Mathematics – Optimization and Control
Scientific paper
2011-10-10
Mathematics
Optimization and Control
26 pages, one figure. Partial results appeared in an IFAC conference (World Congress, Milan, Italy, 2011)
Scientific paper
The theory of dissipativity has been primarily developed for controllable systems/behaviors. For various reasons, in the context of uncontrollable systems/behaviors, a more appropriate definition of dissipativity is in terms of the dissipation inequality, namely the {\em existence} of a storage function. A storage function is a function such that along every system trajectory, the rate of increase of the storage function is at most the power supplied. While the power supplied is always expressed in terms of only the external variables, whether or not the storage function should be allowed to depend on unobservable/hidden variables also has various consequences on the notion of dissipativity: this paper thoroughly investigates the key aspects of both cases, and also proposes another intuitive definition of dissipativity. We first assume that the storage function can be expressed in terms of the external variables and their derivatives only and prove our first main result that, assuming the uncontrollable poles are unmixed, i.e. no pair of uncontrollable poles add to zero, and assuming a strictness of dissipativity at the infinity frequency, the dissipativities of a system and its controllable part are equivalent. We also show that the storage function in this case is a static state function. We then investigate the utility of unobservable/hidden variables in the definition of storage function: we prove that lossless autonomous behaviors require storage function to be unobservable from external variables. We next propose another intuitive definition: a behavior is called dissipative if it can be embedded in a controllable dissipative {\em super-behavior}. We show that this definition imposes a constraint on the number of inputs and thus explains unintuitive examples from the literature in the context of lossless/orthogonal behaviors.
Abdulrazak Rihab
Belur Madhu N.
Karikalan Selvaraj
No associations
LandOfFree
Dissipative systems: uncontrollability, observability and RLC realizability does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Dissipative systems: uncontrollability, observability and RLC realizability, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dissipative systems: uncontrollability, observability and RLC realizability will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-147661