Squaring Down LTI Systems: A Geometric Approach

In this paper, the problem of reducing a given LTI system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory. First, it will be shown how an LTI system can be turned into a left invertible system, thus preserving key system properties like stabilizability, phase minimality, right invertibility, relative degree and infinite zero structure. Moreover, the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane. By duality, the scheme of a right inverter will be derived straightforwardly. Moreover, the squaring down problem will be addressed. In fact, when the left and right reduction procedures are applied together, a system with an unequal number of inputs and outputs is turned into a square and invertible system. Furthermore, as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems.