Sensor and Actuator Placement for Large-Scale Systems: A Projection-Based Formulation

The problem of sensor and actuator placement aims at achieving near-optimal control performance with only a small number of sensors or actuators. In this paper we present a new formulation and design for the problem of sensor and actuator placement as a structural-constrained H₂ optimization. The structural constraint is defined by two projections which indicate the placement of sensors and actuators respectively. The advantage of this formulation is that the associated structural constraint always satisfies quadratic invariance condition, and allows convex reformulation for the constrained H₂ optimization in model matching form. We provide an optimality gap between the constrained H₂ problem with selected sensors and actuators and the unconstrained H₂ problem with full feedback, and subsequently propose the minimization problem to tighten the gap. The overall computational complexity for our proposed design is shown to be O(n³), which can be very scalable for large-scale applications. We illustrate our design with simulations of a IEEE prototype power system model.

I. Introduction

With the advances in cyber-physical technology, the physical systems facilitating modern world have become more and more connected, while the growth in scales and dimensionalities in these systems have made the control and monitoring more difficult. Conventional control mechanisms utilize all available inputs and outputs from system specifications. The problem is that as the number of inputs or outputs scales up with system size, implementing the control would necessitate a large volume of sensors and actuators for measuring the outputs and feeding the control inputs, as well as demand an expensive bandwidth for all-to-all communication between sensors and actuators. Moreover, large-scale network systems such as power systems often exhibit a low-rank coherent dynamics. In this sense, employment of full outputs or inputs may not be necessary for the control objective. This has led to the problem of sensor and actuator placement, that is to employ only a limited number or preferably a small number of sensors or actuators to achieve a near optimal performance of a controller as of full feedback.