Robust H2 Control with Polytopic Observer for MIMO
Descriptor System
M2011MM026 Keisuke ITAKURA
Adviser：Isao TAKAMI

1Introduction

In general, linear system is studied to include uncertain parameters[1],[2]. Recently, linear MIMO system
which is described polytopic system is reported. Thenin the real system, we cannot obtain the all states. The
synthesis of an observer and the controller corresponding to uncertain parameter is necessary.
This paper presents a robust H2 control system with
polytopic observer for MIMO system which as describedas descriptor form. Generally, the performance degradation is expected to happen in case that uncertainty
which is considered in the robust observer for velocity estimate design[3],[4] and robust control synthesis
process[5],[6],[7]. Robust observer theory have potential
to improve performance and estimate velocity in control system. In this study, we focus on this characteristics of robust observer algorithms into usual robust
control system. The proposed system is synthesized by
two-step approach. First, polytopic observer is designedthat minimizing an upper bound on a given quadratic
cost function derived[3]. Second, robust H2 controller issynthesized through solving some LMI conditions.
In this study, LMI based stability analysis method forthe combined with system of attached minimal convex
polytope algorithm[8],[9],[10],[11],[12] and MIMO system described as descriptor form. Our approach for
MIMO descriptor systems are extension of polytopic observer which is designed minimizing an upper bound on
a given quadratic cost function. In this study, we showthat Lyapnov stability is guaranteed. Next, we synthesized the robust controller. Finally, the eﬀectiveness ofthe proposed procedure is veriﬁed by simulations andexperiments with 3-DOF helicopter.
We show the whole system which is combined with thefull order observer in Fig.1.

2

Robust observer synthesis

It is diﬃcult to deal with uncertainties in state space
representation whose dependency is not aﬃne. In thispaper, we avoid this diﬃculty with using descriptor representation. Consider a continuous time multi-input
multi-output system described by:(E+k∑)
δi Ei ẋ(t) =i=1

(A+k∑)(δi Ai x(t) + B +i=1k∑)δi Bi u(t)i=1

y = Cx(t)(1)

where E,Ei ,A,Ai ∈ ℜ,B,Bi ∈ ℜ,C ∈ ℜ .Eq(1)has aﬃne perturbation in each coeﬃcient matrices. Additionally, δi ∈ ℜ is perturbation elements which satisfy|δi | ≤ 1. For simplicity E(δ),A(δ) and B(δ) matrices aredeﬁned as:
n×n

E(δ) = E +k∑

δi Ei , A(δ) = A +i=1B(δ) = B +k∑n×ml×nk
∑

δi Ai

i=1

δi Bi

i=1

In this section, polytopic observer and quadratic stability analysis for full order observer is discussed. Oya
etal [3] developed a LMI-based stability analysis methodthat quadratic cost function for SISO system.
Consider the MIMO system described as descriptor system. Let descriptor variable is x̂(t) = [xT ẋT ]T thenEq(1) is described as follows.
ˆ = Â(δ)x̂(t) + B̂ û(t), y = Ĉ x̂(t)Ê ẋ(t)(2)

TÊ = diag{I, 0}, Ĉ = [ C 0 ][][]0I0Â(δ) =, B̂ =A(δ) −E(δ)B(δ)
Where Eq(2) is redundant descriptor state space equation as estimated mechanism. The full order observer
for argued system is describe as follows.
˙Ê x̂(t)= Â(δ)T x̂(t)+ Ĉ T u(t)+H(y(t)− B̂(δ)T x̂(t)) (3)Matrix H is observer gain. We consider to minimizing
the following cost function.
∫ ∞J=x̂(t)T Qx̂(t) + u(t)T Ru(t)dt(4)
Figure 1 Robust H2 control system with full order observer

0

Where Q ∈ ℜ> 0,R ∈ ℜm×m are given weighting
matrices. The observer gain H is synthesized by optimazing Eq(4). Estimation error e(t) ≜ x − x̂ is deﬁned.
Estimation error system is as follow.
n×nė(t) = (A(δ) − HC)e(t)+ (Ae − HCe − Be Kx )x̂(5)

Where Ae = A(δ) − A,Be = B(δ) − B,Ce = C(δ) − C.
In this study, the system which omitted x̂.
ė(t) = (A(δ) − HC)e(t)(6)

We consider an observer gain is guaranteed e(t) → 0 ofEq(6).

