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On the Exponential Number of Connected Components for the Feasible Set of Optimal Decentralized Control Problems | IEEE Conference Publication | IEEE Xplore

On the Exponential Number of Connected Components for the Feasible Set of Optimal Decentralized Control Problems


Abstract:

The optimal decentralized control (ODC) problem is known to be NP-hard and many sufficient tractability conditions have been derived in the literature for its convex refo...Show More

Abstract:

The optimal decentralized control (ODC) problem is known to be NP-hard and many sufficient tractability conditions have been derived in the literature for its convex reformulations or approximations. To better understand the root cause of the non-existence of efficient methods for solving ODC, we propose a measure of problem complexity in terms of connectivity, and show that there is no polynomial upper bound on the number of connected components for the set of static stabilizing decentralized controllers. Specifically, we present a subclass of problems for which the number of connected components is exponential in the order of the system and, in particular, any point in each of these components is the unique solution of the ODC problem for some quadratic objective functional. The results of this paper have two implications. First, the recent effort in machine learning advocating the use of local search algorithms for non-convex problems, which has also been successful for the optimal centralized control problem, fails to work for ODC since it needs an exponential number of initializations. Second, no reformulation of the problem through a smooth change of variables can reduce the complexity since it maintains the number of connected components. On the positive side, we show that structural assumptions can reduce the connectivity complexity of ODC, one such structure is the system being highly damped.
Date of Conference: 10-12 July 2019
Date Added to IEEE Xplore: 29 August 2019
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Conference Location: Philadelphia, PA, USA

I. Introduction

Classical state-space solutions to optimal centralized control problems do not scale well as the dimension increases [1]. Moreover, structural constraints such as locality and delay are ubiquitous in real-world controllers. The optimal decentralized control problem (ODC) has been proposed in the literature to bridge this gap. On the one hand, ODC can have nonlinear optimal solutions even for linear systems and is NP-hard in the worst case [2], [3]. On the other hand, the existence of dynamic structured feedback laws is completely captured by the notion of fixed modes [4], and several works have discovered structural conditions on the system and/or the controller under which the ODC problem admits tractable solutions. The conditions include spatially invariance [5], partially nestedness [6], positiveness [7], and quadratic invariance [8]. More recently, the System Level Approach [9] has convexified structural constraints at the expense of working with a series of impulse response matrices. Promising approximation [10]–[12] and convex relaxation techniques [13]–[16] also exist in the literature.

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References

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