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- 1. Practice Article Modeling and control approach to a distinctive quadrotor helicopter$ Jun Wu a,b , Hui Peng a,c,n , Qing Chen d , Xiaoyan Peng e a School of Information Science & Engineering, Central South University, Changsha, Hunan 410083, China b School of Electrical & Information Engineering, Changsha University of Science & Technology, Changsha, Hunan 410004, China c Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, Changsha, Hunan 410083, China d China Machinery International Engineering Design & Research Institute, Changsha, Hunan 410007, China e College of Mechanical and Automobile Engineering, Hunan University, Changsha, Hunan 410082, China a r t i c l e i n f o Article history: Received 19 February 2012 Received in revised form 15 August 2013 Accepted 15 August 2013 Available online 7 September 2013 This paper was recommended for publication by Prof. A.B. Rad. Keywords: ARX model Nonlinear system Physical model Quadrotor helicopter RBF-ARX model a b s t r a c t The referenced quadrotor helicopter in this paper has a unique conﬁguration. It is more complex than commonly used quadrotors because of its inaccurate parameters, unideal symmetrical structure and unknown nonlinear dynamics. A novel method was presented to handle its modeling and control problems in this paper, which adopts a MIMO RBF neural nets-based state-dependent ARX (RBF-ARX) model to represent its nonlinear dynamics, and then a MIMO RBF-ARX model-based global LQR controller is proposed to stabilize the quadrotor's attitude. By comparing with a physical model-based LQR controller and an ARX model-set-based gain scheduling LQR controller, superiority of the MIMO RBF-ARX model-based control approach was conﬁrmed. This successful application veriﬁed the validity of the MIMO RBF-ARX modeling method to the quadrotor helicopter with complex nonlinearity. & 2013 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Quadrotor helicopter is a kind of multicopter that is lifted and propelled by four rotors. Compared with the classical helicopter which only has a main rotor and a tail rotor, it is much easier to be constructed because it does not require mechanical linkages to vary rotor angle of attack as they spin. The vehicle motion control is easier too, which can be achieved by varying the relative speed of each rotor. Quadrotor helicopters are commonly designed to be micro- unmanned aerial vehicles (UAVs). With their small size and agile maneuverability, quadrotor helicopters are capable of small-area monitoring and exploration. Recent years, researchers are trying to expand its applications both in commercial ﬁelds and in industrial ﬁelds. A lot of smart quadrotor helicopters appended with all kinds of special mechanical equipments were designed to accomplish many complicated tasks such as gripping and perching. In near future, quadrotor helicopters may even be used as human carrying transportation devices. Undoubtedly, quadrotor helicopter is a useful class of ﬂying robots, and it is also a typical multivariable nonlinear plant. Generally speaking, we have two ways to improve their control performances, building more complete models and designing con- trollers that do not need an accurate model. Recent researches were mainly focused on nonlinear controllers design. The application of two different control techniques (PID and LQ) to OS4 (Omnidirec- tional Stationary Flying Outstretched Robot) were presented in [1]. The results of two model-based control techniques were shown. Tayebi and McGilvray [2] provided a PD2 feedback structure, which had the exponential convergence property due to the compensation of the Coriolis and gyroscopic torques. Bouchoucha et al. [3] presented an approach which is based on the combination of a backstepping technique and a robust nonlinear PI controller to stabilize the quadrotor attitude. A switching function was con- structed to achieve a robust behavior for the overall control law, but the choice of the PI gains would be a restriction of this method. In [4], dynamic inversion was applied, which is effective in the control of both linear and nonlinear systems, to tackle the coupling in quadrotor dynamics. Unlike standard dynamic inversion, the linear controller gains are chosen uniquely to satisfy the tracking perfor- mance. Guenard et al. [5] presented a visual servo control using backstepping techniques for stabilization of a quadrotor. Alexis et al. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.08.010 ☆ This work was supported by the International Science & Technology Cooperation Program (2011DFA10440), and the National Natural Science Foundation of China (71271215, 70921001). n Corresponding author. Tel./Fax: +86 731 88830642. E-mail addresses: jun.wu@csu.edu.cn (J. Wu), huipeng@mail.csu.edu.cn (H. Peng), coolchen302@qq.com (Q. Chen), xiaoyan_p@126.com (X. Peng). ISA Transactions 53 (2014) 173–185
- 2. [6] presented a switching model predictive attitude controller for an unmanned quadrotor helicopter subject to atmospheric distur- bances. The switchings among the piecewise afﬁne models are ruled by the rate of the rotation angles. To attenuate time-varying and non-vanished disturbance torques, Zhang et al. [7] designed a feedback controller with a sliding mode term to stabilize the attitude of the quadrotor. Some literatures discussed neural net- work (NN) based controller design to stabilize an aircraft against modeling error and considerable wind disturbance [8–11]. However, dynamic models built or adopted in recent researches are very similar and may be classiﬁed into three types according to different simpliﬁcations from quadrotor dynamics. The ﬁrst type is shown in [12], which neglected the air friction and the gyroscopic effects resulting from both the rigid body rotation and the propellers rotation. The second type neglected the gyroscopic effect resulting from the four propellers rotation [5,9,13–15]. The third type ideally