Milad Siami

Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations

This paper presents two different stabilization methods based on the fractional-calculus theory. The first method is proposed via using the fractional differentiator, and the other is constructed based on using the fractional integrator. It has been shown that the proposed techniques can be used to suppress chaotic oscillations in 3-D chaotic systems. To show the practical capability of the methods, some experimental results on the control of chaos in chaotic circuits are presented.

More Details on Analysis of Fractional-order Van der Pol Oscillator:

This paper is devoted to the analysis of fractional order Van der Pol system studied in the literature. Based on the existing theorems on the stability of incommensurate fractional order systems, we determine parametric range for which a fractional order Van der Pol system with a specific order can perform as an undamped oscillator. Numerical simulations are presented to support the given analytical results. These results also illuminate a main difference between oscillations in a fractional order Van der Pol oscillator and its integer order counterpart. We show that contrary to integer order case, trajectories in a fractional Van der Pol oscillator do not converge to a unique cycle.

Fundamental Limits and Tradeoffs on Disturbance Propagation in Linear Dynamical Networks

We investigate performance deterioration in linear consensus networks subject to external stochastic disturbances. The expected value of the steady state dispersion of the states of the network is adopted as a performance measure. We develop a graph-theoretic methodology to relate structural specifications of the coupling graph of a linear consensus network to its performance measure. We explicitly quantify several inherent fundamental limits on the best achievable levels of performance and show that these limits of performance are emerged only due to the specific interconnection topology of the coupling graphs. Furthermore, we discover some of the inherent fundamental tradeoffs between notions of sparsity and performance in linear consensus networks.

Stability Preservation Analysis for Frequency-Based Methods in Numerical Simulation of Fractional Order Systems

In this paper, the frequency domain-based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation. First, the stability boundary curve is exactly determined for these methods. Then, this boundary is analyzed and compared with an accurate (ideal) boundary in different frequency ranges. Also, the critical regions in which the stability does not preserve are determined. Finally, the analytical achievements are confirmed via some numerical illustrations.

Fundamental limits on robustness measures in networks of interconnected systems

We investigate robustness of interconnected dynamical networks with respect to external distributed stochastic disturbances. In this paper, we consider networks with linear time-invariant dynamics. The ℋ2 norm of the underlying system is considered as a robustness index to measure the expected steady-state dispersion of the state of the entire network. We present new tight bounds for the robustness measure for general linear dynamical networks. We, then, focus on two specific classes of networks: first- and second-order consensus in dynamical networks. A weighted version of the ℋ2 norm of the system, so called LQ-energy of the network, is introduced as a robustness measure. It turns out that when LQ is the Laplacian matrix of a complete graph, LQ-energy reduces to the expected steady-state dispersion of the state of the entire network. We quantify several graph-dependent and graph-independent fundamental limits on the LQ-energy of the networks. Our theoretical results have been applied to two application areas. First, we show that in power networks the concept of LQ-energy can be interpreted as the total resistive losses in the network and that it does not depend on specific structure of the underlying graph of the network. Second, we consider formation control with second-order dynamics and show that the LQ-energy of the network is graph-dependent and corresponds to the energy of the flock.

Systemic measures for performance and robustness of large-scale interconnected dynamical networks

In this paper, we develop a novel unified methodology for performance and robustness analysis of linear dynamical networks. We introduce the notion of systemic measures for the class of first-order linear consensus networks. We classify two important types of performance and robustness measures according to their functional properties: convex systemic measures and Schur-convex systemic measures. It is shown that a viable systemic measure should satisfy several fundamental properties such as homogeneity, monotonicity, convexity, and orthogonal invariance. In order to support our proposed unified framework, we verify functional properties of several existing performance and robustness measures from the literature and show that they all belong to the class of systemic measures. Moreover, we introduce new classes of systemic measures based on (a version of) the well-known Riemann zeta function, input-output system norms, and etc. Then, it is shown that for a given linear dynamical network one can take several different strategies to optimize a given performance and robustness systemic measure via convex optimization. Finally, we characterized an interesting fundamental limit on the best achievable value of a given systemic measure after adding some certain number of new weighted edges to the underlying graph of the network.

Maximum Number of Frequencies in Oscillations Generated by Fractional Order LTI Systems

In this paper, relation between the inner dimension of a fractional order LTI system and the maximum number of frequencies which exist in oscillations generated by the system is investigated. The considered system is defined in pseudo state space form and the orders of its involved fractional derivatives are rational numbers between zero and one. First, an upper bound is derived for the maximum number of frequencies. Then, using the restricted difference bases concept, a new method is introduced to design a multifrequency oscillatory fractional order system. Finally, based on the proposed method some lower bounds are derived for the maximum number of frequencies obtainable in solutions of a fractional order system having a fixed inner dimension.

Characterization of hard limits on performance of autocatalytic pathways

In this paper, we develop some basic principles to study autocatalytic networks and exploit their structural properties in order to characterize their existing hard limits and essential tradeoffs. In a dynamical system with autocatalytic structure, the systems output is necessary to catalyze its own production. We consider a simplified model of glycolysis as our motivating example. We study the hard limits of the ideal performance of such pathways. First, for a simple two-state model of glycolysis we explicitly derive the hard limit on the minimum L2-gain disturbance attenuation and the hard limit of its minimum output energy. Then, we generalize our results to higher dimensional model of autocatalytic pathways. Finally, we show that how these resulting hard limits lead to some fundamental tradeoffs between transient and steady-state behavior of the network and its net production.

Schur-convex robustness measures in dynamical networks

We investigate robustness of networks with linear time-invariant dynamics under external stochastic disturbances. We propose a partial ordering on this class of dynamical networks and exploit their structural properties to characterize their robustness properties and fundamental limits. Then, we show that several existing and widely used scalar robustness measures are indeed Schur-convex functions on the spectrum of the Laplacian of the networks. We show that certain robustness features of the first-order consensus networks can be formulated as Schur-convex functions of their Laplacian eigenvalues. Specifically, we define the uncertainty volume and entropy based on the minimum-volume covering ellipsoid of the steady-state output points of the excited network. It is shown that the uncertainty volume is directly related to the number of spanning trees of the underlying graph of the network. Furthermore, we show that for networks with regular lattice interconnection topologies this measure scales asymptotically with network size. Finally, we propose an optimization-based method to improve robustness in linear dynamical networks.

Stability preservation analysis in direct discretization of fractional order transfer functions

In this paper, a class of the direct methods for discretization of fractional order transfer functions is studied in the sense of stability preservation. The stability boundary curve is exactly determined for these discretization methods. Having this boundary helps us to recognize whether the original system and its discretized model are the same in the sense of stability. Finally, some illustrative examples are presented to evaluate achievements of the paper.