A Structured Systems Approach for Optimal Actuator-Sensor Placement in Linear Time-Invariant Systems

TL;DR

This paper presents a polynomial-time method for designing minimal dedicated actuator configurations to ensure structural controllability in linear time-invariant systems, with extensions to sensor placement for observability.

Contribution

It introduces a polynomial complexity approach to determine minimal dedicated input configurations for structural controllability, including all possible minimal solutions.

Findings

Minimum number of dedicated inputs for controllability determined

All minimal input configurations characterized

Method extends to sensor placement for observability

Abstract

In this paper we address the actuator/sensor allocation problem for linear time invariant (LTI) systems. Given the structure of an autonomous linear dynamical system, the goal is to design the structure of the input matrix (commonly denoted by $B$) such that the system is structurally controllable with the restriction that each input be dedicated, i.e., it can only control directly a single state variable. We provide a methodology that addresses this design question: specifically, we determine the minimum number of dedicated inputs required to ensure such structural controllability, and characterize, and characterizes all (when not unique) possible configurations of the \emph{minimal} input matrix $B$. Furthermore, we show that the proposed solution methodology incurs \emph{polynomial complexity} in the number of state variables. By duality, the solution methodology may be readily extended to the structural design of the corresponding minimal output matrix (commonly denoted by $C$) that ensures structural observability.