Robust Finite-time Stabilization Research Articles

Robust Finite-Time Control of Linear System with Non-Differentiable Time-Varying Delay

Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is difficult to form a perfect mathematical model for describing behavior of the practical systems since there may be some information (uncertainties) that is not observed. Thus, in this article, we studied the stability of an uncertain linear system with a non-differentiable time-varying delay. We constructed Lyapunov-Krasovskii functionals (LKFs) containing several symmetric positive definite matrices to obtain robust finite-time stability (RFTS) and stabilization (RFTU) of the uncertain linear system. With the controller and uncertainties in the considered system, there exist nonlinear terms occurring in the formulation process. Past research handled these nonlinear terms as new variables but this led to some difficulty from a computation point of view. Instead, we applied a novel approach via Cauchy-like matrix inequalities to handle these difficulties. In the end, we present three numerical simulations to show the effectiveness of our proposed theory.

Robust finite-time fault-tolerant control for networked control systems with random delays: A Markovian jump system approach

In this paper, the problems of robust finite-time fault-tolerant stochastic stability and stabilization of networked control systems (NCSs) in the presence of random delays and actuator faults are investigated. First, sensor-to-controller and controller-to-actuator random delays are modeled as a Markov chain. Since in the network environment, the accurate access to the transition probabilities (TPs) of Markov chain is hard or even impossible, TPs information is limited. Then, by considering actuator fault indicator matrix and employing the state augmentation technique, the resulting closed-loop system is transformed into a Markovian jump system (MJS). Based on this model, the sufficient conditions are developed to ensure the finite-time fault-tolerant stochastic stability of the faulty NCS, and the fault-tolerant controllers are designed by solving a feasibility problem in terms of linear matrix inequalities (LMIs). The effectiveness of the proposed approach is illustrated during the simulation of a numerical example. Moreover, the faulty behavior of a continuous-stirred tank reactor (CSTR) is investigated and the applicability of the proposed method is demonstrated.

Robust Finite-Time Stabilization for Uncertain Discrete-Time Linear Singular Systems

This paper considers the robust finite-time stabilization problem for a class of discrete-time linear singular systems with norm-bounded parameter uncertainties. With the introduction of an additional matrix to describe the algebraic relationship between the slow subsystem and fast subsystem of a discrete-time linear singular system, a matrix inequality condition is first given for a discrete-time linear singular system to be regular, causal and finite-time stable. With this condition, the robust finite-time stability and robust finite-time stabilization problems are also resolved, and the explicit expression of the desired state feedback control law is also given in terms of a set of matrix inequalities. A numerical example is given to show the effectiveness of the proposed method.

Robust finite-time stability of uncertain neutral nonhomogeneous fractional-order systems with time-varying delays

This article addresses the problem of finite-time stability for uncertain neutral nonhomogeneous fractional-order systems with time-varying delays where a stability test procedure is suggested. Based on the extended form of the generalized Gr?nwall inequality, a new sufficient condition for robust finite-time stability of such systems is established. Finally, a numerical example is given to show the effectiveness of the obtained result.

Partial Finite-Time Stabilization of Perturbed Nonlinear Systems Based on the Novel Concept of Nonsingular Terminal Sliding Mode Method

Abstract In this article, a new technique to design a robust controller to achieve finite-time partial stabilization for a class of nonlinear perturbed systems is proposed. Indeed the system is partially stabilized in a finite time, based on the novel concept of the nonsingular terminal sliding mode (TSM) control method. In the first step, the nonlinear dynamical system is divided into two subsystems based on their required stability properties of the system's states (where finite-time stability is only desired for the first subsystem). Then, using a partial diffeomorphism map to transform the first subsystem into the normal form, the control law is designed. Indeed, by introducing this new concept of the TSM method, robust finite-time stability of only a part of the system's state is guaranteed. Subsequently, simulation results demonstrate the effectiveness of the proposed method, and the results are compared with the existing methods.

