input-state linearization - IEEE Xplore

Proceedings of the 2006 IEEE International Conference on Robotics and Biomimetics December 17 - 20, 2006, Kunming, China

Feedback control for yaw angle with input nonlinearity via input-state linearization Zhe Jiang, Juntong Qi, Xingang Zhao, He Wang Robotics Laboratory, Shenyang Institute ofAutomation, CAS Nanta Street 114#, Shenyang China Graduate School, Chinese Academy of Sciences Beijing, China {zhjiang, qijt, zhaoxingang, wanghe}@sia.xn

Jianda Han, Yuechao Wang Robotics Laboratory, Shenyang Institute ofAutomation, CAS Nanta Street 114#, Shenyang China { jdhan, ycwang}(sia.cn

works deal with the exact linearization problem, while dynamic solutions were considered in. John Chiasson uses the differential-geometric approach to realize dynamic feedback linearization for the induction motor system [5]. E.ArandaBricaire built a so-called infinitesimal Brunovsky form which exhibits m controllable blocks whose dimensions play the role of Kronecker controllability indices in the linear case [4]. [6] proposed an algorithm for computing a dynamic controller based on prolongations, which renders a given system linearizable via regular static feedback. [7] presents an inputstate linearization method to allow design of state-feedback controllers for single-input linear systems, and this method does not apply the inverse of the nonlinear function of the systems. Some simulations are presented to illustrate the validity of this method. The paper is organized as follows. Section II describes yaw dynamic of a small-scale helicopter mounted on an experimental platform. Section III presents a systematic method to design controller for yaw dynamics system with input nonlinearity. Applying the derivative of the nonlinear function, the original system is to be extended a new system with a pseudostate variable. This approach makes it possible to avoid using the inverse of the nonlinear function and reduces the calculation load. A numerical simulation is performed to show the feasibility of the proposed approach for yaw control in Section IV. A brief conclusion is given in Section V.

Abstract -This paper discusses the yaw control of small-size unmanned helicopter. The yaw dynamics of helicopter involve input nonlinearity, time-varying parameters and the couplings between main and tail rotor. With respect to such a complicated dynamics, the normal PID control is difficult to realize good tracking performance while maintaining stability and robustness simultaneously. In this paper, a valid control is proposed by applying the derivative of the nonlinear function, the original system is to be extended a new system with a pseudostate variable. This approach makes it possible to avoid using the inverse of the nonlinear function and reduces the calculation load. The simulation results further demonstrate the improvements of the proposed algorithm. Index Terms - Dynamics feedback linearization; Input nonlinearity; Helicopter; Yaw angle. I. INTRODUCTION

Nonlinear control techniques currently available in the literature [1-3] are applicable to plants that are under the assumption of linear or affine linear inputs. For most practical control systems, however, there often exists input nonlinearity in the control input. However, due to physical limitation, there exist nonlinearities in the control input and their effect cannot be ignored in analysis of controller design and realization. The Dynamics of an unmanned helicopter is strongly nonlinear, inherently unstable, highly coupled and forms a multiple input multiple output (MIMO) non-minimum phase system with time varying parameters. During the last years, an urgent demand for helicopter, specially, unmanned helicopter has grown strongly in the area such as national defence, disaster rescue and Anti-terrorism. Researching on reliability and robustness of the nonlinear control methods to improve the performance of the flight control system has been an important focus in the control area [12]. The flight control system design of unmanned helicopter consists of dynamics and control methods. And the flight control design has been dominated by classical control techniques. But, recent years have seen a growing interest in applications of nonlinear control theory when a nonlinear model is deployed for the controller design. Furthermore, yaw control is the guarantee of safe flight and very important when helicopter takes off and lands. So designing suitable controller is the chief problem of unmanned helicopter. The feedback linearization problem has been widely studied both in continuous and in discrete time. The first

1-4244-0571-8/06/$20.00 C)2006 IEEE

II. YAW DYNAMICS

A. Modeling yaw dynamics In this paper a framework of the simulation model for the helicopter-platform (see Fig.1) is set up using rigid body equations of motion of the helicopter fuselage. In this way the effect of the aerodynamic forces and moments acting on the helicopter are described. The total aerodynamic forces and moments acting on a helicopter can be calculated by summing up the contributions of all components on the helicopter, which include main rotor, tail rotor, fuselage, horizontal stabilizer and vertical fin. So, the yaw dynamics has the form:

Of=r

{I

=

Nm

+Ntr +Nfus +N +Nvf

(1)

where (o and r are the yaw angle and angular velocity of the helicopter respectively; Izz is the inertia around z-axis; N

323

presents the torque acted on the helicopter; the subscripts of mr, tr,fus, hs and vf present respectively, main rotor, tail rotor, fuselage, horizontal and vertical fin.

with = vl /(IQr), Cl = aa, Cd=

+{8CfiQJpvoR2(2C3m +C2 C4C42 +4C308r )(R3-R3)

OIQ=r

+6 cd2 CjR

tIJz = -Qmr + Ttrltr + blr + b2So

+4Cd44/pfzR2(2C3m

V

Q tr)drr ~~~~~tr

2tr2pA PA

with

I

Tr =COtr +2C2(C2 +

(3) (4)

I

C+4CiOtr)

tr 1P""tr "tr Similarly, the force of the main rotor is:

Tmr = C3 Omr + 2 C44 +

where,

C3

=

IC42 + 4C38 (R2- R2)

4C308, )(R3- R3)

R2 ) +3Cd2C (R2- R2)}

2

Qmr MkQ Hmr +kQ Hmr +kQo

(9)

Where kQ2, kQl and kQo depend on the shape of the blades and the speed of main rotor Qmr. O Qmr fit Qmr

0.07 0.06

(5) -

E

C1=patrbtrctrQtr 2(Rtr3 -R3o)

C2 =8Qpatrutrctr

c

where R and 0mr are respectively, radial and pitch angle of main rotor. B. Simplified model From (2) we can see that there exist couplings between main rotor torque Qmr and tail rotor thrust Ttr. And (3) and (8) further demonstrate that the models are highly nonlinear and too complex to be used for control design. Instead of the dynamics described by (3) and (8), a simplified model is proposed for control design. By plotting the torque vs. pitch angle, we can find that relation between Qmr and Omr approximated with quadratic polynomial (see Fig.2)

where p atr bt tr 6tr r Vtri and Atr are respectively, density of air, slope of the lift curve, number of the rotor, chord of the blade, speed of the tail rotor, pitch angle, radial distance, induced speed of the tail rotor and area of the tail rotor disc. Combing (3) with (4), we have I

+C42 C42 +

+6CdOpQ2R2(R4 R)+3aC4(R

where Qmr is the torque of main rotor, Tr is the thrust of tail rotor, 1, is the distance between the tail rotor and z-axis, b, and b2 are damping constants. The brief presentation of the forces and torques computing can be obtained by using the blade element method [1 1-13].

Tt,

2R ) +6aCR- R2) + 6Cdl pzQ2R 2(R 4 4R)}

+{3Cd2C4 C+ 4C3 O., (R2-R2)+ 3aC4

(2)

(8)

C42+4C36, )(R33 R3)

+4aQ opvzR 2 (2C3+C42 C4

In hovering and low-velocity flight, the torque generated by main and force generated by tail rotor is dominant. By simplifying the fuselage and vertical fin damping, the yaw dynamics can be rewritten as:

Vt

+ Cdla+Cd2a2

where a , a , r , c , 0 , v1 and Q are respectively slope of the lift curve, the angle of attack of the blade element, speed radial distance, chord of the blade, inflow angle, induced speed and rotor speed of the main rotor. After integral manipulation with the help of Maple, we obtain Q = 1 C CQ2C(R4-R4) 2

Fig. 1 Helicopter coordination

Ttr = 2patrbtrctrQt | uO (ttr

CdO

0.05 0.04

a 0.03

-Rtro)

43mr

0.02 0.01 t 0.02

(6)

0.06

0.08

0.1

Omr ( rad )

0.12

0.14

0.16

Fig. 2 Torque of main rotor with Quadratic Polynomial Fitting

Similarly, the lift of tail rotor, Ttr(see Fig.3), can be written:

6 pabcQ2 (R3- R )

Ttr

C4 = 8 pabcQ 2 / p;zR2 (R2 - R )

k

26r + kTOtr + kT0

(10)

Where kT2, kTl and km depend on the shape of the blades and the speed of tail rotor Q,tr

The torque generated by main rotor is:

(PQ2r2ClcCo ++ pQ2r2Cdc)rdr QQmr =IR 2

0.04

(7)

324

|

0.8

B.

