Modeling and control approach to a distinctive quadrotor helicopter

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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 School of Electrical & Information Engineering, Changsha University of Science & Technology, Changsha, Hunan 410004, China 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 b c

art ic l e i nf o

a b s t r a c t

Article history: Received 19 February 2012 Received in revised form 15 August 2013 Accepted 15 August 2013 This paper was recommended for publication by Prof. A.B. Rad.

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.

Keywords: ARX model Nonlinear system Physical model Quadrotor helicopter RBF-ARX model

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 microunmanned 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

☆ 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: [email protected] (J. Wu), [email protected] (H. Peng), [email protected] (Q. Chen), [email protected] (X. Peng).

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 controllers 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 (Omnidirectional 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 constructed 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 performance. Guenard et al. [5] presented a visual servo control using backstepping techniques for stabilization of a quadrotor. Alexis et al.

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

Please cite this article as: Wu J, et al. Modeling and control approach to a distinctive quadrotor helicopter. ISA Transactions (2013), http: //dx.doi.org/10.1016/j.isatra.2013.08.010i

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[6] presented a switching model predictive attitude controller for an unmanned quadrotor helicopter subject to atmospheric disturbances. 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 network (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 applications 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

Fig. 1. The testbed for modeling research, 3DOF are locked.

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 models 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 understanding 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 AutoRegressive eXogenous input (NARX) model, neural network with internal memory known as Memory Neuron Networks (MNN), and Recurrent 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 flying

Please cite this article as: Wu J, et al. Modeling and control approach to a distinctive quadrotor helicopter. ISA Transactions (2013), http: //dx.doi.org/10.1016/j.isatra.2013.08.010i

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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-setbased 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 statefeedback 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 demonstrated 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 RBFARX 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 comparison 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 F x 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 anticlockwise rotation round Y-axis is defined to be positive. The roll is defined to be the angle circled around the X-axis and the anticlockwise rotation round X-axis is defined to be positive. The yaw is defined to be the angle circled around the Z-axis and the anticlockwise rotation round Z-axis is defined to be positive as well.

3

Fig. 2. Coordinate of the quadrotor helicopter.

Table 1 Symbols and definitions. Symbol

Definition

J p;r;y p r y Fx Kfc Vx Lc;f ;a;b

Body inertia Pitch angle Roll angle Yaw angle Thrust force Thrust factor Motor input Lever

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]: 8 J p″ ¼ ðF l þF r ÞLc F f Lf > < p J r r″ ¼ F r La F l La ð1Þ > : J y″ ¼ F L : b b y And other known conditions of the quadrotor helicopter are as follows pffiffi Lc ¼ 12 Lf ¼ 33La ð2Þ Lb ¼ Lf F x ¼ f ðV x Þ ¼ K fc V x

ð3Þ 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 8   K fc 1 > > > > p″ ¼ 2ðV l þ V r ÞV f Lf J > > p > > pffiffiffi > < 3K fc Lf r″ ¼ ðV r V l Þ ð5Þ > 2J r > > > > K fc Lb > > > > : y″ ¼ J y V b : In addition, if the quadrotor helicopter is stabilized at a steady state where the output is Y s ¼ ½ps r s ys T and the input is Us ¼ ½V fs V rs V ls V bs T , then according to Eq. (5) we can also obtain the

Please cite this article as: Wu J, et al. Modeling and control approach to a distinctive quadrotor helicopter. ISA Transactions (2013), http: //dx.doi.org/10.1016/j.isatra.2013.08.010i

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equations below: 8   K fc 1 > > > ðV Þ″ ¼ V þ V V ÞV þV lf ðpp r rs l ls f fs s > > 2 JP > > > pffiffiffi > < 3K fc lf ðrr s Þ″ ¼ ðV r V rs V l þ V ls Þ 2J r > > > > > K fc lb > > > ðV b V bs Þ: ðyys Þ″ ¼ > : Jy

where the constant matrix P Z0 is the solution of the following algebraic Riccati equation: PA þ AT PPBR1 BT P þ Q ¼ 0: ð6Þ

UðtÞ ¼ Us þ ΔUðtÞn :

A state-space equation model can be built by defining the state vector as  Z xðtÞ ¼ pps ðpps Þ′ ðpps Þ rr s ðrr s Þ′ T

Z

Z 

ðrr s Þ yys ðyys Þ′

ðyys Þ

And the inputs and outputs are 8 ΔUðtÞ ¼ UUs > < ¼ ½V f V fs V r V rs V l V ls V b V bs T : > : ΔYðtÞ ¼ YYs ¼ ½pps rrs yys T

:

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

1

0

0

0

0

0

0 0

1

K fc lf 2J P

0 0 pffiffi

3K fc lf 2J r

0 0 0

0 3 0 0 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 7 0 7 7 K fc lb 7 Jy 5 0

0

3.1. ARX model-set of the quadrotor ð8Þ

07 7 7 07 7 07 7 7 07 7 7 07 7 07 7 7 05 0

0 0

0 0 0

0 0

0 0

3 0 7 05

0

0

1

0

0

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 Z 1 1 T min J ¼ ½x ðtÞQxðtÞ þ ΔUT ðtÞR ΔUðtÞ dt ð11Þ 2 0 ΔUðtÞ where the Q 40 and R 4 0 are the diagonal weighing matrix for xðtÞ and ΔUðtÞ respectively. It was verified that ðA; BÞ is stabilizable and ðA; Q 1=2 Þ is detectable, thus ΔUðtÞn exists and be unique

ΔUðtÞn ¼ R 1 BT PxðtÞ;

When the system goes to a steady state, then x-0 so that ΔY-0, and the tracking goal is achieved. Notice that Y s 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

:

ð14Þ

ð7Þ

Therefore, the state-space model of this linear time-invariant (LTI) system is ( _ ¼ AxðtÞ þ BΔUðtÞ xðtÞ ð9Þ ΔYðtÞ ¼ CxðtÞ where 8 2 0 1 0 0 > > > > 60 0 0 0 > > 6 > > 6 > > 61 0 0 0 > > 6 > > 60 0 0 0 > > 6 > > 6 > > > A¼6 > 60 0 0 0 > > 6 > > 60 0 0 1 > > 6 > > 60 0 0 0 > > 6 > > 6 > > 40 0 0 0 > > > > > 0 0 0 0 > > > 2 > > 0 0 > > > < K fc lf 6 K fc lf 6 JP 2J P 6 > 6 0 > 0 > 6 > > 6 > > 0 6 0 > > pffiffi 6 > > 3K fc lf > >B¼6 0 6 > 2J r > 6 > > 6 0 > 0 > 6 > > 6 > > 6 > 0 0 > > 6 > > 6 0 > 0 > 4 > > > > > 0 0 > > 2 > > > 1 0 0 0 > > > 6 > > C¼40 0 0 1 > > > > : 0 0 0 0

The state feedback control law is ( ΔUðtÞn ¼ KxðtÞ ¼ R 1 BT PxðtÞ

ð13Þ

ð12Þ

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 applications 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 8     > < ξ ¼ fix p2  4 þ fix r5 10 7:5 ð15Þ > : p A ð2; 42Þ; r A ð5; 35Þ 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: YðtÞ þ A1 Yðt1Þ þ ⋯ þ Any Yðtny Þ ¼ Y 0 þ B1 Uðtnk Þ þ ⋯ þ Bnu Uðtnk nu þ1Þ þ ΞðtÞ

ð16Þ

Please cite this article as: Wu J, et al. Modeling and control approach to a distinctive quadrotor helicopter. ISA Transactions (2013), http: //dx.doi.org/10.1016/j.isatra.2013.08.010i

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where 8 2 a11;k > > > > 6 > a > A ¼ 21;k > > k 4 > > a31;k < 2 b11;k > > > > 6b > > B ¼ 21;k 4 > k > > > : b31;k

2 a12;k

a13;k

6 aii;2

3

a22;k

a23;k 7 5

a32;k

a33;k

_

aii;1

αii ¼ 6 6 4

b14;k

3

b12;k

b13;k

b22;k

b23;k

b24;k 7 5

b32;k

b33;k

b34;k

ð17Þ

2

and YðtÞ ¼ ½pðtÞ rðtÞ yðtÞ are the outputs which include the pitch angle, the roll angle and the yaw angle; UðtÞ ¼ ½V f ðtÞ V r ðtÞ V l ðtÞ V b ðtÞT are the inputs that denote the voltage of 4 propellers; nu, ny and nk 4 0 are the system orders; Ak ðk ¼ 1; 2; …; ny Þ and Bk ðk ¼ 1; 2; …; nu Þ are the coefficient matrixes; ΞðtÞ is the modeling residual. Assuming that at a steady state, the input and output are Us and Y s , respectively. From model (16) one can has the following ARX model:

