Strict Lyapunov Function Research Articles

Compensation of Time-Varying Input and State Delays for Nonlinear Systems

We consider general nonlinear systems with time-varying input and state delays for which we design predictor-based feedback controllers. Based on a time-varying infinite-dimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization.

Every ordinary differential equation with a strict Lyapunov function is a gradient system

We explain and prove the statement from the title. This allows us to formulate a new type of gradient inequality and to obtain a new stabilization result for gradient-like ordinary differential equations.

Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise

The control of human heart rate during exercise is an important problem that has implications for the development of protocols for athletics, assessing physical fitness, weight management, and the prevention of heart failure. Here we provide a new stabilization technique for a recently-proposed nonlinear model for human heart rate response that describes the central and peripheral local responses during and after treadmill exercise. The control input is the treadmill speed, and the control objective is to make the heart rate track a prescribed reference trajectory. We use a strict Lyapunov function analysis to design new state and output feedback tracking controllers that render the error dynamics globally exponentially stable. This allows us to prove input-to-state stability properties for our feedback stabilized systems under actuator errors. This robustness property quantifies the effects of variations of the treadmill speed from the controller values. We illustrate our feedback design through simulations.

Global trajectory tracking through output feedback for robot manipulators with bounded inputs

In this work, a globally stabilizing output feedback scheme for the trajectory tracking of robot manipulators with bounded inputs is proposed. It achieves the motion control objective avoiding input saturation and excluding velocity measurements. Moreover, it is not defined using a specific sigmoidal function, but any one on a set of saturation functions. Consequently, the proposed scheme actually constitutes a family of globally stabilizing output feedback bounded controllers. Furthermore, the control gains are not tied to satisfy any saturation-avoidance inequality and may consequently take any positive value, which may be considered beneficial for performance adjustment/improvement purposes. Further, a class of desired trajectories that may be globally tracked avoiding input saturation and excluding velocity measurements is completely characterized. Global asymptotic stabilization of the closed-loop system solutions towards the pre-specified desired trajectory is proved through a strict Lyapunov function. The efficiency of the proposed scheme is corroborated through experimental results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Strict Lyapunov functions for semilinear parabolic partial differential equations

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains

We study adaptive tracking problems for nonlinear systems with unknown control gains. We construct controllers that yield uniform global asymptotic stability for the error dynamics, and hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrate our work using a brushless DC motor turning a mechanical load. We quantify the effects of time-varying uncertainties on the motor electric parameters.

Strict Lyapunov Function Constructions Under LaSalle Conditions With an Application to Lotka-Volterra Systems

We provide new techniques for building explicit global strict Lyapunov functions for broad classes of periodic time varying nonlinear systems satisfying LaSalle conditions. We illustrate our work using the Lotka-Volterra model, which plays a fundamental role in bioengineering. We use our strict Lyapunov constructions to prove robustness of the Lotka-Volterra tracking dynamics to uncertainty in the death rates.

Global Trajectory Tracking Through Static Feedback for Robot Manipulators With Bounded Inputs

In this work, two globally stabilizing bounded control schemes for the tracking control of robot manipulators with saturating inputs are proposed. They may be seen as extensions of the so-called PD+ algorithm to the bounded input case. With respect to previous works on the topic, the proposed approaches give a global solution to the problem through static feedback. Moreover, they are not defined using a specific sigmoidal function, but any one on a set of saturation functions. Consequently, each of the proposed schemes actually constitutes a family of globally stabilizing bounded controllers. Furthermore, the bound of such saturation functions is explicitly considered in their definition. Hence, the control gains are not tied to satisfy any saturation-avoidance inequality and may consequently take any positive value, which may be considered beneficial for performance-adjustment/improvement purposes. Further, a class of desired trajectories that may be globally tracked avoiding input saturation is completely characterized. For both proposed control laws, global uniform asymptotic stabilization of the closed-loop system solutions towards the prespecified desired trajectory is proved through a strict Lyapunov function. The efficiency of the proposed schemes is corroborated through experimental results.

Uniform Global Asymptotic Stability of a Class of Adaptively Controlled Nonlinear Systems

We give a new explicit, global, strict Lyapunov function construction for the error dynamics for adaptive tracking control problems, under an appropriate persistency of excitation condition. We then allow time-varying uncertainty in the unknown parameters. In this case, we construct input-to-state stable Lyapunov functions under suitable bounds on the uncertainty, provided the regressor also satisfies an affine growth condition. This lets us quantify the effects of uncertainties on both the tracking and the parameter estimation. We illustrate our results using Rossler systems.

Stabilization in a Two-Species Chemostat With Monod Growth Functions

We design feedback controllers for two species chemostats so that an equilibrium with arbitrary prescribed positive species concentrations becomes globally asymptotically stable. We use a new global explicit strict Lyapunov function construction, which allows us to quantify the effects of disturbances using the input-to-state stability paradigm. We assume that only a linear combination of the species concentrations is known. We illustrate our approach using a numerical example.

