International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal | www.ijtsrd.com
ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov – Dec 2018
Global Exponential Stabilization for a Class of
Uncertain Nonlinear Control Systems Via Linear Static Control
Yeong-Jeu Sun
Professor, Department of Electrical Engineering
Engineering, I-Shou University, Kaohsiung,
Kaohsiung Taiwan
ABSTRACT
In this paper, the robust stabilization for a class of
uncertain nonlinear systems is investigated. Based on
the Lyapunov-like
like approach with differential
inequalities, a simple linear static control is offered to
realize the global exponential stability of such
uncertain nonlinear systems. Meanwhile, the
guaranteed exponential convergence rate can be
correctly estimated. Finally, some numerical
simulations are given to demonstrate the feasibility
and effectiveness of the obtained results.
Key Words: Global exponential stabilization,
uncertain nonlinear systems, exponential convergence
rate, linear static control
1. INTRODUCTION
Control design with implementation of uncertain
nonlinear dynamical systems is one of the most
challenging areas in systems and control theory. It is
well known that uncertainties and nonlinearities
onlinearities often
appear in various physical systems and aall physical
systems are essentially nonlinear in nature. Nonlinear
control has been active for many years, and many
results concerning with nonlinear control have been
proposed. However,
owever, it is still difficult to implement
nonlinear controllers for practical systems.
Recently, there have several well
well-developed
techniques and methodologies for analyzing uncertain
nonlinear systems, such as back stepping approach
approach,
fuzzy adaptive control
ol approach, feedback
linearization, H-infinity
infinity control approach, sliding
mode control methodology, LMI approach, singular
perturbation method,, Lyapunov approach, Riccati
equation approach, center
enter manifold theorem
theorem, adaptive
sliding mode control, adaptive fuzzy-neural
neural-network,
and others; see, for example, [1-8]] and the references
therein.
In this paper, the stabilizability for a class of uncertain
nonlinear systems will been considered.
considered Based on the
Lyapunoe-like approach with differential inequality, a
linear static control will be established to realize the
global exponential stability of such uncertain systems.
Moreover, the guaranteed exponential convergence
rate can be correctly
orrectly calculated.
calculate Several numerical
simulations will also be provided to illustrate
illustr the use
of the main results.
The layout of the rest of this paper is organized as
follows. The problem formulation, main result, and
controller design are presented in Section 2. In
Section 3,, numerical simulations with circuit
realization are given to
o illustrate the effectiveness of
the developed results. Finally, some conclusions are
drawn in Section 4. In what follows, ℜn denotes the
n-dimensional real space, x denotes the Euclidean
norm of the vector x ∈ ℜn , a denotes the absolute
value of a real number a, and AT denotes the transport
of the matrix A.
2. PROBLEM
ROBLEM FORMULATION AND MAIN
RESULTS
In this paper, we explore the following uncertain
nonlinear systems:
x&1 = ∆a1 x1 + ∆a2 x2 + ∆a3 x4 + ∆a4 x3 x4 , (1a)
x& 2 = ∆a5 x1 + ∆a6 x2 + ∆a7 x3 + ∆a8 x4
+ ∆a9 x1 x3 + ∆a10u1 ,
(1b)
x&3 = ∆a11 x2 + ∆a12 x3 − ∆a9 x1 x2 ,
(1c)
x& 4 = ∆a13 x1 + ∆a14 x2 − ∆a4 x1 x3 + ∆a15u2 , (1d)
[x1 (0)
= [x10
x2 (0) x3 (0) x4 (0)]
T
x20
x30
x40 ] ,
T
(1e)
where x(t ) := [x1 (t ) x2 (t ) x3 (t ) x4 (t )]T ∈ ℜ4 is the state
vector, u (t ) := [u1 (t ) u 2 (t )]T ∈ℜ 2 is the system control,
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec
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Page: 1227
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
2456
[x10
x40 ]
T
is the initial value, and
∆ai , ∀ i ∈ {1,2, L ,15} indicate uncertain parameters of the
system. The hyper-chaotic
chaotic Pan system is a special
case of systems (1) with u (t ) = 0, ∆a2 = −∆a1 = 10 ,
x20
x30
∆a5 = 28 , ∆a8 = −∆a9 = 1
, ∆a12 =
∆ai = 0, ∀ i ∈ {3,4,6,7,10,11,13,15} .
−8
, ∆a14 = −10 , and
3
with δ1 > 0 and δ 2 > 0 . In this case, the guaranteed
exponential convergence rate is given by
 − a1
, δ1 ,
 3
α := min 
− a12

