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Efficient Synthesis of Control Lyapunov-Value Functions for High-Dimensional Nonlinear Systems using System Decomposition and Admissible Control Sets


Core Concepts
A method is proposed to efficiently synthesize Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems by decomposing the system into low-dimensional subsystems, reconstructing the full-dimensional CLVF, and providing sufficient conditions for when the reconstruction is exact.
Abstract
The paper presents a method to efficiently synthesize Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems. The key ideas are: System Decomposition: The original high-dimensional system is decomposed into low-dimensional subsystems with potentially shared states or controls. Subsystem CLVF Computation: The CLVF is computed for each subsystem independently using standard techniques. CLVF Reconstruction: The full-dimensional CLVF is reconstructed from the subsystem CLVFs. If there are no shared controls, the reconstruction is guaranteed to be exact. If there are shared controls, a sufficient condition is provided for when the reconstruction is exact. Inexact Reconstruction: When the exact reconstruction is not possible, the subsystem CLVFs and their "admissible control sets" are used to generate a Lipschitz continuous CLF for the original system. Optimal Controller Synthesis: A quadratic program is used to synthesize an optimal controller that guarantees exponential stabilization of the system. The method is validated through several numerical examples, including 2D, 3D, and 10D systems, demonstrating significant computational efficiency compared to directly computing the CLVF for the high-dimensional system.
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Deeper Inquiries

How can the proposed method be extended to handle more general system structures beyond the coupled nonlinear form considered in this work

To extend the proposed method to handle more general system structures beyond the coupled nonlinear form considered in this work, we can explore the concept of system decomposition and admissible control sets in a broader context. One approach could involve adapting the decomposition technique to systems with more complex interconnections between subsystems. By identifying shared states and controls among multiple subsystems, we can develop a decomposition strategy that effectively separates the system into manageable components. Additionally, incorporating advanced mathematical tools such as graph theory or network analysis may help in identifying the underlying structure of interconnected systems and guiding the decomposition process. By enhancing the decomposition methodology to handle diverse system architectures, we can apply the proposed method to a wider range of high-dimensional systems with intricate relationships between subsystems.

What are the theoretical guarantees on the suboptimality of the reconstructed CLVF compared to the original CLVF when exact reconstruction is not possible

When exact reconstruction of the Control Lyapunov-Value Function (CLVF) is not possible, there are theoretical guarantees on the suboptimality of the reconstructed CLVF compared to the original CLVF. In such cases, the reconstructed CLVF may deviate from the original CLVF, leading to a Lipschitz continuous CLF instead of an exact match. The suboptimality arises due to the inability to precisely reconstruct the CLVF in the original state space, especially when shared controls or states are present in the system decomposition. However, despite this suboptimality, the reconstructed CLF still provides a viable solution for stabilizing the system within a certain domain. The guarantees ensure that even though the reconstructed CLVF may not be identical to the original CLVF, it remains effective in achieving stability and control objectives within the specified region of interest.

Can the ideas of system decomposition and admissible control sets be applied to other control synthesis problems beyond CLVF computation, such as optimal control or model predictive control

The ideas of system decomposition and admissible control sets can indeed be applied to various other control synthesis problems beyond Control Lyapunov-Value Function (CLVF) computation. For instance, in optimal control problems, the concept of system decomposition can help break down complex systems into more manageable subsystems, enabling the synthesis of optimal control strategies for each component. By leveraging admissible control sets, optimal control policies can be formulated to ensure system stability and performance while adhering to constraints on control inputs. Similarly, in model predictive control (MPC), system decomposition can facilitate the implementation of predictive control schemes for large-scale systems by dividing them into interconnected subsystems. Admissible control sets can then be utilized to generate feasible control trajectories that optimize system behavior over a finite prediction horizon. Overall, the principles of system decomposition and admissible control sets offer a versatile framework for addressing a wide range of control synthesis challenges beyond CLVF computation.
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