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Chaos, Solitons and Fractals 37 (2008) 308–315

www.elsevier.com/locate/chaos

Adaptive high order differential feedback control foraffine nonlinear system

Guoyuan Qi a,*, Zengqiang Chen b, Zhuzhi Yuan b

a Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, PR Chinab Department of Automation, Nankai University, Tianjin 300071, PR China

Accepted 5 September 2006

Communicated by Prof. Ji-Huan He

Abstract

In this paper, we present an adaptive high order differential feedback controller (HODFC) which is a kind of model-free control diagram. The proposed HODFC achieves the objective that the system output and its high differentialsapproximate the given reference input and its high order differentials, respectively. Stability, convergence, and robust-ness of the closed-loop system are also studied. Moreover, a kind of linearized decoupling control is obtained forMIMO system by a multivariable HODFC.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

The important characteristic of modern control theory is that estimator and controller depends on the model of thecontrolled plant. It is the same for some chaos controlling schemes. For example, the techniques proposed in [1–3] relyon the known model of the Rossler hyperchaotic system. Based on the models of the Lorenz system, the generalizedLorenz system and the Chen system, several kinds controllers have been proposed in [4–6]. At present, some controlapproaches for chaotic systems considered uncertainties, such as Jang et al. [2], Lu et al. [7] used sliding mode control,and Tsai et al. [3] utilized variable structure control approach. But their robust controllers are mainly based on knownnormal models in which unknown uncertainties are relatively smaller as compared to the known normal models.

In fact, there always exist the cases that the model of the plant is completely unknown or partially unknown. More-over, the model-depending technique is very complicated for model identification. Some engineers would rather use thesimple PID control, since it does not depend on the model but the error and its differential of the system. However, PIDis hard to reach ideal control effect for high order systems, because PID only takes into account the first order differ-ential of the error signal, rather than its second order differential or other high order ones. However, the controlledplant is always high order nonlinear uncertain system. In view of the invert system, the controller should also be a func-tion of the error between the given input and the output and their differentials up to the nth order. In general, these

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.09.027

* Corresponding author.E-mail addresses: qi_gy@yahoo.com.cn (G. Qi), chenzq@nankai.edu.cn (Z. Chen).

G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315 309

differentials were estimated by a high gain observer [8,9] which is independent on the system model. However, unlike theobserver designed in [9], the controller still mainly depended on the known function model of the system, and only theunknown parameters were identified online. Furthermore, these results [9,10] only dealt with SISO affine system ratherthan MIMO system. We proposed a higher-order differentiator (HOD), which can extract differentials up to the nthorder with higher accuracy and better filtering property under noises [11].

In this paper, based on the extracted differentials of the output and the given input by the HOD, adaptive high orderdifferential feedback controllers (HODFC) are presented for SISO and MIMO affine nonlinear systems, respectively,which do not depend on the system models. The analysis shows that the closed-loop systems under the HODFC areasymptotically stable and robust for the disturbances and functional variances. And the outputs can converge toany given smooth objectives. Numerical experiments show the validity of the control performed from the proposedtheory.

2. Problem statement

Differential equation of SISO affine nonlinear system is described as

yðnÞ ¼ f ðx; tÞ þ dðtÞ þ u: ð1Þ

where u is the control input, y is the output, x = [x1,x2, . . .,xn]T = [y,y(1), . . .,y(n)]T denotes the output differential vector,and is also system state vector, y(i) denotes the ith differential of y, f(Æ) is an unknown time-varying bounded smoothnonlinear function, and d(t) is a bounded disturbance.

Usually, controllers of the system mostly based on the knowledge of the nonlinear function f(Æ) and state x. Obvi-ously, the controlled system is the function of the control input and the output and their differentials up to nth order.The high order differentials y(1), . . .,y(n) in Eq. (1) are very significant. We will fully utilize all the information rather thanthe model function.

