Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: Case study in twin rotor helicopter
Introduction
Reducing undesired oscillations and rapid changes of the control signal is one of the common control objectives. Filtering reference input is an efficient method to reduce undesired oscillations and prevent rapid control signal changes in practical systems [1], [2], [3], [4], [5], [6]. In recent years, fractional order pre-filters have been increasingly used in dynamical systems [7], [8], [9], [10]. The existence of the tunable order in the structure of the fractional order pre-filters makes these filters more flexible in comparison with the classical pre-filters [7].
This is due to the fact that the fractional calculus has a great potential to improve the traditional methods in different fields of control systems; such as controller design [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and system identification [21], [22], [23], [24], [25]. Fractional order concepts are employed in simple and advanced control methodologies; such as set-point weighted fractional order PID (SWFOPID) controller [11], [19], phase lead and lag compensator [14], [26], internal model based fractional order controller [27], Smith predictor based fractional order controller [18], [21], optimal fractional order controller [28], [29], robust fractional order control systems [30], [31], [32], fuzzy fractional order controller [33], fractional order sliding mode controller [34], fractional order switching systems [35], and adaptive fractional order controller [17], [36]. It has been shown that the fractional order controllers are more effective in some industrial applications [12], [15], [20]. Also, this type of controllers have been applied in a wide range of practical applications; such as control of magnetic flux in tokamak machine [19], heat flow platform [20], hexapod robot [37], permanent magnetic synchronous motors [38], and power electronic buck converters [39].
Fractional order models have been used in modelling different physical systems; such as modelling of lead-acid battery [40], human arm [41], dynamic backlash [42], thermal system [43], and large three-dimensional RC networks [44]. One significant motivation for applying fractional order models is that these models can well approximate the dynamics of high order classical models [45]. In this paper, it is shown that the fractional order counterpart of a first order transfer function without time delay can be a good candidate for characterizing the behaviour of the CE 150 helicopter which is inherently a high order plant.
The purpose of this paper is to propose an analytical method for tuning fractional order PID (FOPID) controllers based on the set-point weighted control structure. The set-point weighted structure is a simple and widely used control strategy for compensating stable, integrative, and unstable systems [11], [46], [47]. The advantage of the set-point weighted control structure is that the system responses to disturbances and set-point changes can be adjusted separately [47]. Different strategies are available for tuning the parameters of the set-point weighted PI and PID controllers [46], [47], [48]. Also in recent years, different methods have been proposed for designing the parameters of the SWFOPID controllers [11], [19]. In [11], an effective way is proposed to tune the set-point weights when the FOPID controller is designed to optimize the disturbance rejection performance based on the standard fractional set-point weighted control structure. In [19], a simple analytical method is proposed for tuning the parameters of the SWFOPID controller based on the standard fractional set-point weighted control structure.
The main contribution of this paper is to propose a simple analytical method for designing FOPID controllers based on the filtered fractional set-point weighted (FFSW) control structure. In this method, a fractional order pre-filter is employed to achieve a desired closed-loop performance. In order to evaluate the proposed method, the designed SWFOPID controller is implemented on a laboratory scale CE 150 helicopter platform and the results are compared with those of applying a filtered fractional order PI (FFOPI) controller in a similar structure. Also, a double feedback loop method is used to achieve stability and better system performance. Effectiveness and simplicity are key features of the presented method.
This paper is organized as follows. Section 2 presents some preliminaries on fractional calculus, FOPID controllers, and stability of fractional order transfer functions. The analytical design procedure of the FOPID controller based on the FFSW control structure is described in Section 3. In Section 4, the analytical design procedure of the FFOPI controller is described. The sensitivity functions of the designed control systems are compared in Section 5. In Section 6, the CE 150 helicopter platform is introduced and experimental results by applying the designed FOPID controller based on the FFSW control structure in comparison with the FFOPI controller are provided. Finally, the paper is concluded in Section 7.
Section snippets
Preliminaries
Caputo fractional derivative is the most frequently used definition of fractional order derivative [49]. The Caputo definition of the fractional order derivative of order , where is a positive non-integer number is given byIn (1), is the Gamma function. Also, and are the lower and the upper terminals of the integral [49]. The Laplace transform of the Caputo fractional derivative is given by
Fractional set-point weighted PID controller design
The general schematic of a set-point weighted control structure is shown in Fig. 2. In this structure, is the process transfer function. The process is identified by a simple three-parameter fractional order model. The control scheme studied in this paper is the FFSW structure shown in Fig. 3. In this structure to achieve a desired closed-loop response, an FOPID controller is employed. The set-point filter is used to reduce undesired oscillations. In the next subsection, an
Filtered fractional order PI controller design
In this section, an analytical method is proposed to design FFOPI controllers based on the control structure shown in Fig. 5. The closed-loop response of the proposed method in this section is similar to the results of applying the SWFOPID controller described in Section 3. At the end of the next section, it is shown that disturbance rejection of the SWFOPID controller is always better than the FFOPI controller.
Comparison of the designed controllers
In order to explain the difference of two proposed method in this paper, the sensitivity functions of the designed control systems are compared in this section.
Based on relations (16), (17), (20), (21), by applying the SWFOPID controller in the control structure shown in Fig. 4, the sensitivity function and the complementary sensitivity function are respectively obtained asand
Also according to
Experimental results
In this section, the CE 150 laboratory scale helicopter is selected for experimental application of the designed fractional order controllers. The system is shown in Fig. 6 and different control strategies have been applied to this laboratory helicopter [54], [55], [56], [57], [58]. In [55], [57], [58], high order classical models have been used for modelling the CE 150 helicopter. As previously mentioned, fractional order models can be good candidate for describing the dynamics of systems
Conclusions
In this paper, a simple analytical method for tuning the parameters of FOPID controller based on the FFSW control structure was proposed. The parameters of the FOPID controller, set-point weight, and constant coefficient of the fractional order set-point filter were tuned to achieve the desired closed-loop transfer function. Also, it was shown that disturbance rejection by applying the SWFOPID controller is always better done in comparison with the designed SWPID, FFOPI and FPI controllers in
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