INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 11(2):224-232, April 2016.

A Taboo Search Optimization of the Control Law
of Nonlinear Systems with Bounded Uncertainties

A. Gharbi, M. Benrejeb, P. Borne

Amira Gharbi*, Mohamed Benrejeb, Pierre Borne
LARA, Ecole Nationale d’Ingénieurs de Tunis
Tunisie, BP 37, Le Belvédére 1002 Tunis
and
CRIStAL, Ecole Centrale de Lille
France, Cité scientifique BP 48-59651, Villeneuve d’Ascq Cedex
merkarim@gmail.com, mohamed.benrejeb@enit.rnu.tn, pierre.borne@ec-lille.fr
*Corresponding author: merkarim@gmail.com

Abstract: The aim of this paper is to propose a method to determine among the
eligible controls of a nonlinear system, with bounded perturbations, the one which
minimizes the final error. The approach is based on the implementation of aggregation
techniques using vector norms in order to determine a comparison system used to
calculate an attractor in view of its minimization by implementation of metaheuristics.
Keywords: Attractor, aggregation technique, vector norm, optimization, Taboo
search.

1 Introduction

In the presence of uncertainties in modeling, that increase the complexity of the stability
study [1], it is not always possible to obtain a control law ensuring the stability of the process
with respect to a chosen objective. It is then necessary to estimate the maximum deviation
from this target, an operation which can be performed by determining an attractor [2]- [4]
corresponding to the vicinity of the target for which the local stability cannot be guaranteed,
[5], [7], [6], [8], [9], [10], [11], [12]. In case of uncertain or poorly defined problems, possibly
subject to random perturbations or for which the search for solutions might evolve towards
the combinatorial explosion, the exact methods are very unlikely to provide solutions in an
acceptable period of time. The method presented in this paper corresponds to a law finding,
if we do not obtain the optimal solution of the problem, we obtain at least a good solution in
an acceptable run time. The heuristic methods that can be implemented on a computer are
referred to metaheuristics. They rely on the following basic principle: the search for optimum is
simulating either the behaviour of a biologic system or the evolution of a natural phenomenon,
including an intrinsic optimization mechanism. For this reason, a new optimization branch
has been developed in the past 20 years, inspired by nature. Almost all numerical algorithms
designed as metaheuristics are included into this class of optimization techniques [13]. In general,
all metaheuristics are using a pseudo-random engine to select some parameters or operations
that yield to the estimation of an optimal solution. The procedures to generate pseudo-random
(numerical) sequences of optimization are crucial in metaheuristics design. We have two classes
of metaheuristic approaches: global approaches and local approaches, such as the Taboo search
which is one of the easiest to implement. In this paper, the determination of the attractor, when
the process is submitted to uncertainties, is achieved by using aggregation techniques and the
Borne-Gentina stability criteria, with the use of vector norms and of comparison systems [14], [15].
In the following section 2, we propose the determination of the control law of a nonlinear process
submitted to bounded uncertainties with a view to minimize the effect of these uncertainties.
In section 3 we use the taboo search to realize the optimization. An application is presented in
section 4 to illustrate the proposed method.

Copyright © 2006-2016 by CCC Publications



A Taboo Search Optimization of the Control Law
of Nonlinear Systems with Bounded Uncertainties 225

2 Attractor determination

Let us consider the system (S) whose evolution is described by the following state equation

ẋ = f(x,.) + g(x,.)u + δ(.) (1)

y = h(x) (2)

x is the state vector and y is the output, x ∈ Rn,y ∈ Rm,u ∈ Rl
δ ∈ Rn characterizes the disturbances and/or perturbations acting on the system and u is the
control law:

u = u(x,θ) (3)

where θ ∈ Rν is a vector of the adjustable parameters of the control law. A new representation
of system (S) characterized by (1) and (3) can be defined by

ẋ = A(x,θ, .)x + δ(.) (4)

with
|δ(.)| ≤ δM (5)

A = f(x,.) + g(x,.)u(x,θ) (6)

and a comparison system of this system can be determined using the vector norm p(x) defined
by

p(x) = [|x1| , |x2| , . . . , |xn|]T (7)
By noting M(A(x,θ, .)) an overvaluing matrix of A(x,θ, .) related to the vector norm p(x) it
comes

d

dt
p(x) ≤ M(A(x,θ, .))p(x) + N(.) (8)

