# Distributed-Order Dynamic Systems: Stability, Simulation, by Zhuang Jiao

By Zhuang Jiao

Distributed-order differential equations, a generalization of fractional calculus, are of accelerating significance in lots of fields of technology and engineering from the behaviour of complicated dielectric media to the modelling of nonlinear platforms.

This short will expand the toolbox on hand to researchers drawn to modeling, research, keep watch over and filtering. It comprises contextual fabric outlining the development from integer-order, via fractional-order to distributed-order structures. balance matters are addressed with graphical and numerical effects highlighting the basic variations among constant-, integer-, and distributed-order remedies. the ability of the distributed-order version is confirmed with paintings at the balance of noncommensurate-order linear time-invariant structures. usual functions of the distributed-order operator persist with: sign processing and viscoelastic damping of a mass–spring organize.

A new common method of discretization of distributed-order derivatives and integrals is defined. The short is rounded out with a attention of most likely destiny learn and purposes and with a few MATLAB® codes to minimize repetitive coding initiatives and inspire new employees in distributed-order systems.

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**Example text**

2010), and the system is considered in frequency domain, however, this method is not very helpful because of the complicated procedures. In current section, sufficient and necessary condition for fractional-order system with double noncommensurate orders is proposed first. 1), we have s β1 X (s) + s β2 X (s) = AX (s) + BU (s). Assume that D = 0. 1) is H (s) = C (s β1 + s β2 )I − A −1 B. Similar to the BIBO stability for traditional control systems, we have the following definition. 2 Stability Analysis of Some Special Cases of DOLTIS Fig.

8 2 Example 2 Consider a distributed-order system with Case 1 described with parame2 2 1 ters given as A = ,B= , C = 2 1 and D = 0. 1 that this distributed-order system is not bounded-input bounded-output stable. Using MATLAB to derive numerically, the states of impulse response with null initiations are shown in Figs. 12, respectively. 5 Numerical Examples 23 Fig. 5 Fig. 8 2 5000 impulse response 4000 3000 2000 1000 0 −1000 0 2 4 6 8 10 time axis Example 3 Consider a distributed-order system with Case 2 described with parame1 3 1 ters given as A = ,B= , C = 2 1 and D = 0.

2) under the assumption of zero initial conditions is n H1 (s) = C −1 βi s I−A B. i=1 We have the following parallel result. 2) is BIBO stable if and only if all the eigenvalues of A lie on the left of curve l5 := li l j , where li and l j are symmetrical with respect to the real axis, and li is defined as: li := {x + i y |x = xω , y = yω , ω ∈ (0, ∞) } with xω and yω defined as xω := n i=1 ωβi cos β2i π, yω := n i=1 ωβi sin βi 2 π. 1. 3 Noncommensurate Constant Orders as Special Cases of DOLTIS Fig.