Showing papers on "Lyapunov function published in 2001"

Virtual leaders, artificial potentials and coordinated control of groups

Abstract: We present a framework for coordinated and distributed control of multiple autonomous vehicles using artificial potentials and virtual leaders. Artificial potentials define interaction control forces between neighboring vehicles and are designed to enforce a desired inter-vehicle spacing. A virtual leader is a moving reference point that influences vehicles in its neighborhood by means of additional artificial potentials. Virtual leaders can be used to manipulate group geometry and direct the motion of the group. The approach provides a construction for a Lyapunov function to prove closed-loop stability using the system kinetic energy and the artificial potential energy. Dissipative control terms are included to achieve asymptotic stability. The framework allows for a homogeneous group with no ordering of vehicles; this adds robustness of the group to a single vehicle failure.

Stabilization of stochastic nonlinear systems driven by noise of unknown covariance

Abstract: This paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems. For an exemplary class of stochastic strict-feedback systems with vanishing nonlinearities, where the equilibrium is preserved in the presence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance. Next, we introduce a control Lyapunov function formula for stochastic disturbance attenuation. Finally, we address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution.

Stability tests for constrained linear systems

Abstract: This paper is yet another demonstration of the fact that enlarging the design space allows simpler tools to be used for analysis. It shows that several problems in linear systems theory can be solved by combining Lyapunov stability theory with Finsler’s Lemma. Using these results, the differential or difference equations that govern the behavior of the system can be seen as constraints. These dynamic constraints, which naturally involve the state derivative, are incorporated into the stability analysis conditions through the use of scalar or matrix Lagrange multipliers. No a priori use of the system equation is required to analyze stability. One practical consequence of these results is that they do not necessarily require a state space formulation. This has value in mechanical and electrical systems, where the inversion of the mass matrix introduces complicating nonlinearities in the parameters. The introduction of multipliers also simplify the derivation of robust stability tests, based on quadratic or parameter-dependent Lyapunov functions.

Continuous-time analysis, eigenstructure assignment, and H/sub 2/ synthesis with enhanced linear matrix inequalities (LMI) characterizations

Abstract: This note describes a new framework for the analysis and synthesis of control systems, which constitutes a genuine continuous-time extension of results that are only available in discrete time. In contrast to earlier results the proposed methods involve a specific transformation on the Lyapunov variables and a reciprocal variant of the projection lemma, in addition to the classical linearizing transformations on the controller data. For a wide range of problems including robust analysis and synthesis, multichannel H/sub 2/ stateand output-feedback syntheses, the approach leads to potentially less conservative linear matrix inequality (LMI) characterizations. This comes from the fact that the technical restriction of using a single Lyapunov function is to some extent ruled out in this new approach. Moreover, the approach offers new potentials for problems that cannot be handled using earlier techniques. An important instance is the eigenstructure assignment problem blended with Lyapunov-type constraints which is given a simple and tractable formulation.

Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems

Abstract: Motivated by control Lyapunov functions and Razumikhin theorems on stability of time delay systems, we introduce the concept of control Lyapunov-Razumikhin functions (CLRF). The main reason for considering CLRFs is construction of robust stabilizing control laws for time delay systems. Most existing universal formulas that apply to CLFs, are not applicable to CLRFs. It turns out that the domination redesign control law applies, achieving global practical stability and, under an additional assumption, global asymptotic stability. This additional assumption is satisfied in the practically important case when the quadratic part of a CLRF is itself a CLRF for the Jacobian linearization of the system. The CLRF based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin. While, in general, construction of CLRFs is an open problem, we show that for several classes of time delay systems a CLRF can be constructed in a systematic way.

Absolute Stability of Nonlinear Systems of Automatic Control

Abstract: The problem of absolute stability of an ?indirect control? system with a single nonlineartiy ia investlgated by using a method which differs from the second method of Lyapunov. The main condition of the obtained criterion of absolute stability is expressed in terms of the transfer function of the system linear part. It is also shown that by fonning the standard Lyapunov function -?a quadratic form plus the integral of the nonlinearity? It is not possible In the case considered here to obtain a wider stability domain than the one obtained from the presented criterion. Graphical criteria of absolute continuinty are also given by means of the phase-amplitude characteristic or by what is known as ?the modified phase-amplitude characteristic? of the system linear part.

Uniform asymptotic stability of impulsive delay differential equations

Abstract: In this paper, criteria on uniform asymptotic stability are established for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques. It is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behavior. Some examples are also discussed to illustrate the theorems.