Let us denote:
A(.) = {aij(.)} (9)

and M(θ) = {mij(θ)} the matrix such that:{
mii(θ) = maxaii(x,θ, .) ∀i = 1,2, . . .n
mij(θ) = max |aij(x,θ, .)| ∀i ̸= j

(10)

We can define a comparison system by:

z ∈ n/ż(t) = M(θ)z(t) + δM (11)

If M(θ) is the opposite of an M-matrix, it exists an attractor Dθ asymptotically stable such that

Dθ =
{
x ∈ Rn;p(x) ≤−M−1(θ)δM = pM(θ)

}
(12)

3 Taboo search optimization

3.1 Principe of Taboo search

The metaheuristic described in this section belongs to greedy descent local methods. For this
type of methods, the search starts from an admissible solution θi of S. The strategy is then to
focus on a local vicinity V (θi), in order to find another solution θj that can improve the criterion
current performance. The vicinity V (θi) corresponds to the set of all accessible solutions after
applying a single admissible movement, displacement or transition from θi . Usually, this set is
a hyper-cube or a hyper-sphere including the current solution θi.



226 A. Gharbi, M. Benrejeb, P. Borne

3.2 Taboo search method

Based on the principle of local search for minimizing a criterion, by this method, there is the
possibility to jump out from the capturing vicinity and to explore a different zone of the research
area. Here after, the term of movement stands for any modification allowing the search to focus
on vicinity in the neighborhood of the current vicinity. As usual, the search starts from some
initial point (solution), θi in the vicinity V (θi) but it is permitted to relocate the exploitation
around another point (solution) θj ∈ V (θi), even if this choice degrades the criterion to optimize.
This actually is a movement towards another zone of interest. However, in order to avoid
infinite search loops, once a solution is focused on, it will never be focused on again in the future
iterations. Thus, the NT last focused solutions belonging to a Taboo list Tki become untouchable,
"taboo" [16], [17]. Starting from the solution θi, a set of possible movements, say Mk,j, can be

built, during the k- th iteration. Let δθ ∈ Mk,j be such a movement. By convention,θi
δθ→θj

stands for the transition from solution θi to a new point θj as result of movement δθ. Then

Vk(θi) =
{
θj ∈ V (θi)/∃ δθ ∈ Mkj,θi

δθ→θj & θj /∈ Tki
}

(13)

The new solution which is the best non taboo one is added to the last taboo list and the oldest
one is removed from it. The chosen criterion is for this problem the minimisation of a scalar norm
of pM(θ) The optimization of the control law consists to determine the value of θ minimizing a
scalar norm of pM . In the following we use the Euclidian norm ∥pM∥. The optimisation algorithm
corresponds in this paper to the taboo search with NT number of elements of the taboo list and
NS the maximum number of iterations without improvement of the solution to stop the research.

4 Application to a second order system

Let us consider the nonlinear system of second order with uncertainties such that

ẋ = A(x,t)x + B(x)u(x,θ) + δ(.) (14)

y = h(x) (15)

with
u(θ,x) = −(θ1y + θ2x2) (16)

and
h(x) = x1 + (1−e−x

2
1)x2 (17)

with x(t) ∈ R2,B(.) ∈ R2,A(.) a 2×2 matrix and θ ∈ R2 such that

A(x,t) =

[
−2 + cost + cosx1 4−e−x

2
1(1 + sinx1)

4 + cosx2 −8 + sinx1 + e−x
2
1

]
(18)

B(x) =

[
3 + 0.5 cosx1

2

]
(19)

we can write (14) as
ẋ = A(x,t,θ)x + δ(.) (20)

with
A(x,t,θ) = A(x,t)−B(x)[θ1,θ1((1−e−x

2
1)) + θ2] (21)



A Taboo Search Optimization of the Control Law
of Nonlinear Systems with Bounded Uncertainties 227

it comes

A(x,t,θ) =

[
a11 a12

a21 a22

]
(22)

with

a11 = −2 + costcosx1 −θ1(3 + 0.5 cosx1)
a12 = 4−ex

2
1(1 + sinx1)− (3 + 0.5 cosx1)(θ1(1−ex

2
1) + θ2)

a21 = 4 + cosx2 −2θ1
a22 = −8 + ex

2
1 + sinx1 −2[θ1(1−ex

2
1) + θ2]

(23)

