Numerical controllability of fractional dynamical systems

In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α,β ≤ 1. Necessary and sufficient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufficient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders fixed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory.

This paper is concerned with the controllability of fractional neutral stochastic dynamical systems with Poisson jumps in the finite dimensional space. Sufficient conditions for controllability results are obtained by using Krasnoselskii’s fixed point theorem. The controllability Grammian matrix is defined by Mittag-Leffler matrix function.

In this paper, we discuss the trajectory controllability of linear and nonlinear fractional Langevin dynamical systems represented by the Caputo fractional derivative by using the Mittag–Leffler function and Gronwall–Bellman inequality. For the nonlinear system, we assume Lipschitz-type conditions on the nonlinearity. Examples are given to illustrate the theoretical results.

In this paper, we study the controllability results for nonlinear fractional order stochastic dynamical systems with distributed delayed control and Poisson jumps in finite dimensional space. New set of sufficient conditions are derived based on Schauder’s fixed point theorem and the controllability Grammian matrix is defined by Mittag-Leffler matrix function. Finally, a numerical example has been given to validate the efficiency of the proposed theoretical results.

Nonlinear [formula omitted]-gain verification for nonlinear systems

In this paper, a set of equivalent conditions for observability and controllability of linear fractional dynamical systems represented by the fractional differential equation in the sense of Caputo fractional derivative of order α ϵ (0,1] are established by using the tools of linear bounded operators. Examples are included to illustrate the theoretical results.

In this paper, the controllability of nonlinear impulsive fractional dynamical system with delays in state and control is analyzed by using a delayed Mittag Leffler (M-L) function. Controllability Grammian matrix is used to establish the controllability of linear system. The sufficient conditions of the considered nonlinear system are derived by utilizing the fixed point techniques. Finally, an example is provided for the illustration of the obtained result.

This paper discusses the exact controllability of linear and nonlinear impulsive Caputo fractional systems. The exact controllability of a linear impulsive system is studied using the concept of generators and functional analysis. In contrast, the controllability of a nonlinear system is discussed using nonlinear functional analysis. An example is provided in the paper to support the results.

In this paper an automatic learning based adaptive approach elaborated for the control of nonlinear dynamic systems is analyzed from various points of view concerning the imprecision and incompleteness of the available system model used by the controller. The proposed approach is compared to the most elaborated classical techniques using Lyapunov functions and dynamic models that are exact in their mathematical form but imprecise in their parameters, yield globally and asymptotically stable solutions but do not allow the presence of permanent external perturbations. It is shown that the novel control allows both numerical imprecision and enduring external disturbances unknown by the controller, but generally cannot guarantee global stability. It is also shown that its simple structure makes it a prospective candidate for the control of fractional order dynamic phenomena in which the conventional techniques based on the application of integer order time-derivatives of quadratic Lyapunov functions have great difficulties. The possible implementation of the proposed method is mathematically tackled and expounded from various backgrounds as the application of Cauchy Sequences in complete, linear, normed metric (Banach) spaces, and the use of coupled differential equations that may also be obtained from simple quadratic Lyapunov functions the decreasing nature of which generally can be guaranteed only within bounded regions. The operation of the proposed method is illustrated by simulation examples made for integer and fractional order dynamical systems as examples.KeywordsLyapunov FunctionAdaptive ControlFractional OrderFractional DerivativeInput Multiple OutputThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results.

This paper deals with the controllability of nonlinear fractional damped dynamical system with control delay which involves fractional Caputo derivatives. Using fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional damped dynamical system with control delay are established. These conditions are easily checked. Two examples are given to illustrate the main results.

NONLINEAR SELFTUNING CONTROLLER BASED UPON LAGUERRE SERIES REPRESENTATION

Adaptive control methods are widely used for the control of both linear and nonlinear dynamical systems. Among the adaptive control approaches, Model Reference Adaptive Control (MRAC) method has a systematic design feature in which response of a stable reference model is targeted to be followed by the dynamical system. In general, Linear Time Invariant (LTI) Reference Models are considered in the MRAC design and the adaptation rule is formulated accordingly. In this paper, the linear reference model is altered with a stable nonlinear one and the adaptation rule is re-formulated to cope with the nonlinearity. The stable nonlinear reference model is generated by using the State Dependent Riccati Equation (SDRE) approach and the adaptation rule is also updated by using the same SDRE idea. The effects of altering the linear reference model to the nonlinear counterpart are shown in the MRAC formulations. In addition to the new MRAC formulations, the proposed MRAC algorithm is applied to a 3 DoF helicopter test bed to reveal the advantages of the nonlinear reference model.

In this study, we are currently investigating the controllability of nonlinear fractional differential control systems with delays in the state function. The solution representations of fractional delay differential equations have been established by using the delayed Mittag-Leffler function. Firstly we obtain the result of the controllability of a linear fractional control system with delay. Then, for the controllability criteria of nonlinear fractional delay system, we establish the set of sufficient conditions of nonlinear fractional differential systems with delay in their state function by using Schauder’s fixed point theorem. In the end, a numerical example is constructed to support the results.

This paper deals with generalised Caputo fractional proportional linear time-invariant systems in a finite-dimensional space. The Laplace transformation method ensures the analytical solution of the desired fractional linear time-invariant systems. The present article presents the necessary and sufficient conditions for controllability and observability of the generalised Caputo proportional fractional linear time-invariant systems. These two properties can play a more fundamental role in system analysis before controller and observer designs are engaged. Moreover, we have acquired the criterion for generalised Caputo proportional fractional linear time-invariant systems as Kalman rank conditions. Some numerical examples are presented to show the applicability of the paper to demonstrate our findings. Finally, we derive the necessary and sufficient controllability conditions for the generalised Caputo proportional fractional-order nonlinear Chua's electric circuit.