Showing papers on "Lyapunov function published in 1993"

Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.

Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

Abstract: In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

The general topology of dynamical systems

Abstract: Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of ""mathematical sophistication"", Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work.

Explicit construction of quadratic lyapunov functions for the small gain, positivity, circle, and popov theorems and their application to robust stability. part II: Discrete-time theory

Abstract: The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati-like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure-Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust stability. In the case of the Popov criterion, the Lyapunov function is a parameter-dependent quadratic Lyapunov function.

Lyapunov stability theory of nonsmooth systems

Abstract: This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filippov's differential inclusion and Clarke's generalized gradient (1983) are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control. >

The global dynamics of discrete semilinear parabolic equations

Abstract: A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $ The dynamical properties of various finite difference and finite element schemes for the equations are analysed The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied

Internal force-based impedance control for cooperating manipulators

Abstract: An internal force-based impedance control scheme for cooperating manipulators is introduced which controls the motion of the objects being manipulated and the internal force on the objects. The controller enforces a relationship between the velocity of each manipulator and the internal force on the manipulated objects. Each manipulator is directly given the properties of an impedance by the controller; thus, eliminating the gain limitation inherent in the structure of previously proposed schemes. The controller uses the forces sensed at the robot end effectors to compensate for the effects of the objects' dynamics and to compute the internal force using only kinematic relationships. Thus, knowledge of the objects' dynamics is not required. Stability of the system is proven using Lyapunov theory and simulation results are presented validating the proposed concepts. The effect of computational delays in digital control implementations is analyzed vis-a-vis stability and a lower bound derived on the size of the desired manipulator inertia relative to the actual manipulator endpoint inertia. The bound is independent of the sample time.

Topics in Control Theory

Abstract: 1 The problem of output regulation.- 1.1 Introduction.- 1.2 Problem statement.- 1.3 Output regulation via full information.- 1.4 Output regulation via error feedback.- 1.5 A particular case.- 1.6 Well-posedness and robustness.- 1.7 The construction of a robust regulator.- 2 Disturbance attenuation via H?-methods.- 2.1 Introduction.- 2.2 Problem statement.- 2.3 A characterization of the L2-gain of a linear system.- 2.4 Disturbance attenuation via full information.- 2.5 Disturbance attenuation via measured feedback.- 3 Full information regulators.- 3.1 Problem statement.- 3.2 Time-dependent control strategies.- 3.3 Examples.- 3.4 Time-independent control strategies.- 3.5 The local case.- 4 Nonlinear observers.- 4.1 Problem statement.- 4.2 Time-dependent observers.- 4.3 Error feedback regulators.- 4.4 Examples.- 5 Nonlinear H?-techniques.- 5.1 Introduction.- 5.2 Construction of the saddle-point.- 5.3 The local scenario.- 5.4 Disturbance attenuation via linearization.- A Matrix equations.- A.l Linear matrix equations.- A.2 Algebraic Riccati equations.- B Invariant manifolds.- B.l Existence theorem.- B.2 Outflowing manifolds.- B.3 Asymptotic phase.- B.5 A special case.- B.6 Dichotomies and Lyapunov functions.- C Hamilton-Jacobi-Bellman-Isaacs equation.- C.l Introduction.- C.2 Method of characteristics.- C.3 The equation of Isaacs.- C.4 The Hamiltonian version of Isaacs' equation.

Robust stability of uncertain time-delay systems and its stabilization by variable structure control

Abstract: In this paper, two main results for uncertain time-delay systems are derived. The first result is the presentation of a new robust stability criterion for uncertain time-delay systems. The second result is the successful application of the concept of variable structure control to the stabilization problem of uncertain time-delay systems. The robustness and stability degree of perturbed systems in the sliding mode are also discussed. Last, some examples are included to illustrate our results.

