Global Exponential Stabilization for a Class of Uncertain Nonlinear Control Systems Via Linear...

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Global Exponential Stabilization Uncertain Nonlinear Control Systems

Professor, Department of Electrical Engineering

ABSTRACT In this paper, the robust stabilization for a class of uncertain nonlinear systems is investigated. Based on the Lyapunov-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 nonlinear dynamical systems is one of the most challenging areas in systems and control theory. well known that uncertainties and nonlinearitiesappear in various physical systems and asystems are essentially nonlinear in nature.control has been active for many years, and many results concerning with nonlinear control have been proposed. However, it is still difficult to implement nonlinear controllers for practical systems. Recently, there have several welltechniques and methodologies for analyzing nonlinear systems, such as back stepping approachfuzzy adaptive control approach, linearization, H-infinity control approach, mode control methodology, LMI approach, perturbation method, Lyapunov approach, equation approach, center manifold theoremsliding mode control, adaptive fuzzy-neuraland others; see, for example, [1-8] and the references therein.

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ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov

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Global Exponential Stabilization for a Class Uncertain Nonlinear Control Systems Via Linear Static Control

Yeong-Jeu Sun Electrical Engineering, I-Shou University, Kaohsiung

stabilization for a class of uncertain nonlinear systems is investigated. Based on

like approach with differential inequalities, a simple linear static control is offered to realize the global exponential stability of such

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.

Global exponential stabilization, nonlinear systems, exponential convergence

of uncertain nonlinear dynamical systems is one of the most challenging areas in systems and control theory. It is

onlinearities often and all physical

in nature. Nonlinear control has been active for many years, and many results concerning with nonlinear control have been

owever, it is still difficult to implement nonlinear controllers for practical systems.

several well-developed for analyzing uncertain

stepping approach, ol approach, feedback

infinity control approach, sliding methodology, LMI approach, singular

, Lyapunov approach, Riccati enter manifold theorem, adaptive

neural-network, ] and the references

In this paper, the stabilizability for a class of uncertain nonlinear systems will been consideredLyapunoe-like approach with differential inequality, a linear static control will be establishedglobal exponential stability of such Moreover, the guaranteed exponential convergence rate can be correctly calculatesimulations will also be provided to illustrof 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. Section 3, numerical simulations realization are given to illustrate the effectiveness of the developed results. Finally,drawn in Section 4. In what follows, n-dimensional real space, x

norm of the vector nx ℜ∈ ,

value of a real number a, and of the matrix A. 2. PROBLEM FORMULATION AND MAIN

RESULTS In this paper, we explore the following nonlinear systems:

4344322111 xxaxaxaxax ∆+∆+∆+∆=&

,110319

483726152

uaxxa

xaxaxaxax

∆+∆+∆+∆+∆+∆=&

,2193122113 xxaxaxax ∆−∆+∆=&

153142141134 axxaxaxax ∆+∆−∆+∆=&

( ) ( ) ( ) ( )[ ][ ] ,

0000

40302010

4321

T

T

xxxx

xxxx

=

where ( ) ( ) ( ) ( )[ 321:= txtxtxtx

vector, ( ) ( ) ( )[ ] 221: ℜ∈= Ttututu

Research and Development (IJTSRD)

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1 | Nov – Dec 2018

Dec 2018 Page: 1227

Class of Linear Static Control

Kaohsiung, Taiwan

the stabilizability for a class of uncertain nonlinear systems will been considered. Based on the

approach with differential inequality, a established to realize the

of such uncertain systems. the guaranteed exponential convergence

orrectly calculated. Several numerical simulations will also be provided to illustrate the use

The layout of the rest of this paper is organized as The problem formulation, main result, and

controller design are presented in Section 2. In , numerical simulations with circuit

o illustrate the effectiveness of Finally, some conclusions are

In what follows, nℜ denotes the denotes the Euclidean

a denotes the absolute TA denotes the transport

ROBLEM FORMULATION AND MAIN

the following uncertain

,4 (1a)

(1b)

