Showing papers on "Control-Lyapunov function published in 1997"
TL;DR: In this paper, the authors present a straightforward and reliable continuous method for computing the full or partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations.
Abstract: We present a straightforward and reliable continuous method for computing the full or partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram - Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when where is the kth Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.
111 citations
TL;DR: A continuous feedback control law with time-periodic terms is derived for the control of nonholonomic systems in power form by Lyapunov design from a homogeneous LyAPunov function.
Abstract: In this paper a continuous feedback control law with time-periodic terms is derived for the control of nonholonomic systems in power form. The control law is derived by Lyapunov design from a homogeneous Lyapunov function. Global asymptotic stability is shown by applying the principle of invariance for time-periodic systems. Exponential convergence follows since the vector fields are homogeneous of degree zero.
110 citations
10 Dec 1997
TL;DR: New sets of linear continuous-time constant systems having a common quadratic Lyapunov function are presented, whose system matrices are transformed into complex triangular matrices by a common complex matrix.
Abstract: The common Lyapunov function problem studied here is to find a set of linear systems, which has a common quadratic Lyapunov function V(x)=x/sup T/Px. This problem comes from the stability analysis of diverse fields of control systems. We present new sets of linear continuous-time constant systems having a common quadratic Lyapunov function. The classes obtained consist of systems whose system matrices are transformed into complex triangular matrices by a common complex matrix, or are close to them. We also show that the obtained classes include those presented by Narendra et al. (1994) and Mori et al. (1996).
96 citations
TL;DR: In this article, it was shown that a storage function can always be written as a quadratic function of the state of an associated linear dynamical system, which is obtained by combining the dynamics of the original system with the dynamic of the supply rate.
95 citations
TL;DR: This paper combines inverse optimality with backstepping to design a new class of adaptive controllers for strict-feedback systems, i.e. obtaining transient performance bounds that include an estimate of control effort, which is the first such result in the adaptive control literature.
86 citations
TL;DR: In this article, the authors studied the stability of affine in control stochastic differential systems and provided sufficient conditions for the existence of control Lyapunov functions leading to stabilizing feedback laws.
Abstract: The purpose of this paper is to study the asymptotic stability in probability of affine in the control stochastic differential systems. Sufficient conditions for the existence of control Lyapunov functions leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium point of the system, are provided.
81 citations
TL;DR: In this article, a feedback control design procedure is proposed, which can rearrange all the Lyapunov exponents of the controlled system according to the user's desire, namely, to make them positive, zero, and/or negative in any desired order, for any given n-dimensional discrete-time smooth nonlinear dynamical system that could be originally nonchaotic or even asymptotically stable.
Abstract: A simple, yet mathematically precise and rigorous, feedback control design procedure is suggested in this paper, which can rearrange all the Lyapunov exponents of the controlled system according to the user's desire, namely, to make them positive, zero, and/or negative in any desired order, for any given n-dimensional discrete-time smooth nonlinear dynamical system that could be originally nonchaotic or even asymptotically stable. The argument used is purely algebraic and the design procedure is completely schematic, without using any approximations throughout the derivation. A numerical example is included to visualize the anticontrol.
75 citations
TL;DR: In this article, a generalized feedback control law design methodology is presented that applies to systems under control saturation constraints, and Lyapunov stability theory is used to develop stable saturated control laws.
Abstract: A generalized feedback control law design methodology is presented that applies to systems under control saturation constraints. Lyapunov stability theory is used to develop stable saturated control laws that can be augmentedtoanyunsaturatedcontrollawthattransitionscontinuouslyatatouchpointonthesaturationboundary. The time derivative of the Lyapunov function, an error energy measure, is used as the performance index, which provides a measure that is invariant to the system dynamics. Lyapunov stability theory is used constructively to establish stability characteristics of the closed-loop dynamics. Lyapunov optimal control laws are developed by minimizing the performance index over the set of admissible controls, which is equivalent to forcing the error energy rate to be as negative as possible.
65 citations
TL;DR: It is asserted that the partial state which remains in the time derivative of the Lyapunov function converges to zero asymptotically.
Abstract: Asymptotic behavior of a partial state of a coupled ordinary and/or partial differential equation is investigated. It is specifically shown that if a signal x(t) is a solution to a dynamic system existing for all t/spl ges/0 in an abstract Banach space and pth (p/spl ges/1) power integrable, then x(t)/spl rarr/0 as t/spl rarr//spl infin/. The system is allowed to be nonautonomous and assumes the existence of a Lyapunov function. Since the derivative of the Lyapunov function is negative semidefinite, stability or uniform stability in the sense of Lyapunov would be concluded. However, this paper further asserts that the partial state which remains in the time derivative of the Lyapunov function converges to zero asymptotically.
55 citations
04 Jun 1997
TL;DR: In this paper, an adaptive tracking control Lyapunov function whose existence guarantees the solvability of the inverse optimal problem is proposed. But this approach does not lead to optimality of the controller with respect to the overall plant-estimator system, even though both the estimator and the controller may be optimal as separate entities.
Abstract: We pose and solve an "inverse optimal" adaptive tracking problem for nonlinear systems with unknown parameters. A controller is said to be inverse optimal when it minimizes a meaningful cost functional that incorporates integral penalty on the tracking error state and the control, as well as a terminal penalty on the parameter estimation error. The basis of our method is an adaptive tracking control Lyapunov function whose existence guarantees the solvability of the inverse optimal problem. The controllers designed in this paper are not of certainty-equivalence type. Even in the linear case they would not be a result of solving a Riccati equation for a given value of the parameter estimate. Our abandoning of the CE approach is motivated by the fact that, in general, this approach does not lead to optimality of the controller with respect to the overall plant-estimator system, even though both the estimator and the controller may be optimal as separate entities. Our controllers, instead, compensate for the effect of parameter adaptation transients in order to achieve optimality of the overall system. We combine inverse optimality with backstepping to design a new class of adaptive controllers for strict-feedback systems. These controllers solve a problem left open in the previous adaptive backstepping designs-getting transient performance bounds that include an estimate of control effort.
54 citations
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TL;DR: In this paper, the Lyapunov level curve of a nonlinear dynamical system with a chaotic attractor is determined, such that, whenever the state of the system is within this level curve, the feedback controller will drive the nonlinear system to the desired equilibrium solution.
Abstract: A nonlinear dynamical system with a chaotic attractor will produce motion on the attractor which has random-like properties. This observation leads to a very simple algorithm for bringing a discrete or continuous nonlinear dynamical system to a fixed point. Suppose that the system to be controlled is either naturally chaotic or that chaotic motion can be produced by means of open-loop control. Suppose also that a neighborhood of the desired fixed point can be found, such that, under state variable feedback control, the system is guaranteed to be driven to the fixed point. If this neighborhood has points in common with a chaotic attractor, it may be used as a "controllable target" for the fixed point. The control algorithm consists of first using, if necessary, open-loop control to generate chaotic motion and then waiting for the system to move into the controllable target. At such a time the open-loop control is turned off and the appropriate closed-loop control applied. A basic requirement with this approach is to determine a large enough controllable target so that one does not have to wait too long for the system to reach it. The following method is used here: the system is first linearized about the desired fixed-point solution. If necessary, a feedback controller is then designed so that this reference solution has suitable stability properties. Then a Lyapunov function is obtained based on this stable linear system and a level curve for the Lyapunov function is determined, such that, whenever the state of the nonlinear system is within this level curve, the feedback controller will drive the nonlinear system to the desired equilibrium solution. Such a level curve defines a controllable target provided that it actually does contain points on the chaotic attractor. A multiple step approach for determining the Lyapunov level curve is presented which helps in finding large controllable targets for discrete systems.
10 Dec 1997
TL;DR: In this article, an excitation controller for synchronous machines is presented, where the control law is derived using the control Lyapunov function concept and an equilibrium tracking mechanism, resulting in a dynamic state feedback controller.
Abstract: Presents an excitation controller for synchronous machines. The control law is derived using the control Lyapunov function concept and an equilibrium tracking mechanism, resulting in a dynamic state feedback controller. Both damping improvement and enlargement of the stability region are achieved, and the knowledge of the operating point is not required. Simulation results for a case study are presented showing the significant improvement obtained both in transient stability and dynamic performance.
TL;DR: Brief reports as discussed by the authors are accounts of completed research which do not warrant regular articles or the priority handling given to Communications; however, the same standards of scientific quality apply, and page proofs are sent to authors.
Abstract: Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Communications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Repor no longer than four printed pages and must be accompanied by an abstract. The same publication schedule as for regular a followed, and page proofs are sent to authors.
TL;DR: In this article, the Lyapunov stability of impulsive differential systems is considered and sufficient conditions for the LyAPunov function to be nonincreasing along each subsequence of switching and bounded within two switches are derived.
10 Dec 1997
TL;DR: Based on integral Lyapunov inequality associated to discrete-time dynamics, some preliminary results on control LyAPunov design are set.
Abstract: Based on integral Lyapunov inequality associated to discrete-time dynamics, some preliminary results on control Lyapunov design are set.
01 Jul 1997
TL;DR: In this article, a control Lyapunov function for the error system is proposed, which gives a time-varying state feedback which meets saturation constraints and ensures the global uniform asymptotic stability and local exponential stability of the state reference.
Abstract: We are concerned with the problem of the asymptotic tracking of a given state reference for a system which can be written in a feedforward form. The state reference is assumed to be given as a particular bounded solution of the system. Our solution relies on the construction of a control Lyapunov function for the error system. It gives a time-varying state feedback which meets saturation constraints and ensures the global uniform asymptotic stability and the local exponential stability of the state reference. A practical example illustrates our central result.
TL;DR: In this paper, the authors show that a backstepping-style control Lyapunov function, which grows unbounded on the set where the control law becomes unbounded, has level sets that always remain in the feasibility region.
04 Jun 1997
TL;DR: In this article, an alternative to gain scheduling for stabilizing a class of nonlinear systems is proposed, where the computation times required to find stability regions for a given control Lyapunov function vary polynomially with the state dimension for a fixed number of scheduling variables.
Abstract: We propose an alternative to gain scheduling for stabilizing a class of nonlinear systems. The computation times required to find stability regions for a given control Lyapunov function vary polynomially with the state dimension for a fixed number of scheduling variables. Control Lyapunov functions to various trim points are used to expand the stability region, and a Lyapunov based synthesis formula yields a control law guaranteeing stability over this region. Robustness to bounded disturbances is easily handled, and the optimal stability margin, defined as a Lyapunov derivative, is recovered asymptotically. We apply the procedure to an example.
10 Dec 1997
TL;DR: A Lyapunov-based optimal adaptive control-system design problem for nonlinear uncertain systems with exogenous L/sub 2/ disturbances is considered in this article, where an inverse optimal adaptive nonlinear control framework is developed to explicitly characterize globally stabilizing disturbance rejection adaptive controllers that minimize a nonlinear-nonquadratic performance functional for systems with parametric uncertainty.
Abstract: A Lyapunov-based optimal adaptive control-system design problem for nonlinear uncertain systems with exogenous L/sub 2/ disturbances is considered. Specifically, an inverse optimal adaptive nonlinear control framework is developed to explicitly characterize globally stabilizing disturbance rejection adaptive controllers that minimize a nonlinear-nonquadratic performance functional for nonlinear systems with parametric uncertainty. It is shown that the adaptive control Lyapunov function guaranteeing closed-loop stability is a solution to the Hamilton-Jacobi-Isaacs equation for the controlled system and thus guarantees both optimality and robust stability. Additionally, the adaptive control Lyapunov function is dissipative with respect to a weighted input-output energy supply rate guaranteeing closed-loop disturbance rejection.
04 Jun 1997
TL;DR: In this article, an optimality-based framework for designing controllers for discrete-time nonlinear cascade systems is presented. And the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady state Bellman equation for the controlled system and thus guarantees both optimality and stability.
Abstract: We develop an optimality-based framework for designing controllers for discrete-time nonlinear cascade systems. Specifically, using a nonlinear-nonquadratic optimal control framework we develop a family of globally stabilizing backstepping-type controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Bellman equation for the controlled system and thus guarantees both optimality and stability.
10 Dec 1997
TL;DR: In this article, a finite-dimensional linear matrix inequality (LMI) formulation for the stability and induced L/sub 2/ performance analysis of linear parameter-varying (LPV) systems is presented.
Abstract: This paper presents new finite-dimensional linear matrix inequality (LMI) formulations for the stability and induced L/sub 2/ performance analysis of linear parameter-varying (LPV) systems. The approach is based on the nonsmooth, dissipative system theory using a continuous, piecewise affine parameter-dependent Lyapunov function (PAPDLF). The new method is shown to be less conservative than previously published techniques that are based on either affine parameter-dependent Lyapunov functions or robust control techniques. Conservatism is reduced with this new approach because the analysis is performed over several, smaller subregions of the parameter space rather than the entire region. The new analysis approach also uses a more general class of parameter-dependent Lyapunov functions. In contrast to the gridding approach typically used to develop a computationally feasible algorithm, this proposed approach guarantees the analysis result. While computationally intensive, we show that the numerical results using our approach can be used to develop many new insights into the potential conservatism of various classes of Lyapunov functions for LPV systems.
01 Jul 1997
TL;DR: In this article, it was shown that global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.
Abstract: This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.
04 Jun 1997
TL;DR: In this article, a control Lyapunov function approach is used to design globally stabilizing feedback laws that have desirable optimality properties, compared to the performance of previously developed proportional-derivative type control laws and shown to be superior.
Abstract: The problem of globally stabilizing the attitude of a rigid body is considered. A control Lyapunov function approach is used to design globally stabilizing feedback laws that have desirable optimality properties. Their performance is compared to the performance of previously developed proportional-derivative type control laws and shown to be superior.
TL;DR: In this article, the authors proposed a method for the homogenization of differential operators in non-homogeneous elastic media, and showed that it is possible to solve the problem of homogenizing differentially-operators in nonhomogeneous media.
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TL;DR: In this paper, an optimality-based disturbance rejection control framework for nonlinear cascade systems with bounded energy (square-integrable) L disturbances was developed, and a family of globally stabilizing generalized backstepping controllers parametrized by the cost functional was developed.
Abstract: In this paper we develop an optimality-based disturbance rejection control framework for nonlinear cascade systems with bounded energy (square-integrable) L disturbances. Specifically, using a nonlinear-non-quadratic disturbance rejection optimal control framework we develop a family of globally stabilizing generalized backstepping controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closedloop stability is a solution to the steady-state Hamilton-Jacobi-Isaacs equation for the controlled system and thus guarantees both optimality and stability. In addition, the resulting optimal controller guarantees that the closed-loop system is non-expansive (gain bounded).
TL;DR: In this article, the authors developed a unified framework to address the problem of optimal nonlinear-nonquadratic robust control for systems with nonlinear time-invariant real parameter uncertainty.
Abstract: In this paper we develop a unified framework to address the problem of optimal nonlinear-nonquadratic robust control for systems with nonlinear time-invariant real parameter uncertainty. Specifically, we transform a given robust nonlinear control problem into an optimal control problem by modifying the performance functional to account for the system uncertainty. Robust stability of the closed loop nonlinear system is guaranteed by means of a parameter-dependent Lyapunov function composed of a fixed (parameter-independent) and variable (parameterdependent) part. The fixed part of the Lyapunov function can clearly be seen to be the solution to the steady-state Hamilton-Jacobi-Bellman equation for the nominal system. The overall framework generalizes the classical Hamilton-JacobiBellman conditions to address the design of robust optimal controllers for uncertain nonlinear systems via parameter-dependent Lyapunov functions and provides the foundation for extending robust linear-quadratic controller synthesis...
TL;DR: In this paper, the second Lyapunov method is used to model the dynamics of a double-link unit with Lagrange's equations of the second kind and a control law is constructed which enables the system to be transferred from an arbitrary initial state to a given final state in finite time using a force of bounded modulus.
10 Dec 1997
TL;DR: In this paper, an optimality-based nonlinear control framework for nonlinear systems with time-invariant sector-bounded memoryless input nonlinearities is developed.
Abstract: We develop an optimality-based nonlinear control framework for nonlinear systems with time-invariant sector-bounded memoryless input nonlinearities. Specifically, using an optimal nonlinear control framework we develop a family of globally stabilizing controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability over a prescribed set of input nonlinearities is a solution to the steady-state Hamilton-Jacobi-Bellman equation for the controlled system and thus guarantees stability and performance.
04 Jun 1997
TL;DR: In this paper, the authors developed a method for computing the regions of state space over which a control Lyapunov function decreases along trajectories of a closed loop system under an appropriate control law.
Abstract: Nonlinear control systems can be stabilized by constructing control Lyapunov functions and computing the regions of state space over which such functions decrease along trajectories of the closed loop system under an appropriate control law. For systems whose dynamics are nonlinear in only a few state variables, we develop a method for computing such a region based on a given polytopic control Lyapunov function. The procedure is computationally tractable, in the sense that computation times vary polynomially with the state dimension for a fixed number of "nonlinear states". Control constraints and robustness to bounded disturbances are easily incorporated into this framework.