Showing papers on "Lyapunov function published in 2000"

Finite-Time Stability of Continuous Autonomous Systems

Abstract: Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

Fair end-to-end window-based congestion control

Abstract: In this paper, we demonstrate the existence of fair end-to-end window-based congestion control protocols for packet-switched networks with first come-first served routers. Our definition of fairness generalizes proportional fairness and includes arbitrarily close approximations of max-min fairness. The protocols use only information that is available to end hosts and are designed to converge reasonably fast. Our study is based on a multiclass fluid model of the network. The convergence of the protocols is proved using a Lyapunov function. The technical challenge is in the practical implementation of the protocols.

Perspectives and results on the stability and stabilizability of hybrid systems

Abstract: This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.

An integral inequality in the stability problem of time-delay systems

Abstract: An integral inequality is derived, and applied to the stability problem of time-delay systems using discretized Lyapunov functional formulation. As the result, a simpler stability criterion is derived.

Output feedback control of nonlinear systems using RBF neural networks

Abstract: An adaptive output feedback control scheme for the output tracking of a class of continuous-time nonlinear plants is presented. An RBF neural network is used to adaptively compensate for the plant nonlinearities. The network weights are adapted using a Lyapunov-based design. The method uses parameter projection, control saturation, and a high-gain observer to achieve semi-global uniform ultimate boundedness. The effectiveness of the proposed method is demonstrated through simulations. The simulations also show that by using adaptive control in conjunction with robust control, it is possible to tolerate larger approximation errors resulting from the use of lower order networks.

Robust control design using H-∞ methods

Abstract: 1. Introduction.- 1.1 The concept of an uncertain system.- 1.2 Overview of the book.- 2. Uncertain systems.- 2.1 Introduction.- 2.2 Uncertain systems with norm-bounded uncertainty.- 2.2.1 Special case: sector-bounded nonlinearities.- 2.3 Uncertain systems with integral quadratic constraints.- 2.3.1 Integral quadratic constraints.- 2.3.2 Integral quadratic constraints with weighting coefficients.- 2.3.3 Integral uncertainty constraints for nonlinear uncertain systems.- 2.3.4 Averaged integral uncertainty constraints.- 2.4 Stochastic uncertain systems.- 2.4.1 Stochastic uncertain systems with multiplicative noise.- 2.4.2 Stochastic uncertain systems with additive noise: Finitehorizon relative entropy constraints.- 2.4.3 Stochastic uncertain systems with additive noise: Infinite-horizon relative entropy constraints.- 3. H? control and related preliminary results.- 3.1 Riccati equations.- 3.2 H? control.- 3.2.1 The standard H? control problem.- 3.2.2 H? control with transients.- 3.2.3 H? control of time-varying systems.- 3.3 Risk-sensitive control.- 3.3.1 Exponential-of-integral cost analysis.- 3.3.2 Finite-horizon risk-sensitive control.- 3.3.3 Infinite-horizon risk-sensitive control.- 3.4 Quadratic stability.- 3.5 A connection between H? control and the absolute stabilizability of uncertain systems.- 3.5.1 Definitions.- 3.5.2 The equivalence between absolute stabilization and H? control.- 4. The S-procedure.- 4.1 Introduction.- 4.2 An S-procedure result for a quadratic functional and one quadratic constraint.- 4.2.1 Proof of Theorem 4.2.1.- 4.3 An S-procedure result for a quadratic functional and k quadratic constraints.- 4.4 An S-procedure result for nonlinear functionals.- 4.5 An S-procedure result for averaged sequences.- 4.6 An S-procedure result for probability measures with constrained relative entropies.- 5. Guaranteed cost control of time-invariant uncertain systems.- 5.1 Introduction.- 5.2 Optimal guaranteed cost control for uncertain linear systems with norm-bounded uncertainty.- 5.2.1 Quadratic guaranteed cost control.- 5.2.2 Optimal controller design.- 5.2.3 Illustrative example.- 5.3 State-feedback minimax optimal control of uncertain systems with structured uncertainty.- 5.3.1 Definitions.- 5.3.2 Construction of a guaranteed cost controller.- 5.3.3 Illustrative example.- 5.4 Output-feedback minimax optimal control of uncertain systems with unstructured uncertainty.- 5.4.1 Definitions.- 5.4.2 A necessary and sufficient condition for guaranteed cost stabilizability.- 5.4.3 Optimizing the guaranteed cost bound.- 5.4.4 Illustrative example.- 5.5 Guaranteed cost control via a Lyapunov function of the Lur'e-Postnikov form.- 5.5.1 Problem formulation.- 5.5.2 Controller synthesis via a Lyapunov function of the Lur'e-Postnikov form.- 5.5.3 Illustrative Example.- 5.6 Conclusions.- 6. Finite-horizon guaranteed cost control.- 6.1 Introduction.- 6.2 The uncertainty averaging approach to state-feedback minimax optimal control.- 6.2.1 Problem Statement.- 6.2.2 A necessary and sufficient condition for the existence of a state-feedback guaranteed cost controller.- 6.3 The uncertainty averaging approach to output-feedback optimal guaranteed cost control.- 6.3.1 Problem statement.- 6.3.2 A necessary and sufficient condition for the existence of a guaranteed cost controller.- 6.4 Robust control with a terminal state constraint.- 6.4.1 Problem Statement.- 6.4.2 A criterion for robust controllability with respect to a terminal state constraint.- 6.4.3 Illustrative example.- 6.5 Robust control with rejection of harmonic disturbances.- 6.5.1 Problem Statement.- 6.5.2 Design of a robust controller with harmonic disturbance rejection.- 6.6 Conclusions.- 7. Absolute stability, absolute stabilization and structured dissipativity.- 7.1 Introduction.- 7.2 Robust stabilization with a Lyapunov function of the Lur'e-Postnikov form.- 7.2.1 Problem statement.- 7.2.2 Design of a robustly stabilizing controller.- 7.3 Structured dissipativity and absolute stability for nonlinear uncertain systems.- 7.3.1 Preliminary remarks.- 7.3.2 Definitions.- 7.3.3 A connection between dissipativity and structured dissipativity.- 7.3.4 Absolute stability for nonlinear uncertain systems.- 7.4 Conclusions.- 8. Robust control of stochastic uncertain systems.- 8.1 Introduction.- 8.2 H? control of stochastic systems with multiplicative noise.- 8.2.1 A stochastic differential game.- 8.2.2 Stochastic H? control with complete state measurements.- 8.2.3 Illustrative example.- 8.3 Absolute stabilization and minimax optimal control of stochastic uncertain systems with multiplicative noise.- 8.3.1 The stochastic guaranteed cost control problem.- 8.3.2 Stochastic absolute stabilization.- 8.3.3 State-feedback minimax optimal control.- 8.4 Output-feedback finite-horizon minimax optimal control of stochastic uncertain systems with additive noise.- 8.4.1 Definitions.- 8.4.2 Finite-horizon minimax optimal control with stochastic uncertainty constraints.- 8.4.3 Design of a finite-horizon minimax optimal controller.- 8.5 Output-feedback infinite-horizon minimax optimal control of stochastic uncertain systems with additive noise.- 8.5.1 Definitions.- 8.5.2 Absolute stability and absolute stabilizability.- 8.5.3 A connection between risk-sensitive optimal control and minimax optimal control.- 8.5.4 Design of the infinite-horizon minimax optimal controller.- 8.5.5 Connection to H? control.- 8.5.6 Illustrative example.- 8.6 Conclusions.- 9. Nonlinear versus linear control.- 9.1 Introduction.- 9.2 Nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty.- 9.2.1 Problem statement.- 9.2.2 Output-feedback nonlinear versus linear control.- 9.2.3 State-feedback nonlinear versus linear control.- 9.3 Decentralized robust state-feedback H? control for uncertain large-scale systems.- 9.3.1 Preliminary remarks.- 9.3.2 Uncertain large-scale systems.- 9.3.3 Decentralized controller design.- 9.4 Nonlinear versus linear control in the robust stabilizability of linear uncertain systems via a fixed-order output-feedback controller.- 9.4.1 Definitions.- 9.4.2 Design of a fixed-order output-feedback controller.- 9.5 Simultaneous H? control of a finite collection of linear plants with a single nonlinear digital controller.- 9.5.1 Problem statement.- 9.5.2 The design of a digital output-feedback controller.- 9.6 Conclusions.- 10. Missile autopilot design via minimax optimal control of stochastic uncertain systems.- 10.1 Introduction.- 10.2 Missile autopilot model.- 10.2.1 Uncertain system model.- 10.3 Robust controller design.- 10.3.1 State-feedback controller design.- 10.3.2 Output-feedback controller design.- 10.4 Conclusions.- 11. Robust control of acoustic noise in a duct via minimax optimal LQG control.- 11.1 Introduction.- 11.2 Experimental setup and modeling.- 11.2.1 Experimental setup.- 11.2.2 System identification and nominal modelling.- 11.2.3 Uncertainty modelling.- 11.3 Controller design.- 11.4 Experimental results.- 11.5 Conclusions.- A. Basic duality relationships for relative entropy.- B. Metrically transitive transformations.- References.

Advances in linear matrix inequality methods in control

Abstract: Preface Notation Part I. Introduction. Robust Decision Problems in Engineering: A linear matrix inequality approach L. El Ghaoui and S.-I. Niculescu Part II. Algorithms and Software: Mixed Semidefinite-Quadratic-Linear Programs J.-P. A. Haeberly, M. V. Nayakkankuppam and M. L. Overton Nonsmooth algorithms to solve semidefinite programs C. Lemarechal and F. Oustry sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure S.-P. Wu and S. Boyd Part III. Analysis: Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach M. Fu and S. Dasgupta Optimization of Integral Quadratic Constraints U. Jonsson and A. Rantzer Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons F. Paganini and E. Feron Part IV. Synthesis. Robust H2 Control K. Y. Yang, S. R. Hall and E. Feron Linear Matrix Inequality Approach to the Design of Robust H2 Filters C. E. de Souza and A. Trofino Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings C. W. Scherer Advanced Gain-Scheduling Techniques for Uncertain Systems P. Apkarian and R. J. Adams Control Synthesis for Well-Posedness of Feedback Systems T. Iwasaki Part V. Nonconvex Problems. Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints K. M. Grigoriadis and E. B. Beran Bilinearity and Complementarity in Robust Control M. Mesbahi, M. G. Safonov and G. P. Papavassilopoulos Part VI. Applications:Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization J. Oishi and V. Balakrishnan Multiobjective Robust Control Toolbox for LMI-Based Control S. Dussy Multiobjective Control for Robot Telemanipulators J. P. Folcher and C. Andriot Bibliography Index.

Contractive model predictive control for constrained nonlinear systems

Abstract: This paper addresses the development of stabilizing state and output feedback model predictive control (MPC) algorithms for constrained continuous-time nonlinear systems with discrete observations. Moreover, we propose a nonlinear observer structure for this class of systems and derive sufficient conditions under which this observer provides asymptotically convergent estimates. The MPC scheme proposed consists of a basic finite horizon nonlinear MPC technique with the introduction of an additional state constraint, which has been called a contractive constraint. The resulting MPC scheme has been denoted contractive MPC. This is a Lyapunov-based approach in which a Lyapunov function chosen a priori is decreased, not continuously, but discretely; it is allowed to increase at other times. We show in this work that the implementation of this additional constraint into the online optimization makes it possible to prove strong nominal stability properties of the closed-loop system.

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

Abstract: In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network modeling of physical systems, is to embed the system into a larger Hamiltonian system for which a series of Casimir functions can be easily constructed. Interestingly enough, for linear systems the resulting Lyapunov function is the incremental energy; thus our derivations provide a physical explanation to it. An easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given. Some examples that illustrate the method are given.

Nonlinear control of interior permanent-magnet synchronous motor

Abstract: This paper presents a novel speed control technique for an interior permanent-magnet synchronous motor (IPMSM) drive based on newly developed adaptive backstepping technique. The proposed stabilizing feedback law for the IPMSM drive is shown to be globally asymptotically stable in the context of Lyapunov theory. The adaptive backstepping technique takes system nonlinearities into account in the control system design stage. The detailed derivations of the control laws have been given for controller design. The complete IPMSM drive incorporating the proposed backstepping control technique has been successfully implemented in real-time using digital signal processor board DS1102 for a laboratory 1-hp motor. The performance of the proposed drive is investigated both in experiment and simulation at different operating conditions. It is found that the proposed control technique provides a good speed tracking performance for the IPMSM drive ensuring the global stability.

A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions

Abstract: We consider dierential inclusions where a positive semidenite function of the solutions satises a class-KL estimate in terms of time and a second positive semidenite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-KL estimate, exists if and only if the class-KL estimate is robust, i.e., it holds for a larger, perturbed dierential inclusion. It remains an open question whether all class-KL estimates are robust. One sucient condition for robustness is that the original dierential inclusion is locally Lipschitz. Another sucient condition is that the two positive semidenite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for dierential equations and dierential inclusions that have appeared in the literature.