(1c) ,215u (1d)

(1e)

( )] 44 ℜ∈Ttx is the state

is the system control,

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


[ ]Txxxx 40302010 is the initial value, 15,,2,1, L∈∀∆ iai indicate uncertain parameters of the

system. The hyper-chaotic Pan systemcase of systems (1) with ( ) ,0=tu

285 =∆a , 198 =∆−=∆ aa , 3

812

−=∆a , 14 =∆a

15,13,11,10,7,6,4,3,0 ∈∀=∆ iai . The global exponential stabilizationexponential convergence rate of the system (1) are defined as follows. Definition 1. The uncertain systems (1) are said to be exponentially stable if there exist a control positive number α satisfying

( ) ( ) 0,0 ≥∀⋅≤ − textx tα .

In this case, the positive number α exponential convergence rate. The aim of this paper is to find a simple linear control such that the global exponential stabiliof uncertain systems (1) can be guaranteed. Meanwhile, an estimate of the exponential convergence rate of such stable systems is also explored. Throughout this paper, we make thassumption: (A1) There exist constants ia and

15,,2,1, L∈∀≤∆≤ iaaa iii , with b

15,10,0 ∈∀> iai and .12,1,0 ∈∀< iai

Now we present the main result for exponential stabilization of uncertain systems (1) via Lyapunov-like theorem with the differential and integral inequalities. Theorem 1 The uncertain systems (1) with (A1) globally exponential stabilization under the static control

[ ] [ ]TT xkxkuuu 422121 −−== , (2a) where

( ) ( ),1

24

3

1

1612

2117

1

252

101

++++−++−

×≥

δba

bb

a

bb

ak

(2b)

( ) ( ),

44

312

2148

1

2133

152

++++−≥ δbb

a

bb

ak (2c)



the initial value, and parameters of the

chaotic Pan system is a special 1012 =∆−=∆ aa ,

10−= , and

exponential stabilization and the rate of the system (1) are

said to be globally exponentially stable if there exist a control u and

is called the

simple linear static such that the global exponential stabilization

(1) can be guaranteed. , an estimate of the exponential

rate of such stable systems is also

the following

and ia such that

iii aab ,max:= ,

Now we present the main result for the globally uncertain systems (1) via

the differential and

with (A1) realize the under the linear

with 01 >δ and 02 >δ . In this exponential convergence rate is

−−= 2

121

1 ,2

,,3

min: δδα aa .

Proof. Let ( )( ) ( ) ( ) ( ) ( )txtxtxtxtxV 2

423

22

21: +++= .

The time derivative of ( )( )txV the closed-loop systems (1) given by

( )( )44332211 2222 xxxxxxxx

txV

&&&&

&

+++=

((

)( )( 3142141134

2193122113

2110319

83726152

344322111

2

2

2

2

xxaxaxax

xxaxaxax

xkaxxa

xaxaxaxax

xaxaxaxax

∆−∆−∆+∆+∆−∆+∆+

∆−∆+∆+∆+∆+∆+

∆+∆+∆+∆=

24215

1442144113

321923123211

221103219

327226215

413212211

2

222

222

22

2222

2222

xka

xxaxxbxxb

xxxaxaxxb

xkaxxxa

xxbxbxxb

xxbxxbxa

−

∆−++

∆−++

−∆+

++++

∆+++≤

( )

( ) (

( )

( ) ( )

( ) ( )2

2148

1

2133

612

2117

1

252

23

121242148

722641133

15221

111

44

32

24

32

2222

222

2333

2

bb

a

bb

ba

bb

a

bb

xaa

xxbb

bbxbxxbb

xxbbxaaa

++++−−

+

+−+

+−−

++++

+++++

++

++≤

δ

( ) (

( ) (

( ) (

( ) ( )

