Author’s reply: A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

This author’s reply addresses the comment given in the note mentioned in the title. Theorem 3 given in Tan et al. (2009) [1] uses zero exclusion principle for the stability analysis of Fractional Order Interval Polynomial (FOIP). We show that the constant degree assumption is exist in the definition of zero exclusion principle. Although it has not been clearly stated in Tan et al. (2009) [1] that FOIP of Eq. (1) is a constant degree polynomial, this condition is implicit in the zero exclusion principle. Therefore, Theorem 3 is true under the constant degree assumption which is a requirement for zero exclusion principle.     

Robust stability analysis of fractional order interval polynomials

The paper deals with the robust stability analysis of a Fractional Order Interval Polynomial (FOIP) family. Some new results are presented for testing the Bounded Input Bounded Output (BIBO) stability of dynamical control systems whose characteristic polynomials are fractional order polynomials with interval uncertainty structure. It is shown that the Kharitonov theorem is not applicable for this type of polynomial. A procedure is given for computation of the value set of FOIP. Based on the value set, an algorithm is presented for testing the stability of FOIP. The results presented in the paper are useful for the analysis and design of Fractional Order Interval Control Systems (FOICS). Examples are given to show how the proposed method can be used to assess the effects of parametric variations on the stability in feedback loops with fractional order interval transfer functions.     

Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations

With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems.     

Probabilistic robust stabilization of fractional order systems with interval uncertainty

This study investigates effects of fractional order perturbation on the robust stability of linear time invariant systems with interval uncertainty. For this propose, a probabilistic stability analysis method based on characteristic root region accommodation in the first Riemann sheet is developed for interval systems. Stability probability distribution is calculated with respect to value of fractional order. Thus, we can figure out the fractional order interval, which makes the system robust stable. Moreover, the dependence of robust stability on the fractional order perturbation is analyzed by calculating the order sensitivity of characteristic polynomials. This probabilistic approach is also used to develop a robust stabilization algorithm based on parametric perturbation strategy. We present numerical examples demonstrating utilization of stability probability distribution in robust stabilization problems of interval uncertain systems.     

Robust stability and stabilization of fractional order linear systems with positive real uncertainty

This paper investigates the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observer-based controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results.     

A novel fractional order PID navigation guidance law by finite time stability approach

In this paper, a novel fractional order proportional–integral–differential navigation guidance law utilizing finite time stability approach is presented in order to achieve robust performance for intercepting incoming targets. The proposed guidance law is designed following three-loop guidance and control scheme, considering the interceptor’s nonlinear 6 degrees-of-freedom model. In the outer loop, normal acceleration commands are generated by the proposed guidance law. In the intermediate loop, these commands are converted into equivalent body rate commands, which are tracked by dynamic inversion based autopilot in the inner loop. A fractional order circle criterion is developed for the finite time stability analysis of this proposed guidance law, whose stability conditions give an analytical bound for the flight up time in which stability can be insured. Extensive 6 degrees-of-freedom simulations and a variety of comparison studies against maneuvering targets are implemented to demonstrate the effectiveness of the proposed guidance law. The simulation results show that the proposed guidance law has better performance when comparing with the proportional navigation and proportional–integral–differential navigation guidance laws.     

Reply to comment by Maurice et al. in response to “Bragg’s Law Diffraction Simulations for Electron Backscatter Diffraction Analysis”

A reply to Maurice et al.'s comment on "Bragg's Law Diffraction Simulations for Electron Backscatter Diffraction" is presented. A new method for microscope geometry calibration is briefly presented. Also, evidence that simple diffraction simulations can be profitable tools for absolute elastic strain measurements in crystalline materials is reiterated.     

Comments on the paper “Bragg’s law diffraction simulations for electron backscatter diffraction analysis” by Josh Kacher,Colin Landon,Brent L. Adams & David Fullwood

This comment on the paper "Bragg's Law diffraction simulations for electron backscatter diffraction analysis" by Kacher et al. explains the limitations in determining elastic strains using synthetic EBSD patterns.     

Microscopic study on the concretion of ceramics in the “Nanhai I” shipwreck of China,Southern Song Dynasty (1,127–1,279 A.D.)

“Nanhai I” shipwreck of China Southern Song Dynasty is the oldest and the most integrally preserved shipwreck in the world. The related conservation and archeological research have caught great attention of different experts all over the world. In this study, different types of concretion covered on the surface of the ceramics in “Nanhai I” shipwreck were analyzed by X‐ray diffractometer, micro‐Raman spectrometer, and scanning electron microscope equipped with energy dispersive spectroscopy. Based on the analyses, we found that the grey concretion was mainly composed of quartz, aragonite, and calcite while the reddish concretion was mainly composed of pyrite and quartz. Our study indicated that the formation process of the grey concretion probably included the crystallization and transformation of aragonite, while the corrosion of iron implements and crystallization of pyrite were highly involved in the formation of reddish concretion.