Controller design Assume yt(u) is differentiable by u any number of times and each of them hold nonzero values. Our objective is to find a state-feedback controller that stabilizes the system (12) and satisfies its control requirement. One solution for this problem is the following method. If the nonlinear function yt is a nonzero function, then the system (12) can be linearized. Because (12) and (14) show that the right hand of (12) can be decomposed into linear part Ax and nonlinear part byf(u), it can be linearized [8] Applying the method in [8] to this system, the variables and the input are converted as follows:

~~~1'

T

T

T

Ttr 0 fit Ttr

0.6 E 0.4

0.2

0l 0.02

0.04

0.06

0.08

0.1

Otr ( rad )

0.12

0.14

0.16

Fig. 3 Thrust of tail rotor with Quadratic Polynomial Fitting

Substituting (9) and (10) into (2), then we can obtain the following nonlinear model:

{

=r

zr=

(kQ2 Omr + kQ1 Omr + kQo )(

1)

L +(kT2 tr +kTi6tr + kTo)ltr + bl + b2r Where, bi and b2 are damping constants. (1 1) indicates: 1) The yaw dynamics for small scale helicopter can be described second order time-varying system with input nonlinearity. 2) The input nonlinearities include unknown parameter, which depend on the main rotor collective, the speed of main rotor and the speed of tail rotor. 3) There are couples between the main rotor torque and the tail rotor force. The main rotor torque changes while the main rotor collective and the speed of main rotor change, which is a time-varying disturb. 111. FEEDBACK CONTROL DESIGN In this section, we use a dynamics feedback linearization method to design control law for yaw dynamics [7]. A. Preliminary For (11), define X=[x1,x2]T =[p,r]T as a measurable state vector of the system, u = 0 is the control input, as follows:

r]

{X2 (t) blq+ b2r yf(u)

(12)

x = Ax+bybt(u)

(13)

=

i.e. where

A w(u) = (kT2ORtr2 ++kT

W(u)

b

+

b [7]

X1

Z2

=

X2

(16)

blX+ b2X2 y(U) {zl

1Z2

=

Z2

(17)

=V

In this case, the actual input u is calculated from the virtual input v. u = _'(v - b1x1 - b2x2) (18) Here we use a method that does not apply the inverse function directly but uses dynamic nonlinear state feedback. The resulting controller in this framework stabilizes the closed-loop system, and the controller itself is stable. To use this dynamic nonlinear state-feedback method, we introduce the following extended system to realize the linearization of the system (12):

{ x= Ax+byb(x3) l3

=-ax3

(19)

+ /3u

where x3 is a pseudostate variable and u is a pseudocontrol variable. Equation (19) can be written in the following form:

is the system output. So (o1) be expressed

Jxl (t) = X2 (t)

=

+ + IV = and we obtain the following linear system:

o

andsy4

z1

xc = f (xC) + gu

where

(20)

xL x=

X2

LX3J

f(x)=

X2

Ax+

-I-aX3

L+

9P =

-I(u)j

+

-aOX3

(14)

+TO)tr (kQ 67r + kQ 6Rmr + ka ) (15)

b2X2

~~~~~~(21)

As is described in [2], the coordinate transformation is obtained as

express the input nonlinear characteristic of the controller output u, and second order input nonlinear term with unknown parameters.

325

(zi

X1

qZ2

X2

To show that the set of {g, adf g, ad g} is involutive, let this

hl, h2 }. Then, Lie brackets of hi and hj become [h, h = Vhhi -Vh, hJ

(30)

rank[g, adfg] = rank[g, adfg,[h,,hj]]= 2

(31)

set be

(22)

{Z3 =bIx1 +b2x2 +W(x3)

Because the input transformation is u = (v-Lf+'z) l(LgLfz,), we obtain

=(//)[v

-

+b) x2

Note that all elements on the right-hand side have the property that all elements in the first row are 0. These yields

(23)

If /3 . 0 and yV'(x3)

0 , the set is involutive. n So in this case, if /3 . 0 and yJ(x3) . 0, the system can be linearized for all x3 [ 1].