ΔYðtÞ þ A1 ΔYðt1Þ þ ⋯ þ Any ΔYðtny Þ ð18Þ

where ΔUðtÞ ¼ UUs , ΔYðtÞ ¼ YY s . ARX model (18) can be transformed into a state-space model by defining the state vector as follows: 8 " t > > 1 1 1 1 > > XðtÞ ¼ x1;t x2;t x3;t ∑ x1;i ; > > i¼0 > > > > > t > > > x21;t x22;t x23;t ∑ x21;i ; > > > > i¼0 > > > #T > > t > > > > x31;t x32;t x33;t ∑ x31;i > > > i¼0 > > > > > 1 2 > x ¼ pðtÞp ; x ¼ rðtÞr s ; x31;t ¼ yðtÞys > 1;t s 1;t > > > > n þ 1k 3 < l _ xk;t ¼ ∑ ∑ alj;k þ i1 xj1;t1 ⋯ ð19Þ i¼1 j¼1 > > > > > n þ 1k 4 _ > > > þ ∑ ∑ blj;k þ i1 uj;t1 > > > i¼1 j¼1 > > > > > n ¼ maxðn > y ; nu þ nk 1Þ > > ( > > a > _ ij;k ; k rny > > aij;k ¼ > > 0; k 4ny > > > >  > > _ > bij;k ; nk r k r nu þ nk 1 > > b ¼ > > > ij;k 0 else > > > : k ¼ 2; 3; …; n; l ¼ 1; 2; 3

_

1

0

0

1

0 1 0

⋮7 5 0

0

1

⋮ 0

0

⋱ ⋯

1

0

⋯

⋯

_

0

0

0 ⋮

0

0

⋮7 7

⋮

⋮ aii;n

aij;1

3

⋯

_

6 a_ij;2 α~ ij ¼ 6 6 ⋮ 4 _

T

^ ðtÞ ¼ B1 ΔUðtnk Þ þ ⋯ þ Bnu ΔUðtnk nu þ 1Þ þ Ξ

_

5

⋯

0

⋱

⋮ 0

ðn þ 1Þðn þ 1Þ

0

⋮7 7 ⋮7

aij;n

0

0

⋯

0

⋮

0

⋯

⋯

⋯

⋯

0

_

3 5 ðn þ 1Þðn þ 1Þ

_

βij ¼ ½bij;1 bij;2 ⋯ bij;n 0Tðn þ 1Þ1 χ ¼ ½1 0 ⋯ 01ðn þ 1Þ 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 ¼ 1, nk ¼2 and n ¼ maxðny ; nu þ nk 1Þ 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.

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 ( Xðt þ 1Þ ¼ AXðtÞ þ BΔUðtÞ ð20Þ ΔYðtÞ ¼ CXðtÞ þ Ξ^ ðtÞ where 2 8 α11 > > > >A ¼6 ~ 21 > α 4 > > > > > α~ 31 > > > 2 > > > β11 > > < 6 B ¼ 4 β21 > > β31 > > > 2 > > > χ 0 > > > >C¼60 χ > 4 > > > > > 0 0 :

α~ 12 α22 α~ 32 β12 β22 β32 0

3

3

α~ 13 α~ 23 7 5 α33 3 β13 β14 β23 β24 7 5 β33 β34

07 5

χ

Fig. 3. Comparison of actual outputs and the ARX model outputs.

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Table 3 Standard deviations of modeling residuals of the physical model. Region num.

Pitch (deg)

Roll (deg)

Yaw (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.27540 0.24629 0.24464 0.25461 0.21382 0.20383 0.19917 0.18682 0.17178 0.15259 0.16282 0.17483 0.19508 0.18419 0.19212 0.19985

0.51759 0.30584 0.28601 0.43836 0.54285 0.36895 0.30248 0.50068 0.52032 0.41090 0.35907 0.49034 0.54313 0.54569 0.51051 0.43155

0.15524 0.12272 0.12955 0.14071 0.11951 0.12767 0.11814 0.12215 0.11861 0.13090 0.12595 0.12552 0.14181 0.15156 0.15662 0.14126

Fig. 4. Outputs error of the ARX model.