3Robust controller synthesis

Generally, it is diﬃcult to analyze the system stability directly whose E(δ) matrix has uncertainty parameters.However through adopting descriptor variables asx̆(t) := [xT ẋT uT ]T ,uncertainties in each coeﬃcient matrices are integrated into matrix Ă.
˘ = Ă(δ)x̆ + B̆u, y = C̆ x̆
Ĕ ẋ

Lemma1: Consider the following e ≜ x − x̂.
Letting xe = [x̂T eT ]T as extended system. then Eq(7)is described as follows.
ẋe = Aϕ (δ)xe[Aϕ =Aϕ11Aϕ21HCA(δ) − HC(7)Aϕ11 = A + HCe − BKx

ATϕ P̂ + P̂ Aϕ + Θ < 0 f or ∀ ∈ δ][][P110
Q + KxT RKx Q
Θ=, P̂ =0 P22QQ(9)In this approach, the full order observer which guaranteed asymptotically stability. The robust controller
which is combined with robust observer is discussed.
Then, Eq(9) satisﬁes Kx ,H.If there exists e(t) →0,x̂(t) → 0, we obtain x(t) → 0. The whole system
is performed asymptotic stability.
We obtained following relationship:V̇ (e) < 0 → lim e(t) = 0

˙Ẽ x̃(t)= Ãx̃(t) + B̃ ũ(t)[[Cw =(11)Co = [ C 0 ]][]Q 00, Dw =0 0
R

Ad = ATo ∈ ℜn×n , Bd = CoT ∈ ℜn×m , Cd = BoT ∈ ℜl×n[]
X110X=, Y = [ Y11 0 ]X21 X22
minimize γ
X11 > 0He[Admin X + Bd Y ]Cw X − Dw Y

(Cw X − Dw Y )T−I

He[Admax X + Bd Y ] (Cw X − Dw Y )TCw X − Dw Y−I< 0 (12)]< 0 (13)

Y11 = Kd X11(14)[H2 controller synthesis, Ã =, B̃ =(16)0B̆(δ)][, Br =IO]

The plant is described as follows:
˙Ẽ x̃(t)= Ãx̃(t) + B̃ ũ(t) + Br rz(t) = C̃ x̃(t) + D̃u(t)[][]We0 000 Wx 0 , D̃ =C̃ =Wu
00 0(17)

where We ,Wx and Wu are weights for the integration oferror, the state, and the input. For the redundant descriptor system, we have already obtained the following
lemma in the previous research[5],[6],[7].
Lemma2 If there exist X11 > 0, X, Y such thatEq(18),(19) hold, then the closed loop system with the−1state feedback u = K11 x := Y11 X11x is stable.
[]
X1100X = X21 X22 X23 , Y = [ Y11 0 0 ]X31 X32 X33
minimize γ[X11 > 0He[Ãmin X + B̃Y ]C̃X − D̃Y(C̃X − D̃Y )T−I

He[Ãmax X + B̃Y ] (C̃X − D̃Y )TC̃X − D̃Y−I[ 2]γ
Br I>0BrT X11
K = [ K11Observer gain H is obtained by Eq(12),Eq(13).
H = −KdT−CĂ(δ)

00]

3.2

[]

[I0Robust observer design

Eq(11) is the control plant for dual system.
{Ed ẋd (t) = Ad xd (t) + Bd ud (t)yd (t) = Cd xd (t)][][0Ao = Â(δ) B̂(δ) , Bo =I000Ĕ(δ)]

Ẽ =(10)t→∞[Extended system

One integrator is added into the closed loop system. For the plant model Eq(15),let y and r areoutput, reference, respectively.Letting state as x̃(t) =[ ∫]Tr − y x̆ u,We ﬁnally obtain Eq(16) for theaugmented system with integrator.

If there exists P̂ > 0, such that Eq(9) hold.

[Note that Ĕ is independent from uncertainty parameters
and only Ă linearly depends on uncertainty.
3.1

Aϕ21 = Ae + HCe − Be Kx

2.1TĔ = diag{I, 0, 0}, C̆ = [ C 0 0 ][][]0I00Ă(δ) = A(δ) −E(δ) B(δ) , B̆ = 0
00−II]
(8)

(15)

0 ]

]
<0(18)<0(19)](20)(21)

Furthermore, through maximaizing the trace of X11 .
Synthesized controller is divided into integration gain
Ki ∈ ℜm×m and state gain Kx ∈ ℜm×n as K = [Ki Kx ].

