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Observer-based adaptive stabilization of a class ofuncertain nonlinear systemsMohammad M. Arefi

a, Jafar Zarei

b& Hamid R. Karimi

c

aDepartment of Power and Control Engineering, School of Electrical and Computer

Engineering, Shiraz University, 71348-51154 Shiraz, IranbDepartment of Control Engineering, School of Electrical and Electronics Engineering, Shir

University of Technology, 7155713876 Shiraz, IrancDepartment of Engineering, Faculty of Engineering and Science, University of Agder,

N-4898 Grimstad, NorwayPublished online: 09 May 2014.

To cite this article:Mohammad M. Arefi, Jafar Zarei & Hamid R. Karimi (2014) Observer-based adaptive stabilization of aclass of uncertain nonlinear systems, Systems Science & Control Engineering: An Open Access Journal, 2:1, 362-367, DOI:

10.1080/21642583.2014.913510

To link to this article: http://dx.doi.org/10.1080/21642583.2014.913510

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Systems Science & Control Engineering: An Open Access Journal, 2014

Vol. 2, 362367, http://dx.doi.org/10.1080/21642583.2014.913510

Observer-based adaptive stabilization of a class of uncertain nonlinear systems

Mohammad M. Arefia

,Jafar Zareib

and Hamid R. Karimic

aDepartment of Power and Control Engineering, School of Electrical and Computer Engineering, Shiraz University, 71348-51154

Shiraz, Iran; bDepartment of Control Engineering, School of Electrical and Electronics Engineering, Shiraz University of Technology,7155713876 Shiraz, Iran; cDepartment of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad,

Norway

(Received 25 December 2013; final version received 7 April 2014 )

In this paper, an adaptive output feedback stabilization method for a class of uncertain nonlinear systems is presented. Sincethis approach does not require any information about the bound of uncertainties, this information is not neededa priorianda mechanism for its estimation is exploited. The adaptation law is obtained using the Lyapunov direct method. Since all thestates are not measurable, an observer is designed to estimate unmeasurable states for stabilization. Therefore, in the designprocedure, first an observer is designed and then the control signal is constructed based on the estimated states and adaptationlaw with the -modification algorithm. The uniformly ultimately boundedness of all signals in the closed-loop system isanalytically shown using the Lyapunov method. The effectiveness of the proposed scheme is shown by applying to a unified

chaotic system.

Keywords: output feedback; adaptive control; uniformly ultimately boundedness; Lyapunov-based design

1. Introduction

The problem of the robust output feedback regulation of

uncertain nonlinear systems is the design of a feedback con-

trol law for such systems in a way that the boundedness of

signals in the closed-loop is guaranteed (Chen & Huang,

2005a,2005b;Huang, & Chen, 2004). However, in many

practical applications, measurement of all the states is not

possible. Therefore, observer design is an essential step in

this approach.

The design of observers has received a great deal ofattention recently (Liu, 2009). However, there are two

main restrictive conditions in the design of observer-based

controllers. First, the nonlinearities are only functions of

measurable signals, which is a common assumption in the

literature. Moreover, it is assumed that the unknown non-

linearity is bounded by output injection terms and this

unknown nonlinearity satisfies a global Lipschitz condi-

tion, which is the second restriction (Liu, 2009). In practical

cases, these conditions are not held.

In order to make the design procedure more practical,

we should consider a mechanism to relax these condi-

tions. These conditions were further relaxed recently inLiu

(2009),Choi and Lim (2005),Alimhan and Inaba (2006)andHou, Wu, and Duan (2009). InLiu and Zheng (2009), a

Fuzzy logic systemis employed to estimate theupper bound

of nonlinear uncertainties. This procedure could be done

by using other approximation tools such as neural networks

(NNs) (Arefi & Jahed-Motlagh, 2011;Du & Chen, 2009).

Corresponding author. Email:arefi@shirazu.ac.ir

However, the design of fuzzy logic system and NNs needs

to incorporate the knowledge of an expert. In this paper,

these limitations are thoroughly relaxed by the estimation

of upper bound of uncertainties using an adaptive control

strategy. Compared withLiu and Zheng (2009)andDu and

Chen (2009), this method needs neither any information

about the bound of uncertainties nor an experts or system

specialist. Moreover, in the presented method in Liu and

Zheng (2009)andDu and Chen (2009),semi-global results

are obtained, while our proposed approach provides glob-ally uniformly ultimately boundedness (UUB) for all the

signals in the closed-loop system.

This paper is organizedas follows: theproblem formula-

tion of output feedback stabilization of uncertain nonlinear

systems is presented in the second section. Adaptive out-

put feedback, observer design and the stability analysis

of the algorithm are presented in Section 3.In Section4,

an uncertain unified chaotic system is adopted to evalu-

ate the effectiveness of the proposed method. Finally, the

conclusions are given in Section5.

