Showing papers on "Lyapunov function published in 1996"
TL;DR: A design methodology is developed that expands the class of nonlinear systems that adaptive neural control schemes can be applied to and relaxes some of the restrictive assumptions that are usually made.
Abstract: Based on the Lyapunov synthesis approach, several adaptive neural control schemes have been developed during the last few years. So far, these schemes have been applied only to simple classes of nonlinear systems. This paper develops a design methodology that expands the class of nonlinear systems that adaptive neural control schemes can be applied to and relaxes some of the restrictive assumptions that are usually made. One such assumption is the requirement of a known bound on the network reconstruction error. The overall adaptive scheme is shown to guarantee semiglobal uniform ultimate boundedness. The proposed feedback control law is a smooth function of the state.
1,255 citations
TL;DR: These LMI-based tests are applicable to constant or time-varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations, and they often compare favorably with /spl mu/ analysis for time-invariant parameter uncertainty.
Abstract: This paper presents new tests to analyze the robust stability and/or performance of linear systems with uncertain real parameters. These tests are extensions of the notions of quadratic stability and performance where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to a linear matrix inequality (LMI) problem and hence is numerically tractable. These LMI-based tests are applicable to constant or time-varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations. They also avoid the frequency sweep needed in real-/spl mu/ analysis, and numerical experiments indicate that they often compare favorably with /spl mu/ analysis for time-invariant parameter uncertainty.
999 citations
TL;DR: In this article, a converse Lyapunov function theorem motivated by robust control analysis and design is presented, which is based upon, but generalizes, various aspects of well-known classical theorems.
Abstract: This paper presents a converse Lyapunov function theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.
877 citations
TL;DR: This work uses a bounding technique based on a parameter-dependent Lyapunov function, and then solves the control synthesis problem by reformulating the existence conditions into a semi-infinite dimensional convex optimization.
Abstract: A linear, finite-dimensional plant, with state-space parameter dependence, is controlled using a parameter-dependent controller. The parameters whose values are in a compact set, are known in real time. Their rates of variation are bounded and known in real time also. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced L2 norm. Our approach uses a bounding technique based on a parameter-dependent Lyapunov function, and then solves the control synthesis problem by reformulating the existence conditions into a semi-infinite dimensional convex optimization. We propose finite dimensional approximations to get sufficient conditions for successful controller design.
798 citations
TL;DR: A Lyapunov statement and proof of the recent nonlinear small-gain theorem for interconnected input/state-stable (ISS) systems is provided.
613 citations
TL;DR: In this paper, a new criterion for the global stability of equilibria is derived for nonlinear autonomous ODEs in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the $C^1 $ closing lemma of Pugh.
Abstract: A new criterion for the global stability of equilibria is derived for nonlinear autonomous ordinary differential equations in any finite dimension based on recent developments in higher-dimensional generalizations of the criteria of Bendixson and Dulac for planar systems and on a local version of the $C^1 $ closing lemma of Pugh. The classical result of Lyapunov is obtained as a special case.
539 citations
TL;DR: The proposed approach provides feedback laws with several degrees of freedom which can be exploited to tackle design constraints and serves as a basic tool to be used, in a recursive design, to deal with more complex systems.
Abstract: Our study relates to systems whose dynamics generalize x/spl dot/=h(y,u), y/spl dot/=f(y,u), where the state components x integrate functions of the other components y and the inputs u. We give sufficient conditions under which global asymptotic stabilizability of the y subsystem (respectively, by saturated control) implies global asymptotic stabilizability of the overall system (respectively, by saturated control). It is obtained by constructing explicitly a control Lyapunov function and provides feedback laws with several degrees of freedom which can be exploited to tackle design constraints. Also, we study how appropriate changes of coordinates allow us to extend its domain of application. Finally, we show how the proposed approach serves as a basic tool to be used, in a recursive design, to deal with more complex systems. In particular the stabilization problem of the so-called feedforward systems is solved this way.
475 citations
Book•
01 Jan 1996
TL;DR: In this article, the authors present a technique for quantifying the stability of large-scale uncertain dynamic systems in terms of two measures, i.e., mean value theorem and Consequences.
