Robust Stabilization for 3-DOF Helicopter
Including Unmeasurable States using Output Feedback
M2013SC014 Takuma SUGINOSupervisor：Isao TAKAMI

1INTRODUCTION

When a person and a cargo are hanged by a wire and arope from a helicopter, and the helicopter does a rescue operation, it is desirable to prevent swinging a suspended load forsafety. About the oscillation suppression control of the suspended load, it is often the helicopter does not have a sensor
to measure a swing angle of the load. We may not completelyacquire a state because we are not able to use a sensor fromreasons such as the mechanism of the system and cost in areal system. In this paper, we reproduce this situation using
a 3-DOF helicopter which is experiment machine in imitation of a helicopter of tandem rotor type.
Some results are reported about such oscillation suppression control. Sonobe treats each state independently andsuggests the control method to consider each to be plural
SISO system[1]. On the other hand, there is the study using adaptive control method depending on a state estimate of
the suspended load and the rope length by the image processing for the three-dimensional model[2]. Bisgaard considereda restriction condition of motion of the helicopter and thesuspended load. A study using adaptive control method isreported about the oscillation suppression control in case of
the swing angle of the suspended load is not able to measureusing the 3-DOF helicopter[3].
The purpose of the study is designing a controller for attitude control of the 3-DOF helicopter and oscillation suppression control of the suspended load while being aﬀectedby the wind disturbance and motion of the helicopter. The
state of the suspended load of the 3-DOF helicopter is notable to measure, but we pay attention to the point that is detectable and stabilizable. Using this, in this study, we consider stabilizing closed loop system by output feedback control theory. Also mass of the suspended load may not be
the same all the time. Therefore we treat the mass of thesuspended load as varying parameter and design a controller
for robust stabilization in the range of the varying parameter.
There is a study of the robust stabilization using polytope
representation for the parameter that the range is decided onlike this time[4]. And it is easy to treat the problem using
polytope representation because it is relatively easy. However, we are not able to design a controller for robust stabilization by polytope representation using output feedback because its possible solution condition is not able to arrived atLMIs(Linear Matrix Inequalities)[5]. Therefore in this study,
we treat the problem of the varying parameter as robust stabilization problem using H∞ control theory[6]. And we design
the controller which is guaranteed the stabilization of the helicopter and the suspended load for the parameters variation
range. We design the H∞ controller using output feedback
by solving the LMIs using MATLAB[7]. Then, we choose
penalty weights so that good control performance is provided
by simulation using simulink.

2MODELING OF THE PLANT

2.1

State equations

The schematic figure of the 3-DOF helicopter is illustrated
in Figure 1. The support beam AB can rotate on point O.
Let ϵ(t)[rad] is the angle in vertical plane and λ(t)[rad] is theangle in horizontal plane. The support beam CD can rotate invertical direction on point B. Let ρ(t)[rad] is the pitch angle.
In addition, a suspended load of mass M p [kg] is hung by arope from the helicopter. Let l[m] is the rope length. Letθ(t)[deg] is the swing angle of the suspended load. Now, 3DOF helicopter has sensors to measure the states ϵ(t), λ(t)and ρ(t), but does not have a sensor for θ(t). So, the state θ(t)is not able to be measured directly.

FbBack Rotor

Pitch axis zDBFfρ(t)Front Rotor

Mb gCLhyl
θ(t)LaTravel axisOMf gLhλ(t)ε(t)xElevation axis

Mp・gALw

Counter Weight

Mw g

Figure 1 Schematic drawing of the 3-DOF helicopter
We derive the state equations that show the movement ofthe 3-DOF helicopter using Euler-Lagrange equation basedon Figure 1. A state variable of the plant is defined as x p (t) =[ϵ(t) ρ(t) λ(t) θ(t) ϵ̇(t) ρ̇(t) λ̇(t) θ̇(t)]T . An input is defined asu(t) = [u f (t) ub (t)]T . The state equation is given as Eq(1).
Where u f (t)[V] and ub (t)[V] are each the input voltage of thefront rotor and the back rotor.
ẋ p (t) = A p x p (t) + B p u(t)Now, A p and B p are given as follows.

(1)

A p = E0−1 A0 , B p = E0−1 B0






E0 = 



00000Jρ
0010000000010000000010000000010000000000Jλ + M p La2M p lLa

0000Jϵ + M p La2000000000M p lLa
M p l2


[B0 =000000U0000000000000000M p gl000000001000−Bϵ000K f La
K f La01000−Bρ
00001000−Bλ0

K f Lh−K f Lh000 0 
0 1 0 0 0 000]TServo system

We synthesis a controller that follows a provided reference.
In order to remove a steady-state error, a servo system isused. The error between the observed output y(t) and the reference r(t) is e(t). w̃(t) is [rϵ (t) rλ (t)]T ∫. The state∫ variable ofttthe servo system satisfy x(t) = [x p (t) 0 eϵ (t)dt 0 eλ (t)dt]T .
The servo system is expressed as follows.
ẋ(t) = Ax(t) + B̃1 w̃(t) + B2 u(t)(2)z̃(t)= C̃1 x(t) y(t) = C2 x(t)Where matrices A, B̃1 , B2 , C̃1 and C2 are given as follows.
[][]
A0 01 0 0 0 0 0 0 0
A = −C 0 , Ce = 0 0 1 0 0 0 0 0e[ ][][0BB̃1 = I , B2 = 00 , C̃1 = 0

 1
 0

C2 =  0 0
02.30100000
1000000000000000000000000000I000103.1 Robust stabilization problem
Mass of the suspended load may not be the same all thetime. Where ∆m (s) is uncertainty of the multiplication of thenominal plant and the perturbated plant. We apply a small
gain theorem and derive a frequency weight Wt satisfyingσ̄{∆m ( jω)} < |Wt ( jω)|. Where matrix Wt is given as follows.
[]A B
Wt = Ct Dtt

Where Jϵ , Jρ and Jλ [kg·m2 ] are each moment of inertia.
2.2

3 CONTROLLER SYNTHESIS]0-20
0 0 0 0 1-60

-80-1200

10frecency (rad/sec)

Figure 2 σ̄{∆m ( jω)} < |Wt ( jω)|

We consider to realize stability for the swing angle θ(t) ofthe suspended load that is not able to be measured directly.
Where wθ (t) is disturbance for θ(t). w(t) is [wθ (t) w̃(t)]. The
extended system is expressed as follows.
ẋ(t) = Ax(t) + B1 w(t) + B2 u(t)z(t) = C1 x(t) + D12 u(t)(3)
 y(t) = C2 x(t)Where matrices B1 , C1 and D12 are given as follows.
T 0 0 0 1 0 0 0 0 0 0 

B1 =  0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 100000-40-100

Extended system 0
 0

C1 =  0
 0
0t

Singular plots of the frequency weight and uncertainty of themultiplication are shown in Figure2. Now, the range of thevarying parameter M p is 2.85≤ M p ≤ 3.15. We choose thefrequency weight to cover each uncertainty of the multiplication. In Figure2, a dotted line is the frequency weight Wt .
The generalized plant G(s) is expressed as follows.

singular plot (dB)0 0
 0
0
A0 = 
 0
 0

 000 00 00 Wθ
0 00 00000000000[WD12 = 0u000000000000 Weϵ000 0 0 0 Weλ]Where Wθ , Weϵ , Weλ and Wu are each weighting constants forthe states and the input.

 A Bt C3
G(s) =  C1
 Dt C3
C20At0Ct
0B10D110D21







B20D1200(4)Where matrix C3 is given as follows.
 0C3 =  003.20000001000000000000000100 0 
1H∞ controller synthesis using output feedback

Now, Eq(4) is expressed as Eq(5). We design the output feedback controller K expressed as Eq(6) using H∞ controller synthesis for the generalized plant Eq(5).
ẋ(t) = AG x(t) + BG1 w(t) + BG2 u(t)



z(t)
= CG1 x(t) + DG11 w(t) + DG12 u(t)
(5)
G:


 y(t) = CG2 x(t) + DG21 w(t){K:ẋK (t) = AK xK (t) + BK y(t)u(t) = C K xK (t) + DK y(t)(6)

[XIIY300
reference
traveling

250
200

λ[deg]

There is a constant value γ∞ > 0. If X ∈ Sn++ , Y ∈ Sn++ , ÂK ∈Rn×n , B̂K ∈ Rn×ny , Ĉ K ∈ Rnu ×n and D̂K ∈ Rnu ×ny which satisfy
following LMIs(7), (8) exist, the closed loop system is stable.
Also the output feedback control gain which realize ∥ G ∥∞ <γ∞ is given as Eq(6).

150100]50≻0(7)0-500
He(AG X + BG2 )Ĉ K Â + (A + B D̂ C )TKG
G2 K G2
TT TT

+DB
G1G21 D̂K BG2CG1 X + DG12Ĉ K∗
He(Y A + B̂K CG2 )TT
BG1Y + DG21
B̂TK
CG1 + DG12 D̂K CG2

204060time[sec]

80

100

Figure 4 traveling(adding disturbance)
2swing angle

∗
∗
2−γ∞
I
DG11 + DG12 D̂K DG21

∗
∗
∗−I

 ≺ 0

θ[deg]

1(8)

0-1-2

We derive desired H∞ controller using X, Y, ÂK , B̂K , Ĉ K andD̂K satisfying the LMIs(7), (8). Where matrices I − XY andM, N ∈ Rn×n are regular. They satisfy I − XY = MN T .

