Robust Control with Adaptive Law for the System
Including Unobserved States
M2012MM013 Naruya KATSURAYAMA
Adviser: Isao TAKAMI

1Introduction

Model Reference Adaptive Control (MRAC) is consists of a reference model, a plant model, and adaptive law. It is made to follow the target system tothe reference model by fixed gain. As study example,
Yang developed an adaptive control for SISO state spacesystem[1]. Yang’s method[1] was not discussed about
the performance of the reference model. MRAC depends on performance of the reference model. It was
important to consider the reference model. We adopted
closed-loop system that was stabilized by Robust LQ
Control for the reference model[2]. We extended Yang’s
method to descriptor system, and verified by 2-DOF
helicopter[2]. The system depends rationally on uncertain parameters, it is diﬃcult to deal with the system.
We adopted redundant descriptor representation to linear uncertainty.
If a system has unobserved states, robust controller
can t be synthesized. Therefore, a reference model andan plant model are designed separately. Error between
the actual plant and the reference model is canceled byadaptive law. The proposed method is applied to helicopter. For example problem, mass of helicopter ischanged by luggage or people. Helicopter is oscillatedby the weight. As a result, Crash accident is caused bythis problem. It is deal with in the framework of MRAC.
3-DOF helicopter is used as experimental machine[3],[4], [5]. The eﬀectiveness of our approach is verified by
3-DOF helicopter.

2

Proposed Method

If a system has unobserved states, robust controller
can not be synthesized. Therefore, an plant model anda reference model are designed separately. Proposedsystem shows Fig.1.

2.1

Robust LQ Controller Synthesis

If state space system depends rationally on uncertain
parameters, redundant descriptor system depends linearly on uncertainties. It is diﬃcult to deal with uncertainties in state space representation whose dependency
is not aﬃne. In this paper, It is avoided this diﬃcultly
by using redundant descriptor representation. A continuous time multi-input multi-output system is describedby Eq.(1).
(Ep +k∑)δi Ei(ẋp =Ap +i=1k∑)
δi Aixp + Bp up

i=1

y = Cp xp

E(δ) = Ep +k∑(1)

δi Ei , A(δ) = Ap +

i=1k
∑

δi Ai

i=1

Where Ep , Ei , Ap , Ai ∈ ℜn×n , Bp ∈ ℜn×m , Cp ∈ ℜp×n .
Eq.(1) has aﬃne perturbation in each coeﬃcient matrix.
Additionally, δi ∈ ℜ is perturbation elements which satisfy |δi | ≤ 1. If Eq.(1) is transformed to state space system, it depends on rational uncertain parameter. Generally, it is diﬃcult to analyze the system stability directly
whose E(δ) matrix has uncertain parameters. When thesystem depends rationally on uncertain parameters, theexpression is not a polytopic type. Redundant descriptor representation is adopted to reduce rational uncertainty to linear one. Through adopting descriptor variables as xd := [xTp ẋTp ]T , uncertainties in each coeﬃcientmatrices are integrated into matrix A(δ).
Ed ẋd = Ad (∆)xd + Bd up , y = Cd xd
Ed = diag{I, 0}, Cd = [Cp 0][][]0I0Ad (∆) =, Bd =A(δ) −E(δ)Bp(2)

One integrator is added into the closed loop system.
For the∫ plant model Eq.(2), let y, r, ep := r − y,z = ep dt are observable output, reference, error andintegrated value of ep , respectively. Letting state as
xdk = [z T xd T ]T , we finally obtain Eq.(3) for the augmented system with integrator.
Fig. 1 Proposed System
Proposed method is consists of 3 approach: 1.Controller K1 is designed by observed states in state feedback. 2.Reference model including unobserved states,controller K2 is synthesized for reference model. 3.Errorof the actual plant and the reference model is canceledby adaptive law. If the system depends rationally on
uncertain parameters, the expression is not a polytopictype. Redundant descriptor representation and Linear
Fractional Transformation (LFT) are adopted to reduce
rational uncertainty to linear one.

