A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes.

In this paper first the Multi-term Fractional-Order PID (MFOPID) whose transfer function is equal to [Formula: see text] , where kj and αj are unknown and known real parameters respectively, is introduced. Without any loss of generality, a special form of MFOPID with transfer function kp+ki/s+kd1s+kd2sµ where kp, ki, kd1, and kd2 are unknown real and µ is a known positive real parameter, is considered. Similar to PID and TID, MFOPID is also linear in its parameters which makes it possible to study all of them in a same framework. Tuning the parameters of PID, TID, and MFOPID based on loop shaping using Linear Matrix Inequalities (LMIs) is discussed. For this purpose separate LMIs for closed-loop stability (of sufficient type) and adjusting different aspects of the open-loop frequency response are developed. The proposed LMIs for stability are obtained based on the Nyquist stability theorem and can be applied to both integer and fractional-order (not necessarily commensurate) processes which are either stable or have one unstable pole. Numerical simulations show that the performance of the four-variable MFOPID can compete the trivial five-variable FOPID and often excels PID and TID.

The effect of purmorphamine on differentiation of endometrial stem cells into osteoblast-like cells on collagen/hydroxyapatite scaffolds.

We assessed the effect of purmorphamine along with collagen/hydroxyapatite scaffold in inducing osteogenesis of human endometrial stem cells (hEnSCs). The adhesion, viability, proliferation, and differentiation of cells on scaffold were assayed with SEM, MTT, real time-PCR, and ALP assay, respectively. The results were shown good integration of cells with scaffold. Also, qRT-PCR of differentiated cells shows that osteoblast cell markers are expressed after 21d in 2D and scaffold groups while in the scaffold group the expression of these markers were higher than the 2D group. Based on our findings, collagen/hydroxyapatite scaffold with PMA has the potential role in osteogenic differentiation of hEnSCs.

Efficient method for time-domain simulation of the linear feedback systems containing fractional order controllers.

One main approach for time-domain simulation of the linear output-feedback systems containing fractional-order controllers is to approximate the transfer function of the controller with an integer-order transfer function and then perform the simulation. In general, this approach suffers from two main disadvantages: first, the internal stability of the resulting feedback system is not guaranteed, and second, the amount of error caused by this approximation is not exactly known. The aim of this paper is to propose an efficient method for time-domain simulation of such systems without facing the above mentioned drawbacks. For this purpose, the fractional-order controller is approximated with an integer-order transfer function (possibly in combination with the delay term) such that the internal stability of the closed-loop system is guaranteed, and then the simulation is performed. It is also shown that the resulting approximate controller can effectively be realized by using the proposed method. Some formulas for estimating and correcting the simulation error, when the feedback system under consideration is subjected to the unit step command or the unit step disturbance, are also presented. Finally, three numerical examples are studied and the results are compared with the Oustaloup continuous approximation method.

An efficient numerical algorithm for stability testing of fractional-delay systems.

This paper presents a numerical algorithm for BIBO stability testing of a certain class of the so-called fractional-delay systems. The characteristic function of the systems under consideration is a multi-valued function of the Laplace variable s which is defined on a Riemann surface with finite number of Riemann sheets where the origin is a branch point. The stability analysis of such systems is not straightforward because there is no universally applicable analytical method to find the roots of the characteristic equation on the right half-plane of the first Riemann sheet. The proposed method is based on the Rouche's theorem which provides the number of the zeros of a given function in a given simple closed contour. One advantage of the proposed method over previous works is that it gives the number and the location of the unstable poles. The algorithm has a reliable result which is illustrated by several examples.

Extension of the root-locus method to a certain class of fractional-order systems.

In this paper, the well-known root-locus method is developed for the special subset of linear time-invariant systems commonly known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multi-valued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis and breakaway points are extended to the fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Moreover, the effect of perturbation on the root loci is discussed. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.