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 statespace model (20) in each divided working region, the objective function in discrete form is designed as minn J ¼

ΔUðkÞ

1 1 ∑ ½XT ðkÞQXðkÞ þ ΔUT ðkÞR ΔUðkÞ: 2k¼0

ð21Þ

By solving the discrete Riccati equation P ¼ Q þ AT PðI þ BR 1 BT PÞ1 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 8 n > < ΔUðkÞ ¼ KXðkÞ ¼ R1 BT AT ðPQ ÞXðkÞ ð23Þ > : UðkÞ ¼ U þ ΔUðkÞn :

Fig. 5. Histograms of the residuals of the ARX model.

s

Table 2 Standard deviations of modeling residuals of the 16 ARX models. Region num.

Pitch (deg)

Roll (deg)

Yaw (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.08409 0.08137 0.07995 0.07387 0.08155 0.07119 0.06803 0.07525 0.07910 0.09002 0.07823 0.08965 0.09584 0.09611 0.10190 0.10145

0.15942 0.14798 0.14044 0.13977 0.15697 0.11337 0.12433 0.13354 0.13902 0.09742 0.11515 0.14741 0.12968 0.09736 0.10607 0.13660

0.05988 0.06425 0.06481 0.05696 0.05336 0.05543 0.06166 0.04781 0.05717 0.05270 0.05529 0.05134 0.06395 0.05857 0.05354 0.06416

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

When the system goes to a steady state, XðtÞ-0, and the states x11;t , x21;t and x31;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 4 0 and R 4 0 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 respectively. 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

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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 realtime 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 changing 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

7

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. 6. Structure of the control system.

Fig. 10. The control result based on the physical model.

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.

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,

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Fig. 11. Inputs of the control result based on the physical model.

Fig. 13. Inputs of the control result based on the ARX model.

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

Fig. 12. The control result based on the ARX model.

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 performances, 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 approximate the quadrotor helicopter's global nonlinear dynamics with several LTI models identified in different operating points. However, 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

The quadrotor helicopter is a multiple-outputs multiple-inputs system, and its MIMO RBF-ARX structure is given as follows: 8 ny > > > YðtÞ ¼ Cðwðt1ÞÞ þ ∑ Ak ðwðt1ÞÞYðtkÞ > > > k¼1 > > > > nu þ nk 1 > > > þ ∑ Bk ðwðt1ÞÞUðtkÞ þ ΞðtÞ > > > > k ¼ nk > > > > 1 2 3 > > Cðwðt1ÞÞ ¼ ½ϕ0 ðwðt1ÞÞ ϕ0 ðwðt1ÞÞ ϕ0 ððwðt1ÞÞÞT > > > > h > > i > ϕ ðwðt1ÞÞ ¼ ci0 þ ∑ cim expfJ wðt1ÞZY;m J 2λY;m g > > > 0 > m ¼1 > > 2 3 > > ðwðt1ÞÞ ⋯ a13;k ðwðt1ÞÞ a > 11;k > > > 6 7 < A ðwðt1ÞÞ ¼ 4 ⋮ ⋱ ⋮ 5 k ð24Þ ðwðt1ÞÞ ⋯ a ðwðt1ÞÞ a > 31;k 33;k > > > 2 3 > > b11;k ðwðt1ÞÞ ⋯ b14;k ðwðt1ÞÞ > > > > 7 > ⋮ ⋯ ⋮ > Bk ðwðt1ÞÞ ¼ 6 4 5 > > > > ðwðt1ÞÞ ⋯ b ðwðt1ÞÞ b > 31;k 34;k > > > > > h > > ij ij > > aij;k ðwðt1ÞÞ ¼ ck;0 þ ∑ ck;m expf J wðt1ÞZY;m J 2λY;m g > > > m¼1 > > > > h > > > bij;k ðwðt1ÞÞ ¼ dijk;0 þ ∑ dijk;m expfJ wðt1ÞZU;m J 2λ g > U;m > > m¼1 > > > : Zj;m ¼ ½zj;m;1 … zj;m;dimðwðt1ÞÞ ; j ¼ Y; U where UðtÞ ¼ ½V f ðtÞ V r ðtÞ V l ðtÞ V b ðtÞT 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 ij

of Gaussian RBF networks; c1m , c2m , c3m , cijk;m 's, and dk;m 's are the weighting coefficient matrices of suitable dimensions; J x J 2λ^ 9 Þ, and fλ^ λ^ ⋯ λ^ g are the scalxT λ^ x, λ^ ¼ diagðλ^ λ^ … λ^ 1

