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

By Fu Lin, Makan Fardad, M. Jovanovi'c. IEEE Transactions on Automatic Control, 2011

496 citations recorded by the source corpus.

We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the H2 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.

Subfields: Computational Mechanics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/485aa5923b4753cfa4e76d29b950146cbe8efe29

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Citations in the corpus

Dynamical Modeling and Distributed Control of Connected and Automated Vehicles: Challenges and Opportunities (2017, methodology): “This captures the notion of coherence [40], [54].”
A Splitting Method for Optimal Control (2013, methodology): “Therefore, the total cost of finding the proximal step is no more than order T (n + m)3 flops, executed on a single processor; we can reduce this by employing multiple processors and carrying out the proximal optimizations in parallel.”
A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems (2017, methodology): “Particularly, sparsity-promoting approaches have been applied very successfully: the 1-regularized H2-control approach developed in [169] has been applied to WAC by means of PSS [282]–[284] and HVDC links [285], and it has also been adopted for pricing in WAC [286].”
Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding (2013, methodology): “These techniques have been used in a broad range of applications including imaging [20, 41, 61], control [3, 49, 57, 62], estimation [82], signal processing [22, 23, 25, 86], finance [9], distributed optimization [47, 65], and many others.”
A survey of machine learning for big data processing (2016, methodology): “However, vast experimental evidence in the literature supports empirical convergence and good performance of ADMM [74].”
Dynamic Mode Decomposition with Control (2014, background): “Sparse measurements have recently been leveraged in a variety of complex systems, some for control [3], [32], [25], [12].”
Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems (2014, background): “parse zero variance discriminant analysis [42], polynomial optimization [43], tensor decomposition [44], matrix separation [45], matrix completion [46], asset allocation [47], sparse feedback control [48] and so on. However, to the best of our knowledge, existing convergence analysis of ADMM for non”
Input-Output Analysis and Decentralized Optimal Control of Inter-Area Oscillations in Power Systems (2015, methodology): “To cope with these additional structural constraints, the sparsity-promoting approach of [23] has been recently augmented in [24].”