Optimal actuator placement for linear systems with limited number of actuators

A completely controllable linear dynamical system can be steered from any given initial state to any specified final state with an application of input control energy. The input control energy is provided through a combination of actuators. It is desirable to have a limited number of actuators, which also presents the possibility of multiple actuator combinations that render the system completely controllable. Hence, the optimal actuator placement problem very important in system design. Previous studies have been mainly focused on solving the optimal actuator placement problem using greedy heuristic methods which can provide a sub-optimal solution. In this work, the optimal actuator placement problem is presented as a 0/1-mixed integer semidefinite programming problem, and is solved using the branch-and-bound procedure. The problem formulation can be applied to both stable and unstable systems, and the solution procedure does not require an initial controllable actuator combination (starting point). Although no theoretical guarantees regarding optimality of the computed solution is provided in this work, numerical simulations performed on two examples yield the global optimal solution for the optimal actuator placement problem.