Sensitivity analysis of CRA based controllers in fractional order systems

Abstract

This paper focuses on robust performance analysis of a closed loop fractional order system through a sensitivity approach. The characteristic ratio assignment method is selected to attain a desired closed loop transient response. Then, we compute the sensitivity of such a desired transfer function with respect to its characteristic ratio and we explore its specifications. The relation between the coefficient diagram shape and the relative stability of the closed loop system is discussed. Also, the closed loop poles variations due to the changes in the characteristic ratios are investigated. Finally, we study a pseudo second order process to verify the robust performance of the characteristic ratio assignment approach with RST control structure.

Introduction

Benefits of employing fractional order operators in modeling, identification, and control, encourage scientists to investigate different fields of fractional calculus [1], [2], [3], [4]. Most of the physical processes could be represented better with fractional order models, especially those including viscoelasticity, diffusion, and thermoelasticity [5], [6], [7], [8]. Designing appropriate controllers for fractional order models is a main research subject in this regard. A lot of control strategies have been proposed in the literature to improve the performance of a fractional order system [9], [10], [11], [12], [13], [14], [15]. Among them, characteristic ratio assignment (CRA) method is a novel analytical approach to control the transient response of such systems [15]. In this method, the characteristic ratios which could be represented in terms of characteristic equation coefficients are assigned to gain a non overshooting step response. The speed adjustment of the transient response could be independently performed by selecting generalized time constant in accordance with the time scaling property. The change in the generalized time constant only scales the transient response without any effect on its damping or overshoot.

Designing a robust control system which is less sensitive to changes in the process parameters is one of the main goals in control theory. The sensitivity of such a system would be low with respect to perturbation in the process parameters. Thus, the sensitivity analysis of a control structure could help to investigate the robustness of its closed loop system. Sensitivity analysis of a CRA based fractional order controller is the main contribution of this paper. Some useful relations to compute the sensitivity of a closed loop transfer function and its poles due to variations in the characteristic ratios and characteristic equation coefficients are presented. Through this analytical approach, some important qualitative results are derived which could help to design a robust control system. Coefficient diagram for the proposed characteristic equation is plotted and its relation to the relative stability is illustrated. The sensitivity of the closed loop dominant poles to changes in the process parameters is discussed, as well. To verify the obtained results in a commonly used closed loop system, a pseudo second order process with uncertain parameters is considered. Based on the CRA method, an RST control structure is build to attain the desired closed loop transfer function. The robustness of the proposed controller is checked through the sensitivity analysis and results are confirmed based on computer simulations of the controller.

This paper is organized as follows. Section 2 gives a review on the CRA method and its properties for fractional order systems. Some relations to calculate the sensitivity of an all-pole fractional order system to its characteristic ratios are given in Section 3. Section 4 deals with the sensitivity analysis of a desired closed loop transfer function obtained through the CRA method to its characteristic ratios variation. The general shape of the coefficient diagram for the proposed characteristic ratio pattern and its relation to the relative stability is discussed. Robust performance verification of a closed loop system under parametric uncertainties in a case study process is given in Section 5. Section 6 concludes the paper.