Linear Fractional-order Systems Research Articles

Analytical Methods for the Solution of Linear Fractional Order Systems

Analytical Methods for the Solution of Linear Fractional Order Systems

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

Robust asymptotic stability analysis for fractional-order systems with commensurate time delays: The 1 < β ≤ 2 case

The robust asymptotic stability problem for fractional-order linear systems with commensurate time delays (FOSCDs) and structured perturbations is investigated in this manuscript. Based on the properties of Kronecker products and μ-analysis, the necessary and sufficient conditions are given to guarantee the asymptotic stability of FOSCDs. Then, the necessary and sufficient conditions for the robust asymptotic stability of FOSCDs with structured perturbations are investigated. To reduce the computing complexity and deal with the problems of systems with larger dimensions, a sufficient condition on the robust asymptotic stability of FOSCDs with structured perturbations is given based on the properties of Kronecker products and 2-norm transformations. Finally, several examples are illustrated to test the effectiveness of the proposed results.

Examining reachability of fractional dynamical systems with delays in control utilizing ψ-Hilfer pseudo-fractional derivative

This research focuses on the reachability criteria of fractional dynamical systems with delays in control of order ϑ ∈ (0, 1) and type ϱ ∈ [0, 1] in the context of control theory. The study introduces the utilization of the ψ-Hilfer pseudo fractional derivative to describe the dynamics of systems with delays in control. We obtained the necessary and sufficient conditions for the reachability criteria of fractional order linear dynamical systems by employing the positive definiteness of Gramian matrices. We arrived the sufficient conditions for the reachability criteria of fractional order nonlinear dynamical systems using Banach’s fixed point theorem. A few numerical examples are made available for a better understanding of the theoretical conclusions.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.

Solutions For Fractional Linear Systems With Hattaf Derivative

This paper deals with the analytical solutions of the fractional-order linear systems characterized by DX(t) = AX(t)fractional differential equations. Here D, is the fractional derivative in the Hattaf sense and A is the (2x2)-typenon-singular real coefficient matrix. We present some results related to the existence of analytical solutions of the linearequation systems of the Hattaf fractional derivative. These solutions are provided for three different fractional-orderlinear systems.

A negative imaginary lemma and state-feedback negative imaginary synthesis for commensurate fractional-order systems

This article focuses on negative imaginariness of commensurate fractional-order linear time-invariant systems. Enlightened by the generalized Kalman–Yakubovic–Popov lemma, minimal state-space realization based necessary and sufficient conditions are developed to characterize negative imaginariness of commensurate fractional-order systems for both 0<α<1 and 1<α<2 cases. Meanwhile, the problem of negative-imaginary state-feedback controller synthesis for fractional-order systems is addressed. A fractional-order viscoelastic system, a fractional-order RLC circuit network and a numerical example are used as illustrative examples to demonstrate the main developed theory.

LMI based stability condition for delta fractional order system with sector approximation

This paper concerns the stability of linear time invariant delay delta fractional order systems. Due to the complexity of the stable region, it is difficult or even impossible to develop a sufficient and necessary linear matrix inequality (LMI) condition. After analyzing the feature of the suggested stable region, a sector region is constructed to be the proper subset of true stable region. To control the angle of the sector region, a variable θ is introduced which improves the degree of freedom. Afterwards, the equivalent LMI condition corresponding to the sector region is developed. Notably, both the cases of α∈(0,1) and α∈(1,2) are considered. Besides the stability analysis, the developed method also be applied in controller design. Finally, the validity and efficacy of the elaborated method are illustrated by simulation study.

Optimal control problems with a fixed terminal time in linear fractional-order systems

The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.

Novel Stability and Stabilization Conditions of Linear Fractional-Order Time-Delay Systems Using Free Matrix Approach

The stability and stabilization problem of linear fractional-order time-delay systems is investigated. By introducing a free matrix, the linear fractional-order time-delay systems is transformed to a delay-free system interconnected with uncertainties. New stability and stabilization criteria are derived based on the small gain theorem. The criteria are formulated as linear matrix inequalities and are computationally efficient. Since the free matrix is utilized, the new conditions are less conservative than the existing ones. Two numerical examples indicate that the proposed criteria are less conservative than existing ones.

Fault Detection and Reliable Controller Design for Fractional-Order Systems Based on Dynamic Observer

In this paper, the problem of the fractional-order linear systems’ simultaneous fault detection and reliable control (SFDC) in the finite frequency domain is investigated. A dynamic observer aimed at detecting faults is designed, which also generates state estimation signals. In particular, it is found that the proposed dynamic-observer-based controller can achieve a better H∞ performance. We derived the detector and controller’s design conditions to achieve H− fault sensitivity performance and H∞ interference attenuation performance in the limited frequency domain based on the generalized KYP lemma, which can achieve a better disturbance attenuation performance and fault sensitivity performance. Finally, the simulation results show the designed method’s effectiveness.

