Dissipative systems have played an important role in the analysis and synthesis of dynamical systems. The commonly used definition of dissipativity often requires an assumption on the controllability of the system. In this paper we use a definition of dissipativity that is slightly different (and less often used in the literature) to study a linear, time-invariant, possibly uncontrollable dynamical system. We provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not "mixed"; i.e., no pair of uncontrollable poles is symmetric about the imaginary axis. This condition is known to be related to the solvability of a Lyapunov equation; we link Lyapunov functions for autonomous systems to storage functions of an uncontrollable system. The set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature. We show that for an uncontrollable system the set of storage functions is unbounded, and that the unboundedness arises precisely due to the set of Lyapunov functions for an autonomous linear system being unbounded. Further, we show that stabilizability of a system results in this unbounded set becoming bounded from below. Positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive. In this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions. In most of the results in this paper, we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis; we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity.
Olfaction is not as well-understood as vision or audition, nor technologically addressed. Here, Chemical Graph Theory is shown to connect the vibrational spectrum of an odorant molecule, invoked in the Vibration Theory of Olfaction, to its structure, which is germane to the orthodox Shape Theory. Atomistic simulations yield the Eigen-VAlue (EVA) vibrational pseudo-spectra for 20 odorant molecules grouped into 6 different ‘perceptual’ classes by odour. The EVA is decomposed into peaks corresponding to different types of vibrational modes. A novel secondary pseudo-spectrum, informed by this physical insight—the Peak-Decomposed EVA (PD-EVA)—has been proposed here. Unsupervised Machine Learning (spectral clustering), applied to the PD-EVA, clusters the odours into different ‘physical’ (vibrational) classes that match the ‘perceptual’, and also reveal inherent perceptual subclasses. This establishes a physical basis for vibration-based odour classification, harmonizes the Shape and Vibration theories, and points to vibration-based sensing as a promising path towards a biomimetic electronic nose.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.