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Simplified fuzzy ‐Padé controller for attitude control of quadrotor helicopters

2017; Institution of Engineering and Technology; Volume: 12; Issue: 2 Linguagem: Inglês

10.1049/iet-cta.2017.0584

1751-8652

Taleb Abdollahi, Sepideh Salehfard, Caihua Xiong, Jiang‐Feng Ying,

Fuzzy Logic and Control Systems

IET Control Theory & ApplicationsVolume 12, Issue 2 p. 310-317 Brief PaperFree Access Simplified fuzzy -Padé controller for attitude control of quadrotor helicopters Taleb Abdollahi, Taleb Abdollahi State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorSepideh Salehfard, Sepideh Salehfard State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorCai-Hua Xiong, Corresponding Author Cai-Hua Xiong chxiong@hust.edu.cn State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorJiang-Feng Ying, Jiang-Feng Ying State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this author Taleb Abdollahi, Taleb Abdollahi State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorSepideh Salehfard, Sepideh Salehfard State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorCai-Hua Xiong, Corresponding Author Cai-Hua Xiong chxiong@hust.edu.cn State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this authorJiang-Feng Ying, Jiang-Feng Ying State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, Hubei, 430074 People's Republic of ChinaSearch for more papers by this author First published: 01 January 2018 https://doi.org/10.1049/iet-cta.2017.0584Citations: 9AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a simplified fuzzy-Padé controller (FPC) for attitude control of quadrotor helicopters is proposed, which is non-linear, fractional, and model independent. To determine some of the unknown coefficients of the Padé approximant, a proportional–derivative-type fuzzy logic controller (FLC) is first designed. The responses of the simplified FPC due to step set-point manoeuvres are compared with those of the FLC in terms of several performance indices such as rise time, settling time, percentage overshoot, integral absolute error, integral of time multiplied by AE, central processing unit time, and energy consumption. Simulation results demonstrate that in each case the proposed FPC presents an improved performance over the FLC. The asymptotic stability of the simplified FPC is proved with Lyapunov stability theory and La Salle's invariant set theorem. Unlike the FLC, the proposed FPC is simple enough to be implemented on the on-board electronics of the quadrotor. Flight experiments in hovering conditions are given to demonstrate the effectiveness of the simplified FPC. 1 Introduction One of the most interested unmanned aerial vehicles is the quadrotor, which has recently received extensive attention by many researchers and academics. It possesses four independent motors that are often situated on a plus (+) or cross (×) configuration. The stabilising control and guidance of this helicopter are carried out by changes in the speed of the motors. The quadrotor is a low-cost and uncomplicated vehicle that performs many manoeuvres such as vertical take-off and landing, hovering flight, path tracking, and stationary spinning. Accordingly, it can accomplish diverse civil and military applications such as circumnavigation, surveillance, disaster relief, rescue missions, firefighting, crop dusting, film-making, reconnaissance, and anti-terrorism. Nonetheless, it possesses some disadvantages such as coupled non-linear dynamics highly susceptible to disturbances and uncertainties, underactuated characteristic, and open-loop instability [1–3]. In recent years, plentiful researches have been conducted on the design of guidance and attitude control schemes for quadrotors. These studies can be arranged into two major branches: linear and non-linear control structures. Linear controllers are valid in a neighbourhood of equilibrium points, where a linearisation of the system can be carried out. Ordinary linear controllers used for helicopters are proportional–derivative (PD) and linear quadratic regulation ones [4–6]. As a non-linear system gets away from the equilibrium point, the accuracy of linear controllers may degrade significantly. Non-linear controllers can compensate non-linearities in large operation ranges. A non-linear feedback linearisation position controller was presented in [7]. Although it could handle non-linearities in large motions, it was not robust to parameter variations and sensor noise. Two appropriate non-linear methods to deal with uncertainties and disturbances are robust and