Showing papers on "Lyapunov function published in 1992"

Sliding Modes in Control and Optimization

Abstract: I. Mathematical Tools.- 1 Scope of the Theory of Sliding Modes.- 1 Shaping the Problem.- 2 Formalization of Sliding Mode Description.- 3 Sliding Modes in Control Systems.- 2 Mathematical Description of Motions on Discontinuity Boundaries.- 1 Regularization Problem.- 2 Equivalent Control Method.- 3 Regularization of Systems Linear with Respect to Control.- 4 Physical Meaning of the Equivalent Control.- 5 Stochastic Regularization.- 3 The Uniqueness Problems.- 1 Examples of Discontinuous Systems with Ambiguous Sliding Equations.- 1.1 Systems with Scalar Control.- 1.2 Systems Nonlinear with Respect to Vector-Valued Control.- 1.3 Example of Ambiguity in a System Linear with Respect to Control ..- 2 Minimal Convex Sets.- 3 Ambiguity in Systems Linear with Respect to Control.- 4 Stability of Sliding Modes.- 1 Problem Statement, Definitions, Necessary Conditions for Stability ..- 2 An Analog of Lyapunov's Theorem to Determine the Sliding Mode Domain.- 3 Piecewise Smooth Lyapunov Functions.- 4 Quadratic Forms Method.- 5 Systems with a Vector-Valued Control Hierarchy.- 6 The Finiteness of Lyapunov Functions in Discontinuous Dynamic Systems.- 5 Singularly Perturbed Discontinuous Systems.- 1 Separation of Motions in Singularly Perturbed Systems.- 2 Problem Statement for Systems with Discontinuous control.- 3 Sliding Modes in Singularly Perturbed Discontinuous Control Systems.- II. Design.- 6 Decoupling in Systems with Discontinuous Controls.- 1 Problem Statement.- 2 Invariant Transformations.- 3 Design Procedure.- 4 Reduction of the Control System Equations to a Regular Form.- 4.1 Single-Input Systems.- 4.2 Multiple-Input Systems.- 7 Eigenvalue Allocation.- 1 Controllability of Stationary Linear Systems.- 2 Canonical Controllability Form.- 3 Eigenvalue Allocation in Linear Systems. Stabilizability.- 4 Design of Discontinuity Surfaces.- 5 Stability of Sliding Modes.- 6 Estimation of Convergence to Sliding Manifold.- 8 Systems with Scalar Control.- 1 Design of Locally Stable Sliding Modes.- 2 Conditions of Sliding Mode Stability "in the Large".- 3 Design Procedure: An Example.- 4 Systems in the Canonical Form.- 9 Dynamic Optimization.- 1 Problem Statement.- 2 Observability, Detectability.- 3 Optimal Control in Linear Systems with Quadratic Criterion.- 4 Optimal Sliding Modes.- 5 Parametric Optimization.- 6 Optimization in Time-Varying Systems.- 10 Control of Linear Plants in the Presence of Disturbances.- 1 Problem Statement.- 2 Sliding Mode Invariance Conditions.- 3 Combined Systems.- 4 Invariant Systems Without Disturbance Measurements.- 5 Eigenvalue Allocation in Invariant System with Non-measurable Disturbances.- 11 Systems with High Gains and Discontinuous Controls.- 1 Decoupled Motion Systems.- 2 Linear Time-Invariant Systems.- 3 Equivalent Control Method for the Study of Non-linear High-Gain Systems.- 4 Concluding Remarks.- 12 Control of Distributed-Parameter Plants.- 1 Systems with Mobile Control.- 2 Design Based on the Lyapunov Method.- 3 Modal Control.- 4 Design of Distributed Control of Multi-Variable Heat Processes.- 13 Control Under Uncertainty Conditions.- 1 Design of Adaptive Systems with Reference Model.- 2 Identification with Piecewise-Continuous Dynamic Models.- 3 Method of Self-Optimization.- 14 State Observation and Filtering.- 1 The Luenberger Observer.- 2 Observer with Discontinuous Parameters.- 3 Sliding Modes in Systems with Asymptotic Observers.- 4 Quasi-Optimal Adaptive Filtering.- 15 Sliding Modes in Problems of Mathematical Programming.- 1 Problem Statement.- 2 Motion Equations and Necessary Existence Conditions for Sliding Mode.- 3 Gradient Procedures for Piecewise Smooth Function.- 4 Conditions for Penalty Function Existence. Convergence of Gradient Procedure.- 5 Design of Piecewise Smooth Penalty Function.- 6 Linearly Independent Constraints.- III. Applications.- 16 Manipulator Control System.- 1 Model of Robot Arm.- 2 Problem Statement.- 3 Design of Control.- 4 Design of Control System for a Two-joint Manipulator.- 5 Manipulator Simulation.- 6 Path Control.- 7 Conclusions.- 17 Sliding Modes in Control of Electric Motors.- 1 Problem Statement.- 2 Control of d. c. Motor.- 3 Control of Induction Motor.- 4 Control of Synchronous Motor.- 18 Examples.- 1 Electric Drives for Metal-cutting Machine Tools.- 2 Vehicle Control.- 3 Process Control.- 4 Other Applications.- References.

Gaussian networks for direct adaptive control

Abstract: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture uses a network of Gaussian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibit by the nonlinear functions, the algorithm is proven to be globally stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with example systems. >

Adaptive nonlinear regulation: estimation from the Lyapunov equation

Abstract: A stabilizing adaptive controller for a nonlinear system depending affinely on some unknown parameters is presented. It is assumed that this system is feedback stabilizable. A key feature of the method is the use of the Lyapunov equation to design the adaptive law. A result on local stability, two different conditions for global stability, and a local result where the initial conditions of the state of the system only are restricted are given. >

Adaptive control of nonlinear systems using neural networks

Abstract: Layered networks are used in a nonlinear adaptive control problem. The plant is an unknown feedback-linearizable discrete-time system, represented by an input-output model. A state space model of the plant is obtained to define the zero dynamics, which are assumed to be stable. A linearizing feedback control is derived in terms of some unknown nonlinear functions. To identify these functions, it is assumed that they can be modelled by layered neural networks. The weights of the networks are updated and used to generate the control. A local convergence result is given. Computer simulations verify the theoretical result.

On the robust control of robot manipulators

Abstract: A simple robust nonlinear control law for n-link robot manipulators is derived using the Lyapunov-based theory of guaranteed stability of uncertain systems. The novelty of this result lies in the fact that the uncertainty bounds needed to derive the control law and to prove uniform ultimate boundedness of the tracking error depend only on the inertial parameters of the robot. In previous results of this type, the uncertainty bounds have depended not only on the inertia parameters but also on the reference trajectory and on the manipulator state vector. The presented result also removes previous assumptions regarding closeness in norm of the computed inertia matrix to the actual inertial matrix. The design used thus provides the simplest such robust design available to date. >

Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates

Abstract: A model of exploitative competition of n species in a chemostat for a single, essential, nonreproducing, growth-limiting resource is considered. S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760–763] applies LaSalle’s extension theorem of Lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by Michaelis–Menten dynamics. G. J. Butler and G. S. K. Wolkowicz [SIAM J. Appl. Math., 45 (1985), pp. 138–151], on the other hand, allow more general response functions (including monotone and nonmonotone functions), but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate, and hence can be ignored. By means of Lyapunov stability theory, the global dynamics of the model for a large class of response functions are studied, including both monotone and nonmonotone functions (though it is not as general as the class studied by Butler and Wolkowicz) and the results in ...

Robust stability of delay dependence for linear uncertain systems

Abstract: A sufficient condition for the stability of the linear uncertain time-delay systems is presented. The result obtained can include information on the size of the delay, and therefore can be a delay-dependent stability condition. The Lyapunov method is employed to investigate the problem. The result is demonstrated by an illustrative example. >

Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters

Abstract: New results for an established for the global asymptotic stability of the equilibrium x=0 of nth order discrete-time systems with state saturations, x(k+1)=sat(Ax(k)), utilizing a class of positive definite and radially unbounded Lyapunov functions, v. When v is a quadratic form, necessary and sufficient conditions are obtained under which positive definite matrices H can be used to generate a Lyapunov function v(w)=w/sup T/Hw with the properties that v(Aw(k)) is negative semidefinite, and that v(sat(w)) >

Vector norms as Lyapunov functions for linear systems

Abstract: A unified theory of quadratic and piecewise-linear Lyapunov functions for continuous and discrete-time linear systems is presented. The key to this work is the description of these Lyapunov functions by vector norms. The main results are sufficient and necessary conditions for a vector norm to be a Lyapunov function as well as a method (based on these conditions) of constructing such Lyapunov functions. >

Robust adaptive decentralized control of robot manipulators

Abstract: The authors proposes a robust adaptive decentralized control algorithm for trajectory tracking of robot manipulators. The controller is designed based on a Lyapunov method, which consists of a PD (proportional plus derivative) feedback part and a dynamic compensation part. It is shown that, without any prior knowledge of manipulator or payload parameters and possibly under deterioration of parameter variation with time or state-independent input disturbances, the tracking error is bound to converge to zero asymptotically. In particular, the algorithm does not require explicit system parameter estimation and therefore makes the controller structurally simple and computationally easy. Moreover, the controller is implemented in a decentralized manner, i.e. a subcontroller is independently and locally equipped at each joint servoloop. To illustrate the performance of the controller, a numerical simulation example is provided. >

Modelling, Identification and Stable Adaptive Control of Continuous-Time Nonlinear Dynamical Systems Using Neural Networks

Abstract: Several empirical studies have demonstrated the feasibility of employing neural networks as models of nonlinear dynamical systems. This paper develops the appropriate mathematical tools for synthesizing and analyzing stable neural network based identification and control schemes. Feedforward network architectures are combined with dynamical elements, in the form of stable filters, to construct a general recurrent network configuration which is shown to be capable of approximating a large class of dynamical systems. Adaptive identification and control schemes, based on neural network models, are developed using the Lyapunov synthesis approach with the projection modification method. These schemes are shown to guarantee stability of the overall system, even in the presence of modelling errors. A crucial characteristic of the methods and formulations developed in this paper is the generality of the results which allows their application to various neural network models as well as other approximators.

An approach to stability analysis of second order fuzzy systems

Abstract: The stability of fuzzy systems can be discussed by the theorem of K. Tanaka and M. Sugeno (1990). However, it is difficult to find the common positive definite matrix P which is introduced in the theorem, and satisfies, for example, two Lyapunov inequalities A/sub 1//sup T/PA/sub 1/-P >

Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors

Abstract: This paper surveys results of the authors and others conceming estimates for the Hausdorff dimension of strange attractors, particularly in the case of (generalized) Lorenz systems and Rossler systems. A key idea is the interpretation of Hausdorff measure as an analogue of a Lyapunov function.

Lyapunov exponents for one-dimensional cellular automata

Abstract: In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).

Stable Controllers for Instantaneous Optimal Control

Abstract: Recently, instantaneous optimal control algorithms were proposed and developed for applications to control of seismically excited linear, nonlinear, and hysteretic structural systems. In particular, these control algorithms are suitable for aseismic hybrid control systems, for which the linear quadratic optimal control theory is not applicable. Within the framework of instantaneous optimal control, the weighting matrix Q should be assigned to guarantee the stability of the controlled structure. A systematic way of assigning the weighting matrix by use of the Lyapunov direct method is investigated. Based on the Lyapunov method, several possible choices for the weighting matrix are presented, and their control performances are examined and compared for active and hybrid control systems under seismic loads. It is shown that the performance of the stable controllers presented herein are remarkable.

Passivity and global stabilization of cascaded nonlinear systems

Abstract: An alternative stability analysis is presented for recent results on global stabilization of a nonlinear system in cascade with a linear system. The analysis is carried out using passivity arguments. The relationship between passivity and an important class of Lyapunov function is also presented. >

Lyapunov function-based control laws for revolute robot arms: tracking control, robustness, and adaptive control

Abstract: A new class of joint level control laws for all-revolute robot arms is introduced. The analysis is similar to an energy-like Lyapunov function approach, except that the closed-loop potential function is shaped in accordance with the underlying joint space topology. This approach gives way to a much simpler analysis and leads to a new class of control designs which guarantee both global asymptotic stability and local exponential stability. When Coulomb and viscous friction and parameter uncertainty are present as model perturbations, a sliding mode-like modification of the control law results in a robustness-enhancing outer loop. Adaptive control is formulated within the same framework. A linear-in-the-parameters formulation is adopted and globally asymptotically stable adaptive control laws are derived by simply replacing unknown model parameters by their estimates. >

Space Station attitude control and momentum management - A nonlinearlook

Abstract: Nonlinear design procedures are presented for obtaining attitude and angular momentum control laws for the Space Station. These are based on Lyapunov's second method for stability analysis. In the absence of disturbances, there exist four stable equilibrium points. Only one of these is desired and its stability boundary is estimated from the gravitational dynamic potential. Simulation results are presented for the current Space Station configuration for large initial deviations from the local-vertical-local-horizontal frame. The control laws obtained under the zero disturbance assumption are tested in the presence of constant disturbance torques. It is shown that the Space Station can be stabilized about a torque-equilibrium attitude and the momentum of the control moment gyroscopes is bounded, provided the disturbance is smaller than the maximum gravity gradient torque that can be produced by the Space Station.

On the exponential stability of singularly perturbed systems

Abstract: This paper establishes some results and properties related to the exponential stability of general dynamical systems and, in particular, singularly perturbed systems. For singularly perturbed systems it is shown that if both the reduced-order system and the boundary-layer system are exponentially stable, then, provided that some further regularity conditions are satisfied, the full-order system is exponentially stable for sufficiently small values of the perturbation parameter $\mu $, and its rate of convergence approaches that of the reduced-order system $(\mu = 0)$ as $\mu $ approaches zero. Exponentially decaying norm bounds are given for the “slow” and “fast” components of the full-order system trajectories. To achieve this result, a new converse Lyapunov result for exponentially stable systems is presented.

Robust feedback linearization approach to autopilot design

Abstract: The feasibility of autopilot design for highly maneuverable bank-to-turn (BTT) missiles using feedback-linearization-based approaches is investigated. Two schemes, namely, feedback linearization and robust feedback linearization, are designed and compared based on a full-scale six-degree-of-freedom HAVE DASH II terminal homing missile model. Although the feedback linearization controller is quite satisfactory in general, a sizable coupling between the longitudinal motion and lateral motion for large maneuvers is observed. This is attributed to the uncertainties arising from the approximation of the aerocoefficients and model simplification. The second scheme, which adds a robust outer-loop design based on Lyapunov's second method to the first scheme in order to account for this uncertainty, shows a significant improvement over the first scheme. >