ROBUST SIMPLE ADAPTIVE H
∞
MODEL FOLLOWING
CONTROL DESIGN BY LMIS
R. Ben Yamin
Tel Aviv University, 69978, Tel Aviv, Israel
I. Yaesh
Tel Aviv University, 69978, Tel Aviv, Israel
U. Shaked
Tel Aviv University, 69978, Tel Aviv, Israel
Keywords:
Simple adaptive control, SAC, H
∞
Model following.
Abstract:
An output-feedback direct adaptive control problem is considered for MIMO linear systems with polytopic-
type parameter uncertainties and disturbances. The objective is to make the system output follow the output of
a system model and to attain guaranteed H
∞
performance of the proposed adaptive control scheme. Sufficient
conditions for closed-loop stability, model following performance, and achieving a prescribed bound on the
H
∞
disturbance attenuation level are derived, in terms of linear matrix inequalities. A numerical example,
taken from the field of flight control, demonstrates the proposed method.
1 INTRODUCTION
A class of direct adaptive controller schemes for
continuous-time systems, known as Simple Adaptive
Control (SAC), has received considerable attention
in the literature (Kaufman et al., 1998)-(Peaucelle
and Fradkov, 2008). Robustness of SAC controllers
facing polytopic uncertainties has been established
(Kaufman et al., 1998)-(Yaesh and Shaked, 2006) al-
lowing application to real engineering problems (see
e.g. reference (Barkana, 2005)). The stability of
continuous-time SAC is related to the Strictly Posi-
tive Real (SPR) property of the controlled plant. The
stability of closed-loop SAC is related to the Almost
Strictly Positive Real (ASPR) property of the con-
trolled plant. Namely, if a plant is ASPR there ex-
ists a static output-feedback gain (possibly parameter-
dependent) which stabilizes the plant and makes it
SPR. In such a case, SAC stabilizes the closed-loop
dynamics and consequently leads to zero tracking er-
rors.
The existing SPR or ASPR results are developed
for systems with equal number of inputs and outputs
(square systems). The concepts of passivity and passi-
fiability (feedback passivity) are introduced in (Frad-
kov, 2003) to non-square systems. The latter passifi-
cation results will be used in this paper.
In (Ben-Yamin et al., 2008), a framework for the
combination of optimal H
∞
control and SAC model
following has been developed. The idea is to use SAC
while satisfying some H
∞
-norm bound on the distur-
bance attenuation level, and sufficient conditions have
been derived for the stability of the closed-loop dy-
namics of the SAC scheme with a prescribed distur-
bance attenuation level γ. These sufficient conditions
are expressed in terms of Bilinear Matrix Inequalities
(BMI), which in many cases are difficult to solve.
A breakthrough achieved in (Peaucelle and Frad-
kov, 2008) is the formulation of a solution to the reg-
ulation problem, for robust adaptive L
2
-gain control
of polytopic MIMO LTI systems by LMIs rather then
by BMIs. Note that in (Peaucelle and Fradkov, 2008)
measurement noise was not considered.
The present paper applies and extends the method
of (Peaucelle and Fradkov, 2008) in order to solve
the problems considered in (Ben-Yamin et al., 2008)
by LMI’s, including the MIMO case which was not
solved in (Ben-Yamin et al., 2008). As in (Ben-Yamin
338
Ben Yamin R., Yaesh I. and Shaked U..
ROBUST SIMPLE ADAPTIVE H8 MODEL FOLLOWING CONTROL DESIGN BY LMIS.
DOI: 10.5220/0003515803380343
In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 338-343
ISBN: 978-989-8425-74-4
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