221

23

1221

1

2148

4214822

732117

23

12

341133

21

1

22152

21

1

22

23

2

42

222

4

3

32

4

3

32

xxa

xa

bbxxbbx

bxxbbx

a

bxxbbx

a

bxxbbx

a

δ −−

−−

−−

+++−−

−++−−−

−++−−−

−++−−−=

221

23

1221

1 22

23

2 xxa

xa δ −−

−−

−−≤

( )24

23

22

212 xxxx αααα +++−≤

.0,2 ≥∀−= tVα

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

Dec 2018 Page: 1228

this case, the guaranteed is given by

(3)

. (4) along the trajectories of

s (1) with (2) and (A1), is

)

)4215

4

43

xka

x

x

∆

43

428

4314

2

xx

xxb

xxxa∆

)

24

2216

23

3211

2

1

x

x

xxb

x

++ δ

)

)

)

242

22

22

12

2117

24

1

2133

22

1

252

2

2

4

4

x

x

xa

b

xa

b

xa

b

δ−

+

+

+

2422 xδ−



Hence, one has

[ ].0,0

2 222

≥∀≤

⋅=⋅+⋅

t

Vedt

dVeVe ttt ααα α&

It results

( )( )[ ]( )( ) ( )( )02

0

2

xVtxVe

dtxVed

d

t

t

−⋅=

⋅∫α

ατ ττ

.0,000

≥∀=≤ ∫ tdt

τ (5)

From (4) and (5), it follows ( ) ( )( ) ( )( )

( ) .0,0

022

22

≥∀=

≤=−

−

txe

xVetxVtx

t

t

α

α

As a consequence, we conclude that ( ) ( ) .0,0 ≥∀≤ − txetx tα

This completes the proof. 3. NUMERICAL SIMULATIONS Consider the uncertain systems (1) with

,109,1011 21 ≤∆≤−≤∆≤− aa (6a) ,10,2827 85 ≤∆≤≤∆≤ aa (6b)

,3

83,11 129

−≤∆≤−≤∆≤− aa (6c)

,2,1,910 151014 ≤∆∆≤−≤∆≤− aaa (6d) 13,11,7,6,4,3,0 ∈∀=∆ iai . (6e)

By comparing (A1) and (6) with parameters of

( ) ,1,3

8,1,10,,, 1512101

−−=aaaa

( ) ( ),0,0,0,28,10,,,, 611752 =bbbbb ( ) ( ),10,1,0,0,,, 148133 =bbbb (A1) is evidently satisfied. With the choice in (2), it can be obtained that

[ ] [ ] .32111 4221TT xxuuu −−== (7)

As a consequence, by Theorem 1, we conclude that the uncertain systems (1) with (6) exponentially stable under the linear static (7). Besides, from (3), the guaranteed exponential convergence rate is given by 1=α . The typical strajectories of the uncontrolled system and the feedback-controlled system are depicted in Figand Figure 2, respectively. In addition, signals and the electronic circuit to realize such a control law are depicted in Figure 3 respectively.



with selecting the

With the choice 121 == δδ

we conclude that ) are globally static control of

guaranteed exponential The typical state system and the

controlled system are depicted in Figure 1 In addition, the control

and the electronic circuit to realize such a and Figure 4,

4. CONCLUSION In this paper, the robust stabilization for a class of uncertain nonlinear systems has been exploredon the Lyapunov-like approachinequalities, a simple linear presented to realize the global exponential stability of such uncertain nonlinear systems. guaranteed exponential convergence rate can be correctly calculated. Finally, some numerical simulations have been offered feasibility and effectiveness of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106MOST 106-2813-C-214-025-EE-214-030. Furthermore, the authorChair Professor Jer-Guang Hsieh comments. REFERENCES 1. A. Bayas, I. Škrjanc, and

fuzzy robust control strategies for a distributed solar collector field,” Applied Soft Computinvol. 71, pp. 1009-1019, 201

2. Z.M. Li and X.H. Chang, “networked control systems with randomly occurring uncertainties: ObserverISA Transactions, vol. 83, pp.

3. R.X. Zhong, C. Chen, Y.P. Huang, A. Sumand D.B. Xu, “Robust perimeter control for two urban regions with macroscopic fundamental diagrams: A controlapproach,” Transportation Research Part B: Methodological, vol. 117, pp.