Abstract: Preface. 1: 1.0. Introduction. 1.1. Measure Chains and Time Scales. 1.2. Differentiation. 1.3 Mean Value Theorem and Consequences. 1.4. Integral and Antiderivative. 1.5. Notes. 2: 2.0. Introduction. 2.1. Local Existence and Uniqueness. 2.2. Dynamic Inequalities. 2.3. Existence of Extremal Solutions. 2.4. Comparison Results. 2.5. Linear Variation of Parameters. 2.6. Continuous Dependence. 2.7. Nonlinear Variation of Parameters. 2.8. Global Existence and Stability. 2.9. Notes. 3: 3.0. Introduction. 3.1. Comparison Theorems. 3.2. Stability Criteria. 3.3. A Technique in Stability Theory. 3.4. Stability of Conditionally Invariant Sets. 3.5. Stability in Terms of Two Measures. 3.6. Vector Lyapunov Functions and Practical Stability. 3.7. Notes. 4: 4.0. Introduction. 4.1. Monotone Iterative Technique. 4.2. Method of Quasilinearization. 4.3. Monotone Flows and Stationary Points. 4.4. Invariant Manifolds. 4.5. Practical Stability of Large-Scale Uncertain Dynamic Systems. 4.6. Boundary Value Problems. 4.7. Sturmian Theory. 4.8. Convexity of Solutions Relative to the Initial Data. 4.9. Invariance Principle. 4.10. Notes. References. Subject Index.
438 citations
TL;DR: In this paper, the robust control Lyapunov function (Bf rclf) is introduced, and it is shown that the existence of such a function is equivalent to robust stabilizability via continuous state feedback.
Abstract: The concept of a robust control Lyapunov function ({\Bf rclf}) is introduced, and it is shown that the existence of an {\Bf rclf} for a control-affine system is equivalent to robust stabilizability via continuous state feedback. This extends Artstein's theorem on nonlinear stabilizability to systems with disturbances. It is then shown that every {\Bf rclf} satisfies the steady-state Hamilton--Jacobi--Isaacs (HJI) equation associated with a meaningful game and that every member of a class of pointwise min-norm control laws is optimal for such a game. These control laws have desirable properties of optimality and can be computed directly from the {\Bf rclf} without solving the HJI equation for the upper value function.
425 citations
TL;DR: It is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant, and sufficient conditions for the existence of parameter-dependent Lyapunov functions are given in terms of a criterion reminiscent of Popov's stability criterion.
Abstract: In this paper, the problem of robust stability of systems subject to parametric uncertainties is considered. Sufficient conditions for the existence of parameter-dependent Lyapunov functions are given in terms of a criterion which is reminiscent of, but less conservative than, Popov's stability criterion. An equivalent frequency-domain criterion is demonstrated. The relative sharpness of the proposed test and existing stability criteria is then discussed. The use of parameter-dependent Lyapunov functions for robust controller synthesis is then considered. It is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant. A possible synthesis procedure and a numerical example are then discussed.
415 citations
TL;DR: In this article, the authors derived a new class of globally asymptotically stabilizing feedback control laws for the complete dynamics and kinematics attitude motion of rotating rigid bodies.
Abstract: In this paper we present some recent results on the description and control of the attitude motion of rotating rigid bodies We derive a new class of globally asymptotically stabilizing feedback control laws for the complete i e dynamics and kinematics attitude motion We show that the use of a Lyapunov function which involves the sum of a quadratic term in the angular velocities and a logarithmic term in the kinematic parameters leads to the design of linear controllers We also show that the feedback control laws for the kinematics minimize a quadratic cost in the state and control variables for all initial conditions For the complete system we construct a family of exponentially stabilizing control laws and we investigate their optimality characteristics The proposed control laws are given in terms of the classical Cayley Rodrigues parameters and the Modi ed Rodrigues parameters
01 Oct 1996
TL;DR: It is formally proved that the overall closed-loop system composed by the full nonlinear robot dynamics and the controller is Lyapunov stable and it is demonstrated that the Controller is capable to yield an asymptotically stable system which is robust against radial lens distortions and uncertainty in the camera orientation.
Abstract: In this paper we address the visual servoing of planar robot manipulators under fixed-camera configuration. The control goal is to place the robot end-effector over a desired static target by using a vision system equipped with a fixed camera to 'see' the robot end-effector and target. We analyze an image-based controller, whose implementation requires the robot Jacobian, the gravitational torque vector, and the camera orientation. Further, the robot manipulator is not treated as an ideal positioning device but modeled by the Lagrangian dynamics. We formally prove that the overall closed-loop system composed by the full nonlinear robot dynamics and the controller is Lyapunov stable. Also, we demonstrate that the controller is capable to yield an asymptotically stable system which is robust against radial lens distortions and uncertainty in the camera orientation. Simulations on a two degrees of freedom arm are presented to illustrate the controller performance.
TL;DR: It is shown that, in the case of input-to-state stability (ISS)—the output is the state itself-ISS and exp—ISS are in fact equivalent properties, and it is proposed that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin.