-b2y(X33) + a (x3)]

Then, substituting (22) and (23) into (20) and (21) results

in the linearized system given here:

_11 =Z2 Z2 Z3

D. Stability of the controller

(24)

Theorem 3.1 Consider the system consisting of (20) and controller (26). If constant /3 . 0 and y (x3) . 0 , the system can be full state linearized and the controller is stable. Proof. Now consider the following Lyapunov function candidate V 2I 2 for 3 .0 (32) Its derivative with respect to time is

V

The system (24) is state controllable, and so we can design the controller using linear control theories v = -Kz = -klZ1 (25) k2Z2 -k3 Z3 Although the control law designed in this manner uses pseudostate variables z1 Z2,Z3 in the linearized state space, it can be implemented as a simple state-feedback controller using the actual state variables x as described next. Using (22),

(33) Consider the expression in (27) (34) x3 = -(klxl - k2x2)ly- (b2 + k3)Vr / r with input terms x1, x2. Rewriting this equation without the input terms yields

(23), and (25), the control variable u is given by

U* = -k1x1 k2X2 (b2 +k3)y + ay/X3 -

-

(26)

Summarizing the second equation of (20) and (26), the dynamic characteristic of the controller is (27) x3 = -(klxl - k2x2 ) I - (b2 + k3 )V 1 V Thus, the controller can solve (27) using x1, x2 and outputs the signal x3 to the actuator.

x3 = -(b2 + k3)y,(x3)/y,(x3) Using (33), we obtain

V

C.

V = V(X3Ak(X3) =

Where (35) has introduced. Therefore if

b2 > -k3

(37)

V~= Vr(X3)r(X3) < °

(38)

Then

calculation of the rank of the matrix. The controllability matrix is

According to Lyapunov stability theory,The controller is stable. Thus the proof is achieved completely. a

(28)

IV. SIMULATIONS In this section, the control algorithm is verified by the simulation model obtained from the helicopter-on-arm platform, as shown in Fig.4. A small-scale electrical helicopter is mounted at the end of a two-DOF arm, while the weight of the helicopter is perfectly balanced at the other side of the arm. First, the parameters of the nonlinear yaw dynamic model

From this equation, if , . 0 and yJ'(x3) . 0,

rank g, adf g, adf g = 3

'(36)

-Vy(x3)yV'(x3)(b2 + k3)yJ(x3) / y'(x3) = -(b2 + k3)*2(x3)

it is sufficient to show that the vector field {g, adf g, adfg} is linearly independent and that a set of {g, adf g, adfg} is involutive. The controllability is directly shown by the

]

Vr(X3)V (x3)x3

=

Possibility ofLinearization To prove that the system (20) can be full state linearized,

[g,adg,ad]g] = r

(35)

(29)

for all x3. The controllability is proved.

326

are identified by least square method, and the followings are the result:

The initial states are x1 (0) = 30, x2 (0) = 10. Here, b2=-2.9912 >-k3=-4, so the controller is stable. To verify the robustness of our method for the model parameter and disturbance, in the simulation, the disturbance is designed to change according to the step-changing and continuous sine changing, i.e. (a), (b). (a) A sine change at t=lOs, i.e.,

770 + A,7sin(w 7(t-1O)) tl1Os (b) A step change at t= l Os, i.e., tlOs

F1g.4 Helicopter-on-arm Platform

Where the initial value of the model parameter is 1o0=0, the constant change value in lOs is A -5, the amplitude of the sine change is A -2, and the angular frequency is -2rad/s.

2(39)

tr=b(o+ b2r +k3Ottr + k46, +kkQmr0tr + 77

{.L

L

L

) 3

With, b1=-1.36, b2=-2.9912, k3=49.1359, k4=6.7012, k5=0.0958. Qmr = 1200, 77 are disturbances. It is obviously that + mrktr is a nonlinear function with respect to k3r + k kt the control input 6tr .Fig.5 demonstrates the fitness of identified model of (39) with respect to the measure performance, from which we can see that the simulation model mat with system very well.

40 _ 30 T x

20 _

b

10 _

0

_

-10_ 0

2

4

6

8

0

2

4

6

8

2

4

6

8

2

4

6

8

10 time, s

12

14

16

18

20

10

12

14

16

18

20

10

12

14

16

18

20

10

12

14

16

18

20

10

1b

0

100

50

(41)

25

4

-10

' -20

-30

(3)

-5u

-40

-100

time, s

0.4 1:I

-150 -200. 0

4

8

12

16

20

24

28

32

36

40

time,s --- Identification results Measurement data Fig.5 Comparison of the flight test data and computed from the identified model

0.2 -0.4

0

In the simulation, the PID controller and our method use the model (39). The parameters of PID controller are: k =0.01, ki=0.000025, kd-0.025, which are real parameters for our

3-

Next our method is applied to the yaw control of unmanned helicopter. The parameters of the extended system

0-

2

experiments.