Sliding-mode controller design for spacecraft attitude tracking maneuvers

Abstract: A robust sliding-mode control law that deals with spacecraft attitude tracking problems is presented. Two important natural properties related to the spacecraft model of motion are discussed. It is shown that by using these properties and the second method of Lyapunov theory, the system stability in the sliding mode can be easily achieved. The success of the sliding-mode controller and its robustness relating to uncertainties are illustrated by an example of multiaxial attitude tracking maneuvers. >

Control of Uncertain Nonlinear Systems

Abstract: This paper describes some of my research in the analysis and control of nonlinear uncertain systems in which the uncertainties are modeled deterministically rather than stochastically. The main applications are to mechanical/aerospace systems, such as robots and spacecraft; the underlying theoretical approach is based on Lyapunov theory

Stability Analysis in Terms of Two Measures

Abstract: Part 1 Basic theory: definitions of stability basic Lyapunov theory comparison method converse theorem boundedness and Lagrange stability invariance principle. Part 2 Refinements: several Lyapunov functions perturbations of Lyapunov functions method of vector Lyapunov functions perturbed systems integral stability method of higher derivatives cone-valued Lyapunov functions. Part 3 Extensions: delay differential equations impulsive differential systems stabilization of control systems impulsive integro-differential systems discrete systems random differential systems dynamic systems on time scales. Part 4 Applications: holomorphic mechanical systems motion of winged aircraft models from economics motion of a length-varying pendulum population models angular motion of rigid bodies.

Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators

Abstract: A Lyapunov function for the phase-locked state of the Kuramoto model of non-linearly coupled oscillators is presented. It is also valid for finite-range interactions and allows the introduction of thermodynamic formalism such as ground states and universality classes. For the Kuramoto model, a minimum of the Lyapunov function corresponds to a ground state of a system with frustration: the interaction between the oscillators,XY spins, is ferromagnetic, whereas the random frequencies induce random fields which try to break the ferromagnetic order, i.e., global phase locking. The ensuing arguments imply asymptotic stability of the phase-locked state (up to degeneracy) and hold for any probability distribution of the frequencies. Special attention is given to discrete distribution functions. We argue that in this case a perfect locking on each of the sublattices which correspond to the frequencies results, but that a partial locking of some but not all sublattices is not to be expected. The order parameter of the phase-locked state is shown to have a strictly positive lower bound (r ⩾ 1/2), so that a continuous transition to a nonlocked state with vanishing order parameter is to be excluded.

Shunting inhibitory cellular neural networks: derivation and stability analysis

Abstract: A class of biologically inspired cellular neural networks (CNNs) is introduced that possess lateral interactions of the shunting inhibitory type only; hence, they are called shunting inhibitory cellular neural networks (SICNNs). Their derivation and biophysical interpretation are presented along with a stability analysis of their dynamics. In particular, it is shown that the SICNNs are bounded input bounded output stable dynamical systems. Furthermore, a global Lyapunov function is derived for symmetric SICNNs. Using the LaSalle invariance principle, it is shown that each trajectory converges to a set of equilibrium points; this set consists of a unique equilibrium point if all inputs have the same polarity. >

Quadratic stochastic operators, lyapunov functions, and tournaments

Abstract: A class of quadratic stochastic operators acting in a finite-dimensional simplex is distinguished that has trajectory at any point of the simplex behaving in a nonregular fashion as a rule. For the discrete dynamical systems generated by such operators the existence of a Lyapunov function of the form is proved, and an algorithm for finding the numbers is indicated. Upper estimates are obtained for the set of limit points of the trajectories. It is proved that the negative trajectories exist and converge. The question of the number of isolated fixed points of the operators in the distinguished class is considered. The connection between discrete dynamical systems and the theory of tournaments is also studied.Bibliography: 12 titles.

Generalized spectral decompositions of mixing dynamical systems

Abstract: We introduce a method for the explicit computation of the eigenvalue problem of the evolution operator of mixing dynamical systems. The method is based on the subdynamics decomposition of the Brussels–Austin groups directed by Professor I. Prigogine. We apply the method to three different representatives of mixing systems, namely, the Renyi maps, baker's transformations, and the Friedrichs model. The obtained spectral decompositions acquire meaning in suitable rigged Hilbert spaces that we construct explicitly for the three models. The resulting spectral decompositions show explicitly the intrinsic irreversibility of baker's transformations and Friedrichs model and the intrinsically probabilistic characters of the Renyi maps and baker's transformations. The dynamical properties are reflected in the spectrum because the eigenvalues are the powers of the Lyapunov times for the Renyi and baker systems and include the lifetimes for the Friedrichs model. © 1993 John Wiley & Sons, Inc.