included the gyroscopic effects resulting from both the rigid body rotation in space and the four propellers' rotation [3,4,8,16,17]. But the relation between the rotor thrust and the rotor input voltage was simpliﬁed. If accurate parameters of quadrotor can be obtained and its nonlinearities are clearly known, some physical model-based methods can achieve very good control performances. However, in some cases the quadrotor structure could be very complex so that it is difﬁcult to obtain its accurate physical model. First, the physical model needs some accurate physical quantities, such as the moment of inertia and the motor force constant, which are not easy or even unable to obtain in real application. Some groups had to try to construct their own prototype to get the accurate quantities and the symmetric structure [17], but in most applica- tions it is unfeasible to reconstruct the controlled objects. Second, in order to establish the moment equilibrium equations, we need to know the thrust forces generated by propellers. It is obvious that the relation between the thrust force and the control voltage is complicated and related to many factors, such as blade area, density of air and radius of blade. Hoffmann et al. [18] introduced a method to measure the thrusts and the torques using a load cell, but developing a thrust test stand is a big challenge itself. In addition, some quadrotors may have different conﬁgurations and some of them may be difﬁcult to be taken apart (see Fig. 1). Third, in different ﬂying postures, especially in the condition of large operating angle, the coupling dynamics among the outputs are also varying, and uncertain nonlinear terms like aerodynamic friction and blade ﬂapping can hardly be taken into account. Besides, in different applications, there might be some complex mechanical equipments appended to the quadrotor helicopters for their special purposes, these equipments could also bring some uncertain nonlinearities. Sometimes, the simpliﬁed physical mod- els may be rough and inaccurate so that the control performance would be degraded accordingly. To overcome the restrictions of the physical models, system identiﬁcation in control engineering was proposed for under- standing and controlling those unknown nonlinear dynamical systems. In [19], neural networks with linear ﬁlter also known as Narendra's model and recurrent neural networks with internal memory (Memory Neuron Networks) are used for the purpose. From the simulation studies it is shown that MNN approach is better than Narendra's approach. In [20], a comparative study and analysis of different Recurrent Neural Networks (RNN) for the identiﬁcation of helicopter dynamics using ﬂight data is presented. Three different RNN architectures, namely Nonlinear AutoRegres- sive eXogenous input (NARX) model, neural network with internal memory known as Memory Neuron Networks (MNN), and Recur- rent MultiLayer Perceptron (RMLP) networks, are used to identify dynamics of the helicopter at various ﬂight conditions. Based on the results, the practical utility, advantages and limitations of the three models are critically appraised and it is found that the NARX model is most suitable for the identiﬁcation of helicopter dynamics. In this paper, a novel NARX model is proposed to handle the modeling and control problems to an unknown nonlinear dynamical quadrotor helicopter. The referenced quadrotor helicopter in this paper is a testbed shown in Fig. 1. It is a very useful experimental equipment to test and verify all kinds of modeling and control methods to the quadrotor helicopter. Three degrees of freedom were locked in order to reduce control complexity and avoid system damage. The issue we are concerned with is obviously attitude stabilization, which is very important for control of a quadrotor helicopter since it allows the vehicle to maintain a desired orientation and results in lateral or sideways motion [2]. It has 4 propellers, 3 of which are horizontally mounted to control its pitch and roll rotation while the last one is vertically mounted to control its yaw rotation. Therefore, it has the advantage of classical helicopters in yaw motion control and also has the advantage of quadrotors in pitch and roll motion control [18]. Therefore, this quadrotor helicopter has 3 outputs and 4 inputs. The outputs are the pitch angle, roll angle and yaw angle, and the inputs are the control voltages of the 4 propellers' motors equipped at the 4 ends of the quadrotor helicopter. It is easier to be controlled compared with the traditional underactuated quadrotor which only has two inputs [21]. The quadrotor helicopter's motion is captured by a 3D universal joint and decoded to extract absolute orientation information. An intelligent data acquisition card on a PCI slot of the upper computer was used to collect real-time data and send orders to the testbed. Thanks to the Real-Time Workshop (RTW) of Matlab, we can construct real-time control system based on Matlab/Simulink environment to implement many control strategies conveniently. In this paper, we handle its modeling and control problems step by step. In Section 2, we shall ﬁrst brieﬂy introduce its physical model-based LQR controller design, from which we can clearly see that the quadrotor helicopter is a complex system whose accurate physical model can hardly be obtained. Therefore, in Section 3 the second method using system identiﬁcation technique was presented for the ﬁrst time. According to the ﬂyingFig. 1. The testbed for modeling research, 3DOF are locked. J. Wu et al. / ISA Transactions 53 (2014) 173–185174