2. Problem formulation

Consider the following uncertain nonlinear system:

x(t)= Ax(t)+Bf(x)+Bu,

y= CTx,(1)

2014 The Author(s). Published by Taylor & Francis.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The moral rights of the named author(s) have been

asserted.

mailto:arefi@shirazu.ac.irmailto:arefi@shirazu.ac.ir

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Systems Science & Control Engineering: An Open Access Journal 363

wherex = [x1, . . . ,xn]T n corresponds to the state vec-

tor of the system, y m is the system output, u=

[u1, . . . , um]T m is the input vector of the plant, f(x)=

[f1(x) . . . , fm(x)]T m is the unknown nonlinear function

vector, A nn, B nm, andC nm are constant

matrices with appropriate dimensions. In the system (1),

not allxiare assumed to be measurable and only the system

outputyis assumed to be available for measurement. In the

controller design, we need the following assumptions.

Assumption1 For the nonlinear function f(x), there

exists a positive constant such that

f(x) .

Remark 1 Although Assumption1is restrictive, one can

suppose that is large enough so that this assumption is

satisfied.

Assumption2 (A, B) is controllable and (A, CT) is

detectable.The main goal is to design an output feedback con-

troller such that the states of(1) in the presence of unknown

nonlinear function are bounded.

Iff(x)is known and the state vectorx is available, the

controller can be chosen as

u= f(x)kTcx, (2)

wherekc is the state feedback gain matrix. By substituting

Equation(2)into Equation(1)we have

x(t)= (ABkTc )x(t). (3)

Since the pair (A, B) is controllable, the gain matrix kc

in Equation(3)can be chosen such that the characteristicpolynomial of ABkTc is strictly Hurwitz. Then, it can

be shown thatlimtx(t)= 0. However, becausef(x)is

unknown and only the system outputy is measurable, this

controller cannot be implemented in practice. A solution is

to estimate the upper bound of the known function using the

adaptive control strategy and design a suitable observer to

estimate the state vectorx using the measurable outputy.

3. Adaptive output feedback controller and observer

design

In the previous section we assumed that only the system

output is measurable and other states cannot be used inthe controller design. So, we need to design an observer

to estimate the unmeasurable states. Suppose that x is the

estimation of state vectorx. Thefollowing observer is given

as

x(t)= (ABkTc )x(t)+ko(yCTx),

y= CTx.(4)

Since the pair(A, CT) is observable, the gain matrixko

nm in Equation (4)can be chosen so that AkoCT is

strictly Hurwitz. Let the estimation errore = x x. From

Equations(1) and(4), we have

e= (AkoCT)e+BkTcx+Bf(x)+Bu,

e= cTe.(5)

Assumption3 There exist positive-definite matrices P

andQ satisfying

(AkoCT)TP+P(AkoC

T)+Q = 0,

PB= C.(6)

Remark 2 Ifko can be chosen such that the triple (A

KoCT, B, C) is strictly positive real (SPR), one can use

the KalmanYakubovichPopov lemma (Slotine & Li,

1991), which guarantees the existence of positive-definite

symmetric matricesP andQ in Equation(6).

Theorem1 Consider the nonlinear system (1) and the

observer given in Equation (4). Under Assumptions 13,

construct the following adaptive controller:

u= kTcx e2

e + , (7)

and the adaptation law is as follows:

= e , (8)

where >0, >0, (t0) >0, and >0 are the design

parameters.

Then all the signals in the closed-loop system are

UUB. Furthermore, the estimation error can approach an

arbitrarily small value by choosing the design parameter

appropriately.

Proof Choose the following continuously differentiable

function as a Lyapunov candidate

V = 1

2

eTPe+

1

2

, (9)

where = and is the adaptation gain given in

Equation(8). Thederivativeof Equation (9), usingEquation

(5) is

V = 1

2eT(ATo P+PAo)e+e

TPBkTcx+eTPBf(x)

+eTPBu+ 1

, (10)

where Ao = A koCT. Accordingto Equation (6),wehave

eTPB= eTC= eT. (11)

By using Equations (6),(7), (10), and(11) we have

V = 1

2eTQe+ eTf

e2 2

e + +

1

. (12)

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364 M.M. Arefiet al.

Now, using the fact that eTQe min(Q) e 2,

where min(Q) is the minimum eigenvalue of Q and

regarding to Assumption1we have

V 1

2min (Q) e

2 + e e2 2

e + +

1

.

(13)

Furthermore, the following inequality is true for the thirdterm of the right-hand side of inequality (13)

e2 2

e + = e

1+

e +

e + . (14)

Considering inequality(14) we have

V 1

2min(Q) e

2 e + + 1

. (15)

Finally, by substituting the adaptation law (8) into

Equation(15), we obtain

V 1

2min (Q) e

2 +,

1

2min (Q) e

2 +2 +| |.

