Minimal Actuator Placement with Bounds on Control Effort (1409.3289v5)

Abstract: We address the problem of minimal actuator placement in linear systems so that the volume of the set of states reachable with one unit or less of input energy is lower bounded by a desired value. 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(logn), 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 to maximize the volume of the set of states reachable with one unit or less of input energy. While this is also an NP-hard problem, we use our proposed algorithm to efficiently approximate its solutions as well.

• The paper establishes that finding minimal actuator sets under control effort bounds is NP-hard and proposes efficient approximation algorithms with provable guarantees.
• It leverages supermodular function theory and approximation algorithms to solve the minimal placement and cardinality-constrained optimization problems with optimal approximation factors up to O(log n).
• The research has practical implications for multi-agent systems, smart grids, robotics, and sensor networks, suggesting future work on distributed implementations and dynamic networks.

Minimal Actuator Placement with Bounds on Control Effort: An Expert Overview

This paper explores the intricate problem of actuator placement in linear control systems, specifically focusing on achieving minimal actuator configuration while maintaining bounds on control effort. The authors address two key problems: determining the minimal set of actuators ensuring a desired controllability energy bound and optimizing actuator placement given cardinality constraints to minimize control effort.

The problem of actuator placement is anchored in the concept of controlling large-scale systems where input energy efficiency is crucial. The paper begins by establishing that the task of finding a minimal number of actuators to maintain system controllability, while adhering to specified energy bounds, is NP-hard. This aspect underscores the computational complexity associated with actuator placement tasks in large-scale settings.

The authors proceed to develop approximation algorithms tailored to address these challenges. By leveraging a supermodular function approach, they propose efficient algorithms that provide provable guarantees of solution quality. For the minimal actuator placement problem subject to energy bounds, they introduce an auxiliary problem formulation using an $\epsilon$-perturbation to the energy metric. The algorithm efficiently approximates solutions to this auxiliary problem, ensuring that resultant actuator sets are controllable and meet the desired energy criteria.

One of the significant contributions of the paper is the derivation of an upper bound on the volume of reachable states by the selected actuator set, ensuring the control energy does not exceed certain bounds. The approximation factor achieved is shown to be optimal up to $O(\log n)$ in polynomial time, where $n$ represents the network size. This is noteworthy as it parallels the complexity considerations inherent in submodular set coverage problems.

For the second problem, concerning cardinality-constrained actuator placement for minimum control effort, the authors propose a bisection-type algorithm. By repeatedly executing their approximation algorithm for varying bounds, they efficiently determine the actuator set within cardinality constraints, ensuring the minimal control effort.

The practical implications of this research are manifold. In multi-agent networked systems, effective actuator placement ensures both the controllability and efficiency of control operations. The algorithms proposed could be beneficial in domains such as smart grids, robotics, and distributed sensor networks, where energy constraints and system controllability are critical considerations.

This paper sets the stage for further explorations into distributed implementations and the impact of network topology on actuator placement. Moreover, the supermodular function approach adopted here could inspire new techniques for similar NP-hard problems in control systems and beyond.

The paper's contributions are methodical, grounded in robust mathematical formulations and optimization strategies, making it a noteworthy reference point for researchers in control systems and networked system management. Future developments might consider extending these algorithms to account for dynamic environments or networks with evolving configurations, thereby enhancing their applicability in real-world settings.