Stability Of Fractional-order Linear Systems Research Articles

Robust H[formula omitted]-PID control Stability of fractional-order linear systems with Polytopic and two-norm bounded uncertainties subject to input saturation

This paper deals a type of H∞ proportional–integral–derivative (PID) control mechanism for a type of structural uncertain fractional order linear systems by convex Polytopic and two-norm bounded uncertainties subject to input saturation which mainly focuses on the case of a fractional order α such that 0<α<1. The Gronwall–Bellman lemma and the sector condition of the saturation function are investigated for system stability analysis and stabilization. The main strategy of the presented strategy is to restore fractional order PID controller design under input saturation problem from static output feedback controller design. Unlike existing strategies, non-iterative strategy is used to get optimal output feedback based on the LMI. On the premise of a linear matrix inequality algorithm, the SOF control laws can be obtained. After that, the fractional-order PID controller is recovered from the SOF controller. A numerical example is provided in order to show the validity and superiority of the proposed method.

A Generalization of Routh–Hurwitz Stability Criterion for Fractional-Order Systems with Order α ∈ (1, 2)

Based on the generalized Routh–Hurwitz criterion, we propose a sufficient and necessary criterion for testing the stability of fractional-order linear systems with order α∈1,2, called the fractional-order Routh–Hurwitz criterion. Compared with the existing criterion, ours involves fewer and simpler expressions, which is significant for analyzing the robust stability of high-dimensional uncertain systems. All these expressions are explicit ones about the coefficients of the characteristic polynomial of the system matrix, so the stable parameter region of fractional-order systems can be described directly. Some examples show the effectiveness of our method.

Dynamic robust stabilization of fractional-order linear systems with nonlinear uncertain parameters: an LMI approach

ABSTRACT This paper presents a dynamic output feedback controller with determined order for the stabilization of a class of fractional-order system with nonlinear uncertain parameters with fractional order . Using stability theories of fractional-order systems and linear matrix inequalities (LMIs), some sufficient conditions in the LMI form are deduced to guarantee the robustness and asymptotic stabilization of the system. Designing a dynamic robust controller, along with all its useful features, leads to more unknown parameters in comparison with a static controller and makes controller design procedure more difficult due to more complex constraints that must be solved. In this paper, using proper lemmas and theorems, LMI techniques, and suitable solvers and parsers the difficulty of designing such controllers has been overcome. Simulation results of three different numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems.

Delay-Dependent and Order-Dependent Stability and Stabilization of Fractional-Order Linear Systems With Time-Varying Delay

Stability of fractional-order (FO) delayed systems remains a formidable problem. In this brief, asymptotic stability and stabilization of FO linear systems with time-varying structured uncertainties and time-varying delay are discussed. By employing FO Razumikhin theorem, two new delay-dependent and order-dependent stability criteria in form of LMI for FO nominal systems and FO uncertain sytems are first formulated, respectively. Moreover, state feedback controllers of stabilization are also derived with the help of the proposed stability criteria. The numerical simulation demonstrates the effectiveness of the theoretical formulation.

Robust stability and stabilization of fractional-order linear systems with polytopic uncertainties

Robust stability and stabilization of fractional-order uncertain linear systems with order α:0<α<1 and 1⩽α<2 are considered in the paper. A new polytopic type uncertain state-space model for fractional-order linear systems is addressed, which allows second-order uncertain parameters. The uncertainty in the fractional-order model appears in terms of a polytope of matrices. Some sufficient criteria for the robust asymptotical stable and stabilization for such fractional-order uncertain linear systems are derived. All the results are obtained in terms of linear matrix inequalities (LMIs). Numerical examples are presented to demonstrate the validity and feasibility of the obtained results.

Asymptotical stabilization of fractional-order linear systems in triangular form

In this paper, both state and output feedback stabilization controllers are designed for triangular fractional-order linear time-invariant (FO-LTI) systems with fractional-order 0<α<2. By introducing appropriate transformations of coordinates, the problems of control design are converted into the problems of finding some parameters, which can be certainly obtained by solving relevant matrix inequalities. Contrary to many existing linear matrix inequalities (LMIs)-based approaches, all the matrix inequalities involved for the systems under study always have feasible solutions. A simulation example is given to demonstrate the effectiveness of the proposed controllers.

Robust asymptotical stability of fractional-order linear systems with structured perturbations

This paper considers the robust asymptotical stability problem of fractional-order linear systems with structured perturbations, which are more general than the investigated fractional-order interval systems. Based on the Kronecker product and μ-analysis, necessary and sufficient conditions for the robust asymptotical stability are established by transforming such a problem into checking the nonsingularity of a class of uncertain matrices. Furthermore, the robustness bounds with respect to parametric perturbations to preserve the asymptotical stability are given in terms of the structured singular values. Finally, illustrative examples are given to show the effectiveness of the proposed approach.

Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties

Abstract This paper considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems with convex polytopic uncertainties. The stability condition of the fractional-order linear time-invariant systems without uncertainties is extended by introducing a new matrix variable. The new extended stability condition is linear with respect to the new matrix variable and exhibits a kind of decoupling between the positive definite matrix and the system matrix. Based on the new extended stability condition, sufficient conditions for the above robust stability and stabilization problems are established in terms of linear matrix inequalities by using parameter-dependent positive definite matrices. Finally, numerical examples are provided to illustrate the proposed results.

Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach

This paper investigates the stability and stabilization problem of fractional-order linear systems with nonlinear uncertain parameters, which allow second-order uncertain parameters. The uncertainty in the fractional-order model appears in the form of a combination of additive uncertainty and multiplicative uncertainty. It is shown that the fractional-order model has a strong practical background. Sufficient conditions for the stability and stabilization of such fractional-order model are presented in terms of linear matrix inequalities. Two examples are given to show the effectiveness of the proposed results.