Strong Lyapunov Functions Research Articles

Strong Lyapunov Functions for Linear Time-Varying Systems Under Persistency of Excitation

Strong Lyapunov Functions for Linear Time-Varying Systems Under Persistency of Excitation

Strong Lyapunov functions for two classical problems in adaptive control

Strong Lyapunov functions for two classical problems in adaptive control and parameter identification are presented. These Lyapunov functions incorporate in their structure the classical persistency of excitation conditions, allowing to show global uniform asymptotic stability of the associated adaptive systems under sufficient and necessary conditions.

Convexity and the Hele–Shaw Equation

Walter Craig’s seminal works on the water-wave problem established the importance of several exact identities: Zakharov’s hamiltonian formulation, shape derivative formula for the Dirichlet-to-Neumann operator, and normal forms transformations. In this paper, we introduce several identities for the Hele–Shaw equation which are inspired by his nonlinear approach. First, we study convex changes of unknowns and obtain a large class of strong Lyapunov functions; in addition to be non-increasing, these Lyapunov functions are convex functions of time. The analysis relies on a new, simple compact elliptic formulation of the Hele–Shaw equation, which is of independent interest. Then, we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh–Taylor coefficient a (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function $$\varphi (x)=x\log x$$ and find that $$\varphi (1/\sqrt{a})$$ is a sub-solution of a well-posed equation.

Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators

Conicyt grants PIA AFB-170001 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1151003 1190012 Conicyt-Redes 150040 Mathamsud 17-MATH-06 Conicyt-Pcha/Doctorado Nacional/2014-63140104

Higher order sliding mode control for active suspension systems subject to actuator faults and disturbances

As the possibility of faults in active suspension actuators are higher and more severe compared to other components, this study presents a fault-tolerant control approach based on the second-order sliding mode control method. The aim of the controller is to improve riding comfort, guarantee handling stability, and provide adequate suspension stroke in the presence of disturbances and actuator faults. A nonlinear full-vehicle suspension system and hydraulic actuator with nonlinear characteristics are adopted for accurate control. Firstly, a nonlinear sliding manifold based on a nonsingular fast terminal sliding mode controller is introduced to suppress the sprung mass heave, pitch, and roll motions arising from road disturbances. Secondly, a second-order sliding mode-based super twisting controller is utilized to track the desired forces generated by the nonsingular fast terminal sliding mode controller with actuator faults and uncertainties. The controllers are robust against disturbances, uncertainties, and faults. Moreover, the stability of the super twisting controller is proved by the strong Lyapunov functions. Finally, numerical simulations are performed to demonstrate the effectiveness of the controller. Four different conditions, random road profile, bump road excitation, single-wheel bump excitation, and partial faults are considered. The main contributions of this study are: (1) combination of the above algorithms to deal with actuator faults and improve active suspension performance; (2) the controller proposed in this study has a simple structure. Simulation results indicate that the nonsingular fast terminal sliding mode super twisting controller can guarantee the performance of the closed-loop system under both faulty and healthy conditions.

Strong Morse–Lyapunov functions for Morse decompositions of attractors of random dynamical systems

In this paper, we first construct strong Lyapunov functions for random attractors. On this basis, we then establish strong Morse–Lyapunov functions for Morse decompositions of the random attractors. Finally, the stability of the Morse decompositions is studied through the Morse–Lyapunov functions.

Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm.

Constrained mechanical systems and gradient systems with strong Lyapunov functions

The characteristics of stationary and non-stationary gradient systems with strong Lyapunov functions are studied. The conditions under which holonomic mechanical systems, generalized Birkhoff systems and generalized Hamilton systems can be considered as gradient systems are given. The characteristics of the gradient systems can be used to study the stability of the mechanical systems. Some examples are given to illustrate the application of the results.