LPV system analysis via quadratic separator for uncertain implicit systems

Abstract: This paper considers a class of linear systems containing time-varying parameters whose behavior is not known exactly. We assume that the parameters vary within known intervals and there are known bounds on their rates of variation. Our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations. We first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model. We then develop a tool, called the quadratic separator, for stability analysis of uncertain implicit systems. Using this tool, a sufficient stability condition is obtained for the original linear parameter-varying system. Moreover, we show that the stability condition obtained is equivalent to the existence of a Lyapunov function that depends on the parameters in a linear fractional manner. Finally, the computational aspects of the proposed stability conditions are addressed.

Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty

Abstract: Simple modifications to the representation of the bounded real lemma (BRL) in the linear continuous time-invariant case are introduced. These modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic-type uncertainties. A different Lyapunov function accompanies each of the vertices of the uncertainty polytope, thus eliminating the need for quadratic stability or stabilizability. The advantages of these new representations are demonstrated by way of two examples.

Hybrid control of formations of robots

Abstract: We describe a framework for controlling a group of nonholonomic mobile robots equipped with range sensors. The vehicles are required to follow a prescribed trajectory while maintaining a desired formation. By using the leader-following approach, we formulate the formation control problem as a hybrid (mode switching) control system. We then develop a decision module that allows the robots to automatically switch between continuous-state control laws to achieve a desired formation shape. The stability properties of the closed-loop hybrid system are studied using the Lyapunov theory. We do not use explicit communication between robots; instead we integrate optimal estimation techniques with nonlinear controllers. Simulation and experimental results verify the validity of our approach.

Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions

Abstract: It is demonstrated that many previously reported Lyapunov-based stability conditions for time-delay systems are equivalent to the robust stability analysis of an uncertain comparison system free of delays via the use of the scaled small-gain lemma with constant scales. The novelty of this note stems from the fact that it unifies several existing stability results under the same framework. In addition, it offers insights on how new, less conservative results can be developed.

Robust and reduced-order filtering: new LMI-based characterizations and methods

Abstract: This paper addresses several challenging problems of robust filtering. We derive new linear matrix inequality (LMI) characterizations of minimum variance or H/sub 2/ performance and demonstrate that they allow the use of parameter-dependent Lyapunov functions while preserving tractability of the problem. The resulting conditions are less conservative than earlier techniques, which are restricted to fixed (not parameter-dependent) Lyapunov functions. The remainder of the paper discusses reduced-order filter problems. New LMI-based nonconvex optimization formulations are introduced for the existence of reduced-order filters, and several efficient optimization algorithms of local and global optimization are proposed. Nontrivial and less conservative relaxation techniques are presented as well. The viability and efficiency of the proposed approaches are then illustrated through computational experiments and comparisons with existing methods.

Nonlinear predictive control : theory and practice

Abstract: * Part I * Chapter 1: Review of nonlinear model predictive control applications * Chapter 2: Nonlinear model predictive control: issues and applications * Part II * Chapter 3: Model predictive control: output feedback and tracking of nonlinear systems * Chapter 4: Model predictive control of nonlinear parameter varying systems via receding horizon control Lyapunov functions * Chapter 5: Nonlinear model-algorithmic control for multivariable nonminimum-phase processes * Part III * Chapter 6: Open-loop and closed-loop optimality in interpolation MPC * Chapter 7: Closed-loop predictions in model based predictive control linear and nonlinear systems * Chapter 8: Computationally efficient non linear predictive control algorithm for control of constrained nonlinear systems * Part IV * Chapter 9: Long-prediction-horizon nonlinear model predictive control * Chapter 10: Nonlinear control of industrial processes * Chapter 11: Nonlinear model based predictive control using multiple local models * Chapter 12: Neural network control of a gasoline engine with rapid sampling

A fuzzy Lyapunov approach to fuzzy control system design

Abstract: This paper discusses the stability of Takagi-Sugeno fuzzy models via the so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. Based on a fuzzy Lyapunov approach, we gives the stability conditions for open-loop fuzzy systems. All the conditions derived here are represented in terms of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Hence, the treatment of the upper bounds play an important and effective role in system analysis and design. In addition, relaxed stability conditions are also derived by considering the property of the time derivative of premise membership functions. Several analysis and design examples illustrate the utility of the fuzzy Lyapunov approach.