(16)

Using a2 +ab= a2/2 12

(ab)2 +b2/2 a2/2

+b2/2 for anya, bwe can write

V 1

2min(Q) e

2 +2

2+

||2

2. (17)

From Equations (9) and (17)andmin(P) e 2eTPe

max(P) e 2 we have

V cV+ +||2

2, (18)

where

c= min

min(Q)

max(P),

. (19)

Solving inequality (18)gives

0 V(t) +||2/2

c

+

V(t0)

+||2/2

c

ect t 0.

(20)

Thus, V(t) max{V(t0), (+ (||2/2)/c}, t 0.

From the definition ofV(t) in Equation(9), the error vector

e(t), is bounded by

e(t)

max{V(t0), (+ (||2/2))/c}

min (P). (21)

Equation (21) means that V(t) is eventually bounded by

(+ (||2/2))/c. Thus, all signals of the closed-loop sys-

tem, i.e. e(t), are UUB (Khalil, 2002). Besides, since

Equation (17) implies that V T. Since the characteristic polynomial ofABkTc is

strictly Hurwitz, it can be concluded from Equation(4) that

x is bounded. Then according to e= x x, we can also

conclude that x is also bounded. In addition, based on the

definition of u in Equation (7), u is also bounded. This

completes the proof.

4. Simulation results

To show theproficiency of thepresented algorithm, thesim-ulation results are presented in this section. A class of more

general nonlinear systems is studied in this section. For

example, the following unified chaotic system is considered

(Liu & Zheng, 2009):

x1 = (25+10)(x2x1),

x2 = (2835)x1 x1x3+ (291)x2+ u2,

x3 = x1x2+ 8

3x3+ u3,

(23)

where [0, 1]. When [0,0.8), it is a Lorenz chaotic

system, = 0.8 is the Lu chaotic system, and (0.8, 1]is Chens chaotic system. The system can be easily trans-

formed into the canonical form of Equation (1) with the

following parameters:

A=

2510 25+ 10 02835 291 0

0 0 +83

,

B=

0 01 0

0 1

,

f(x)=

x1x3x1x2

, C=

0 1 00 0 1

T.

It is worth mentioning that Assumption 1,which imposes

an upper bound for f(x), is simply satisfied in chaotic

systems due to the boundedness of trajectories in these

systems (Arefi & Jahed-Motalgh, 2012). The simulation is

carried out with initial conditions x0 = [2,1, 1]T, x0 =

[3,2,1]T. It is straightforward to verify that the triple

(AKoCT, B, C) can be made SPR by the choice of the

observer gain.

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Systems Science & Control Engineering: An Open Access Journal 365

Besides, the observer and state feedback gain are

selected as

kc =

0 200 50

50 10 100

T, ko =

60 10 15

30 100 15

T.

Furthermore, the parametersof the controller and adaptation

law are as follows:

= 0.01, =0.5, =0.001, (0)= 0.1.

Figure1 shows the chaotic behavior of system(23) with

u= 0 and = 0.5.

The state trajectories of the system by applying the con-

troller (7) with = 0.5 are shown in Figure 2.Itcanbeseen

from the simulations that the adaptive output feedback con-

troller (7)makes the state estimations tend the actual states

precisely. As this figure shows, the proposed controller sta-

bilizes the system in the presence of unknown nonlinear

uncertainties.

The time responses of the control input and adaptation

parameter are shown in Figures3and4,respectively.

We see that the control input is smooth and imple-

mentable. Furthermore, the value of is bounded.

0 5 10 15 20 25 3020

0

20

40

x1

0 5 10 15 20 25 3050

0

50

x2

0 5 10 15 20 25 3050

0

50

100

time [S]

x3

Figure 1. The chaotic behavior of system withu = 0 where is chosen as = 0.5.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52

0

2

4(a)

(b)

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

0

1

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

2

0

2

Time [S]

Figure 2. (a)x1 (solid line) andx1 (dashed line); (b) x2 (solid line) andx2 (dashed line); (c)x3 (solid line) andx3 (dashed line).

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366 M.M. Arefiet al.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5600

400

200

0

200

u1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5300

200

100

0

100

Time [S]

u2

Figure 3. Control inputs of the system (a)u1 and (b)u2.

Figure 4. Adaptation parameter.

5. Conclusion

In this paper, an adaptive output feedback stabilization

strategy for a class of uncertain nonlinear systems was pro-

posed. Since all the states are not measurable, an observerwas presented to estimate unmeasurable states. The design

method is based on the Lyapunov stability Theorem, and

it was shown that all signals in the closed-loop system are

UUB. Additionally, in this method, the mere knowledge of

boundedness of uncertain term is sufficient.

Simulation results for the stabilization of a unified

chaotic system show that the proposed approach has a fast

response in stabilization. Moreover, the norm of the estima-

tion errorsis bounded, while thecontrol signalis completely

smooth.

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