Suboptimal control law for a multi fractional high order linear quadratic regulator system in the presence of disturbance

A new method for finding a suboptimal control law for a multi fractional spring–mass system as a fractional high order linear quadratic regulator system in the presence of known disturbance is introduced. The method is presented in a general form for solving practical multi fractional linear quadratic regulator systems with disturbances. After defining a closed-loop controller, formulations for finding the suboptimal controller are derived. In the proposed method, there is no need to solve the fractional vector differential equation. Simulation studies demonstrate the applicability of the method for practical purposes. Some new types of a spring–mass–damper system are considered.

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.

Условие оптимальности типа принципа максимума Понтрягина в задаче управления линейными разностными уравнениями дробного порядка

The optimal control problem of fractional order linear two-dimensional difference equations sys-tems is considered. It is assumed that the control function is included in the boundary condition, and the functional is linear. A necessary and sufficient optimality condition is proved in the discrete maximum principal form. A sufficient optimality condition is proved in the nonlinear but convex cost functional case.

Non-asymptotic and robust estimation for fractional order pseudo-state space model using an algebraic parametric method

This paper mainly investigates the derivative estimation problem for a class of fractional order linear systems. After applying the Laplace transform to the considered pseudo-state space representation model, a series of multiplications and derivations are applied to eliminate unknown initial conditions, which define an annihilator. Then, through strict mathematical derivation, an input-output integral equation is deduced when returning into the time domain. Based on the obtained equation and thanks to the fractional Leibniz formulas, algebraic integral formulas are provided, which can non-asymptotically and robustly estimate the fractional derivatives of the system output. Moreover, error analysis in discrete noisy cases is given to study the performance of the proposed differentiator. Finally, numerical simulations are provided to show the effectiveness of the proposed method.

Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems

We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterward, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists of defining an attainable set, which allows us to prove exact and weak controllability. Moreover, the main results include not only those of controllability but also a powerful Hilbert uniqueness method, which allows us to solve the minimum energy optimal control problem. More precisely, an explicit control is obtained that drives the system from an initial given state to a desired regional state with minimum energy. Two examples are given to illustrate the obtained theoretical results.

Parameter estimation of linear fractional-order system from laplace domain data

A novel parameter estimation method based on the Laplace transform and the response sensitivity method has been presented to recognize the parameters of linear fractional-order systems (FOS) rapidly. The proposed method consumes two orders of magnitude computational resources lower than the traditional time-domain method. Fractional-order operators are increasingly widely used in control and synchronization, epidemiology, viscoelastic material modelling and other emerging disciplines. It is difficult to measure the fractional-order α and system parameters directly in real engineering applications. This paper’s main work include: Firstly, a general linear fractional differential equation is transformed into an algebraic equation by the Laplace transform, and the parameter sensitivity analysis concerning the unknown parameters is also deduced. Then, the parameter estimation problem of the linear FOS is established as a nonlinear least-squares optimization in the Laplace domain, and the enhanced response sensitivity method is adopted to resolve this nonlinear minimum optimization equation iteratively. In addition, the Tikhonov regularization is employed to cope with the potential ill-posed situations, and the trust-region restriction is also introduced to improve the convergence. Finally, taking a differential system with two types of fractional-order operators, a multi-degree-of-freedom FOS with external excitation and an actual piezoelectric actuator model as examples, the specific implementation process is demonstrated in detail to test the robustness and validity of the proposed approach.

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.

Complete Robust Stability Domain of Fractional-Order Linear Time-Invariant Single Parameter-Dependent Systems With the Order 0 &lt; α &lt; 2

The complete robust stability domain of fractional-order linear time-invariant single parameter-dependent systems with the system order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0,2)$ </tex-math></inline-formula> is investigated in this brief. Firstly, the complete robust stability domain of such parameter-dependent systems is given using the Kronecker sum and the Bialternate sum respectively. The complexity of the algorithm can be reduced with Bialternate sum. This method can be easily extended to the case of multiple parameters. The final numerical examples are used to demonstrate the method proposed in this brief is effective and less conservative than the previous results.

Mikhailov stability criterion for fractional commensurate order systems with delays

In this paper, we present the extension of the Mikhailov stability criterion to linear fractional commensurate order systems with delays of the retarded type. The extension is obtained by generalizing the Mikhailov stability criterion of fractional commensurate order and integer order delay systems. The validity of the results is illustrated by means of several examples.