adaptive controllers. Sliding mode control [8–10], backstepping method [11–13], robust compensation technique [14–16], and non-linear control [17, 18] are some robust methods, which have been developed to deal with the tracking problem of quadrotors. Adaptive control structures based on immersion and invariance approach [19], projection adaptation [3], and disturbance observers [20] are some examples, which have been presented to compensate disturbances and uncertainties in helicopters. Fuzzy logic controllers (FLCs) provide a convenient approach to construct non-linear controllers by embedding experience of a skilled human. Unlike conventional controllers, which are based on the mathematical model of the system, FLCs require minimum modelling of the system. Furthermore, they are validated to be robust against disturbances, and their performances are less sensitive to model variations than conventional controllers. Therefore, FLCs are suitable for non-linear systems which dynamics is hard to model and susceptible to disturbances and uncertainties [21]. To improve the performance of FLCs, they are usually combined with other control approaches. Incorporation of them with linear control methods such as P–integral (PI), PD, or PID controllers leads to a considerable improvement in the performance of the system [22]. PI-type FLCs are known to possess satisfactory performance for first-order processes. Unlike PI-type FLCs, PD-types are appropriate for higher-order non-linear systems with large dead time or integrating elements. They have been developed and implemented for various practical systems. By contrast, PID-type FLCs are seldom employed owing to the hardships related to the generation of an effective rule base and the adjustment of the huge number of the parameters [21]. Similarly, inserting FLCs to non-linear control approaches can improve the stability and robustness of the system. In [23], an attitude control scheme for an aerial vehicle was introduced by combining a PD-type FLC with a sliding mode controller. In [24], a fuzzy adaptive backstepping control approach was presented to control the position of an aerial robot. A quadrotor is a second-order non-linear system, whose dynamics is complicated and hard to define. Moreover, it is an open-loop unstable vehicle, which is highly sensitive to disturbances and uncertainties. Furthermore, it possesses some dead time due to time delays of actuators and sensors. Therefore, employing a PD-type FLC is a suitable approach to deal with the tracking control of a quadrotor. However, they produce more computational load than conventional controllers. A PD-type FLC was presented in [25] for trajectory tracking control of a quadrotor. Since the implementation of the FLC was out of the computing power of the quadrotor, the control commands were generated by a computer and transferred to the vehicle using a wireless transmitter. An online self-tunable FLC was described in [26] for path control of a quadrotor, without any experimental tests. In [27], an adaptive fuzzy gain-scheduling sliding mode control scheme was proposed for attitude regulation of quadrotors and verified just with simulations. In addition to the high computational complexity of FLCs, another problem with them is that their stability has not been proved, except for very few situations. A simple approach to decrease the computational burden of an FLC and exploit the heuristic knowledge embedded in it is the use of fuzzy-Padé controllers (FPCs). In this methodology, fuzzy singleton rules extracted from fuzzy rules are employed to determine unknown parameters in a Padé approximant. Since the fuzzy singleton rules are classical (crisp) rules, inference engine, fuzzification, and defuzzification levels are eliminated in FPCs. Accordingly, their stability can more easily be proved by employing common methods such as Lyapunov theorem. Furthermore, FPCs are proved to be robust to disturbances and uncertainties. Besides, they do not require extensive modelling of the system. Moreover, they are considerably fast and straightforward which can easily be implemented in empirical applications where FLCs are not applicable [22]. In this research, a simplified FPC for attitude control of quadrotor helicopters is proposed, which is non-linear, fractional, and model independent. To determine the unknown coefficients of the Padé approximant, a PD-type FLC is first designed. Since the number of the fuzzy singleton rules derived from the FLC is less than the number of the unknown coefficients, some of them cannot be determined by the rules. For convenience, the