Robust Finite-Time Stability and Stabilization of a Class of Fractional-Order Switched Nonlinear Systems

This paper deals with the problem of finite-time boundedness and finite-time stabilization boundedness of fractional-order switched nonlinear systems with exogenous inputs. By constructing a simple Lyapunov-like function and using some properties of Caputo derivative, the authors obtain some new sufficient conditions for the problem via linear matrix inequalities, which can be efficiently solved by using existing convex algorithms. A constructive geometric is used to design switching laws amongst the subsystems. The obtained results are more general and useful than some existing works, and cover them as special cases, in which only linear fractional-order systems were presented. Numerical examples are provided to demonstrate the effectiveness of the proposed results.

Finite-Time Stabilization of Descriptor Time-Delay Systems with One-Sided Lipschitz Nonlinearities: Application to Partial Element Equivalent Circuit

This paper deals with the design of a stabilizing state feedback controller for a class of nonlinear time-delay descriptor system over the finite-time interval. The considered system is exposed to parameter uncertainties and time-varying disturbances. To provide a wider and more general class of nonlinear time-delay descriptor systems, the nonlinear parts of the system are assumed to be a function of state variables, delayed state variables, input and also delayed input and to satisfy the one-sided Lipschitz condition. The purpose of finite-time control is to propose a robust control law such that the resulting closed-loop system is stable, impulse free and regular for all admissible model uncertainties over the considered time interval. Additionally, the designed controller ensures the $$ H_{\infty } $$ disturbance attenuation of the closed-loop descriptor system. In order to achieve these goals, two theorems are given. The sufficient conditions are gained based on the delay-dependent analyses. These conditions then are converted to solvable linear matrix inequalities (LMIs). Finally, simulations are provided a practical example, to confirm the theoretical achievements. The practical example is a partial element equivalent circuit which is a neutral time-delay system. This system is transformed into a time-delay descriptor system. Simulation outcomes illustrate the effective performance of the given method in robust finite-time stabilization.

Finite-time stabilization of nonlinear time delay systems using LQR based sliding mode control

In this paper, we discussed the robust finite-time stability of conic type nonlinear systems with time varying delays. Some novel conditions are derived to design a linear quadratic regulator (LQR) based sliding mode control (SMC) by proposing an integral switching surface. The sufficient conditions are derived for the considered nonlinear system using Lyapunov–Krasovskii stability theory and linear matrix inequality (LMI) approach. The proposed conditions can be solved using some standard numerical packages. Finally, a practical example is provided to validate the advantages and effectiveness of the proposed results.

Robust Finite-Time Stabilizers for a Connected Chain of Nonlinear Double-Integrator Systems

In this paper, the robust finite-time stabilization problem is addressed for a wide group of fully actuated nonlinear systems containing dependent double-integrator subsystems in the presence of unbounded uncertainties. Based on the nonsingular terminal sliding mode control, three types of stabilizers are proposed to steer the system states to zero within adjustable convergence finite times. Moreover, several inequalities are obtained to determine the conservative upper bounds for these convergence finite times. To clarify the applicability of the suggested stabilization methods, a finite-time tracking problem for robotic manipulators is discussed and handled by applying three introduced stabilizers. Finally, a numerical simulation related to this tracking problem is provided.

Finite-time guaranteed cost controller design for uncertain linear continuous-time singular systems

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

Robust finite-time stabilization for positive delayed semi-Markovian switching systems

Robust finite-time stabilization is discussed for positive semi-Markovian switching systems (S-MSSs), in which semi-Markovian process, time-varying delay, external disturbance, and transient performance in finite-time level are all considered in a unified framework. In the system under consideration, finite-time problem can describe transient performance of practical control process. Firstly, some finite-time boundedness and L1 finite-time boundedness criteria for positive delayed S-MSSs are proposed by constructing the stochastic semi-Markovian Lyapunov–Krasovskii functional with mode-dependent integral term. Then, a developed L1 finite-time feedback controller design method is presented to reduce some constraints of input matrices, which guarantees the resulting closed-loop system achieves positivity, finite-time boundedness, and has a prescribed L1 noise attenuation performance index in a novel standard linear programming. Finally, a practical example is illustrated to validate the proposed results by applying a virus mutation treatment model.

