Energy is one of the fundamental concepts in science and engineering practice, where it is common to view dynamical systems as energy-transformation devices. This perspective is particularly useful in studying complex nonlinear systems by decomposing them into simpler subsystems that, upon interconnection, add up their energies to determine the full system's behavior. The action of a controller may also be understood in energy terms as another dynamical system. The control problem can then be recast as finding a dynamical system and an interconnection pattern such that the overall energy function takes the desired form. This energy-shaping approach is the essence of passivity-based control (PBC), a controller design technique that is very well known in mechanical systems. Our objectives in the article are threefold. First, to call attention to the fact that PBC does not rely on some particular structural properties of mechanical systems, but hinges on the more fundamental (and universal) property of energy balancing. Second, to identify the physical obstacles that hamper the use of standard PBC in applications other than mechanical systems. In particular, we show that standard PBC is stymied by the presence of unbounded energy dissipation, hence it is applicable only to systems that are stabilizable with passive controllers. Third, to revisit a PBC theory that has been developed to overcome the dissipation obstacle as well as to make the incorporation of process prior knowledge more systematic. These two important features allow us to design energy-based controllers for a wide range of physical systems.

In the literature on dynamical systems analysis and the control of systems with complex behavior, the topic of synchronization of the response of systems has received considerable attention. This concept is revisited in the light of the classical notion of observers from (non)linear control theory,.

It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannon's statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a compact topological space. Using similar open cover techniques, the notion of topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a noncompact topological space with inputs generates stability information. It is then proven that a topological dynamical plant can be stabilized into a compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants in Euclidean space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.

Motivated by transient stability analysis of power systems, a framework for study of Lyapunov stability of equilibria in differential/algebraic (DA) systems is presented. Following a basic result on existence and uniqueness of solutions, it is easy to state general stability results. Several useful stability criteria for special DA structures are derived. One result for a Hamiltonian-type structure is applied to the study of undamped power systems.link_to_subscribed_fulltex