coefficients not specified are found by the values that construct the simplest form of the Padé approximant. Since all of the coefficients are not determined by the fuzzy singleton rules, this simplest form of the Padé approximant is called a simplified FPC. The asymptotic stability of the simplified FPC is proved with Lyapunov stability theory. The responses of the proposed FPC due to step set-point manoeuvres are compared with those of the FLC in terms of several performance indices such as rise time, settling time, percentage overshoot, integral AE (IAE), integral of time multiplied by AE (ITAE), central processing unit (CPU)-time, and energy consumption. Simulation results demonstrate that in each case the FPC presents an improved performance over the FLC. Unlike the FLC, the proposed FPC is simple enough to be implemented on the on-board electronics of the quadrotor. Flight experiments in hovering conditions are given to demonstrate the effectiveness of the FPC. The rest of this paper is organised as follows: Section 2 presents the non-linear mathematical model of the quadrotor. Section 3 describes the design procedure of the FLC and FPC with the stability analysis. In Section 4, the FLC is compared with the FPC using simulations. Experimental results are shown in Section 5. Concluding remarks are given in Section 6. 2 Model description As depicted in Fig. 1, the rotation speed of each rotor is presented by , where . The thrust force () and the torque () generated by each propeller can be obtained by and , where and are the force and torque coefficients, respectively. denotes an earth-fixed inertial frame and defines a body-fixed frame with the origin located at the quadrotor centre of mass. Let represent the position of the origin of the frame B with respect to the inertial coordinate. Moreover, let indicate Euler angles, representing roll, pitch, and yaw, respectively. The rotation matrix relating the frame B to the inertial coordinate is given by (1). Here, and denote and , respectively [28] (1) Fig. 1Open in figure viewerPowerPoint Quadrotor coordinate frames, rotation speed of rotors, thrust forces, and torques generated by propellers Let denote the angular velocity of the quadrotor with respect to the inertial coordinate, expressed in the frame B. The Euler angle rates () can be derived by (2), where denotes [28] (2) Assumption 1.The pitch angle is required to be to avoid the singularity in (2). Supposing as the external torques acting on the quadrotor, as the angular momentum of the vehicle, and as the symmetric positive definite constant inertia matrix of the body, Euler's moment equation can be written as (3) in the body-fixed frame [28] (3) For simplifying the vector product (×), the notation is presented which is a skew-symmetric matrix such that for any vector . The angular momentum of the vehicle () is split up into the momentum of the stationary components () and the momentum of the moving parts (). Supposing as the inertia of the moving parts of each motor, can be calculated by (4) (4) The external torques acting on the helicopter () can be expressed by (5), where is the horizontal distance from each rotor centre to the quadrotor centre of mass (5) Since under Assumption 1, the matrix is invertible, from (2) and (3), one can obtain the following equation [29]: (6)where (7) (8) (9) By Newton equation, the linear acceleration of the helicopter in the inertial frame () is proportional to the external forces acting on it including the thrust and gravity forces, as presented in (10) (10) Here, , , and m is the mass of the vehicle. Consequently, (6) and (10) indicate the dynamic model of the quadrotor. In the following sections of this paper, the attitude control problem will be investigated. The control outputs for three Euler angles are denoted by . Furthermore, is defined to control the altitude of the helicopter. Remark 1.In this paper, is assigned a positive value to generate enough lift to fix the altitude of the vehicle. The attitude controllers set appropriate values for the control outputs , which determine the rotors speed . Remark 2.Furthermore, a velocity controller is required to manage the rotors speed. Since the convergence of the rotors speed to the desired speed is much faster than the convergence of the attitude tracking error to zero, this velocity controller is ignored in this research. 