Strict Lyapunov Functions Research Articles

Explicit convergence relations for a class of discrete LTV systems and its application to performance analysis of deterministic learning

A class of linear time-varying (LTV) systems commonly appears in classical adaptive control and identification, in which the accurate identification of parameters is highly related to their exponential stability. However, there is limited research on explicit convergence relations for discrete LTV systems. In this article, the explicit convergence relation of a class of discrete LTV systems is first established, in which strict Lyapunov functions are constructed by considering the convergence properties of the interconnected unforced subsystems. Next, based on the derived explicit convergence relation, a performance analysis of deterministic learning under the sampling-data framework is established. We show that the learning speed and learning accuracy increase with the persistent excitation (PE) level and decrease with the identifier gain. Moreover, an optimal learning gain exists related to the identifier gains. To illustrate the results, simulation studies are included.

A modified Runge–Kutta method for increasing stability properties

In control theory, we are interested mainly in investigating asymptotic stability of systems. In this paper explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear ODEs with known strict Lyapunov functions. In order to increase stability properties, a modified version of explicit Runge–Kutta methods is presented. The modified Runge–Kutta method is explicit. It is different from the projection methods in literature which are implicit. Comparing with the standard Runge–Kutta method, the modified Runge–Kutta one has better stability properties. Numerical examples are provided to illustrate the effectiveness of the modified Runge–Kutta method.

Finite-Time Regulation of Robot Manipulators in the Task-Space

This paper proposes a control scheme in order to attain finite-time regulation of robots manipulators in the task-space. The stability analysis is carried out by means of strict Lyapunov functions since the well-known results of homogeneous systems cannot be applied directly. Simulations results verify the performance of the proposed control scheme.

A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs

A generalised proportional-derivative type scheme for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with input constraints is developed. Its generalised design permits the choice among multiple saturating structures. Suitable functions are involved for the appropriate shaping of every error correction term, and complementarily of their addition, to achieve the concerned type of convergence (in addition to the input saturation avoidance). Such a shaping is carried out through control parameters that act as exponential weights on the gained configuration and generalised velocity errors and on their joint action. Compared to previous finite-time approaches, such exponential weights are required to satisfy generalised comparative conditions, avoiding a fixed equivalence relation among any of them. This permits to give rise to a wider spectrum of finite-time convergent closed-loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one). The resulting extended conditions on the exponential weights and the generalised saturating structure enlarge the controller potential for performance improvement or adjustment. The generalised design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other control objectives. The study is further supported through experimental tests on a 3-degree-of-freedom robot manipulator.

Non-homogeneous observer-based-output feedback scheme for the double integrator

In this paper we propose a non-homogeneous observer-based-output feedback scheme for the double integrator system. The control scheme is composed by two non-homogeneous structures. It resembles a non-homogeneous state feedback controller, where the unmeasurable state is replaced by the that provided by a non-homogeneous observer. Compared to other reported schemes, the closed-loop system is not homogeneous of negative degree, so it is not possible to use the results established for homogeneous systems. To overcome this problem, the finite-time stability analysis is carried out by means of (local) strict Lyapunov functions.

Strict Smooth Lyapunov Functions and Vaccination Control of the SIR Model Certified by ISS

This article addresses analysis and control of the SIR model of infectious diseases in the framework of input-to-state stability (ISS) with respect to the net flow of susceptible individuals into a region in both disease-free and epidemic situations. The key development is the construction of a continuously differentiable strict Lyapunov function. First, this article clarifies that a continuously differentiable Lyapunov function whose derivative is nonpositive can deduce asymptotic stability on the whole state space. Second, it is demonstrated that a new idea of rendering the derivative strictly negative allows one to detect a margin proving ISS of the SIR model. The accomplishment is based on the pursuit of a Lyapunov function that is not entirely separable into components. It contrasts with previously studied and popular Lyapunov functions that are proven to be incapable of assessing robustness properties such as ISS. Third, two types of feedback control laws are proposed for mass vaccination of immigrants and inhabitants by making use of the strict Lyapunov functions. One type modifies the other type by focusing on the reduction of peaks of the infected population within the ISS guarantees.

A more exhaustive study on the finite‐time and exponential tracking continuous control for constrained‐input mechanical systems: Improved design and experiments

AbstractA more exhaustive study on the finite‐time/exponential tracking continuous control for mechanical systems with input constraints is developed. The resulting control scheme is characterized by the inclusion of position‐error and velocity‐error correction terms which are suitably shaped to achieve the concerned types of convergence. Such a shaping is carried out through control parameters that act as exponential weights on the gained position and velocity errors. As an innovative aspect of the proposal, such exponential weights are not restricted to satisfy a fixed equivalence relation among them but rather a wider comparative one. This gives rise to a wider spectrum of finite‐time convergent closed‐loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one), which enlarges the controller potential for performance adjustment. This is further potentiated through a more generalized saturating structure on the error correction terms, that includes the usual saturating‐proportional saturating‐derivative type one as a particular case. The improved design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other (more complex) control objectives. The study is further supported through experimental tests on a 3‐degree‐of‐freedom robot manipulator.

Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination

<p style='text-indent:20px;'>We provide a new global strict Lyapunov function construction for a susceptible, infected, and recovered (or SIR) disease dynamics that includes quarantine of infected individuals and mass vaccination. We use the Lyapunov function to design feedback controls to asymptotically stabilize a desired endemic equilibrium, and to prove input-to-state stability for the dynamics with a suitable restriction on the disturbances. Our simulations illustrate the potential of our feedback controls to reduce peak levels of infected individuals.</p>

Strict Lyapunov Functions for Dynamic Consensus in Linear Systems Interconnected Over Directed Graphs

We study dynamic consensus for general networked (homogeneous) linear autonomous systems, that is, it is only assumed that they are stabilizable. Dynamic consensus pertains to a general form of consensus in which, as a result of the systems’ interactions, they exhibit a rich collective dynamic behavior. This generalizes the classical consensus paradigm in which case all systems stabilize to a common equilibrium point. Our main statements apply to systems interconnected over generic directed connected graphs and, most significantly, the proofs are constructive. Indeed, even though our controllers are reminiscent of others previously used in the literature, to the best of our knowledge, we provide for the first time in the literature strict Lyapunov functions for fully distributed consensus over generic directed graphs.

Trajectory-tracking in finite-time for robot manipulators with bounded torques via output-feedback

In this paper we propose a novel output-feedback controller for robot manipulators that exhibit actuator saturation. This proposal ensures trajectory-tracking in finite-time provided that the desired velocity and acceleration are sufficiently slow. The scheme consists of a continuous dynamic nonlinear Proportional-Derivative (PD)-like control plus a feed-forward compensation term and global uniform finite-time stability of the tracking error is established designing novel strict Lyapunov functions. Further, in presence of small uncertainties in the dynamic model of the robot, we establish global uniform ultimate boundedness of the trajectory tracking error. Simulations are shown to validate the theoretical findings.

Output‐feedback finite‐time and exponential tracking continuous control for mechanical systems with constrained inputs

AbstractAn output‐feedback finite‐time/exponential tracking continuous control scheme for mechanical systems with constrained inputs is proposed. The developed controller has a velocity‐free saturating‐proportional saturating‐derivative type structure, inducing damping through a dirty‐derivative based auxiliary dynamics. Special well‐defined functions that coat the error correction actions of the control scheme give the required shaping for the concerned type of convergence and input saturation avoidance. The type of convergence is determined through the choice on the values of specific control parameters. Differently to similar previous approaches for state‐feedback tracking or output‐feedback regulation, the control parameters that determine the trajectory convergence are not restricted to satisfy a fixed equivalence relation but rather a wider comparative one. This expands the spectrum of finite‐time convergent trajectories that may be induced, and further gives the option of an additional (unconventional) type of exponential stability (in addition to the conventional one). The closed‐loop analysis is supported through suitable strict Lyapunov functions. Experimental tests on a 3‐degree‐of‐freedom robot manipulator further support the contributed result.

Edge-based strict Lyapunov functions for consensus with connectivity preservation over directed graphs

In this paper we address the edge-agreement problem with preserved connectivity for networks of first and second-order systems under proximity constraints and interconnected over a class of directed graphs. We provide a strict Lyapunov function that leads to establishing uniform asymptotic stability of the consensus manifold with guaranteed connectivity preservation. Furthermore, robustness of the edge-agreement protocol, in the sense of input-to-state stability with respect to external input disturbances, is also demonstrated. These results hold for directed-spanning-tree and directed-cycle topologies, which are notably employed, respectively, in leader–follower and cyclic-pursuit control.

Control of underactuated marine crafts with matched disturbances

ABSTRACT Stabilisation of marine craft in the presence of disturbances is important to many offshore activities. The process of stabilising a marine craft using the actuators is known as dynamic positioning, and is challenging for underactuated craft. In our work, we develop a robust dynamic positioning controller for an underactuated marine craft that ensures input-to-state-stability with respect to matched disturbances. Using the strict Lyapunov function approach of Malisoff and Mazenc [(2009). Constructions of strict Lyapunov functions. Springer-Verlag], we show that the proposed controller ensures uniform global asymptotic stability of the desired equilibrium when there are no disturbances. We also show that the closed-loop system is input-to-state stable with respect to matched environmental force and moment disturbances. Finally, we illustrate the performance of the proposed controller through simulations.

