Indirect Lyapunov Approach Research Articles

Finite-time reliable nonfragile control for fractionalorder nonlinear systems with asymmetrical saturation and structured uncertainties

This paper investigates the finite-time stabilization problem of fractional-order nonlinear differential systems via an asymmetrically saturated reliable control in the sense of Caputo’s fractional derivative. In particular, an asymmetrical saturation control problem is converted to a symmetrical saturation control problem by using a linear matrix inequality framework criterion to achieve the essential results. Specifically, in this paper, we obtain two sets of sufficient conditions under different scenarios of structured uncertainty, namely, norm-bounded parametric uncertainty and linear fractional transformation uncertainty. The uncertainty considered in this paper is a combination of polytopic form and structured form. With the help of control theories of fractional-order system and linear matrix inequality technique, some sufficient criteria to ensure reliable finite-time stability of fractional-order differential systems by using the indirect Lyapunov approach are derived. As a final point, the derived criteria are numerically validated by means of examples based on financial fractional-order differential system and permanent magnet synchronous motor chaotic fractional-order differential system.

Robust observer‐based‐controller design for uncertain fractional‐order time‐varying‐delay systems

AbstractThis article focuses on the robust observer‐based controller (‐ROBC) for Uncertain Fractional‐Order Systems with Time‐Varying‐Delay (DU‐FOS‐TVD). First, the existence conditions of ‐ROBC are given. Then, based on the diffusive representation of the fractional‐order derivative and the indirect Lyapunov approach, a new sufficient condition is obtained and presented as a convex optimization problem with a Linear Matrix Inequality (LMI) constraint, which guarantees the robust stability of the estimation errors and the stabilization of the DU‐FOS‐TVD simultaneously. All observer matrices gains and control laws can be computed by solving the given LMI. A Numerical example is given to illustrate the validity of the proposed method.

Observer-based synchronization of fractional-order Markovian jump multi-weighted complex dynamical networks subject to actuator faults

This paper is concerned with an observer-based robust H∞ synchronization problem for a class of fractional-order multi-weighted complex dynamical networks subject to Markovian jump parameters, actuator faults and time-varying coupling delay by using fault-tolerant control scheme. The considered actuator fault model is a general one, which consisting of both linear and nonlinear fault terms. Further, by employing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, a new set of sufficient conditions is derived to ensure the asymptotic synchronization of the addressed system in the mean square sense with a prescribed H∞ performance index. In particular, by constructing monochromatic Lyapunov functional and utilizing extended Wirtinger-based integral inequality the required sufficient conditions are developed in terms of linear matrix inequalities. Finally, two numerical examples including financial model are provided to illustrate the effectiveness of the obtained results.

Sensor fault diagnosis in fractional-order singular systems using unknown input observer

ABSTRACTThis paper investigates the design of an unknown input observer for sensor fault diagnosis in linear fractional-order singular systems. The considered system is rectangular in general form. The necessary and sufficient conditions for the existence of the proposed observer are derived, and a systematic design approach is presented. The designed observer is nonsingular and uses only the original coefficient matrices to reconstruct the sensor faults. The proposed diagnosis method can decouple both the unknown inputs appearing in the system dynamics and the output equation, using only the available inputs and measurable output signals. The asymptotic stability conditions of the designed observer are obtained in terms of linear matrix inequalities. Moreover, the proposed approach is developed for sensor fault diagnosis in fractional-order singular one-sided Lipschitz systems. The convergence conditions of the designed nonlinear observer are derived in terms of linear matrix inequalities by introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach. Finally, the proposed approach is applied to a machine infinite bus system and a numerical example to demonstrate its effectiveness.

Fault-Tolerant Resilient Control For Fuzzy Fractional Order Systems

This paper investigates the problem of resilient fault-tolerant control for a class of Takagi–Sugeno fuzzy model which is described by fractional order differential equations. In particular, the system input under consideration is subject to actuator saturations, actuator faults, and nonlinearities. The main purpose of this paper is to put forward a generalized nonlinear resilient fault-tolerant controller that can make the considered system robustly asymptotically stable. Utilizing the concept of continuous frequency distribution, a new set of linear matrix inequalities, which are the sufficient conditions for robust asymptotic stability of the closed-loop system, are derived by using the indirect Lyapunov approach. Two numerical examples are included to illustrate the feasibility and effectiveness of the proposed method.

Design of unknown input fractional order proportional–integral observer for fractional order singular systems with application to actuator fault diagnosis

This study proposes the design of an unknown input fractional order proportional–integral observer for fault diagnosis of linear fractional order singular (FOS) systems. The considered system is rectangular in general form. The necessary and sufficient conditions for the existence of the proposed observer are derived, and a systematic design approach is presented. The proposed observer is non-singular, and uses only the original coefficient matrices to estimate the norm-bounded actuator faults. Also, the unknown input appearing in measurement is considered and the effects of unknown inputs are decoupled from observer dynamics. By introducing a continuous frequency distributed model and using indirect Lyapunov approach, the convergence conditions of the proposed observer are derived in terms of linear matrix inequalities. Furthermore, for a class of non-linear FOS systems with both output disturbances and input uncertainties, a non-linear observer is developed based on the proposed design approach. The existence and convergence of this observer are proved. Finally, the effectiveness of the proposed method is illustrated via two examples.