et al., 2008), a combination of SAC model follow-
ing and optimal H
∞
control is applied. The objec-
tive is to use SAC while satisfying some H
∞
-norm
bound γ. Sufficient conditions are derived for the sta-
bility and model following of the closed-loop dynam-
ics of the SAC scheme with disturbance attenuation
level γ. These sufficient conditions are expressed in
terms of Linear Matrix Inequalities (LMI), which can
be solved using Matlab’s LMI Toolbox (Gahinet et al.,
1995). A numerical flight control example is given
which illustrates the method.
1.1 Notation
Throughout the paper the superscript ‘T’ stands for
matrix transposition, R
n
denotes the n dimensional
Euclidean space, R
n×m
is the set of all n× m real ma-
trices, and the notation P > 0, for P ∈ R
n×n
, means
that P is symmetric and positive definite. T
zw
denotes
the transference from the exogenous disturbance w to
the objective function z, kT
zw
k
∞
is its H
∞
-norm and
kT
zw
k
2
is its H
2
-norm. col{a, b} stands for [a
T
b
T
]
T
and tr{H} denotes the trace of the matrix H.
2 PRELIMINARIES
2.1 Ideal Strictly Proper System and
Ideal Control
Consider the following continuous-time linear sys-
tem:
˙x
∗
(t) = Ax
∗
(t) + B
2
u
∗
(t), x
∗
(0) = 0
y
∗
(t) = C
2
x
∗
(t)
(1a,b)
where x
∗
(t) ∈ R
n
is the system state, y
∗
(t) ∈ R
l
is the
plant output which can be measured and u
∗
(t) ∈ R
m
is
the control input. A, B
2
and C
2
are constant matrices
of appropriate dimensions.
The output of the plant (1) is required to follow
the output of the asymptotically stable model:
˙x
m
(t) = A
m
x
m
(t) + B
m
u
m
(t), x
m
(0) = 0
y
m
(t) = C
m
x
m
(t)
(2a,b)
where x
m
(t) ∈ R
q
is the system state, y
m
(t) ∈ R
l
is
the plant output, u
m
(t) ∈ R
m
is the control input and
A
m
, B
m
and C
m
are constant matrices of appropriate
dimensions. The reference model (2) is used to define
the desired input-output behavior of the plant. It is
important to note that the dimension of the reference
model state may be less than the dimension of the
plant state. However, since y
∗
(t) is to track y
m
(t), the
number of the model outputs must be equal to number
of the plant outputs.
2.2 Hyper Minimum Phase Systems
Following (Fradkov, 2003), we introduce the follow-
ing notation:
δ(s) = det(sI
n
− A), G(s) = C
2
(sI
n
− A)
−1
B
2
.
Let T ∈ R
m×l
be off full row-rank and define
Ψ(s) = δ(s)det(TG(s)), Λ = TC
2
B
2
.
Definition 1. (Fradkov, 2003) The system (1) is
called minimum phase if the polynomial Ψ(s) is Hur-
witz (its zeros belong to the open left half-plane). It
is called Strictly Minimum Phase (SMP) if it is mini-
mum phase and detΛ 6= 0, and Hyper Minimum Phase
(HMP) if it is minimum phase and Λ > 0.
Remark 1. HMP is closely related to the ASPR for
square systems. For strictly proper square systems,
the conditions which define the ASPR property are
similar, and in such cases any minimum phase system
of m inputs and m outputs satisfying also C
2
B
2
> 0 is
ASPR (Kaufman et al., 1998).
Suppose that (1) closed with the feedback
u
∗
(t) = Ky
∗
(t) + v(t)
(3)
where K ∈ R
m×l
and v(t) is an auxiliary input. The
proof of the following Theorem can be found in
(Fradkov, 2003).
Theorem 1. The system (1) is strictly passifiable by
the output feedback (3) with fixed matrix K iff the sys-
tem (1) is HMP.
We will see in the sequel, as in (Kaufman et al.,
1998), that the HMP property allows applying a class
of direct adaptive controllers referred to as “simple
adaptive controllers”. In the sequel we assume that
the system (1) is HMP.
2.3 Perfect Following
Perfect Following (PF) is defined as following with
zero tracking error, namely
y
∗
(t) = y
m
(t)
The next lemma determines the relation that exists be-
tween the plant’s and the model’s state vectors.
Lemma 1. There exist F(t) ∈ R
n×q
and G(t) ∈
R
n×m
such that the trajectories of (1) are of the form:
x
∗
(t) = F(t)x
m
(t) + G(t)u
m
(t)
(4)
Proof: Equation (4) describes n equations with
n× (q + m) variables, thus the existence of F(t) and
G(t) is guaranteed for all 0 ≤ t < ∞. QED
ROBUST SIMPLE ADAPTIVE H8 MODEL FOLLOWING CONTROL DESIGN BY LMIS
339

Remark 2. Note that F(t) and G(t) are not actually
used; only their existence is required.
Since the system (1) is HMP, Λ = TC
2
B
2
> 0 so
that Λ
−1
exists. Define
K
∗
x
(t) ≡ Λ
−1
(TC
m
A
m
− TC
2
AF(t))
K
∗
u
(t) ≡ Λ
−1
(TC
m
B
m
− TC
2
AG(t)).
(5a,b)
If the ideal control u
∗
(t), defined as:
u
∗
(t) = K
∗
x
(t)x
m
(t) + K
∗
u
(t)u
m
(t),
(6)
is substituted in (1), we obtain that y
∗
(t) = y
m
(t). The
ideal control signal u
∗
(t) thus achieves PF.
3 PROBLEM FORMULATION
Consider the following continuous-time linear sys-
tem:
˙x(t) = Ax(t) + B
1
w(t) + B
2
u(t), x(0) = x
0
y(t) = C
2
x(t) + D
21
w(t)
(7a,b)
where x(t) ∈ R
n
is the system state, y(t) ∈ R
l
is the
plant output which can be measured, w(t) ∈ R
m
is the
exogenous disturbance which is energy bounded and
w(t) ∈ L
2
and u(t) ∈ R
m
is the control input. A, B
1
,
B
2
, C
2
and D
21
are constant matrices of appropriate
dimensions.
The output of plant (7) is required to follow the
output of the asymptotically stable model (2). We de-
fine the following objective vector:
z(t) = C
1
e
y
(t) + D
12
e
u
(t)
(8)
where following (Kaufman et al., 1998), we define
e
y
(t) = y
m
(t) − y(t) = y
∗
(t) − y(t)
(9)
e
u
(t) = u
∗
(t) − u(t)
(10)
The matrices C
1
and D
12
are weights used to shape
the control objective (8). It is required to assure that
the plant (7) follows the output of the asymptotically
stable model (2) so that the standard H
∞
cost J satis-
fies
J
∆
= ||z||
2
2
− γ
2
||w||
2
2
< 0
(11)
for any w(t) 6= 0 and w(t) ∈ L
2
, by employing a SAC
controller.
4 SOLUTION
4.1 Control Law
We consider a controller of the form (Kaufman et al.,
1998),(Ben-Yamin et al., 2008):
u(t) = K
∗
(t)r(t) − eu(t)
(12)
where:
K
∗
(t) =

K
e
(t) K
∗
x
(t) K
∗
u
(t)

(13)
r(t) = col{e
y
(t), x
m
(t), u
m
(t)}
(14)
and where K
e
(t) ∈ R
m
is a stabilizing gain which is
calculated in the sequel,K
∗
x
(t)∈R
m×q
andK
∗
u
(t)∈R
m
are defined in (5), and where eu(t) is an auxiliary in-
put signal which will be defined later. Note that when
e
y
(t) = 0, the controller (12-14) reduces to (6), for
eu(t) = 0. This control, however, requires calculation
of F(t) and G(t) for all 0 ≤ t < ∞ and explicit knowl-
edge of the system dynamics.
Instead, we use the direct adaptive control scheme
known as the Simplified Adaptive Control (SAC)
(Kaufman et al., 1998) to calculate the gains which
lead, in the steady state, to the same control signal
that would have been achieved by K
e
(t), K
∗
x
(t) and
K
∗
u
(t). The application of SAC requires neither ex-
plicit knowledge of the gains matrix nor exact knowl-
edge of the system dynamics or the exogenous distur-
bance w(t).
4.2 Simple Adaptive Control Law
Consider the following SAC scheme (Kaufman et al.,
1998):
u(t) = K(t)r(t)
(15)
where:
K(t) =

