The trajectory tracking problem for repetitive nonlinear
systems has been important subject research in the control
theory field. In fact, Iterative Learning Control (ILC) is the
best technique to deal with this problem [1], [2], [3], [4],
[5]. The basic idea of this approach is to use the information
from the previous iterations to generate a better controller
through the iterations. In general, there are two approaches to
studying the asymptotic stability of nonlinear systems based
on the ILC scheme. The first is the theory of Lyapunov,
which based on the construction of a scalar energy-like
function and showing that to be monotonically decreasing
under the control design scheme [6], [7], [8]. The second
approach is the λ-norm. This method has been defined and
used in the first publication of the ILC technique [9], and
many works employ this method, see for example [10],
[11], [12], [13]. In our work, we use λ-norm to prove the
asymptotic stability of the closed loop system.
In addition, in accordance with the learning action type
in the ILC schemes, we have mainly four types of ILC: P-
type ILC [14], [15], D-type ILC [16], PD-type ILC [17]
and PID-type ILC [18]. Indeed, numerical methods can be
applied to obtain the error derivative in the implementation
of a derivative action. However, the numerical error differ-
entiation might be a source of several noises if the output
is contaminated with measurement noise. In fact, due to
this action (derivative), the measurement noise amplification
accumulates and increases through iterations for repetitive
systems. Thus, it is preferred to use only proportional action
in the controller scheme. In our paper, we use P-type ILC.
On the other hand, and according to the information used in
the controller, the ILC is classified into two types: one order
and high order. For the one order type, the information used
in the controller comes only from one iteration. For the high
order type, the information used in the controller comes from
several iterations. In the literature, some studies have been
done to compare the two types [19], [20]. It is shown that the
performances of the systems are better with the high order
type. In our work, we use the high order type, in which
the information obtained from previous and current trials are
used to improve the control input for next trial to achieve a
fast convergence rate.
Furthermore, the most systems that existed in the industry
can be affected by nonsmooth and non-affine input uncertain-
ties such as the saturation. The presence of saturation may
limit the performance of the system and even worse may
lead the system unstable. However, owing to the difficulty
of guaranteeing the stability of the closed-loop system in
the presence of input saturation, there are little studies based
on the ILC method to deal with this problem. For example,
an iterative learning control for single input single output
systems with input saturation is presented in [21], [22], [23].
There are other works on the ILC that have studied the
input saturation for linear time-invariant system [24] and
for differential and discrete linear systems [25]. Differently
to these studies, we proposed a high order P-type ILC for
multi-input multi-output nonlinear systems in the presence
of unknown input saturation.
In this paper, we present a simple high order P-type
ILC scheme to solve the trajectory tracking problem of
multi input multi output (MIMO) non-linear systems with
unknown input saturation. To achieve a fast convergence rate,
the information obtained from previous and current trials
are used to improve the control input for next trial. The
asymptotic stability of the closed system under unknown
input saturation over the whole finite time is guaranteed by
using the λ-norm method. Finally, an illustrative example is
presented to demonstrate the effectiveness of the proposed
controller. The rest of this paper is organized as follows: the
problem formulation of the nonlinear systems with unknown
input saturation is presented in Section II. The high order
A High Order P-type Iterative Learning Control Scheme for Unknown
Multi Input Multi Output Nonlinear Systems with Unknown Input
Saturation
1TAREK BENSIDHOUM, 2FARAH BOUAKRIF, 3MICHEL ZASADZINSKI
1University of Bordj Bou Areridj, ALGERIA
2University of Jijel, ALGERIA
3CRAN, University of Lorraine, FRANCE
Abstract: In this paper, a new high order P-type iterative learning control scheme is presented to solve the trajectory tracking problem
of Multi Input Multi Output (MIMO) nonlinear systems with unknown input saturation. It is well known that most systems can be
affected by input uncertainties such as saturation. This undesirable input has the potential to destabilize the system. Thus, it is important
to mitigate the effect of saturation. In this paper, we take this problem into account. In addition, the controller scheme is very simple, in
which the control input in each trial is adjusted by using the tracking error signals obtained from previous and current trials. As the
iterations continue, the control system eventually learns the task and follows the desired trajectory with little or no errors. The asymptotic
stability of the closed loop system under unknown input saturation is guaranteed over the whole finite time by using the λ-norm method.
