Solution of a specific class of nonlinear fractional optimal control problems including multiple delays

This research provides a new framework based on a hybrid of block‐pulse functions and Legendre polynomials for the numerical examination of a special class of scalar nonlinear fractional optimal control problems involving delay. The concepts of the fractional derivative and the fractional integral are employed in the Caputo sense and the Riemann‐Liouville sense, respectively. In accordance with the notion of the Riemann‐Liouville integral, we derive a new integral operator related to the proposed basis called the operational matrix of fractional integration. By employing two significant operators, namely, the delay operator and the integral operator connected to the hybrid basis, the system dynamics of the primal optimal control problem converts to another system involving algebraic equations. Consequently, the optimal control problem under study is reduced to a static optimization one that is solved by existing well‐established optimization procedures. Some new theoretical results regarding the new basis are obtained. Various kinds of fractional optimal control problems containing delay are examined to measure the accuracy of the new method. The simulation results justify the merits and superiority of the devised procedure over the existing optimization methods in the literature.     

Biorthogonal multiwavelets on the interval for solving multidimensional fractional optimal control problems with inequality constraint

This article proposes a new numerical approach for solving fractional optimal control problems including state and control inequality constraints using new biorthogonal multiwavelets. The properties of biorthogonal multiwavelets are first given. The Riemann-Liouville fractional integral operator for biorthogonal multiwavelets is utilized to reduce the solution of optimal control problems to a nonlinear programming one, to which existing, well-developed algorithms may be applied. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. The method is computationally very attractive and gives very accurate results.     

Fractional Chebyshev pseudospectral method for fractional optimal control problems

In this paper, we introduce and apply a fractional pseudospectral method for indirectly solving a generic form of fractional optimal control problems. By employing the fractional Lagrange interpolating functions and discretizing the necessary optimality conditions at Chebyshev‐Gauss‐Lobatto points, the problem is converted into an algebraic system. By solving this system, the optimal solution of the main fractional optimal control problem is approximated. Finally, in some numerical examples, we show the applicability, efficiency, and accuracy of the proposed method comparing with some other methods.     

Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals, derivatives of arbitrary order. The strength of derivatives of non‐integer order is their ability to describe real situations more adequately than integer order derivatives, especially when the problem has memory or hereditary properties. This paper is mainly concerned with the square‐mean almost automorphic mild solutions to a class of fractional neutral stochastic integro‐differential equations with infinite delay driven by Poisson jumps. The existence of square‐mean almost automorphic mild solutions of the previous fractional dynamical system is proved by using the method of successive approximation. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. Further, the existence of optimal control of the proposed problem is also presented. An example is provided to illustrate the developed theory. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

In this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.     

Pseudospectral method for fractional infinite horizon optimal control problems

Up to now, several numerical methods have been presented to solve finite horizon fractional optimal control problems by researchers, while solving fractional optimal control problems on infinite domain is a challenging work. Hence, in this article, a numerical method is proposed to solve fractional infinite horizon optimal control problems. At the first stage, a domain transformation technique is used to map the infinite domain to a finite horizon. Also, fractional derivative defined on an unbounded domain is converted into an equivalent derivative on a finite domain. Then, a new shifted Legendre pseudospectral method is utilized to solve the obtained finite problem and a nonlinear programming problem is suggested to approximate the optimal solutions. Finally, some numerical examples are given to show the efficiency of the method.     

On the optimal control for fractional multi‐strain TB model

In this paper, optimal control of a general nonlinear multi‐strain tuberculosis (TB) model that incorporates three strains drug‐sensitive, emerging multi‐drug resistant and extensively drug‐resistant is presented. The general multi‐strain TB model is introduced as a fractional order multi‐strain TB model. The fractional derivatives are described in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pontryagin maximum principle. Four controls variables are proposed to minimize the cost of interventions. Two simple‐numerical methods are used to study the nonlinear fractional optimal control problem. The methods are the iterative optimal control method and the generalized Euler method. Comparative studies are implemented, and it is found that the iterative optimal control method is better than the generalized Euler method. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust combined feedback/feedforward control for fractional FOPDT systems

In this paper, a combined feedback/feedforward design methodology is proposed for fractional systems in order to cope with model uncertainty and to minimize performance degradation. Based on a fractional commensurate uncertain model, a parametric robust controller is first designed. Then, a parametric command signal for the unity feedback loop is designed. Finally, an optimal set of tuning parameters is found by solving a constrained min–max optimization problem in order to minimize the worst‐case settling time. Simulation results show the effectiveness of the methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control problems with Atangana‐Baleanu fractional derivative

In this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.     

Improving integral square error performance with implementable fractional‐order PI controllers

In this paper, an algebraic rule for tuning the integer realizations of fractional‐order PI controllers is developed, with an integral square error performance index, which outperforms that of an optimal ordinary PI controller. To this end, the PI^λ control structure is used in conjunction with a third‐order integer approximating filter to provide a three parameter fixed‐structure extension of the ordinary PI controller. Next, the extra degree of freedom in setting the order of integration λ is leveraged to introduce a steepest descent direction in the extended controller parameter space. It is then stated that shifting the parameters of an ordinary PI controller along the proposed descent direction will result in a fractional‐based three parameter controller with a performance index, which is superior to that of the original PI controller. The stability of the controller parameters derived in this manner is then analyzed, and examples and simulation results are offered to verify the theoretical expectations and analyses. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison of PID and FOPID controllers tuned by PSO and ABC algorithms for unstable and integrating systems with time delay

This paper deals with optimization and design of an integer order–based and fractional order–based proportional integral derivative (PID) controller tuned by particle swarm optimization (PSO) and artificial bee colony (ABC) algorithms. These algorithms were used to find the best parameters for the best controller performance. A comparative study has been made to highlight the advantage of using ABC‐based controller over a PSO‐based controller. The validity of the controller tuning algorithms was tested in 2 different systems with time delay and a nonminimum phase zero used commonly in process control. The optimal tuning process of the PID and fractional order PID controllers has also been performed with 3 different cost functions. From the perspectives of time‐domain performance criteria, such as settling time, rise time, overshoot, and steady‐state error, the controller tuned by ABC gives better dynamic performances than controllers tuned by the PSO. Moreover, the results obtained from robustness analysis showed that the parameters of controller tuned by ABC are quite robust under internal and external disturbances.     