Suppose the given objective yr has differentials up to nth order, and yðnÞr is continuous. Otherwise, we can soften yr tosatisfy these conditions. Let

r ¼ yv; . . . ; yðn�1Þr

� �T; �r ¼ rT; yðnÞr

� �T; �x ¼ x; yðnÞ

� �T

be the given input differential vector, the given input extended differential vector and the output extended differentialvector, respectively. Let

e ¼ r� x ¼ ½e1; e2; . . . ; en�T ¼ e; eð1Þ; . . . ; eðn�1Þ� �T; �e ¼ �r� �x ¼ eT; eðnÞ

� �T; ð2Þ

be the error differential vector, and the error extended differential vector, where e = yr � y.In general, the output y and the given input yr are known, but �x and �r are unknown. Let

�x ¼ y; yð1Þ; . . . ; yðnÞ� �T

; �r ¼ yr; yð1Þr ; . . . ; yðnÞr

� �T; ð3Þ

denote the estimate of �x and �r, respectively.In [4], to extract differentials of y and yr, we proposed the HOD which is described by the no order dynamic system

(4) with n + 1 order algebraic Eq. (5).

X _zi ¼ ziþ1 þ aiðy � z1Þ; 1 6 i 6 no � 1;

_zno ¼ anoðy � z1Þ;

�ð4Þ

y ¼ z1;

yðiÞ ¼ ziþ1 þ aiðy � z1Þ; i ¼ 1; . . . ; n:

�ð5Þ

where no is the order of the system, satisfying no P n + 1, z1; . . . ; zn0are the states, ai are the parameters, and

ai ¼ KCi�1no�1ai�1; K ¼ nno

o a=ðno � 1Þn0�1;Ci

j denotes the combination expression.We can obtain z1; . . . ; zn0

based on the measured signal y via (4), furthermore, calculate the estimated differentialsy; . . . ; yðnÞ via (5).

Remarks [4]:.

(1) The HOD does not depend on the model of system (1), and it is an additional system based on the signal y or yr.(2) The HOD is an asymptotically stable system.(3) The HOD holds convergence, i.e.,

310 G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315

limt!1

yðiÞ ¼ yðiÞ; i ¼ 1; . . . ; n; ð6Þ

(4) All extracted differentials yðiÞ, yðiÞr , i = 1, . . .,n are smooth. SinceP

has no (no P n + 1) integrators, which meanseven if y or yr is not smooth, both y and yr are no order differentiable. In fact, the HOD itself is an higher accuracyfilter.

(5) To overcome peaking impulse phenomenon in initial process, we modify the HOD by adding a restraint device in(6), which is amended as

y ¼ x1; yðiÞ ¼ xiþ1 þ aiðy � x1ÞriðtÞ; i ¼ 1; . . . ; n;

riðtÞ ¼ 1�exp ð�bt2iÞ1þexp ð�bt2iÞ ;

(ð7Þ

where b is a large positive constant.

3. High order differential feedback control

3.1. HODFC for SISO nonlinear system

Assumption 1. The extended output differential vector �x and extended given input differential vector �r are known, andy(n) is yðnÞr continuous.

Theorem 1. For the time-varying nonlinear affine system (1) with unknown model, the HODFC is described by

u ¼ K�eþ u; ð8Þ

where K ¼ ½kn; . . . ; k1; 1� makes the polynomial sn + k1sn�1+� � �+kn be a Hurwitz polynomial, and u denotes the filtering

signal of the control u, satisfying

_u ¼ �kuþ ku; ð9Þ

or other filtering equation, where k is a large positive constant. Then the HODFC has the following properties:

(1) The HODFC makes the closed-loop system asymptotically stable, and satisfies the following convergent property

limt!1

limk!1

x ¼ r: ð10Þ

(2) All system variables are bounded.

(3) The controller is strongly robust for the function f(Æ) and bounded disturbance d(t).

Proof. From Eq. (1) and the definition of ei in Eq. (2), We have

_ei ¼ eiþ1; i ¼ 1; . . . ; n� 1;

_en ¼ yðnÞr � yðnÞ ¼ yðnÞr � yðnÞ þ yðnÞ � ðf ðx; tÞ þ dðtÞ þ uÞ:

�ð11Þ

Setting K = [kn,kn�1, . . .,k1] 2 R1·n, Am = A + bK

A ¼

0 1 0 � � � 0

0 0 1 � � � 0

..

. . ..

0 0 0 � � � 1

0 0 0 � � � 0

26666664

37777775

n�n

; b ¼

0

0

0

0

1

26666664

37777775

n�1

;

from (11) and definition of vectors e and �e in Eq. (2), we have

_e ¼ Aeþ b yðnÞr � yðnÞ þ yðnÞ � ðf ðx; tÞ þ dðtÞ þ uÞ� �

¼ Ameþ b Keþ yðnÞ � ðf ðx; tÞ þ dðtÞ þ uÞ� �

; ð12Þ

K makes 1sn + k1sn�1+� � �+kn be a Hurwitz polynomial. Equivalently, K makes Am be a Hurwitz matrix. Let

G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315 311

K�eþ yðnÞ � ðf ðx; tÞ þ dðtÞ þ uÞ ¼ 0: ð13Þ

We have the stable control law

u ¼ K�eþ yðnÞ � ðf ðx; tÞ þ dðtÞÞ: ð14Þ

If the sum item f(x, t) + d(t) is unknown, the control law is unable to be realized. In other words, this is a model-depen-dent control law. From system (1), We have

yðnÞ � ðf ðx; tÞ þ dðtÞÞ ¼ u: ð15Þ

But u is an unsolved control law, so it is still unable to be realized. Considering the lag property of the filtering in (11),replacing u by u, We will obtain

u � yðnÞ � ðf ðx; tÞ þ dðtÞÞ: ð16Þ

Substituting (16) into (14), We obtain the controller (8). Substituting (8) into (13), and using (15), we have

_e ¼ Ameþ bðu� uÞ: ð17Þ

From (9), the filtering u is realized via integrator, so the filtering u is necessarily continuous no matter whether u iscontinuous or not. Furthermore, from Assumption 1, y(n) and yðnÞr are continuous, so y(i) and yðiÞr ði ¼ 0; . . . ; n� 1Þmustbe continuous, too, which means that �e is continuous. Therefore, from (8), the control law u must be continuous. From(9) and the continuity of u, we obtain

limk!1

u ¼ u: ð18Þ

From (17) and (18) and Am being a Hurwitz matrix, the closed-loop control system is asymptotically stable and satisfies

limt!1

limk!1

e ¼ 0: ð19Þ

Obviously, r is bounded. And from (10), x is bounded. From Assumption 1 and (8), the control law u is bounded. Fur-thermore, controller (8) does not depend on the system model, which implies that the controller is strongly robust withrespect to the function f(Æ) and bounded disturbance d(t). h

Remarks.

(1) Here, k!1 is just a rigorous expression for mathematics meanings, in general, k 2 [5,50]. The filter (9) is notunique. The filtering u can be replaced by other filtering equation, for instance, the HOD.

(2) From (10), the system output y and its differentials of the output up to (n � 1)th-order can track the given inputyr and its corresponding differentials, respectively, which is different from the general control objective that thesystem output can only tracks given input.

(3) The control law has two terms: u eliminates the sum term f(x,t) + d(t), and K�e ensures that the closed-loop systemis asymptotically stable. Therefore, the HODFC is different from parameters tuning in PID control.

(4) When the particular form is described as a non-minimum phase form:

yðnÞ ¼ f ðx; tÞ � uþ dðtÞ: ð20Þ

System (20) can be controlled by the way that the control law of the HODFC is converted into

u ¼ �K�eþ u: ð21Þ

From the Assumption 1, the HODFC is non-adaptive for �e ¼ �x� �xr. If �e is unknown, we can estimate it via the HODto obtain �e ¼ �x� �xr by error e = yr � y. From remarks (4) in Section 2, yðnÞ and yðnÞr are continuous, so all conditions ofAssumption 1 can be realized via HOD. The adaptive controller can be written as

u ¼ K�eþ u: ð22Þ

Hence, we yield an adaptive HODFC.

3.2. HODFC for MIMO affine nonlinear system

Consider a MIMO nonlinear time-varying system with unknown model

312 G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315

yðniÞi ¼ fiðX; tÞ þ ui þ diðtÞ; i ¼ 1; . . . ; p; ð23Þ

where X ¼ ½XT1 ;X

T2 ; . . . ; XT

p �T is the output differential vector, Xi ¼ ½yi; y

ð1Þi ; . . . ; yðni�1Þ

i �T is the ith output differentialvector, yi and ui are the ith output and input, and di(t) is the unknown disturbance. Assuming that fi(Æ) is an unknowntime-varying smooth function.