Parameterized LMIs in Control Theory

Abstract: A wide variety of problems in control system theory fall within the class of parameterized linear matrix inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. Such problems, though convex, involve an infinite set of LMI constraints and hence are inherently difficult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impact of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, $\mu$ analysis, and linear parameter-varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

Stability and stabilization of piecewise affine and hybrid systems: an LMI approach

Abstract: In this paper we present various algorithms both for stability analysis and state-feedback design for discrete-time piecewise affine systems. Our approach hinges on the use of piecewise quadratic Lyapunov functions that can be computed as the solution of a set of linear matrix inequalities. We show that the continuity of the Lyapunov function is not required in the discrete-time case. Moreover, the basic algorithms are made less conservative by exploiting the switching structure of piecewise affine systems and by using relaxation procedures.

Optimal design of CMAC neural-network controller for robot manipulators

Abstract: This paper is concerned with the application of quadratic optimization for motion control to feedback control of robotic systems using cerebellar model arithmetic computer (CMAC) neural networks. Explicit solutions to the Hamilton-Jacobi-Bellman (H-J-B) equation for optimal control of robotic systems are found by solving an algebraic Riccati equation. It is shown how the CMAC can cope with nonlinearities through optimization with no preliminary off-line learning phase required. The adaptive-learning algorithm is derived from Lyapunov stability analysis, so that both system-tracking stability and error convergence can be guaranteed in the closed-loop system. The filtered-tracking error or critic gain and the Lyapunov function for the nonlinear analysis are derived from the user input in terms of a specified quadratic-performance index. Simulation results from a two-link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances.

Lyapunov Characterizations of Input to Output Stability

Abstract: This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated.

A separation principle for dynamic positioning of ships: theoretical and experimental results

Abstract: Presents a globally asymptotically stabilizing (GAS) controller for regulation and dynamic positioning of ships, using only position measurements. It is assumed that these are corrupted with white noise hence a passive observer which reconstructs the rest of the states is applied. The observer produces noise-free estimates of the position, the slowly varying environmental disturbances and the velocity which are used in a proportional-derivative (PD)-type control law. The stability proof is based on a separation principle which holds for the nonlinear ship model. This separation principle is theoretically supported by results on cascaded nonlinear systems and standard Lyapunov theory, and it is validated in practice by experimentation with a model ship scale 1:70.

Non-linear iterative learning by an adaptive Lyapunov technique

Abstract: We consider the iterative learning control problem from an adaptive control viewpoint. It is shown that some standard Lyapunov adaptive designs can be modified in a straightforward manner to give a solution to either the feedback or feedforward ILC problem. Some of the common assumptions of non-linear iterative learning control are relaxed: e.g. we relax the common linear growth asssumption on the non-linearities and handle systems of arbitrary relative degree. It is shown that generally a linear rate of convergence of the MSE can be achieved, and a simple robustness analysis is given. For linear plants we show that a linear rate of MSE convergence can be achieved for non-minimum phase plants.

A receding horizon generalization of pointwise min-norm controllers

Abstract: Control Lyapunov functions (CLFs) are used in conjunction with receding horizon control to develop a new class of receding horizon control schemes. In the process, strong connections between the seemingly disparate approaches are revealed, leading to a unified picture that ties together the notions of pointwise min-norm, receding horizon, and optimal control. This framework is used to develop a CLF based receding horizon scheme, of which a special case provides an appropriate extension of Sontag's formula. The scheme is first presented as an idealized continuous-time receding horizon control law. The issue of implementation under discrete-time sampling is then discussed as a modification. These schemes are shown to possess a number of desirable theoretical and implementation properties. An example is provided, demonstrating their application to a nonlinear control problem. Finally, stronger connections to both optimal and pointwise min-norm control are proved.

Input-Output-to-State Stability

Abstract: This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function As corollaries, one shows the existence of "norm-estimators," and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates

Stable neural controller design for unknown nonlinear systems using backstepping

Abstract: We propose, from an adaptive control perspective, a neural controller for a class of unknown, minimum phase, feedback linearizable nonlinear system with known relative degree. The control scheme is based on the backstepping design technique in conjunction with a linearly parametrized neural-network structure. The resulting controller, however, moves the complex mechanics involved in a typical backstepping design from off-line to online. With appropriate choice of the network size and neural basis functions, the same controller can be trained online to control different nonlinear plants with the same relative degree, with semi-global stability as shown by the simple Lyapunov analysis. Meanwhile, the controller also preserves some of the performance properties of the standard backstepping controllers. Simulation results are shown to demonstrate these properties and to compare the neural controller with a standard backstepping controller.