, δ2  .
2

(3)
Proof. Let
V (x(t )) := x12 (t ) + x22 (t ) + x32 (t ) + x42 (t ) .
The global exponential stabilization and the
exponential convergence rate of the system (1) are
defined as follows.
Definition 1.
The uncertain systems (1) are said to be globally
exponentially stable if there exist a control u and
positive number α satisfying
x(t ) ≤ x(0) ⋅ e −αt , ∀ t ≥ 0 .
In this case, the positive number α is called the
exponential convergence rate.
(4)
The time derivative of V (x(t )) along the trajectories of
the closed-loop systemss (1) with (2) and (A1), is
given by
V& (x(t ))
= 2 x1 x&1 + 2 x 2 x& 2 + 2 x3 x& 3 + 2 x 4 x& 4
= 2 x1 (∆a1 x1 + ∆a2 x2 + ∆a3 x4 + ∆a4 x3 x4 )
+ 2 x2 (∆a5 x1 + ∆a6 x2 + ∆a7 x3 + ∆a8 x4
+ ∆a9 x1 x3 − ∆a10 k1 x2 )
+ 2 x3 (∆a11 x2 + ∆a12 x3 − ∆a9 x1 x2 )
+ 2 x4 (∆a13 x1 + ∆a14 x2 − ∆a4 x1 x3 − ∆a15 k 2 x4 )
≤ 2a1 x12 + 2b2 x1 x2 + 2b3 x1 x4 + 2∆a4 x1 x3 x4
+ 2b5 x1 x2 + 2b6 x22 + 2b7 x2 x3 + 2b8 x2 x4
The aim of this paper is to find a simple linear static
control such that the global exponential stabili
stabilization
of uncertain systems (1) can be guaranteed.
Meanwhile,, an estimate of the exponential
convergence rate of such stable systems is also
explored.
the following
Throughout this paper, we make th
assumption:
(A1) There exist constants ai and ai such that
ai ≤ ∆ai ≤ ai , ∀ i ∈ {1,2, L ,15} ,
with
{
}
bi := max ai , ai ,
ai > 0, ∀ i ∈ {10, 15} and ai < 0, ∀ i ∈ {1, 12}.
Now we present the main result for the globally
exponential stabilization of uncertain systems (1) via
Lyapunov-like theorem with the differential and
integral inequalities.
Theorem 1
The uncertain systems (1) with (A1) realize the
globally exponential stabilization under the linear
static control
u = [u1 u2 ] = [− k1 x2
T
− k 2 x4 ]
T
,
(2a)
where
k1 ≥
1
×
a10
 − 3(b2 + b5 )2 − (b7 + b11 )2

+
+ b6 + δ1 + 1,

4
a
2
a
1
12


k2 ≥
2

(b + b )2
1  − 3(b3 + b13 )
+ 8 14 + δ 2 ,

a15 
4a1
4

(2b)
(2c)
+ 2∆a9 x1 x2 x3 − 2a10 k1 x22
+ 2b11 x2 x3 + 2a12 x32 − 2∆a9 x1 x2 x3
+ 2b13 x1 x4 + 2b14 x2 x4 − 2∆a4 x1 x3 x4
− 2a15 k 2 x42
a a a 
≤ 2 1 + 1 + 1  x12 + 2(b2 + b5 ) x1 x2
3 3 3
+ 2(b3 + b13 ) x1 x4 + 2b6 x22 + 2(b7 + b11 ) x2 x3
a 
a
+ 2(b8 + b14 ) x2 x4 + 2 12 + 12  x32
2
2 