K
e
(t) K
x
(t) K
u
(t)

˙
K
e
(t) = T
e
e
y
(t)e
T
y
(t) − φ(t), K
e
(0) = 0
˙
K
x
(t) = T
x
e
y
(t)x
T
m
(t), K
x
(0) = 0
˙
K
u
(t) = T
u
e
y
(t)u
T
m
(t), K
u
(0) = 0
(16a-d)
where T
e
, T
x
and T
u
are constant weighting matrices
and where
φ(t) = σ(tr{K
e
(t)K
e
(t)
T
})K
e
(t). (17)
σ is a scalar function such that:
σ(µ) = {
µ−α
αβ−µ
if α < µ < αβ
0 if otherwise
(18)
where α > 0 and β > 1.
The next two Lemma, which will be required to
assure model following of (7) with a disturbance at-
tenuation level γ, are proved in (Peaucelle and Frad-
kov, 2008).
Lemma 2. (Peaucelle and Fradkov, 2008)
tr{K
e
(t)K
e
(t)
T
} < αβ if e
y
(t) is bounded for all
t ≥ 0.
Lemma 3. (Peaucelle and Fradkov, 2008) For all F,
K
e
(t) satisfying tr{F
T
F}≤α and tr{K
e
(t)K
e
(t)
T
}<αβ,
the inequality tr{φ(t)(K
e
(t) − F)
T
}≥0 holds.
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
340

We define δ(t)=K
∗
(t)−K(t), that is the difference
between the ideal gain K
∗
(t) and the current SAC gain
K(t). The control law of (15) can now be expressed
by the following choice of the auxiliary control signal
eu(t) of (12):
eu(t) = δ(t)r(t).
(19)
We define the state errors:
e
x
(t) = x
∗
(t) − x(t)
and using (10) , (12) and (6), we obtain that e
u
(t) of
(10) is given by
e
u
(t) = −K
e
(t)e
y
(t) + eu(t)
(20)
which, after simple algebraic manipulations, leads to:
˙e
x
(t) = Ae
x
(t) + B
1
w(t) + B
2
e
u
(t)
e
y
(t) = C
2
e
x
(t) + D
21
w(t)
z(t) = C
1
e
x
(t) + D
11
w(t) + D
12
e
u
(t)
(21a-c)
where C
1
∆
= C
1
C
2
, D
11
∆
= C
1
D
21
and D
12
are weights
used to shape the control objective (8).
In order to establish the desired model following
of (7) with a disturbance attenuation level γ when (11)
is satisfied, the asymptotic stability of the error sys-
tem (21a-b) with the objective vector (21c) should
be proven. Stability will be proven here by apply-
ing the fact that passivity implies stability (Peaucelle
and Fradkov, 2008). To this end, define the signal
y
p
(t) = T
e
e
y
(t) + De
u
(t), where D will be defined be-
low, and
b
β = αβ. We are now in a position to state the
main result of this section.
Theorem 2. If there exist two scalars α > 0 and
b
β > α and two matrices F and D, and three matri-
ces P > 0, R > 0, G > 0 such that the following LMI
conditions hold

−R C
T
2
T
T
e
− PB
2
∗ −I

≤ 0. (22)

G F
T
F I

≥ 0 , tr(G) ≤ α (23)




L PB
2
−C
T
2
T
T
e
PB
1
+C
T
2
ΓD
21
C
T
1
∗ −D− D
T
−T
e
D
21
D
T
12
∗ ∗ −γ
2
I + D
T
21
ΓD
21
D
T
11
∗ ∗ ∗ −I




≤ 0
(24)
where
L = A
T
P+ PA+C
T
2
ΓC
2
+ R,
Γ = (
b
βI + T
T
e
F + F
T
T
e
)
then the adaptive scheme consisting of the plant (7),
the control law (15) and the gain adaptation formula
(16) satisfy the following
i) It strictly passifies the system (21) with respect to
signals eu(t) andy
p
(t)in case of zero disturbance w=0.
ii) Achieve a disturbance attenuation level γ, for zero
initial conditions when eu(t) = 0.
In such a case, the controller is given by (15)-(18),
where β =
b
β
α
.
Proof: We consider the radially-unbounded Lya-
punov function candidate
V(e
x
(t),K
e
(t)) =
1
2
e
T
x
(t)Pe
x
(t)+
1
2
tr{(K
e
(t) − F)(K
e
(t) − F)
T
}.
(25)
Note that V(0,F) = 0 and V(e
x
(t),K
e
(t)) > 0
for all {e
x
(t),K
e
(t)} 6= {0,F}. Note also that
V(e
x
(t),K
e
(t)) → ∞ if ke
x
(t)k → ∞ or kK
e
(t)k → ∞.
Using (20), the derivative of (25) along the trajecto-
ries of (21) is given by
˙
V(t) = e
T
x
(t)P(Ae
x
(t) + B
1
w(t) − B
2
K
e
(t)e
y
(t)+
B
2
eu(t)) + tr{
˙
K
e
(t)(K
e
(t) − F)
T
}.
(26)
Using Schur complement argument, we can rewrite
the inequalities of (24) as:


L+C
T
1
C
1
PB
2
−C
T
2
T
T
e
+C
T
1
D
12
Ψ
1
∗ −D− D
T
+ D
T
12
D
12
Ψ
2
∗ ∗ Ψ
3


≤ 0
where
Ψ
1
= PB
1
+C
T
2
ΓD
21
+ 2C
T
1
D
11
Ψ
2
= −T
e
D
21
+ 2D
T
12
D
11
Ψ
3
= −γ
2
I + D
T
11
D
11
+ D
T
21
ΓD
21
Pre and post multiply this inequality by

e
T
x
(t) e
T
u
(t) w
T
(t)

and its transpose respec-
tively, to get
2e
T
x
(t)P(Ae
x
(t) + B
1
w(t) + B
2
eu(t))
−γ
2
w(t)
T
w(t) + z
T
(t)z(t)
+e
T
y
(t)Γe
y
(t) − 2y
T
P
(t)eu(t) + e
T
x
(t)Re
x
(t) ≤ 0
and hence
˙
V(t) ≤ e
T
x
(t)PB
2
K
e
(t)e
y
(t) + y
T
P
(t)eu(t)
+
1
2
(γ
2
w(t)
T
w(t) − z
T
(t)z(t))
−
1
2
e
t
y
(t)Γe
y
(t) −
1
2
e
T
x
(t)Re
x
(t))
+tr{
˙
K
e
(t)(K
e
(t) − F)
T
}
Pre and post multiplying (22) by

e
T
x
(t) −e
T
y
(t)K
T
e
(t)

and its transpose, respec-
tively, the following is obtained:
e
x
(t)
T
PB
2
K
e
(t)e
y
(t) −
1
2
e
x
(t)
T
Re
x
(t) ≤
e
y
(t)
T
K
e
(t)
T
T
e
C
2
e
x
(t) +
1
2
e
y
(t)
T
K
e
(t)
T
K
e
(t)e
y
(t)
Combining the last two inequalities, we find that the
derivative of V(t) satisfies the following:
˙
V(t) ≤ y
T
P
(t)eu(t) +
1
2
(γ
2
w(t)
T
w(t) − z
T
(t)z(t))
+
1
2
e
T
y
(t)(K
e
(t)
T
K
e
(t) −
b
βI)e
y
(t)+
e
T
y
(t)(K
e
(t) − F)
T
T
e
e
y
(t)+
tr{
˙
K
e
(t)(K
e
(t) − F)
T
}
(27)
ROBUST SIMPLE ADAPTIVE H8 MODEL FOLLOWING CONTROL DESIGN BY LMIS
341