Finally, to illustrate the effectiveness of the proposed method, simulation results are presented.
Keywords: Nonlinear Systems, Control, Simulation, Nonlinear Control
Received: June 17, 2022. Revised: May 16, 2023. Accepted: June 15, 2023. Published: July 10, 2023.
1. Introduction
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.4
Tarek Bensidhoum,
Farah Bouakrif, Michel Zasadzinski
E-ISSN: 2769-2477
27
Volume 3, 2023
P-type ILC scheme for the system and its convergence
analysis is proposed in Section III. An illustrative example
is presented in Section IV and conclusions are discussed in
Section V.
Now, we consider the unknown MIMO nonlinear system
with unknown input saturation:
˙xk=fxk(t), t+Btsat(uk(t), u∗)(1)
where: t∈[0, T ], k denotes the iteration index. xk∈IRnis
the state of the system.f(xk)∈IRnis an unknown function,
B(t)∈IRn×nis an unknown input matrix. sat(uk(t), u∗)is
the vector-valued of the saturation function, which can be
defined as follows
sat(uk(t), u∗) =
uk(t)|uk(t)|6u∗
sgn(uk(t))u∗else (2)
Lemma 1: [26] For saturation function w= sat(u, m) + d,
we can obtain
sat(w, m)−w)T(sat(w, m)−w)6dTd(3)
where w, u, m and d∈IRm
Lemma 2: [27] Let x(t) = [x1(t), x2(t),...,xn(t)]T∈IRn
be defined for t∈[0, T ], then we have
Zt
0
kx(τ)kdτe−λt 61
λkx(t)kλ, λ > 0.(4)
Lemma 3: [Gronwall-Bellman] [26] Suppose that f(t)and
g(t)>0are real and locally integrable scalar functions, Lis
a constant. If f(t)satisfies
f(t)6L+Zt
0
g(τ)f(τ)dτ, t ∈[a, b](5)
then, on the same interval, f(t)satisfies
f(t)6Lexp Zt
0
g(τ)dτ.(6)
The following assumptions for system (1) are made.
Assumption 1: The function f(xk(t), t)satisfies the Lips-
chitz condition in x for t∈[0, T ]that means
kf(x1(t), t)−f(x2(t), t)k6αkx1(t)−x2(t)k(7)
where αis the Lipschitz constant.
Assumption 2: The identical initialisation condition is sat-
isfied, i.e., xk(0) = xd(0).
Our objective in this work is to find a sequence of updating
control along with iteration such that, the real state trajectory
xk(t)follows exactly the desired trajectory xd(t)when the
number of iteration ktends to infinity.
The high order P-type ILC scheme at the (k+ 1)th iteration
is developed as follows
uk+1(t) = sat(uk(t), u∗) + P1ek(t) + P2ek+1 (t)(8)
where P1and P2are positive gains, ek(t) = xd(t)−xk(t)and
ek+1(t) = xd(t)−xk+1 (t).
Theorem 1: Applying the controller law (8) to the MIMO
nonlinear systems (1). Under Assumptions 1 and 2, we get,
lim
k→∞
xk(t) = xd(t),∀t∈[0, T ].
