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.

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 [

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, [

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 [

The paper is organized as follows. In Section

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 [

The Laplace transform of the differintegral is given by [

The (time-delay-free) fractional order transfer function can be written as [

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 form

Then, the family of polynomials is [

The family of polynomials (

Under assumption of a family of polynomials (

It means that

The zero exclusion condition for Hurwitz stability of family of continuous-time polynomials (

Authors of [

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.

Consider a fractional order time-delay plant given by

More specifically, the controlled system is described, for example, as

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 [

More information on integer order approximations of fractional order systems can be found, for example, in [

Comparison of step responses of original (FO) model (

The control responses for the loops with controller (

Comparison of control responses of original (FO) model (

Nevertheless, the approximation was done only for the sake of IO controller choice. The robust stability of the closed-loop control system will be investigated by means of the family of its characteristic quasipolynomials, which contains the true FO model (

First, only the gain is supposed to be uncertain while the time constant and time-delay term remain fixed; that is, scenario (

The straight-line value sets computed for the corresponding family of closed-loop characteristic quasipolynomials for the range of frequencies from 0 to 15 with the step 0.05 are depicted in Figure

Value sets for controller (

Value sets for controller (

Now, the time constant is going to be the only uncertain parameter according to (

Value sets for controller (

Value sets for controller (

Next simulation scenario is given by (

Value sets for controller (

Value sets for controller (

Finally, the controlled plant with all three varying parameters is assumed—see (

Value sets for controller (

The family definitely contains a stable member and the zero point is excluded from the value sets (as can be seen from the zoomed Figure

Value sets for controller (

In addition to all robustly stable cases shown in the previous parts, one can very easily obtain the family which is robustly unstable. For example, assume the uncertain gain case (

Value sets for controller (

Value sets for controller (

Nevertheless, even if the stability test using the zero exclusion condition is visually very simple, one has to be careful about fulfillment of all given preconditions, for example, the existence of at least one stable member of the analyzed family. If they are ignored, it can lead to the incorrect results. For example, consider again the same controlled plant with gain (

Value sets for controller (

Value sets for controller (

Since the zero is obviously excluded from the value sets, it could (wrongly) indicate the robust stability of the family. However, the family does not have any stable member and so the zero exclusion condition is not fulfilled actually. In fact, all members of the family are unstable which is the reason why the stability border is not crossed at all and why the zero point is not included. All in all, the family is not robustly stable and the assumed control loop would be unstable even for all possible values of

Whereas the previous examples from Section

Water is indispensable element for life and it is becoming the most valuable resource all over the world. Nowadays, irrigation is reported as the major water consuming activity [

In [

Two controllers were designed in [

The maximal assumed variations of parameters from [

The corresponding families of closed-loop characteristic quasipolynomials are

The sampling of frequency and parameters for the sake of the value sets visualization has been chosen as

The value sets for the family of quasipolynomials (

Value sets for controller (

Value sets for controller (

The value sets for the family of quasipolynomials (

Value sets for controller (

Value sets for controller (

The main aim of the paper was to present a graphical approach to robust stability analysis and its application to fractional order time-delay feedback control loops consisting of a family of fractional order time-delay plants and either integer order or fractional order controller. The robust stability was verified through visualization of the value sets of a closed-loop characteristic quasipolynomial family and subsequent application of the zero exclusion condition for various combinations of uncertain parameters. Despite the fact that the presented computational examples from Section

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was performed with financial support of research project NPU I no. MSMT-7778/2014 by the Ministry of Education of the Czech Republic and also by the European Regional Development Fund under the Project CEBIA-Tech no. CZ.1.05/2.1.00/03.0089.