4.1 Determination of a comparison system

For the vector norm p(x) = [|x1| , |x2|]T , we obtain the overvaluation defined by

d

dt
p(x) ≤ M(A(x,θ, .))p(x) + N(.) (24)

z ∈ Rn/ż(t) = M(.)z(t) + N(.) (25)

with

M(A(x(t))) =

[
a11 |a12|
|a21| a21

]
(26)

and
|N(.)| ≤ δM (27)

In our example δ(.) is assumed to be by bounded by

δ1 =

[
−0.2
0.3

]
≤ δ(.) ≤ δ2 =

[
0.1

0.5

]
(28)

then

δM =

[
0.2

0.5

]
(29)

and by overvaluation, for the process without feedback, for θ = (θ1,θ2) = (0,0) we obtain the
linear comparison system ż = Mz + N

ż =

[
0 2

5 −6

]
z +

[
0.2

0.5

]
(30)

after application of stability conditions we have

det(M) < 0 (31)

it appears that M is not stable and so is not the opposite of an M-matrix which needs the
determination of a suitable feedback optimized in order to limit the influence of the uncertainties.



228 A. Gharbi, M. Benrejeb, P. Borne

4.2 Attractor optimization with taboo search

For this taboo search we choose NT = NS = 4.
Starting from the solution θ1 = 2 and θ2 = 0 a set possible movements, can be built, during
the k-th iteration. Let δθl ∈ Mk,j be such a movement, with |δθ1| = 0.2 and |δθ2| = 0.1 . By
convention, θi1

δθ1→θj1 , θi2
δθ2→θj2 stands for the transition from solution θli to a new point θlj

with l = {1,2} as result of movement δθl . Then for θ1 = 2 and θ2 = 0 the overvaluing system
for the vector norm p(x) = [|x1| , |x2|]T is defined by (22) with

M(x,t,2,0) =


 −8 + cost

∣∣∣∣∣ −2−e
−x21(−5 + sinx1 − cosx1)

−cosx1

∣∣∣∣∣
|cosx2| −12 + sinx1 + 5e−x

2
1


 (32)

and

N =

[
0.2

0.5

]
(33)

then the linear comparison system is the following

ż =

[
−7 3
1 −7

]
z +

[
0.2

0.5

]
(34)

The stability conditions for matrix M can be written{
−7 < 0(
−12

)
det(M) > 0

(35)

as M is the opposite of M-matrix, we have

p(x) ≤−M−1N =

[
0.0630

0.0804

]
= pM(2,0) (36)

The strategy is then to focus on a local vicinity V (θi) in order to find the best non taboo solution
θi the chosen criterion being the Euclidian norm of pM(θ).
For this, eight solutions ⋆ will be tested starting from θ = (2,0)

θ = (θ1,θ2 + δθ2),θ = (θ1,θ2 − δθ2),θ = (θ1 + δθ1,θ2),θ = (θ1 − δθ1,θ2),θ = (θ1 + δθ1,θ2 + δθ2),
θ = (θ1 + δθ1,θ2 − δθ2),θ = (θ1 − δθ1,θ2 + δθ2),θ = (θ1 − δθ1,θ2 − δθ2)

 

2θ  

1θ  

(2,0)
 

Figure 1: The vicinity of θ = (2,0) solution

for



A Taboo Search Optimization of the Control Law
of Nonlinear Systems with Bounded Uncertainties 229

θ = (2,0.1) ⇒ p(x) ≤ pM(2,0.1) =

[
0.0579

0.0775

]
,

θ = (2,−0.1) ⇒ p(x) ≤ pM(2,−0.1) =

[
0.0668

0.0787

]
,

θ = (2.2,0) ⇒ p(x) ≤ pM(2.2,0) =

[
0.0572

0.0763

]
,

θ = (1.8,0) ⇒ p(x) ≤ pM(1.8,0) =

[
0.0727

0.0860

]
,

θ = (2.2,0.1) ⇒ p(x) ≤ pM(2.2,0.1) =

[
0.0528

0.0738

]
,

θ = (2.2,−0.1) ⇒ p(x) ≤ pM(2.2,−0.1) =

[
0.0621

0.0790

]
,

θ = (1.8,0.1) ⇒ p(x) ≤ pM(1.8,0.1) =

[
0.0664

0.0824

]
,

θ = (1.8,−0.1) ⇒ p(x) ≤ pM(1.8,−0.1) =

[
0.0796

0.0899

]
,

The best non taboo solution minimizing ∥p(x)∥ : pM(2.2,0.1) is obtained for θ = (2.2,0.1), and
the solution for θ = (2,0) becomes "taboo".
Now the strategy is then to focus on a local vicinity of this solution in order to find the best one
which does not belong to the taboo list. So, we test other solutions that are neighbouring the
current one’s
θ = (2.2,0),θ = (2.2,0.2),θ = (2,0.1),θ = (2,0.2),

θ = (2.4,0),θ = (2.4,0.1),θ = (2.4,0.2).

