Constrained Dynamical Systems and Invariants

In this paper, which is largely review, I will discuss dissipative behavior in mechanical systems which preserve energy. The paper will encompass both the classical and quantum domains. In the classical context I will consider almost Poisson systems; systems which which have a Poisson braket which fails the Jacobi identity. This class of systems includes nonholonomic mechanical systems: systems with nonintegrable constraints such as rolling constraints. Such systems may either preserve or fail to preserve a natural measure. I will discuss also pure Hamiltonian systems such as the Toda lattice which can exhibit dissipative behavior in certain contexts as well as infinite-dimensional systems exhibiting radiative damping. In the quantum context I will discuss systems of quantum oscillators coupled to a heat bath, which also exhibit natural dissipative behavior.

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrodinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.

ABSTRACTIn this paper, we address the robust control design problem for nonlinear dynamical systems tracking unreliable reference signals. Specifically, we present robust model reference adaptive control laws that guarantee uniform ultimate boundedness of the trajectory tracking error for nonlinear plants that are affected by matched, unmatched, and parametric uncertainties, and are subject to constraints on the state space and the measured output. These control laws guarantee satisfactory results even in case the reference trajectory or the reference output signal do not verify the given constraints and hence, may draw the plant's trajectory or measured output outside their constraint sets. A numerical example involving the attitude control of a spacecraft illustrates the feasibility of the theoretical results presented.

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

Recently there has been increased research interest in the study of the hybrid dynamical systems (Sun & Ge, 2005) and (Li et al., 2005). These systems involve the interaction of discrete and continuous dynamics. Continuous variables take the values from the set of real numbers and the discrete variables take the values from finite set of symbols. The hybrid systems have the behaviour of an analog dynamic system before certain abrupt structural or operating conditions are changed. The event driven dynamics in hybrid control systems can be described using different frameworks from discrete event systems (Cassandras & Lafortune, 2008) such as timed automata, max-plus algebra or Petry nets. For dynamic systems whose component are dominantly discrete event, main tools for analysis and design are representation theory, supervisory control, computer simulation and verification. From the clasical control theory point of view, hybrid systems may be considered as a switching control between analog feedback loops. Generally, hybrid systems can achieve better performance then non-switching controllers because they can to reconfigure and reorganize their structures. For that is necessery correct coordination of discrete and analog control variables. The mathematical model for real process, generally, has the Hammerstein-Wiener form (Crama & Atkins, 2001) and (Zhao & Chen, 2006). It means that on the input and output of the process are present nonlinear elements (actuator and sensor). Here we will consider Hammerstein model which has the input saturation as nonlinear element. That is the most frequent nonlinearity encountered in practice (Hippe, 2006). Also, unmodeled dynamics with matching condition is present. As a control strategy will be used switching control. The switched systems can be viewed as higher abstraction of hybrid systems. The design of switching controllers having guaranted stability, known as the picewise linear LQ control (PLC), is first considered in (Wredenhagen & Belanger, 1994). The picewise linear systems are systems that have different linear dynamics in different regions of the continuous state space (Johansson, 2003). The PLC control has the associated switching surfaces in form of positively invariant sets and yields a relatively low-gain controller. In the LHG (low-and-high gain) design a low gain feedback law is first designed in such a way

The Hamiltonian structure for dynamic free boundary problems

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called “naive” approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis.

Carathèodory’s classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton–Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.

The development of an adaptive control system for the control of an uncertain aeroelastic system, in the presence of constraints on the leading- and trailing-edge control surface deflections, is the subject of this paper. The model describes the nonlinear plunge and pitch dynamics of a prototypical wing section. The open-loop system exhibits limit-cycle oscillations beyond a critical freestream velocity. It is assumed that all the system parameters, except the signs of principal minors of the high-frequency gain matrix, are unknown, and external disturbance inputs are present. The objective is to design a saturating control system for suppressing the limit-cycle oscillations. An adaptive control system is designed for tracking reference pitch angle and plunge displacement trajectories. For avoiding singularity in the adaptive law, a decomposition of the input matrix is used for the design. For the analysis of the effect of input saturation, an auxiliary dynamic system is introduced. By the Lyapunov stability analysis, it is shown that the trajectories of the closed-loop system are uniformly ultimately bounded. Simulation results are presented that show suppression of oscillatory responses despite symmetric and nonsymmetric input constraints, uncertainties in the system parameters, and wind gust.

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

The problem of defining Poisson brackets for a degenerate Lagrangian without introduction of canonical variables is discussed. More generally, we introduce and give a complete geometrical description of a class of Poisson brackets on a presymplectic manifold. The construction is illustrated by both finite-dimensional and field-theoretic examples.

The formulation of Action Principles in Physics, and the introduction of the Hamiltonian framework, reduced dynamics to bracket algebra of observables. Such a framework has great potentialities, to understand the role of symmetries, or to give rise to the quantization rule of modern microscopic Physics. Conservative systems are easily algebrized via the Hamiltonian dynamics: a conserved observable H generates the variation of any quantity f via the Poisson bracket {f,H}. Recently, dissipative dynamical systems have been algebrized in the scheme presented here, referred to as metriplectic framework: the dynamics of an isolated system with dissipation is regarded as the sum of a Hamiltonian component, generated by H via a Poisson bracket algebra; plus dissipation terms, produced by a certain quantity S via a new symmetric bracket. This S is in involution with any other observable and is interpreted as the entropy of those degrees of freedom statistically encoded in friction. In the present paper, the metriplectic framework is shown for two original “textbook” examples. Then, dissipative Magneto-Hydrodynamics (MHD), a theory of major use in many space physics and nuclear fusion applications, is reformulated in metriplectic terms.

A paradigm for joined Hamiltonian and dissipative systems

In the opening chapters of the book, we saw how a variety of conservative phenomena could be described through the Poisson bracket of classical physics. As already mentioned, the majority of systems with which engineers and physicists must deal reveal dissipative phenomena inherent to their nature. In the last chapter we offered a concise overview of nonequilibrium thermodynamics as traditionally applied to describe, close to equilibrium, dissipative dynamic phenomena. In this chapter, we lay the groundwork for the incorporation of non-conservative effects into an equation of Hamiltonian form, and show several well-known examples in way of proof that such a thesis is, in fact, tenable. The unified formulation of conservative and dissipative processes based on the fundamental equations of motion is not a new idea. Among the most complete treatments is the one offered by the Brussels school of thermodynamics [Prigogine et al., 1973; Prigogine, 1973; Henin, 1974] based on a modified Liouville/von Neumann equation. This is a seminal work where, using the mathematical approach of projection operators and relying only on first principles, it is demonstrated how large isolated dynamic systems may present dissipative properties in some asymptotic limit. A key characteristic of their theory is that dissipative phenomena arise spontaneously without the need of any macroscopic assumptions, including that of local equilibrium [Prigogine et al., 1973, p. 6]. The main value, however, is mostly theoretical, demonstrating the compatibility of dissipative, irreversible, processes with the reversible dynamics of elemental processes through a “symmetry-breaking process.” The value of the theory in applications is limited since it relies on a quantum mechanical equation for the density matrix ρ which is, in general, very difficult to solve except for highly simplified problems [Henin, 1974]. In a nutshell, we hope to offer with the present work a macroscopic equivalent of the Brussels school theory which, at the expense of the introduction of the local equilibrium assumption, attempts to unify the description of dynamic and dissipative phenomena from a continuum, macroscopic viewpoint. The main tool to achieve that goal is the extension of the Poisson bracket formalism, analyzed in chapter 5, to dissipative continua.

Dynamically orthogonal field equations for continuous stochastic dynamical systems