Theory Of Dissipative Systems Research Articles

Minimum theorems in incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. ( Nemat-Nasser et al., 1980; Nguyen, 2000; Maugin, 1992; Bourdin et al., 2008; Salvadori, 2008). This approach is here further advanced, by noting that Stress Intensity Factors (SIFs) asymptotic expansion ( Amestoy et al., 1986; Amestoy and Leblond, 1992) enjoys a Colonnetti’s decomposition ( Colonnetti, 1918; Colonnetti, 1950) interpretation. As a consequence, minimum theorems are derived in terms of crack tip “quasi static velocity”. They are reminiscent of Ceradini’s theorem ( Ceradini, 1965; Ceradini, 1966) in plasticity.

Optimal Robust Stabilization and Dissipativity Synthesis by Behavioral Interconnection

Given a nominal plant, together with a fixed neighborhood of this plant, the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood (in an appropriate sense). If a controller achieves this design objective, we say that it robustly stabilizes the nominal plant. In this paper we formulate the robust stabilization problem in a behavioral framework, with control as interconnection. We also formulate a relevant behavioral $\mathcal{H}_{\infty}$ synthesis problem, which will be instrumental in solving the robust stabilization problem. We use both rational and polynomial representations for the behaviors under consideration. Necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems. We will also find the optimal stability radius, i.e., the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller. This smallest upper bound is expressed in terms of certain storage functions associated with nominal control system.

A note on Artificial Intelligence techniques and Dissipativity-based Approach in Traffic Signal Control For an Over-Saturated Intersection

AbstractThe main motivation of this study is to take full advantage of the capabilities of neural networks techniques and dissipative system theory to design a powerful state feedback real-time controller for an over-saturated intersection. The developed controller makes a realtime decisions about whether to increase or decrease (and how much) the current green time in order to get out the over-saturation situation. First, a discrete-time model that describes the evolution of the queue lengths at signalized intersection is presented. Then, the control problem is formulated and solved by using the dissipative system theory. Moreover, the proposed intersection controller needs to be provided with a real-time input traffic flow data. To achieve this goal, we use the artificial neural networks technique as input traffic flow data predictor. The results of the simulations indicate that our control strategy guarantees a high degree of control benefit.

Quantum Dissipative Systems and Feedback Control Design by Interconnection

The purpose of this paper is to extend J.C. Willems' theory of dissipative systems to open quantum systems described by quantum noise models. This theory, which combines ideas from quantum physics and control theory, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of external systems, called exosystems, in terms of feedback networks of interconnected open quantum systems. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using network models for open quantum systems, which implements Willems' “control by interconnection” for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

A simple framework to justify linear response theory

The use of linear response theory for forced dissipative stochastic dynamical systems through the fluctuation dissipation theorem is an attractive way to study climate change systematically among other applications. Here, a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems is developed. The main results are formulated in an abstract setting and apply to suitable systems, in finite and infinite dimensions, that are of interest in climate change science and other applications.

Energy-Based Control For an Over-Saturated Three-phase Intersection

Using the energy concept, the traffic flow controlled by traffic lights on an oversaturated single intersection is addressed in this paper. By regarding the flow entry of vehicles as the energy stored in the system, we show that the dissipative system theory can be applied. As a result, a state feedback control law is found in terms of the solution of an associated dissipation equation that renders the system non-accumulative, Le., to get out the over-saturation situation. The simulation results indicate that our control strategy guarantees a high degree of control benefit.

On the work of Dolgopyat on partial and nonuniform hyperbolicity

This paper is a nontechnical survey and aims to illustrate Dolgopyat'sprofound contributions to smooth ergodic theory. I will discuss some ofDolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity withemphasis on the interaction between the two-the class of dynamical systemswith mixed hyperbolicity. On one hand, this includes uniformlypartially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in thecenter direction. The study of their ergodic properties has provided analternative approach to the Pugh-Shub stable ergodicity theory for bothconservative and dissipative systems. On the other hand, ideas of mixedhyperbolicity have been used in constructing volume-preservingdiffeomorphisms with nonzero Lyapunov exponents on any manifold.

Dissipative stochastic differential systems with risk-sensitive storage function and control design problems

We consider stochastic control-affine systems defined by a differential Ito-type equation. A new notion of dissipativity with risk-sensitive storage function is introduced, and corresponding theory of dissipative systems is developed. Problems of finding storage function and control which ensures dissipativity are explicitly solved. In a linear-quadratic case the final results are expressed in terms of linear matrix inequalities. Connections to risk-sensitivity, differential (stochastic and deterministic) games, and deterministic ℋ∞-control theory are established. An example is given.

Dissipativity preserving model reduction by retention of trajectories of minimal dissipation

We present a method for model reduction based on ideas from the behavioral theory of dissipative systems, in which the reduced order model is required to reproduce a subset of the set of trajectories of minimal dissipation of the original system. The passivity-preserving model reduction method of Antoulas (Syst Control Lett 54:361-374, 2005) and Sorensen (Syst Control Lett 54:347-360, 2005) is shown to be a particular case of this more general class of model reduction procedures.

Canonical active Brownian motion

Active Brownian motion is the complex motion of active Brownian particles. They are "active" in the sense that they can transform their internal energy into energy of motion and thus create complex motion patterns. Theories of active Brownian motion so far imposed couplings between the internal energy and the kinetic energy of the system. We investigate how this idea can be naturally taken further to include also couplings to the potential energy, which finally leads to a general theory of canonical dissipative systems. Explicit analytical and numerical studies are done for the motion of one particle in harmonic external potentials. Apart from stationary solutions, we study nonequilibrium dynamics and show the existence of various bifurcation phenomena.