Abstract: We consider nonlinear systems with input-to-output stable (IOS) un- modeled dynamics which are in the "range" of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small-gain condi- tion is satisfied, we propose a first control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem (JTP, Theorem 2.1). However, this control law exhibits a high-gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with different characteristics in this respect. This con- troller requires more a priori knowledge on the unmodeled dynamics, as it is dynamic and incorporates a signal bounding the unmodeled effects. However, this is only possible by restraining the IOS property into the exp-IOS property. Never- theless, we show that, in the case of input-to-state stability (ISS)--the output is the state itself--ISS and exp-ISS are in fact equivalent properties.
TL;DR: In this article, the Poincare-Bendixson theorem holds for cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property.
TL;DR: A global stabilization procedure for nonlinear cascade and feedforward systems which extends the existing stabilization results and gives conditions for continuous differentiability of the Lyapunov function and the resulting control law.
Abstract: We present a global stabilization procedure for nonlinear cascade and feedforward systems which extends the existing stabilization results. Our main tool is the construction of a Lyapunov function for a class of (globally stable) uncontrolled cascade systems. This construction serves as a basis for a recursive controller design for cascade and feedforward systems. We give conditions for continuous differentiability of the Lyapunov function and the resulting control law and propose methods for their exact and approximate computation.
TL;DR: In this paper, the authors discuss some design methods for the control of a class of underactuated mechanical systems including gymnastic robots, such as the Acrobot, as well as the classical cart-pole system.
TL;DR: In this article, an analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours, where the dimension of nonlocal problems is determined by the three different integers: the geometrical dimension dg, the critical dimension de, and the Lyapunov dimension dL.
Abstract: An analogue of the center manifold theory is proposed for non-local bifurcations of homo and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension de which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong unstable foliations, and the Lyapunov dimension d L which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values de and dL. A connection with the problem of hyperchaos is discussed.
11 Dec 1996
TL;DR: In this article, the authors considered the stability and robustness issues for hybrid systems and proposed stronger conditions for stability, and formulated the search for Lyapunov functions as a linear matrix inequality problem.
Abstract: Stability and robustness issues for hybrid systems are considered in this paper. Present stability results, that are extensions of classical Lyapunov theory, are not straightforward to apply in general due to two reasons. First, existing theory do not unveil how to find needed Lyapunov functions. Secondly, at some time instants it is necessary to know the values of the continuous trajectory. Because of these drawbacks, stronger conditions for stability are suggested. The search for Lyapunov functions can then be formulated as a linear matrix inequality problem. Additionally, it is shown how to obtain robustness properties. An example illustrates the results.
TL;DR: Model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation on a repeated scalar uncertainty structure and a related necessary and sufficient condition for the exact reducibility of stable uncertain systems are presented.
Abstract: Model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure. These methods involve a complete generalization of balanced realizations, balanced Gramians, and balanced truncation model reduction with guaranteed error bounds, based on solutions to a pair of linear matrix inequalities which generalize Lyapunov equations. The resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems.
TL;DR: In this paper, it was shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every $L^p$-norm, and explicit bounds on closed-loop gains were obtained.
Abstract: This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every $L^p$-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without saturation.
These results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple (size $>1$) Jordan blocks, contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state-space sense; this is shown in particular for the double integrator.
TL;DR: This paper shows how adaptive controllers can be used to adjust the parameters of the systems such that the two systems will synchronize, and shows how it is related to Huberman-Lumer adaptive control and the LMS adaptive algorithm.
Abstract: In this paper, we study the synchronization of two coupled nonlinear, in particular chaotic, systems which are not identical. We show how adaptive controllers can be used to adjust the parameters of the systems such that the two systems will synchronize. We use a Lyapunov function approach to prove a global result which shows that our choice of controllers will synchronize the two systems. We show how it is related to Huberman-Lumer adaptive control and the LMS adaptive algorithm. We illustrate the applicability of this method using Chua's oscillators as the chaotic systems. We choose parameters for the two systems which are orders of magnitude apart to illustrate the effectiveness of the adaptive controllers. Finally, we discuss the role of adaptive synchronization in the context of secure and spread spectrum communication systems. In particular, we show how several signals can be encoded onto a single scalar chaotic carrier signal.
TL;DR: An adaptive control technique, using dynamic structure Gaussian radial basis function neural networks, that grow in time according to the location of the system's state in space is presented for the affine class of nonlinear systems having unknown or partially known dynamics.
