Smooth Lyapunov Function Research Articles

MIMO Super-Twisting Controller using a passivity-based design

A novel MIMO Super-Twisting Algorithm is proposed in this paper for nonlinear systems with relative degree one, having a time and state-varying uncertain control matrix. The uncertainty is represented by a time and state-varying but unknown left matrix factor. Sufficient conditions for stability with full-matrix control gains are established, in contrast to the usual scalar gains. For this a smooth Lyapunov function, based on a passivity interpretation, is used. Moreover, continuous and homogeneous approximations of the classical discontinuous Super-Twisting algorithm are obtained, using a unified analysis method. Moreover, the proposed Super-Twisting algorithm is of the discontinuous type, in contrast to the usual unitary type.

A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation

The stochastic approximation (SA) method stands as the foundational mathematical tool for modern large-scale optimization and machine learning. Therefore, gaining a theoretical understanding of SA algorithms is of fundamental interest. In their paper titled “A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation,” Chen et al. present a unified Lyapunov framework for the finite-sample analysis of a Markovian SA algorithm under a contractive operator with respect to an arbitrary norm. The key novelty lies in the construction of a smooth Lyapunov function called the generalized Moreau envelope. The authors demonstrate the effectiveness of their SA results in the context of reinforcement learning (RL), specifically through popular algorithms such as variants of temporal difference (TD) learning and Q-learning. As byproducts, the results provide theoretical insights into the efficiency of bootstrapping in TD learning with eligibility traces and the bias-variance tradeoff in off-policy learning.

Introducing some classes of stable systems without any smooth Lyapunov functions

This paper proves that certain classes of stable nonlinear systems do not admit any smooth Lyapunov functions. In fact, the stability analysis of two different classes of nonlinear dynamical systems is provided, and it is demonstrated that their stability properties cannot be established by any convex Lyapunov functions or smooth Lyapunov functions. In this regard, we begin by studying our first class of autonomous dynamical systems and prove that, despite stability, there are no convex Lyapunov functions in the form of V(x) or V(t,x) to establish the stability properties. For the second class, we consider a much more general form of nonlinear autonomous dynamical systems with a stable origin, and prove that this class, too, does not admit any smooth Lyapunov functions in the form of V(x) or V(t,x). Furthermore, another general class of stable non-autonomous dynamical systems is presented, and it is demonstrated that there is no smooth Lyapunov function in the form of V(x) to establish the stability properties. Finally, for the second class of more general stable non-autonomous systems, it is proved that the class does not admit even a continuous Lyapunov function in the form of V(x) or V(t,x).

Energy Function for Direct Products of Discrete Dynamical Systems

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

High‐order sliding‐mode functional observers for multiple‐input multiple‐output (MIMO) linear time‐invariant systems with unknown inputs

AbstractFor arbitrary multiple‐input multiple‐output linear time invariant systems with unknown inputs this article provides sufficient conditions to estimate linear functionals of the state variables. When the unknown input is uniformly bounded these conditions are strictly weaker than the classical conditions for functional unknown input observers, well‐known in the literature, and generalize previous results using discontinuous differentiators. Furthermore, a general methodology is proposed to design functional observers, that are able to estimate the functionals, when possible, exactly and in finite‐time or fixed‐time. Instead of using a cascade of Luenberger observers and high‐order sliding‐mode differentiators, standard in the literature for this problem, a bi‐homogeneous observer of reduced order is proposed in the article. Proofs of the convergence are provided using smooth Lyapunov functions and an academic example illustrates the behavior of the proposed observer for a system not tractable with the available methods.

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

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.

Smooth converse Lyapunov-barrier theorems for asymptotic stability with safety constraints and reach-avoid-stay specifications

Stability and safety are two important aspects in safety-critical control of dynamical systems. It has been a well established fact in control theory that stability properties can be characterized by Lyapunov functions. Reachability properties can also be naturally captured by Lyapunov functions for finite-time stability. Motivated by safety-critical control applications, such as in autonomous systems and robotics, there has been a recent surge of interests in characterizing safety properties using barrier functions. Lyapunov and barrier functions conditions, however, are sometimes viewed as competing objectives. In this paper, we provide a unified theoretical treatment of Lyapunov and barrier functions in terms of converse theorems for stability properties with safety guarantees and reach-avoid-stay type specifications. We show that if a system (modeled as a dynamical system with measurable perturbations) possesses a stability with safety property, then there exists a smooth Lyapunov function to certify such a property. This Lyapunov function is shown to be defined on the entire set of initial conditions from which solutions satisfy this property. A similar but slightly weaker statement is made for reach-avoid-stay specifications. We show by a simple example that the latter statement cannot be strengthened without additional assumptions. We further extend the results for systems with control inputs and prove existence of converse Lyapunov-barrier functions for reach-and-avoid specifications. One clear limitation of the results of this paper is that the converse Lyapunov-barrier theorems are not constructive, as with classical converse Lyapunov theorems. We believe, however, that the unified necessary and sufficient conditions with a single Lyapunov-barrier function are of theoretical interest and can hopefully shed some light on computational approaches.

Design of Saturating State Feedback With Sign-Indefinite Quadratic Forms

In this article, we propose a novel class of piecewise smooth Lyapunov functions leading to linear-matrix-inequality-based stability/performance analysis and control design for linear systems with saturating inputs. We provide conditions for global properties, and also conditions for local properties and guaranteed estimates of the basin of attraction. The backbone of our result consists in using quadratic forms with constant matrices that are not necessarily sign definite, thereby providing additional degrees of freedom. Using generalized sector conditions involving the dead-zone nonlinearity and its derivative, we formulate convex optimization conditions to verify their positivity in the region of interest. Several numerical examples with connections to existing results illustrate the potential behind our novel construction.

On the Existence of Smooth Lyapunov Functions for Arbitrary Closed Sets

On the Existence of Smooth Lyapunov Functions for Arbitrary Closed Sets

Dynamical properties of direct products of discrete dynamical systems

A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established in the paper is that the direct product of diffeomorphisms holds hyperbolic structure on the direct product of hyperbolic sets. It is known that if a diffeomorphism has a hyperbolic chain-recurrent set, then this mapping is Ω-stable. Therefore, it follows from the results of the paper that the direct product of Ω-stable diffeomorphisms is also Ω-stable. Another question which is raised in the article concerns the existence of an energy function for the direct product of diffeomorphisms which already have such functions (recall that energy function is a smooth Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system). Authors show that in this case the function can be found as a weighted sum of energy functions of initial diffeomorphisms.

Enhanced Force Estimation for Electromechanical Brake Actuators in Transportation Vehicles

In transportation vehicles, the installation of force sensors for electromechanical brake systems causes cost issues and implementation difficulties. The elimination of the sensor is highly demanded. The main vulnerability of the existing force estimators is that the algorithms either rely on shifting characteristic curves or empirical data to determine the major quantities. Hence, they are less capable of dealing with initialization offset errors, fast-changing dynamics, and uncertainties. This article presents a novel super-twisting observer to accurately estimate the clamping force on the basis of a dynamic torque-balance model. In this observer, an extended state is defined on top of a super-twisting algorithm. The observer gains are computed through a smooth Lyapunov function to guarantee the fast convergence. It is experimentally validated that the observer enhances the performance required by the antilock braking systems in comparison with conventional methods.

Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle

Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle

A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

Existence of an energy function for three-dimensional chaotic "sink-source" cascades.

This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any A-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of a chaotic one-dimensional canonically embedded surface attractor and repeller.

On necessary and sufficient conditions for output finite-time stability

Output global finite-time stability of locally Lipschitz continuous autonomous systems is characterized by means of smooth Lyapunov functions. The so-called output-Lagrange stable systems are studied with details. Influence of a kind of continuity of the settling-time function is considered. Necessary and sufficient conditions of output finite-time stability are presented. The theoretical results are supported by academic examples and numerical simulations.

Discontinuous integral action for arbitrary relative degree in sliding-mode control

For uncertain minimum phase systems with arbitrary and well-defined relative degree, we propose the implementation of a sliding-mode control through a homogeneous discontinuous integral action. The output (sliding) variable will track, exactly and in finite-time, any sufficiently smooth reference, while rejecting smooth perturbations with an absolutely continuous control signal. This controller generalizes the well-known super-twisting algorithm to arbitrary order. Continuous and homogeneous integral controllers are also proposed approximating the behaviour of the discontinuous controller. Smooth Lyapunov functions are derived to show the convergence and robustness properties of all controllers. The performance is illustrated by means of experimental results on a magnetic suspension system.

The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

Homogeneous integral controllers for a magnetic suspension system

In this work a discontinuous integral controller is applied to a magnetic suspension system to robustly track a time-varying reference, while rejecting non-vanishing time-varying perturbations or uncertainties. Moreover, a family of continuous integral controllers, including a linear one, are derived and its performance is compared with the discontinuous algorithm using both, theoretical arguments, simulations and experiments, performed at a laboratory scale suspension system. The theoretical results are established for the family of controllers using smooth Lyapunov functions. It is shown experimentally that the discontinuous integral controller outperforms the continuous controllers, including the linear one, because it can achieve a high tracking precision for constant and time-varying references in presence of time-varying perturbations and despite of the sensor noise. Moreover, it avoids the limit cycle oscillation caused by the dry friction on the system controlled by the linear algorithm.

Adaptive region tracking control with prescribed transient performance for autonomous underwater vehicle with thruster fault

This paper investigates the problem of region tracking control with prescribed transient performance for autonomous underwater vehicles (AUVs) in presence of ocean current and thruster fault. An adaptive region tracking control scheme with prescribed transient performance is proposed. On the basis of prescribed performance control concept, the proposed control scheme transforms the tracking error into a virtual error by combining with a prescribed performance function. And then the control law is derived by constructing a type piecewise and smooth Lyapunov function in the framework of backstepping technique. Moreover, the compensation of ocean current, modelling uncertainty and thruster fault is achieved by estimating the bound of the general uncertainty term and the variation of the thruster distribution matrixes. Finally, the proposed region tracking control scheme is applied on ODIN AUV, and the simulation results verify the effectiveness of the proposed method.