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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2016; 26:2314–2337Published online 27 August 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3414

Active disturbance rejection control approach to stabilization oflower triangular systems with uncertainty

Zhi-Liang Zhao1 and Bao-Zhu Guo2,3,*,†

1School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, Shaanxi, China2Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Academia Sinica,

Beijing 100190, China3School of Computer Science and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg,

South Africa

SUMMARY

In this paper, we apply the active disturbance rejection control (ADRC) to stabilization for lower triangularnonlinear systems with large uncertainties. We first design an extended state observer (ESO) to estimatethe state and the uncertainty, in real time, simultaneously. The constant gain and the time-varying gain areused in ESO design separately. The uncertainty is then compensated in the feedback loop. The practicalstability for the closed-loop system with constant gain ESO and the asymptotic stability with time-varyinggain ESO are proven. The constant gain ESO can deal with larger class of nonlinear systems but causes thepeaking value near the initial stage that can be reduced significantly by time-varying gain ESO. The natureof estimation/cancelation makes the ADRC very different from high-gain control where the high gain is usedin both observer and feedback. Copyright © 2015 John Wiley & Sons, Ltd.

Received 17 June 2014; Revised 19 July 2015; Accepted 31 July 2015

KEY WORDS: active disturbance rejection control; nonlinear systems; uncertainty

1. INTRODUCTION

Dealing with uncertainty is a key issue in modern control theory. The active disturbance rejec-tion control (ADRC), different from many other robust control design approaches, is a new controlstrategy to cope with uncertainty. This approach was first proposed by Han (e.g., [1]) based on anSISO nonlinear system with large uncertainty. The central idea of ADRC is that the ‘total distur-bance’ that includes unknown system dynamics and external disturbance can be estimated by outputthrough extended state observer (ESO). The ‘total disturbance’ is then compensated (canceled) inthe feedback loop by ESO-based output feedback control. The ADRC consists basically of threeparts. The first part is the tracking differentiator that is an independent topic in control theory; see,for instance, [2] and the references therein. The convergence of nonlinear tracking differentiator inADRC is proven in our previous works [3, 4]. The second part of ADRC is the ESO that is used toestimate not only the state but also the ‘total disturbance’. The convergence of nonlinear ESO forSISO systems is proven in our paper [5]. The ESO, like state observer in control theory, is also anindependent topic that has many applications such as fault diagnosis [6]. The last part of ADRC isthe ESO-based output feedback control. The convergence of ADRC for MIMO nonlinear systemsis available in our previous work [7]. A similar idea of linear ADRC can also be found in [8]. The

*Correspondence to: Bao-Zhu Guo, Key Laboratory of Systems and Control, Academy of Mathematics and SystemsScience, Academia Sinica, Beijing 100190, China.

†E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

ACTIVE DISTURBANCE REJECTION CONTROL APPROACH TO TRIANGULAR SYSTEMS 2315

ADRC can deal with at least stabilization and output regulation for very general nonlinear uncertainsystems. Very recently, the ADRC has also been applied to deal with systems described by partialdifferential equations in [9–11].

In the last two decades, the ADRC has been successfully applied to many engineering controls,and its merits of energy saving and capability of coping with large uncertainty have been witnessedby many engineering control practices, for instance, hysteretic systems [12], power converter [13],superconducting RF cavities [14], noncircular tuning process [15], flight vehicles [16], chemicalprocesses [17], and vibrational MEMS gyroscopes [18]. The ADRC control has been tested in ParkerHannifin Parflex hose extrusion plant and across multiple production lines for over 8 months. Theproduct performance capability index (Cpk) is improved by 30%, and the energy consumption isreduced over 50% [19].

In this paper, we consider stabilization for the following lower triangle SISO nonlinear systemswith large uncertainties:

8̂̂̂ˆ̂̂̂̂<̂ˆ̂̂̂̂ˆ̂̂̂:

Px1.t/ D x2.t/C �1.x1.t//;

Px2.t/ D x3.t/C �2.x1.t/; x2.t//;

:::

Pxn.t/ D f .t; x.t/; �.t/; w.t//C b.t; x.t/; �.t/; w.t//u.t/;

P�.t/ D f0.t; x.t/; �.t/; w.t//;

x.0/ D .x10; : : : ; xn0/>; �.0/ D .�10; : : : ; �s0/

>;

(1.1)

where x.t/ D .x1.t/; : : : ; xn.t// 2 Rn is the state and �.t/ 2 Rs is the state of the zero dynamics;the function �i .�/ 2 C.Ri ;R/ is known, while f .�/ 2 C.RnCsC2;R/ and f0.�/ 2 C.RnC2Cs;R/are possibly unknown system functions; y.t/ D x1.t/ is the output (measurement); u.t/ is thecontrol (input); w.t/ 2 C.Œ0;1/;R/ is the external disturbance; and b.�/ 2 C.RnCsC2;R/ isthe control coefficient with some uncertainty. We assume that there is a known function b0.�/ 2C.RnC1;R/ serving as a nominal function of b.�/.

It is indicated in [20] that any uniform observable affine SISO nonlinear system can be trans-formed into the lower triangular form (1.1). The state observer design and control for lowertriangular systems have been extensively studied; see, for instance, [21–24] and the referencestherein. In [22, 24], a state observer is designed for an uncertain lower triangular system withoutestimating the system uncertainty. In [21], an output tracking problem for a lower triangular systemis studied by high-gain approach, where the uncertainty is also not estimated. In [23], an unknownconstant in control is estimated on stabilization for lower triangular nonlinear systems. In this paper,we first focus on the design of ESO to estimate uncertainty. The estimation for uncertainty allowsus to design an ESO-based output feedback control without using high gain in the feedback loop,while for system with general uncertainty, the high-gain dominated method must use (generally)high gain in both observer and feedback except some very special cases as shown numerically in[20]. The uncertainty estimation/cancelation strategy can save control energy significantly in prac-tice. This is the most advantage of ADRC. Our ESO can estimate the state of x-subsystem in (1.1),and the ‘total disturbance’ xnC1.t/ that contains un-modeled system dynamic, external disturbance,and uncertainty caused by the deviation of control parameter b.�/ from its nominal value b0.�/:

xnC1.t/ , f .t; x.t/; �.t/; w.t//C .b.t; x.t/; �.t/; w.t// � b0.t; Ox.t///u.t/; (1.2)

where Ox.t/ denote the estimation of state x.t/, obtained by ESO. On the basis of ESO, we designan ESO-based output feedback control to stabilize x-subsystem in (1.1).

We design two kinds of nonlinear ESO for system (1.1). The first one is using the constant high-gain ESO, similar to Guo and Zhao [5]. The main problem for constant high-gain ESO, like manyother high-gain designs, is the peaking value problem. However, the peaking problem can be signifi-cantly reduced through saturation function design by making use of priori bounds of initial state and

Copyright © 2015 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2016; 26:2314–2337DOI: 10.1002/rnc

2316 Z.-L. ZHAO AND B.-Z. GUO

external disturbance. The practical convergence of the closed loop under the ESO-based output feed-back is proved. The advantage of constant high-gain ESO lies in its easy implementation in practiceand allowing a large class of unknown functions f .�/ in (1.1). However, the feedback design by theconstant high-gain ESO is complicated and requires additional information of the priori bounds forinitial state and external disturbance. If the priori bounds estimated are too small, the closed loopmight diverge, and if the estimated bounds are too large, the peaking value becomes large as well.In addition, by the constant high-gain ESO, only practical convergence can be achieved.

To overcome the problems caused by constant high-gain ESO, we design a second kind of ESO byusing time-varying gain, where the tuning gain is a function of time rather than a constant number.The properly selected gain function can not only reduce peaking value of ESO dramatically butalso simplifies the feedback design without priori assumption. More importantly, the asymptoticconvergence instead of practical convergence can be achieved by time-varying gain ESO.

On the other hand, the time-varying gain degrades the ability of ESO to filter the high-frequencynoise, while the constant high-gain ESO does not. A recommended strategy is to apply time-varyinggain ESO in the initial stage and then apply constant high-gain ESO after the observer error reachesa reasonable level. The main contributions of this paper are as follows:

� The class of systems that can be stabilized by ADRC is extended to the lower triangular systemswith large uncertainty.� A time-varying gain method is proposed in the ADRC design.� Instead of practical convergence, we achieve asymptotical stability by the time-varying gain

method.

The remaining part of the paper is organized as follows. In the next section, Section 2, we designa constant high-gain ESO and ESO-based feedback control for system (1.1). The practical stabilityof the closed loop is proved. The asymptotic stability for the closed loop with a special case that thederivative of external disturbance satisfying Pw.t/! 0 as t !1 is also discussed, which covers theproblem discussed in [23]. In Section 3, we propose a time-varying gain ESO-based closed loop forsystem (1.1). The asymptotic stability is proved. Finally, in Section 4, we present some numericalsimulations for illustration of the convergence and the peaking value reduction.