- 3. posture of the quadrotor helicopter, we divide its working states into 16 regions averagely and a set of ARX models is identiﬁed in each region to approximate its global nonlinear behaviors, which may outperform the physical model. Then the ARX model-set- based state-feedback control law is calculated using LQR. After debugging in each region to get the optimal parameters, a gain scheduling controller integrating all the regional models and state- feedback control laws is designed, which shows good control performance in the full range of the quadrotor's attitude control. The essence of the second method is to represent the quadrotor helicopter's nonlinear characteristics with several linear ARX models which were obtained in some different working regions. For this kind of nonlinear process whose dynamic behavior can be represented by several local linear models at each time-varying operating point, one can use the Gaussian radial basis function (RBF) neural networks-based local linearization autoregressive with eXogenous (ARX) model (RBF-ARX model) [22] to effectively characterize such nonlinear system. In other words, the RBF-ARX model is a type of hybrid pseudo-linear time-varying model. The SISO RBF-ARX modeling method and its nonlinear MPC design method had been investigated in both simulation studies and real applications [23,24]. Furthermore, some stability conditions on the ofﬂine identiﬁed RBF-ARX model-based NMPC were investigated in [25]. On the basis of that the MIMO RBF-ARX model-based MPC controller had been also proposed and its effectiveness were demon- strated by a simulation study on thermal power plant [26], but it does not mean that the MIMO RBF-ARX model works for the quadrotor helicopter as well. However, it is a good motivation to try it. In Section 4 we proposed a MIMO RBF-ARX model-based global LQR control strategy (a kind of inﬁnite horizon predictive controller) for the ﬁrst time in order to improve the control performance of the quadrotor helicopter further. The essence of the last method is to describe the quadrotor helicopter's nonlinear characteristics via the global MIMO RBF- ARX model. By using a set of RBF networks to approximate the coefﬁcients of a state-dependent ARX model, the RBF-ARX model is yielded, which has the advantage of the state-dependent ARX model in the description of nonlinear dynamics. It also has the advantage of RBF networks in function approximation [27,28]. In general, a RBF-ARX model uses far fewer RBF centers compared with a single RBF network model, because the complexity of the model is dispersed into the lags of the autoregressive parts of the model. The RBF-ARX model is identiﬁed ofﬂine, which avoids potential problems led by online identiﬁcation and high costs for real-time computation. Moreover, it provides enough time for analysis and optimizations. Based on the MIMO RBF-ARX model, an inﬁnite horizon predictive controller was designed. The com- parison of the real-time control results of the three methods given in this paper showed the advantages of the MIMO RBF-ARX model-based method and conﬁrmed the validity of the MIMO RBF-ARX modeling method to this class of fast nonlinear systems. 2. The physical model-based LQR controller The coordinate of the quadrotor helicopter is shown in Fig. 2, where Fx ðx ¼ f ; l; r; bÞ denotes the thrust forces generated by 4 propellers, and its sufﬁxes mean its locations which are front, left, right, and back. From Fig. 2, we can see that the pitch is deﬁned to be the angle circled around the Y-axis, and the anti- clockwise rotation round Y-axis is deﬁned to be positive. The roll is deﬁned to be the angle circled around the X-axis and the anti- clockwise rotation round X-axis is deﬁned to be positive. The yaw is deﬁned to be the angle circled around the Z-axis and the anti- clockwise rotation round Z-axis is deﬁned to be positive as well. In order to reduce control complexity and avoid system damage, three degrees of freedom of the translations subsystem were locked. The structure is assumed to be symmetrical, the origin is assumed to coincide with the quadrotor's centroid. According to the torque equilibrium equation of each axis, three differential equations can be formulated as follows [29]: Jpp″ ¼ ðFl þFrÞLcÀFf Lf Jrr″ ¼ FrLaÀFlLa Jyy″ ¼ FbLb: 8 >< >: ð1Þ And other known conditions of the quadrotor helicopter are as follows Lc ¼ 1 2 Lf ¼ ﬃﬃ 3 p 3 La ð2Þ Lb ¼ Lf ð3Þ Fx ¼ f ðVxÞ ¼ KfcVx ðx ¼ f ; l; r; bÞ: ð4Þ Deﬁnitions of symbols are detailed in Table 1. Notice that the relation between Fx and Vx in Eq. (4) is assumed to be linear and time-invariant. Substituting Eqs. (2)–(4) to Eq. (1), we obtain p″ ¼ 1 2 ðVl þVrÞÀVf Lf Kfc Jp r″ ¼ ﬃﬃﬃ 3 p KfcLf 2Jr ðVrÀVlÞ y″ ¼ KfcLb Jy Vb: 8 : ð5Þ In addition, if the quadrotor helicopter is stabilized at a steady state where the output is Ys ¼ ½ps rs ysT and the input is Us ¼ ½Vfs Vrs Vls VbsT , then according to Eq. (5) we can also obtain the Fig. 2. Coordinate of the quadrotor helicopter. Table 1 Symbols and deﬁnitions. Symbol Deﬁnition Jp;r;y Body inertia p Pitch angle r Roll angle y Yaw angle Fx Thrust force Kfc Thrust factor Vx Motor input Lc;f ;a;b Lever J. Wu et al. / ISA Transactions 53 (2014) 173–185 175
- 4. equations below: ðpÀpsÞ″ ¼ 1 2 ðVlÀVls þVrÀVrsÞÀVf þVfs lf Kfc JP ðrÀrsÞ″ ¼ ﬃﬃﬃ 3 p Kfclf 2Jr ðVrÀVrsÀVl þVlsÞ ðyÀysÞ″ ¼ Kfclb Jy ðVbÀVbsÞ: 8 : ð6Þ A state-space equation model can be built by deﬁning the state vector as xðtÞ ¼ pÀps ðpÀpsÞ′ Z ðpÀpsÞ rÀrs ðrÀrsÞ′ Â Z ðrÀrsÞ yÀys ðyÀysÞ′ Z ðyÀysÞ T : ð7Þ And the inputs and outputs are ΔUðtÞ ¼ UÀUs ¼ ½Vf ÀVfs VrÀVrs VlÀVlsVbÀVbsT ΔYðtÞ ¼ YÀYs ¼ ½pÀps rÀrsyÀysT : 8 : ð8Þ Therefore, the state-space model of this linear time-invariant (LTI) system is _xðtÞ ¼ AxðtÞþBΔUðtÞ ΔYðtÞ ¼ CxðtÞ ( ð9Þ where A ¼ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 B ¼ 0 0 0 0 À Kfclf JP Kfclf 2JP Kfclf 2JP 0 0 0 0 0 0 0 0 0 0 ﬃﬃ 3 p Kfclf 2Jr ﬃﬃ 3 p Kfclf 2Jr 0 0 0 0 0 0 0 0 0 0 0 0 Kfclb Jy 0 0 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 C ¼ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 6 4 3 7 5 : 8 : ð10Þ On the basis of the above state-space model, we can design an inﬁnite-time state regulator for this LTI system. The objective function incorporating the states and the control efforts is min ΔUðtÞ J ¼ 1 2 Z 1 0 ½xT ðtÞQxðtÞþΔUT ðtÞRΔUðtÞ dt ð11Þ where the Q 40 and R40 are the diagonal weighing matrix for xðtÞ and ΔUðtÞ respectively. It was veriﬁed that ðA; BÞ is stabilizable and ðA; Q1=2 Þ is detect- able, thus ΔUðtÞn exists and be unique ΔUðtÞn ¼ ÀRÀ1 BT PxðtÞ; ð12Þ where the constant matrix PZ0 is the solution of the following algebraic Riccati equation: PAþAT PÀPBRÀ1 BT PþQ ¼ 0: ð13Þ The state feedback control law is ΔUðtÞn ¼ ÀKxðtÞ ¼ ÀRÀ1 BT PxðtÞ UðtÞ ¼ Us þΔUðtÞn : ( ð14Þ When the system goes to a steady state, then x-0 so that ΔY-0, and the tracking goal is achieved. Notice that Ys and Us are desired output and input. In the control strategy presented above, Us is always equal to 0, because 3DOF are locked and the gravity factor is not taken into account in the physical model. The LQR control results based on the model will be presented later (see Figs. 10 and 11). 