Finite-time Lyapunov Function Research Articles

Observer-Based Fuzzy Adaptive Finite-Time Containment Control of Nonlinear Multiagent Systems With Input Delay.

This article is concerned with the finite-time containment control problem for nonlinear multiagent systems, in which the states are not available for control design and the control input contains time delay. Fuzzy-logic systems (FLSs) are used to approximate the unknown nonlinear functions and a novel distributed fuzzy state observer is proposed to obtain the unmeasured states. Under the framework of cooperative control and finite-time Lyapunov function theory, an observer-based adaptive fuzzy finite-time output-feedback containment control scheme is developed via the adaptive backstepping control design algorithm and integral compensator technique. The proposed adaptive fuzzy containment control method can ensure that the closed-loop system is stable and all followers can converge to the convex hull built by the leaders in finite time. A simulation example is provided to confirm the effectiveness of the proposed control method.

Finite-Time Lyapunov Functions and Impulsive Control Design

In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.

Adaptive Finite-Time Control for High-Order Nonlinear Systems With Multiple Uncertainties and its Application

The globally finite-time control issue is concerned in this note for high-order nonlinearly parameterized systems with unknown control gain and external disturbances. A novel control strategy combining adaptive control technique with sign function can well deal with serious uncertainties and unknown control gain. Without any nonlinear growth assumptions, a unified and systematic design procedure is employed to derive an adaptive state-feedback controller with the help of the adding a power integrator method and backstepping technique. Then, by finite-time stability analysis and Lyapunov functions, the proposed controller ensures that the closed-loop system is globally practically finite-time stable (PFTS). Two simulation examples, including a mass-spring mechanical system and a numerical example, are applied to verify the performance and effectiveness of the designed schemes.

INPUT-TO-STATE STABILITY FINITE-TIME LYAPUNOV FUNCTIONS FOR CONTINUOUS-TIME SYSTEMS

In this paper we propose an input-to-state stability (ISS) criterion for continuous–time systems based on a finite–time decrease condition for a positive definite function of the norm of the state. This yields a so–called ISS finite–time Lyapunov function, which allows for easier choice of candidate functions compared to standard ISS Lyapunov functions. An alternative converse ISS theorem in terms of ISS finite– time Lyapunov functions is also provided. Moreover, we prove that ISS finite–time Lyapunov functions are equivalent with standard ISS Lyapunov functions using a Massera–type construction. The developed ISS framework can be utilized in combination with Sontag’s “universal” stabilisation formula to develop input–to–state stabilizing control laws for continuous–time nonlinear systems that are affine in the control and disturbance inputs, respectively. MSC: 93C10, 93D09, 93D30, 93D15

Non-Transformed Prescribing Performance Function and Finite-Time RISE-Based Tracking Control for Euler-Lagrange Systems

This study addressed the issues of providing an enhanced prescribed performance control technique and a finite–time convergence of the conventional robust integral of the sign of the error (RISE) control method and for Euler-Lagrange systems. An improved RISE control strategy blended with a novel sine hyperbolic function as the prescribed function technique was designed based on the finite-time convergence theorem. The unknown uncertainty due to unknown dynamics of Euler-Lagrange dynamics was estimated by the adaptive law comprised by the command trajectory signals. The stability of the closed-loop finite-time RISE control system was proved via the constructive finite-time Lyapunov function method. The outperformed results of the proposed control strategy over the conventional controls were demonstrated by the experimental verification for an articulated manipulator.

Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

비선형 동적시스템에 대한 비선형 미분 외란관측기를 이용한 유한시간 Dynamic Surface 제어기 설계

This study proposes a finite-time dynamic surface control (DSC) combined with a nonlinear differentiator disturbance observer for nonlinear systems, in which dynamics are partially known. The nonlinear differentiator disturbance observer, virtual stabilizing controllers, and final controller were designed based on finite-time convergent theorem via the recursive steps in conventional DSC system. In conclusion, the study results show that the filtered output error due to the first-order filter in a conventional DSC, which is the source of controller design complexity and stability, can be bypassed; this is because stability is provided by a finite-time Lyapunov function comprised of newly defined variables instead of the tracking error variables used in conventional DSCs. Thus, the controller design procedure and stability analysis can be more intuitive than those of a conventional DSC. Another important study result is the design of a nonlinear differentiator disturbance observer for nonlinear systems to estimate uncertainty in finite-time. The study shows that very competitive results can be achieved using the proposed method, as shown by the simulation results for an articulated manipulator system.

Application of finite-time control Lyapunov function in low-power PMSG wind energy conversion systems for sensorless MPPT

This paper discusses the problem of observer-based finite-time control (FTC) of a grid-connected wind turbine and a low-power permanent magnet synchronous generator (PMSG). An adaptive nonlinear observer is employed to estimate the mechanical variables. Maximum power point tracking (MPPT) is obtained using the estimation of rotor speed and torque from the adaptive observer, and excluding the wind speed sensor. Improvement of the MPPT technique through the designed FTC is investigated. The proposed controller stabilizes the WECS and tracks the reference trajectories in a short pre-known time alternative to common nonlinear controllers with large settling time. The suggested controller is also robust against uncertainties in WECS parameters. Parameters’ variations are compensated by robust control design. Finite time stability and robustness of the proposed WECS controller is mathematically proved. Moreover, the advanced performance of the suggested FTC is demonstrated by simulation and is compared to a conventional asymptotic convergent controller (ACC). The proposed FTC provides fast and robust rotor speed regulation and thus enhances the sensorless MPPT. The proposed FTC improves the WECS performance for tracking of ramp references and robustness against parameter uncertainties. Furthermore, advanced control of the grid-side converter yields improved resiliency and reliability.

Adaptive Finite-Time Controller Design for T-S Fuzzy Systems.

This paper studies the adaptive finite-time stabilization problem for a class of nonlinear systems described by Takagi-Sugeno (T-S) fuzzy dynamic models with parametric uncertainties. A novel adaptive state feedback control scheme for the T-S fuzzy systems is proposed, and the scheme is developed based on finite-time Lyapunov theorem and adaptive backstepping-like method. Augmented dynamics are introduced in the design of finite-time stabilization controllers to construct suitable finite-time Lyapunov functions. It is shown that finite-time convergence of the closed-loop adaptive control system can be achieved, and the potential controller singularity problem caused by the augmented dynamics can be avoided. In addition, constructive procedures to obtain such an adaptive finite-time controller are given. Convergence time as a transient performance specification is also taken into account, and a finite upper bound on the convergence time is estimated. Finally, two numerical examples are provided to illustrate the effectiveness and practicality of the proposed adaptive control approach.

Computation of Lyapunov functions for nonautonomous systems on finite time-intervals by linear programming

We present an algorithm for numerically computing Lyapunov functions for nonautonomous systems on finite time-intervals. The algorithm relies on a linear optimization problem and delivers a continuous and piecewise affine function on a compact set. The level-sets of such a Lyapunov function give concrete bounds on the time-evolution of the system on the time-interval and for time-periodic systems they deliver an ultimate bound on solutions. Four examples of computed finite-time Lyapunov functions are given.

On input–to–state stability analysis of discrete–time systems via finite–time Lyapunov functions

This paper considers the problem of inherent robustness analysis for nonlinear discrete–time systems using the concept of a finite–time Lyapunov function. The main contribution is to prove that, for sufficiently continuous dynamics and finite–time Lyapunov functions, inherent global input–to–state stability to general disturbances can be established for nominally stable nonlinear systems. Moreover, under mere continuity of the finite–time Lyapunov function and of the dynamics, inherent input–to–state stability on a compact set is obtained.

