Bilal Senol

A numerical investigation for robust stability of fractional-order uncertain systems

This study presents numerical methods for robust stability analysis of closed loop control systems with parameter uncertainty. Methods are based on scan sampling of interval characteristic polynomials from the hypercube of parameter space. Exposed-edge polynomial sampling is used to reduce the computational complexity of robust stability analysis. Computer experiments are used for demonstration of the proposed robust stability test procedures.

Investigation of robust stability of fractional order multilinear affine systems: 2q-convex parpolygon approach

Abstract This paper discusses the robust stability problem of fractional order systems with the multi-linear affine uncertainty structure. The 2 q -convex parpolygon approach has been extended to compute the value set of the fractional order uncertain system and to investigate the robust stability via zero exclusion principle. An illustrative example is included for fractional order multi-linear affine system to present the advantages of the 2 q -convex parpolygon approach over classical value set computation methods in the stability investigation.

Computation of the value set of fractional order uncertain polynomials: A 2q convex parpolygonal approach

Fractional order models are frequently used to describe real processes especially in the last decades. Uncertainties in this processes mostly yield to some bad results and brings computational complexity. So this comes up as a new problem waiting to be analyzed. In this paper, a 2q convex parpolygonal approach is applied to the computation of value set of fractional order uncertain polynomials to reduce the computational complexity. The analysis steps are given and the results are shown via graphical examples. It is shown that this approach is an effective way of analyzing fractional order uncertain polynomials and can be used to investigate the stability.

Tuning of fractional order PID with master-slave stochastic multi-parameter divergence optimization method

This paper presents master-slave stochastic multi-parameters divergence optimization (SMDO) method for tuning of fractional order proportional-integral-derivative (FOPID) controllers. Bodes ideal control loop (BICL) is used as a reference model that guides the optimization process. Master-slave SMDO performs optimization of FOPID and BICL parameters together. In the first stage called as locking, step responses of FOPID and BICL approximate each other. In the second stage called as drifting, BICL guides tuning of the FOPID towards a desired step response. One of the advantages of presented optimization approach is that both optimization of the reference model (BICL) and controller (FOPID) cooperates and the tuning of FOPID is governed by BICL in the drifting stage. Operation and performance of the proposed algorithm are demonstrated by simulation examples.

An experimental investigation for error-cube PID control:

This experimental study investigates the practical benefits and drawbacks of error-cube control for closed-loop PID control structures. The error-cube control approach employs the cube power of the error signal for controllers and this causes variability in control characteristics due to the non-linearity of the cube power operation. The error-cube signal introduces attenuated and magnified error regions. These two characteristic error regions result in a tight control regime and a slack control regime, depending on magnitude of the error signal. The study presents a discussion on non-linear error signals in a practical aspect and demonstrates the effects of non-linear error signals on the step response of closed-loop PID control systems via simulation results and experimental measurements. An enhanced error-cube controller was proposed to improve the control performance of the error-cube control and results are discussed.

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.

Analysis of fractional order polynomials using Hermite-Biehler theorem

This paper presents some results for stability analysis of fractional order polynomials using the Hermite-Biehler theorem. The possibilities of the extension of the Hermite-Biehler theorem to fractional order polynomials is investigated and it is observed that the Hermite-Biehler theorem can be an effective tool for the stability analysis of fractional order polynomials. Variable changing has been applied to the fractional order polynomial to transform it into an integer order one. Roots of this polynomial are found and verified with the roots obtained using the Hermite-Biehler theorem. Stability analysis has been done investigating the interlacing property of the polynomial. Results are verified with the Radwan procedure. The method is clarified via illustrative examples.

Two approaches to description and robust stability analysis of fractional order uncertain systems

The principal goal of this contribution is to present two different approaches to description and robust stability analysis of continuous-time fractional order uncertain systems. The first approach uses fractional order models with parametric uncertainty and robust stability of corresponding closed-loop control systems is investigated by using the value set concept in combination with the zero exclusion condition. On the other hand, the second approach is based on fractional order models with unstructured multiplicative uncertainty created by choosing a nominal plant and a suitable weight function. In this case, the graphical robust stability test utilizes the envelopes of the Nyquist diagrams. Both methods are illustrated by means of the joint example.

Filter approximation and model reduction comparison for fractional order systems

This paper presents a comparison study for filter approximations and model reduction techniques for control systems of fractional orders. Oustaloups recursive filter and a refined Oustaloups filter are represented to obtain higher integer order approximations of fractional order systems. Then these higher integer order systems are exposed to model reduction with some existing techniques preserving some of the dominant eigenvalues. Original system and reduced order systems are compared on a Bode diagram and average errors of the reduced systems are computed. Then, a Matlab toolbox is developed which one can easily enter the fractional order system and apply filter approximations and model reductions. Average errors of each technique for desired fractional order system can be computed using the toolbox. Thus, this study is thought to be useful for the related area of research.

Fractional order stability of systems

This paper investigates and offers some stability analysis methods for systems of non-integer orders. Well known analysis methods such as Hurwitz, interlacing property, monotonic phase increment property are reconsidered in a fractional order way of thinking. A method to find the roots of the so-called fractional order polynomials is proposed and Hurwitz-like stability of the pseudo-polynomials is investigated. Effectiveness of the interlacing property and outcomes of the monotonic phase increment property for fractional order case is shown. Results are comparatively proved and illustrated clearly.