3. ARX model-set-based gain scheduling LQR controller 3.1. ARX model-set of the quadrotor Though the physical model had been given, it was obviously an inaccurate model. For one thing, the quadrotor helicopter (see Fig. 2) has a complex structure, so that some components can hardly be taken into account when calculating the inertia. And the ﬁxed rotational pivot may deviate from its actual centroid because of its unideal symmetric structure. This would result in the inaccuracy of the physical quantities. Actually, in many real applica- tions there might be all kinds of mechanisms, such as cameras and claws, appended to the quadrotors for accomplishing different tasks. This would make their structures very complex so that the accurate physical models can hardly be obtained. For another, many nonlinear factors or unmodeled dynamics had to be simpliﬁed or totally ignored. For example, in Eq. (4) the relation between the voltage and the thrust force was simpliﬁed as a linear relation. Additional dynamics introduced by the ﬁxed rotational pivot, such as the damping rotations due to friction, had been totally ignored. And the gyroscopic effects resulting from both the rigid body rotation and the propellers rotation were totally ignored. So we can see that sometimes it is difﬁcult or even unable to obtain the accurate physical model, and the control performance would be degraded accordingly. Therefore, a set of identiﬁed ARX models is proposed in this section to approximate the global nonlinear dynamics of the quadrotor helicopter, which may outperform the physical model. Deng et al. [30] presented a discrete-time linear time-invariant (LTI) model to approximate an actual continuous-time nonlinear system and based on the identiﬁed model an output feedback LQR regulator was designed. In this paper, a set of ARX models is identiﬁed, each of them describes a local dynamics, and all the models may represent the global nonlinear behaviors well. The nonlinear characteristics is chieﬂy determined by the ﬂying posture, which is mainly related with the pitch and roll angles. Thus, according to the range of pitch and roll angle, we divide the working state into 16 regions as follows, and ξ denotes the region number ξ ¼ fix pÀ2 10 Â 4þfix rÀ5 7:5 pAð2; 42Þ; rAð5; 35Þ 8 : ð15Þ where fixðαÞ is an operator that rounds α to the nearest integer towards zero. When p¼22 and r¼20, the quadrotor helicopter is horizontally postured. In each region one ARX model is identiﬁed ofﬂine. Because the quadrotor helicopter is a multiple-output multiple-input (MIMO) system, its locally linear ARX structure is designed as follows: YðtÞþA1YðtÀ1Þþ⋯þAny YðtÀnyÞ ¼ Y0 þB1UðtÀnkÞþ⋯þBnu UðtÀnkÀnu þ1ÞþΞðtÞ ð16Þ J. Wu et al. / ISA Transactions 53 (2014) 173–185176
- 5. where Ak ¼ a11;k a12;k a13;k a21;k a22;k a23;k a31;k a32;k a33;k 2 6 4 3 7 5 Bk ¼ b11;k b12;k b13;k b14;k b21;k b22;k b23;k b24;k b31;k b32;k b33;k b34;k 2 6 4 3 7 5 8 : ð17Þ and YðtÞ ¼ ½pðtÞ rðtÞ yðtÞT are the outputs which include the pitch angle, the roll angle and the yaw angle; UðtÞ ¼ ½Vf ðtÞ VrðtÞ VlðtÞ VbðtÞT are the inputs that denote the voltage of 4 propellers; nu, ny and nk 40 are the system orders; Ak ðk ¼ 1; 2; …; nyÞ and Bk ðk ¼ 1; 2; …; nuÞ are the coefﬁcient matrixes; ΞðtÞ is the modeling residual. Assuming that at a steady state, the input and output are Us and Ys, respectively. From model (16) one can has the following ARX model: ΔYðtÞþA1ΔYðtÀ1Þþ⋯þAny ΔYðtÀnyÞ ¼ B1ΔUðtÀnkÞþ⋯þBnu ΔUðtÀnkÀnu þ1Þþ ^ΞðtÞ ð18Þ where ΔUðtÞ ¼ UÀUs, ΔYðtÞ ¼ YÀYs. ARX model (18) can be transformed into a state-space model by deﬁning the state vector as follows: XðtÞ ¼ x1 1;t x1 2;t x1 3;t ∑ t i ¼ 0 x1 1;i; x2 1;t x2 2;t x2 3;t ∑ t i ¼ 0 x2 1;i; x3 1;t x3 2;t x3 3;t ∑ t i ¼ 0 x3 1;i #T x1 1;t ¼ pðtÞÀps; x2 1;t ¼ rðtÞÀrs; x3 1;t ¼ yðtÞÀys xl k;t ¼ ∑ nþ 1Àk i ¼ 1 ∑ 3 j ¼ 1 a _ lj;k þiÀ1 xj 1;tÀ1⋯ þ ∑ nþ 1Àk i ¼ 1 ∑ 4 j ¼ 1 b _ lj;k þ iÀ1 uj;tÀ1 n ¼ maxðny; nu þnkÀ1Þ a _ ij;k ¼ Àaij;k; krny 0; k4ny ( b _ ij;k ¼ bij;k; nk rkrnu þnkÀ1 0 else k ¼ 2; 3; …; n; l ¼ 1; 2; 3 8 : ð19Þ aij;k and bij;k are the elements in Ak and Bk of Eq. (17). The state-space equation corresponding to model (18) may then be given by Xðtþ1Þ ¼ AXðtÞþBΔUðtÞ ΔYðtÞ ¼ CXðtÞþ ^ΞðtÞ ( ð20Þ where A ¼ α11 ~α12 ~α13 ~α21 α22 ~α23 ~α31 ~α32 α33 2 6 4 3 7 5 B ¼ β11 β12 β13 β14 β21 β22 β23 β24 β31 β32 β33 β34 2 6 4 3 7 5 C ¼ χ 0 0 0 χ 0 0 0 χ 2 6 4 3 7 5 8 : αii ¼ Àa _ ii;1 1 0 ⋯ 0 0 Àa _ ii;2 0 1 ⋮ ⋮ ⋮ ⋮ ⋱ 1 ⋮ Àa _ ii;n 0 0 ⋯ 0 0 1 0 ⋯ ⋯ 0 1 2 6 6 6 4 3 7 7 7 5 ðn þ1ÞÂðnþ 1Þ ~αij ¼ Àa _ ij;1 0 0 ⋯ 0 0 Àa _ ij;2 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 ⋮ Àa _ ij;n 0 0 ⋯ 0 ⋮ 0 ⋯ ⋯ ⋯ ⋯ 0 2 6 6 6 4 3 7 7 7 5 ðnþ1ÞÂðnþ 1Þ βij ¼ ½b _ ij;1 b _ ij;2 ⋯ b _ ij;n 0T ðnþ 1ÞÂ1 χ ¼ ½1 0 ⋯ 01Âðn þ1Þ There are many ways to determine the orders [31], such as Loss Function, Akaike Final Prediction Error and Akaike Information Criterion (AIC). We adopt the AIC to select the orders of the ARX models. To start identiﬁcation, the valid data is needed. Under the physical model, a controller had been designed. Although the control performance is not good, it does not affect the data acquisition. First, one made the quadrotor helicopter stabilized in one divided region, and then a noise signal was added to make it swing a little but not to exceed the divided region. By sampling a length of inputs and outputs, the data for identiﬁcation was obtained. Using the same method, we sampled 16 groups of local data for identifying all 16 local ARX models. Under the different orders, the identiﬁed models' AICs were calculated. By taking the trade-off between the smallest AIC and the complexity of the model, also considering the real-time control performance, the system orders are selected as ny ¼3, nu ¼1, nk ¼2 and n ¼ maxðny; nu þnkÀ1Þ ¼ 3. We use the least square method to estimate coefﬁcients of the ARX models. Figs. 3–5 showed a comparison between the actual outputs and the one-step-ahead prediction of the local ARX model in three degrees of freedom when ξ ¼ 15. Figs. 3–5 revealed that the ARX model can represent the local dynamics of the quadrotor helicopter very well. By using the same method, we obtained and tested 16 local ARX models one by one, and Table 2 shows the standard deviations of the one-step-ahead predictive errors in 16 local regions. The global dynamics of the quadrotor helicopter could be approximated by these local ARX models. The 16 groups of actual data sampled in 16 local regions can also be used to test the physical model presented in Section 2. Fig. 3. Comparison of actual outputs and the ARX model outputs. J. Wu et al. / ISA Transactions 53 (2014) 173–185 177
- 6. Table 3 shows the standard deviations of the one-step-ahead predictive errors of the physical model in 16 local regions. By comparing the predictive errors shown in Tables 2 and 3, one can see that the ARX model-set modeling method improved modeling accuracy considerably. Therefore, it is meaningful to use system identiﬁcation technologies to build a better model for the quadrotor. 3.2. Gain scheduling LQR controller Using the same method shown in Section 2, we can design an inﬁnite-time quadratic regulator based-on the time-invariant state- space model (20) in each divided working region, the objective function in discrete form is designed as min ΔUðkÞn J ¼ 1 2 ∑ 1 k ¼ 0 ½XT ðkÞQXðkÞþΔUT ðkÞRΔUðkÞ: ð21Þ By solving the discrete Riccati equation P ¼ Q þAT PðIþBRÀ1 BT PÞÀ1 A; ð22Þ the state-feedback gain matrix K based on the ARX model at a ﬂying posture can be obtained, and the state-feedback optimal control law at this posture is given by ΔUðkÞn ¼ ÀKXðkÞ ¼ ÀRÀ1 BT AÀT ðPÀQÞXðkÞ UðkÞ ¼ Us þΔUðkÞn : 8 : ð23Þ When the system goes to a steady state, XðtÞ-0, and the states x1 1;t, x2 1;t and x3 1;t also go to zero. This means the achievement of the tracking goal. We model the quadrotor helicopter at 16 operating regions, and the corresponding 16 linear ARX models are then built to describe the nonlinear characteristics of the quadrotor helicopter. In each divided region, an ARX model is identiﬁed and a state-feedback control law is obtained, which is debugged to work well in its region. A desired control performance may be easily achieved by adjusting the weighing matrix Q 40 and R40 in Eq. (21) appropriately. Q and R in Eq. (21) represent the relative importance to be assigned to the structural response and the control effort respec- tively. A relative larger weight would impose a higher penalty on the corresponding term for optimization of the cost function. Hence, if the reduction of the structural response is the prime concern irrespective of the cost of control or even at the expense of higher cost of control, a lower weighting force should be assigned to the term associated with the calculation of the control effort and vice versa. Because the ARX model is a local model, when the quadrotor helicopter's working region is exceeded its modeling range, a good control performance cannot be guaranteed. There is a necessity to design a global controller which integrates all the models and Table 2 Standard deviations of modeling residuals of the 16 ARX models. Region num. Pitch (deg) Roll (deg) Yaw (deg) 1 0.08409 0.15942 0.05988 2 0.08137 0.14798 0.06425 3 0.07995 0.14044 0.06481 4 0.07387 0.13977 0.05696 5 0.08155 0.15697 0.05336 6 0.07119 0.11337 0.05543 7 0.06803 0.12433 0.06166 8 0.07525 0.13354 0.04781 9 0.07910 0.13902 0.05717 10 0.09002 0.09742 0.05270 11 0.07823 0.11515 0.05529 12 0.08965 0.14741 0.05134 13 0.09584 0.12968 0.06395 14 0.09611 0.09736 0.05857 15 0.10190 0.10607 0.05354 16 0.10145 0.13660 0.06416 Table 3 Standard deviations of modeling residuals of the physical model. Region num. Pitch (deg) Roll (deg) Yaw (deg) 1 0.27540 0.51759 0.15524 2 0.24629 0.30584 0.12272 3 0.24464 0.28601 0.12955 4 0.25461 0.43836 0.14071 5 0.21382 0.54285 0.11951 6 0.20383 0.36895 0.12767 7 0.19917 0.30248 0.11814 8 0.18682 0.50068 0.12215 9 0.17178 0.52032 0.11861 10 0.15259 0.41090 0.13090 11 0.16282 0.35907 0.12595 12 0.17483 0.49034 0.12552 13 0.19508 0.54313 0.14181 14 0.18419 0.54569 0.15156 15 0.19212 0.51051 0.15662 16 0.19985 0.43155 0.14126 Fig. 4. Outputs error of the ARX model. Fig. 5. Histograms of the residuals of the ARX model. J. Wu et al. / ISA Transactions 53 (2014) 173–185178
- 7. operates well in the full range. Therefore, a gain scheduling controller is proposed, which can change its state feedback gain properly according to the posture of the quadrotor helicopter. To avoid the possible instability caused by repeatedly switching from one regional model to another at the fringes, the controller uses a gain switch law like a pivoted loop of relay. With appropriate thresholds, the problems mentioned above can be eliminated. The structure of the system is given in Fig. 6. After accomplishing all the tasks mentioned above, the real- time control is carried out. The sample period is 0.1 s. In practice, ﬁrstly the debugging was undertaken in each region to choose the optimal parameters, and then the global controller integrated all the state feedback gain matrices K. The real-time control results of the ARX model-set-based gain scheduling LQR controller are shown in Figs. 7–9. From Figs. 7 and 8 we can see that the ARX model-set-based LQR controller can stabilize the quadrotor helicopter in any given postures when the 3 outputs were changing one by one. Fig. 9 showed that the region number (or the model index) was chan- ging with the varying of the quadrotor's postures. The comparison between the new method and the former one has been illustrated in Figs. 10–13. All the parameters have been adjusted to be optimal already. The controllers accomplished the same task, which is: the quadrotor helicopter goes to a given posture, and stabilizes for a while, then returns to the original posture and stabilizes again. Notice that the 3 outputs were changing simultaneously, which is different from the control results shown in Fig. 7. From the real-time control results, one can see that the new method is feasible, and the control performance is pretty good. From the structure of the quadrotor helicopter, one can easily see that the yaw is comparably easier to be controlled than the pitch and roll, because the yaw is little coupled with pitch and roll, and is mainly affected by the back propeller. Meanwhile, the pitch and roll, which mainly represent the ﬂying posture, are mostly related with the front, left and right propellers, so the complex couplings exist, which make them more difﬁcult to be controlled relatively. In Fig. 10, we can see that there are big oscillations when the quadrotor helicopter was stabilized in the horizontal posture, Fig. 7. Outputs of real-time control based on identiﬁed ARX models. Values in square bracket denote the desired values of pitch, roll, and yaw angle; the real line represents the pitch angle, the dashed line represents the roll angle, and the dotted line represents the yaw angle. Fig. 8. Inputs of real-time control based on identiﬁed ARX models. Fig. 9. The index in (15) in real-time control based on identiﬁed ARX models, which reﬂects the region of the quadrotor helicopter posture. Fig. 10. The control result based on the physical model. Fig. 6. Structure of the control system. J. Wu et al. / ISA Transactions 53 (2014) 173–185 179
- 8. because the physical model is just a simpliﬁed model and it could not perform well at all postures. From Figs. 7 and 12 one can see that the second method shows a comparably better control perfor- mances, and it can stabilize the quadrotor helicopter at any given working point quickly without large oscillations. In Fig. 12, from the 20th second to the 30th second, one can see that the transient procedure is not smooth, because the simultaneously changing of the pitch and roll resulted in the rapid changing of the quadrotor's working regions. In order to avoid the jumping of the state feedback gain matrix K and make the transient procedure smooth, it is needed to identify the local ARX models as many as possible. 4. MIMO RBF-ARX model-based global LQR controller The essence of the second method in Section 3 is to approx- imate the quadrotor helicopter's global nonlinear dynamics with several LTI models identiﬁed in different operating points. How- ever, identifying so many local models is really a tough work and sometimes even unable to realize in real applications. The MIMO RBF-ARX model is constructed as a global model. It treats a nonlinear process by splitting the state space up into a large number of small segments, and regards the process as locally linear within each segment. Therefore the quadrotor helicopter's global nonlinear dynamics can be represented by using only one MIMO RBF-ARX model. The RBF-ARX model may be estimated by the structured nonlinear parameter optimization method (SNPOM) [26]. 4.1. RBF-ARX model of the quadrotor The quadrotor helicopter is a multiple-outputs multiple-inputs system, and its MIMO RBF-ARX structure is given as follows: YðtÞ ¼ CðwðtÀ1ÞÞþ ∑ ny k ¼ 1 AkðwðtÀ1ÞÞYðtÀkÞ þ ∑ nu þnkÀ1 k ¼ nk BkðwðtÀ1ÞÞUðtÀkÞþΞðtÞ CðwðtÀ1ÞÞ ¼ ½ϕ1 0ðwðtÀ1ÞÞ ϕ2 0ðwðtÀ1ÞÞ ϕ3 0ððwðtÀ1ÞÞÞT ϕi 0ðwðtÀ1ÞÞ ¼ ci 0 þ ∑ h m ¼ 1 ci m expfÀJwðtÀ1ÞÀZY;m J2 λY;m g AkðwðtÀ1ÞÞ ¼ a11;kðwðtÀ1ÞÞ ⋯ a13;kðwðtÀ1ÞÞ ⋮ ⋱ ⋮ a31;kðwðtÀ1ÞÞ ⋯ a33;kðwðtÀ1ÞÞ 2 6 4 3 7 5 BkðwðtÀ1ÞÞ ¼ b11;kðwðtÀ1ÞÞ ⋯ b14;kðwðtÀ1ÞÞ ⋮ ⋯ ⋮ b31;kðwðtÀ1ÞÞ ⋯ b34;kðwðtÀ1ÞÞ 2 6 4 3 7 5 aij;kðwðtÀ1ÞÞ ¼ cij k;0 þ ∑ h m ¼ 1 cij k;m expfÀJwðtÀ1ÞÀZY;m J2 λY;m g bij;kðwðtÀ1ÞÞ ¼ d ij k;0 þ ∑ h m ¼ 1 d ij k;m expfÀJwðtÀ1ÞÀZU;m J2 λU;m g Zj;m ¼ ½zj;m;1 … zj;m;dimðwðtÀ1ÞÞ; j ¼ Y; U 8 : ð24Þ where UðtÞ ¼ ½Vf ðtÞ VrðtÞ VlðtÞ VbðtÞT are the inputs, which denote the voltage of 4 propellers; YðtÞ ¼ ½pðtÞ rðtÞ yðtÞT are the outputs, which are the pitch angle, the roll angle and the yaw angle respectively; ny, nu, nk and h are the orders; zj;m's are the centers of Gaussian RBF networks; c1 m, c2 m, c3 m, cij k;m 's, and d ij k;m's are the weighting coefﬁcient matrices of suitable dimensions; JxJ2 ^λ 9 xT ^λx, ^λ ¼ diagð^λ1 ^λ2 … ^λdimðxÞÞ, and f^λ1 ^λ2 ⋯ ^λdimðxÞg are the scal- ing factors; ΞðtÞAR3 denotes noise usually regarded as Gaussian white noise. The signal wðtÀ1Þ in Eq. (24) is the index of the model, which is the variable causing nonlinearity. wðtÀ1Þ could be a system variable causing the operation-point of the system to change with time. wðtÀ1Þ has direct or indirect relation with inputs or outputs of the system, in some cases probably being just Fig. 12. The control result based on the ARX model. Fig. 13. Inputs of the control result based on the ARX model.Fig. 11. Inputs of the control result based on the physical model. J. Wu et al. / ISA Transactions 53 (2014) 173–185180