On finite element analysis of fluid flows fully coupled with structural interactions

Abstract: The solution of fluid flows, modeled using the Navier-Stokes or Euler equations, fully coupled with structures/solids is considered. Simultaneous and parti- tioned solution procedures, used in the solution of the coupled equations, are briefly discussed, and advantages and disadvantages of their use are mentioned. In addi- tion, a simplified stability analysis of the interface equa- tions is presented, and unconditional stability for certain choices of time integration schemes is shown. Further- more, the long-term dynamic stability of fluid-structure interaction systems is assessed by the use of Lyapunov characteristic exponents, which allow differentiating be- tween a chaotic and a regular system behavior. Some state-of-the-art numerical solutions are also presented to indicate the type of problems that can now be solved us- ing currently available techniques.

Nonlinear control of hydraulic robots

Abstract: This paper addresses the control problem of hydraulic robot manipulators. The backstepping design methodology is adopted to develop a novel nonlinear position tracking controller. The tracking errors are shown to be exponentially stable under the proposed control law. The controller is further augmented with adaptation laws to compensate for parametric uncertainties in the system dynamics. Acceleration feedback is avoided by using two new adaptive and robust sliding-type observers. The adaptive controllers are proven to be asymptotically stable via Lyapunov analysis. Simulation and experimental results performed with a hydraulic Stewart platform demonstrate the effectiveness of the approach.

Neural network approaches to dynamic collision-free trajectory generation

Abstract: In this paper, dynamic collision-free trajectory generation in a nonstationary environment is studied using biologically inspired neural network approaches. The proposed neural network is topologically organized, where the dynamics of each neuron is characterized by a shunting equation or an additive equation. The state space of the neural network can be either the Cartesian workspace or the joint space of multi-joint robot manipulators. There are only local lateral connections among neurons. The real-time optimal trajectory is generated through the dynamic activity landscape of the neural network without explicitly searching over the free space nor the collision paths, without explicitly optimizing any global cost functions, without any prior knowledge of the dynamic environment, and without any learning procedures. Therefore the model algorithm is computationally efficient. The stability of the neural network system is guaranteed by the existence of a Lyapunov function candidate. In addition, this model is not very sensitive to the model parameters. Several model variations are presented and the differences are discussed. As examples, the proposed models are applied to generate collision-free trajectories for a mobile robot to solve a maze-type of problem, to avoid concave U-shaped obstacles, to track a moving target and at the same to avoid varying obstacles, and to generate a trajectory for a two-link planar robot with two targets. The effectiveness and efficiency of the proposed approaches are demonstrated through simulation and comparison studies.

Control Lyapunov Functions for Controllable Series Devices

Abstract: Summary form only given as follows. Controllable series devices (CSD), i.e. series-connected flexible AC transmission systems (FACTS) devices, such as unified power controller (UPFC), controllable series capacitor (CSC) and quadrature boosting transformer (QBT) with a suitable control scheme can improve transient stability and help to damp, electromechanical oscillations. For these devices, a general model, which is referred to as the injection model, is used. This model is valid for load flow and angle stability analysis and is helpful for understanding the impact of the CSD on power system stability. Also, based on Lyapunov theory a control strategy for damping of electromechanical power oscillations in a multi-machine power system is derived. Lyapunov theory deals with dynamic systems without inputs. For this reason, it has traditionally been applied only to closed-loop control systems, that is, systems for which the input has been eliminated through the substitution of a predetermined feedback control. However, in this paper, we use Lyapunov function candidates in feedback design itself by making the Lyapunov derivative negative when choosing the control. This control strategy is called control Lyapunov function (CLF) for systems with control inputs.

Global stability of relay feedback systems

Abstract: For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions.

Flight control design using extended state observer and non-smooth feedback

Abstract: This paper proposes a novel nonlinear approach for high performance flight control design. The dynamic linearization is accomplished via a kind of unknown input observer, called extended state observer. A nonsmooth feedback law is employed to achieve the desirable dynamic performances. A Lyapunov function is constructed for the proposed method.

A control Lyapunov function approach to multi-agent coordination

Abstract: In this paper, the multiagent coordination problem is studied. This problem is addressed for a class of robots for which control Lyapunov functions can be found. The main result is a suite of theorems about formation maintenance, task completion time, and formation velocity. It is also shown how to moderate the requirement that, for each individual robot, there exists a control Lyapunov function. An example is provided that illustrates the soundness of the method.