DK := D̂KC K := (Ĉ K − DK CG2 X)M −T


−1(9)
BK := N ( B̂K − Y BG2 DK )AK := N −1 (ÂK − NBK CG2 X − Y BG2C K M T −Y(A + B D C )X)M −T4G2K

20

40
60
time[sec]

80

100Figure 5 swing angle(adding disturbance)
28uf26

ubinput[V]G0G22422SIMULATION
20

The output feedback gain is obtained by solving theLMIs(7), (8). Then, D11 = 0, D21 = 0, and matrices M andN is derived by LU decomposition.
4.1

180

Simulation adding disturbance

The weighting constants Weϵ , Weλ and Wu are chosen bytrial and error. We set the reference of ϵ(t)[deg] and λ(t)[deg]to 20[deg] and 200[deg] respectively. Then, we add the wind
disturbance for the suspended load at 30[sec] to the simulation. The simulation results are shown in Figure3, Figure4,Figure5 and Figure6.
40
referenceelevation

30

204060time[sec]80100Figure 6 input(adding disturbance)

4.2 Simulation using nominal controller
We simulate to verify the robust stability. At first, we use
the controller of not considering the robust stability. Then,mass of the suspended load M p is 0.15[kg] heavier than thenominal load. The simulation results are shown in Figure7,Figure8, Figure9 and Figure10.

ε[deg]

201040referenceelevation

030

0204060
time[sec]

80

100

ε[deg]-10

20100Figure 3 elevation(adding disturbance)-10In Figure3 and Figure4, it can be seen that results of thesimulation of the elevation and the traveling converge in thereference. Thus, we were able to design the controller for position control of the 3-DOF helicopter. In Figure5, it can beseen that the simulation of the swing angle of the suspended
load suppresses the wind disturbance at 30[sec]. Thus, wewere able to design the controller for stabilization of the statethat is not able to be measured directly by output feedback.

0204060
time[sec]

80100Figure 7 elevation(nominal controller)

In Figure7, Figure8, Figure9 and Figure10, the signals are
not able to be stabilized when we used the nominal controller.

40

reference
traveling600

referenceelevation30

λ[deg]

400
ε[deg]

20200

1000-10-2000204060time[sec]801000Figure 8 traveling(nominal controller)

20

4060time[sec]

80

100

Figure 11 elevation(robust controller)
300

20

reference
traveling

swing angle
250

10λ[deg]θ[deg]

200

0150
100
50-100-20-50

020

4060
time[sec]

80

1000204060time[sec]

80

100

Figure 12 traveling(robust controller)

Figure 9 swing angle(nominal controller)
228swing angle

uf
26
241

θ[deg]

input[V]

ub

220-1

20
18-20204060time[deg]

80

100

0204060time[sec]

80

100

Figure 13 swing angle(robust controller)

Figure 10 input(nominal controller)28

ufSimulation using robust controller

Next, we use the controller which guaranteed the robust
stability on a condition same as 4.2. The simulation results
are shown in Figure11, Figure12, Figure13 and Figure14.
In Figure11, Figure12, Figure13 and Figure14, the signals
are able to be stabilized when we used the proposed controller. Thus, the controller which is guaranteed the robust
stability for varying parameter by H∞ control using output
feedback is eﬀective.

5input[V]4.326ub24

2220

1802040
60
time[sec]

80100

Figure 14 input(robust controller)

CONCLUSION

In this paper, we show the stabilization for the unmeasurable state using output feedback. In addition, we realized
robust stability by H∞ control using output feedback for thecontrol object which has a varying parameter including thestate space representation.

References[1] Sonobe, M., Miwa, M., Hino, J.: Study on simple vibration control for small-size helicopter with slung load system (Experiment based studies of the vibration controlon hovering), Transactions of the JSME (in Japanese),
vo.78, no.789(2012)[2] Bisgaard, M., Bendsten, J., and Cour-Harbo, A.：Modeling of generic slung load system, AIAA Modeling andsimulation technologies conference and exhibit(2006)[3] Katsurayama, N.：Robust Control with Adaptive Law for
the System Including Unobserved States, Masters thesis,
Nanzan University(2014)[4] Sugino, T., Nagaya, S., Morikawa, T., Takami, I., Chen,G.：Robust stabilization of the 3DOF-helicopter using descriptor representation and polytope systems, SICE annual conference(2013)[5] Asai, T.: Analysis and synthesis of linear robust control
systems based on linear matrix inequalities, (2003)[6] Yoshikawa, T., Imura, J.: Modern control theory, shokodo(2009)[7] Ebihara, Y.: System control by linear matrix inequalities,Morikita(2012)