A Homotopy Theorem for Incremental Stability

Incremental stability is the notion that the distance between every pair of solutions to the system has stable behavior and approaches zero asymptotically. This paper introduces this notion for a class of hybrid systems. In particular, we define incremental stability as well as incremental partial stability, and study their properties. The approach used to derive our results consists of recasting the incremental stability problem as a set stabilization problem, for which the tools for asymptotic stability of hybrid systems are applicable. In particular, we propose an auxiliary hybrid system to study the stability of the diagonal set, which relates to incremental stability of the original system. The proposed notions are illustrated in examples throughout the paper.

The analysis of incremental stability typically involves measuring the distance between any two solutions of a given dynamical system at the same time instant, which is problematic when studying hybrid dynamical systems. Indeed, hybrid systems generate solutions defined with respect to hybrid time instances (that consists of both the continuous time elapsed and the discrete time, which is the number of jumps experienced so far), and two solutions of the same hybrid system may not be defined at the same hybrid time instant. To overcome this issue, we present novel definitions of incremental stability for hybrid systems based on graphical closeness of solutions. As we will show, defining incremental asymptotic stability with respect to the hybrid time yields a restrictive notion, such that we also investigate incremental asymptotic stability notions with respect to the continuous time only or the discrete time only, respectively. In this manner, two (effectively dual) incremental stability notions are attained, called jump- and flow incremental asymptotic stability. To present Lyapunov conditions for these two notions, in both cases, we resort to an extended hybrid system and we prove that the stability of a well-defined set for this extended system implies incremental stability of the original system. We can then use available Lyapunov conditions to infer the set stability of the extended system. Various examples are provided throughout this paper, including an event-triggered control application and a bouncing ball system with Zeno behavior, that illustrate incremental stability with respect to continuous time or discrete time, respectively.

This paper is concerned with global asymptotic output synchronization in networks of identical feedback systems. Using an operator theoretic approach based on an incremental small gain theorem, the method reformulates the synchronization problem as one of achieving incremental stability using a coupling operator that plays the role of an incrementally stabilizing feedback. In this way, conditions on static or dynamic coupling operators that achieve output synchronization of nodes of arbitrary structure are derived. These conditions lead to a methodology for the construction of coupling architectures that ensure output synchronization of a wide range of systems. The result is illustrated for a network of biochemical oscillators.

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

Constructive analysis of control system stability

A basic knowledge of stability of linear systems has been assumed throughout the previous chapters. Stability was associated with the location of the poles of the system in the left half plane. In chapter 6, we saw that the poles are the eigenvalues of the system matrix A when the system is written in state variable form. In chapter 7, we examined the Nyquist criterion for closed-loop stability of a SISO system; we concluded on the stability of the closed-loop system G(1 + G)-1 from the number of encirclements of —1 by the open-loop transfer function G(s). In this chapter, we examine the salient results of Liapunov’s theory of stability; it is attractive for mechanical systems, because of its exceptional physical meaning and its wide ranging applicability, especially for the analysis of nonlinear systems, and also in controller design. We will conclude this chapter with a class of collocated controls that are especially useful in practice, because of their guaranteed stability, even for nonlinear systems; we will call them energy absorbing controls. The following discussion will be restricted to time invariant systems (also called autonomous), but most of the results can be extended to time varying systems. As in the previous chapters, most of the general results are stated without demonstration and the discussion is focussed on vibrating mechanical systems; a deeper discussion can be found in the references.KeywordsEquilibrium PointAsymptotic StabilityPhase PortraitVibration ControlNegative Real PartThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In order to solve the system instability caused by a large number of constant power load access in the DC microgrid, an active damping control method based on the DC current feed forward of the grid connected converter is proposed. The mathematical model of DC-AC converter and the control strategy of active damping method suitable for DC-AC converter are studied in this paper, and a small signal mathematical model of DC microgrid is established. Based on the principle of system dominant pole distribution and impedance matching, the adverse effects of constant power load on the stability of the system and the improvement of the active damping method on the stability of the system are analyzed, and the design principle of the active damping parameters is given. Simulation and experimental results show that the proposed active damping method can improve the stability of DC microgrid system with constant power load and high permeability.

For any positive integer the author constructs a continuous mapping of the -dimensional Menger compactum onto itself that is universal in the class of mappings between -dimensional compacta, i.e., for any continuous mapping between -dimensional compacta there exist imbeddings of and in such that the restriction of to is homeomorphic to . The mapping plays the same role in the theory of Menger -dimensional manifolds as the projection plays in the theory of -manifolds ( is the Hilbert cube). It can be used to carry over the classical theorems in the theory of -manifolds to the theory of -manifolds:Stabilization Theorem. For any -manifold and any imbedding of in the space is homeomorphic to .Triangulation Theorem. For any -manifold there exists an -dimensional polyhedron such that the space is homeomorphic to for every imbedding of in .Bibliography: 20 titles.

Short-circuit faults in a receiving-end power system can lead to blocking events of the feed-in high-voltage direct-current (HVDC) systems, which may further result in system instability. However, security assessment methods based on the transient stability (TS) simulation can hardly catch the fault propagation phenomena between AC and DC subsystems. Moreover, effective emergency control strategies are needed to prevent such undesired cascading events. This paper focuses on power systems with multi-infeed HVDCs. An on-line security assessment method based on the electromagnetic transient (EMT)-TS hybrid simulation is proposed. DC and AC subsystems are modeled in EMTDC/PSCAD and PSS/E, respectively. In this way, interactions between AC and DC subsystems can be well reflected. Meanwhile, high computational efficiency is maintained for the on-line application. In addition, an emergency control strategy is developed, which coordinates multiple control resources, including HVDCs, pumped storages, and interruptible loads, to maintain the security and stability of the receiving-end system. The effectiveness of the proposed methods is verified by numerical simulations on two actual power systems in China. The simulation results indicate that the EMT-TS hybrid simulation can accurately reflect the fault propagation phenomena between AC and DC subsystems, and the coordinated emergency control strategy can work effectively to maintain the security and stability of systems.

In this work, mathematical models of drug delivery are presented. We are particularly interested in studying the stability of these models to release the drug in a polymer matrix and detect its transfer to the overall biological tissue. These results are illustrated on two models. In order to gain protocol treatment, instead of using the optimal control theory, Lyapunov’s stability theorem is used to study the stability of the first nonlinear system. For the second, we proceed by establishing the properties of the equilibrium point (which is strongly related to the stability of the nonlinear system) by modifying the system into a canonical equation and studying the spectrum of its Jacobian matrix to show that the system is stable.

The purpose of this article is to highlight the role of the Basel 3 proposals forthe prevention of banking crises to improve the stability of the financial system in thelight of financial globalization, where the banking system is one of the most the mostimportant financial parts of the system, and the stability of the system is linked to thestability of the banking system, and among the reasons that led The instability of thebanking system is the globalization of financial markets and the emergence of manyfinancial innovations. The growing trend towards the application of globalization andfinancial liberalization and the associated integration of markets and the free flow ofcapital to make the issue of financial stability stand on the list of countries' concerns, andundergoing total banking supervision is of great importance, including the BaselConvention 3, by experiences with the application of the policy of financial liberalizationThe banker confirmed to the monetary authorities and the supervisory bodies of thefinancial system that this policy should be combined with effective supervision of thebanking system to avoid financial crises.

In modern science technology and many motion process of engineering fields, such as neural network activity, movement of missile and spacecraft, control of robot and so on, there are many phenomena which are suddenly changed when their movement is disturbed in some time. The mathematical model of the instantaneous phenomena which we referred is impulsive differential equation. More and more control experts and mathematicians pay close attention to impulsive system, because the study of impulsive has the wide actual background and the value of the application. The robust control problems of the uncertain linear impulsive delay system are studied. Based on the stability of the impulsive delay system, we take advantage of the stability theory of Lyapunov, Lyapunov function and the technology of linear matrix inequality to design a robust feedback controller in order to eliminate the impact of pulse and time delay on the stability of the system. The controller can be got by the way of solving linear matrix inequality by using MATLAB toolbox. And some sufficient conditions of the robust exponential stability are proposed so that the system contains robust feedback control.

This paper reexamines the stability of uncertain closed-loop systems resulting from the nonsequential (NS) MIMO QFT design methodology. By combining the effect of satisfying both the robust stability and robust performance specifications in a NS MIMO QFT design, a proof for the stability of the uncertain closed-loop system is derived. The stability theorem proves that, subject to the satisfaction of a critical necessary and sufficient condition, the original NS MIMO QFT design methodology will provide a robustly stable closed-loop system. This necessary and sufficient condition provides a useful existence test for a successful NS MIMO QFT design. The results expose the salient features of the NS MIMO QFT design methodology. Two 2×2 MIMO design examples are presented to illustrate the key features of the stability theorem.

This study is concerned with stability analysis of non‐linear time‐varying systems by using Lyapunov function‐based approach. The classical Lyapunov stability theorems are generalised in the sense that the time‐derivative of the Lyapunov functions are allowed to be indefinite. The stability analysis is accomplished with the help of the scalar stable functions introduced in the author's previous study. Both asymptotic stability and input‐to‐state stability are considered. Particularly, for asymptotic stability, several concepts such as uniform and non‐uniform asymptotic stability, and uniform and non‐uniform exponential stability are studied. The effectiveness of the proposed theorems is illustrated by several numerical examples.

For networked sensor systems (NSSs) with hard and soft sensors including five uncertainties, two universal approaches of solving the robust fusion estimation problems are presented. It includes an integrated sequential covariance intersection (SCI) fusion minimax robust Kalman filtering approach with cross‐covariance information and a generalized Lyapunov equation approach with four pairs of Lyapunov equations. Applying them, the robust local and SCI fused time‐varying and steady‐state Kalman filters are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds. The equivalent batch SCI fusers are also presented. Their robustness and accuracy relations are proved, and the sensitivity of the SCI fuser with respect to the fused orders of sensors is analyzed. Applying the dynamic error system analysis method and the dynamic variance error system analysis method, a new convergence and absolute asymptotic stability theory of robust fusion Kalman filtering is presented. The classical Kalman filtering convergence and stability theory is developed. Compared with the original covariance intersection fuser, they significantly reduced the computational complexity and burden. Compared with the optimal and conservative SCI fusers, they significantly improved the robust accuracies. They are suitable to deal with asynchronous or random delayed data and are suitable for real‐time applications. A simulation applied to the two‐mass spring damper mechanical system shows their effectiveness.