Explicit Lyapunov Function Research Articles

A two-galectin network establishes mesenchymal condensation phenotype in limb development

Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction–diffusion–adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle–node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.

Adaptive delay compensation for consensus control under switching topologies by observer–predictor

This paper considers observer–predictor feedback for delay compensation in consensus control under undirected switching graphs. Each node involved in the network is described by a linear system with an actuator delay of uncertain length, and the connection graph is allowed to be unknown and switch arbitrarily. An observer is proposed to estimate the disagreement state using locally available information, and a delay-adaptive consensus protocol is proposed via observer–predictor feedback. With the employment of a “backstepping” state transformation involving the observer state and the estimation delay state, an explicit Lyapunov function is constructed to prove the consensus and stability properties of the resultant closed-loop system. Numerical simulations are presented to demonstrate the effectiveness of the theoretic results.

A Lyapunov-Based Approach for Recursive Continuous Higher Order Nonsingular Terminal Sliding-Mode Control

A recursive continuous higher order nonsingular terminal sliding-mode (TSM) controller is proposed in this article for nonlinear systems. A new integral TSM manifold is constructed in a recursive manner by modifying the tool of adding a power of integrator instead of exploring nonrecursive design directly. A super-twisting like reaching law is designed to achieve continuous control action without sacrificing disturbance rejection specification as that in boundary-layer approaches. By the new Lyapunov-based design, the proposed control method admits the following new features: first, rather than imposing some existence condition for nonrecursive design, the proposed method admits the certainty for chosen fractional power to guarantee the finite-time stability of the closed-loop system; second, an explicit Lyapunov function approach is proposed to establish finite-time stability of the closed-loop system; and, third, the proposed method is shown to be tunable to exhibit desired transient performance and control energy restriction.

Finite-time trajectory control for a quadrotor aircraft using disturbance observer

In this article, we investigate the problem of finite-time trajectory tracking control for a quadrotor aircraft with unknown external disturbances. To improve convergence rate and disturbance rejection performance, a new composite controller is proposed by integrating finite-time control design and disturbance estimation attenuation technique. Explicit Lyapunov function is given to ensure the finite-time stability of the closed-loop control system. Numerical simulations also show the effectiveness of the proposed method.

입력지연을 갖는 이산시간 선형시스템을 위한 의사예측기 제어기의 안정성 해석

We establish a sufficient condition to guarantee the global exponential stability when the existing pseudo-predictor controller is applied to discrete-time linear systems with constant input delays. By modeling the constant input delay with a first-order PdE (Partial difference Equation) and applying a backstepping transformation, we build a target system that enables the construction of an explicit Lyapunov function. By constructing the explicit Lyapunov function that covers the entire state variables, we derive a sufficient condition to guarantee the global exponential stability. In addition, the proposed analysis method facilitates the analysis of transient state performance because it allows the estimation of an exponential convergence rate. The numerical example demonstrates the validity of the sufficient condition for the global exponential stability and the usefulness of the proposed method for the design and analysis of transient state performance.

Levant's Arbitrary-Order Exact Differentiator: A Lyapunov Approach

We propose a family of smooth explicit Lyapunov functions (LF) for Levant's differentiator, which is a discontinuous real-time robust and exact differentiator. These LF allow the analysis of the convergence and performance properties and the design of appropriate gains for the differentiator.

Robust Model Predictive Control of Linear Systems With Predictable Disturbance With Application to Multiobjective Adaptive Cruise Control

This paper presents a novel robust model predictive control (RMPC) concept for linear time-invariant systems with a predictable additive disturbance and linear constraints on the state and the input. Major properties of the approach are that: 1) available knowledge of the disturbance is considered in the optimization and 2) the robustness and the performance are addressed separately. As a result, the control performance is optimized while a less conservative condition on constraint satisfaction and recursive feasibility compared to the existing RMPC schemes is obtained. Traditionally, the Lyapunov function is chosen as the optimum of the objective function which must usually be quadratic in terms of the state and the input and contain a terminal cost term. These standard assumptions for the stability may restrict the flexibility of the optimization problem formulation and, thus, limit the applicability of the related RMPC strategies. To overcome this limitation, this paper proposes an explicit Lyapunov function and ensures the input-to-state stability (ISS) with a quadratic constraint, allowing to use any arbitrary convex objective function. To evaluate the novel RMPC concept, a multiobjective adaptive cruise control (ACC) is proposed and a simulation study using measured velocity profiles for the leading vehicle on a highway is presented. In the evaluation, a less restrictive constraint tightening and a larger terminal constraint set compared to the classical RMPC policies could be found and multiple objectives including driving comfort, energy efficiency, stability, and robust recursive fulfillment of safety, velocity, and further physical constraints could be achieved with the novel RMPC concept.

Magnetic attitude control of bias momentum spacecraft by bounded linear feedback

This paper addresses the three-axis magnetic attitude control of small spacecraft with momentum bias configuration by bounded linear feedback. Necessary and sufficient conditions are derived to guarantee that the linearized dynamics and kinematics are stable in the Lyapunov sense. Explicit solutions to some Lyapunov equations associated with the considered open-loop system are obtained. Based on the explicit solutions to the Lyapunov equation, both explicit saturated linear state feedback controller and function observer based output feedback controller are designed. Globally asymptotic stability of the closed-loop systems is proved by constructing explicit Lyapunov functions. Simulations show the effectiveness of the proposed approach.

