Linear Time Invariant Fractional Order Research Articles

Handling a Commensurate, Incommensurate, and Singular Fractional-Order Linear Time-Invariant System

From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems in light of the Adomian decomposition method. Accordingly, we also generate another general solution of the singular FoLTI system with the use of the same methodology. Several more numerical examples are given to illustrate the core points of the perturbations of the considered singular FoLTI systems that can ultimately generate a variety of corresponding solutions.

A new guardian map and boundary theorems applied to the stabilization of initialized fractional control systems

In the present work, some generalizations of the boundary crossing theorem and the zero exclusion principle are given for fractional order initialized systems. A new guardian map for the maximal stability interval is proposed as well as its relation with the boundary theorems is established. An extension of the technique for linear time invariant fractional order initialized systems to determinate the maximal stability interval is derived from the guardian map defined as a monoparametric pseudo‐polynomials family. This technique is essentially the task of locating the zeros of the family of characteristic pseudo‐polynomials of a linear fractional order system inside the stability region. Finally, an illustrative example is also provided to show the technique.

Parameter-disturbance-robust model predictive control of input-saturated MIMO fractional systems

This paper primarily discusses the solution to multi-degree of freedom linear time invariant fractional order dynamic systems with fractional actuators in detail. This solution is of significant importance due its ease of usage and compatibility with digital systems. Subsequently, a modified model predictive control (MMPC) scheme is introduced that is an infusion of a novel discrete-time robust sliding mode control and model predictive control. The proposed modified model predictive control algorithm is proven to illustrate robustness against uncertainty in system’s parameters and external disturbance while abstaining from the undesirable characteristics of sliding mode control such as chattering. By employing the MMPC scheme, it is possible to constrain the control inputs to be less than pre-determined bounds and by this mean, take a step toward making this algorithm more suitable for practical applications. A general and illustrious plant, represented by the famous Torvik’s three parameter model, is selected and the proposed control scheme is applied to further elucidate the merits, qualities and capabilities of MMPC for fractional systems. The corresponding MATLAB code for the simulation section is also presented in the appendix of the paper. A discussion is also included that examines the effectiveness of MMPC and provides insights for future works.

Commensurate and Non-Commensurate Fractional-Order Discrete Models of an Electric Individual-Wheel Drive on an Autonomous Platform.

This paper presents integer and linear time-invariant fractional order (FO) models of a closed-loop electric individual-wheel drive implemented on an autonomous platform. Two discrete-time FO models are tested: non-commensurate and commensurate. A classical model described by the second-order linear difference equation is used as the reference. According to the sum of the squared error criterion (SSE), we compare a two-parameter integer order model with four-parameter non-commensurate and three-parameter commensurate FO descriptions. The computer simulation results are compared with the measured velocity of a real autonomous platform powered by a closed-loop electric individual-wheel drive.

High-performance modeling and discrete-time sliding mode control of uncertain non-commensurate linear time invariant MIMO fractional order dynamic systems

This paper primarily proposes an analytical solution to the set of coupled non-commensurate linear time invariant fractional order differential equations (representing Multi input Multi output (MIMO) fractional order dynamic systems). The proposed solution is reviewed and accurately truncated to have a short finite-memory and to demonstrate further compatibility for practical applications. It is also stated that this discrete-time approach is explicitly superior to any approach in continuous-time in terms of the possibility of implementation. Standards are provided for adequate truncation of the response with respect to the application. A novel discrete-time enhanced sliding mode control (ESMC) scheme is proposed for MIMO fractional order dynamic systems that can cope with parameter uncertainty while illustrating robustness against disturbance. Due to its structure, ESMC shows no sign of chattering in its control signal and employs no fractional operator in its structure; hence, computational cost of ESMC is minuscule. A novel mathematical method is also proposed for acquiring the stability bounds and solving linear matrix inequalities that arise in this problem. A famous MIMO fractional order dynamic system is presented and controlled under considerable disturbance and parameter uncertainty to render the merits of ESMC.

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

Robust stabilisation of fractional-order interval systems via dynamic output feedback: an LMI approach

ABSTRACTThis paper addresses the problem of robust dynamic output stabilisation of FO-LTI interval systems with the fractional order , in terms of linear matrix inequalities (LMIs). Our purpose is to design a robust dynamic output feedback controller that asymptotically stabilises interval fractional-order linear time-invariant (FO-LTI) systems. Sufficient conditions are obtained for designing a stabilising controller with a predetermined order, which can be chosen to be as low as possible. The LMI-based procedures of designing robust stabilising controllers are preserved in spite of the complexity of assuming the most complete model of linear controller, with direct feedthrough parameter. Finally, some numerical examples with simulations are presented to demonstrate the effectiveness and correctness of the theoretical results.

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

AbstractStability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 < α < 1 and 1 ≤ α < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.

The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, which depends on introducing a method of decomposing their dynamic states from their static states. The solution is formulated by converting the singular system of regular pencils into a recursive form using the sequence of transformations, which separates the dynamic variables from the algebraic variables. The main idea of this work is demonstrated via numerical examples.

Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations

With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems.

Finite-time stability and finite-time boundedness of fractional order linear systems

In this paper, the finite-time stability (FTS) and the finite-time boundedness (FTB) for the fractional order linear time invariant (LTI) systems with 0<α<1 are studied. First, some conditions to guarantee the FTS and the FTB for a class of fractional order LTI systems are derived by combining a new property for Caputo fractional derivatives, the generalized Gronwall inequality and the Laplace transform. Moreover, based on these conditions, the method for the design of state feedback controllers is presented. Finally, illustrative examples are provided to show the effectiveness of the results.

Stabilization of autonomous linear time invariant fractional order derivative switched systems with different derivative in subsystems

Abstract In this paper, the stabilization problem of a autonomous linear time invariant fractional order (LTI-FO) switched system with different derivative order in subsystems is outlined. First, necessary and sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems based on the convex analysis and linear matrix inequality (LMI) for two subsystems is presented and proved. Also, sufficient condition for stability of an LTI-FO switched system with different derivative order in subsystems for more than two subsystems is proved. Then a sliding sector is designed for each subsystem of the LTI-FO switched system. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the norm of the switched system. Simulation results are given to show the effectiveness of the proposed variable structure controller.

On the Stabilization of Linear Time Invariant Fractional Order Commensurate Switched Systems

AbstractIn this paper, the stabilization problem of a linear time invariant fractional order (LTI‐FO) switched system is outlined. First, the sufficient condition for stability of a commensurate LTI‐FO switched system based on the convex analysis and linear matrix inequality (LMI) is presented. Then, a single Lyapunov function is constructed based on the optimization method. Then, a sliding sector is designed for each subsystem of the LTI‐FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the optimization method is decreasing. Finally, a switching control law is designed to switch the LTI‐FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed variable structure controller.

On the positive invariance of polyhedral sets in fractional-order linear systems

In this paper, we derive conditions for a given polyhedral set to be a positively invariant set with respect to fractional-order linear time-invariant (FO-LTI) systems with fractional order of 0<α<1. FO-LTI systems are described using Riemann–Liouville operator with initialization response. Then, the conditions are obtained from Farkas’ lemma and the definition of the Mittag-Leffler function. Furthermore, we apply these conditions to the constrained stabilization.

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.

Oscillations in fractional order LTI systems: Harmonic analysis and further results

This paper studies undamped oscillations generated by marginally stable fractional order linear time invariant (LTI) systems. In this study, an analytical approach is proposed to be used for harmonic analysis of such oscillations in marginally stable commensurate order LTI systems. Also, it is shown that the Q-semi norm of the limit sets for a trajectory of these systems can be analytically determined based on the Q-semi norm of the initial condition, where Q is a specific matrix. Moreover, this result is extended to the wider class of rational order systems. Finally, some numerical examples are presented to demonstrate the use of the paper results.

Effect of Initialization on a Class of Fractional Order Systems: Experimental Verification and Dependence on Nature of Past History and System Parameters

The objectives of this work are (i) to verify experimentally the theoretical claim that neither Riemann–Liouville nor Caputo fractional derivative can be used to predict the time response of fractional order systems without proper correction for the system’s past history in terms of an initialization function and (ii) to study quantitatively how the error incurred due to ignoring initialization depends on the nature of the past history and the system parameters. The entire analysis is restricted to a special class of single input single output linear time invariant fractional order system which can be realized by a simple electrical circuit consisting of a resistance and a fractance in series. This work involves two different realizations of fractances or constant phase elements whose characteristic parameters are first determined based on their respective impedance frequency responses and then used to simulate the time responses of the circuit with the input same as the one used for experimentation using a numerical method for two cases: (i) taking past history into account and (ii) without taking past history into account. Thereafter, an integral square error criterion is presented and variation of the same is studied with respect to system parameters and nature of the history function to have a relative idea of how much partial past history in the absence of a complete one should suffice in practical applications.

Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0

In this paper we consider a class of fractional order linear time invariant (FO-LTI) interval systems with linear coupling relationships among the fractional order, the system matrix and the input matrix. We present the sufficient conditions for the robust stability and stabilization of such coupling FO-LTI interval systems with the fractional order α satisfying 0<α<1. All the results are proposed in terms of linear matrix inequalities (LMI). Two numerical examples show that our results are effective for checking the robust asymptotical stability and designing the stabilizing controller for FO-LTI interval systems.

Stability of fractional-order linear time-invariant systems with multiple noncommensurate orders

Bounded-input bounded-output stability conditions for fractional-order linear time-invariant (LTI) systems with multiple noncommensurate orders have been established in this paper. The orders become noncommensurate orders when they do not have a common divisor. Sufficient and necessary conditions of stability for a fractional-order LTI system with multiple noncommensurate orders are presented in two cases. Based on the numerical inverse Laplace transform technique, time-domain responses for a fractional-order system with double noncommensurate orders are presented to illustrate the obtained stability results.

Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers

This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1 < q < 2 by a single Lyapunov function whose derivative is negative and bounded by a quadratic function within the activation regions of each subsystem. The switching law is extracted based on the variable structure control with a sliding sector. First, a sufficient condition for the stability of an LTI-FO switched system with order 1 < q < 2 based on the convex analysis and linear matrix inequality (LMI) is presented and proved. Then a single Lyapunov function, whose derivative is negative, is constructed based on the extremum seeking method. A sliding sector is designed for each subsystem of the LTI-FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the extremum seeking control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller.