Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 847210, 9 pages

http://dx.doi.org/10.1155/2015/847210

## Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach

Centre for Security, Information and Advanced Technologies (CEBIA-Tech), Faculty of Applied Informatics, Tomas Bata University in Zlín, nám. T. G. Masaryka 5555, 760 01 Zlín, Czech Republic

Received 16 March 2015; Accepted 14 June 2015

Academic Editor: Haranath Kar

Copyright © 2015 Radek Matušů and Roman Prokop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper deals with a graphical approach to investigation of robust stability for a feedback control loop with an uncertain fractional order time-delay plant and integer order or fractional order controller. Robust stability analysis is based on plotting the value sets for a suitable range of frequencies and subsequent verification of the zero exclusion condition fulfillment. The computational examples present the typical shapes of the value sets of a family of closed-loop characteristic quasipolynomials for a fractional order plant with uncertain gain, time constant, or time-delay term, respectively, and also for combined cases. Moreover, the practically oriented example focused on robust stability analysis of main irrigation canal pool controlled by either classical integer order PID or fractional order PI controller is included as well.

#### 1. Introduction

Recently, the fractional order calculus (FOC) and its engineering applications represent attractive research field with rapidly growing amount of related scientific works. This progress is understandable since the use of differentiation and integration under an arbitrary real or even complex number of the operations provides efficient tool for many real-life problems and since the knowledge of suitable and relatively comprehensible mathematical instruments for fractional order issues has increased lately. The principal sources for studying the FOC are, for example, the monographs [1–3] and possibly also [4] or [5]. The FOC has already been useful in areas such as bioengineering, viscoelasticity, electronics, robotics, control theory, and signal processing [6, 7]. The examples of several useful control-oriented works can be seen in [8–12]. Obviously, the FOC has influenced also analysis and control of time-delay systems which represent usually complicated but relatively frequent controlled objects [13–17].

Models with parametric uncertainty are popular and effective way to uncertainty modelling and consequently to description of too complicated, nonlinear, or varying real-life systems by means of linear models. In such systems, the structure (model order) is supposed or known, but the parameters are bounded somehow. Typically, they lie within given intervals. One of the related principal tasks consists in robust stability analysis, that is, in investigation of keeping the stability under all possible variations of uncertain parameters. Some authors have already tried to combine the issue of robust stability of systems affected by parametric uncertainty with fractional order systems, for example, [18–28].

This paper is focused on a graphical approach to robust stability analysis and especially on its application to fractional order time-delay control systems. More specifically, the control loop studied in the computational examples consists of a fractional order time-delay plant with uncertain parameters and standard integer order PID controller. The robust stability is tested via plotting the value sets of a closed-loop characteristic quasipolynomial and application of the zero exclusion condition. The presented examples include the typical shapes of the value sets for a fractional order controlled system with uncertain gain, time constant, or time-delay term, respectively, and then also for the case of all uncertain parameters together. Moreover, the final process-control-oriented example deals with robust stability analysis for main irrigation canal pool controlled by either classical PID or fractional order PI controller. This paper is the significantly extended version of the conference contribution [29].

The paper is organized as follows. In Section 2, basic theoretical background and description of fractional order systems are provided. Section 3 then presents the robust stability analysis for integer order and fractional order systems with parametric uncertainty with especial emphasis on the value set concept and the zero exclusion condition. Next, a number of computational examples and visualizations of the value sets for closed loop containing a fractional order time-delay plant with various uncertain parameters are shown in the extensive Section 4. Further, Section 5 contains more specific and practically oriented example motivated by control of main irrigation canals with variable parameters. And finally, Section 6 offers some concluding remarks.

#### 2. Fractional Order Systems

The FOC is grounded in generalization of differentiation and integration to an arbitrary (rational, irrational, or even complex) order. This generalization has resulted in the introduction of basic continuous differintegral operator [1, 2, 4, 6]:where is the order of the differintegration (ordinarily ) and is a constant related to initial conditions. The differintegral can be defined in various ways. The three most common ones are Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions.

The Laplace transform of the differintegral is given by [4, 9] where integer lies within .

The (time-delay-free) fractional order transfer function can be written as [3, 5]where with and with denote constants and with and with are arbitrary real numbers. According to [4, 5], one can assume inequalities and without loss of generality. In this paper, the controlled time-delay system is supposed generally as

#### 3. Robust Stability Analysis under Parametric Uncertainty

The stability of the closed-loop system will be tested via stability of its characteristic polynomial (or quasipolynomial in the case of this paper).

The continuous-time fractional order uncertain polynomial can have the formwhere is the vector of uncertainty and for are coefficient functions. Besides, the characteristic quasipolynomial (for closed control loop with time-delay plant) would contain the term .

Then, the family of polynomials is [30]where is the uncertainty bounding set (frequently, it is a multidimensional box).

The family of polynomials (6) is robustly stable if and only if is stable for all . The choice of technique for investigation of robust stability depends primarily on the structure of uncertainty. Generally, the higher level of relation among coefficients entails more complex robust stability analysis which requires more sophisticated tools. However, one graphical method seems to be unique from the viewpoint of its universality and applicability. It is based on combination of the value set concept and the zero exclusion condition [30]. It can be applied for a wide range of uncertainty structures, from the simplest to the very complicated ones. Moreover, it is applicable also for various regions of stability (robust -stability). The detailed information on parametric uncertainty and robust stability analysis as well as examples of the typical value sets can be found in [30] and subsequently, for example, in [31, 32]. And finally, [18–21] have extended the idea of the value set concept also to fractional order uncertain polynomials.

Under assumption of a family of polynomials (6), the value set at frequency is given by [30]

It means that is the image of under . Practical construction of the value sets can be accomplished by substituting for , fixing , and letting the vector of uncertain parameters range over the set .

The zero exclusion condition for Hurwitz stability of family of continuous-time polynomials (6) is defined as follows [30]: assume invariant degree of polynomials in the family, pathwise connected uncertainty bounding set , continuous coefficient functions for , and at least one stable member . Then the family is robustly stable if and only if the complex plane origin is excluded from the value set at all frequencies ; that is, is robustly stable if and only if

Authors of [18–21] construct the value sets of the fractional order families of polynomials mainly on the basis of the fact that the fractional power of can be written asand on the consequent analysis of vertices and exposed edges.

In this work, the value sets are plotted for quasipolynomials (closed-loop characteristic quasipolynomials of the feedback circuits with the uncertain time-delay fractional order plant and fixed integer order or fractional order controller) and their visualization is based on sampling the uncertain parameters and on computation of partial points of the value sets for a considered frequency range. Thanks to the applied sampling (brute-force) method, the value sets of quasipolynomials can be easily computed and consequently the robust stability can be investigated with the assistance of standard zero exclusion condition. The technique itself should be clear from the following examples.

#### 4. Computational Examples: Typical Shapes of Value Sets

Consider a fractional order time-delay plant given bywhere is a gain, stands for a time constant, is a real number representing the fractional order of the dynamics, and is a time-delay term. One or more of the parameters , , and are uncertain and they can vary within given intervals.

More specifically, the controlled system is described, for example, aswhere either one of the parameters is uncertain:or all of them can lie within supposed bounds:

In all cases, the nominal system used for the controller design is assumed with the fixed (average) values:

The PID controller for this plant could be obtained, for example, with the assistance of the FOMCON Toolbox for MATLAB [33, 34] and its routine “iopid_tune.” More specifically, the Oustaloup filter based [35] approximation leads to the integer order model:The selected controller for this plant has the form

More information on integer order approximations of fractional order systems can be found, for example, in [36]. The comparison between step responses of the fractional order (FO) model and its integer order (IO) approximation can be seen in Figure 1. It is still obtained through the FOMCON Toolbox.