Edk ẋdk = Adk xdk + Bdk up[][I 00Edk =, Adk =0 Ed0[]T
Bdk = 0 Bd T

(3)
−CdAd]

To derive a stabilizing state feedback up = K1 xp , consider to minimize Eq.(4).
∫ ∞J=(xdk T Qxdk + uTp Rup )dt(4)0Where Q ∈ ℜn×n > 0 and R ∈ ℜm×m > 0 are given
weighting matrices. For the redundant descriptor system, we have already obtained the following lemma inthe previous research[6]. If there exist X11 > 0, Xd , Ydsuch that Eq. (5) hold, then the closed loop system with−1
the state feedback u = K1 xp := Y X11xp is stable.
11

He[Adk Xd − Bdk Yd ]1
Q 2 Xd1
R 2 Yd[Xd =X11X21
X310X22X32

XdT (Q 2 )T−I00X23X33]

YdT(R 2 )T0−I, Yd = [ Y < 0(5)Êd ė = Am e + B̂d Ŵ (t)T ϕ(x̂d )(10)Ŵ (t) are updated using Eq. (11) as adaptive law withσ-modification[1], [8], [9].

0 0 ]˙Ŵ (t) = −γϕ(x̂d )eT P̂ B̂d − σ Ŵ (t)Furthermore, through maximizing the trace of X11 , J
is guaranties J < trace(X11 )−1 . Synthesized controller
is divided into integration gain Kr1 ∈ ℜm×m and stategain Kx1 ∈ ℜm×n as K1 = [Kr1 Kx1 ]. Nominal input
using robust LQ state feedback is given as follows.
∫unom = Kx1 xp + Kr1 (r(t) − y)dt
(6)2.2

Ŵm (t) − Wm (t)) is the estimation error. Let B̂di =[B̂d1 · · · B̂dn ] ∈ ℜ2n×m
Finally the error between Eq.(7) and Eq.(9) is obtained as Eq.(10).

(11)

Where γ > 0 is adaptive gain and σ > 0 is σmodification gain.
The matrix P̂ > 0 in Eq.(11) satisfies following LMI
condition Eq.(12) by reference[2].
[ATm P̂ T + P̂ Am + 2ρÊd P̂ < 0,P̂ =P11T
P12P12P22](12)Adaptive Law Synthesis

In this section, adaptive law and quadratic stability
analysis for adaptive control loop are discussed. Yang etal [1] developed a LMI-based stability analysis methodthat employs σ-modification for SISO system. In thisstudy, we expand Yang et al’s method to MIMO descriptor system.
Consider the MIMO system described as descriptor
system. Let descriptor variable is x̂d = [x̂p x̂˙ p ]T then it
is described as Eq(7).
Êd x̂˙ d = Âd x̂d + B̂d (u + W T ϕ(x̂d )), y = Ĉd x̂d[Êd =B̂d =[I00][0IÂ(δ) −Ê(δ)][]T, Ĉd = Ĉp 0B̂pT00, Âd =(7)Ē ζ̇ = Āζ + B̄σW
Ā = 
−σIN0..
.

0(13)T−γϕ(x̂d )B̂d1P̂..
.
,−γϕ(x̂d )B̂ T P̂ 0000−σINdn
B̂d1 ϕ(x̂d ) · · · B̂dn ϕ(x̂d )
Am[]
[] []
I 0−IN ×nI 0B̄ =, Ē ==00 00 Êd],W T (t)ϕ(x̂d )is a matched system uncertainty. WhereW (t) = [W1 (t) · · · Wm (t)] ∈ ℜ2n×m , Wi (t) ∈ ℜ2n×1 is
uncertain parameter matrix and ϕ(x̂d ) ∈ ℜ2n×1 is aknown set of smooth basis functions. Actual input ufor the argued system is described as Eq(8).
u = unom + uad , uad = Ŵ (t)T ϕ(x̂d )TLet ζ = [W̃1T · · · W̃me]T as error dynamics variables,
the consolidated error dynamics whose descriptor variable is consist of the tracking error and weight estimation error is described as Eq.(13).

(8)

Where unom is the nominal input for reference model
derived in the previous section and uad is the adaptivesignal. uad functions as canceling matched uncertainty
Ŵ T ϕ(x̂d ) through estimating the uncertain parameter
matrix W (t) with Ŵ (t) = [Ŵ1 (t) · · · Ŵm (t)] ∈ ℜ2n×m ,Ŵi (t) ∈ ℜ2n×1 . Reference model which generates ideal
output for Eq.(7) is described as Eq.(9)∫Ed ẋm = Am xm + Bm (r(t) − y)dt
(9)
Where Am and Bm as Am = Âd − B̂d [Kx2 0], Bm =B̂d Kr2 .
Let e = xm − x̂d is tracking error and W̃ (t) =Ŵ (t) − W (t) (W̃1 (t) = Ŵ1 (t) − W1 (t) · · · W̃m (t) =Where ϕ(x̂d ) = [ϕ1 (x̂d ), . . . , ϕN (x̂d )]T is a set of basis functions. Each vertex of the uncertainty region isdefined as: ϕi ∈ [ϕi , ϕi ]For the uncertainties, let polytope Ā as Eq. (14).
Ā =n∑
i=1ai Āi ,n∑ai = 1, ai ≥ 0(14)i=1