Let yri, i = 1, . . .,p be the ith given input, X ri ¼ yri; . . . ; yðn1�1Þri

h iT

be the ith given input differential vector,

�X ri ¼ XTri; y

ðniÞri

h iT

, �Xi ¼ X Ti ; y

ðniÞi

h iT

be the ith given input extended differential vector, and the output extended differ-

ential vector, respectively. Let

E i ¼ ½Ei1; . . . ; Eini �T ¼ X ri � Xi ¼ ei; . . . ; eðni�1Þ

i

h iT

; Ei ¼ �X ri � �X i ¼ ETi ; e

ðniÞi

h iT

; ð24Þ

be the ith error differential vector, and the ith error extended differential vector, respectively, where ei = yri � yi. Assumeyri has bounded differentials up to nith order.

Assumption 2. Suppose that X i and X ri are completely measurable. We have the following theorem.

Theorem 2. For the time-varying MIMO affine nonlinear system (23) with unknown model and unknown disturbance, the

multivariable HODFC (MHODFC) is described by

ui ¼ K iEi þ ui; i ¼ 1; . . . ; p: ð25Þ

where K i ¼ ½kini ; . . . ; ki1; 1� makes sni þ ki1sni�1 þ � � � þ kini be Hurwitz polynomial, and ui is the filtering value of the control

ui. The MHODFC has following properties:

(1) It realizes linearized decoupling control.

(2) It makes the closed-loop system asymptotically stable, and satisfies the following convergent property

limt!1

X i ¼ X ri ð26Þ

(3) All system variables are bounded.

(4) It is strongly robust for the function f(Æ) and bounded disturbance d(t).

Proof. Substituting (25) into (23), we have

yðniÞi ¼ fiðX; tÞ þ K iEi þ ui þ diðtÞ; ð27Þ

From Eq. (27) and Eq. (23), we have

yðniÞi þ ki1yðni�1Þ

i þ � � � þ kini yi

� �¼ yðniÞ

ri þ ki1yðni�1Þri þ � � � þ kini yri

� �þ di; i ¼ 1; . . . ; p: ð28Þ

where, di ¼ ui � ui. From (28), we obtain p linear decoupling differential equations. From Eq. (28) and the definition ofscalars ei and Eik in Eq. (24), we have

eðniÞi ¼ �ki1e

ðni�1Þi � � � � � kiniei þ di ð29Þ

It implies

_Eini ¼ �ki1Eini � � � � � kini Ei1 þ di ð30Þ

From Eq. (30) and the definition of vector Ei in Eq. (24), we obtain the following important equation:

_E i ¼ Ami E i þ bidi ð31Þ

where Ami 2 Rni�ni is a controllable normal matrix with parameters kini ; . . . ; ki1, and bi ¼ ½0; . . . ; 0; 1�T 2 Rni�1. Similar tothe Proof of Theorem 1, we can obtain that di! 0, and sni þ ki1sni�1 þ � � � þ kini is a Hurwitz polynomial, from (31), weobtain

limt!1

E i ¼ 0: ð32Þ

Meanwhile, according to Assumption 2 and the second property, control input ui is bounded. Therefore, all variablesare bounded. Since the MHODFC does not rely on the system model, the MHODFC has strong robustness for var-iance of function fi(Æ) and disturbance di(t).

G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315 313

If Ei ¼ Xri � X i in (24) is unknown, we obtain the estimating vector �Ei ¼ Xri � X i using the HOD based onei = yri � yi. Therefore we obtain the following adaptive MHODFC

ui ¼ ðK iE i þ uiÞ; i ¼ 1; . . . ; p: � ð33Þ

4. Simulation study

Example 1. Consider the following nonlinear system

yð3Þ ¼ f0ð�Þ þ 2hf1ð�Þ þ g _u ð34Þ

where f0ð�Þ ¼ uþ y � €y, f1ð�Þ ¼ y _y þ _y2 þ y€y, g = 1, and y is the system output.

