Time-Optimal Control of Controllable Linear Systems: A Novel State-Centric Necessary Condition Based on Augmented Switching Laws

Core Concepts

A novel state-centric necessary condition for time-optimal control of controllable linear systems is established, which only requires information on states and controls without dependence on costate information.

Abstract

The paper establishes a theoretical framework for time-optimal control of controllable linear systems, proposing the concept of "augmented switching law" to represent the input control and the feasibility of the trajectory in a compact form.
Key highlights:

The augmented switching law fully characterizes the control and the feasibility of the Bang-Singular-Bang trajectory, enabling a novel variational approach and a local optimization method with feasibility assurance.
A novel state-centric necessary condition is proposed, stating that the Jacobian matrix of the augmented switching law should not be full row rank for an optimal trajectory. This necessary condition only requires information on states and controls, without dependence on costate information.
The proposed state-centric necessary condition is applied to high-order chain-of-integrators systems with full box constraints, contributing to conclusions that are challenging to prove using traditional costate-based necessary conditions.

The paper develops a theoretical framework focusing on the control of each unconstrained/constrained arc, the constraints induced by the connections of adjoining arcs, the feasibility of the disturbed trajectory near the constrained boundary, and the optimality of a given feasible trajectory. The proposed state-centric necessary condition provides a novel approach to determining the optimality of a given feasible trajectory and optimizing it further.

Stats

Most existing necessary conditions for optimal control require both state information and costate information, yet the lack of costates for a given feasible trajectory in practice impedes the determination of optimality.
The proposed state-centric necessary condition only requires information on states and controls, without dependence on costate information.
The proposed state-centric necessary condition is applied to high-order chain-of-integrators systems with full box constraints, contributing to conclusions that are challenging to prove using traditional costate-based necessary conditions.

Quotes

"Given a planned feasible trajectory, it is challenging to determine whether the trajectory is optimal based on state information since most existing necessary conditions require costate information."
"The necessary conditions are usually applied to rule out some forms of non-optimal controls in practice. In some simple cases, the optimal control can be fully solved based on some necessary conditions, while in other cases, the necessary conditions are applied to guide the design of suboptimal trajectories."

Key Insights Distilled From

A Novel State-Centric Necessary Condition for Time-Optimal Control of Controllable Linear Systems Based on Augmented Switching Laws

by Yunan Wang,C... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08943.pdf

Deeper Inquiries

How can the proposed state-centric necessary condition be extended to more general nonlinear systems beyond controllable linear systems

The proposed state-centric necessary condition for time-optimal control of controllable linear systems can be extended to more general nonlinear systems by incorporating techniques from nonlinear control theory. One approach is to utilize Lyapunov-based methods to analyze the stability and optimality of nonlinear systems. By formulating Lyapunov functions and studying their properties, it is possible to derive necessary conditions for optimality in nonlinear systems. Additionally, techniques such as feedback linearization, sliding mode control, and adaptive control can be employed to handle the nonlinearities present in the system dynamics. By adapting the augmented switching law representation to capture the dynamics of nonlinear systems, the state-centric necessary condition can be extended to ensure optimality in a broader class of systems.

What are the potential applications of the augmented switching law representation beyond time-optimal control, such as in the analysis and design of switched systems

The augmented switching law representation proposed for time-optimal control can find applications beyond just optimizing control trajectories. One potential application is in the analysis and design of switched systems, where the switching law can be used to determine the optimal switching sequences for different modes of operation. By incorporating constraints and feasibility considerations into the augmented switching law, it can guide the switching decisions in a switched system to ensure stability, performance, and safety. Furthermore, the representation can be utilized in fault detection and isolation systems, where the switching law can help in identifying and isolating faulty components or subsystems based on their behavior and interactions within the system.

Can the proposed framework and necessary condition be integrated with machine learning techniques to enable data-driven optimal control of complex systems

The proposed framework and necessary condition can be integrated with machine learning techniques to enable data-driven optimal control of complex systems. By leveraging machine learning algorithms such as reinforcement learning, neural networks, and deep learning, the framework can learn the optimal control policies from data and experience. The augmented switching law representation can be used as input features for the machine learning models, allowing them to capture the complex relationships between system states, controls, and optimality criteria. This integration can enable the development of adaptive and intelligent control systems that can adapt to changing environments, uncertainties, and disturbances, leading to improved performance and efficiency in controlling complex systems.