Stochastic stability analysis for continuous-time fault tolerant control systems

Abstract: Active fault tolerant control systems are feedback control systems that reconfigure the control law in real time based on the response from an automatic failure detection and identification (FDI) scheme. The dynamic behaviour of such systems is characterized by stochastic differential equations because of the random nature of the failure events and the FDI decisions. The stability analysis of these systems is addressed in this paper using stochastic Lyapunov functions and supermartingale theorems. Both exponential stability in the mean square and almost-sure asymptotic stability in probability are addressed. In particular, for linear systems where the coefficients of the closed loop system dynamics are functions of two random processes with markovian transition characteristics (one representing the random failures and the other representing the FDI decision behaviour), necessary and sufficient conditions for exponential stability in the mean square are developed.

Robust input-output feedback linearization

Abstract: To make input-output feedback linearization a practical and systematic design methodology for single-input nonlinear systems, two problems need to be addressed One is to handle systematically the difficulties associated with the internal dynamics or zero-dynamics when the relative degree is less than the system order The other is to account for the effect of model uncertainties in the successive differentiations of the output of interest While the first problem has recently received considerable attention, the second has been largely unexplored This paper represents a preliminary study of a systematic methodology to account robustly for parametric uncertainties in the original system model The approach is based on combining sliding control ideas with the recursive construction of a closed-loop Lyapunov function, and is illustrated with a simple example

Control System Analysis and Synthesis via Linear Matrix Inequalities

Abstract: A wide variety of problems in systems and control theory can be cast or recast as convex problems that involve linear matrix inequalities (LMIs). For a few very special cases there are "analytical solutions" to these problems, but in general they can be solved numerically very efficiently. In many cases the inequalities have the form of simultaneous Lyapunov or algebraic Riccati inequalities; such problems can be solved in a time that is comparable to the time required to solve the same number of Lyapunov or Algebraic Riccati equations. Therefore the computational cost of extending current control theory that is based on the solution of algebraic Riccati equations to a theory based on the solution of (multiple, simultaneous) Lyapunov or Riccati inequalities is modest. Examples include: multicriterion LQG, synthesis of linear state feedback for multiple or nonlinear plants ("multi-model control"), optimal transfer matrix realization, norm scaling, synthesis of multipliers for Popov-like analysis of systems with unknown gains, and many others. Full details can be found in the references cited.

Stability of queueing networks and scheduling policies

Abstract: Usually, the stability of queueing networks is established by explicitly determining the invariant distribution. However, except for product form queueing networks, such explicit solutions are rare. We develop here a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady-state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. For an open re-entrant line, we show that all stationary nonidling policies are stable for all load factors less than one. This includes the well known first-come-first-served (FCFS) policy. We determine a subset of the stability region for the Dai-Wang example (1991), for which they have shown that the Brownian approximation does not hold. In another re-entrant line, we show that the last-buffer-first-served (LBFS) and first-buffer-first-served (FBFS) policies are stable for all load factors less than one. Finally, for the Rybko-Stolyar example (1992), for which a subset of the instability region has been determined by them under a certain buffer priority policy, we determine a subset of the stability region. >

Nonquadratic cost and nonlinear feedback control

Abstract: Nonlinear controllers offer significant advantages over linear controllers in a variety of circumstances. Hence there has been significant interest in extending linear-quadratic synthesis to nonlinear-nonquadratic problems. The purpose of this paper is to review the current status of such efforts and to present, in a simplified and tutorial manner, some of the basic ideas underlying these results. Our approach focuses on the role of the Lyapunov function in guaranteeing stability for autonomous systems on an infinite horizon. Sufficient conditions for optimality are given in a form that corresponds to a steady-state version of the Hamilton-Jacobi-Bellman equation. These results are used to provide a simplified derivation of the nonlinear feedback controller obtained by Bass and Webber (1966)38 and to obtain a deterministic variation of the stochastic nonlinear feedback controller developed by Speyer (1976).45.