 

2θ  

1θ  

(2,0) : solution taboo
 

(2.2,0.1)

θ

Figure 2: The vicinity of θ = (2.2,0.1) solution

θ = (2.4,0) ⇒ p(x) ≤ pM(2.4,0) =

[
0.0523

0.0729

]
, θ = (2.4,0.1) ⇒ p(x) ≤ pM(2.4,0.1) =[

0.0528

0.0727

]
, θ = (2.4,0.2) ⇒ p(x) ≤ pM(2.4,0.2) =

[
0.0540

0.0690

]
, θ = (2,0.2) ⇒ p(x) ≤

pM(2,0.2) =

[
0.0531

0.0747

]
,

θ = (2.2,0.2) ⇒ p(x) ≤ pM(2.2,0.2) =

[
0.0536

0.0719

]
,

The best non taboo solution minimizing ∥p(x)∥ :
[
0.0540 0.0690

]T
is obtained for θ =



230 A. Gharbi, M. Benrejeb, P. Borne

(2.4,0.2) then the solution θ = (2.2,0.1) becomes "taboo".
Now we continue the iteration starting from this new solution θ = (2.2,0.2),θ = (2.2,0.3),
θ = (2.4,0.1),θ = (2.4,0.3),θ = (2.6,0.1),θ = (2.6,0.2),θ = (2.6,0.3).

The best non taboo solution minimizing ∥p(x)∥ :
[
0.0558 0.0673

]T
is obtained for θ =

(2.4,0.3), then the solution θ = (2.4,0.2) becomes "taboo". Now we will test the solutions
in the neighbourhood of θ = (2.4,0.3)
The best non taboo solution minimizing ∥p(x)∥ :

[
0.0575 0.0656

]T
is obtained for θ =

(2.4,0.4) then the solution θ = (2.4,0.3) becomes "taboo". Now we will test the vicinity of
this solution
The best non taboo solution minimizing ∥p(x)∥ :

[
0.059 0.064

]T
is obtained for θ = (2.4,0.5),

then the solution becomes "taboo".

 

2θ  

1θ  

(2,0) : solution taboo
 

(2.2,0.1) : solution taboo

θ

(2.4,0.2) : solution taboo

 

(2.4,0.3) : solution taboo
 

(2.4,0.4) : solution taboo
 

(2.4,0.5)
 

Figure 3: The vicinity of θ = (2.4,0.5) solution

At the next iteration the best non taboo solution minimizing ∥p(x)∥ :
[
0.0605 0.0625

]T
is

obtained for θ = (2.4,0.6). For the two following iterations the best non-taboo solutions corre-
spond to pM(2.4,0.7) and pM(2.4,0.8), but ∥pM(2.4,0.4)∥ = ∥pM(2.4,0.5)∥ = ∥pM(2.4,0.6)∥ =
∥pM(2.4,0.7)∥ = ∥pM(2.4,0.8)∥ = 0.870, so as we have had 4 iterations without improvement
we can stop the research. The control law defined by θ = (2.4,0.4), corresponds to the best
solution. Hence the evolution of the state vector, and its evolution of the state vector in the
attractor defined in figure 4.

0.022 0.024 0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04
0.0632

0.0633

0.0634

0.0635

0.0636

0.0637

0.0638

0.0639

0.064

x
2

x1

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0. 1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x
2

x1

Figure 4: Evolution of the state vector in the attractor

5 Conclusion

The approach proposed here consists, having defined the attractor of the process for a control
law depending of parameters to minimize the size of this attractor by implementation of a
metaheuristic to determine the optimal values of these parameters. The method presented in
this paper is applied, with success, for a second order nonlinear complex system using the concept



A Taboo Search Optimization of the Control Law
of Nonlinear Systems with Bounded Uncertainties 231

of vector norm for the determination of the comparison system. The minimization of the norm
of the vector defining the limits of the attractor is realized by using a taboo search method.

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