Generic pole assignability, structurally constrained controllers and unimodular completion

In this paper we assume dynamical systems are represented by linear differential-algebraic equations (DAEs) of order possibly higher than one. We consider a structured system of DAEs for both the to-be-controlled plant and the controller. We model the structure of the plant and the controller as an undirected and bipartite graph and formulate necessary and sufficient conditions on this graph for the structured controller to generically achieve arbitrary pole placement. A special case of this problem also gives new equivalent conditions for structural controllability of a plant. Use of results in matching theory, and in particular, 'admissibility' of edges and 'elementary bipartite graphs', make the problem and the solution very intuitive. Further, our approach requires standard graph algorithms to check the required conditions for generic arbitrary pole placement, thus helping in easily obtaining running time estimates for checking this. When applied to the state space case, for which the literature has running time estimates, our algorithm is faster for sparse state space systems and comparable for general state space systems. The solution to the above problem also provides a necessary and sufficient condition for the following matrix completion problem. Given a structured rectangular polynomial matrix, when can it be completed to a unimodular matrix such that the additional rows that are added during the completion process are constrained to have zeros at certain locations. (C) 2013 Elsevier Inc. All rights reserved

Fast modes in the set of minimal dissipation trajectories

In this paper, we study the set of trajectories satisfying both a given LTI system's laws and also laws of the corresponding 'adjoint' system: in other words, trajectories in the intersection of the system's behavior and that of the adjoint system. This intersection has important system theoretic significance: for example, it is known that the trajectories in this intersection are the ones with minimal 'dissipation'. Underlying the notion of adjoint is that of a power supply: it is with respect to this supply rate that the trajectories in the intersection are known to be 'stationary'. In this paper, we deal with half-line solutions to the differential equations governing both the system and its adjoint. Analysis of half-line solutions plays a central role for example in initial value problems and in well-posedness studies of an interconnection. We interpret the set of half-line trajectories allowed by a system and its adjoint as an interconnection of these two systems, and thus address issues about well-posedness/ill-posedness of the interconnection. We formulate necessary and sufficient conditions for this intersection to be autonomous. For the case of an ill -posed interconnection and resulting autonomous system, we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system. We link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system. We show that absence of impulsive initial conditions is equivalent to the well-known subspace iteration algorithms for these subspaces converging in one step. (C) 2016 Elsevier B.V. All rights reserved