Abstract: An adaptive control technique, using dynamic structure Gaussian radial basis function neural networks, that grow in time according to the location of the system's state in space is presented for the affine class of nonlinear systems having unknown or partially known dynamics. The method results in a network that is "economic" in terms of network size, for cases where the state spans only a small subset of state space, by utilizing less basis functions than would have been the case if basis functions were centered on discrete locations covering the whole, relevant region of state space. Additionally, the system is augmented with sliding control so as to ensure global stability if and when the state moves outside the region of state space spanned by the basis functions, and to ensure robustness to disturbances that arise due to the network inherent approximation errors and to the fact that for limiting the network size, a minimal number of basis functions are actually being used. Adaptation laws and sliding control gains that ensure system stability in a Lyapunov sense are presented, together with techniques for determining which basis functions are to form part of the network structure. The effectiveness of the method is demonstrated by experiment simulations.
TL;DR: It is shown that not all work-conserving policies are stable for such networks; however, all workThe stability and instability of Kelly-type networks are shown to be stable in a ring network.
Abstract: Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab.5 49--77. which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served FBFS and Last-Buffer-First-Served LBFS disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.
TL;DR: In this paper, a discrete (integer-valued) Lyapunov function V for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition was defined, and the values of V were derived from the real parts of the Floquet multipliers for such linear periodic systems.
TL;DR: This work considers a single-input/single-output (SISO) nonlinear system which has a well-defined normal form with asymptotically stable zero dynamics and designs an output feedback controller which regulates the output to a constant reference.
Abstract: We consider a single-input/single-output (SISO) nonlinear system which has a well-defined normal form with asymptotically stable zero dynamics. We allow the system's equation to depend on constant uncertain parameters and disturbance inputs which do not change the relative degree. Our goal is to design an output feedback controller which regulates the output to a constant reference. The integral of the regulation error is augmented to the system equation, and a robust output feedback controller is designed to bring the state of the closed-loop system to a positively invariant set. Once inside this set, the trajectories approach a unique equilibrium point at which the regulation error is zero. We give regional as well as semiglobal results.
01 Sep 1996
TL;DR: In this article, three types of optimal, continuously time-varying sliding modes for robust control of second-order uncertain dynamic systems subject to input constraint are presented, two of them incorporate straight sliding lines, and the third uses the so-called terminal slider, that is a curve that guarantees system error convergence to zero in finite time.
Abstract: Three types of optimal, continuously time-varying sliding mode for robust control of second-order uncertain dynamic systems subject to input constraint are presented. Two of the modes incorporate straight sliding lines, and the third uses the so-called terminal slider, that is a curve that guarantees system error convergence to zero in finite time. At first, all three lines adapt themselves to the initial conditions of the system, and afterwards they move in such a way that, for each of them, the integral of the absolute value of the systems error is minimised over the whole period of the control action. By this means, insensitivity of the system to external disturbances and parameter uncertainties is guaranteed from the very beginning of the proposed control action, and the system error convergence rate can be increased. Performance of the three control algorithms is compared, and the Lyapunov theory is used to prove the existence of a sliding mode on the lines.
TL;DR: A quadratic-type Lyapunov function for the flow of a competitive neural system with fast and slow dynamic variables as a global stability method and a modality of detecting the local stability behavior around individual equilibrium points is presented.
Abstract: The dynamics of complex neural networks must include the aspects of long-and short-term memory. The behavior of the network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. The main idea of this paper is to apply a stability analysis method of fixed points of the combined activity and weight dynamics for a special class of competitive neural networks. We present a quadratic-type Lyapunov function for the flow of a competitive neural system with fast and slow dynamic variables as a global stability method and a modality of detecting the local stability behavior around individual equilibrium points.
11 Dec 1996
TL;DR: In this paper, a control design method for nonlinear systems based on control Lyapunov functions and inverse optimality is analyzed, which is shown to recover the LQ optimal control when applied to linear systems.
Abstract: A control design method for nonlinear systems based on control Lyapunov functions and inverse optimality is analyzed. This method is shown to recover the LQ optimal control when applied to linear systems. More generally, it is shown to recover the optimal control whenever the level sets of the control Lyapunov function match those of the optimal value function. The method can be readily applied to feedback linearizable systems, and the resulting inverse optimal control law is generally much different from the linearizing control law. Examples in two dimensions are given to illustrate both the strengths and the weaknesses of the method.
TL;DR: A kind of global controller design method can be developed, and thus the disadvantage of using fixed P in the Lyapunov function can be overcome and a constructive algorithm is developed to obtain the stabilizing feedback control law.
Abstract: This paper presents a design method for fuzzy control systems. The method is based on a fuzzy state-space model A suitable piecewise smooth quadratic (PSQ) Lyapunov function is used to establish asymptotic stability of the closed-loop system. With the PSQ Lyapunov function a kind of global controller design method can be developed, and thus the disadvantage of using fixed P in the Lyapunov function can be overcome. Furthermore, a constructive algorithm is developed to obtain the stabilizing feedback control law. The controller design algorithm involves solving a set of certain algebraic Riccati equations. An example is given to illustrate the application of the method