Efficiency and duality for a vector of quotients of curvilinear functionals on the first‐order jet bundle

In the present work, we focus on the multitime multiobjective nondifferentiable fractional variational problems in the first‐order jet bundle. The problem minimizes a vector of quotients of the form of curvilinear integral type containing the support functions with respect to some partial differential equations and inequations. The necessary and sufficient optimality conditions for the problem have been established under (H,δ)‐convexity assumptions. Further, we have introduced a parametric multitime multiobjective nondifferentiable variational dual problem and studied the duality relations considering the functionals involved to be (H,δ)‐convex.     

Parametric approach to multitime multiobjective fractional variational problems under (F,ρ)‐convexity

In this paper, we consider a multitime multiobjective fractional variational problem of minimizing a vector of quotients of path‐independent curvilinear integral functionals subject to certain partial differential equations and inequations. Using the so‐called parametric approach, we establish necessary and sufficient optimality conditions for the considered class of multitime multiobjective fractional variational problems under both (F,ρ)‐convexity and generalized (F,ρ)‐convexity. Further, the parametric multiobjective variational dual problem is formulated for the considered multitime multiobjective fractional variational problem, and several duality results are established under (generalized) (F,ρ)‐convexity. Copyright © 2015 John Wiley & Sons, Ltd.     

Fractional Chebyshev functional link neural network-optimization method for solving delay fractional optimal control problems with Atangana-Baleanu derivative

In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control-state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu. The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two-point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using single-layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided.     

Successive approximation and optimal controls on fractional neutral stochastic differential equations with Poisson jumps

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations with Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations with Poisson jumps is also presented. An example is given to illustrate the theory. Copyright © 2015 John Wiley & Sons, Ltd.     

Fractional linear control systems with Caputo derivative and their optimization

In this paper, a fractional linear control system, containing Caputo derivative, with an integral performance index is studied. First, the existence and uniqueness of a solution to the mentioned control system is obtained. The main result is a theorem on the existence of optimal solutions to considered optimal control problems. Moreover, in order to find these solutions, the necessary optimality conditions (Pontryagin maximum principle) are derived. Our considerations consist of two parts: first, we consider starting a problem with zero initial condition and, next, with nonzero initial condition. All results are obtained by using results of such a type for equivalent fractional optimal control problem containing a Riemann–Liouville derivative. Copyright © 2014 John Wiley & Sons, Ltd.     

An analysis and design method for fractional‐order linear systems subject to actuator saturation and disturbance

In this paper, two methods are proposed for investigating stability of fractional‐order systems under saturated linear feedback. The first stability condition can be used for fractional‐order linear systems with nonlinear element that is Lipschitz in state. The second stability condition has been achieved by exploring the special property of saturation using an auxiliary feedback matrix. For comparison of the two analyses, domains of attraction are applied using the proposed optimizations whose conditions can be expressed as Linear Matrix Inequalities in terms of all the varying parameters hence being appropriate for controller synthesis. Furthermore, the stability condition is achieved with regard to persistence disturbance. It is employed for determining the invariant sets of the given system by the proposed optimization problems. The results of the proposed analysis and methods are illustrated by three examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Solving fractional optimal control problems by new Bernoulli wavelets operational matrices

In this article, a new numerical method based on Bernoulli wavelet basis has been applied to give the approximate solution of the fractional optimal control problems. The new operational matrices of multiplication and fractional integration are constructed. The proposed method is applied to reduce the problem to the solution of a system of algebraic equations. The fractional derivative is considered in the Caputo sense. Convergence of the algorithm is proved and some results concerning the error analysis are obtained. Approximate solutions are given and in the cases when we have an exact solution, a comparison with the exact solution is presented to demonstrate the validity and applicability of the proposed method. In addition, we compare the obtained results with the results of other methods. Comparison shows the more accuracy of presented technique in comparison to other published methods.     

Fractional Buffer Layers: Absorbing Boundary Conditions for Wave Propagation

We develop fractional buffer layers (FBLs) to absorb propagating waves without reflection in bounded domains. Our formulation is based on variable-order spatial fractional derivatives. We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain. In particular, we first design proper FBLs for the one-dimensional one-way and two-way wave propagation. Then, we extend our formulation to two-dimensional problems, where we introduce a consistent variable-order fractional wave equation. In each case, we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time. We compare our results with a finely tuned perfectly matched layer (PML) method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain. We also demonstrate that our formulation is more robust and uses less number of equations.     

Robust preview control for uncertain discrete‐time systems based on LMI

For a class of uncertain discrete‐time systems, a preview controller based on linear matrix inequality is proposed. A new method is derived to construct an augmented error system instead of taking the difference of the error signal and the system equation. The new approach avoids applying the difference operator to the time‐varying matrix and can simplify the augmented error system. For the augmented error system of the uncertain system, state feedback is introduced. The sufficient condition of asymptotic stability of the closed‐loop system is derived for the performance index by using the relevant theorems of robust control theory. The condition can be realised by solving a linear matrix inequality optimization problem. By incorporating the controller obtained into the original system, we obtain the preview controller. Moreover, introducing an integrator allows the closed‐loop system to robustly track the desired tracking signal without steady‐state error.