1-

a=l,fi=1.

I

0

X1 =X2 2 k3tr +k46tr + k5Qmr6tr First, this system is extend as

=X2

time, s

time, s

Fig.6 Simulation result when the disturbance satisfies a sine change, xl and x2 are the result of tracking yaw angle and angle velocity X3 is a pseudo state variable and u* is a pseudo control variable

X2 =y(X3),3

= -X3+U Applying the coordinate transformation (22), the linearized system (24) can be obtained. Now, we can design a controller for the linearized system using pole placement techniques. Here , K = [4 6 4]

327

REFERENCES [1] Khalil, H. K., Nonlinear Systems, 3rd ed., Prentice-Hall, Upper SaddleRiver, NJ, 2002. [2] Slotine, J. E., and Li, W., Applied Nonlinear Control, PrenticeHall,Englewood Cliffs, NJ, 1991. [3] G. Tao and P. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York: Wiley, 1996. [4] E. Aranda-Bricaire, C.H. Moog, J.-B. Pomet, A linear algebraic framework for dynamic feedback linearization, IEEE Trans. Automat. Control, 40 (1995), pp.127-132. [5] John Chiasson, A new approach to dynamic feedback linearization control of an induction motor. IEEE Transaction on automatic control, Vol. 43, NO. 3, 1998, pp.391-397. [6] Stefano Battilotti, Claudia Califano, A constructive condition for dynamic feedback linearization, Systems & Control Letters, 52 (2004), pp.329-338 [7] Tetsujiro Ninomiya and Isao Yamaguchi, Feedback Control of Plants Driven by Nonlinear Actuators via Input-State Linearization, Journal of guidance control and dynamics, Vol. 29, No. 1, 2006, pp. 20-24. [8] Su, R., "On the Linear Equivalents of Nonlinear Systems," Systems andControl Letters, Vol. 2, No. 1, 1982, pp. 48-52. [9] J.G. Leishman, Principles of Helicopter Aero Dynamics. Cambridge University Press, 2000. [lO]Prouty, R. W, Helicopter performance, stability and control. New York: Krieger Publishing Co, 1995. [1 I]Bernard Mettler, Identification Modelling and Characteristics of Miniature Rotorcraft. Massachusetts, Kluwer Academic Publishers, 2003. [1 2]Zhe Jiang, Jianda Han, Enhanced LQR Controlfor Unmanned Helicopter in Hover, 1st International Symposium on Systems and Control in Aerospace and Astronautics, pp, 1438-1344, Harbin, China, January 2006.

40 _ 30 a x-1

20 10 _

0_ -10

0

2

4

6

8

10 time, s

12

14

16

18

20

2

4

6

8

10 time, s

12

14

16

18

20

2

4

6

8

10 time, s

12

14

16

18

20

10 _ _\s

-20 -30 -40

0

0.4

CO

x

3-

-

-

2

-

--

-

--

-

I-I

- -

-

-2

0

-4 -

--

2

4

I 1-- -

- - - - - -

- -

X- -I

6

8

- -

-1I-

I-I - - - - - -

- -

10 time, s

-

4 -

--

12

14

- -

- -

- - -

X- -I

- -

16

18

-1

20

Fig.7 Simulation result when the disturbance satisfies a step change, x1 and x2 are the result of tracking yaw angle and angle velocity X3 is a pseudo state variable and u* is a pseudo control variable

Simulated results are demonstrated in Figs. 6-7. In these simulations, it can be easily seen that using our the proposed method can work effectively, even if the disturbance change according to the step-changing and continuous sine changing in 10 seconds. Summarizing these simulations, it is noted that the proposed feedback control via input-state linearization design method can improve the system performance in the presence of the disturbance. V. CONCLUSION

In this paper, we used a new input state linearization design approach for the yaw control of a small-scale helicopter mounted on an experimental platform. This method can work effectively, even if the disturbance change according to the step-changing and continuous sine changing. ACKNOWLEDGMENT

This work was supported in part by the National High Technology Research and Development Program under the grant 2003AA421020. Additional support was provided by the creativity encouragement foundation of Shenyang Institute of Automation, Chinese Academy of Sciences.

328