A Non–conservative Small-Gain Theorem for GAS Discrete–Time Systems

This paper makes use of the concept of a finite–time Lyapunov function to derive a non– conservative small-gain theorem for stability analysis of interconnected discrete–time nonlinear systems. Firstly, it is shown that the existence of a global finite–time Lyapunov function is equivalent to global asymptotic stability (GAS) of the overall interconnected system. Secondly, it is indicated that existence of Lyapunov–type functions for each subsystem, together with a small-gain condition implies GAS of the interconnected system. Thirdly, the main result of this paper establishes that GAS of the interconnected system always yields a set of Lyapunov–type functions that satisfy the small-gain condition for a rather general class of GAS nonlinear systems. A simple example demonstrates the non–conservatism of the proposed small-gain theorem.

Non–conservative and tractable stability tests for general linear interconnected systems with an application to Power Systems

Existing Lyapunov methods for verifying stability of linear interconnected systems provide either non–conservative but non–tractable global conditions, or tractable but conservative local conditions. In this paper we provide non–conservative and tractable stability tests for general linear interconnected systems. Firstly, we exploit the concept of a finite–time Lyapunov function to derive a global stability test that can be implemented efficiently by parallelization. Secondly, the same concept is further employed to derive local, dissipativity–type conditions, that can be formulated as a set of distributed linear matrix inequalities. Thirdly, an assessment of all stability tests for linear interconnected systems in terms of complexity, scalability and applicability is performed. The usefulness of the developed stability tests for real–life applications is illustrated on benchmark power system networks.

Finite-Time H ∞ Inverse Optimal Control of Affine Nonlinear Systems

This paper is focused on the problem of the finite-time H ∞ inverse optimal control for affine nonlinear systems. Based on the finite-time control Lyapunov function, we derive a sufficient condition for the existence of time-invariant, continuous, finite-time stabilizing and inverse optimal state feedback control law, and propose a universal formula for constructing the finite-time H ∞ inverse optimal control law. We investigate the relationship between the finite-time stabilization and the finite-time H ∞ inverse optimal control. Finally, some examples are given to illustrate the effectiveness of the presented results.

Construction of a finite-time Lyapunov function by meshless collocation

We consider a nonautonomous ordinary differential equation of the form $\dot{x}=f(t,x)$, $x\in \mathbb{R}^n$ over a finite-time interval $t\in [T_1,T_2]$. The basin of attraction of an attracting solution can be determined using a finite-time Lyapunov function.  &nbsp In this paper, such a finite-time Lyapunov function is constructed by Meshless Collocation, in particular Radial Basis Functions. Thereto, a finite-time Lyapunov function is characterised as the solution of a second-order linear partial differential equation with boundary values. This problem is approximately solved using Meshless Collocation, and it is shown that the approximate solution can be used to determine the basin of attraction. Error estimates are obtained and verified in examples.

Finite time stability and stabilization of hybrid dynamic systems

Finite time stability and stabilization are studied for hybrid dynamic systems. By combining multiple Lyapunov function and finite time Lyapunov function, a sufficient condition of finite time stability is given for the system. Compared with the previous works, our results have less conservativeness. Furthermore, based on the state partition of continuous and resetting parts of system, a hybrid feedback controller is constructed, which stabilizes the closed-loop systems in finite time. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.

A Finite-time State Feedback Stabilization Method for a Class of Second Order Nonlinear Systems

The finite-time feedback stabilization problem of a class of second-order nonlinear systems is discussed. Three kinds of global finite-time feedback stabilization approaches based on continuous state feedback are given. First, using homogeneous properties of nonlinear homogeneous systems, a state feedback controller is designed to guarantee that the closed loop system is asymptotically stable and satisfies negative homogeneity. Sec-ond, using backstepping constructive approach based on finite-time Lyapunov function, a finite-time controller is developed. Finally, using nonsingular terminal sliding mode technology, a feedback stabilizing controller which can make the closed loop system converge to equilibrium in finite time is given. Simu-lation results are provided to demonstrate the effectiveness of these approaches.