3 Controller design In this section, a simplified FPC for attitude control of the quadrotor is presented. To determine the unknown coefficients of the FPC, a PD-type FLC is first designed. For evaluating the performance of the controllers, an identical block diagram is used for both of them, as presented in Fig. 2. The desired Euler angles are denoted by . Define and as actual attitude tracking error and error rate, respectively. Fig. 2Open in figure viewerPowerPoint Block diagram of the FLC and FPC Since all membership functions (MFs) for the inputs and output of the controllers are defined on the common interval [−1, 1], input scaling matrixes (SMs) and are employed to normalise the values of and to the range of [−1, 1]. Therefore, and represent normalised attitude tracking error and error rate, respectively. To impose upper and lower values of and in the range of [−1, 1], saturation blocks are also applied. Since the output of the controllers is normalised in the range of [−1, 1], an output SM is used to convert it to the respective actual output . Input and output SMs are defined as , where , and , , and are strictly positive constant scaling factors (SFs). 3.1 FLC design The designed FLC is a knowledge-based controller, formed on our experience in manipulating the controller output based on and to decrease the attitude tracking error to zero. Unlike conventional PD controllers, the PD-type FLC is a non-linear function of and to improve the controller performance. The FLC incorporates this non-linearity by a limited number of IF–THEN rules. The inference engine of the FLC employs these rules to develop a mapping from the input fuzzy sets to the output fuzzy sets. To transfer each element of the inputs ( and ) to the input fuzzy sets, identical MFs () have been used, as shown in Fig. 3 a. There are three MFs for the input sets as negative (N), zero (Z), and positive (P). The fuzzy sets applied to each element of the output () are displayed in Fig. 3 b. Fig. 3Open in figure viewerPowerPoint MFs used for the inputs and output of the FLC (a) MFs for each element of the inputs, (b) MFs for each element of the output There are five MFs for the output sets as negative big (NB), negative (N), zero (Z), positive (P), and positive big (PB). The MFs become more narrow and compact near the origin to provide higher sensitivity around it. All MFs in Fig. 3 are symmetric Gaussian functions with the standard deviation () equal to 0.3, excepting the two MFs at the extreme ends, which are Gaussian combination functions with . The designed rule base for computing based on and is shown in Table 1, which . This table demonstrates that when the normalised attitude tracking error and error rate of an angle are both P or N, the error is large and increasing. Therefore, the normalised control output of that angle should be large (PB or NB) to decrease the error fast. When either of the error or error rate is P or N, if the other one is Z, the error is large and constant or oscillating around the origin. Consequently, the output should be small (P or N) to reduce the error slowly. Table 1. Fuzzy rules ofthe FLC N Z P N NB N Z Z N Z P P Z P PB When or is P, if the other is N, the error is naturally decreasing. Hence, the output should be Z. When and are both Z, the error is small around the origin. Accordingly, the output should be Z. Finally, Mamdani-type inferencing and centroid method of defuzzification are used, to complete the FLC. Choosing appropriate values for the SMs is required to accomplish the satisfactory control performance. Since there is no well-defined tuning approach for FLCs such as Ziegler–Nichols for conventional controllers, the SMs are tuned through trial and error [21]. For this end, and are first adjusted so that and almost cover the whole domain [−1, 1] to use the entire rules completely. Then, is tuned to make the performance of the controller as efficient as possible. The selected values for the SFs are presented in Table 2, which are applied for both of the controllers. Table 2. Input and output SFs used for the controllers SF Value SF Value SF Value 1.15 0.17 2 1.15 0.17 2 1.15 0.23 1 3.2 FPC design A simple approach to approximate a function of variables is the use of Padé approximant, which often provides larger convergence region and faster convergence rate than Taylor series. An Padé approximant for is a rational fraction composed of an m th-order polynomial of in the numerator and an n th-order polynomial of in the denominator. In most cases, these two orders are supposed the same that results in an Padé approximant. Without losing the generality, supposing , (11) gives a general [2, 2] Padé approximant for [22] (11) To be confident that the denominator in (11) is always non-zero, applying the following constraint is required [22]: (12)To construct an FPC, the