, while a valuable performance metric in the context of optimal control for discrete-time systems, does have several potential limitations and drawbacks:

Sensitivity to Disturbances: The ε-norm primarily focuses on the size of the ellipsoidal approximations of reachable and hardly observable sets. However, it may not adequately capture the system's performance in the presence of specific types of disturbances, particularly those that are non-bounded or exhibit high-frequency characteristics. This could lead to suboptimal control strategies in practical applications.

Lack of Time-Domain Information: The ε-norm is a static measure that does not account for the dynamic behavior of the system over time. As a result, it may overlook important transient responses and stability characteristics that are critical in control applications.

Potential for Over-Simplification: By focusing solely on the size of the ellipsoidal sets, the ε-norm may oversimplify the complexities of system dynamics, leading to control strategies that do not fully exploit the system's capabilities or address its limitations.

To improve the ε-norm as a performance metric, it could be combined with other measures such as:

Time-Domain Performance Metrics: Incorporating time-domain metrics like settling time, overshoot, and steady-state error could provide a more comprehensive assessment of system performance. This would allow for a better understanding of how the system behaves over time, particularly during transient phases.

Robustness Measures: Integrating robustness metrics, such as H-infinity or H2 norms, could enhance the ε-norm by ensuring that the control strategy remains effective under a wider range of disturbances and uncertainties.

Multi-Objective Optimization: Employing a multi-objective optimization framework that considers both the ε-norm and other performance criteria could lead to more balanced control strategies that optimize for various aspects of system performance simultaneously.
By addressing these limitations and combining the ε-norm with complementary performance metrics, the overall effectiveness and applicability of the control approach can be significantly enhanced.

What are the potential applications of the optimal output-feedback control solution beyond the discrete-time systems considered in this paper, and how could it be adapted to different problem settings?

The optimal output-feedback control solution presented in this paper has a wide range of potential applications beyond the discrete-time systems specifically considered. Some of these applications include:

Robotics and Automation: In robotic systems, optimal output-feedback control can be utilized to enhance the precision and stability of robotic movements, particularly in environments with bounded disturbances such as external forces or sensor noise. The approach can be adapted to real-time control of robotic arms, drones, and autonomous vehicles, where maintaining performance under varying conditions is crucial.

Aerospace Systems: The proposed control solution can be applied to aerospace systems, including aircraft and spacecraft, where disturbances such as wind gusts or atmospheric turbulence are common. By adapting the control strategy to account for these disturbances, the stability and performance of flight control systems can be significantly improved.

Networked Control Systems: In networked control systems, where multiple interconnected systems operate collaboratively, the optimal output-feedback control can be adapted to manage communication delays and packet losses. This adaptation would involve modifying the control laws to ensure robust performance despite the uncertainties introduced by the network.

Power Systems: The control approach can be applied to power systems for optimal regulation of voltage and frequency in the presence of disturbances such as load changes or generation fluctuations. By implementing the output-feedback control solution, power systems can achieve better stability and reliability.

Manufacturing Processes: In manufacturing, the optimal output-feedback control can be utilized to enhance process control, ensuring that production systems remain stable and efficient despite variations in material properties or external conditions. This could involve real-time adjustments to control inputs based on output measurements.

To adapt the optimal output-feedback control solution to these different problem settings, several strategies can be employed:

Modeling and System Identification: Accurate modeling of the specific application is essential. Techniques such as system identification can be used to develop models that capture the dynamics of the application, allowing the control solution to be tailored accordingly.

Integration with Existing Control Frameworks: The proposed solution can be integrated with existing control frameworks, such as PID control or state-space control, to enhance performance while leveraging established methodologies.

Real-Time Implementation: For applications requiring real-time control, the computational efficiency of the proposed approach can be further optimized, ensuring that the control laws can be executed within the necessary time constraints.
By leveraging these strategies, the optimal output-feedback control solution can be effectively adapted to a variety of applications, enhancing performance and robustness in diverse operational environments.