Fractional-order Discrete-time Research Articles

Discrete-Time Fractional Order Terminal Sliding Mode Tracking Control for Linear Motor

This paper presents a practical discrete-time fractional order terminal sliding mode (DFOTSM) control strategy for high-precision tracking tasks based on a linear motor. In particular, the practical parametric uncertainties involving sliding friction, uncertain payload, and disturbance in tracking tasks are considered in this paper. Combining Grunwald–Letnikov fractional order definition and terminal sliding mode technique, the proposed method synthesizes a novel DFOTSM control law to drive the sliding mode dynamics into the stable region in finite steps theoretically, even though the system is suffering from uncertainties and disturbances, and the motion on the surface can guarantee higher tracking precision than the conventional discrete-time terminal sliding surface by selecting suitable controller parameters. The theoretical analyses give out the guideline of parameter selection, and experiments are carried out on the linear-motor-based test platform to demonstrate that the proposed controller is easily implemented and can achieve high-precision tracking, fast response, and considerable robustness to uncertainties.

Fractional discrete-time consensus models for single- and double-summator dynamics

ABSTRACTThe leader–following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the systems, interactions between opinions are defined like in Krause and Cucker–Smale models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear systems. In this paper, we investigate fractional-order models of opinions for the single- and double-summator dynamics of discrete-time by analytical methods as well as by computer simulations. The necessary and sufficient conditions for the leader–following consensus are formulated by proposing a consensus control law for tracking the virtual leader.

Discrete-time generalized mean fractional order controllers

This paper studies the performance of discrete-time fractional order controllers. The fractional derivatives and integrals are numerically implemented by means of a generalized mean of discrete generating functions. The two additional degrees of freedom provided by the method, namely the averaging order and the weight of the generating functions, are tuned for increasing the performance of the closed-loop system. Experiments with a fractional order PID controller reveal the benefits of the approach.

Closed-Form Discretization of Fractional-Order Differential and Integral Operators

This paper introduces a closed-form discretization of fractional-order differential and integral Laplacian operators. The proposed method depends on extracting the angle information from the phase diagram. The magnitude frequency response follows directly due to the symmetry of the poles and zeros of the finite ztransfer function. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a 1st, 2nd,3rd, and 4th-order discrete-time operators (DTO) that are stable and of minimum phase. The proposed method depends only on the order of the Laplacian operator. The resulting discrete-time operators enjoy a flat phase response over a wide range of the discrete-time frequency spectrum. The closed-form DTOs allow us to identify the stability regions of fractional-order discrete-time systems and to design discrete-time fractional-order PIλDµ controllers. The effectiveness of this work is demonstrated via several numerical simulation.

Leader-following consensus of discrete-time fractional-order multi-agent systems

Leader-following consensus of fractional order multi-agent systems is investigated. The agents are considered as discrete-time fractional order integrators or fractional order double-integrators. Moreover, the interaction between the agents is described with an undirected communication graph with a fixed topology. It is shown that the leader-following consensus problem for the considered agents could be converted to the asymptotic stability analysis of a discrete-time fractional order system. Based on this idea, sufficient conditions to reach the leader-following consensus in terms of the controller parameters are extracted. This leads to an appropriate region in the controller parameters space. Numerical simulations are provided to show the performance of the proposed leader-following consensus approach.

A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.

State estimation strategy for fractional order systems with noises and multiple time delayed measurements

The fractional order calculus can represent systems with high-order dynamics and complex non-linear phenomena using fewer coefficients. In this study the extended fractional Kalman filter is developed for the class of non-linear discrete-time fractional order systems using observations with multiple delays contaminated by additive white noise. For a wide class of practical applications, the time delay cannot be neglected. In the time delay integer order systems, a common approach is partial differential equation. This method is very difficult to solve. The authors’ approach is applied the reorganised innovation technique. As a result of the reorganised innovation sequence, the Kalman filtering problem with multiple delayed measurements is converted to filter of a delay-free system. In the rest of paper, the covariance matrix of the prediction error of the states is presented in the form of the novel Riccati equations. Finally, the solution of the Kalman filtering problem is obtained by applying the re-organised innovation sequence and Riccati equations. A numerical example is given to illustrate the effectiveness of the proposed scheme.

Stabilising model predictive control for discrete-time fractional-order systems

In this paper, a model predictive control scheme is proposed for constrained fractional-order discrete-time systems. We prove that constraints are satisfied and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system. A finite-dimensional approximation of the original infinite-dimensional dynamics is employed for which the approximation error can become arbitrarily small. The approximate dynamics is used to design a tube-based model predictive controller which steers the system state to a neighbourhood of the origin of controlled size. Stability conditions are finally derived for the MPC-controlled system which are computationally tractable and account for the infinite dimensional nature of the fractional-order system and the state and input constraints. The proposed control methodology guarantees asymptotic stability of the discrete-time fractional order system, satisfaction of the prescribed constraints and recursive feasibility.