4. F. Yue and X. Li, “Robust adaptbackstepping control for optosystem based on modified LuGre friction modelISA Transactions, vol. 80, pp.

5. N. Adhikary and C. Mahantacontrol of position commanded robot manipulators,” Control Engineering Practice81, pp. 183-198, 2018.

6. A. Ayadi, M. Smaoui, S. Farza, “Adaptive sliding mode control with moving surface: Experimental validation for electropneumatic system,” and Signal Processing, vol.


Dec 2018 Page: 1229

In this paper, the robust stabilization for a class of has been explored. Based

approach with differential linear static control has been

to realize the global exponential stability of systems. Meanwhile, the

guaranteed exponential convergence rate can be Finally, some numerical

have been offered to demonstrate the ffectiveness of the obtained results.

Ministry of Science and of Republic of China for supporting this

work under grants MOST 106-2221-E-214-007, E, and MOST 107-2221-

, the author is grateful to Guang Hsieh for the useful

and D. Sáez, “Design of fuzzy robust control strategies for a distributed

Applied Soft Computing, , 2018.

, “Robust H∞ control for networked control systems with randomly occurring uncertainties: Observer-based case,”

, pp. 13-24, 2018.

R.X. Zhong, C. Chen, Y.P. Huang, A. Sumalee, Robust perimeter control for two

urban regions with macroscopic fundamental diagrams: A control-Lyapunov function

Transportation Research Part B: 7, pp. 687-707, 2018.

Robust adaptive integral backstepping control for opto-electronic tracking system based on modified LuGre friction model,”

, pp. 312-321, 2018.

Mahanta, “Sliding mode control of position commanded robot

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7. Y. Solgi and S. Ganjefar, “Variable structure fuzzy wavelet neural network controller for complex nonlinear systems,” Applied Soft Computing, vol. 64, pp. 674-685, 201

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Figure 1: Typical state trajectories of the system (1) with (6) and .0=u

Figure 2: Typical state trajectories of the feedbackcontrolled system of (1) with (6) and (7)

0 5 10 15 20 25 30-50

0

50

100

t (sec)

x1(t

); x

2(t)

; x3

(t);

x4(

t)

x1: the deep blue curve

x2: the green curve

x3: the red curvex4: the light blue curve

0 0.5 1 1.5 2 2.5 3 3.5 4-5

-4

-3

-2

-1

0

1

2

3

4

5

t (sec)

x1(t

); x

2(t)

; x3

(t);

x4(

t)


x2: the green curve

x3: the red curve

x4: the light blue curve



Variable structure fuzzy wavelet neural network controller for

Applied Soft , 2018.

and H.G. Han, tive fuzzy neural network control of

wastewater treatment process with multiobjective 275, pp. 383-

Pourseifi, and H. Derili, Optimal control in teleoperation systems with

A singular perturbation approach,” Journal of Computational and Applied

, 2018.

Robust feedback control of the underactuated Inertia Wheel Inverted Pendulum under parametric uncertainties

d subject to external disturbances: LMI Journal of the Franklin Institute, vol.

Typical state trajectories of the system (1)

f the feedback-

and (7).

Figure 3: Control signals

The uncertain systems (1)

with (6)1u

2u

2R

4R

Figure 4: The diagram of implementation of controller, where 131 == RR

324 = kR

35 40

x3: the red curvex4: the light blue curve

4.5 5


x2: the green curve

x3: the red curve

x4: the light blue curve

0 0.5 1 1.5 2 2.5-600

-500

-400

-300

-200

-100

0

100

200

t (sec)

u1(t

); u

2(t)


Dec 2018 Page: 1230

Control signals of ( )tu1 and ( )tu2 .

he uncertain systems (1)

2x

4x

1R

3R

The diagram of implementation of

,1 Ωk ,1112 Ω= kR and .Ωk

3 3.5 4 4.5 5t (sec)

u1: the Blue Curve

u2: the Green Curve

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