Path-Tracking Control of Stochastic Quadrotor Aircraft in Three-Dimensional Space

Despite the fact that environmental loads (forces and moments) induced by wind on quadrotor vertical take-off and landing (VTOL) aircraft consist of both deterministic and stochastic components, all existing works on controlling the aircraft either ignore these loads or treat them as deterministic. This ignorance or treatment deteriorates the control performance in a practical implementation. This paper presents a constructive design of controllers for a quadrotor aircraft to track a reference path in three-dimensional (3D) space under both deterministic and stochastic disturbances. A combination of Euler angles and unit-quaternion for the attitude representation of the aircraft is used to result in an effective control design, and to achieve path-tracking control results. Weak and strong nonlinear Lyapunov functions are introduced to overcome difficulties caused by underactuation and Hessian terms induced by stochastic differentiation rule. To overcome the inherent underactuation of the aircraft, the roll and pitch angles of the aircraft are considered as immediate controls. Potential projection functions are introduced to design estimates of the deterministic components and covariances of the stochastic components. Simulations illustrate the results.

Lyapunov-Based Small-Gain Theorems for Hybrid Systems

Constructions of strong and weak Lyapunov functions are presented for a feedback connection of two hybrid systems satisfying certain Lyapunov stability assumptions and a small-gain condition. The constructed strong Lyapunov functions can be used to conclude input-to-state stability (ISS) of hybrid systems with inputs and global asymptotic stability (GAS) of hybrid systems without inputs. In the absence of inputs, we also construct weak Lyapunov functions nondecreasing along solutions and develop a LaSalle-type theorem providing a set of sufficient conditions under which such functions can be used to conclude GAS. In some situations, we show how average dwell time (ADT) and reverse average dwell time (RADT) “clocks” can be used to construct Lyapunov functions that satisfy the assumptions of our main results. The utility of these results is demonstrated for the “natural” decomposition of a hybrid system as a feedback connection of its continuous and discrete dynamics, and in several design-oriented contexts: networked control systems, event-triggered control, and quantized feedback control.

Uniform Robust Exact Differentiator

The differentiators based on the Super-Twisting Algorithm (STA) yield finite-time and theoretically exact convergence to the derivative of the input signal, whenever this derivative is Lipschitz. However, the convergence time grows unboundedly when the initial conditions of the differentiation error grow. In this technical note a Uniform Robust Exact Differentiator (URED) is introduced. The URED is based on a STA modification and includes high-degree terms providing finite-time, and exact convergence to the derivative of the input signal, with a convergence time that is bounded by some constant independent of the initial conditions of the differentiation error. Strong Lyapunov functions are used to prove the convergence of the URED.

Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem

The classical Matrosov theorem concludes uniform asymptotic stability of time-varying systems via a weak Lyapunov function (positive definite, decrescent, with negative semi-definite derivative along solutions) and another auxiliary function with derivative that is strictly nonzero where the derivative of the Lyapunov function is zero (Mastrosov in J Appl Math Mech 26:1337–1353, 1962). Recently, several generalizations of the classical Matrosov theorem have been reported in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005). None of these results provides a construction of a strong Lyapunov function (positive definite, decrescent, with negative definite derivative along solutions) which is a very useful analysis and controller design tool for nonlinear systems. Inspired by generalized Matrosov conditions in Loria et al. (IEEE Trans Autom Control 50:183–198, 2005), we provide a construction of a strong Lyapunov function via an appropriate weak Lyapunov function and a set of Lyapunov-like functions whose derivatives along solutions of the system satisfy inequalities that have a particular triangular structure. Our results will be very useful in a range of situations where strong Lyapunov functions are needed, such as robustness analysis and Lyapunov function-based controller redesign. We illustrate our results by constructing a strong Lyapunov function for a simple Euler-Lagrange system controlled by an adaptive controller and use this result to determine an ISS controller.

Strong Lyapunov Functions for Systems Satisfying the Conditions of La Salle

We present a construction of a (strong) Lyapunov function whose derivative is negative definite along the solutions of the system using another (weak) Lyapunov function whose derivative along the solutions of the system is negative semidefinite. The construction can be carried out if a Lie algebraic condition that involves the (weak) Lyapunov function and the system vector field is satisfied. Our main result extends to general nonlinear systems the strong Lyapunov function construction presented in a previous paper that was valid only for homogeneous systems.