Control of fractional periodic discrete-time linear systems by parametric state feedback matrices

In this article stabilization of fractional-order periodic discrete-time linear system by a complete nonlinear parametric approach for pole assignment with respect to stability and reachability of system via state feedback is proposed. To modify the dynamic response of linear system we should change the poles of state feedback matrices. Nonlinear parametric feedback matrix makes some freedoms in conditions such as minimum norm of feedback matrix. But it makes large monodromy matrix and changing all of eigenvalues makes some problems. Reassigning a part of bad spectrums, leaving the rest of the spectrums invariant, we have lower-ordered matrix to modify the dynamic response of linear system and we can make nonlinear parametric for this new lower-ordered matrix by less expenses and better stability conditions. Numerical examples illustrate the effectiveness of the proposed approaches.

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

ABSTRACTThe concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.

Fractional discrete-time of Hegselmann–Krause's type consensus model with numerical simulations

The leader-following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the system, interactions between opinions are defined like in the classical Hegselmann–Krause models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear system. In the paper we investigate various models for the dynamics of discrete-time fractional order opinions by analytical methods as well as by computer simulations.

Control of Fractional Periodic Discrete-Time Linear Systems by Partial Eigenvalue Assignment of State Feedback Matrices

In this article to control of fractional-order periodic discrete-time linear system a special form of state feedback matrix is proposed to assign suitable eigenvalues to closed-loop monodromy matrix of the periodic discrete-time system. It makes large monodromy matrix and changing all of eigenvalues make some problems. By reassigning a part of bad spectrums of monodromy matrix, leaving the rest of the spectrums invariant, we have lower oredered matrix to modify the dynamic response of linear system and we can lie the poles of this systems in the unit circle by less expenses. Controlling the alternating states by other controllers such that the controllability of the resulting system is preserved. The effectiveness of our algorithm is illustrated by an example.

Stability of fractional-order periodic discrete-time linear systems

Stability of fractional-order periodic discrete-time linear systems

Dynamic Programming Problem for Fractional Discrete-Time Dynamic Systems. Quadratic Index of Performance Case

Dynamic programming problem for discrete-time fractional order systems with quadratic performance index has been formulated and solved. A new method for numerical computation of optimal dynamic programming problem has been presented. The efficiency of the method has been demonstrated on numerical example and illustrated by graphs. Graphs also show the differences between the fractional and classical (standard) systems theory. Results for other cases of the coefficient $$\alpha $$ ? , and not illustrated with numerical examples, have been obtained through a computer algorithm written for this purpose.

Discretization of Fractional-Order Differentiators and Integrators

This paper introduces a closed form discretization method of fractional-order differentiators or integrators. Unlike the continued fraction expansion technique, or the infinite impulse response of second-order IIR-type filters, the proposed technique generalizes the Tustin operator to derive a stable and minimum 1st and 2nd-order discrete-time operators (DTO) that discretize continuous fractional-order differintegral operators. Such DTOs exploits the phase properties of the DTOs over a wide range of the frequency spectrum, which depend only on the order of the continuous operators. Moreover, the closed-form DTOs enable one to identify the stability regions of fractional-order discrete-time systems. The effectiveness of this work is demonstrated via several numerical examples.

Local controllability of nonlinear discrete-time fractional order systems

Abstract The Riemann-Liouville, Caputo and Gr¨unwald-Letnikov fractional order difference operators are discussed and used to state and solve the controllability problem of a nonlinear fractional order discrete-time system. It is shown that independently of the type of fractional order difference, such a system is locally controllable in q steps if its linear approximation is globally controllable in q steps

Discretization algorithm for fractional order integral by Haar wavelet approximation

A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm.

Stability of Discrete Fractional Order State-space Systems

In this article, the stability problem for discrete-time fractional order systems is considered. The discrete-time fractional order state-space model introduced by the authors in earlier works is recalled in this context. The proposed stability definition is adopted from one used for infinite dimensional systems. Using this definition, the main stability result is presented in the form of a simple stability condition for the fractional order discrete state-space system. This is one of the first few attempts to give the stability conditions for this type of system. The condition presented is conservative1 the method gives only sufficient conditions, and the stability areas obtained when using it are smaller than those obtained from numerical solutions of the system. The relationship between the eigenvalues of the system matrix and the poles of the fractional-order system transfer function is also discussed. The main observation in this respect is that a set of L poles is related to every eigenvalue of the system matrix.

Efficient Generation of $1/f^{\beta}$ Noise Using Signal Modeling Techniques

This paper presents a new procedure for generating 1/fbeta noise sequences by filtering Gaussian white noise through a recursive filter approximating a discrete-time fractional order integrator. Power series expansion and deterministic signal modeling techniques were combined to determine a rational transfer function to approximate the ideal discrete fractional transfer function resulting from Al-Alaoui's rule raised to the power of the fractional order of integration. The proposed approach is computationally more efficient and more accurate than infinite impulse response truncation. Numerical results are provided to demonstrate the performance of the proposed method.

Controllability and Observability of Linear Discrete-Time Fractional-Order Systems

Controllability and Observability of Linear Discrete-Time Fractional-Order SystemsIn this paper we extend some basic results on the controllability and observability of linear discrete-time fractional-order systems. For both of these fundamental structural properties we establish some new concepts inherent to fractional-order systems and we develop new analytical methods for checking these properties. Numerical examples are presented to illustrate the theoretical results.