Since tr{M
1
M
2
} = tr{M
2
M
1
}, we obtain that:
e
T
y
(t)(K
e
(t)−F)
T
T
e
e
y
(t)=tr{T
e
e
y
(t)e
T
y
(t)(K
e
(t)−F)
T
}.
Therefore, using (16b), one obtains
e
T
y
(t)(K
e
(t) − F)
T
T
e
e
y
(t) + tr{
˙
K
e
(t)(K
e
(t) − F)
T
} =
−tr{φ(t)(K
e
(t) − F)
T
}
which is negativedue to Lemma 2. Moreover, Lemma
1 guarantees that tr(K
T
e
K
e
) ≤ αβ, and using the fact
that
b
β = αβ, hence K
e
(t)
T
K
e
(t) −
b
βI ≤ 0. The deriva-
tive of the Lyapunov function along the closed-loop
trajectories is therefore, for all t ≥ 0, bounded by:
˙
V(t) ≤ y
T
P
(t)eu(t) +
1
2
(γ
2
w(t)
T
w(t) − z
T
(t)z(t)).
(28)
For w(t) = 0, taking the integral of (28) over time
proves strict passivity of the system. For eu(t) = 0 and
zero initial conditions, taking the integral over time
leads to the standard interpretation of γ as a bound on
the H
∞
norm of the system. From the definition of
eu(t) it follows that eu(t) = 0 if
(K
∗
x
(t) − K
x
(t))x
m
(t) + (K
∗
u
(t) − K
u
(t))u
m
(t) = 0.
(29)
Note that model following does not require K
∗
(t) =
K(t); it suffices that the LHS of (29) vanishes. Note
also that K
∗
(t) may not be unique. QED
Remark 3. The parameters γ, α and
b
β = αβ appear
distinctly in LMIs (22-24). The H
∞
performance of
the closed-loop is represented by γ, whereas
b
β− α is
the allowed dynamic range of the SAC gain. Since
a larger gain range (intuitively) can cope with larger
system performance variations, it may be said that, in
principle, we are faced with a Pareto optimal perfor-
mance versus robustness problem. The minimization
of γ traded the maximization of
b
β− α. ( In fact, sepa-
rater minimization of α and maximization of
b
β)
Remark 4. LMI’s (22-24) are affine in the system
matrices, therefore Theorem 1 can be used to derive a
criterion that will guarantee the stability in the case
where the system matrices are not exactly known and
they reside within a given polytope. Denoting
Ω =

A B
1
B
2

(30)
where Ω ∈ Co{Ω
j
, j = 1,...N}, namely,
Ω =
N
∑
j=1
f
j
Ω
j
for some 0 ≤ f
j
≤ 1,
N
∑
j=1
f
j
= 1
(31)
where the vertices of the polytope are described by
Ω
j
=
n
A
( j)
B
( j)
1
B
( j)
2
o
, j = 1,2...,N. (32)
Multiplying (22-24) by f
j
and summing over j =
1,2,...,N, it is readily obtained that the stability and
performance conditions are satisfied over Ω.
5 NUMERICAL EXAMPLES -
MIMO LATERAL CONTROL
FOR A 747 JET TRANSPORT
In this section we present a numerical example to
demonstrate the application of the theory developed
above. Consider a modified version of a 747 air-
craft using the classical control design features in the
Control System Toolbox of MATLAB (Mathworks,
1995). The example is modified to include distur-
bances and deals with bank angle and yaw rate control
(MIMO case) of the airplane.
The example describes the dutch roll mode of a
747 jet transport. A simplified trim model of the air-
craft during cruise flight at MACH = 0.8 and H =
40,000 ft has four states: sideslip angle [rad], bank
angle [rad] , yaw rate [rad/sc], roll rate [rad/sec].
The plant inputs are rudder [rad] and aileron [rad] de-
flections. We assume that the bank angle, the yaw
rate and the roll rate are measured. The plant of (7) is
described by the following matrices:
A=