2

dimðxÞ

1

2

dimðxÞ

ing factors; ΞðtÞ A R3 denotes noise usually regarded as Gaussian white noise. The signal wðt1Þ in Eq. (24) is the index of the model, which is the variable causing nonlinearity. wðt1Þ could be a system variable causing the operation-point of the system to change with time. wðt1Þ has direct or indirect relation with inputs or outputs of the system, in some cases probably being just

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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 wðt1Þ at time t1. 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 ; Bk g at a local mean ϕ0 , all of which depend on the ‘working-point’ of the process at time t1. Thus the structure of MIMO RBF-ARX resembles the ARX in application. Assuming that, at a steady state, Us , Y s , and ws are the values of the related variables, then from the MIMO RBF-ARX model (24) yields ny

nu þ nk 1

k¼1

k ¼ nk

Y s ¼ Cðws Þ þ ∑ Ak ðws ÞY s þ

∑

Bk ðws ÞUs

ð25Þ

may then be given by ( Xðt þ 1Þ ¼ AXðtÞ þBΔUðtÞ ΔYðtÞ ¼ CXðtÞ þ Ξ^ ðtÞ

4 2

k¼1

þ

nu þ nk 1

∑

k ¼ nk

^ ðtÞ Bk ðws ÞΔUðtkÞ þ Ξ

ð26Þ

where 8 > < ΔYðtÞ ¼ YðtÞY s

ΔUðtÞ ¼ UðtÞUs > : ΔCðwðt1ÞÞ ¼ Cðwðt1ÞÞCðw Þ: s

The increment MIMO RBF-ARX model (26) can be transformed into a state-space equation model by defining state vector as follows: 8 " t > > > XðtÞ ¼ x1 x12;t ⋯ x1n;t ∑ x11;i ; > 1;t > > i¼0 > > > > > t > > > > x21;t x22;t ⋯ x2n;t ∑ x21;i ; > > > i ¼ 0 > > > > #T > > t > 3 > 3 3 3 > x ⋯ x ∑ x x > 1;i n;t 1;t 2;t > > i¼0 > > > > > 1 1 1 > > x ¼ pðtÞp  ϕ ðwðtÞÞ þ ϕ ðw sÞ > 1;t s 0 0 > > > 2 2 > 2 > x1;t ¼ rðtÞr s ϕ0 ðwðtÞÞ þ ϕ0 ðws Þ > > > > > 3 3 3 < x1;t ¼ yðtÞys ϕ0 ðwðtÞÞ þ ϕ0 ðws Þ > n þ 1k 3 _ > > l > ∑ alj;k þ i1 xj1;t1 ⋯ > > xk;t ¼ i ∑ > ¼1 j¼1 > > > > n þ 1k 4 _ > > > > þ ∑ ∑ blj;k þ i1 uj;t1 > > > i¼1 j¼1 > > > > > n ¼ maxðny ; nu þ nk 1Þ > > ( > > > aij;k ; k rny > >_ > aij;k ¼ > > 0; k 4ny > > > >  > > _ ; d r k r nu þ d1 b > ij;k > > bij;k ¼ > > > 0 else > > > : k ¼ 2; 3; …; n; l ¼ 1; 2; 3 where aij;k and bij;k are the elements in Ak and Bk of Eq. (24). Notice that ΔCðwðt1ÞÞ in Eq. (26) is included in the state variables, so state-space equation corresponding to model (26)

_

aii;1

6 aii;2

αii ¼ 6 6

ny

ΔYðtÞΔCðwðt1ÞÞ ¼ ∑ Ak ðws ÞΔYðtkÞ

ð27Þ

where 2 3 8 α11 α~ 12 α~ 13 > > > 6 7 > > A ¼ 4 α~ 21 α22 α~ 23 5 > > > > ~ ~ > α 31 α 32 α33 > > > 2 3 > > > β β12 β13 β14 > 11 > < 6β 7 B ¼ 4 21 β22 β23 β24 5 > > β31 β32 β33 β34 > > > 2 3 > > > χ 0 0 > > > > 6 7 > C¼40 χ 05 > > > > > χ 0 0 : 2

from which we can easily get the Us with a desired Y s . And an increment equation around the steady state may be derived from Eq. (24) and (25) as follows:

