Overtaking Optimal Control for Linear Time-Invariant Systems with Unknown Disturbances: A Near-Optimal Approach

Concepts de base

This paper proposes a novel near-optimal control strategy for linear time-invariant systems subject to unknown disturbances, aiming to achieve both optimal steady-state performance and improved transient performance characterized by overtaking optimality.

Résumé

Bibliographic Information:

Li, M., Wang, Z., Liu, F., Cao, M., & Yang, B. (2024). Optimal Control in Both Steady State and Transient Process with Unknown Disturbances. IEEE Transactions on Automatic Control, [Under Review].

Research Objective:

This paper addresses the challenge of designing a control strategy for linear time-invariant (LTI) systems with unknown disturbances that can simultaneously optimize both steady-state and transient performance, with the latter characterized by overtaking optimality.

Methodology:

The authors first derive an overtaking optimal controller for the system assuming known disturbances, which serves as a benchmark for comparison. To handle unknown disturbances, they propose a near-optimal controller inspired by primal-dual dynamics and the structure of the overtaking optimal controller. The stability of the closed-loop system under the proposed controller is rigorously proven. Finally, the transient performance of the proposed near-optimal controller is analyzed and compared with the overtaking optimal controller.

Key Findings:

• The derived overtaking optimal controller for known disturbances is a superposition of linear state feedback and a constant term related to the optimal steady state.
• The proposed near-optimal controller, independent of disturbance information, guarantees global asymptotic stability of the closed-loop system and convergence to the optimal steady state.
• The transient performance gap between the near-optimal controller and the overtaking optimal controller is shown to be either zero or inversely proportional to the control gains, depending on the initial state.

Main Conclusions:

The proposed near-optimal controller offers a practical and effective solution for controlling LTI systems with unknown disturbances, achieving both optimal steady-state performance and near-optimal transient performance in the sense of overtaking optimality.

Significance:

This research contributes to the field of optimal control by providing a novel approach to address the often-neglected aspect of transient performance optimization in the presence of unknown disturbances. The theoretical results and the performance analysis provide valuable insights for designing controllers for a wide range of practical systems.

Limitations and Future Research:

The paper focuses on LTI systems. Future research could explore extending the proposed approach to nonlinear systems or systems with time-varying disturbances. Additionally, investigating the impact of different optimization algorithms on the performance of the near-optimal controller could be of interest.

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Citations

Idées clés tirées de

Optimal Control in Both Steady State and Transient Process with Unknown Disturbances

by Ming Li, Zha... à arxiv.org 10-07-2024

https://arxiv.org/pdf/2410.03178.pdf

Questions plus approfondies

How can the proposed near-optimal control strategy be adapted for systems with time-varying dynamics or disturbances?

Adapting the near-optimal control strategy for systems with time-varying dynamics or disturbances presents several challenges and requires modifications to handle the non-stationary nature of the problem. Here's a breakdown of potential approaches:
1. Time-Varying Dynamics:

Time-Varying Gain Scheduling: If the system dynamics vary slowly, a gain scheduling approach can be employed. This involves designing a set of controllers offline, each optimized for a specific operating point or parameter range. Online, the controller parameters are updated based on the current system state or estimated parameters.
Adaptive Control Techniques: For more rapid variations, adaptive control techniques like Model Reference Adaptive Control (MRAC) or Self-Tuning Regulators (STR) can be incorporated. These methods continuously adjust the controller parameters based on real-time system identification and adaptation laws.
Receding Horizon Control (RHC): RHC, also known as Model Predictive Control (MPC), is well-suited for time-varying systems. It involves solving a finite-horizon optimal control problem at each time step, using a model of the system dynamics to predict future behavior. The control input is then updated based on the solution of this optimization problem.
2. Time-Varying Disturbances:

Disturbance Estimation and Rejection: Techniques like disturbance observers or Kalman filters can be used to estimate the time-varying disturbances. The estimated disturbance can then be fed forward to the controller to counteract its effect.
Robust Control Design: Robust control methods aim to design controllers that maintain stability and performance despite uncertainties in the system dynamics or disturbances. Techniques like H-infinity control or sliding mode control can be employed to achieve robustness.
Challenges and Considerations:

Computational Complexity: Adaptive and RHC methods often require significant computational resources, which may limit their applicability in real-time control systems.
Stability and Convergence: Guaranteeing stability and convergence for time-varying systems can be challenging, especially for nonlinear systems.
Parameter Tuning: Adaptive and robust control methods typically involve tuning multiple parameters, which can be a non-trivial task.