Extended LaSalle's Invariance Principle for Full-Range Cellular Neural Networks

The paper develops a Lyapunov method, which is based on a generalized version of LaSallepsilas invariance principle, for studying convergence and stability of the differential inclusions modeling the dynamics of the full-range (FR) model of cellular neural networks (CNNs). The method is applied to yield a rigorous proof of convergence for symmetric FR-CNNs. The proof, which is a direct consequence of the fact that a symmetric FR-CNN admits a strict Lyapunov function, is much more simple than the corresponding proof of convergence for symmetric standard CNNs.

Nonuniform exponential contractions and Lyapunov sequences

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict Lyapunov sequences. We note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. We also obtain “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. We also show how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. Moreover, we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.

Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size.

Global Trajectory Tracking Through Output Feedback for Robot Manipulators With Input Saturations

Abstract In this work, a globally stabilizing output feedback scheme for the motion control of robot manipulators with bounded inputs is proposed. It achieves the trajectory tracking goal avoiding input saturation and excluding velocity measurements. Moreover, it is not defined using a specific sigmoidal function, but any one on a set of saturation functions. Consequently, the proposed algorithm actually constitutes a family of globally stabilizing output feedback bounded controllers. Furthermore, the bound of such saturation functions is explicitly considered in their definition. Hence, the control gains are not tied to satisfy any saturation-avoidance inequality and may consequently take any positive value, which may be considered beneficial for performance adjustment/improvement purposes. Further, a class of desired trajectories that may be globally tracked avoiding input saturation and excluding velocity measurements is completely characterized. Global uniform asymptotic stabilization of the closed-loop system solutions towards the pre-specified desired trajectory is proved through a strict Lyapunov function. The efficiency of the proposed scheme is corroborated through experimental results.

On Lyapunov stability of linearised Saint-Venant equations for a sloping channel

We address the issue of the exponential stability (in $L^2$-norm) of the classical solutions of the linearised Saint-Venant equations for a sloping channel. We give an explicit sufficient dissipative condition which guarantees the exponential stability under subcritical flow conditions without additional assumptions on the size of the bottom and friction slopes. The stability analysis relies on the same strict Lyapunov function as in our previous paper [5]. The special case of a single pool is first treated. Then, the analysis is extended to the case of the boundary feedback control of a general channel with a cascade of $n$ pools.

Strict Lyapunov function for sliding mode control of manipulator using quasi-velocities

This paper describes proposition of sliding mode control using strict Lyapunov function for robot manipulators described in terms of some quasi-velocities (QV). The considered here quasi-velocities contain both kinematical and dynamical parameters of a manipulator. Introducing QV together with joint position leads to one first-order dynamic equations with diagonal mass matrix instead of a second-order differential equation and one first-order equation which describes transformation between joint velocities and QV. The presented controller based on strict Lyapunov function guarantees closed-loop stability and global positioning. Differences between classical sliding mode control scheme and the considered here were shown on 3 d. o. f., 3- D Yasukawa-like robot.

Boundary control with integral action for hyperbolic systems of conservation laws: Stability and experiments

A strict Lyapunov function for boundary control with integral actions of hyperbolic systems of conservation laws that can be diagonalised with Riemann invariants, is presented. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions and the integral control gains. Previous stability results are extended to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state measured only at the boundaries. The control design method is illustrated with a hydraulic application, namely the level and flow regulation in a reach of the Sambre river and in the micro-channel of Valence, respectively through simulations and experimentations.

Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems

We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.

ROBUST GLOBALLY EXPONENTIALLY STABILIZING CONTROL LAW FOR FULLY ACTUATED 6-DOF AUVS

This work deals with the global exponential stabilization of the origin of 6- DOF AUVs. The control law is time-invariant and continuous. Unit quaternions are used as representation of orientation, preventing problems caused by the Euler singularity. No simplifications are made in kinematics or dynamics. A strict Lyapunov function is also developed for the controlled system. The system is proven to be ISS with respect to disturbances. This work also presents some highly useful results regarding unit quaternions, and presents a global diffeomorphism for position that eases controller design. Results are verified with simulations on the HUGIN 1000 AUV.

Robustness of PID-controlled Manipulators vis-à-vis Actuator Dynamics and External Disturbances

We present a robustness analysis for PID-controlled robot manipulators. For robot manipulators under the influence of external disturbances and a wide class of neglected dynamics, we provide establish uniform semiglobal practical asymptotic stability and propose a tuning procedure of the PID gains in order to achieve any given precision from any given bounded set of initial conditions. In contrasts to other works on robust stability of PIDs, we do not use La Salle's principle but provide a strict Lyapunov function. The perturbations that we consider include discontinuous functions of the state, such as coulomb friction. As corollaries of our main results, one may conclude the same stability property for the case of motion control using linear PID.