Padé unknown coefficients in (11) should be determined based on fuzzy singleton rules extracted from an FLC, which are a particular form of fuzzy rules. A fuzzy singleton rule is an IF–THEN rule, which is defined with fuzzy singleton MFs. Instead of fuzzy MFs which span in the range of [0, 1] and have infinite members, fuzzy singleton MFs have only one member with the degree of membership equal to 1. Accordingly, fuzzy singleton rules are classical (crisp) rules, which generate some data pairs. Since the fuzzy singleton rules are crisp, fuzzifier, defuzzifier, and fuzzy inference engine are not required any more in the FPC. Consequently, the computational burden of the FPC is remarkably less than the relative FLC. Since the output of the FLC (), designed in Section 3.1, is a non-linear function of two variables ( and ), we assume in (11). Therefore, (13) gives a [2, 2] Padé approximant for as a function of and , which possesses 11 unknown coefficients (13) For convenience, the extracted fuzzy singleton rules are indicated in Fig. 3 by the numbers on the , , and axes (). For an instance, the fuzzy rule ‘IF is Z and is P THEN is P’ constructs the fuzzy singleton rule ‘IF is 0 and is 0.2 THEN is 0.2’. This fuzzy singleton rule is equivalent to . Table 3 shows the nine fuzzy singleton rules derived from the fuzzy rules described in Table 1. Remark 3.Giving the same importance to each fuzzy rule, we drew out just one fuzzy singleton rule from each fuzzy rule, even where we could extract more than one. Table 3. Fuzzy singleton rules of the FPC −1 0 +1 −1 −1 −0.2 0 0 −0.2 0 +0.2 +1 0 +0.2 +1 To develop a more accurate FPC, a higher-order Padé approximant is required, which needs more fuzzier singleton rules to be determined, whereas the number of the fuzzy singleton rules is limited by the FLC, and it might be less than the required number to construct the accurate FPC. For example, the number of the unknown coefficients in (13) is two more than the number of the fuzzy singleton rules in Table 3. Therefore, two unknown coefficients ( and ) cannot be determined, as shown in (14). According to the constraint defined in (12), the non-zero denominator region () versus the unknown coefficients in (14) is depicted by Fig. 4. (14) Fig. 4Open in figure viewerPowerPoint Non-zero denominator region of the FPC Since an FPC is less complicated than corresponding FLC, approximating an FLC with a simpler FPC may degrade the performance. Consequently, employing an optimisation algorithm can be a useful approach to determine the rest of the unknown coefficients and make the FPC more efficient. In this method, the unknown values subject to the constraint (12) can be found by optimising an objective function such as energy consumption or convergence speed. However, in this research, for convenience in proving the stability of the proposed controller, the simplest form of the Padé approximant given by (15) is used, which is derived from (14) by setting the values of and . In accordance with Fig. 4, these values are consistent with the constraint mentioned in (12) (15) Remark 4.Since all Padé coefficients in the proposed FPC have not been determined by the fuzzy singleton rules, the simplest form of the Padé approximant presented in (15) is called a simplified FPC, which is a non-linear, fractional, and model independent controller. To provide a clear visual representation of the non-linear relationship between the inputs ( and ) and output () of the controllers, the control surface of the FLC and the simplified FPC are illustrated in Figs. 5 a and b, respectively. It is apparent that (15) without the second term in the denominator converts to a conventional PD controller. To depict the effect of this term more clearly, the variation of versus and with and without this term is displayed in Fig. 5 c. To make a better visual depiction of the contrast between the simplified FPC and the relevant PD controller, the control surface in Fig. 5 b has been rotated 90° in Fig. 5 c. From Fig. 5 c, it can be seen that the FPC operates like the PD controller in the vicinity of the origin. However, it provides larger (smaller) control commands than the PD controller as the error and error rate are both large (small). This non-linear characteristic stems from the heuristic knowledge embedded in the FLC, as depicted in Fig. 5 a. Fig. 5Open in figure viewerPowerPoint Control surface of the controllers (a) Control surface of the FLC, (b) Control surface of the simplified FPC, (c) Control surface of the PD controller and simplified FPC In the following section, the performance of the FPC is