- 9. the input or/and output itself. For this quadrotor helicopter, we choose pitch angle and roll angle as the model index, because the nonlinear characteristics may change with the ﬂying postures, which are mainly related with the pitch and roll. It is easy to see that the local linearization of the model (24) is a linear MIMO ARX model by ﬁxing wðtÀ1Þ at time tÀ1. It is natural and appealing to interpret model (24) as a locally linear MIMO ARX model in which the evolution of the process at time t is governed by a set of AR coefﬁcient matrices fAk; Bkg at a local mean ϕ0, all of which depend on the ‘working-point’ of the process at time tÀ1. Thus the structure of MIMO RBF-ARX resembles the ARX in application. Assuming that, at a steady state, Us, Ys, and ws are the values of the related variables, then from the MIMO RBF-ARX model (24) yields Ys ¼ CðwsÞþ ∑ ny k ¼ 1 AkðwsÞYs þ ∑ nu þ nkÀ1 k ¼ nk BkðwsÞUs ð25Þ from which we can easily get the Us with a desired Ys. And an increment equation around the steady state may be derived from Eq. (24) and (25) as follows: ΔYðtÞÀΔCðwðtÀ1ÞÞ ¼ ∑ ny k ¼ 1 AkðwsÞΔYðtÀkÞ þ ∑ nu þ nkÀ1 k ¼ nk BkðwsÞΔUðtÀkÞþ ^ΞðtÞ ð26Þ where ΔYðtÞ ¼ YðtÞÀYs ΔUðtÞ ¼ UðtÞÀUs ΔCðwðtÀ1ÞÞ ¼ CðwðtÀ1ÞÞÀCðwsÞ: 8 : The increment MIMO RBF-ARX model (26) can be transformed into a state-space equation model by deﬁning state vector as follows: XðtÞ ¼ x1 1;t x1 2;t ⋯ x1 n;t ∑ t i ¼ 0 x1 1;i; x2 1;t x2 2;t ⋯ x2 n;t ∑ t i ¼ 0 x2 1;i; x3 1;t x3 2;t ⋯ x3 n;t ∑ t i ¼ 0 x3 1;i #T x1 1;t ¼ pðtÞÀpsÀϕ1 0ðwðtÞÞþϕ1 0ðwsÞ x2 1;t ¼ rðtÞÀrsÀϕ2 0ðwðtÞÞþϕ2 0ðwsÞ x3 1;t ¼ yðtÞÀysÀϕ3 0ðwðtÞÞþϕ3 0ðwsÞ xl k;t ¼ ∑ nþ 1Àk i ¼ 1 ∑ 3 j ¼ 1 a _ lj;k þiÀ1 xj 1;tÀ1⋯ þ ∑ nþ 1Àk i ¼ 1 ∑ 4 j ¼ 1 b _ lj;k þ iÀ1uj;tÀ1 n ¼ maxðny; nu þnkÀ1Þ a _ ij;k ¼ Àaij;k; krny 0; k4ny ( b _ ij;k ¼ bij;k; drkrnu þdÀ1 0 else k ¼ 2; 3; …; n; l ¼ 1; 2; 3 8 : where aij;k and bij;k are the elements in Ak and Bk of Eq. (24). Notice that ΔCðwðtÀ1ÞÞ in Eq. (26) is included in the state variables, so state-space equation corresponding to model (26) may then be given by Xðtþ1Þ ¼ AXðtÞþBΔUðtÞ ΔYðtÞ ¼ CXðtÞþ ^ΞðtÞ ( ð27Þ where A ¼ α11 ~α12 ~α13 ~α21 α22 ~α23 ~α31 ~α32 α33 2 6 4 3 7 5 B ¼ β11 β12 β13 β14 β21 β22 β23 β24 β31 β32 β33 β34 2 6 4 3 7 5 C ¼ χ 0 0 0 χ 0 0 0 χ 2 6 4 3 7 5 8 : αii ¼ Àa _ ii;1 1 0 ⋯ 0 0 Àa _ ii;2 0 1 ⋮ ⋮ ⋮ ⋮ ⋱ 1 ⋮ Àa _ ii;n 0 0 ⋯ 0 0 1 0 ⋯ ⋯ 0 1 2 6 6 6 4 3 7 7 7 5 ðn þ1ÞÂðnþ 1Þ ~αij ¼ Àa _ ij;1 0 0 ⋯ 0 0 Àa _ ij;2 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ 0 ⋮ Àa _ ij;n 0 0 ⋯ 0 ⋮ 0 ⋯ ⋯ ⋯ ⋯ 0 2 6 6 6 4 3 7 7 7 5 ðnþ1ÞÂðnþ 1Þ βij ¼ ½b _ ij;1 b _ ij;2 ⋯ b _ ij;n 0T ðnþ 1ÞÂ1 χ ¼ ½1 0 ⋯ 01Âðn þ1Þ We also adopt the AIC to select the order of the MIMO RBF-ARX models, whose expression on the quadrotor helicopter is deﬁned by AIC ¼ N log jΣjþ2ðð1þhÞð3þ9ny þ12nuÞþ2m dimðwðtÀ1ÞÞþnyÞ ð28Þ where N is the data length; jΣj is the determinant of variance– covariance matrix of modeling residuals. In order to get a global data for identiﬁcation, a set of sine signals is set to make the quadrotor helicopter swing in the full range. And so as to meet the persistent excitation condition, a set of Gauss White Noises with small power is added. By sampling a length of inputs and outputs, the data for identiﬁcation is obtained, which is shown in Fig. 14 where the front 800 data points are used to identify the RBF-ARX model, and the back 600 data points are used to test the model. Under different model orders the identiﬁed model's AICs are calculated. By taking the trade-off between the smallest AIC and the real-time control performance, the system orders are selected as ny ¼3, nu ¼1, nk ¼2, h¼1, and n ¼ max ðny; nu þnkÀ1Þ ¼ 3. The corresponding AIC value is À10 904. The modeling results of the MIMO RBF-ARX model are shown in Figs. 15 and 16. From the ﬁgures, one can see that the MIMO RBF-ARX model has an excellent modeling accuracy. We use the 16 groups of actual data sampled in 16 different working regions to test the MIMO RBF-ARX model too, just the same work as we did in Tables 2 and 3. Table 4 shows the standard deviations of the one-step-ahead predictive errors of the MIMO RBF-ARX model in the 16 working regions. By comparing Tables 2 and 4, we can see that the MIMO RBF- ARX model and the ARX model-set have close modeling accuracy. Both of them show much better modeling accuracy than the physical model does in all 16 local working regions. J. Wu et al. / ISA Transactions 53 (2014) 173–185 181
- 10. 4.2. Global LQR controller Based on the ofﬂine identiﬁed MIMO RBF-ARX model, a global LQR controller is designed, which can self-adjust the LQR gain according to the ﬂying posture of the quadrotor helicopter. The system structure is given in Fig. 17. At any working point, by using the same method introduced in Section 3 we can design an inﬁnite-time quadratic regulator based on the locally linear time-invariant state-space model (27). The objective function of LQR in discrete-time form is given by min ΔUðkÞn J ¼ 1 2 ∑ 1 k ¼ 0 ½XT ðkÞQXðkÞþΔUT ðkÞRΔUðkÞ ð29Þ By solving the discrete Riccati equation (22) at a working-point, the state-feedback matrix K at the working-point can be obtained as follows: K ¼ RÀ1 BT AÀT ðPÀQÞ ð30Þ and the state-feedback optimal control law at this working-point is ΔUðkÞn ¼ ÀKXðkÞ UðkÞ ¼ Us þΔUðkÞn : ( ð31Þ When the system goes to a steady state, i.e. XðtÞ-0, and the states x1 1;t, x2 1;t, and x3 1;t also go to zero, this means the achievement of the tracking goal. The RBF-ARX model-based global LQR controller is a special case of the RBF-ARX model-based predictive controller (RBF-ARX- MPC). Peng et al. had discussed its stability in [25], and this paper would focus on its practical application. After all the tasks mentioned above have been accomplished, the real-time control can be carried out. The control sample period is 0.1 s in which dynamics of quadrotor varies slightly and enough time for control law calculation is guaranteed. The results of real- time control based on the identiﬁed MIMO RBF-ARX model are shown in Figs. 18–21. From Fig. 18 we can see that the quadrotor helicopter can be stabilized at any given point very quickly and smoothly by using the MIMO RBF-ARX model-based global LQR control strategy when the 3 outputs were changing one by one. Compared with the ARX model-set-based gain-scheduling LQR control strategy, it used only one model but obtained better control results. From Fig. 20 one can see that when the pitch and roll changed simultaneously, from the Table 4 Standard deviations of modeling residuals of the RBF-ARX model. Region num. Pitch (deg) Roll (deg) Yaw (deg) 1 0.12069 0.14334 0.07597 2 0.12040 0.15015 0.06463 3 0.10845 0.14612 0.07327 4 0.11264 0.16021 0.07580 5 0.11078 0.16360 0.06646 6 0.09537 0.14166 0.06749 7 0.09811 0.13789 0.06082 8 0.09006 0.15414 0.06874 9 0.12415 0.14965 0.06072 10 0.09894 0.13478 0.07016 11 0.08680 0.14197 0.06472 12 0.09036 0.16699 0.06667 13 0.10610 0.15922 0.07072 14 0.09913 0.16542 0.07845 15 0.09094 0.16232 0.07627 16 0.08848 0.17551 0.06616 Fig. 17. Structure of the control system based on RBF-ARX model. Fig. 15. Residuals of MIMO RBF-ARX model for test data. Fig. 16. Histograms of the residuals of MIMO RBF-ARX model for test data. Fig. 14. Identiﬁcation data. J. Wu et al. / ISA Transactions 53 (2014) 173–185182