A Construction of Strict Lyapunov Functions for A Bilinear Balancing Model

This paper focuses on a two-dimensional compartmental model which is linear except for bilinear terms of balancing kinetics. It is an equation of Lotka-Volterra type, but its structure is not symmetric like the predator-prey equation. This paper elucidates fundamental challenges in non-local stability analysis of the model. In particular, obstacles in constructing strict Lyapunov functions are discussed. A method consisting of tilting and semi-global construction is presented to overcome the challenges. Its benefit is highlighted from the viewpoint of input-to-state stability with respect to inflow perturbation.

Robust trajectory‐tracking in finite‐time for robot manipulators using nonlinear proportional‐derivative control plus feed‐forward compensation

AbstractIn this article, we propose a novel nonlinear control scheme for the trajectory tracking in finite‐time of robot manipulators. Our control proposal is composed of two parts: a proportional‐derivative (PD) nonlinear feedback and a feed‐forward compensation term. The (discontinuous) nonlinear PD term is designed like a twisting sliding mode controller that ensures robust global uniform finite‐time stabilization of the tracking error, in spite of model uncertainties and nonvanishing perturbations. Moreover, in order to decrease the (possible) chattering phenomena, we also propose two alternatives: a variable‐gain version and a continuous version of the PD term. The feed‐forward term allows us to deal intelligently with Coriolis and gravity effects because the compensation for these terms is generated off‐line. In the continuous case, practical stability is guaranteed in the presence of model uncertainties and nonvanishing perturbations. The stability conclusions are established, in both cases, designing novel strict Lyapunov functions.

Strict Lyapunov functions for finite-time control of robot manipulators

Motivated by the energy shaping framework and the properties of homogeneous systems, we introduce a methodology to derive strict Lyapunov Functions (LFs) for a class of global finite-time (FT) controllers for robot manipulators. As usual in the energy shaping methodology, these controllers are described by the gradient of the controller potential energy plus the gradient of (nonlinear) energy dissipation-like functions. Sufficient conditions on the controller potential energy and energy dissipation-like functions are provided in order to obtain, in a straightforward manner, strict LFs ensuring global FT stability at the desired equilibrium. As an important practical outcome, we illustrate the proposed methodology by constructing strict LFs for some particular FT controllers. The construction of such strict LFs allows us to solve the open problem of FT trajectory tracking of robot manipulators with continuous control laws to track any bounded trajectory.

Cascades-Based Leader–Follower Formation Tracking and Stabilization of Multiple Nonholonomic Vehicles

It is known that nonholonomic systems on the plane cannot be stabilized on arbitrary leader's trajectories with piecewise continuous velocities via continuous controllers, even time-varying. In this article, we identify a wide class of smooth bounded reference trajectories (that includes set-points), and we propose a smooth time-varying controller that stabilizes such trajectories, not only for one nonholonomic mobile robot, but also for a network of mobile robots in a leader-follower configuration. We provide new constructions of strict Lyapunov functions with which we establish uniform global asymptotic stability and other strong robustness properties in the formation-error-coordinates space. Thus, our stability statements are unique in the literature and our tools of analysis are original.

A new class of strict Lyapunov functions for nonlinear time-varying systems

This paper focuses on strict Lyapunov functions of nonlinear time-varying systems using homogeneity-based methods. The concept of homogeneous auxiliary system is introduced. Based on the auxiliary system, a new class of strict Lyapunov functions is established. Sufficient conditions of uniform asymptotic stability are provided under the homogeneous framework. The results are extended to deal with robustness problems. It is shown that the nonlinear time-varying systems remain stable in the presence of higher order perturbing terms.

Strict Lyapunov Functions for Model Reference Adaptive Control: Application to Lagrangian Systems

Model reference adaptive control is a well-understood and popular method used in the case when the plant's constant parameters are unknown and must be identified. The related literature is very rich and there exist many proofs of stability and convergence. Lyapunov functions for such systems, having the property that the total derivative is negative definite, are, however, very scarce. In this note, we use the Mazenc construction to design a simple strict Lyapunov function in a rather intuitive manner, based on a first-choice function whose derivative is negative semidefinite. Furthermore, we provide, for the first time in the literature, a Lyapunov function for a popular passivity-based adaptive controller for Lagrangian systems.

Uniform stability of homogeneous time-varying systems

ABSTRACT A sufficient condition for uniform asymptotic stability of nonlinear time-varying systems is proposed within the Narendra-Annaswamy (NA) framework. Based on homogeneity, a class of strict Lyapunov functions is established, in which time-varying terms are included. By using the new Lyapunov functions, concise criterion is provided. Unlike the classical NA approach, divergent time sequence in the criterion is not needed. The utility of our result is illustrated through the study of applications and numerical examples.