Non-equilibrium statistical theory for electromigration damage

A non-equilibrium statistical theory for aging dissipative systems formed by atoms is developed and applied to electromigration damage in conductors. A modification of Black’s mean failure time is presented. We describe the experimental relationship between the initial rate of increase in the functional damage and the electric conductor lifetime. Also, for situations where electromigration damage is mainly due to heat dissipation caused by conduction inside the conductor, it was possible to obtain a mean-field theory which yields expressions (that agree with the experimental data) for the increase of resistance with the current stressing time, in test probes of different materials, and for the total volume of hillocks as a function of the current stressing time.

H∞ control for discrete‐time nonlinear switching systems

AbstractThis paper formulates and solves the robust H∞ control problem for discrete‐time nonlinear switching systems. The H∞ control problem is interpreted as the l2 finite gain control problem and is studied using a dissipative systems theory for switched systems. Both state and measurement feedback control problems are formulated as dynamic games and solved using dynamic programming. The partially observed dynamic game corresponding to the measurement feedback control problem is solved by transforming into a completely observed, full state infinite‐dimensional game problem using information states. Our results are illustrated with an example. Copyright © 2007 John Wiley & Sons, Ltd.

Dissipative Systems Analysis and Control: Theory and Applications (2nd Edition)-[Book review; B. Brogliato, R. Lozano, B. Maschke

This book presents a fully revised and expanded treatment of dissipative systems theory. It examines linear and nonlinear systems with examples of both in each chapter; some infinite-dimensional examples are also included. Throughout emphasis is placed on the use of the dissipative properties of a system for the design of stable feedback control laws. This second edition is substantially reorganized both to accommodate new material and to enhance its pedagogical properties. The main value of the book is that it us an up-to-date reference of dissipative systems in the framework of many system descriptions. It is a good roadmap to the many contributions on the subject. It is a good reference for dissipative systems for advanced engineering students and researchers in control, however quantity was obtained at the cost of pedagogical quality; an opportunity was missed to create an interesting textbook for these categories. The text is recommended for its vast amount of timely information and indications to the literature. However, the latter information could have been better filtered and structured to provide a unified basic deep knowledge of the field of dissipative systems, which can be mentally reconstructed by the student reader.

Algorithms for multidimensional spectral factorization and sum of squares

In this paper, algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates, and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems. These algorithms are based on the calculus of 2n-variable polynomial matrices and their associated quadratic differential forms, and share the common feature that the problems are lifted from the original n-variable polynomial context to a 2n-variable polynomial context. This allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities (LMI’s), to the feasibility of a semialgebraic set or to a linear eigenvalue problem.

Passivity-Based Dynamic Visual Feedback Control for Three-Dimensional Target Tracking: Stability and $L_{2}$-Gain Performance Analysis

This paper investigates vision-based robot control based on passivity for three-dimensional (3-D) target tracking. First, using standard body-attached coordinate frames (the world frame, camera frame, and object frame), we represent the relative position and orientation between a moving target and a camera as an element of SE(3). Using this representation we derive a nonlinear observer to estimate the relative rigid body motion from the measured camera data. We then establish the relationship between the estimation error in a 3-D workspace and in the image plane. We show passivity of the dynamic visual feedback system by combining the passivity of both the visual feedback system and the manipulator dynamics which allows us to prove stability in the sense of Lyapunov for the full 3-D dynamic visual feedback system. The L2 -gain performance analysis, which deals with the disturbance attenuation problem, is then considered via dissipative systems theory. Finally, experimental results are presented to verify the stability and L2-gain performance of the dynamic visual feedback system

Casimir force and complications in the van Kampen theory for dissipative systems

In dissipative media it appears as if there are distinct normal modes with complex valued energies. The summation of the zero-point energies of these modes render a complex valued result. Using the contour integration, resulting from the use of the generalized argument principle, gives a real valued and different result. This paper resolves this contradiction and shows that the true normal modes form a continuum with real frequencies. Furthermore, it suggests a way to obtain an approximate result from a summation of the zero-point energy of modes.

DISSIPATIVITY-BASED CONTROL FOR AN OVER-SATURATED SINGLE INTERSECTION

Using the energy concept, the traffic flow controlled by traffic lights on an over-saturated single intersection is addressed in this paper. By regarding the flow entry of vehicles as the energy stored in the system, we show that the dissipative system theory can be applied. As a result, a state feedback control law is found in terms of the solution of an associated dissipation equation that renders the system non-accumulative, i.e., to get out the over-saturation situation. The results of the simulations indicate that our control strategy guarantees a high degree of control benefit.

Analysis of input-to-state stability for discrete time nonlinear systems via dynamic programming

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L 2 -gain analysis and nonlinear H ∞ theory). As such, the results presented address an obvious gap in the literature.

Adaptive Robust Fuzzy Tracking Control for Pole Balancing Robots using Small Gain Design

Abstract A novel adaptive robust fuzzy tracking control (ARFTC) algorithm is proposed for a class of nonlinear systems with uncertain system and gain functions, which are all the unstructured (or non-repeatable) state-dependent functions coming from modelling errors and external disturbances. The Takagi-Sugeno type fuzzy logic systems are used to approximate uncertain functions and a systematic procedure is developed for synthesis of ARFTC by use of small gain approach and dissipative system theory. The algorithm preserves three advantages: (1) the asymptotic stability of adaptive control in the presence of unstructured uncertainties can be guaranteed, (2) possible controller singularity problem in some of existing adaptive control schemes using feedback linearization techniques can be avoided, (3) the adaptive mechanism with minimal learning parameterizations can be obtained. The effectiveness of the proposed ARFTC for the tracking control ofpole-balancing robots is demonstrated though numerical examples.

Spectral Theory of Time Dispersive and Dissipative Systems

We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system -- which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium.