Showing papers on "Lyapunov function published in 1990"
TL;DR: In this paper, the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency domain results on H/sup infinity / optimization.
Abstract: The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >
1,464 citations
TL;DR: In this paper, necessary and sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite-time Markovian jump linear quadratic (JLQ) problem are established.
Abstract: Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem. >
955 citations
TL;DR: In this paper, a stable discrete sliding mode control insensitive to the choice of sampling interval and not yielding chattering is presented, which is based on a discrete Lyapunov function and a sufficient condition of the control gain to make the system stable is given.
849 citations
Book•
01 Jan 1990
TL;DR: In this paper, a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets is presented, and the manifestations of general Lyapunov's method can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
Abstract: This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
343 citations
Abstract: A linear stabilizable, nonlinear asymptotically stable, cascade system is globally stabilizable by smooth dynamic state feedback if (a) the linear subsystem is right invertible and weakly minimum phase, and, (b) the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output. Both of these conditions are coordinate-free and there is freedom of choice for the linear output variable. This result generalizes several earlier sufficient conditions for stabilizability. Moreover, the weak minimum-phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem.
283 citations
01 Jun 1990
TL;DR: In this article, the fundamental properties of the network elements making up a switched power converter are examined and the nominal open-loop operation of a broad class of such converters is shown to be stable in the large via a Lyapunov argument.
Abstract: The fundamental properties, such as passivity or incremental passivity, of the network elements making up a switched power converter are examined. The nominal open-loop operation of a broad class of such converters is shown to be stable in the large via a Lyapunov argument. The obtained Lyapunov function is then shown to be useful for designing globally stabilizing controls that include adaptive schemes for handling uncertain nominal parameters. Numerical simulations illustrate the application of this control approach in DC-DC converters. >
255 citations
01 Jan 1990
TL;DR: This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems.
Abstract: This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems.
186 citations
TL;DR: In this paper, a rigorous proof is given in the standard map associated with a Frenkel-Kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k > k 0.
178 citations
05 Dec 1990
TL;DR: It is shown that the direct method of Lyapunov can be used to determine sufficient conditions for global stability of a broad class of fuzzy control schemes and a measure of robustness is proposed that can be use to evaluate and possibly redesign a given fuzzy control system so as to enhance the range of its stable operation.
Abstract: A new approach to the stability analysis of fuzzy linguistic control (FLC) systems is presented. Specifically, it is shown that the direct method of Lyapunov can be used to determine sufficient conditions for global stability of a broad class of fuzzy control schemes. Moreover, a measure of robustness is proposed that can be used to evaluate and possibly redesign a given fuzzy control system so as to enhance the range of its stable operation. Finally, the application of the proposed methodology is shown and its implications in terms of control design are demonstrated by means of numeric examples. >
153 citations
TL;DR: In this article, a survey of viability theorems for nonlinear control problems with state constraints and state-dependent control constraints is presented, and the concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closedloop feedbacks and closed-loop dynamical feedbacks.
Abstract: Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature.Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls.For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge.The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks.Viability theorems also allow the extension of Lyapunov’s second meth...
148 citations
TL;DR: The results indicate that the neural network approach functions well in noise, works for linear and nonlinear systems, and can be implemented very efficiently for large-scale systems.
Abstract: A comparison is made of a neural-network-based controller similar to the cerebellar model articulation controller (CMAC) and two traditional adaptive controllers, a self-tuning regulator (STR) and a Lyapunov-based model reference adaptive controller (MRAC). The three systems are compared conceptually and through simulation studies on the same low-order control problem. Results are obtained for the case where the system is linear and noise-free, for the case where noise is added to the system, and for the case where a nonlinear system is controlled. Comparisons are made with respect to closed-loop system stability, speed of adaptation, noise rejection, the number of required calculations, system tracking performance, and the degree of theoretical development. The results indicate that the neural network approach functions well in noise, works for linear and nonlinear systems, and can be implemented very efficiently for large-scale systems. >
TL;DR: In this paper, the authors examined the question of accurately determining Lyapunov exponents for a time series and found that it is advantageous to use local mappings with higher-order Taylor series, rather than linear maps as done earlier.
Abstract: We examine the question of accurately determining Lyapunov exponents for a time series. We find that it is advantageous to use local mappings with higher-order Taylor series, rather than linear maps as done earlier. We demonstrate this procedure for the Ikeda map and the Lorenz system. We present methods for identifying spurious exponents by analyzing data-set singularities and by determining the Lyapunov direction vectors. The behavior of spurious exponents in the presence of noise is also investigated, and found to be different from that of the true exponents.
TL;DR: There is an interesting connection between two, recently popular, methods for finding good approximate solutions to hard optimisation problems, the ‘neural’ approach of Hopfield and Tank and the elastic-net method of Durbin and Willshaw.
Abstract: There is an interesting connection between two, recently popular, methods for finding good approximate solutions to hard optimisation problems, the ‘neural’ approach of Hopfield and Tank and the elastic-net method of Durbin and Willshaw. They both have an underlying statistical mechanics foundation and can be derived as the leading approximation to the thermodynamic free energy of related physical models. The apparent difference in the form of the two algorithms comes from different handling of constraints when evaluating the thermodynamic partition function. If all the constraints are enforced ‘softly’, the ‘mean-field’ approximation to the thermodynamic free energy is just the neural network Lyapunov function. If, on the other hand, half of the constraints are enforced ‘strongly’, the leading approximation to the thermodynamic free energy is the elastic-net Lyapunov function. Our results have interesting implications for the general problem of mapping optimisation problems to ‘neural’ and ‘elastic’ netw...
13 May 1990
TL;DR: Algorithms for continuous-time quadratic optimization of motion control are presented and it is shown that global asymptotic stability holds and how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters.
Abstract: This paper presents algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are found by solving an algebraic matrix equation. The system stability is investigated according to Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution results in natural design parameters in the form of square weighting matrices as known from linear quadratic optimal control The proposed optimal control is useful both for motion control, trajectory planning, and motion analysis.
TL;DR: This paper uses Lyapunov's second method to derive a uniform boundness result for the PD controller and develops two additional controllers that use auxiliary control inputs along with the PD controllers to yield a uniform ultimate boundness property for the tracking error.
Abstract: In this paper we examine the stability of a proportional derivative (PD) controller for the trajectory-following problem of a robot manipulator. We use Lyapunov's second method to derive a uniform boundness result for the PD controller. We show that if the PD controller gains are chosen greater than a specific bound and if the initial tracking error is zero, the velocity and position tracking errors are uniformly bounded. We then develop two additional controllers that use auxiliary control inputs along with the PD controller. Both of these controllers are shown to yield a uniform ultimate boundness property for the tracking error.
TL;DR: A feedback control-design problem involving structured plant parameter uncertainties is considered and the optimal projection approach to fixed-order, dynamic compensation is merged with the guaranteed cost control approach to robust stability and performance to obtain a theory of full- and reduced-order robust control design.
Abstract: A feedback control-design problem involving structured plant parameter uncertainties is considered. Two robust control-design issues are addressed. The Robust Stability Problem involves deterministic bounded structured parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. The optimal projection approach to fixed-order, dynamic compensation is merged with the guaranteed cost control approach to robust stability and performance to obtain a theory of full- and reduced-order robust control design. The principle result is a sufficient condition for characterizing dynamic controllers of fixed dimension which are guaranteed to provide both robust stability and performance. The sufficient conditions involve a system of modified Riccati and Lyapunov equations coupled by an oblique projection and the uncertainty bounds. The full-order result involves a system of two modified Riccati equations and two modified Lyapunov equations coupled by the uncertainty bounds. The coupling illustrates the breakdown of the separation principle for LQG control with structured plant parameter variations.
TL;DR: In this paper, the problem of stabilization and observation of uncertain and/or nonlinear dynamic systems for which matching conditions are not satisfied is investigated, but no statistical description of uncertain elements is assumed.
Abstract: The problem of stabilization and observation of uncertain and/or nonlinear dynamic systems for which matching conditions are not satisfied is investigated. No statistical description of uncertain elements is assumed. The uncertain and/or nonlinear quantities are described only in terms of bounds on their possible sizes. Stabilizing feedback controllers and observers are proposed whose design is based on the constructive use of Lyapunov functions and the Bellman-Gronwall lemma. >
01 Aug 1990
TL;DR: In this paper, a Lyapunov-based continuous-time direct adaptive control algorithm for robot manipulators is presented, which is used for controller design and stability investigation, as well as for rapid continuous time adaptive control.
Abstract: Algorithms for continuous-time direct adaptive control of robot manipulators are presented. Lyapunov theory is used for controller design and stability investigation. Algorithms for rapid continuous-time adaptive control are also presented. >
TL;DR: In this article, a wide variety of quadratic Lyapunov bounds are systematically developed and a unified treatment of several bounds developed previously for feedback control design is provided for robust stability and performance analysis.
Abstract: For a given asymptotically stable linear dynamic system it is often of interest to determine whether stability is preserved as the system varies within a specified class of uncertainties. If, in addition, there also exist associated performance measures (such as the steady-state variances of selected state variables), it is desirable to assess the worst-case performance over a class of plant variations. These are problems of robust stability and performance analysis. In the present paper, quadratic Lyapunov bounds used to obtain a simultaneous treatment of both robust stability and performance are considered. The approach is based on the construction of modified Lyapunov equations, which provide sufficient conditions for robust stability along with robust performance bounds. In this paper, a wide variety of quadratic Lyapunov bounds are systematically developed and a unified treatment of several bounds developed previously for feedback control design is provided.
TL;DR: In this article, under a weaker hypothesis on the input signal, a proof is given by constructing a Lyapunov function that is standard in the control literature, and a simple example is given to show that this approach can yield explicit bounds in certain situations of interest.
Abstract: Recently, M. Kelemen (IEEE Trans. Automat. Contr., vol.AC-31, 766-768, Aug. 1986) presented a stability result that deals with the response of a nonlinear system to slowly varying input signals. In this work a proof is given, under a weaker hypothesis on the input signal, by constructing a Lyapunov function that is standard in the control literature. This proof may be more accessible than the other approaches. For completeness, brief proofs of some intermediate results that are regarded as well known are included. A simple example is given to show that this approach can yield explicit bounds in certain situations of interest. >
23 May 1990
TL;DR: In this article, the stability of a Proportional Derivative (PD) controller for the trajectory following problem of a robot manipulator is examined and two additional controllers that use auxiliary control inputs along with the PD controller are shown to yield a uniform boundness property for the tracking error.
Abstract: In this paper, we examine the stability of a Proportional Derivative (PD) controller for the trajectory following problem of a robot manipulator. We use Lyapunov's second method to derive a uniform boundness result for the PD controller. We show that if the PD controller gains are chosen greater than a specific bound and if the Initial tracking error is zero, the velocity and position tracking errors are uniformly bounded. We then develop two additional controllers that use auxiliary control inputs along with the PD controller. Both of these controllers are shown to yield a uniform ultimate boundness property for the tracking error.
05 Dec 1990
TL;DR: A learning scheme using feedback-error-learning for a neural network model applied to adaptive nonlinear feedback control that can be used for many kinds of controlled objects, such as chemical plants, machines, and robots is proposed.
Abstract: The authors propose a learning scheme using feedback-error-learning for a neural network model applied to adaptive nonlinear feedback control. After the neural network compensates perfectly or partially for the nonlinearity of the controlled object through learning, the response of the controlled object follows the desired set in the conventional feedback controller. This learning scheme does not require the knowledge of the nonlinearity of a controlled object in advance. Using the proposed approach, the actual responses after learning correspond to desired responses. When the desired response in Cartesian space is required, learning impedance control is derived. The convergence properties of the neural networks are provided by the averaged equation and Lyapunov method. Simulation results on this learning approach are presented. The proposed scheme can be used for many kinds of controlled objects, such as chemical plants, machines, and robots. >
TL;DR: In this paper, the chaotic behavior of the Bianchi IX cosmological models was investigated using techniques developed in the study of dynamical systems and chaotic behaviour. And they showed that instead of converging to a constant value, it decreases steadily.
Abstract: In this paper we investigate the chaotic behaviour of the Bianchi IX cosmological models using techniques developed in the study of dynamical systems and chaotic behaviour. We numerically calculate the Lyapunov exponent, λ, and show that instead of converging to a constant value, it decreases steadily. We study this effect further by studying the Lyapunov exponent using short-time averages. We show that the usual method of calculating λ is invalid in the case of a cosmological model.
TL;DR: In this paper, the authors investigate the problem of designing a linear state feedback control to stabilize a class of single-input uncertain linear dynamical systems with time-varying uncertain parameters whose values are unknown but bounded in given compact sets.
Abstract: The author investigates the problem of designing a linear state feedback control to stabilize a class of single-input uncertain linear dynamical systems. The systems under consideration contain time-varying uncertain parameters whose values are unknown but bounded in given compact sets. The method used to establish asymptotical stability of the closed-loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The author first shows that to ensure a stabilizable system some entries of the system matrices must be sign invariant. He then derives necessary and sufficient conditions under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties. The conditions show that all uncertainties can only enter the system matrices in such a way as to form a particular geometrical pattern called an antisymmetric stepwise configuration. For the systems satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example. >
TL;DR: In this paper, the spectral norm of the inverse Lyapunov operator was analyzed for the stable discrete-time Lipschitz equation, and an easy-to-interpret sensitivity measure based on the open-loop state matrix was derived.
Abstract: The sensitivity of the stable discrete-time Lyapunov equation is analyzed through the spectral norm of the inverse Lyapunov operator. This leads to a directly computable easy-to-interpret sensitivity measure, which involves only the open-loop state matrix and also provides insight into the connection between sensitivity, stability radius, and conditioning of the eigenproblem of the open-loop state matrix. These results are an extension, to the discrete-time case, of analogous results for the continuous-time Lyapunov equation. >
TL;DR: In this paper, necessary and sufficient Lyapunov conditions for linear control systems using output feedback were established, generalizing the Artstein's stabilization theorem, by means of a linear output feedback law.
Abstract: The stabilization of control systems using output feedback is addressed. Necessary and sufficient Lyapunov conditions are established, generalizing the Artstein's stabilization theorem. For linear control systems, a necessary and sufficient Lyapunov condition is provided for stabilization by means of a linear output feedback law. >
Book•
30 Mar 1990
TL;DR: In this article, the output feedback control of linear singularly perturbed and weakly coupled linear control systems is studied in the context of differential games and linear stochastic systems.
Abstract: Algebraic lyapunov and riccati equations.- Output feedback control of linear singularly perturbed and weakly coupled systems.- Linear stochastic systems.- Recursive approach to finite time singularly perturbed and weakly coupled linear control systems.- Application to the differential games.- Linear discrete weakly coupled control systems.- Linear discrete singularly perturbed control systems.
TL;DR: In this article, the second method of Lyapunov is proposed as a method for analyzing the stability and tracking of the nonlinear phase-lock loop model, and the ability of loops designed using these techniques to track a phase step is also proven.
Abstract: The second method of Lyapunov is proposed as a method for analyzing the stability and tracking of the nonlinear phase-lock loop model. It is shown that a fairly regular procedure, known as Lyapunov design, which is quite well known in the area nonlinear control theory, is applicable to solving this class of problems. The ability of loops designed using these techniques to track a phase step is also proven. >
TL;DR: In this article, a method to obtain the best delay time for the reconstruction of chaotic attractors in phase space is presented, which uses purely geometrical considerations and guarantees a maximum distance of trajectories.
Abstract: A method to obtain the best delay time for the reconstruction of chaotic attractors in phase space is presented. The procedure uses purely geometrical considerations and guarantees a maximum distance of trajectories. This is tested with numerical (Duffing oscillator) and experimental (Taylor-Couette flow) systems by calculating correlation dimensions and Lyapunov spectra.
TL;DR: The singular system observer theory is extended both by providing a geometric basis for it and showing the best that may be accomplished if the system is not observable.