Internal dynamics of a wheeled mobile robot

Abstract: Since the dynamics of a wheeled mobile robot is nonlinear, the feedback linearization technique is commonly used to linearize the input-output map. The input-output linearized system has a nonlinear internal dynamics. In this paper, the internal dynamics of the mobile robot under the look-ahead control is first characterized. The look-ahead control takes the coordinates of a reference point in front of the mobile robot as the output equation. Using a novel Lyapunov function, the stability of the internal dynamics is then analyzed. In particular, it is shown that the internal motion of the mobile robot is asymptotically stable when the reference point is commanded to move forward, whereas the internal motion is unstable when the reference point moves backward. Simulation and experimental results are provided to verify the analysis.

Parallel Algorithms for Optimal Control of Large Scale Linear Systems

Abstract: One - Theoretical Concepts.- 2. Linear-Quadratic Control Problems.- 2.1 Introduction.- 2.2 Recursive Methods for Singularly Perturbed Linear Continuous Systems.- 2.2.1 Parallel Algorithm for Solving Algebraic Lyapunov Equation.- 2.2.2 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.2.3 Case Study: Magnetic Tape Control Problem.- 2.3 Recursive Methods for Weakly Coupled Linear Continuous Systems.- 2.3.1 Parallel Algorithm for Solving Algebraic LyapIIDov Equation.- 2.3.2 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.4 Approximate Linear Regulator Problem for Continuous Systems.- 2.5 Recursive Methods for Singularly Perturbed Linear Discrete Systems.- 2.5.1 Parallel Algorithm for Solving Algebraic Lyapunov Equation.- 2.5.2 Case Study: An F-8 Aircraft.- 2.5.3 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.6 Approximate Linear Regulator for Discrete Systems.- 2.6.1 Case Study: Discrete Model of An F-8 Aircraft.- 2.7 Recursive Methods for Weakly Coupled Linear Discrete Systems.- 2.7.1 Parallel Algorithm for Solving Discrete Algebraic Lyapunov Equation.- 2.7.2 Case Study: Discrete Catalytic Cracker.- 2.7.3 Parallel Algorithm for Solving Algebraic Riccati Equation.- 2.7.4 Case Study: Discrete Model of a Chemical Plant.- 2.8 Notes and Comments.- 3. Decoupling Transformations.- 3.1 Introduction.- 3.2 Decoupling Transformation for Singularly Perturbed Linear Systems.- 3.3 Decoupling Transformation for Weakly Coupled Linear Systems.- 3.4 New Versions of Decoupling Transformations.- 3.4.1 New Decoupling Transformation for Linear Weakly Coupled System.- 3.4.2 New Decoupling Transformation for Linear Singularly Perturned Systems.- 3.5 Decomposition of the Differential Lyapunov Equations.- 3.6 Boundary Value Problem for Linear Continuous Weakly Coupled System.- 3.7 Boundary Value Problem for Linear Discrete-Time Weakly Coupled System.- 4. Output feedback control.- 4.1. Introduction.- 4.2 Output Feedback for Singularly Perturbed Linear Systems.- 4.3 Case Study: Fluid Catalytic Cracker.- 4.4 Output Feedback for Linear Weakly Coupled Systems.- 4.5 Case Study: Twelve Plate Absorption Column.- 5. Linear Stochastic Systems.- 5.1 Recursive Approach to Singularly Perturbed Linear Stochastic Systems.- 5.2 Case Study: F-S Aircraft LQG Controller.- 5.3 Recursive Approach to Weakly Coupled Linear Stochastic system.- 5.4 Case Study: Electric Power System.- 5.5 Parallel Reduced-Order Controllers for Stochastic Linear Discrete Singularly Perturbed Systems.- 5.6 Case Study: Discrete Steam Power System.- 5.7 Linear-Quadratic Gaussian Control of Discrete Weakly Coupled Systems at Steady State.- 5.8 Case Study: Distillation Column.- Appendix 5.1.- 6. Open-Loop Optimal Control Problems.- 6.1 Open-Loop Singularly Perturbed Control Problem.- 6.2 Case Study: Magnetic Tape Control.- 6.3 Open-Loop Weakly Coupled Optimal Control Problem.- 6.4 Case Study: Distillation Column.- 6.5 Open-Loop Discrete Singularly Perturbed Control Problem.- 6.6 Case Study: F-8 Aircraft Control Problem.- 6.7 Open-Loop Discrete Weakly Coupled Control Problem.- 6.8 Numerical Example.- 6.9 Conclusion.- Appendix 6.1.- Appendix 6.2.- Appendix 6.3.- Appendix 6.4.- 7. Exact Decompositions of Algebraic Riccati Equations.- 7.1 The Exact Decomposition of the Singularly Perturbed Algebraic Riccati Equation.- 7.2 Numerical Example.- 7.3 The Exact Decomposition of the Weakly Coupled Algebraic Riccati Equation.- 7.4 Case Study: A Satellite Control Problem.- 7.5 Conclusion.- Appendix 7.1.- Appendix 7.2.- Appendix 7.3.- 8. Differential and Difference Riccati Equations.- 8.1 Recursive Solution of the Singularly Perturbed Differential Riccati Equation.- 8.2 Case Study: A Synchronous Machine Connected to an Infinite Bus.- 8.3 Recursive Solution of the Riccati Differential Equation of Weakly Coupled Systems.- 8.4 Case Study: Gas Absorber.- 8.5 Reduced-Order Solution of the Singularly Perturbed Matrix Difference Riccati Equation.- 8.6 Case Study: Linearized Discrete Model of an F-8 Aircraft.- 8.7 Reduced-Order Solution of the Weakly Coupled Matrix Difference Riccati Equation.- 8.8 Numerical Example.- Appendix 8.1.- Appendix 8.2.- Appendix 8.3.- Appendix 8.4.- Two - Applications.- 9. Quasi Singularly Perturbed and Weakly Coupled Linear Systems.- 9.1 Linear Control of Quasi Singularly Perturbed Hydro Power Plants.- 9.2 Case Study: Hydro Power Plant.- 9.2.1 Weakly Controlled Fast Modes Structure.- 9.2.2 Strongly Controlled Slow Modes Structure.- 9.2.3 Weakly Controlled Fast Modes and Strongly Controlled Slow Modes Structure.- 9.3 Reduced-Order Design of Optimal Controller for Quasi Weakly Coupled Linear System.- 9.4 Case Studies.- 9.4.1 Chemical Reactor.- 9.4.2 F-4 Fighter Aircraft.- 9.4.3 Multimachine Power System.- 9.5 Reduced-Order Solution for a Class of Linear-Quadratic Optimal Control Problems.- 9.5.1 Numerical Example.- 9.6 Case Studies.- 9.6.1 Case Study 1: L-1011 Fighter Aircraft.- 9.6.2 Case Study 2: Distillation Column.- Notes.- Appendix 9.1.- 10. Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.1 Introduction.- 10.2 Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.3 Case Studies.- 10.3.1 Case Study 1: A Model of Supported Beam.- 10.3.2 Case Study 2: A Satellite Control Problem.- 10.4 Quasi Singularly Perturbed Weakly Coupled Linear Control Systems.- 10.5 Case Studies.- 10.6 Conclusion.- Appendix 10.1.- 11. Stochastic Output Feedback of Linear Discrete Systems.- 11.1 Introduction.- 11.2 Output Feedback of Quasi Weakly Coupled Linear Stochastic Discrete Systems.- 11.3 Case Study: Flight Control System for Aircrafts.- 11.4 Output Feedback of Singularly Perturbed Stochastic Discrete Systems.- 11.4.1 Problem Formulation.- 11.4.2 Slow-Fast Lower Order Decomposition.- 1111.5 Case Study: Discrete Model of a Steam Power System.- 12. Applications to Differential Games.- 12.1 Weakly Coupled Linear-Quadratic Nash Games.- 12.2 Solution of Coupled Algebraic Riccati Equations.- 12.2.1 Zeroth-Order Approximation.- 12.2.2 Solution of Higher Order of Accuracy.- 12.3 Numerical Examples.- Appendix 12.1.- Appendix 12.2.- 13. Recursive Approach to High Gain and Cheap Control Problems.- 13.1 Linear-Quadratic Cheap and High Gain Control Problems.- 13.1.1 High Gain Feedback Control.- 13.1.2 Cheap Control Problem.- 13.1.3 Parallel Algorithm for Solving Algebraic Riccati Equations for Cheap Control and High Gain Feedback.- 13.2 Case Study: Large Space Structure.- 13.3 Decomposition of the Open-Loop Cheap Control Problem.- 13.4 Numerical Example.- 13.5 Exact Decomposition of the Algebraic Riccati Equation for Cheap Control Problem.- 13.6 Numerical Example.- Appendix 13.1.- 14. Linear Approach to Bilinear Control Systems.- 14.1 Introduction.- 14.2 Reduced-Order Open Loop Optimal Control of Bilinear Systems.- 14.3 Reduced-Order Closed Loop Optimal Control of Bilinear Systems.- 14.3.1 Composite Near-Optimal Control of Bilinear Systems.- 14.4 Case Study: Induction Motor Drives.- 14.5 Near-Optimal Control of Singularly Perturbed Bilinear Systems.- 14.6 Optimal Control of Weakly Coupled Bilinear Systems.- 14.6.1 Open-Loop Control of Weakly Coupled Bilinear Systems.- 14.6.2 Closed-Loop Control of Weakly Coupled Bilinear Systems.- 14.7 Case Study: A Paper Making Machine.- 14.8 Conclusion.

Stability robustness of interval matrices via Lyapunov quadratic forms

Abstract: A sufficient condition for the robust stability of a class of interval matrices is derived using the Lyapunov approach. The matrices considered have elements which are nonlinear functions of a vector of independent and bounded parameters. The robust stability condition requires that a quadratic form be positive definite in a finite number of conspicuous points of an enlarged parameter space. >

Higher-dimensional targeting

Abstract: This paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations. Such a procedure is called targeting.'' Previous work on targeting for chaotic attractors has been in the context of one- and two-dimensional maps. Here it is shown that targeting can also be done in higher-dimensional cases. The method is demonstrated with a mechanical system described by a four-dimensional mapping whose attractor has two positive Lyapunov exponents and a Lyapunov dimension of 2.8. The target is reached by making very small successive changes in a single control parameter. In one typical case, 35 iterates on average are required to reach a target region of diameter 10[sup [minus]4], as compared to roughly 10[sup 11] iterates without the use of the targeting procedure.

Position and force control for constrained manipulator motion: Lyapunov's direct method

Abstract: A design procedure for simultaneous position and force control is developed, using Lyapunov's direct method, for manipulators in contact with a rigid environment that can be described by holonomic constraints. Many manipulators that interact with their environment require taking into account the effects of these constraints in the control design. The forces of constraint play a critical role in constrained motion and are, along with displacements and velocities, to be regulated at specified values. Lyapunov's direct method is used to develop a class of position and force feedback controllers. The conditions for gain selection demonstrate the importance of the constraints. Force feedback has been shown not to be mandatory for closed-loop stabilization, but it is useful in improving certain closed-loop robustness properties. >

Computer generated Lyapunov functions for a class of nonlinear systems

Abstract: The problem of constructing Lyapunov functions for a class of nonlinear dynamical systems is considered. The problem is reduced to the construction of a polytope satisfying some conditions. A generalization of the concept of sector condition that it makes possible to evaluate a given nonlinear function by using a set of piecewise-linear functions is proposed. This improvement greatly reduces the conservatism in the stability analysis of nonlinear systems. Two algorithms for constructing such polytopes are proposed, and two examples are shown to demonstrate the usefulness of the results. >