2. ADRC WITH CONSTANT GAIN ESO

The constant gain ESO for (1.1) is designed as follows:

8̂̂̂ˆ̂̂<ˆ̂̂̂̂̂:

POx1.t/ D Ox2.t/C1

rn�1g1.r

n.x1.t/ � Ox1.t///C �1. Ox1.t//;

:::

POxn.t/ D OxnC1.t/C gn.rn.x1.t/ � Ox1.t///C b0.t; Ox1.t/; : : : ; Oxn.t//u.t/;

POxnC1.t/ D rgnC1.rn.x1.t/ � Ox1.t///;

(2.1)

where gi .�/ 2 C.R;R/ is a designed function to be specified later and r 2 RC is the tuningparameter. The main idea of ESO is to choose some appropriate gi .�/ so that when r is large enough,the Oxi .t/ approaches xi .t/ for all i D 1; 2; : : : ; nC 1 and sufficiently large t , where xnC1.t/ is thetotal disturbance defined by (1.2).

The ESO (2.1) based output feedback control is designed as

u.t/ D�u0.satM1.�

n�1 Ox1.t//; satM2.�n�2 Ox2.t//; : : : ; satMn. Oxn.t/// � satMnC1. OxnC1.t//

b0.t; Ox1.t/; : : : ; Oxn.t//;

(2.2)

where � > 0 is a constant, OxnC1.t/ is used to compensate (cancel) the total disturbance, and u0 WRn ! R is chosen so that the following system is globally asymptotically stable:



´.n/.t/ D u0.´; : : : ; ´.n�1//: (2.3)

The continuous differentiable saturation odd function satMi W R ! R is used to avoid the peakingvalue in control, which is defined by (the counterpart for t 2 .�1; 0� is obtained by symmetry)

satMi .�/ D

8̂<:̂�; 0 6 � 6Mi ;

�12�2 C .Mi C 1/� �

1

2M 2i ; Mi < � 6Mi C 1;

Mi C12; � > Mi C 1;

(2.4)

where Mi .1 6 i 6 n/ are constants depending on the bounds of initial value and externaldisturbance to be specified in (2.8) later. It is obvious that j PsatMi .�/j 6 1.

To obtain convergence of the closed-loop system under the ESO-based output feedback control(2.2), we need some assumptions.

Assumption A1 is on functions �i .�/; i D 1; : : : ; n � 1; f .�/; f0.�/, and b.�/.

Assumption A1� j�i .x1; : : : ; xi / � �i . Ox1; : : : ; Oxi /j 6 Lk.x1 � Ox1; : : : ; xi � Oxi /k; L > 0; �i .0; : : : ; 0/ D 0;� f .�/; b.�/ 2 C 1.RnC2Cs;R/; f0.�/ 2 C 1.RnC2Cs;Rs/, and there exists function $.�/ 2C.RnC2Cs;R/ such that

max ¹jf .t; x; �; w/j; krf .t; x; �; w/k; jb.t; x; �; w/j; krb.t; x; �; w/k; kf0.t; x; �; w/k;

krf0.t; x; �; w/kº 6 $.x; �; w/; 8 t 2 R; x 2 Rn; � 2 Rs; w 2 RI and

� There exist positive definite functions V0; W0 W Rs ! R such that Lf0V0.�/ 6 �W0.�/ for all� W k�k > �.x;w/, where � W RnC1 ! R is class K1-function and Lf0V0.�/ denotes the Liederivative of V0.�/ along the vector field f0.�/.

Assumption A2 is on nonlinear functions gi .�/ in (2.1).

Assumption A2gi .�/ 2 C.R;R/; jgi .�/j 6 �i j� j for some �i > 0 and all � 2 R, and there exists a continuous,positive definite, and radially unbounded function V W RnC1 ! R such that

� kek�� @V@e1 .e/; : : : @V

@enC1.e/�� 6 c2V.e/; ˇ̌̌ @V

@enC1.e/ˇ̌̌6 c3V� .e/; 0 < < 1;

�

nXiD1

.eiC1 � gi .e1//@V@i� gnC1.e1/

@V@nC1

.e/ 6 �c1V.e/; e 2 RnC1; c1; c2; c3 > 0; and

� V.e/ 6nXiD1

jei j�i ; i D 1; 2; : : : ; n; �i > 0; e D .e1; e2; : : : ; enC1/ 2 RnC1.

The following assumption is for the nonlinear function u0.�/ in feedback control (2.2).

Assumption A3u0.�/ 2 C 1.Rn;R/; u0.0; : : : ; 0/ D 0, and there exist radially unbounded, positive definitefunctions V.�/ 2 C 1.Rn;R/ and W.�/ 2 C.Rn;R/ such that

n�1XiD1

�iC1@V

@�i.�/C u0.�/

@V@�n.�/ 6 �W.�/;

n�1XiD1

k�k

ˇ̌̌ˇ@V@�i .�/

ˇ̌̌ˇ 6 c4W.�/; 8 � D .�1; : : : ; �n/ 2 Rn:

Set

A1 ,² 2 Rn W V. / 6 max

�2Rn;k�k6dV.�/C 1

³; (2.5)



where d D�Pn

iD1.�n�iˇi C 1/

2�1=2

and ˇi is an upper bound of jxi0j, that is, ˇi > jxi0j wherexi0 is the i th component of initial state x.0/. The continuity and radial unboundedness of V.�/ensures that A1 is a compact set of Rn.

The following assumption is to guarantee that the nominal function b0.�/ is close to b.�/.

Assumption A4b0.�/ 2 C

1.RnC1;R/ satisfies that inf.t;y/2RnC1 jb0.t; y/j > c0 > 0, and all partial derivatives ofb0.�/ are globally bounded. In addition,

^ , supt2Œ0;1/;x2A1;w2B;�2C

jb.t; x; �; w/ � b0.t; x/j 6 min

²c0

2;

c0c1

2c2�nC1

³; (2.6)

where C � Rs is a compact set defined by

C D²& 2 Rs W k&k 6 max

�2A1;�2B�. ; �/

³; B D Œ�B;B� � R; B > 0: (2.7)

The positive constants Mi used in saturation function in (2.2) are chosen so that

Mi > sup ¹j�i j W .�1; : : : ; �n/ 2 A1º ; i D 1; : : : ; n; MnC1 Dc0B1 C^�B2

c0 � ^: (2.8)

Theorem 2.1Let � be chosen so that � > L C c4 C 1. Suppose that w.t/ 2 B; Pw.t/ 2 B for all t > 0 and�.0/ 2 C. Then under Assumptions A1, A2, A3, and A4, the closed-loop system composed of (1.1),(2.1), and (2.2) has following convergence: For any a > 0; � > 0, there exists an r� > 0 such thatfor all r > r�,

� jxi .t/ � Oxi .t/j < � uniformly for t 2 Œa;1/, that is, limr!1 jxi .t/ � Oxi .t/j D 0 uniformlyfor t 2 Œa;1/; 1 6 i 6 nC 1; and� jxj .t/j < � .1 6 j 6 n/ uniformly for t 2 Œtr ;1/, where tr is an r-dependent positive

constant.

ProofLet

�i .t/ D �n�ixi .t/; i .t/ D r

nC1�i .xi .t/ � Oxi .t//: (2.9)

A direct computation shows that8̂̂̂ˆ̂<ˆ̂̂̂̂:

P�1.t/ D ��2.t/C �n�1�1.�1.t/=�

n�1/;

P�2.t/ D ��3.t/C �n�2�2.�1.t/=�

n�1; �2.t/=�n�2/;

:::

P�n.t/ D �u0.satM1.�n�1 Ox1.t//; : : : ; satMn. Oxn.t///C xnC1.t/ � satMnC1. OxnC1.t//;

(2.10)

and8̂̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:̂

P1.t/ D r.2.t/ � g1.1.t///C rn.�1.x1.t// � �1. Ox1.t///;

:::

Pn�1.t/ D r.n.t/ � gn�1.1.t///C r2.�n�1.x1.t/; : : : ; xn�1.t// � �n�1. Ox1.t/; : : : ; Oxn�1.t///;

Pn.t/ D rŒ.nC1.t/ � gn.1.t///C .b0.t; x.t// � b0.t; Ox.t///u�;

PnC1.t/ D �rgnC1.1.t//C PxnC1.t/:(2.11)



Define a compact set A2 � Rn by

A2 ,² 2 Rn W V. / 6 max

�2Rn;k�k6dV.�/

³; (2.12)

where d is the same as in (2.5). It is obvious that A2 � A1. The remainder of the proof is split intothree steps.

Step 1: There exists an r0 > 0 such that ¹�.t/ W t 2 Œ0;1/º � A1 for all r > r0.Since by (2.9), �.0/ D .�nx10; �n�1x20; : : : ; xn0/, and by (2.12), �.0/ 2 A2, we claim that there

exists a T > 0 such that �.t/ 2 A2 for all t 2 Œ0; T �, which shows that the possible escape time for�.t/ from A2 is larger than T . Let

B1 , sup.t;x;�;w/2Œ0;1/�A1�C�B

¹jf .t; x; �; w/j; jb.t; x; �; w/j; krb.t; x; �; w/; krf .t; x; �; w/k;

kf0.t; x; �; w/kº ;

B2 , supjxi j6Mi

¹ju0.x1; : : : ; xn/j; kru0.x1; : : : ; xn/kº ;

B3 , sup.t;x/2RnC1

¹jb0.t; x/j; krb0.t; x/kº ; B4 , supx2A1

kxk:

(2.13)By Assumption A1, Bi < C1 .i D 1; 2; 3; 4/. Before �.t/ escaping from A2; �.t/ 2 A2 �A1. Noticing w.t/ 2 B; �.0/ 2 C, and the input-to-state stable condition on �-subsystem inAssumption A1, if �.t/ 2 A1, then �.t/ 2 C. Because � > 1 and by (2.9), if � 2 A1, then x 2 A1.By (2.10) and (2.13), if � 2 A1, then²

j�n.t/j 6 .2B1 C �B2 C B3 CMnC1/t C jxn0j;

j�i .t/j 6 �.LC 1/B4t C �nC1�i jxi0j:(2.14)

Let

T D min

²1

2B1 C B2 C �B3 CMnC1

;1

.LC 1/B4

³: (2.15)

By (2.14) and (2.15), for any t 2 Œ0; T �,

j�i .t/j 6 �nC1�i jxi0j C 1; 1 6 i 6 n:

This gives

¹�.t/ D .�1.t/; : : : ; �n.t// W t 2 Œ0; T �º � A2: (2.16)

We suppose that the conclusion of step 1 is false and obtain a contradiction. Actually, bycontinuity of �.t/ and (2.16), there exist t2 > t1 > T such that

�.t1/ 2 @A2; �.t2/ 2 @A1; ¹�.t/ W t 2 .t1; t2/º 2 A1 �Aı2; ¹�.t/ W t 2 Œ0; t2�º 2 A1: (2.17)

Finding the derivative of the total disturbance xnC1.t/ with respect to t gives

PxnC1.t/ Dd

dt.f .t; x.t/; �.t/; w.t//C .b.t; x.t/; �.t/; w.t// � b0.t; Ox.t///u.t//

Ddf

dt

ˇ̌̌ˇalong (1.1)

C

db

dt

ˇ̌̌ˇalong (1.1)

�db0

dt

ˇ̌̌ˇalong (2.1)

!u.t/

C .b.t; x.t/; �.t/; w.t// � b0.t; Ox.t///du

dt

ˇ̌̌ˇalong.2:1/

:

(2.18)



Next, computing the derivative of f .�/ along the solution of system (1.1) yields

df

dt

ˇ̌̌ˇalong .1:1/

D@f

@t.t; x.t/; �.t/; w.t//C f0.t; x.t/; �.t/; w.t// �

@f

@�.t; x.t/; �.t/; w.t//

C

n�1XiD1

.xiC1.t/C �i .x1.t/; : : : ; xi .t///@f

@xi.t; x.t/; �.t/; w.t//

C Pw.t/@f

@w.t; x.t/; �.t/; w.t//

C .f .t; x.t/; �.t/; w.t//C b.t; x.t/; �.t/; w.t//u.t//@f

@xn.t; x.t/; �.t/; w.t//:

(2.19)

From the last expression in (2.17), �.t/ 2 A1 for all t 2 Œ0; t2�. This, together with w.t/ 2 B and�.0/ 2 C, gives ¹�.t/ W t 2 Œ0; t2�º � C. By (2.13),ˇ̌̌ˇ̌ dfdt

ˇ̌̌ˇalong .1:1/

ˇ̌̌ˇ̌ 6 B1

�1C B C 2B1 C n.n � 1/.LC 1/B4=2C

�B2 CMnC1

c0

�; 8 t 2 Œ0; t2�:

(2.20)Similarly,ˇ̌̌ˇ̌ dbdt

ˇ̌̌ˇalong .1:1/

ˇ̌̌ˇ̌ 6 B1

�1C B C 2B1 C n.n � 1/.LC 1/B4=2C

�B2 CMnC1

c0

�; 8 t 2 Œ0; t2�:

(2.21)A direct computation shows that

db0

dt

ˇ̌̌ˇalong .2:1/

D@b0

@t.t; Ox.t//C

n�1XiD1

�OxiC1.t/C

1

rn�igi .1.t//C �i . Ox1.t/; : : : ; Oxi .t//

�@b0

@ Oxi.t; Ox.t//C . OxnC1.t/C gn.1.t//C b0.t; Ox.t//u.t//

@b0

@ Oxn.t; Ox.t//:

(2.22)

By Assumptions A1 and A2 and (2.13), for every t 2 Œ0; t2�,ˇ̌̌ˇ̌ db0dt

ˇ̌̌ˇalong .2:1/

ˇ̌̌ˇ̌ 6 B3

0@1CB1 C n.n� 1/.1CL/B4=2C .^CB3/.�B2 CMnC1/=c0 C jnC1.t/j

Cnk1.t/k C

n�1XiD1

�jiC1.t/j CLk1.t/; : : : ; i .t/k C

i

rn�ij1.t/j

�1A :(2.23)

Finding the derivative of u.�/ along the solution of (2.1) yields

du

dt

ˇ̌̌ˇalong .2:1/

D1

b0.t; Ox.t//

n�1XiD1

�OxiC1.t/C

1

rn�igi .1.t//

C �i . Ox1.t/; : : : ; Oxi .t//

!d satMid Oxi

. Oxi .t//@u0

@ Oxi.satM1. Ox1.t//; : : : ; satMn. Oxn.t///

C . OxnC1.t/C gn.1.t//C b0.t; Ox.t//u.t//d satMnd Oxn

. Oxn.t//@u0

@ Oxn.satM1. Ox1.t//; : : : ; satMn. Oxn.t///C rgnC1.r

n.x1.t/ � Ox1.t///

d satMnC1d OxnC1

. OxnC1.t//

�

��u0.satM1.�

n�1 Ox1.t//; : : : ; satMn. Oxn.t/// � satMnC1. OxnC1.t//

b20.t; Ox.t//

db0

dt

ˇ̌̌ˇalong.2:1/

:

(2.24)



Similarly with (2.23), we have

ˇ̌̌ˇ̌ dudt

ˇ̌̌ˇalong .2:1/

ˇ̌̌ˇ̌ 6 B2

c0.1C B1 C n.n � 1/.1C L/B4=2C .^C B3/.�B2 CMnC1/=c0

CjnC1.t/jC�nk1.t/kC

n�1XiD1

�jiC1.t/j C Lk1.t/; : : : ; i .t/kC

�i

rn�ij1.t/j

�!

C�nC1r

c0j1.t/j C

�B2 CMnC1

c20

ˇ̌̌ˇ̌ db0dt

ˇ̌̌ˇalong.2:1/

ˇ̌̌ˇ̌ ; 8 t 2 Œ0; t2�:

(2.25)By (2.18), (2.20), (2.21), (2.23), and (2.25), there exist positive constants N1 and N2 depending onBi ; c0; L;Mi , and �i such that

j PxnC1.t/j 6 N1 CN2k.t/k C�nC1 ^ r

c0j1.t/j; 8 t 2 Œ0; t2�: (2.26)

Let V.�/ and W.�/ be the Lyapunov functions satisfying Assumption A2. The derivative of V.�/along the solution of (2.11) is

dVdt

ˇ̌̌ˇalong .2:11/

D

nC1XiD1

Pi .t/@V@i

..t//

D

n�1XiD1

�r.iC1.t/ � gi .1.t///C r

nC1�i .�i .x1.t/; : : : ; xi .t// � �i . Ox1; : : : ; Oxi //�

@V@i

..t//C rŒnC1.t/ � gn.1.t//C .b0.t; x.t// � b0.t; Ox.t///u�@V@n

..t//

C .�rgnC1.1.t//C PxnC1.t//@V@nC1

..t//:

(2.27)This, together with Assumption A2 and (2.26), gives

dVdt


6 �c1rV..t//C c2�.n � 1/LCN2 C

.�B2 CMnC1/B3

c0

�V..t//

Cc2�nC1 ^ r

c0V..t//C c2c3N1V� ..t//; 8 t 2 Œ0; t2�:

(2.28)

Let r > 4c2..n � 1/LCN2 C .�B2 CMnC1/B3=c0/. By Assumption A4, it follows that

dVdt


6 �c1r4

V..t//C c3N1V� ..t//; 8 t 2 Œ0; t2�: (2.29)

Furthermore, if .t/ ¤ 0, then

d

dt

�V1�� ..t//

�6 �c1.1 � /

4V1�� ..t//C c3N1; 8 t 2 Œ0; t2�: (2.30)

By comparison principle in ordinary differential equation, we have

V1�� ..t// 6 e�c1.1��/r

4 tV1�� ..0//C c3N1Z t

0

e�c1.1��/r

4 .t�s/ds; 8 t 2 Œ0; t2�; (2.31)



where

.0/ D�rn.x10 � Ox10/; r

n�1.x20 � Ox20/; : : : ; x.nC1/0 � Ox.nC1/0�; (2.32)

and . Ox10; Ox20; : : : ; Ox.nC1/0/ is the initial value of (2.1). By Assumption A2,

ˇ̌̌V1�� ..0//

ˇ̌̌6

nXiD1

ˇ̌rnC1�i .xi0 � Oxi0/

ˇ̌�i!1��:

Notice that t1 � T and for any t 2 Œt1; t2�,

e�c1.1��/r

4 tV1�� ..0// 6 e�c1.1��/r

4 T

nXiD1

ˇ̌rnC1�i .xi0 � Oxi0/

ˇ̌�i!1��! 0; r !1: (2.33)

Because V.�/ is continuous, positive definite, and radially unbounded, it follows from lemma 4.3of Khalil [25, p.145] that there exists continuous class K1-function � W NRC ! NRC such thatV./ > �.kk/ for all 2 RnC1. Let

ı D min

²1

2;Mi

2;

min�2A1 W. /

2�nB2 C 3

³: (2.34)

By (2.33), there exists an r�1 > 0 such that

ˇ̌̌ˇe� c1.1��/r4 tV1�� ..0//

ˇ̌̌ˇ 6 .�.ı//1��2

; 8 r > r�1 ; t 2 Œt1; t2�: (2.35)

The second term on the right-hand side of (2.31) satisfies

ˇ̌̌ˇc3N1

Z t

0

e�c1.1��/r

4 .t�s/ds

ˇ̌̌ˇ 6 4c3N1

c1.1 � /r: (2.36)

By (2.33) and (2.36), for any r > r�2 , max®�; r�1 ; .8c3N1/=.c1.1 � /.�.ı//

1�� /¯;V..t// 6

�.ı/ uniformly in t 2 Œt1; t2�. This, together with (2.9), yields

j�n�ixi .t/ � �n�i Oxi .t/j 6 k.�n�1.x1.t/ � Ox1.t//; : : : ; xn.t/ � Oxn.t//k

6 k.rn.x1.t/ � Ox1.t//; : : : ; r.xn.t/ � Oxn.t///k 6 k.t/k 6 ı;8 t 2 Œt1; t2�; r > r

�2 :

(2.37)

From (2.17), jxi .t/j 6 j�n�ixi .t/j D j�i .t/j 6Mi for all t 2 Œt1; t2�, i D 1; 2; : : : ; n. This, togetherwith (1.2) and Assumption A4, yields

jxnC1.t/j 6 B1 C^�B2 CMnC1

c0DMnC1; 8 t 2 Œt1; t2�:

If j�n�i Oxi .t/j 6Mi ; then �n�i Oxi .t/�satMi .�n�i Oxi .t// D 0; i D 1; : : : ; nC1. If j�n�i Oxi .t/j > Mi

and �n�i Oxi .t/ > 0, because ı 6Mi=2, we have �n�i Oxi > Mi . Hence,

j�n�i Oxi .t/ �Mi j D �n�i Oxi .t/ �Mi 6 �n�i Oxi .t/ � �n�ixi .t/ 6 ı 6

1

2; 8 t 2 Œt1; t2�: (2.38)



This, together with (2.4), gives

j�n�i Oxi .t/ � satMi .�n�i Oxi .t//j D � Oxi .t/C

1

2.� Oxi .t//

2 � .Mi C 1/� Oxi C1

2M 2i

D.� Oxi �Mi /

2

2<ı2

2< ı; 8 t 2 Œt1; t2�:

(2.39)

Similarly, (2.39) also holds true when j�n�i Oxi .t/j > Mi and �n�i Oxi .t/ < 0. Therefore, j�n�i Oxi .t/�satMi .�

n�i Oxi .t//j 6 ı for all t 2 Œt1; t2�.Let V.�/ and W.�/ be the Lyapunov functions satisfying Assumption A3. Finding the derivative

of V.�.t// along (2.10) gives

dV

dt


D P�i .t/@V

@�i.�.t//

D

n�1XiD1

��iC1.t/C �

n�i�i .�1.t/=�n�1; : : : �i .t/=�

n�i /� @V@�i.�.t//

C��u0.satM1.�

n�1 Ox1.t//; : : : ; satMn. Oxn.t///C xnC1.t/ � satM1. OxnC1.t//�

@V

@�n.�.t//:

(2.40)By Assumption A1, (2.37), and (2.39),

ju0.satM1.�n�1 Ox1.t//; : : : ; satMn. Oxn.t/// � u0.�.t//j

6 ju0.satM1.�n�1 Ox1.t//; : : : ; satMn. Oxn.t/// � u0.�

n�1 Ox1.t/; : : : ; Oxn.t//j

C ju0.�n�1 Ox1.t/; : : : ; xn.t// � u0.�

n�1x1.t/; : : : ; xn.t//j 6 2nB2ı:(2.41)

Let N3 D sup�2A1

@V

@�n.�/. By Assumption A3, (2.40), (2.41), and (2.34),

dV

dt


D P�i .t/@V

@�i.�.t//

D

n�1XiD1

��iC1.t/C �

n�i�i .�1.t/=�n�1; : : : �i .t/=�

n�i /� @V@�i.�.t//

C��u0.satM1.�

n�1 Ox1.t//; : : : ;satMn. Oxn.t///CxnC1.t/�satM1. OxnC1.t//� @V@�n

.�.t//

6 ��W.�.t//C Ln�1XiD1

k�.t/k

ˇ̌̌ˇ @V@xi .�.t//

ˇ̌̌ˇC 2.�nB2 C 1/N3ı

6 �W.�.t//C 2.�nB2 C 1/N3ı < 0; 8 t 2 Œt1; t2�:(2.42)

This shows that V.�.t// is monotone decreasing in t 2 Œt1; t2�. But by (2.17), (2.5), and (2.12),V.�.t2// D V.�.t1// C 1, which is a contradiction. Therefore, ¹�.t/ W t 2 Œ0;1/º � A1 for allr > r�2 . The claim of step 1 then follows.

Step 2: .t/! 0 uniformly in t 2 Œa;1/ as r !1.By step 1, �.t/ 2 A1 for all t 2 Œ0;1/ as r > r�2 . Similar to (2.26), we can obtain that

j PxnC1.t/j 6 N1 CN2k.t/k C�nC1 ^ r

c0j1.t/j; 8 t 2 Œ0;1/; r > r

�2 : (2.43)



Similar with (2.31), (2.43) implies that

V1�� ..t// 6 e�c1.1��/r

4 tV1�� ..0//C c3N1Z t

0

e�c1.1��/r

4 .t�s/ds; t 2 Œ0;1/; r > r�2 : (2.44)

Because for any a > 0; � > 0, and t 2 Œa;1/,

e�c1.1��/r

4 tV1�� ..0// 6 e�c1.1��/r

4 aV1�� ..0//! 0 as r !1; (2.45)

there exists an r�3 such that when r > r�3 ; e�c1.1��/r

4 tV1�� ..0// < �2

uniformly in t 2 Œa;1/. Let

r�4 D max

²r�2 ; r

�3 ;

8c2c3N1

c1.1 � /�1��

³:

By (2.44) and (2.45), jxi .t/� Oxi .t/j 6 kk 6 � uniformly in t 2 Œa;1/ as r > r�4 . This shows thatlimr!1 jxi .t/ � Oxi .t/j D 0 uniformly in t 2 Œa;1/.

Step 3: �.t/! 0 as t !1 and r !1.For the positive definite and radially unbounded Lyapunov function W.�/, it follows from lemma

4.3 of Khalil [25, p.145] that there exits class K1-function ~ W NRC ! NRC such that ~.k k/ 6 W. /for all 2 Rn. For any given � > 0, by step 2, there exists an r� > r�4 such that k.t/k < �1 D~.�/=.3.�nB2 C 1/N3/ uniformly in t 2 Œa;1/ as r > r�. Similar to (2.42), we have

dV

dt


6 �W.�.t//C 2.�nB2 C 1/N3�1

6 �~.k�.t/k/C 2.�nB2 C 1/N3�1; 8 t 2 Œa;1/:(2.46)

So, if k�.t/k > � , then

dV

dt


6 �~.�/C 2.�nB2 C 1/N3�1 D �1

3~.�/ < 0: (2.47)

Therefore, there exists tr such that k�.t/k 6 � for all t 2 Œtr ;1/. By (2.9) and � > 1, we havejxj .t/j < � .1 6 j 6 n) uniformly in t 2 Œtr ;1/. This completes the proof of the theorem. �

The simplest example of ADRC satisfying conditions of Theorem 2.1 is the linear ADRC; thatis, gi .�/; i D 1; : : : ; nC 1 in ESO (2.1) and u0.�/ in feedback control are linear functions. Let

gi .�/ D ki�; u0.y1; : : : ; yn/ D ˛1y1 C � � � C ˛nyn: (2.48)

Define the matrices K and A as follows:

K D

0BBBB@�k1 1 0 � � � 0�k2 0 1 � � � 0:::

::::::: : :

:::

�kn 0 0 � � � 1�knC1 0 0 � � � 0

1CCCCA.nC1/�.nC1/

A D

0BBBB@0 1 0 � � � 00 0 1 � � � 0:::

::::::: : :

:::

0 0 0 � � � 1˛1 ˛2 0 � � � ˛n

1CCCCAn�n

: (2.49)

Let �max.P / and �min.P / be the maximal and minimal eigenvalues of matrix P that is the uniquepositive definite matrix solution of the Lyapunov equation PK C K>P D �I.nC1/�.nC1/ for.nC 1/-dimensional identity matrix I.nC1/�.nC1/.

Corollary 2.1Let � D L C 1. Suppose that w.t/ 2 B and Pw.t/ 2 B for a compact set B , Œ�B;B� � R, andmatrices K and A in (2.49) are Hurwitz. Then, under Assumptions A1 and A4 with �nC1 D knC1

and D 12; c2 D

2�max.P /�min.P /

, the closed-loop system composed of (1.1), (2.1), and (2.2) with linear

function (2.48) has the following convergence: For any a > 0, there exists an r� > 0 such that forany r > r�,



� jxi .t/� Oxi .t/j 6 1rnC2�i

uniformly in t 2 Œa;1/, where�2 > 0 is an r-independent constant,that is, limr!1 jxi .t/ � Oxi .t/j D 0 uniformly in t 2 Œa;1/; 1 6 i 6 nC 1; and� jxj .t/j <

2r

uniformly in t 2 Œtr ;1/, where �2 > 0 is an r-independent constant and tr isan r and initial value dependent positive constant.

ProofLet the Lyapunov functions V W RnC1 ! R and V;W W Rn ! R be defined as V./ D >P for 2 RnC1; V .�/ D �>Q�; W.�/ D k�k2 for � 2 R2. Let Q be the unique positive definite matrixsolution of the Lyapunov equation QAC A>Q D �In�n for n-dimensional identity matrix In�n.Then it is easy to verify that all conditions of Assumptions A2 and A3 are satisfied. The results thenfollow directly from Theorem 2.1. �

We point out that in Theorem 2.1, we have only practical stability by the constant high-gain ESO.Now, we indicate a special case where the derivative of the external disturbance converges to 0 astime goes to infinity:

limt!1

Pw.t/ D 0: (2.50)

This case covers the problem studied in [23] as a typical example. For this special case, we have theasymptotic stability. To this purpose, we need the following assumption.

Assumption A1�. There exist continuous functions �n W Rn ! R; Nf W Rn ! R and a positiveconstant L1 > 0 such that

� f .t; x; �; w/ D �n.x/C Nf .�; w/; 8 t 2 Œ0;1/; x 2 Rn; � 2 Rs; w 2 R;� �i .�/; i D 1; 2; : : : ; n and Nf .�/ are continuous differentiable, kr�i .x1; : : : ; xi /k 6Lk.x1; : : : ; xik; �i .0; : : : ; 0/ D 0; kr Nf .�; w/k 6 L, x 2 Rn; � 2 Rs; w 2 R;� kf0.t; x; �; w/k 6 Lkxk; 8 t 2 Œ0;1/; x 2 Rn; � 2 Rs; w 2 R; and� b.t; x; �; w/ � b0; b0 ¤ 0 is a constant real number.

For this case, we also allow the nonlinear function �i .�/ to be unknown but satisfies some prioriconditions in Assumption A1�. The nonlinear ESO that is independent of �i .�/ for this case isdesigned as follows:8̂̂

ˆ̂̂̂̂ˆ̂̂̂<ˆ̂̂̂̂ˆ̂̂̂̂̂:

POx1.t/ D Ox2.t/C1

�n�1g1.�

n.x1.t/ � Ox1.t///;

POx2.t/ D Ox3.t/C1

�n�2g2.�

n.x1.t/ � Ox1.t///;

:::

POxn.t/ D OxnC1.t/C gn.�n.x1.t/ � Ox1.t///C b0u.t/;

POxnC1.t/ D �gnC1.�n.x1.t/ � Ox1.t///;

(2.51)

where the nonlinear functions gi .�/s are chosen to satisfy the following assumption.

Assumption A2�. gi .�/ 2 C.R;R/, and there exists continuous, positive definite, and radiallyunbounded functions V; W W RnC1 ! R such that

�

nXiD1

.eiC1 � gi .e1//@V@ei� gnC1.e1/

@V@enC1

.e/ 6 �W.e/; and

� kek2 C krV.e/k2 CnC1XiD1

jgi .e1/j2 6 c1W.e/;

ˇ̌̌ˇgnC1.e1/ @V

@enC1.e/

ˇ̌̌ˇ 6 �W.e/;8 e D

.e1; : : : ; enC1/> 2 RnC1; c1 > 0.



The ESO (2.51) based output feedback control is designed as

u.t/ Du0.�

n˛ Ox1.t/; �.n�1/˛ Ox2.t/; : : : ; �

˛ Oxn.t// � OxnC1.t/

b0; (2.52)

where ˛ 2 .n=.n C 1/; 1/ and the nonlinear function u0 W Rn ! R is supposed to satisfy thesucceeding assumption.

Assumption A3�. u0.�/ 2 C 1.Rn;R/; kru0k 6 L; u0.0; : : : ; 0/ D 0, and there exist radiallyunbounded, positive definite functions V.�/ and W.�/ 2 C 1.Rn;R/ such that

�

n�1XiD1

xiC1@V

@xi.x/C u0.x/

@V@xn

.x/ 6 �W.�/; and

� k�k2 C krV.�/k2 6 c2W.�/; c2 > 0; � D .�1; : : : ; �n/ 2 Rn.

For the high-gain parameter � in (2.51) and (2.52), we need the following assumption.

Assumption A4�.

� > max

´1; 2c1LC 4c1L

2; 4

�nc2LC

Lc2

2

� 1˛

;�pc1c2.1C nL/

� 21�˛

μ:

Theorem 2.2If limt!1 Pw.t/ D 0 and Assumptions A1�, A2�, A3�, and A4� are satisfied, then the closed-loopsystem composed of (1.1), (2.51), and (2.52) is convergent in the sense that

limt!1jxi .t/ � Oxi .t/j D 0; i D 1; 2; : : : ; nC 1; lim

t!1jxj .t/j D 0; j D 1; 2; : : : ; n;

where the total disturbance xnC1.t/ , Nf .�.t/; w.t//.

ProofLet .t/ and �.t/ be defined as

�i .t/ D �.nC1�i/˛xi .t/; i D 1; 2; : : : ; n; j .t/ D �

nCj�1.xj .t/ � Oxj .t//; j D 1; 2; : : : ; nC 1:(2.53)

A direct computation shows that �i .t/ and j .t/ satisfy8̂̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:̂

P�1.t/ D �˛�2.t/C �

n˛�1.�1.t/=�n˛/;

P�2.t/ D �˛�3.t/C �

.n�1/˛�2.�1.t/=�n˛; �2.t/=�

.n�1/˛/;

:::

P�n.t/ D �˛u0.�

n˛ Ox1.t/; : : : ; �˛ Oxn.t//

C �˛�n.�1.t/=�n˛; : : : ; �n.t/=�

˛/C �˛.xnC1.t/ � OxnC1.t//;

(2.54)

and 8̂̂̂ˆ̂̂̂<ˆ̂̂̂̂ˆ̂:

P1.t/ D �.2.t/ � g1.1.t///C �n�1.�1.t/=�

n˛/;

:::

Pn�1.t/ D �.n.t/ � gn�1.1.t///C �2�n�1.�1.t/=�

n˛; : : : ; �n.t/=�2˛/;

Pn.t/ D �.nC1.t/ � gn.1.t///C ��n.�1.t/=�n˛; : : : ; �n.t/=�

˛/;

PnC1.t/ D ��gnC1.1.t//C PxnC1.t/:

(2.55)



Let V.�/; W.�/; V .�/, and W.�/ be the Lyapunov functions satisfying Assumptions A2� and A3�,respectively. Define V ; W W RnC1 ! NRC as

V .�; / D V.�/C V./; W .�; / D W.�/CW./; � 2 Rn; 2 RnC1: (2.56)

Finding the derivative of V .�/ along the solutions of the error Equations (2.54) and (2.55) to obtain

dV

dt

ˇ̌̌ˇalong .2:54/ and .2:55/

D

nXiD1

P�i .t/@V

@�i.�.t//C

nC1XjD1

Pj .t/@V

@j .t/

D

n�1XiD1

��˛�iC1.t/C �

.nC1�i/˛�i

��1.t/

�n˛; : : : ;

�i .t/

�.nC1�i/˛

��@V

@�i.�.t//

C �˛u0.�1.t/; : : : ; �n.t//@V

@�n.�.t//C

��˛�n

��1.t/

�n˛; : : : ;

�n.t/

�˛

�C�˛nC1.t/

C .u0 .�n˛ Ox1.t/; : : : ; �

˛ Oxn.t//� u0 .�n˛x1.t/; : : : ; �

˛xn.t////@V

@�n.�.t//

C

nXiD1

��.i .t/� gi .1.t///C �

nC1�i�i

��1.t/

�n˛; : : : ;

�i .t/

�.nC1�i/˛

��@V@i

..t//

� �gnC1.1.t//@V

@nC1..t//

C

Pw.t/

@ Nf

@w.�.t/;w.t//C f0.t; x.t/; �.t/;w.t// �

@ Nf

@�.�;w/

!@V

@nC1..t//:

(2.57)

By Assumptions A1�, A2�, and A3�, we obtain further that

dV

dt

ˇ̌̌ˇ.2:54/;.2:55/

6 ��˛W.�.t//� �W..t//

CL

nXiD1

k�.t/k

ˇ̌̌ˇ @V@�i .�.t//

ˇ̌̌ˇCLk.t/k

ˇ̌̌ˇ @V@�n .�.t//

ˇ̌̌ˇ

C �˛k.t/k

ˇ̌̌ˇ @V@�n .�.t//

ˇ̌̌ˇCL

nXiD1

�.nC1�i/.1�˛/k�.t/k

ˇ̌̌ˇ @V@i ..t//

ˇ̌̌ˇ

CL2k.t/k

ˇ̌̌ˇ @V@nC1

..t//

ˇ̌̌ˇCLj Pw.t/j

ˇ̌̌ˇ @V@nC1

.�.t//

ˇ̌̌ˇ

6 ��˛W.�.t//� �W..t//CN1W.�.t//CN2W..t//

CN3�˛pW.�.t//

pW..t//CLj Pw.t/j

ˇ̌̌ˇ @V@nC1

.�.t//

ˇ̌̌ˇ

6 ��˛

2W.�.t//� �W..t//CN1W.�.t//CN2W..t//

CN23�˛

2W ..t//CLj Pw.t/j

ˇ̌̌ˇ @V@nC1

.�.t//

ˇ̌̌ˇ

6 ��˛

4W .�.t/; .t//C

pc2Lj Pw.t/j

pW .�.t/; .t//;

(2.58)

where

N1 D nc2LCLc2

2; N2 D

c1L

2C c1L

2; N3 Dpc1c2.1C nL/:

By hypothesis (2.50), there exists t1 > 0 such that j Pw.t/j < �˛=�8pc2L

�for all t > t1. For V .�/

and W .�/ defined in (2.56), there exist continuous class K1-functions �i .i D 1; 2/ W NRC ! NRC

satisfying

�1.V .e// 6W .e/ 6 �2.V .e//; 8 e 2 R2nC1: (2.59)



This, together with (2.58), shows that for any t > t1, if V .�.t/; .t// > ��11 .1/, thenW .�.t/; .t// > 1 and

dV

dt

ˇ̌̌ˇalong .2:54/ and .2:55/

6 ��˛

4W .�.t/; .t//C

�˛

8

pW .�.t/; .t// 6 ��

˛

8< 0: (2.60)

So, there exists t2 > t1 such that V ..t// 6 ��11 .1/ and W ..t// 6 �2 ı ��11 .1/ for all t 2 .t2;1/.For any given � > 0, by (2.50), there exists t3 > t2 such that j Pw.t/j 6 �˛�1.�/

8Lpc2�2ı�

�11.1/

for all

t3 > t2. Hence, for t > t3, if V ..t// > � , then

dV

dt

ˇ̌̌ˇalong .2:54/ and .2:55/

6 ��˛

4W .�.t/; .t//C L

pc2 �2 ı �

�11 .1/j Pw.t/j 6 ��

˛�1.�/

8: (2.61)

So, there exists t4 > t3 such that V ..t// 6 � for all t > t4, that is, limt!1 V .�.t/; .t// D 0.Because V .�/ is continuous and positive definite, there exists class K1-function O� W RC ! RC

such that k.�.t/; .t//k 6 O�.V .�.t/; .t///. Therefore, limt!1k.�.t/; .t//k D 0. This completes the

proof of the theorem. �Same with Corollary 2.1, for this example, the linear ESO, that is, the functions gi .�/ and u0.�/ in

(2.51) and (2.52) chosen as that in (2.48) with Hurwitz matricesK and A defined by (2.49), satisfiesall conditions of Theorem 2.2.

3. ADRC WITH TIME-VARYING GAIN ESO

In this section, we propose a time-varying gain ESO for system (1.1) as follows:8̂̂ˆ̂̂̂̂ˆ̂̂̂<ˆ̂̂̂̂ˆ̂̂̂̂:̂

POx1.t/ D Ox2.t/C1

%n�1.t/g1.%

n.t/.x1.t/ � Ox1.t///C �1. Ox1.t//;

POx2.t/ D Ox3.t/C1

%n�2.t/g2.%

n.x1.t/ � Ox1.t///C �2. Ox1.t/; Ox2.t//;

:::

POxn.t/ D OxnC1.t/C gn.%n.t/.x1.t/ � Ox1.t///C b0u.t/;

POxnC1.t/ D %.t/gnC1.%n.t/.x1.t/ � Ox1.t///;

(3.1)

where gi .�/s are nonlinear functions satisfying Assumption A2� and %.�/ 2 C.Œ0;1/! RC/ is thegain function to be required to satisfy the following assumption. For simplicity, we assume that b0is a constant nominal value of b.�/ in this section.

Assumption A5%.�/ 2 C 1

�NRC; NRC

�; %.t/ > 0; P%.t/ > a > 0, and

ˇ̌̌P%.t/%.t/

ˇ̌̌6M for all t > 0, where a > 0;M > 0.

The ESO (3.1) based output feedback control is designed as follows:

u.t/ D�u0.�

n�1 Ox1.t/; : : : ; Oxn.t// � OxnC1.t/

b0; (3.2)

where u0.�/ is a nonlinear function satisfying Assumption A3�, Oxi .t/ is the state of ESO (3.1), andOxnC1.t/ is used to compensate (cancel) the total disturbance xnC1.t/ defined by

xnC1.t/ , f .t; x.t/; �.t/; w/C .b.t; w/ � b0/u.t/: (3.3)

To obtain the convergence, we need the following assumption on the nonlinear functionsf .�/; f0.�/, and �i .�/ in system (1.1), which is stronger than Assumption A1.



Assumption A1��. jf .t; x; �; w/j C kf0.t; x; �; w/k 6 Lk.x;w/kI krf .t; x; �; w/k 6Lkwk; k�i .x1; : : : ; xi / � �i . Ox1; : : : ; Oxi /k 6 L;L > 0; �i .0; : : : ; 0/ D 0.

The following assumption is the conditions on b.�/ and its nominal value b0.

Assumption A4��. b.t; x; �; w/ D b.t; w/; jb.t; w/j C krb.t; w/k 6 L, and its nominal valueb0 2 R satisfy jb.t; w/ � b0j 6 ^ , ^�=.2b0/ for all .t; w/ 2 R2.

Theorem 3.1Let � be chosen so that � > .nLC3/c2

2C 1. Suppose that w.t/ and Pw.t/ are uniformly bounded and

Assumptions A1��, A2�, A3�, A4��, and A5 are satisfied. Then, the closed loop composed of(1.1), (3.1), and (3.2) is asymptotically stable:

limt!1k.t/k D 0; lim

t!1k�.t/k D 0: (3.4)

ProofBy Assumptions A1��, A2�, and A3�, we can estimate the derivative of the total disturbancexnC1.t/ given in (3.3) as

j PxnC1.t/j 6 C .1C k.t/k C k�.t/k/Cˇ̌̌ˇb.t; w.t// � b0b0

ˇ̌̌ˇ %.t/jgnC1.1.t//j; (3.5)

where C is an r-independent constant. Let

�i .t/ D �n�ixi .t/; i .t/ D %

nC1�i .t/.xi .t/ � Oxi .t//: (3.6)

A straightforward calculation shows that

8̂̂̂ˆ̂̂<ˆ̂̂̂̂:̂

P�1.t/ D ��2.t/C �n�1�1.�1.t/=�

n�1/;

P�2.t/ D ��3.t/C �n�2�2.�1.t/=�

n�1; �2.t/=�n�2/;

:::

P�n.t/ D �u0.�n�1 Ox1.t/; : : : ; Oxn.t//C xnC1.t/ � OxnC1.t/;

(3.7)

and

8̂̂̂ˆ̂̂̂̂̂ˆ̂̂̂̂ˆ̂<ˆ̂̂̂̂̂ˆ̂̂̂̂ˆ̂̂̂̂:

P1.t/ D %.t/.2.t/ � g1.1.t///C %n.t/.�1.x1.t// � �1. Ox1.t///C

n P%.t/

%.t/1.t/;

:::

Pn�1.t/ D %.t/.n.t/ � gn�1.1.t///C.n � 1/ P%.t/

%.t/2.t/

C %2.t/.�n�1.x1.t/; : : : ; xn�1.t// � �n�1. Ox1.t/; : : : ; Oxn�1.t///;

Pn.t/ D %.t/.nC1.t/ � gn.1.t///CP%.t/

%.t/n.t/;

PnC1.t/ D �%.t/gnC1.1.t//C PxnC1.t/:

(3.8)

Let V.�/;W.�/, and V.�/;W.�/ be the Lyapunov functions satisfying Assumptions A2� and A3�,respectively. Let V ;W W R2nC1 ! NRC be the same as that in (2.56). Finding the derivative of V .�/



along the solutions of (3.7) and (3.8) gives

dV

dt

ˇ̌̌ˇalong .3:7/ and .3:8/

D

n�1XiD1

��iC1.t/C �

n�i�i

��1.t/

�n�1; : : : ;

�i .t/

�.n�i/˛

��@V

@�i.�.t//

C �u0.�1.t/; : : : ; �n.t//@V

@�n.�.t//C nC1.t/

@V

@�n.�.t//

C��u0.�

n�1 Ox1.t/; : : : ; Oxn.t// � �u0.�n�1x1.t/; : : : ; xn.t//

� @V@�n

.�.t//

C

n�1XiD1

�%.t/.iC1.t/ � gi .1.t///C%

nC1�i .t/.�i .x1.t/; : : : ; xi .t//��i . Ox1; : : : ; Oxi //� @V@i

..t//

C

nXiD1

.nC 1 � i/ P%.t/ji .t/j

%.t/

ˇ̌̌ˇ @V@i ..t//

ˇ̌̌ˇ

C .%.t/.nC1.t/ � gn.1.t///@V@n

..t//C .�%.t/gnC1.1.t//C PxnC1.t//@V@nC1

..t//:

(3.9)

By Assumption A5, there exists a t1>0 such that %.t/ > max¹�; 2.Cc1CC 2c1CnLc1Cn.nC1/Mc1/º for all t > t1. This, together with (3.9), gives

dV

dt


6 ��W.�.t// � %.t/W..t//C

n�1XiD1

Lk�.t/k

ˇ̌̌ˇ@V@�i .�.t//

ˇ̌̌ˇC 2k.t/k

ˇ̌̌ˇ @V@�n .�.t//

ˇ̌̌ˇ

C

n�1XiD1

Lk.t/k

ˇ̌̌ˇ @V@i ..t//

ˇ̌̌ˇC %.t/jgnC1.1.t//j

ˇ̌̌ˇb.t; w.t// � b0b0

ˇ̌̌ˇˇ̌̌ˇ @V@nC1

..t//

ˇ̌̌ˇ

C C

1C k.t/k C k�.t/k C

nXiD1

jgi .1.t//j

! ˇ̌̌ˇ @V@nC1

..t//

ˇ̌̌ˇ

CM

nXiD1

.nC 1 � i/ji .t/j

ˇ̌̌ˇ @V@i ..t//

ˇ̌̌ˇ

6 ��W.�.t// � %.t/W..t//CnLc2

2W.�.t//C c1W..t//C c2W.�.t//C

nLc1

2W..t//

C %.t/^�

b0W..t//C C

pc1pW..t//C Cc1W..t//C

C 2c1

2W..t//C

c2

2W.�.t//

6 ��

.nLC 3/c2

2

�W.�.t//C C

pc1pW..t//

�

�%.t/ � %.t/

^�

b0CnLC 2C 2C C 2C 2 C n.nC 1/M

2c1

�W..t//

6 �W.�.t// � %.t/4

W..t//C Cpc1pW..t//:

(3.10)



Now, we show that W.�.t// C W..t// is uniformly ultimately bounded. In fact, ifW.�.t// C W..t// > 16max

®1; c1C

2¯, then either W..t// > 8max

®1; c1C

2¯

orW..t// 6 8max

®1; c1C

2¯

and W.�.t// > 8max®1; c1C

2¯. In the first case,

dV

dt

ˇ̌̌ˇalong .3:7/;.3:8/

6 �%.t/4

W..t//C Cpc1pW..t//

6pW..t//

�

pW..t//

4C Cpc1

!6 �C 2c1; t > t1:

(3.11)

In the second case,

dV

dt

ˇ̌̌ˇalong .3:7/;.3:8/

6 �W.�.t//C Cpc1pW..t// 6 �

�8 � 2

p2�C 2c1; t > t1: (3.12)

Hence, there exists positive constant t2 > t1 such thatW.�.t//CW..t// 6 16max®1; c1C

2¯

forall t > t2. This, together with (3.5), produces

j PxnC1.t/j 6 D C^

b0%.t/jgnC1.1.t//j; t > t2; D > 0: (3.13)

Finding the derivative of V.�/ along the solution of (3.8) yields

dVdt

ˇ̌̌ˇalong .3:8/

6 �%.t/W..t//C

�D C

^

b0%.t/jgnC1.1.t//j

� ˇ̌̌ˇ @V@nC1

..t//

ˇ̌̌ˇ

6 �%.t/2

W..t//C c1DpW..t//; 8 t > t2:

(3.14)

For positive definite and radial unbounded Lyapunov functions V.�/ and W.�/, there exist classK1 functions �ij .�/ .i; j D 1; 2/ such that,

�11.k k/ 6 V. / 6 �12.k k/; �21.k k/ 6W. / 6 �22.k k/; 2 RnC1: (3.15)

From Assumption A5, for any � > 0, there exists a positive constant t3 > t2 such that %.t/ >4c1D

��21 ı �

�112 ı �11.�/

��1=2. This, together with (3.15) and (3.14), yields that if V..t// > � ,

then

dVdt


6 �c1Dp.t/ 6 �c1D

��21 ı �

�112 ı �11.�/

��1=2< 0: (3.16)

Therefore, there exists t4 > t3 such that V..t// 6 �1.�/ for all t > t4, and hence,

k.t/k 6 ��111 .V..t/// 6 �; 8 t 2 Œt4;1/: (3.17)

This shows that limt!1 k.t/k D 0.Finding the derivative of V.�/ along the solution of (3.7) gives

dV

dt

ˇ̌̌ˇalong .3:7/

6 �W.�.t//C c2k.t/kpW.�.t//; 8 t > t3: (3.18)

By positive definiteness and radial unboundedness of V.�/ and W.�/, there exist class K1-functionsQ�ij .�/; i; j D 1; 2 such that

Q�11.k k/ 6 V. / 6 Q�12.k k/; Q�21 6 W. / 6 Q�22.k k/; 2 Rn: (3.19)

Similar to (3.17), there exists t4 > t3 such that



k.t/k 6q�Q�21 ı Q�

�112 ı Q�11

�.�/=.2c2/; 8 t > t4:

This, together with (3.17), shows that if V.�.t// > Q�11.�/, then

dV

dt

ˇ̌̌ˇalong .3:8/

6 ��Q�21 ı Q�

�112 ı Q�11

�.�/

2c2; 8 t > t4: (3.20)

Therefore, there exists a constant t5 > 0 such that V.�.t// 6 Q�11.�/ for all t > t5. By (3.19),k�.t/k 6 Q��111 .V .�.t/// 6 � for t > t5. This shows that limt!1 k�.t/k D 0. The result thenfollows from (3.6). �

Remark 3.1As indicated in Section 1, the time-varying gain ESO degrades the ability of ESO to filter high-frequency noise, while the constant gain ESO does not. In practical applications, we can use time-varying gain %.t/ as follows: (a) Given a small initial value %.0/ > 0; (b) from the constant high gain,we obtain the convergent high-gain value r that can also be obtained by trial-and-error experimentfor practical systems; and (c) our gain function is usually supposed to grow continuously from%.0/ > 0 to r . For instance, P%.t/ D a%.t/; a > 0. In this case, we can compute the switching time asln.r=%.0//=a where a is used to control the convergent speed and the peaking value. The larger a is,the faster the convergence but with larger peaking, while the smaller a is, the lower the convergencespeed and smaller peaking.

4. NUMERICAL SIMULATIONS

In this section, we present several numerical simulations for illustration.

Example 4.1Consider the following system8̂<

:̂Px1.t/ D x2.t/C sin.x1.t//;

Px2.t/ D f .t; x1; x2; �.t/; w.t//C b.t; x1.t/; x2.t/; �.t/; w.t//u.t/;

P�.t/ D f0.t; x1; x2; �.t/; w.t//;

(4.1)

where f .�/ and f0.�/ 2 C.R5;R/ are unknown nonlinear functions, and w.t/ is the externaldisturbance.

As in Corollary 2.1, we first design a linear ESO to estimate the states x1.t/; x2.t/, and the ‘totaldisturbance’ x3.t/ , f .t; x1.t/; x2.t/; �.t/; w.t//C .b.t; x1.t/; x2.t/; �.t/; w.t//�b0/u.t/ whereb0 is a constant nominal value of b.�/, as follows:8̂<

:̂Ox1.t/ D Ox2.t/C 6r.x1.t/ � Ox1.t//C sin Ox1.t/;

Ox2.t/ D Ox3.t/C 11r2.x1.t/ � Ox1.t//;

Ox3.t/ D 6r3.x1.t/ � Ox1.t//:

(4.2)

A linear ESO (4.2) based output feedback control u.�/ is designed to stabilize the x-subsystemof (4.1):

u.t/ D�8sat10. Ox1.t// � 4sat10. Ox2.t// � sat10. Ox3.t//

b0: (4.3)

We suppose that the functions f .�/; f0.�/, and the external disturbance w.t/ (it is seen from(4.2) and (4.3) that these functions are not used to design the ESO (4.2) and feedback (4.3)) fornumerical simulation



´f .t; x1; x2; �; w/ D te

�t C x21 C sin x2 C cos � C w;

f0.t; x1; x2; �; w/ D ��x21 C w

2��; w.t/ D sin.2t C 1/:

(4.4)

The control amplification coefficient is given by

b.t; x1; x2; �; w/ D 2C1

10sin.t C x1 C x2 C w/: (4.5)

The nominal value for b.�/ is chosen as b0 D 2. Set r D 200, integration step h D 0:001. Thenumerical results are plotted in Figure 1, and the local amplifications of Figure 1(b) and (c) areplotted in Figure 2. It is seen from Figures 1(a)–(c) and 2(b) that the stabilization is very satisfactory.Figure 1(c) shows that the total disturbance estimation is also satisfactory. But the large peakingvalues for Ox2.t/ and Ox3.t/ that reach almost 2:5 � 104 near the initial time are observed. However,the saturations of Oxi .t/; i D 1; 2; 3make the control value less than 10 that is seen from Figure 1(c).Because we do not apply high gain in the feedback loop like high-gain control method, there arealmost no peaking phenomena for system states x1.t/ and x2.t/, which is an advantage of ADRCapproach.

Next, we design a nonlinear time-varying gain ESO for system (4.1) as8̂̂<̂ˆ̂̂:Ox1.t/ D Ox2.t/C 6%.t/.x1.t/ � Ox1.t//C

1

%.t/ˆ

�x1.t/ � Ox1.t/

%2.t//

�C sin Ox1.t/;

Ox2.t/ D Ox3.t/C 11%2.t/.x1.t/ � Ox1.t//;

Ox3.t/ D 6%3.t/.x1.t/ � Ox1.t//;

(4.6)

where the nonlinear function ˆ W R! R is given by

ˆ.�/ D

8̂<:̂

14�; � > �=2;

sin �4�; ��=2 6 � 6 �=2;

� 14�; � < ��=2:

(4.7)

Figure 1. The numerical results of system (4.1) under the saturated control (4.3) with constant gain ESO(4.2).

Figure 2. Magnifications of Figure 1 (b) and (c), and control u.



In what follows, we use the time-varying gain %.t/ D e0:6t for the numerical simulation. A linearESO (4.10) based output feedback control u.�/ is designed as follows:

u.t/ D�8 Ox1.t/ � 4 Ox1.t/ � Ox3.t/

b0: (4.8)

The nonlinear function class required by Theorem 2.1 is more general than the class required byTheorem 3.1 where the nonlinear functions are required to be Lipschitz continuous that is not sat-isfied by class (4.4). So, we choose the nonlinear functions f .�/ and f0.�/ for system (4.1) with thetime-varying gain ESO (4.6) as

f .t; x1; x2; �; w/ D te�t C sin x2 C cos � C w;

f0.t; x1; x2; �; w/ D � sin.�.t//jx1.t/j; w.t/ D sin.2t C 1/:(4.9)

The other functions and parameters are the same as Figure 1. It is seen from Figure 3(a)–(c) that thestabilization and ‘total disturbance’ estimation are also very satisfactory. Meanwhile, there are nopeaking values for Ox2.t/ and Ox3.t/.

Generally, the large gain value needs small integration step. Therefore, as recommended inSection 1, in practice, the time-varying gain should be small value in the beginning and graduallyincreases to a large constant high gain for which we choose as8̂<

:̂%.0/ D 1;

P%.t/ D 5%.t/ if %.t/ < 200;

P%.t/ D 0; otherwise;

(4.10)

which means that for t 2 Œ0; ln 200=5�; %.t/ D e5t , and %.t/ D 200 for every t > ln 200=5. Hence,the switch time is ln200

5as indicated in Remark 3.1. The other functions and parameters are the same

as Figure 3. It is seen from Figure 4(a)–(c) that the stabilization and ‘total disturbance’ estimationare also very satisfactory. Meanwhile, there are also no peaking values for Ox2.t/ and Ox3.t/.

Finally, we compute system (4.1) with nonlinear functions given in (4.9) by using constant gainESO and saturated feedback control. Let the feedback control be (4.3), where Oxi .t/ comes from (4.6)with constant gain % � 200, and let the other functions and parameters be the same as in Figure 4.

Figure 3. The numerical results of system (4.1) with (4.9) under feedback control (4.8) and nonlinear ESO(4.6) with time-varying gain %.t/ D e0:6t .

Figure 4. The numerical results of system (4.1) with (4.9) under feedback control (4.8) and nonlinear ESO(4.6) with time-varying gain.



The numerical results are plotted in Figures 5 and 6. It is seen from Figures 5 and 6 that althoughthe stabilization and total disturbance estimation are also satisfactory, the peaking values for Ox2.t/and Ox3.t/ are obviously observed compared with Figure 4.

To end this section, we use a simple example to illustrate the reason that ESO using time-varyinggain can diminish the notorious peaking value problem that occurs by using constant high gain. Forthe following system

Px.t/ D Ax.t/C�0; : : : ; 0; 1

�>.w.t/C u.t//; y.t/ D x1.t/; (4.11)

with matrix A defined in (2.49), the linear ESO is designed as in (3.1) by using linear functionsgi . / D ki ; 2 R, and ki ; i D 1; 2; : : : nC 1 that are constants such that the matrix K definedin (2.49) has nC 1 different negative eigenvalue �1; �2; : : : ; �nC1. Let i .t/; i D 1; 2; : : : ; nC 1 bedefined in (3.6). Then i .t/ satisfies that

P.t/ Dd

dt

0BBB@

1.t/:::

n.t/

nC1.t/

1CCCA D �.t/K.t/C�.t/; �.t/ D

0BBBB@n P�.t/�.t/

1.t/

:::P�.t/�.t/

n.t/

Pw.t/

1CCCCA : (4.12)

First, by constant high gain %.t/ � r , the solution of ESO (3.1) is found to be

Oxi .t/ D �1

rnC1�ii .t/C xi .t/; (4.13)

where 0BBB@

1.t/

2.t/:::

nC1.t/

1CCCA D erKt

0BBB@

rn.x1.0/ � Ox1.0//

rn�1.x2.0/ � Ox2.0//:::

xnC1.0/ � OxnC1.0/

1CCCAC

Z t

0

erK.t�s/

0BBB@

0

0:::

Pw.s/

1CCCA ds: (4.14)

Figure 5. Stabilization of system (4.1) with (4.9) under feedback control (4.3) and nonlinear ESO (4.6) withconstant gain.

Figure 6. Magnifications of Figure 5(b) and (c).



The peaking value is mainly caused by the large initial value of .t/ D .1.t/; : : : ; nC1.t//:

i .t/ D

nC1XkD1

nC1XjD1

djkert�j rnC1�k.xk.0/ � Oxk.0//; (4.15)

where dij are real numbers determined by the matrix K. It is seen that in spite that ert�k decaysvery rapidly to 0 as r goes to infinity for any t > �0; �0 > 0, in the initial time stage, however, ert�kis very close to 1. This is the reason behind for the peaking value problem by constant high gain.Actually, the peaking values for Ox2.t/; : : : ; OxnC1.t/ are the orders of r; r2; : : : ; rn, respectively. Thelarger r is, the larger the peaking values are.

Next, when we apply the time-varying gain and let the gain be relatively small in the initial stage,the initial value of error .t/ is�

%.0/n.x1.0/ � Ox1.0// %n�1.0/.x2.0/ � Ox2.0// : : : xnC1.0/ � OxnC1.0/

�>; (4.16)

which is also small. Actually, if %.0/ D 1, the initial value of error .t/ is

�.x1.0/ � Ox1.0// .x2.0/ � Ox2.0// : : : xnC1.0/ � OxnC1.0/

�>: (4.17)

Because the gain function %.t/ is small in the initial stage, when k.t/k increases with increasingof eigenvalues to some given value, k.t/k stops increasing at some value that is determined by theexternal disturbances, but does not rely on the maximal value r . In what follows, we illustrate thispoint briefly. Let V W RnC1 ! R

Cbe V. / D >PK ; 2 RnC1, and let PK be the positive matrix

solution to the Lyapunov equation K>PK CPK D �InC1. Let the gain function %.t/ initiate from1 and increase continuously to r , which is the constant high gain used in (4.13)–(4.15), for example,

%.t/ D

8̂<:̂eat ; t 6 ln r

a;

r; t >ln r

a;; a > 0: (4.18)

Finding the derivative of V..t// for t 2 .0; ln r=a/ to obtain

dVdt


D %.t/>.t/K>PK.t/C�>.t/PK.t/C %.t/

>.t/PKK.t/C >.t/PK�.t/

6 �.%.t/ � 4an�max.PK//k.t/k2 C 2 sup

t2Œ0;1/

j Pw.t/j�max.PK/k.t/k:

(4.19)

It follows from (4.19) that .t/ may increase at first owing to %.t/ that is small in the initial time.However, if .t/ > 1 and %.t/ increases to 4an�max.PK/ C 2 supt2Œ0;1/ j Pw.t/j�max.PK/, thenV..t// stops increasing. This together with k.t/k 6 V./=�min.P / shows that kk does notincrease any more although %.t/ increases continuously to a large number r . If 4an�max.PK/ C2 supt2Œ0;1/ j Pw.t/j�max.PK/ 6 1;V..t// decreases from the beginning. This is the reason that thepeaking values become much smaller.

5. CONCLUDING REMARKS

In this paper, we apply the ADRC approach to stabilization for a class of lower triangular systemswith large uncertainty. Both constant gain ESO and time-varying gain ESO are investigated. Theconstant gain ESO-based ADRC takes the advantages of simpler tuning in ESO and allows largerclass of system functions to be applicable. The time-varying gain ESO can reduce significantly thepeaking value for ESO and takes a simple form in feedback. In addition, the time-varying gain ESO-based ADRC could reach asymptotic stability rather than the practical stability by constant gainESO-based ADRC. The numerical results validate the efficiency of both design methods. Because



the time-varying gain ESO may track instead of filtering the noise as expected, a recommendedstrategy is to apply time-varying gain ESO in the initial stage and then increase continuously to alarge tuning high-gain value r that is usually the value in constant gain design or can be obtained bytrial-and-error experiment for practical systems then keep this gain value afterwards. The choice oftuning gain value r is the same as gain value choice in constant ESO.

ACKNOWLEDGEMENTS

This work is supported partly by the National Natural Science Foundations of China (no. 61403242) and theFundamental Research Funds for the Central Universities (no. GK201402003).

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