compared with that of the relevant FLC to present its superiority over the FLC. Furthermore, the asymptotic stability of (15) for attitude control of quadrotors is proved in the following theorem. Theorem 1.Consider a system described by (6) under the following control law: (16)where , , , , , , and represents the proposed FPC by (17) Then, given a constant , the equilibrium point () is asymptotically stable. Proof.Let us take the following Lyapunov function candidate: (18)where . Using (8) and (9), one can derive that (19) Since is skew symmetric, (19) indicates the skew symmetry of the matrix (). Differentiating the first term of V in (18) explicitly (20)Expanding the term using (6), one can conclude that (21)Given the skew symmetry of (), one can write the time derivative of the Lyapunov function as (22)Using the control law expressed in (16) and (7), one can obtain that (23)where and I is a 3 × 3 unity matrix. Expanding the second term in the right-hand side of (23), one can get that (24)where and . Consequently, . Invoking La Salle's invariant set theorem, it is only required to show that the system cannot get stuck at a point where while . Since indicates , which in turn implies that , one can conclude that the largest invariant set contains only the equilibrium point (). Consequently, the system converges asymptotically to the desired state . □ 4 Simulation results In this section, the performance of the controllers is compared by performing some simulations using Simulink/MATLAB R2016a. The personal computer (PC) employed for this comparison is equipped with an Intel Core Processor i5 (3.2 GHz) and 8 GB random access memory. All simulations are conducted with a sampling frequency of 200 Hz. The quadrotor parameters are extracted from a faithful computer-aided design model or identified from experimental tests as follows: , , , , , and . To compare the performance of the proposed FPC with that of the FLC, they are tested with step set-point changes. To draw a clear comparison between the controllers, several performance indices are employed. Since rise time () and percentage overshoot () are usually in conflict with each other, they are both considered. In addition to the and , the settling time () is also investigated to explore the convergence speed of the controllers. Besides the mentioned time-domain specifications, the two integral indices IAE and ITAE are used because visual observation of the variation of angles is not sufficient to draw a complete contrast between the controllers. The IAE index integrates the AE over time, whereas the ITAE index integrates the AE multiplied by the time over time. Thereupon, IAE penalises all errors equally over time, while ITAE weights errors that occur late more heavily than those appear early. Consequently, IAE and ITAE are employed to indicate the transient and steady-state attributes of the controllers, respectively. The CPU-time index () is utilised to measure the time spent for the overall simulation by each controller. The energy consumption index (W) is employed to investigate the actuators effort. It is calculated by the integral of the output mechanical power of the motors over time. This power equals for each motor. As illustrated in Fig. 6, the controllers are utilised to stabilise the quadrotor from an initial orientation to the origin. In this figure, each angle converges from the initial value to the origin as two other initial angles are zero to measure the performance of the controllers in stabilising each angle separately. In Fig. 6, simulations are conducted with and without parametric uncertainties to demonstrate the robustness of the controllers. The uncertain parameters are assumed as , , and to make the value of the uncertain parameters in equal to of the measured parameters in . Here, subscript u stands for the uncertainty. Fig. 6Open in figure viewerPowerPoint Stabilisation of the quadrotor using the FPC and FLC (a) Stabilisation of the roll angle, (b) Stabilisation of the pitch angle, (c) Stabilisation of the yaw angle The introduced performance indices related to Fig. 6 are listed in Table 4. Comparison of the indices reveals that the FPC in roll and pitch directions with and without uncertainties and in yaw direction with uncertainties has reduced the rise time and settling time compared with the FLC. However, the yaw FPC with measured parameters exhibits the same and as the FLC. Such as the FLC, the FPC generates no overshoot in stabilising all angles. Therefore, one can conclude that the proposed FPC can stabilise the quadrotor with faster convergence speed than the