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The theory of dissipativity is well developed for controllable systems. A more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function, namely a function such that, along every system trajectory, its rate of change at each time instant is at most the power supplied to the system at that time. However, even when the supplied power is expressible in terms of just the external variables, the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable/hidden from the external variables: this paper investigates the key aspects of both cases, and also proposes another intuitive definition of dissipativity. These three definitions are compared: we show that drawbacks of one definition are addressed by another. Dealing first withobservablestorage functions, under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as frequency tends to infinity, we prove that the dissipativities of a system and its controllable part are equivalent. We use the behavioural approach for formalising key notions: a systembehaviouris the set of all system trajectories. We prove that storage functionshaveto be unobservable for ‘lossless’ uncontrollable systems. It is known, however, that unobservable storage functions result in certain ‘fallacious’ examples of lossless systems. We propose an intuitive definition of dissipativity: a system/behaviour is called dissipative if it can beembeddedin a controllable dissipativesuperbehaviour. We prove embeddability results and use them to resolve the fallacy in the example termed ‘lossless’ due to unobservable storage functions. We next show that, quite unreasonably, the embeddability definition admits behaviours that are both strictly dissipative and strictly antidissipative. Drawbacks of the embeddability definition in the context of RLC circuits are finally related to the inability to realise/synthesise the special one-port electrical network, called the nullator, using only passive electrical components. [ABSTRACT FROM PUBLISHER]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair . In order to do so, we construct a sequence of structured matrices from given system matrices and . Controllability of the pair is equivalent to a condition on the null-space dimension of an appropriate matrix in this sequence. We show that the dimension of the reachability space is also related to the above condition. Further, it is shown that the nearest Structured Low Rank Approximation (SLRA) of this structured matrix corresponds to a nearest uncontrollable system to the pair . [Copyright &y& Elsevier]