4Applying to 3-DOF helicopter

The picture of the 3-DOF helicopter is put on Fig.2.It
has two propellers drew by DC motors at front andback.The helicopter can move in horizontal and vertical direction.The simple ﬁgure of the helicopter is illustrated in Fig.3.The support beam AB can rotate on
point O.Where ϵ(t)[deg] is the angle in vertical plane andλ(t) is the angle in horizontal plane.The support beam
CD can rotate in vertical direction on point B.Where
ρ(t)[deg] is the angle of this time.Where uf [V] and ub [V]are the input voltage of the front rotor and back rotor
each.The counter weights is added to keep the balanceof the helicopter and it saves the energy of the rotors.

4.1

State space equation

The state vector for the 3-DOF helicopter is deﬁned as[]Tx(t) = ϵ(t) ρ(t) λ(t) ϵ̇(t) ρ̇(t) λ̇(t). Input
Tis deﬁned as u(t) = [ uf (t) ub (t) ] . The state space
representation is given as follows:{E ẋ(t) = Ax(t) + Bu(t) + L(22)y(t) = Cx(t)
1
 0
 0
E= 0001 0
 0
A= 000010000010000001000001000000Jϵ2(Mf − Mb )La Lh
0000000000000[C=00000010

000(Mf − Mb )La LhJρ0
 , B = Kf 
00010000000LaLh
La sin ρ(t)00

00000Jλ


000La−LhLa sin ρ(t)]000L= −(Mf + Mb )gLa + Mw gLw
−(Mf − Mb )gLh0

where Jϵ [kg·m2 ],Jρ [kg·m2 ] and Jλ [kg·m2 ] are each moment of inertia. There are given as follows.
Jϵ = (Mf + Mb + Mg )L2a + Mw L2w(23)

Jρ = (Mf + Mb )L2h(24)Jλ = (Mf + Mb )(L2a + L2h ) + Mw L2w(25)

Figure 2 3-DOF helicopter

Now take notice of sin ρ(t) = ρ(t) as sin ρ(t) = ρ(t),thenλ̈(t) =U (t)ρ(t), (U (t) := Kf La (uf (t) + ub (t))) (26)JλEq(26) is nonlinear.Matrices E, A and B are rewritten
by linearization around equilibrium point, U0 = 0 andρ0 = 0.
 1
 0
 0E=
 0

000
1000000
1000000(Mf + Mb )L2a + Mw L2w
2(Mf − Mb )La Lh
0000(Mf − Mb )La Lh(Mf + Mb )L2h0000001


Figure 3 Schematic drawing

MfMbMwgLaLhLw
Kf

Mass of the front rotor
Mass of the back rotor
Mass of the counter weight
Gravitation acceleration
Distance from O to B
Distance from B to C or D
Distance from O to A
Lift coeﬃcient

[kg]
[kg]
[kg]
[m/s2 ]
[m]
[m]
[m]
[N/V]

Table 1 Physical parameters



A=


00000000000U0
Jλ000000100000010000001000

0 0
 0 , B = Kf  La
Lh0000La−Lh0


We verify the proposed method using the simulationand experiments in 3-DOF helicopter. In this simulations and experiments, we add a weight of 50.0[g] at
the helicopter to verify the robust control and observer
performance. We ,Wx and Wu are chosen as follows.
[][]
1 00.5 0
We =, Wu =0 10 0.5

 1 0 0 1
 0 0
Wx = 
 0 0
0 00 00010000 0 0 0 0 0 0 0 0 
2 0 0 

0 0.5 00 0 1

The simulation and experiment results are shown for
the Fig.4, Fig.5 and Fig.6. The step response of 3-DOF
helicopter is veriﬁed by simulations and experiments.
The step response of elevation which is given 30 seconds
later is 0.26[rad](15[deg]).

Figure 4 positioning of elevation(simulation)

Figure 5 positioning of elevation(experiment)

Figure 6 velocity estimation

5Conclusion

In this study, we designed robust H2 control system
which is combined with robust observer. In addition,
we show the method to deal with uncertain parameterswhich is described in rational function by redundant descriptor system. The eﬀectiveness of proposed system
is veriﬁed by the experiments. It can be said that theproposed method, that is the combination of robust controller and robust observer theory, is able to improve therobust control performance.

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