Robust Finite-Time H∞ Controller Design for Uncertain One-Sided Lipschitz Systems with Time-Delay and Input Amplitude Constraints

This paper studies the finite-time stabilization of a class of time-delay nonlinear systems in the presence of uncertainties and external disturbances under the input amplitude constraints. The external disturbances are unknown and energy-bounded and the nonlinear vector function of system satisfies the one-sided Lipschitz condition which is less conservative than the well-known Lipschitz condition. To have a robust finite-time stabilization in the considered system, a robust H∞ controller is designed with respect to a finite-time interval. In this regard, two theorems are presented based on Lyapunov–Krasovskii approach and the sufficient conditions are derived as linear matrix inequalities which guarantee the finite-time boundedness of the resulting uncertain closed-loop system. The effectiveness of the proposed method is illustrated by two examples, one numerical and one practical (time-delay Chua’s circuit) with simulation results.

Formula omitted] finite-time stabilization for positive semi-Markovian switching systems

This paper investigates robust finite-time stabilization scheme for positive semi-Markovian switching systems (S-MSSs). Semi-Markovian process (SMP), external disturbances, and transient performances at a finite-time level may appear in a controlled system. A more general system model for S-MSSs that covers Markovian switching systems (MSSs) as a special case is suitable for describing some complex systems that are subject to random abrupt changes in structure and parameter. The main motivation for this is that finite-time problems can be used to describe the transient performance of practical industrial control processes. First, under the framework of stochastic semi-Markovian Lyapunov functions theory, some sufficient conditions for finite-time boundedness (FTBs) and L1 FTBs criteria for positive S-MSSs are proposed for all admissible disturbances. Then, a novel L1 finite-time controller design method that employs the gain matrix decomposition method is presented to reduce some conservativeness, thereby guaranteeing that the resulting closed-loop system (RLCS) could achieve positivity, FTBs, and attain a prescribed L1 noise attenuation performance index in standard linear programming (LP). Finally, a practical example is introduced to show the effectiveness of the main theory

Robust finite-time stabilizers for five-degree-of-freedom active magnetic bearing system

In this paper, the robust finite-time stabilization problem for a fully suspended five-degree-of-freedom active magnetic bearing system is addressed in the presence of external disturbances and additive uncertainties. By developing the nonsingular terminal sliding mode control and defining new nonlinear sliding surfaces, three separate classes of stabilizers are proposed to regulate and place the suspended rotor in the desired positions of air gaps within adjustable finite times. The suggested nonlinear sliding surfaces and designed control inputs for each class of stabilizers are two major differences between these stabilizers. It is mathematically proven that five control voltages of this system, designed by each class of the suggested stabilizers, are able to locate the suspended rotor at the centers of air gaps in the adjustable finite time which is summation of two reaching and settling finite times. Moreover, several new inequalities are extracted for determining the reaching and settling finite times related to the three classes of stabilizers. These inequalities reveal the dependencies between optional parameters of the proposed stabilizers and the mentioned finite times. Finally, numerical simulations are provided to illustrate the efficiency and good performance of each class of the designed stabilizers.