 − 3(b2 + b5 )2 − (b7 + b11 )2

− 2
+
+ b6 + δ1 + 1 x22
4a1
2a12


 − 3(b3 + b13 )2 (b8 + b14 )2

− 2
+
+ δ 2  x42
4a1
4


2
− a
− 3(b2 + b5 ) 2 
= −2  1 x12 − (b2 + b5 ) x1 x2 +
x2 
4a1

 3
2
− a
− 3(b3 + b13 ) 2 
− 2 1 x12 − (b3 + b13 ) x1 x4 +
x4 
4a1
 3

2
− a
− (b7 + b11 ) 2 
− 2 12 x32 − (b7 + b11 ) x2 x3 +
x2 
2a12
 2


(b + b )2 
− 2 x22 − (b8 + b14 ) x2 x4 + 8 14 x22 
4


−
a
−
a




− 2 1  x12 − 2 12  x32 − 2δ1 x22 − 2δ 2 x42
 3 
 2 
−a 
−a 
≤ −2 1  x12 − 2 12  x32 − 2δ1 x22 − 2δ 2 x42
 3 
 2 
2
2
≤ −2 αx1 + αx2 + αx32 + αx42
= −2αV , ∀ t ≥ 0.
(
)
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
2456
Hence, one has
e
2αt
⋅ V& + e
2αt
[
]
d 2αt
⋅ 2α V =
e ⋅V
dt
≤ 0, ∀ t ≥ 0.
It results
∫ dτ [e
t
d
2ατ
]
⋅ V ( x(t )) dτ
0
= e 2αt ⋅ V (x(t )) − V (x(0 ))
t
≤ ∫ 0 dτ = 0, ∀ t ≥ 0.
(5)
0
4. CONCLUSION
In this paper, the robust stabilization for a class of
uncertain nonlinear systems has been explored.
explored Based
on the Lyapunov-like approach with differential
inequalities, a simple linear static control has been
presented to realize the global exponential stability of
such uncertain nonlinear systems. Meanwhile, the
guaranteed exponential convergence rate can be
correctly calculated. Finally, some numerical
simulations have been offered to demonstrate the
feasibility and effectiveness
ffectiveness of the obtained results.
From (4) and (5), it follows
x(t ) = V ( x(t )) ≤ e −2αtV ( x(0))
2
= e − 2αt x(0) , ∀ t ≥ 0.
2
As a consequence, we conclude that
x(t ) ≤ e −αt x(0) , ∀ t ≥ 0.
This completes the proof.
□
3. NUMERICAL SIMULATIONS
Consider the uncertain systems (1) with
−11 ≤ ∆a1 ≤ −10, 9 ≤ ∆a2 ≤ 10,
(6a)
27 ≤ ∆a5 ≤ 28, 0 ≤ ∆a8 ≤ 1,
(6b)
−8
,
3
−10 ≤ ∆a14 ≤ −9, 1 ≤ ∆a10 , ∆a15 ≤ 2,
− 1 ≤ ∆a9 ≤ 1, − 3 ≤ ∆a12 ≤
(6c)
∆ai = 0, ∀ i ∈ {3,4,6,7,11,13} .
(6d)
(6e)
By comparing (A1) and (6) with selecting the
parameters of
(a , a
)
−8 

, a12 , a15 =  − 10, 1,
, 1,
3 

(b2 , b5 , b7 , b11 , b6 ) = (10,28,0,0,0),
(b3 , b13 , b8 , b14 ) = (0,0,1,10),
1
10
(A1) is evidently satisfied. With the choice δ1 = δ 2 = 1
in (2), it can be obtained that
T
T
u = [u1 u2 ] = [− 111x2 − 32 x4 ] .
(7)
As a consequence, by Theorem 1, we conclude that
the uncertain systems (1) with (6)) are globally
exponentially stable under the linear static control of
(7). Besides, from (3), the guaranteed exponential
convergence rate is given by α = 1 . The typical sstate
trajectories of the uncontrolled system and the
feedback-controlled
controlled system are depicted in Fig
Figure 1
and Figure 2, respectively. In addition, the control
signals and the electronic circuit to realize such a
control law are depicted in Figure 3 and Figure 4,
respectively.
ACKNOWLEDGEMENT
The author thanks the Ministry of Science and
Technology of Republic of China for supporting this
work under grants MOST 106-2221-E-214-007,
106
MOST 106-2813-C-214-025-E
E, and MOST 107-2221E-214-030. Furthermore,, the author is grateful to
Chair Professor Jer-Guang
Guang Hsieh for the useful
comments.
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-500
-600
0
0.5
1
1.5
2
2.5
t (sec)
3
3.5
4
4.5
5
Figure 3: Control signals of u1 (t ) and u2 (t ) .
u2
u1
x
The
he uncertain systems (1) 4
x2
with (6)
100
x1: the deep blue curve
x1(t); x2(t); x3(t); x4(t)
x2: the green curve
x3: the red curve
x4: the light blue curve
R2
R1
50
0
R4
-50
0
5
10
15
20
t (sec)
25
30
35
40
R3
Figure 1: Typical state trajectories of the system (1)
with (6) and u = 0.
5
Figure 4: The diagram of implementation of
controller, where R1 = R3 = 1 kΩ, R 2 = 111 kΩ, and
x1: the deep blue curve
4
x2: the green curve
x1(t); x2(t); x3(t); x4(t)
3
x3: the red curve
2
R 4 = 32 kΩ.
x4: the light blue curve
1
0
-1
-2
-3
-4
-5
0
0.5
1
1.5
2
2.5
t (sec)
3
3.5
4
4.5
5
Figure 2: Typical state trajectories off the feedback
feedbackcontrolled system of (1) with (6) and (7)
(7).
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