Observers for a non-Lipschitz triangular form

We address the problem of designing an observer for triangular non locally Lipschitz dynamical systems. We show the convergence with an arbitrary small error of the classical high gain observer in presence of nonlinearities verifying some Hölder-like condition. Also, for the case when this Hölder condition is not verified, we propose a novel cascaded high gain observer. Under slightly more restrictive assumptions, we prove the convergence of a homogeneous observer and of its cascaded version with the help of an explicit Lyapunov function.

A quadratic Lyapunov function for hyperbolic density–velocity systems with nonuniform steady states

A new explicit Lyapunov function allows to study the exponential stability for a class of physical ‘2 by 2’ hyperbolic systems with nonuniform steady states. In fluid dynamics, this class of systems involves isentropic Euler equations and Saint-Venant equations. The proposed quadratic Lyapunov function allows to analyze the local exponential stability of the system equilibria for suitable dissipative Dirichlet boundary conditions without additional conditions on the system parameters.

Global stabilization of periodic linear systems by bounded controls with applications to spacecraft magnetic attitude control

We study in this paper the three-axis magnetic attitude control of small spacecraft by considering the actuator saturation. By noticing that the linearized dynamics is a neutrally stable linear periodic system, a general problem of global stabilization of linear periodic systems with bounded feedback is formulated and solved by both state feedback and observer based output feedback. Explicit saturated linear feedback control laws are established by using solutions to some Lyapunov differential equations associated with the open-loop systems and global stability of the closed-loop system is proved by constructing an explicit Lyapunov function. To apply the theoretical results on the three-axis magnetic attitude control systems, necessary and sufficient conditions are established to guarantee that the linearized open-loop system is neutrally stable and explicit solutions to the associated Lyapunov equations are obtained consequently. The controllers for the three-axis magnetic attitude control systems are then given in closed-form. Simulations show the effectiveness of the proposed approach.

Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes

The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state weakly interacting N-particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using results from [5] and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.

An alternative converse Lyapunov theorem for discrete-time systems

This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and for globally asymptotically stable systems, Lyapunov functions on a set [a,b] with 0<a<b<∞ are derived. Furthermore, for specific classes of systems, the developed converse Lyapunov theorem can be used to establish non-conservatism of existence of a particular type of Lyapunov functions. Most notably, a proof that the existence of conewise linear Lyapunov functions is non-conservative for GES conewise linear systems is given and, as a by-product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.

입력 지연을 갖는 이산시간 선형 시스템을 위한 예측기 피드백의 지수적 안정성

We consider discrete-time LTI (Linear Time-Invariant) systems with constant input delays. The input delay is modeled by a first-order PdE (Partial difference Equation) and a backstepping transformation is employed to design a predictor feedback controller. The backstepping approach results in the construction of an explicit Lyapunov function, with which we prove the exponential stability of the closed-loop system formed by the predictor feedback. The numerical example demonstrates the design of the predictor feedback controller, and illustrates the validity of the exponential stability.

H 2-stabilization of the Isothermal Euler equations: a Lyapunov function approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H 2-norm. To this end, an explicit Lyapunov function as a weighted and squared H 2-norm of a small perturbation of the stationary solution is constructed. The authors show that by a suitable choice of the boundary feedback conditions, the H 2-exponential stability of the stationary solution follows. Due to this fact, the system is stabilized over an infinite time interval. Furthermore, exponential estimates for the C 1-norm are derived.

LYAPUNOV FUNCTION AND GLOBAL PROPERTIES OF VIRUS DYNAMICS WITH CTL IMMUNE RESPONSE

The stability of infections disease model with CTL immune response in vivo is considered in this paper. Explicit Lyapunov functions for our dynamics model with CTL immune response with nonlinear incidence of the form βVqTpfor the case q ≤ 1 are introduced, and global properties of the model are thereby established.

Further Results on Stabilization of Periodic Trajectories for a Chemostat With Two Species

We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation.

A Lyapunov's direct method for the global robust stabilization of nonlinear cascaded systems

The global robust stabilization problem of cascaded systems with dynamic uncertainty has been approached by the small gain theorem. This method, however, does not produce an explicit Lyapunov function for the closed-loop system. In this paper, we develop a Lyapunov's direct method based recursive approach to solving the global robust stabilization problem for the mentioned systems. This method also produces an explicit Lyapunov function for the closed-loop system which is a superposition of those of individual subsystems. This Lyapunov function is indispensable when the adaptive control of the same class of systems is further considered.

Lyapunov/Passivity-Based Adaptive Control of Relative Degree Two MIMO Systems With an Application to Visual Servoing

This note presents a Lyapunov-based design of model-reference adaptive control (MRAC) for multiple-input-multiple-output (MIMO) systems of uniform relative degree two. It consists of an extension of a well known MRAC Lyapunov-based scheme for single-input-single-output (SISO) plants with relative degree one or two. The case of relative degree one, was completely extended to the MIMO case only recently. The corresponding extension of the relative degree two case remained hitherto unpublished. While MIMO MRAC schemes exist which can deal with arbitrary relative degree, the one presented here has the significant advantage of possessing an explicit Lyapunov function which easily leads to a stability and error convergence proof. It can also be regarded as a passivity-based approach. As a result, it becomes easier to guarantee stability when connecting such adaptive system to other adaptive control systems with similar passivity properties, e.g., in adaptive visual servoing of robot manipulators

Further results on Lyapunov functions for slowly time-varying systems

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-state stable Lyapunov functions for slowly time-varying control systems. We illustrate our findings by constructing explicit Lyapunov functions for a pendulum model, an example from identification theory, and a perturbed friction model.