The following is already obtained for stability analysis
of descriptor systems for Eq.(13) [1]. Quadratic stability
is analyzed by solving Eq.(15) at each vertexes of ϕ(·).
Eq.(13) is quadratically stable for perturbation ϕi ifthere exists X11 > 0 such thatX̄ T ĀTn + Ān X̄ < 0,[]
X110X̄ =X21 X22

n = 1, . . . 2n(15)

Where σ and γ are designed whether satisfy thequadratic stability through solving above LMI at eachvertices.

3

Applying to 3-DOF Helicopter

The eﬀectiveness of the proposed method is verified
with using test scale 3 DOF(Degree-Of-Freedom) helicopter. 3-DOF helicopter is shown Fig.2. The helicopter has parallel rotors at the front and back. Thehelicopter is able to control elevation and traveling byfront and back propeller. Let, the elevation angle isϵ(t)[rad], the pitching angle is ρ(t)[rad], the travelingangle is λ(t)[rad]. The voltage of front rotor is Vf (t)[V],
the voltage of back rotor is Vb (t)[V]. 3-DOF helicopter’s

The oscillation angle θ can’t be observed. A reference model is used to deal the problem. A reference
model and an actual plant are designed separately. Theactual plant is designed in six-dimensional state vector. And, the reference model is designed in eightdimensional state vector. The actual plant follows thereference model.
3.1

The plant model

Let, xp := [ϵ ρ λ ϵ̇ ρ̇ λ̇ ]T as state variable, then plant
dynamics of the actual plant is Eq.(20). Where, Jϵ1 =Jϵ + mw L2a + mw l2 , Jρ1 = Jρ + mw L2h + mw l2 , Jλ1 =Jλ + mw L2h + mw L2a , up = [Vf Vb ].
Ep ẋp = Ap xp + Bp up
 0 0 0100 010 
 0 0 0
 0 0 0001 Ap = 000−B00 ϵ
0 0 00−Bρ
00 0 000−Bλ 1 0 0 000 00 
 0 1 0 0
 0 0 1 000 Ep = 000J00 ϵ1
0 0 0 0 Jρ100 U 0 00 Jλ1[]T0 0 0 Kf La Kf Lh 0Bp =0 0 0 Kf La −Kf Lh 0Fig. 2 3DOF-Helicopter
model is shown Fig.3. Eq.(16), Eq.(17), Eq.(18), andFig. 3 3DOF-Helicopter modelEq.(19) are obtained by lagrange equations of motion.
(Jϵ +mw L2a +mw l2 )ϵ̈ = −Bϵ ϵ̇−mw La lθ̇+Kf La (Vf +Vb )(16)(Jρ + mw L2h + mw l2 )ρ̈ = −Bρ ρ̇ + Kf Lh (Vf − Vb ) (17)(Jλ + mw L2a + mw L2h )λ̈ = −Bλ λ̇ + mw La lθ̇ + U ρ(18)2mw l2 θ̈ = mw glθ̇ − mw La lϵ̇ + mw La lλ̇ − Bθ θ̇

(19)Let, inertia of helicopter body about elevation isJϵ [kg·m2 ], inertia of helicopter body about pitching isJρ [kg·m2 ], inertia of helicopter body about traveling isJλ [kg·m2 ], distance between travel axis to helicopter
body is La [m], distance between pitch axis to helicopter body is Lh [m], propeller force-thrust constant isKf [N/V], viscous friction of elevation is Bϵ [N·m/V], viscous friction of pitching is Bρ [N·m/V], viscous frictionof traveling is Bλ [N·m/V], viscous friction of theta isBθ [N·m/V], mass of the weight is mw [kg], length of thestring is l[m], lift to keep the helicopter body in a horizontal is U [N]. Therefore, controlled plant is describedas the model which depends on mw and l as not simply
aﬃne. Therefore mw and l are considered as the uncertainty parameter in deriving robust controller. The
uncertainty is assumed that true value of mw exists in
between 0 to 30[g] and l exists in between 0 to 7[cm].