−0.0558 −0.9968 0.0802 0.0415
0.5980 −0.1150 −0.0318 0
−3.0500 0.3880 −0.4650 0
0 0.0805 1.0 0



B
1
= B
2
=



0.0073 0
0.4750 0.0077
0.1530 0.1430
0 0



,
C
2
=

0 1 0 0
0 0 0.2 10

,D
21
=

0.06 0
0 0.06

,
The weights matrices of the control objective (8) are
chosen as:
C
1
=

1 0
0 1

,D
12
=

0.001 0
0 0.001

.
Using Matlab’s LMI Toolbox (Gahinet et al., 1995),
we find that the LMI’s (22-24) are feasible for:
D =

0.1 0
0 0.1

,
P =



1.52 −0.05 0.03 0.01
−0.05 1.62 −0.01 0.01
0.03 −0.01 0.05 0.02
0.01 0.01 0.02 0.02



R = 10
6



5.44 −0.01 −0.001 0.002
−0.004 5.450 −0.001 −0.003
−0.001 −0.001 5.456 0.001
0.002 −0.003 0.001 5.445



F = −3.5, G = 12.7,α = 12.8, β = 1.4,γ = 0.6
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
342

The chosen reference system is:
A
m
=



−3 −2.5 0 0
4 0 0 0
0 0 −3 −2.5
0 0 4 0



, B
m
=



2 0
0 0
0 2
0 0



,
C
m
=

0 1.25 0 0
0 0 0 1.25

Our aim is to make the plant outputs track the refer-
ence model outputs for bank angle step response, say,
w(t) =

0.1
1

sin(10t)e
−0.01t
.
Note that aircraft coordinated turns are accompanied
by non-zero yaw rates and the model includes this
feature. The relation between the yaw rate command
(r
com
) and the bank angle command (φ
com
) is:
r
com
=
gtan(φ
com
)
TAS
where g = 9.81 [m/sec
2
] is a the earth’s gravitational
and TAS = 235 [m/sec] is a true air speed of the air-
craft. The simulation results are given in Fig 1-2. Fig.
1a describes the yaw rate and yaw rate command, Fig.
1b the bank angle and bank angel command, Fig 2.a
the rudder command and Fig 2.b the aileron com-
mand. Evidently, the yaw rate and the bank angel
successfully tracks their commands by the proposed
control law (15) and the gain adaptation formula (16).
2 4 6 8 10 12 14
−1
−0.5
0
0.5
1
Yaw rate &
Yaw rate command [deg/sec]
Time [sec]
0 5 10 15
−20
−10
0
10
20
Bank angle &
Bank angle command [deg]
Time [sec]
Plant output
Model input
Figure 1: Simulation result: yaw rate and bank an-
gle(command and measured).
6 CONCLUSIONS
In this note, the existing model following theory of
simple adaptive control for continuous-time systems
is generalized for MIMO systems. The results assure
closed-loop stability and best disturbance attenuation
0 5 10 15
−0.8
−0.6
−0.4
−0.2
0
0.2
rudder [deg]
Time [sec]
0 5 10 15
−10
−5
0
5
10
aileron [deg]
Time [sec]
Figure 2: Simulation results: The rudder and the aileron
[deg].
level γ. The conditins are formulated in LMI (rather
then BMI) form, and are shown to be valid also for
systems with polytopic uncertainties.
The design method is simple and the results are
most encouraging. The results are illustrated via a
numerical example from the field of flight control and
encourage further research of the effects of exoge-
nous disturbances and measurement noise for mea-
surements delayed MIMO systems.
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