As explorers of yore ventured from friendly harbors across unchartered waters, the creative imagination of early cartographers provided, at best, an outline of what might be expected, threatening dragons beyond the boundaries of the known world. The pioneers of discovery steamed ahead, unimpressed by the hidden dangers. They sought to expand their known universe and to create new fertile ground for their fellow travelers. What they brought home was often a mixture of verities and conjectures, but their stories inspired and attracted followings. When speaking of what they had seen, their descriptions varied greatly, causing narratives to bifurcate in directions that appeared to refer to entirely different experiences. And as the confusion threatened to grow further, myth and magic were dispelled only by painstaking and systematic verification and organization. While perhaps a task less glamorous, this was as much a heroic undertaking as the original journey.
And so, as modern-day explorers probe the endless frontiers, not of earth-bound geography, but of science, no significant progress is possible without equally heroic efforts to assign structure and order to their discovery. Such is the purpose of the textbook Hybrid Dynamical Systems, co-authored by a multidisciplinary team in mathematics, aerospace and mechanical engineering, and electrical and computer engineering. Targeted at graduate students in engineering with an interest in dynamics and control, but equally valuable to experienced researchers in the modeling and analysis of applied dynamical systems, this text establishes a systematic framework for conceptualizing hybrid dynamical systems and their solutions. Armed with the rigor of a productive formalism, and not shying away from the analysis of set-valued functions and their graphs, the authors explore weak and strong notions of asymptotic stability, the existence and use of Lyapunov functions, the robustness of solutions to regularization, and the construction of conical approximations that accurately represent local behavior near an operating point.
Many a student will have found the properties of hybrid dynamical systems a fertile opportunity for unexpected discovery. Already in the first year of calculus, students are able to appreciate the paradox of a bouncing ball undergoing infinitely many impacts before coming to rest in finite time! Such Zeno-type behavior, named for the author of the fable about Achilles’s unfortunate struggle with a tortoise, strains our sense of what is proper and reasonable for a physical system. The possible existence of reverse Zeno-type behavior, in which an accumulation point of impacts is found in backward time, is possibly blasphemous, until shown to occur in classical models of rigid-body mechanics with frictional collisions [1]. Equally fascinating are the many bifurcation scenarios that have been shown to be associated with the onset of degenerate interactions of equilibria and periodic orbits with system discontinuities, e.g., in models of biological reset oscillators and power-electronic converters [2].
Conceived as amalgamates of continuous-time and discrete-time dynamics, examples of hybrid systems are common both in the study of natural phenomena, in which a separation of time scales suggests a division between flows and jumps, and in the design of engineered systems, in which discreteness is often overlaid on an underlying flow in order to regulate and stabilize a desired behavior. As with the Scandinavian saying regarding the many names of a dear child, hybrid dynamical systems come in the guise of piecewise-smooth systems, systems with state resets or impulses, switching systems, hybrid automata, quantized control systems, Filippov systems, and so on. That these various modeling frameworks can be collected under a uniform, and pleasingly simple, umbrella is the conclusion of the initial chapter of Hybrid Dynamical Systems.
In their formulation, the authors characterize changes in the system state as either governed by the flow map, a differential inclusion applicable on the flow set, or by the jump map, a difference inclusion applicable on the jump set. A solution to a hybrid dynamical system is then a special case of a hybrid arc—a function on the Cartesian product $\mathbb{R}_{\ge 0}\times\mathbb{N}$, whose domain is a “union of finite or infinite sequences of intervals $[t_j,t_{j+1}]\times\{j\}$ with the last interval (if existent) possibly of the form $[t_j, T)$ with $T$ finite or $T=\infty$”—which flows when it should, and jumps when it should. By counting both the elapsed time along segments in the flow set, as well as the number of jumps from points in the jump set, this solution concept includes Zeno-type arcs with a least upper bound for the values of $t$ that appear in their domain, but no such bound for the corresponding sequence of integers $j$, as well as solutions with bounded domain that are maximal in the sense of not being embeddable in a solution on a larger domain.
The authors proceed to demonstrate how the generalized solution concepts of Hermes and Krasovskii offer a path to exploring the effects of noise on a hybrid dynamical system, in which the flow and jump maps are defined in terms of physical measurements. Here, it is shown that for systems that are invariant under a regularization process, “limits of convergent sequences of solutions under vanishing perturbations are solutions.” Indeed, under such conditions of regularity, the analysis establishes the nominal well-posedness of the hybrid system, with desirable implications to the continuous dependence of solutions on initial conditions, as well as the local boundedness of solutions with initial conditions in a sufficiently small neighborhood of some compact set.
Although there is great value in the structure imposed by Hybrid Dynamical Systems on a topic of diverse and broad applicability, readers should beware that a careful reading will necessitate dedicated time. The content is laid out with great care, precision, and elegance, but specialized terminology and long-distance dependencies between chapters sometimes take a toll on the casual reader. A large collection of example systems provides welcome relief and elucidate an otherwise occasionally terse presentation. The authors should be commended for providing what is likely to serve as an essential fundamental reference to an exciting area of dynamics research that is only expected to increase in importance in the future.
[1] Nordmark, A., Dankowicz, H., Champneys, A., “Friction-induced reverse chatter in rigid-body mechanisms with impacts,” IMA J Appl Math, 2011.
[2] Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P., Piecewise-smooth Dynamical Systems, Springer-Verlag, 2008.