In [9] the error differential vector ½yð1Þr � yð1Þ; yð2Þr � yð2Þ� and yð3Þr were estimated by high gain observer, but thefunction f0, f1 were known except the parameter h to design output feedback controller.

We design adaptive HODFC (22) which only depend on the extended error differential vector

�e ¼ ½e1; e2; e3�T ¼ yr � y; . . . ; yð3Þr � yð3Þ� �T

estimated by the HOD and control filtering uð1Þ, but do not depend on f0, f1. Take no = 5, a = 20, b = 50 in the HODwith restraint, and K ¼ ½24; 26; 9; 1�, k = 10 in the HODFC. The given input yr is the smooth curve of square wavew(t) = sign(cos (0.1t)) with softening transfer function 9/(s2 + 6s + 9). Initiate the system with y(0) = 0.5, y(1)(0) = 0,y(2)(0) = 0, and initial values of the HOD are all zeros. In Fig. 1, the comparing curves are shown for output y, its esti-mate y and given input yr. In Fig. 2, the comparing curves are shown for y(1), yð1Þ and yð1Þr which realizes the stricterobjective. We can see that the control performance is good.

We also consider system (34) with disturbance, which can be described as

yð3Þ ¼ f0ð�Þ þ 2hf1ð�Þ þ g _uþ dðtÞ ð35Þ

where d(t) = 4sign(cos (0.5t)). We can find out that a satisfactory control performance can be obtained if simulatingsystem (35) (the figures are omitted).

Example 2. Consider controlling the Chen chaotic system [12] with multivariable as fellows

_x ¼ aðy � xÞ þ u1;

_y ¼ ðc� aÞx� xzþ cy þ u2;

_z ¼ xy � bzþ u3

ð36Þ

where a = 35, b = 3, c = 28. The given inputs are the variables in Lorenz chaotic system which is described as

_xr ¼ pðyr � xrÞ;_yr ¼ xrzr þ rxr � yr;

_zr ¼ xryr � qzr;

ð37Þ

where p = 10, r = 28, q = 8/3.

Fig. 1. Curves among y, y and yr.

Fig. 2. Curves among y(1), yð1Þ and yð1Þr .

Fig. 3. Comparing curves between x and xr.

314 G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315

The controller is adaptive MHODFC (33), where e1 ¼ xr � x; xð1Þr � xð1Þ� �T

, e2 ¼ yr � y; yð1Þr � yð1Þ� �T

,e3 ¼ zr � z; zð1Þr � zð1Þ

� �T. Take no = 5, a = 15 in the HOD. Consider high frequency property of the Lorenz system,

we use the higher accuracy filter HOD to estimate ui, with the parameters of no = 5, a = 30, and the parameters of thecontroller Ki = [20,1], i = 1,2,3. In Fig. 3, the comparing curves are shown for x and xr. The tracking accuracy is quitehigh. To show the control accuracy sufficiently, we give the phase plane curves indicating that x, y track to xr, yr.Clearly, we magnify the comparing curves 6 · 6 times around one attractor, shown in Fig. 4, which indicating that thetracking accuracy is very high.

Fig. 4. Comparing curves for between x, y and xr, yr.

G. Qi et al. / Chaos, Solitons and Fractals 37 (2008) 308–315 315

We find out that ui is very important in the controller (33), and the filtering accuracy of ui directly influence thetracking accuracy.

On the contrary, the controlled plant is Lorenz system, and the states of the Chen system is as given inputs. Thecontrol performance can be as good as above.

5. Conclusions

In this paper, a model-free diagram i.e adaptive high differentials feedback control are presented for time-varyingnonlinear SISO and MIMO systems with unknown model and unknown bounded disturbance. This HODFC makesthe closed-loop system asymptotically stable, and all system variables are bounded. This controller can realize thatthe system output and its differentials up to (n � 1)th-order asymptotically track the given reference input and its cor-responding differentials, respectively.

Acknowledgements

This research is supported by grants from National Nature Science Foundation of China (60374037) and from Sci-ence and Technology development foundation of Tianjin Colleges (20051528).

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