FLC. The contrast of the integral indices shows that the FPC produces less IAE and ITAE than the FLC in all angles. Moreover, the CPU time of the FPC is more than 50 times less than the FLC. This demonstrates that the proposed FPC is fast enough to be easily implemented in practical applications. The observation of the energy index presents that the proposed FPC reduces the actuators effort in comparison with the FLC. In summary, Fig. 6 and Table 4 demonstrate that the proposed FLC is superior to the corresponding FLC in all performance indices. Table 4. Performance analysis of the controllers utilised in Fig. 6 Controller Tr, s Ts, s %OS IAE, s ITAE, s2 Tc, s W, J with measured parameters roll FPC 0.29 0.56 0 0.099 0.085 0.4 8.280 FLC 0.32 0.61 0 0.106 0.090 21.6 8.331 pitch FPC 0.29 0.57 0 0.101 0.086 0.4 8.283 FLC 0.32 0.62 0 0.107 0.091 21.7 8.333 yaw FPC 0.22 0.65 0 0.135 0.112 0.4 8.068 FLC 0.22 0.65 0 0.145 0.120 21.5 8. 097 with uncertain parameters roll FPC 0.22 0.56 0 0.116 0.100 0.4 2.981 FLC 0.32 0.63 0 0.126 0.107 21.5 2.996 pitch FPC 0.21 0.57 0 0.117 0.100 0.4 2.982 FLC 0.32 0.64 0 0.127 0.107 21.6 2.997 yaw FPC 0.25 0.69 0 0.139 0.117 0.4 2.883 FLC 0.37 0.78 0 0.147 0.121 21.4 2. 910 5 Experimental results In this section, the quadrotor parameter identification method and the experimental results to verify the effectiveness of the proposed controller are presented. The quadrotor is equipped with an inertial measurement unit module to obtain the attitude information, a Bluetooth transceiver to communicate with a PC through an UART interface, and a radio transceiver to receive commands from a radio controller transmitter, as illustrated in Fig. 7 a. Fig. 7Open in figure viewerPowerPoint Configuration of the experimental setups (a) Hardware configuration of the quadrotor, (b) Parameters identification test stand To identify the thrust coefficient, the quadrotor frame equipped with one motor and propeller is fixed at one end of a miniature seesaw, which can freely pivot on a small bearing, as depicted in Fig. 7 b. The thrust created by the propeller is balanced by a force scale at the other side of the seesaw. An optical tachometer and a digital inclinometer are used to measure the speed of the propeller and the required angles, respectively. For measuring the torque coefficient, the quadrotor is rotated 90° and attached again to the seesaw beam as illustrated in the torque setup in Fig. 7 b. To verify the performance of the proposed FPC, the quadrotor is first tethered in a frame and allowed to freely rotate only around roll or pitch axis, as presented in Fig. 7 a. After getting acceptable results, the quadrotor is allowed to fly freely without the tethers outside of the frame. The variations of Euler angles in hovering flight are depicted in Fig. 8. The experiments are carried out with the same sampling frequency used in the simulations. It can be observed that the attitude tracking errors are <5°. Fig. 8Open in figure viewerPowerPoint Hovering flight of the quadrotor using the FPC (a) Response of the roll angle, (b) Response of the pitch angle, (c) Response of the yaw angle 6 Conclusions In this research, a simplified FPC for attitude control of quadrotors was proposed which was non-linear, fractional, and model independent. To determine some of the unknown coefficients of the Padé approximant, a PD-type FLC was first designed. The coefficients not specified by the fuzzy singleton rules were found by the values that constructed the simplest form of the Padé approximant. Since all of the coefficients were not determined by the fuzzy singleton rules, this simplest form of the Padé approximant was called a simplified FPC. The asymptotic stability of the proposed FPC was proved with Lyapunov stability theory and La Salle's invariant set theorem. The responses of the simplified FPC due to step set-point manoeuvres were compared with those of the FLC in terms of several performance indices. Simulation results demonstrated that in each case the FPC remarkably outperformed the FLC. Unlike the FLC, the proposed FPC was simple enough to be implemented on the on-board electronics of the quadrotor. Flight experiments in hovering conditions were given to demonstrate the effectiveness of the FPC. As a future study, it can be interesting to increase the performance of the FPC by optimising the coefficients of the Padé approximant. In this way, the unknown coefficients subject to the constraint (12) can be found by optimising an objective function such as energy consumption or convergence speed. 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