- 11. 20th second to the 25th second, the transition process is very smooth and fast, which is much better than the control results of the ARX model-set-based gain scheduling LQR controller (see Fig. 12). The detailed comparisons of the control results shown in Figs. 10, 12 and 20 are given in Tables 5–7. From Tables 5 and 6, one can see that the ARX model-set-based method and the MIMO RBF-ARX model-based method show a much better control performance compared with the physical model-based one. And the MIMO RBF-ARX model-based method is close to the ARX model-set-based one in general. From Table 7, one can also see that the dynamic process of the MIMO RBF-ARX model-based method is smoother and faster than that of the ARX model-set-based method. We did not compare the control results with those presented in other literatures, because the referenced quadrotor helicopter in this paper has not only a unique conﬁguration but also larger inertia, and it is quite different from those classic quadrotors. Anti-disturbance tests are very important for this kind of vehicles, because it is easily affected by the wind disturbance or encounter sudden collisions in obstacle-dense environments. For testing anti-disturbance performance, the pulse-type disturbance Fig. 19. Inputs of real-time control based on identiﬁed RBF-ARX model. Fig. 20. Control result based on RBF-ARX model. Fig. 21. Inputs of control result based on RBF-ARX model. Table 5 Standard deviation of steady-state errors. Models Pitch (deg) Roll (deg) Yaw (deg) Physical 0.3818 1.0347 0.2397 ARX model-set 0.1957 0.2295 0.0372 MIMO RBF-ARX 0.1465 0.1824 0.1825 Table 6 Overshoot of dynamic transition processes. Models Pitch (deg) Roll (deg) Yaw (deg) Physical 2.02 3.39 1.07 ARX model-set 0.85 2.04 1.29 MIMO RBF-ARX 1.12 0.51 1.43 Table 7 Transient time of dynamic transition processes (75% error band). Models Pitch (s) Roll (s) Yaw (s) Physical 4.3 6.7 5.7 ARX model-set 4.3 4.1 4.5 MIMO RBF-ARX 2.3 1.9 2.7 Fig. 18. Outputs of real-time control based on identiﬁed MIMO RBF-ARX model. Values in square bracket denote the desired values of pitch, roll, and yaw angles; the real line represents the pitch angle, the dashed line represents the roll angle, and the dotted line represents the yaw angle. J. Wu et al. / ISA Transactions 53 (2014) 173–185 183
- 12. with 720 V voltage and 1 s duration time are added on each motor's control input. Fig. 22 shows the disturbance signals (dotted lines) together with the input signals of the physical model-based control approach (dash-dotted lines) and the MIMO RBF-ARX model-based control approach (solid lines). Fig. 23 compares the anti-disturbance results of the physical model-based control approach (dash-dotted lines) and the MIMO RBF-ARX model-based control approach (solid lines). It is clear that the MIMO RBF-ARX model-based LQR controller can stabilize the quadrotor helicopter much faster than the physical model-based one against the sudden disturbance. It is attributed to the remarkable capability of the MIMO RBF-ARX model in the descrip- tion of nonlinear dynamics of the quadrotor helicopter, not only at some working-points but also in the full range of the quadrotor's working area. The MIMO RBF-ARX model has the time-varying AR coefﬁcient matrices and can be locally linearized in any sampling period easily. In other words, one MIMO RBF-ARX model can be regarded as a composition of inﬁnite amount of ARX models. Therefore, the method of using one MIMO RBF-ARX model to represent the global nonlinear dynamics of the quadrotor heli- copter is far better than that of using only 16 ARX models or using one linear physical model. 5. Conclusions The referenced quadrotor helicopter in this paper has a unique conﬁguration. It has 4 propellers, 3 of which are horizontally mounted to control its pitch and roll rotation while the last one is vertically mounted to control its yaw rotation. It is also an unknown nonlinear dynamical system whose physical model is not accurate. By comparing the modeling accuracy of three modeling methods in 16 working regions, it is demonstrated that the ARX model-set and the MIMO RBF-ARX model are much better than the physical model. The MIMO RBF-ARX model has a close modeling accuracy with the ARX model-set, besides, it avoids the tough work of the ARX model-set identiﬁcations. By comparing the real-time control results of the three model-based LQR controller, it is concluded that the MIMO RBF-ARX model-based LQR control strategy presented in this paper is better. The anti-disturbance tests also demonstrated the superiority of the MIMO RBF-ARX model-based method. The validity of the MIMO RBF-ARX modeling method for such kind of plants was conﬁrmed by this successful application. Acknowledgments The authors would like to thank the editors and reviewers for his valuable comments. References [1] Bouabdallah S, Noth A, Siegwart R. PID vs LQ control techniques applied to an indoor micro quadrotor. In: 2004 IEEE/RSJ international conference on intelligent robots and systems, 2004. (IROS 2004). Proceedings, vol. 3. IEEE; 2004. p. 2451–6. ISBN 0780384636. [2] Tayebi A, McGilvray S. 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