(20)

In this study, we focus on mw and l as uncertain parameters in deriving robust control low. By using redundant
descriptor representation, it is enable to deal with theplant model uncertainties in polynomials and to express
more naturally. Generally, it is not easy to directly analyze stability of the system whose matrix Ep contains
uncertainty parameters in Eq.(20). To deal with this difficulty, redundant descriptor representation is adopted[7]. Let xd := [ϵ ρ λ ϵ̇ ρ̇ λ̇ ϵ̈ ρ̈ λ̈ ]T as descriptor variable,
then plant dynamics is Eq.(21).
Ed ẋd = Ad xd + Bd up

(21)


0 0 0100000010000  0 0 0
 0 0 0001000  0 0 0000100 
000010 Ad =  0 0 0
 0 0 0000001 
00Jϵ100 
 0 0 0 −Bϵ 0 0 00−Bρ
00 Jρ10 0 U 000−Bλ 00 Jλ1[]
I6 0
Ed =0 0[]T
0 0 0 0 0 0 Kf La Kf Lh 0
Bd =0 0 0 0 0 0 Kf La −Kf Lh 0
3.2

The reference modelThe model with state variables is designed that can’t
be observed the the reference model. Let x̂p :=[ϵ ρ λ θ ϵ̇ ρ̇ λ̇ θ̇ ]T as state variable, then plant dynamics
of the reference model is Eq.(22).

 0
 0
 0
 0
Âp =  0
 0

00
 1
 0
 0

 0Êp =  0

 0

00
[0B̂p =
0000000
U0000000000000000mw gl1000−Bϵ00−mw La l0100000000100000000100000000
Jϵ100000000Jρ100000000Kf La
Kf La

000000Jλ10Kf Lh−Kf Lh01000−Bρ
00001000−Bλmw L a l

00
0
0
0

0
022mw l]T0 00 0(22)0001Êd x̂˙ d = Âd x̂d + B̂d up

Âd = An + Aδ , Aδ = Bδ ∆(I − Dδ ∆)−1 Cδ (δ)

robust LQ with adaptive law
robust LQ
nominal LQ-0.5-1-1.52530

3540time[s]Fig. 4 Step response of traveling with load weight of110g

4(23)Âd in Eq.(23) can’t be deal in exact polytope representation, because it has uncertain parameters ml, ml2 .
LFT enables us to extract high order terms of uncertainty as aﬃne. Âd is transformed by LFT, then Aδ canbe defined Eq.(24).
(24)∆ = diag(l, l, l)Eq.(25) is equivalent to Eq.(23).
(25)

zδ = Cδ x̂d + Dδ wδ
wδ = ∆zδLet descriptor variable x̂dl := [x̂d zδ ]T then closed loopsystem is obtained as Eq.(26).
Êdl x̂˙ dl = Âdl x̂dl + B̂dl up[]AnBδ ∆Âdl =Cδ (δ) −I + Dδ ∆[]Ê0dÊdl =0 0[]B̂
dB̂dl =0−mw La l 0mw La l−BθRedundant descriptor representation is adopted [7].
Let x̂d := [ϵ ρ λ θ ϵ̇ ρ̇ λ̇ θ̇ ϵ̈ ρ̈ λ̈ θ̈ ]T as descriptor variable, then plant dynamics is Eq.(23).

Êd x̂˙ d = An x̂d + Bδ wδ + B̂d up

Traveling[rad]Êp x̂˙ p = Âp x̂p + B̂p up

0(26)Note that Êdl is independent from uncertainty parameters and only Âdl linearly depends on uncertainty.
The eﬀectiveness of the proposed method is verified
by some experiments. The step response of traveling isr = −1.57[rad](-90 degree). The robust LQ with adaptive law, the robust LQ, and the nominal LQ are compared. In this experiment, the helicopter has the oscillation angle of weight to verify the robust control performance. The weight mw = 110[g] is larger than considered range (0 ≤ mw ≤ 30). The length of the stringl = 8[cm] is larger than considered range (0 ≤ l ≤ 7).
Step responses are shown Fig.4. From Fig.4, response of
the proposed method is the best in the response of three.
Therefore, this results shows the eﬀectiveness of robust
LQ control with adaptive law in 3-DOF helicopter.

Conclusion

In this study, robust LQ control with adaptive law
was extend the system including unobserved states. The
proposed method was applied to 3-DOF helicopter. Theeﬀectiveness of our approach was verified by experimental results of 3-DOF helicopter.

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