Minimal Actuator Placement With Bounds on Control Effort

• Published in: IEEE transactions on control of network systems Vol. 3; no. 1; pp. 67 - 78
• Main Authors: Tzoumas, V., Rahimian, M. A., Pappas, G. J., Jadbabaie, A.
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01.03.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuators; Aerospace electronics; Algorithms; Approximation; Approximation algorithms; Approximation methods; Control systems; Controllability; Controllability Energy Metrics; Input Placement; Leader Selection; Linear systems; Measurement; Minimal Network Controllability; Multi-agent Networked Systems; Networks; Optimization; Placement; multiagent networked systems; leader selection; input placement; Controllability energy metrics; minimal network controllability
• ISSN: 2325-5870; 2372-2533
• DOI: 10.1109/TCNS.2015.2444031

We address the problem of minimal actuator placement in a linear system subject to an average control energy bound. First, following the recent work of Olshevsky, we prove that this is NP-hard. Then, we provide an efficient algorithm which, for a given range of problem parameters, approximates up to a multiplicative factor of O(log n), with n being the network size, any optimal actuator set that meets the same energy criteria; this is the best approximation factor one can achieve in polynomial time in the worst case. Moreover, the algorithm uses a perturbed version of the involved control energy metric, which we prove to be supermodular. Next, we focus on the related problem of cardinality-constrained actuator placement for minimum control effort, where the optimal actuator set is selected so that an average input energy metric is minimized. While this is also an NP-hard problem, we use our proposed algorithm to efficiently approximate its solutions as well. Finally, we run our algorithms over large random networks to illustrate their efficiency.