9

_

0

0

1

0 1 0

⋮7 5 0

0

1

⋮ 0

0

⋱ ⋯

1

0

⋯

⋯

0

0

0

0

_

⋮7 7

⋯

0 0 0 ⋯

⋮

aij;n

⋮ 0

0

⋱ ⋯

0

⋯

⋯

⋯

_

0

⋮

⋮ aii;n

aij;1

3

⋯

_

6 a_ij;2 α~ ij ¼ 6 6 ⋮ 4 _

_

1

ðn þ 1Þðn þ 1Þ

0

3

⋮7 7 ⋮7 5 ⋮

0

ðn þ 1Þðn þ 1Þ

_

βij ¼ ½bij;1 bij;2 ⋯ bij;n 0Tðn þ 1Þ1 χ ¼ ½1 0 ⋯ 01ðn þ 1Þ 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 jΣ j þ 2ðð1 þ hÞð3 þ 9ny þ 12nu Þ þ 2m dimðwðt1ÞÞþ ny Þ ð28Þ where N is the data length; jΣ j 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, h ¼1, and n ¼ max ðny ; nu þ nk 1Þ ¼ 3. The corresponding AIC value is 10 904. 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 RBFARX 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.

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10

Table 4 Standard deviations of modeling residuals of the RBF-ARX model. Region num.

Pitch (deg)

Roll (deg)

Yaw (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.12069 0.12040 0.10845 0.11264 0.11078 0.09537 0.09811 0.09006 0.12415 0.09894 0.08680 0.09036 0.10610 0.09913 0.09094 0.08848

0.14334 0.15015 0.14612 0.16021 0.16360 0.14166 0.13789 0.15414 0.14965 0.13478 0.14197 0.16699 0.15922 0.16542 0.16232 0.17551

0.07597 0.06463 0.07327 0.07580 0.06646 0.06749 0.06082 0.06874 0.06072 0.07016 0.06472 0.06667 0.07072 0.07845 0.07627 0.06616

Fig. 14. Identification data.

Fig. 17. Structure of the control system based on RBF-ARX model.

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 J ¼

ΔUðkÞn

Fig. 15. Residuals of MIMO RBF-ARX model for test data.

1 1 ∑ ½XT ðkÞQXðkÞ þ ΔUT ðkÞR ΔUðkÞ 2k¼0

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 R 1 BT AT ðPQ Þ

ð30Þ

and the state-feedback optimal control law at this working-point is ( ΔUðkÞn ¼ KXðkÞ ð31Þ UðkÞ ¼ Us þ ΔUðkÞn :

Fig. 16. Histograms of the residuals of MIMO RBF-ARX model for test data.

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.

When the system goes to a steady state, i.e. XðtÞ-0, and the states x11;t , x21;t , and x31;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-ARXMPC). 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 realtime 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

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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.

11

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 ARX model-set MIMO RBF-ARX

0.3818 0.1957 0.1465

1.0347 0.2295 0.1824

0.2397 0.0372 0.1825

Table 6 Overshoot of dynamic transition processes. Models

Pitch (deg)

Roll (deg)

Yaw (deg)

Physical ARX model-set MIMO RBF-ARX

2.02 0.85 1.12

3.39 2.04 0.51

1.07 1.29 1.43

Fig. 19. Inputs of real-time control based on identified RBF-ARX model.

Table 7 Transient time of dynamic transition processes ( 7 5% error band).

Fig. 20. Control result based on RBF-ARX model.

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.

Models

Pitch (s)

Roll (s)

Yaw (s)

Physical ARX model-set MIMO RBF-ARX

4.3 4.3 2.3

6.7 4.1 1.9

5.7 4.5 2.7

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

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5. Conclusions

Fig. 22. Input signals of the physical model-based method (dash-dotted lines) and the MIMO RBF-ARX model-based method (solid lines), and the disturbance signals (dotted lines).

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

Fig. 23. Comparison of anti-disturbance tests between the physical model-based LQR control method (dash-dotted lines) and the MIMO RBF-ARX model-based LQR control method (solid lines).

with 7 20 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 description 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 helicopter is far better than that of using only 16 ARX models or using one linear physical model.

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Please cite this article as: Wu J, et al. Modeling and control approach to a distinctive quadrotor helicopter. ISA Transactions (2013), http: //dx.doi.org/10.1016/j.isatra.2013.08.010i