Proof: From (1), the state vector at the (k+1)th iteration
can be written as
xk+1(t) = xk+1 (0)+Zt
0f(xk+1)+B(t) sat(uk+1 , u∗)dτ (9)
Applying (8) and (9), we obtain
xk+1(t) = xk(0) + Zt
0f(xk+1) + B(t)
sat(sat(uk, u∗) + P1ek+P2ek+1, u∗)dτ (10)
Adding and subtracting f(xk) + B(t)(sat(uk, u∗) + P1ek+
P2ek+1), we get
xk+1(t) = xk(0) + Zt
0
(f(xk+1)−f(xk))dτ +Zt
0f(xk)
+B(t)(sat(uk, u∗) + P1ek+P2ek+1)dτ
+Zt
0
B(t)sat(sat(uk, u∗) + P1ek+P2ek+1, u∗)
−(sat(uk, u∗) + P1ek+P2ek+1)dτ (11)
Using Lemma 1, (11) is transformed into
xk+1(t)6xk(0) + Zt
0f(xk+1)−f(xk)dτ
+Zt
0f(xk) + B(t) sat(uk, u∗)dτ
+Zt
0P1ek+P2ek+1dτ
+Zt
0
µP1kekk+P2kek+1kdτ (12)
where µ= max(kB(t)k).
From (1) and using Assumption 2, we have
xk+1(t)6xk(t) + Zt
0f(xk+1)−f(xk)dτ
+Zt
0P1ek+P2ek+1dτ
+Zt
0
µP1kekk+P2kek+1kdτ (13)
Taking the norm of both sides of (13) and using the
Assumption 1, we obtain
kxk+1 −xkk6αZt
0
kxk+1 −xkkdτ
+Zt
0P1kekk+P2kek+1kdτ
+Zt
0
µP1kekk+P2kek+1kdτ (14)
Using Lemma 3, (14) can be rewritten as
kxk+1 −xkk6Zt
0P1+P1µkekk
+P2+P2µkek+1kdτ exp(α)(15)
Multiplying the both side of (15) by e−λt and according
to Lemma 2, we find
kxk+1 −xkkλ6 P1+P1µ
λkeklkλ
+P2+P2µ
λkek+1kλ!expα
λ(16)
Knowing that ek+1 −ek=xk−xk+1 . (13) can be written
2. Problem Formulation
3. Main Results
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.4
Tarek Bensidhoum,
Farah Bouakrif, Michel Zasadzinski
E-ISSN: 2769-2477
28
Volume 3, 2023
as:
−ek+1−Zt
0P1ek+P2ek+1dτ −Zt
0
µP1kekk+P2kek+1kdτ
6−ek−Zt
0f(xk+1)−f(xk)dτ (17)
According to Assumption 1 and Lemma 2, (17) becomes
P1+P1µ
λ!kekkλ+ 1 + P2+P2µ
λ!kek+1kλ
6kekkλ+α
λkxk+1 −xkkλ(18)
Substituting (15) in (18), we get
P2+P2µ
λ+ 1!kek+1kλ6 P1+P1µ
λ−1!kekkλ
+α
λ P1+P1µ
λkekkλ
+P2+P2µ
λkek+1kλ!expα
λ(19)
Equation (19) can be simplified as
kek+1kλ6
1−P1+P1µ
λ
1 + P2+P2µ
λ
kekkλ
+
αexpα
λ
λ1 + P2+P2µ
λ P1+P1µ
λkekkλ
+P2+P2µ
λkek+1kλ!(20)
Choosing λ > 0widely great, we have
kek+1kλ6
1−P1+P1µ
λ
1 + P2+P2µ
λ
kekkλ(21)
This implies
kek+1kλ6ηkekkλ, η < 1(22)
Thus, it is clear that
lim
k→+∞
kek+1(t)kλ= 0,∀t∈[0, T ](23)
This completes the proof.
We consider the nonlinear systems with the following
dynamics [29]:
˙x1k(t)
˙x2k(t)
=
x2k
−J−1
mSsin(x1k)
+
0
−J−1
m
sat(uk(t), u∗)
(24)
where: t∈[0,2],Jm= 14 kg m2,S= 6 kg m and g=
9.8m/s2. Let the desired trajectory be xd(t) = sin(2πt)and
y(t) = x2(t).