Adaptive backstepping control of a class of chaotic systems

Abstract: This paper is concerned with the control of a class of chaotic systems using adaptive backstepping, which is a systematic design approach for constructing both feedback control laws and associated Lyapunov functions. Firstly, we show that many chaotic systems as paradigms in the research of chaos can be transformed into a class of nonlinear systems in the so-called nonautonomous "strict-feedback" form. Secondly, an adaptive backstepping control scheme is extended to the nonautonomous "strict-feedback" system, and it is shown that the output of the nonautonomous system can asymptotically track the output of any known, bounded and smooth nonlinear reference model. Finally, the Duffing oscillator with key constant parameters unknown, is used as an example to illustrate the feasibility of the proposed control scheme. Simulation studies are conducted to show the effectiveness of the proposed method.

Speed tracking control of a permanent-magnet synchronous motor with state and load torque observer

Abstract: This paper is concerned with the speed tracking control problem for a permanent-magnet synchronous motor (PMSM) in the presence of an unknown load torque disturbance. After a brief review of the mathematical model of the PMSM, a speed tracking control law using the exact linearization methodology is introduced. The tracking control algorithm is completed by adding an extended observer which provides, on the one hand, the motor speed and acceleration and, on the other hand, estimates the unknown load torque. The stability of the closed-loop system composed of a nonlinear speed tracking controller and an observer is studied by the way of Lyapunov theory. Furthermore, the decoupling of the state observer and the load torque observer is discussed. Finally, a real-time implementation and the experimental results of the proposed control strategy are presented.

Disturbance Observer Based Tracking Control

Abstract: A disturbance observer based tracking control algorithm is presented in this paper. The key idea of the proposed method is that the plant nonlinearities and parameter variations can be lumped into a disturbance term. The lumped disturbance signal is estimated based on a plant dynamic observer. A state observer then corrects the disturbance estimation in a two-step design. First, a Lyapunov-based feedback estimation law is used. The estimation is then improved by using a feedforward correction term. The control of a telescopic robot arm is used as an example system for the proposed algorithm. Simulation results comparing the proposed algorithm against a standard adaptive control scheme and a sliding mode control algorithm show that the proposed scheme achieves superior performance, especially when large external disturbances are present. @S0022-0434~00!00802-9# Tracking control for uncertain nonlinear systems with unknown disturbances is a challenging problem. To achieve good tracking under uncertainties, one usually needs to combine several or all of the following three mechanisms in the control design: adaptation, feedforward ~plant-inversion!, and high-gain, this paper is no exception. The tracking control of nonlinear systems under plant uncertainties and exogenous disturbances is studied in this paper. However, we will focus on the robotic examples for both literature review and numerical simulations. Many adaptive control schemes for robotic manipulators assume that the structure of the manipulator dynamics is known and/or the unknown parameters influence the system dynamics in an affine manner @1‐5#. There are several inherent difficulties associated with these approaches. First of all, the plant dynamic structure may not be known exactly. Second, it was demonstrated @6,7# that some of these designs may lack robustness against uncertainties. Recently, adaptive control algorithms requiring less model information were proposed @8‐11#. These algorithms adjust the control gains based on the system performance and thus are commonly referred to as performance-based adaptive control. These algorithms require little knowledge of system structures and parameter values. However, the control signal might become quite large. Plant-inversion based methods ~e.g., I/O linearization, backstepping!, roughly speaking, focus on the canceling of unwanted nonlinear dynamics. High-gain approaches such as sliding model controls could guarantee stability but, again, sometimes require very large control signals. While in some cases this may be a viable approach, in many other applications it may not be the best solution. In this paper, a disturbance-estimation based tracking control method is presented. Disturbance observer based control algorithms first appeared in the late 1980s @12#. Since then, they have been applied to many applications @13‐15#. Recently, the H‘ technique has been applied for the design of an optimal disturbance observer @16#. In this paper, we focus on the design for nonlinear systems. The magnitude of the disturbance is estimated based on the state estimation error in a two-step design. The estimated disturbance can then be used to improve the performance of literally any control algorithms. In this paper, a simple computed torque method is selected. The performance of the disturbanceobserver-enhanced method is then compared against those of a simple adaptive control and a simple robust control algorithm.

Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems

Abstract: The paper presents a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems.The effects of full, partial, anti-phase and in-phase-anti-phase chaotic synchronization are investigated via the linear invariant manifolds of the corresponding differential equations. Existence of various invariant manifolds, a self-similar behavior, a hierarchy and embedding of the manifolds of the coupled system are discovered. Sufficient conditions for the stability of the invariant manifolds are obtained via the method of Lyapunov functions. Conditions under which full global synchronization can not be achieved even for the largest coupling constant are defined. The general rigorous results are illustrated through examples of coupled Lorenz-like and coupled Rossler systems.