Uncontrollable dissipative systems: observability and embeddability
Abstract
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 with observable storage 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 system behaviour is the set of all system trajectories. We prove that storage functions have to 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 be embedded in a controllable dissipative superbehaviour. 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 oneport electrical network, called the nullator, using only passive electrical components.
 Publication:

International Journal of Control
 Pub Date:
 January 2014
 DOI:
 10.1080/00207179.2013.823668
 Bibcode:
 2014IJC....87..101K
 Keywords:

 algebraic Riccati equation;
 indefinite linear algebra;
 Lyapunov equation;
 storage function;
 Hamiltonian matrix;
 behavioural approach