By applying the high order P-type ILC (3) to the system
(19), we get the simulation results shown in Figures 1, 2
and 3. where the parameters of the controller are chosen as
P1= 100 ,P2= 25 and u∗= 9.
From Figure 1, it is easily seen that the real tracking
trajectory follows exactly the desired trajectory after just 10th
iterations. Figure 2 presents the variations of the maximum
errors bounds through the iteration numbers, it is clear that
the maximum errors decreasing through the iterations and
tends to 0after only 5th iterations. The control signal input
with saturation is presented in Figure 3, we can see that the
high order P-type ILC satisfy the constraints and can be
adapted gradually into the chosen boundary (u∗= 9) along
with the iteration numbers.
Xd
k= 1
k= 2
k= 10
Time [s]
Position of the joint [rad]
Fig. 1. The real and desired trajectories of the 1st,2nd and 10th iterations
Iteration number
Maximum error bounds
Fig. 2. The Sup-norm tracking errors through the iterations
k= 1
k= 2
k= 10
Time [s]
Control input
Fig. 3. Control signals input through the iterations
4. Simulation Results
4.1 First Example
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.4
Tarek Bensidhoum,
Farah Bouakrif, Michel Zasadzinski
E-ISSN: 2769-2477
29
Volume 3, 2023
We consider the MIMO nonlinear systems with input
iteration [30]:
˙x1k(t)
˙x2k(t)
=
f1xk
f2xk
+
1.5 + t0
0 2 −0.3t
sat(uk, u∗)(25)
where
f1(xk) = (2 + 0.3t)x1,k + (2.2 + 0.5t)x1,kx2,k
+0.8−0.1 cos(4t)x2
1,k (26a)
f2(xk) = (2 + 0.3t) + 0.7 + 0.08 sin(3t)x2
2,k
+ (2.2 + 0.5t)x2,k −x1,k
+0.8−0.1 cos(4t)x1,k −x2,k2(26b)
The desired trajectories are given as
x1d(t)
x2d(t)
=
sin 2πt
cos 2πt
(27)
Applying the high order P-type ILC (3) to the system (20),
where the parameters of the controller are chosen as P1= 18,
P2= 9 and u∗= 5. The simulation results are shown in
Figures 4-7.
Figures 4 and 5 present the maximum tracking error
through the iterations for the first and the second outputs,
respectively. It is clear that the maximum errors decreasing
through the iteration, in which, after 30th iteration the maxi-
mum error for the first and the second outputs are less than
4×10−3and 1.3×10−2, respectively.
The control signal for the first and the second inputs are
presented in Figures 6 and 7, respectively. We can see that
the high order P-type ILC satisfy the constraints and can
be adapted gradually into the chosen boundary (u∗= 5) for
both inputs along with the iteration numbers.
Iteration number
Maximum error bounds for the 1st output
Fig. 4. The Sup-norm of the first tracking errors through the iterations
numbers
From these simulation results, it is clear that the proposed
controller works well.
In this paper, the trajectory tracking problem for MIMO
nonlinear systems with unknown input saturation is solved
by introducing a novel hight order P-type ILC. In order to
Iteration number
Maximum error bounds for the 2nd output
Fig. 5. The Sup-norm of the second tracking error through the iterations
k= 2
k= 10
k= 30
Time [s]
1st control input
Fig. 6. The first control input signals through the iterations
achieve a fast convergence rate, the information obtained
from previous and current trials are used to improve the
control input for next trial. The λ-norm method is used to
guarantee the asymptotic stability of the closed-loop system
over the whole finite time. Finally, in order to evaluate the
effectiveness of the method proposed, simulation results are
presented, in which we conclude that the proposed controller
works well.
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Tarek Bensidhoum,
Farah Bouakrif, Michel Zasadzinski
E-ISSN: 2769-2477
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Volume 3, 2023
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International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.4
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Farah Bouakrif, Michel Zasadzinski
E-ISSN: 2769-2477
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