Non-fragile robust finite-time stabilization and $$H_{\infty }$$H∞ performance analysis for fractional-order delayed neural networks with discontinuous activations under the asynchronous switching

In this paper, the global non-fragile robust finite-time stabilization and $$H_{\infty }$$ performance analysis are investigated for uncertain switched fractional-order neural networks with discontinuous activations, time delay and external disturbance under the asynchronous switching. Firstly, a new inequality, which is concerned with the fractional derivative of the variable upper limit integral for the nonsmooth integrable functional, is developed. Secondly, the non-fragile switched controller with two types of gain perturbations is designed. Under the Filippov fractional differential inclusion framework, the global non-fragile robust finite-time stabilization conditions are addressed in terms of linear matrix inequalities (LMIs) by applying nonsmooth analysis theory, inequality analysis technique, average dwell-time method and Lyapunov functional approach. In addition, the global non-fragile robust finite-time $$H_{\infty }$$ performance analysis is performed, and the global non-fragile robust finite-time stabilization conditions with $$H_{\infty }$$ disturbance attenuation level are also derived in the form of LMIs. Finally, two numerical examples are given to illustrate the feasibility of the proposed non-fragile switched controller and the validity of the theoretical results.

Finite-time robust decentralized control for uncertain singular large-scale systems with exogenous disturbances

AbstractThis paper considers the problem of finite-time robust decentralized control for uncertain continuous-time singular large-scale systems with exogenous disturbances (UCSLSSED). The concept of finite-time robust decentralized stabilization for UCSLSSED is introduced by definition of finite-time stability for uncertain continuous-time large-scale systems with exogenous disturbances. The analyses of finite-time robust bounded and finite-time robust stabilization for UCSLSSED are done. The design approach of finite-time robust decentralized state feedback controller is also given. Finally, an illustrative example is provided to show the effectiveness of the proposed method.

Nonsingular terminal sliding-mode control of nonlinear planar systems with global fixed-time stability guarantees

Abstract This paper proposes the use of a novel nonsingular Terminal Sliding surface for the finite-time robust stabilization of second order nonlinear plants with matched uncertainties. Mathematical characteristics of the proposed surface are such that a fixed bound naturally exists for the settling time of the state variable, once the surface has been reached. A simple redesign of the control input able to ensure the feature of fixed-time reaching of the sliding surface will be also presented, and fixed-time stability will be guaranteed by the proposed Terminal Sliding Mode Control design method. A careful simulation study has been performed using a benchmark system taken from the literature.

New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach

This paper investigates robust finite-time stability problem of fractional-order systems (FOSs) with time-varying delay and nonlinear perturbation. Based on a generalized Gronwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems is established in terms of the Mittag-Leffler function. Compared with the finite-time stability conditions in the literature, our condition is simple and convenient for testing and application. The obtained results are applied to finite-time stability of linear uncertain FOSs and linear non-autonomous FOSs with time-varying delays. A numerical example is given to show the effectiveness of the obtained result.

Asynchronously finite-time H∞ control for morphing aircraft

This paper investigates the problem of asynchronously finite-time H∞ control for morphing aircraft with controller uncertainties. Based on the switched linear parameter-varying model of morphing aircraft, the inherent packet dropouts of system are taken into consideration, which will lead to asynchronous switching. The asynchronous switching means that the switching of controllers will lag behind the switching of corresponding subsystem. Moreover, in order to overcome the undesirable influence of controller uncertainties and guarantee that the corresponding closed-loop system is finite-time bounded with a specified H∞ performance index, the non-fragile finite-time H∞ controller is developed via asynchronously switched control. For the purpose of less conservative controller design, the finite-time robust stability analysis of the system is verified by combining mode-dependent average dwell time method and multiple Lyapunov functional method. The sufficient existing conditions and solutions of the controllers are found by solving a series of linear matrix inequalities. Finally, the method proposed in this paper is demonstrated by an illustrative numerical example.

Robust finite-time stability of singular nonlinear systems with interval time-varying delay

The problem of robust finite-time stability (RFTS) for singular nonlinear systems with interval time-varying delay is studied in this paper. Some delay-dependent sufficient conditions of RFTS are derived in the form of the linear matrix inequalities (LMIs) by using Lyapunov–Krasovskii functional (LKF) method and singular analysis technique. Two examples are provided to show the applications of the proposed criteria.