Could the performance gap between the near-optimal and overtaking optimal controllers be further reduced by incorporating adaptive mechanisms or learning algorithms?

Yes, incorporating adaptive mechanisms or learning algorithms holds significant potential for reducing the performance gap between near-optimal and overtaking optimal controllers. Here's how:
1. Adaptive Mechanisms:

Adaptive Gain Tuning: The performance gap is often influenced by the choice of control gains. Adaptive mechanisms can be used to fine-tune these gains online based on the observed system behavior. This can involve monitoring the transient response characteristics and adjusting the gains to minimize deviations from the overtaking optimal trajectory.
Parameter Estimation: If the system model is not perfectly known, adaptive methods can be used to estimate unknown parameters online. This improved model accuracy can lead to more effective control and a reduced performance gap.
2. Learning Algorithms:

Reinforcement Learning (RL): RL algorithms can be employed to learn an optimal control policy directly from interactions with the system. By defining a reward function that penalizes deviations from the overtaking optimal performance, RL agents can iteratively improve their control strategy.
Iterative Learning Control (ILC): For repetitive tasks, ILC can be used to learn from previous iterations and improve the control performance over time. By storing and analyzing data from past trials, ILC algorithms can update the control input to minimize tracking errors and approach the overtaking optimal trajectory.
Benefits and Considerations:

Improved Performance: Adaptive and learning-based approaches can adapt to system uncertainties and disturbances, potentially achieving performance closer to the overtaking optimal controller.
Reduced Dependence on Model Accuracy: These methods can compensate for model inaccuracies or uncertainties, making them suitable for systems with complex or partially unknown dynamics.
Computational Cost: Implementing adaptive or learning algorithms can increase the computational burden, requiring careful consideration of hardware limitations and real-time constraints.
Stability and Convergence: Guaranteeing stability and convergence for adaptive and learning-based controllers can be challenging, requiring rigorous analysis and design.

What are the potential applications of this research in fields such as robotics, power systems, or process control, where both steady-state accuracy and transient response are crucial?

The research on achieving both steady-state accuracy and optimal transient response has significant implications for various fields where these factors are critical:
1. Robotics:

Trajectory Tracking: In robot manipulators or mobile robots, accurately tracking desired trajectories while minimizing settling time and overshoot is crucial. This research can be applied to optimize the motion control of robots, enabling smoother and more efficient movements.
Force Control: In tasks involving robot-environment interaction, such as grasping or manipulation, precise force control is essential. This research can be used to design controllers that achieve desired contact forces while minimizing oscillations or instability during the transient phase.
2. Power Systems:

Frequency Regulation: Maintaining a stable grid frequency is paramount in power systems. This research can be applied to design controllers for power generators that respond quickly and optimally to disturbances like load changes or generator outages, ensuring frequency stability.
Voltage Control: Regulating voltage levels within acceptable limits is crucial for power system reliability. This research can be used to develop controllers for voltage regulators or reactive power compensators that maintain voltage stability while minimizing transient fluctuations.
Renewable Energy Integration: Integrating renewable energy sources like solar and wind power introduces intermittency and variability into the grid. This research can be applied to design controllers that mitigate these fluctuations and ensure smooth power delivery.
3. Process Control:

Chemical Reactors: In chemical processes, controlling temperature, pressure, and other variables within tight tolerances is essential for product quality and safety. This research can be used to design controllers that achieve desired setpoints quickly and accurately while minimizing overshoot or oscillations.
Manufacturing Processes: In manufacturing, precise control of variables like position, speed, and temperature is crucial for product consistency and efficiency. This research can be applied to optimize the control of manufacturing equipment, leading to faster production rates and reduced defects.
Overall Benefits:

Enhanced Performance and Efficiency: Achieving both steady-state accuracy and optimal transient response leads to improved system performance, faster response times, and increased efficiency.
Improved Stability and Reliability: By minimizing transient oscillations and ensuring quick settling times, the risk of instability or system failures can be reduced.
Optimized Resource Utilization: Optimal control strategies can help minimize energy consumption, material waste, or other resources, leading to cost savings and environmental benefits.