Disturbance rejection of fractional-order T-S fuzzy neural networks based on quantized dynamic output feedback controller

The disturbance rejection approach is employed for the stabilization of fractional-order neural networks described by Takagi-Sugeno fuzzy model with dynamic output feedback controller under quantization. First, an equivalent continuous frequency distributed integral-order system is formulated for the fractional-order neural networks to estimate the system state. Specifically, the dynamic output feedback control with quantization is proposed and the measurement output is quantized by logarithmic quantizer before transmission. By employing an indirect Lyapunov approach and equivalent input disturbance (EID) technique, a set of newly established sufficient conditions with corresponding quantizer’s dynamic parameters is obtained in the shape of LMIs to ensure the asymptotical stability of the considered fractional-order system. Finally, the validity of the considered design method is illustrated through a numerical example.

Observer-based robust control for fractional-order nonlinear uncertain systems with input saturation and measurement quantization

The paper is concerned with the observer-based robust control problem for fractional-order (FO) nonlinear uncertain systems subject to control input saturation and measurement quantization, in which the fractional commensurate order satisfies α∈(0,1). The state measurements of observer are quantized by a logarithmic quantizer. Firstly, by introducing a continuous frequency distributed equivalent model of fractional integrator, sufficient condition for guaranteeing the asymptotic stability of closed-loop FO systems is established via the indirect Lyapunov approach. Then, by using matrix’s singular value decomposition (SVD) and linear matrix inequality (LMI) technique, the co-design problem of desired observer and controller gains are derived, which will be shown that the solution guarantees the stability of closed-loop FO nonlinear uncertain control systems. Finally, a simulation example is given to illustrate the validity of this method.

Correction to the Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems & a New Enhanced LMI Condition

This note considers the work entitled “Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems”. In the above paper [1], the authors gave the existence conditions of such observer, then based on the fractional order Lyapunov stability approach, a sufficient condition for the asymptotic stability of the estimation error have given in a linear matrix inequality (LMI) formulation which is incorrect. In this note, we give the correction of Theorem 7. The new proposed Theorem can be applied to a large kind of delayed fractional-order-system when the delay is time varying or constant, while the above mentioned paper consider only a constant time delay. The proof is based on the diffusive representation of the fractional order derivative and the indirect Lyapunov approach proposed in [2].

Large‐scale fractional‐order systems: stability analysis and their decentralised functional observers design

This study concerns the stability analysis of linear large-scale fractional-order systems and its functional observers design. First, based on the diffusive representation of the fractional-order derivative and the indirect Lyapunov approach, the stability of large-scale fractional-order interconnected systems is studied. Then, a sufficient stability condition of such systems is given in linear matrix inequality (LMI) formulation. In the second part of this work, the existence conditions of functional decentralised observers for this class of systems are given, and the asymptotic stability of estimation errors is investigated. The observers gains matrices are derived by solving the obtained LMI. Numerical examples are given to illustrate the validity of the proposed approach.

Fractional order composite MRAC for MIMO systems based on SDU factorization

Based on factorization of high frequency gain (HFG) matrix Kp, a novel fractional order composite model reference adaptive control (FO-CMRAC) is investigated for a class of fractional order MIMO systems. By introducing Kp = SDU, the stringent symmetry assumption related with the plant high frequency gain matrix is removed. Besides, with the aid of fractional order tracking differentiator (FOTD), the parameter estimation errors is obtained to construct the fractional order parameter updating laws, which can improve the tracking performance. In addition, both the convergence of closed-loop control systems and parameters estimation are proven by indirect Lyapunov approach. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed approach.

An indirect Lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty

The paper is concerned with the problem of the robust stabilization for a class of fractional order linear systems with positive real uncertainty under Riemann–Liouville (RL) derivatives. Firstly, by utilizing the continuous frequency distributed model of the fractional integrator, the fractional order system is expressed as an infinite dimensional integral order system. And via using indirect Lyapunov approach and linear matrix inequality techniques, sufficient condition for robust asymptotic stability of the fractional order systems and design methods of the state feedback controller are presented. Secondly, by using matrixs singular value decomposition technique the static output feedback controller and observer-based controller for asymptotically stabilizing the fractional order systems are derived. Finally, the validity of the proposed methods are demonstrated by numerical examples.

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

AbstractThe paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.

An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks

This paper concerns with the problem of the observer-based robust control for a class of fractional-order complex dynamic networks. First, by introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for robust asymptotic stability of the closed-loop systems via state observer-based control is presented. Next, using matrix׳s singular value decomposition (SVD) and linear matrix inequality (LMI) techniques, the existence condition and method of designing a robust non-fragile observer-based controller are derived. Finally, the validity of the proposed methods are demonstrated by a numerical example.

Non-fragile observer-based robust control for a class of fractional-order nonlinear systems

The paper is concerned with the problem of the observer-based robust control for a class of fractional-order nonlinear systems. First, by introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for robust asymptotic stability of the closed-loop system via state observer-based control is presented, which is less conservative than some existing ones in recent literature. Next, using matrix’s singular value decomposition (SVD) and linear matrix inequality (LMI) techniques, the existence condition and method of designing a robust non-fragile observer-based controller are derived. Finally, the validity of the proposed methods are demonstrated by two numerical examples.

Non-fragile Observer Design for Fractional-order One-sided Lipschitz Nonlinear Systems

This paper is concerned with the problem of the full-order observer design for a class of fractional-order Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full-order observer error dynamic system is presented. The stability condition is obtained in terms of LMI, which is less conservative than the existing one. A numerical example demonstrates the validity of this approach.