Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers (1111.6188v3)

Abstract: We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the $H_2$ norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsity-promoting penalty functions into the optimal control problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize feedback gains subject to structural constraints determined by the identified sparsity patterns. In the first step, the sparsity structure of feedback gains is identified using the alternating direction method of multipliers, which is a powerful algorithm well-suited to large optimization problems. This method alternates between promoting the sparsity of the controller and optimizing the closed-loop performance, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of the sparsity-promoting penalty functions to decompose the minimization problem into sub-problems that can be solved analytically. Several examples are provided to illustrate the effectiveness of the developed approach.

• The paper introduces a two-phase method that first promotes sparsity through penalty functions and then optimizes feedback gains using ADMM.
• Numerical results demonstrate that using only 2% non-zero elements causes just an 8% performance drop compared to a centralized controller.
• The method effectively balances performance and communication cost, offering practical insights for distributed control in large-scale networks.

Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers

This paper addresses the challenge of designing sparse and block sparse feedback gains to minimize variance amplification in distributed systems using the $H_2$ norm. The authors present a two-phased approach to this problem. Initially, they focus on identifying sparsity patterns by incorporating sparsity-promoting penalty functions into the optimal control framework. Subsequently, they optimize feedback gains subject to these identified patterns.

Methodology

The proposed approach involves:

1. Structure Identification: The initial phase employs sparsity-promoting penalty functions, which are added to the optimal control problem to penalize the number of communication links in the controller. By increasing the weight of these penalty terms, the solution transitions from centralized to sparse feedback gains, balancing between performance and sparsity.
2. Optimization with Constraints: Once the sparsity structure is identified, feedback gains are optimized further to improve performance while respecting the identified structural constraints.

Implementation via ADMM

The core computational tool utilized is the Alternating Direction Method of Multipliers (ADMM). This method efficiently handles large-scale optimization problems by alternately focusing on promoting sparsity and optimizing performance. The paper leverages the separability of sparsity-promoting functions to decompose the problem into analytically solvable sub-problems. The following types of penalty functions are compared:

• Cardinality Function: Directly counts non-zero elements; however, it's non-convex.
• Weighted $\ell_1$ Norm: A convex relaxation preferred for its computational tractability.
• Sum-of-Logs: A non-convex alternative that offers a more aggressive means to encourage sparsity.

Numerical Results

The paper provides numerical examples demonstrating the effectiveness of the developed approach. In one scenario involving a mass-spring system, the authors showed that by using only 2% of the non-zero elements, the system's performance was only degraded by 8% compared to a centralized controller. The examples underscore the potential of obtaining significant sparsity with minimal loss of performance.

Implications and Future Directions

The results have practical implications for designing efficient distributed systems where communication cost, energy consumption, and computational load are of concern. The framework, though focused on the $H_2$ performance criterion in this paper, can be adapted to other optimal control settings. Notably, the success in utilizing ADMM in this context suggests broader applicability across various control design problems that require sparsity.

The methodologies presented hold promise for innovative developments in fields involving large-scale networks, such as social networks and biological systems. Potential future research directions could explore the application of these techniques to observer-based control systems or extend them to incorporate additional performance metrics besides the $H_2$ norm.

Overall, this paper contributes a robust framework and efficient methodology to the portfolio of tools available for optimal control system design in distributed settings.