Identification of Fractional-Order Dynamical Systems

The paper focuses on the numerical approximation of nabla fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed models of discrete fractional order system are introduced. Then, resorting the nabla Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is...

Closed-loop neurotechnology requires the capability to predict the state evolution and its regulation under (possibly) partial measurements. There is evidence that neurophysiological dynamics can be modeled by fractional-order dynamical systems. Therefore, we propose to establish a separation principle for discrete-time fractional-order dynamical systems, which are inherently nonlinear and are able to capture spatiotemporal relations that exhibit non-Markovian properties. The...

In this paper we present the mathematical description and analysis of a fractional-order regulated system in the state space. A little historical background of our results in the analysis and synthesis of the fractional-order dynamical regulated systems is given. The methods and results of simulations of the fractional-order system described by a state space equation equivalent to three-member fractional-order differential equation with a fractional-order $PD^{\delta}$ regula...

Time domain identification is studied in this paper for parameters of a continuous-time multi-input multi-output descriptor system, with these parameters affecting system matrices through a linear fractional transformation. Sampling is permitted to be slow and non-uniform, and there are no necessities to satisfy the Nyquist frequency. This model can be used to described the behaviors of a networked dynamic system, and the obtained results can be straightforwardly applied to a...

The difficulty in frequency domain identification is that frequency components of arbitrary inputs and outputs are not related by the system's transfer function if signals are windowed. When rectangular windows are used, it is well known that this difference is related to transient effects that can be estimated alongside the systems' parameters windows. In this work, we generalize the approach for arbitrary windows, showing that signal windowing introduces additional terms in...

The behavior of solution trajectories usually changes if we replace the classical derivative in a system by a fractional one. In this article, we throw a light on the relation between two trajectories $X(t)$ and $Y(t)$ of such a system, where the initial point $Y(0)$ is at some point $X(t_1)$ of trajectory $X(t)$. In contrast with classical systems, trajectories $X$ and $Y$ do not follow the same path. Further, we provide a Frenet apparatus of both trajectories in various cas...

Modelling physical data with linear discrete time series, namely Fractionally Integrated Autoregressive Moving Average (ARFIMA), is a technique which achieved attention in recent years. However, these models are used mainly as a statistical tool only, with weak emphasis on physical background of the model. The main reason for this lack of attention is that ARFIMA model describes discrete-time measurements, whereas physical models are formulated using continuous-time parameter...

Balancing the model complexity and the representation capability towards the process to be captured remains one of the main challenges in nonlinear system identification. One possibility to reduce model complexity is to impose structure on the model representation. To this end, this work considers the linear fractional representation framework. In a linear fractional representation the linear dynamics and the system nonlinearities are modeled by two separate blocks that are i...

We generalize notions of passivity and dissipativity to fractional order systems. Similar to integer order systems, we show that the proposed definitions generate analogous stability and compositionality properties for fractional order systems as well. We also study the problem of passivating a fractional order system through a feedback controller. Numerical examples are presented to illustrate the concepts.

We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves integer derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example.