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Asymptotic rejection of unmatched general periodicdisturbances with nonlinear Lipschitz internal modelsZhengtao Ding aa Control Systems Centre, School of Electrical and Electronic Engineering , University ofManchester , Sackville Street Building, Manchester M13 9PL , UKPublished online: 10 Sep 2012.

To cite this article: Zhengtao Ding (2013) Asymptotic rejection of unmatched general periodic disturbances with nonlinearLipschitz internal models, International Journal of Control, 86:2, 210-221, DOI: 10.1080/00207179.2012.722231

To link to this article: http://dx.doi.org/10.1080/00207179.2012.722231

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International Journal of ControlVol. 86, No. 2, February 2013, 210–221

Asymptotic rejection of unmatched general periodic disturbances with nonlinear

Lipschitz internal models

Zhengtao Ding*

Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester,Sackville Street Building, Manchester M13 9PL, UK

(Received 22 March 2012; final version received 15 August 2012)

This article deals with asymptotic rejection of general periodic disturbances in a class of nonlinear systems byexploiting observer design with Lipschitz output nonlinearities for the internal model design. It is shown that aclass of general periodic distances can be modelled as outputs of linear systems with nonlinear output functions.Based on this observation, a nonlinear observer design with output Lipschitz nonlinearities is investigated andintegrated for the internal model design to estimate the phase and amplitude of the desired feedforward input forasymptotic disturbance rejection in a class of nonlinear systems with input-to-state stable zero dynamics.A number of problems on disturbance rejection can be formulated in the disturbance rejection form shown in thisarticle to which the proposed nonlinear Lipschitz internal model can be applied. Two examples are included todemonstrate the proposed design strategies.

Keywords: disturbance rejection; periodic disturbance; nonlinear systems; internal model design

1. Introduction

Periodic disturbances exist in many engineeringproblems, such as nonlinear vibrations in mechanicalsystems and harmonics in power distribution systems.There are compelling demands to reject undesiredperiodic disturbances, such as vibrations in disc drivesand high-order harmonics in power systems. Methodshave been developed to reject harmonic disturbancesbased on different design principles (Bodson, Sacks,and Khosla 1994; Bodson and Douglas 1997;Ding 2001; Marino, Santosuosso, and Tomei 2003).The key point of the various methods is to generatethe desired feedforward input term to cancel theperiodic disturbances, and this coincides with theinternal model principle. At this point, it is worthpointing out that the output regulation of nonlinearsystems are very closely related to disturbance rejec-tion problems.

Harmonic disturbances can be modelled as outputsof linear dynamic systems. Adaptive control techniquescan be applied to design adaptive internal models toreject harmonic disturbances with an unknown ampli-tude, unknown phase and even with an unknownfrequencies. When the disturbances are nonharmonicperiodic disturbances, their wave profiles are exploitedfor asymptotic rejection in nonlinear systems by con-structing filters for phase and amplitude estimation(Ding 2006a, 2009a). The methods in Ding (2006a,

2009a) require special features of the wave profileswithin one period, and various compensations fordelays in phase estimation are needed for differentprofiles. There have been attempts to reject certainspecified harmonic components in the general periodicdisturbances (Ding 2008a) and to approximate periodicdisturbances using neural networks for rejections (Chen2009; Chen and Tian 2009). In these cases, the rejectionis not asymptotic, that is, there are still certaindisturbance components in the system outputs.

Many general periodic disturbances can be mod-elled as outputs of nonlinear systems, and in particular,as the outputs of linear dynamic systems with non-linear output functions. For the systems with Lipschitznonlinearities, nonlinear observers can be designed andthere are many results in the literature on this kind ofobservers (Rajamani 1998; Zhu and Han 2002; Zhao,Tao, and Shi 2010). Of course, the problem addressedin this article cannot be directly solved by a nonlinearobserver design, not even the state estimation of thedisturbances system, as the disturbance is not mea-sured. However, there is an intrinsic relationshipbetween the observer design and internal modeldesign, as evidenced in the previous results on theoutput regulation of nonlinear systems (Huang andRugh 1990; Isidori and Byrnes 1990) and in someresults for output regulation with nonlinear exosystems(Ding 2006b; Xi and Ding 2007).

*Email: zhengtao.ding@manchester.ac.uk

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2013 Taylor & Francis

http://dx.doi.org/10.1080/00207179.2012.722231

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With the formulation of general periodic distur-bances as the nonlinear outputs of a linear dynamicsystem, the information of the phase and amplitude ofa general periodic disturbance is then embedded in thestate variables and the information of the wave profilein the nonlinear output function. The nonlinear outputfunctions are assumed to be Lipschitz. By formulatingthe general periodic disturbances in this way, we areable to relax the restrictions on the wave profilesrequired in Ding (2006a, 2009a). Estimation of thephase and amplitude of a general periodic disturbanceis then converted to state estimation, and for which thenonlinear observer design of nonlinear systems withLipschitz nonlinearities can be explored. As theobserver design for systems with output nonlinearitiesis not readily available in the literature, we present anobserver design in this article with a condition on theLipschitz constant of the output function, for com-pleteness, although the techniques are very similar tothe case for the observer design with Lipschitznonlinearities in the system dynamics. We will showthat general periodic disturbances can be modelled asnonlinear outputs of a second-order linear system witha pair of pure imaginary poles, which depends on thefrequencies. For this specific system with the Lipschitznonlinear output, a refined condition on the Lipschitzconstant will be given by applying the proposedmethod, and observer gain will be explicitly expressedin terms of the Lipschitz constant and the period orfrequency of the disturbance.

An internal model design is then introduced basedon the proposed Lipschitz output observer for a classof nonlinear systems. Conditions are identified for thenonlinear system and control design is carried outusing the proposed internal model. Different from theresults shown in Ding (2006a, 2009a), the wave profilesare considered in the nonlinear output functions, andthe design procedure is the same for different waveprofiles, provided they satisfy the conditions specified.Two examples are included to demonstrate the pro-posed internal model and control design procedures.These examples also demonstrate that some otherproblems can be converted to the problem addressed inthis article.

2. Problem formulation

We consider a nonlinear system

_y ¼ aðzÞ þ 0ð yÞ þ ð y, vÞ þ bðu� �ðvÞÞ,

_z ¼ f ðz, v, yÞ

�ð1Þ

where y2R is the output, a: Rn!R is a continuous

function, 0: Rn!R is a continuous function, v2R

m

denotes general periodic disturbances, : R�Rm!R

is a continuous function and satisfies the condition that

j ð y, vÞj2 � y � ð yÞ with � being a continuous function,

b is a known constant, u2R is the input, z2Rn is the

internal state variable, f : Rn�R

m�R!R

n is a

continuous function.

Remark 1: For the convenience of presentation, we

only consider the system with relative degree 1 as in (1).

The systems with higher relative degrees can be dealt

with similarly by invoking backstepping. The zero

dynamics of this system are nonlinear in general.

Remark 2: The system in (1) specifies a kind of

standard form for the asymptotic rejection of general

periodic disturbances. For example, the disturbance

rejection problem considered in Ding (2006a) can be

covered to this format. This following class of nonlinear

systems is with more involved nonlinear terms than

the system considered in Ding (2006a), as shown by

_x ¼ Axþ �ð y, vÞ þ bu,

y ¼ cTx,

�ð2Þ

with b, c,2Rn and

A ¼

�a1 1 � � � 0

�a2 0 . ..

0

..

. ... . .

. ...

�an � � � � � � 0

2666664

3777775, b ¼

b1

b2

..

.

bn

266664

377775, c ¼

1

0

..

.

0

266664

377775,

where x2Rn is the state vector, y, u2R are the output

and input, respectively, of each subsystem, v2Rm

denotes general periodic disturbances, �: R�Rm!R

n

is a nonlinear smooth vector field in Rn with �(0, 0)¼ 0.

For this class of nonlinear systems, the asymptotic

disturbance rejection depends on the existence of state

transform to put the systems in the form shown in (1),

and it can be shown that such a transform exists under

some mild assumptions.The previous results for the asymptotic rejection of

general periodic disturbance (Ding 2006a, 2009a) use

delay and half-wave integral operations to obtain

phase and amplitude information together with the

wave profiles to generate the desired feedforward

inputs. A number of assumptions on the wave profiles

are needed, such as half-alternating wave profiles. In

this article, a different strategy is used. The profile

information is used to construct a nonlinear function

as the output function for a linear exosystem, for the

generation of the desired feedforward input. By doing

that, an observer with a nonlinear output function can

be designed, viewing the feedforward input as the

output, and an internal model can then be designed

based on the nonlinear observer. The problem to be

solved in this article is to design a control scheme,

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using a nonlinear observer-based internal model, toasymptotically reject a class of general periodic distur-bances for the system in (1).

3. General periodic disturbances and observers with

nonlinear output functions

In this section, we start with modelling general periodicdisturbance as the outputs of a linear oscillator withnonlinear output functions, and then propose a non-linear observer design for such a system, for thepreparation of internal model design in the nextsection.

Many periodic functions with period T can bemodelled as outputs of a second-order system

_w ¼ Aw, with A ¼0 !

�! 0

� ��ðvÞ ¼ hðwÞ,

8<: ð3Þ

where ! ¼ 2�T . Here, the desired feedforward input �(v)

is modelled as the nonlinear output h(w) of the second-order system. With

eAt ¼cos!t sin!t

� sin!t cos!t

� �,

the linear part of the output h(w), Hw, is always inthe form of a sin(!tþ�) where a and � denote theamplitude and phase, respectively. Hence we can setH¼ [1 0] without loss of generality, as the amplitudeand the phase can be decided by the initial value with

wð0Þ ¼ ½a sinð�Þ a cosð�Þ�T:

Based on the above discussion, the dynamic model forgeneral periodic disturbance is described by

_w1 ¼ !w2,

_w2 ¼ �!w1,

� ¼ w1 þ h1ðw1,w2Þ,

8><>: ð4Þ

where h1(w1, w2) is a Lipschitz nonlinear function witha Lipschitz constant �.

Remark 3: General periodic disturbances can bemodelled as af(tþ�) with a and � for the amplitudeand phase of a disturbance, and the wave profile isspecified by a periodic function f. This form of generalperiodic disturbances are used in Ding (2006a, 2009a)for their asymptotic rejection. In the model shown in(4), the amplitude and phase of the disturbance aredetermined by the system state variables w1 and w2,and the profile is determined by the nonlinear outputfunction. In the previous results, the phase andamplitude are obtained by delay and half-periodintegral operations. Here, we use nonlinear observer

for the estimation of phases and amplitudes of general

periodic disturbances.

For the model shown in (4), the dynamics are

linear, but the output function are nonlinear. Many

results in the literature on the observer design for

nonlinear Lipschitz systems are for the system with

nonlinearities in the system dynamics while the output

functions are linear. Here we need the results for

observer design with nonlinear output functions.

Techniques similar to the observer design of nonlinea-

rities in dynamics can be applied to the case when the

output functions are nonlinear. As the result for

observer design for output nonlinearities is not readily

available in the literature, we present a result here, and

then apply it to the observer design for general periodic

disturbances.Consider a nonlinear system with a nonlinear

Lipschitz output function

_x ¼ Ax,

y ¼ hðxÞ,

�ð5Þ

where x2Rn is the state vector, y2R

m is the output,

A2Rn�n is a constant matrix and h: R

n!R

m is a

continuous function. We can write the nonlinear

function h as h¼Hxþ h1(x) with Hx denoting a

linear part of the output, and the nonlinear part h1with the Lipschitz constant �.

An observer can be designed as

_̂x ¼ Ax̂þ Lð y� hðx̂ÞÞ, ð6Þ

where the observer gain L2Rn�m is a constant matrix.

Lemma 3.1: The observer (6) provides an exponen-

tially convergent state estimate of (1) if the observer gain

L can be chosen to satisfy the following conditions:

L ¼1

�2P�1HT,

PAþ ATP�HTH

�2þ ð1þ �ÞI � 0, ð7Þ

where P is a positive definite matrix and � is a positive

real constant.

Proof: Let ~x ¼ x� x̂. From (5) and (6), we have

_~x ¼ ðA� LHÞ ~xþ Lðh1ðxÞ � h1ðx̂ÞÞ: ð8Þ

Let V ¼ ~xTP ~x, where ~x ¼ x� x̂. We can obtain

_V ¼ ~xT½ðA� LHÞTPþ PðA� LHÞ� ~x

þ 2 ~xTPLðh1ðxÞ � h1ðx̂ÞÞ

� ~xT½ðA� LHÞTPþ PðA� LHÞ� ~x

þ ~xT½Iþ �2PLLTP� ~x

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¼ ~xT ATPþ PA�HTH

�2þ I

� �~x

þ ~xTH

�� �LTP

� �TH

�� �LTP

� �~x

� �� ~xT ~x:

Therefore we can conclude that ~x converges to zero

exponentially.Now we can apply the result to observer design for

the model of general periodic disturbances. For the

model shown in (4), the observer shown in (6) can be

applied with A ¼ 0 !�! 0

� �and H¼ [1 0]. We have the

following lemma for the stability of this observer.

Lemma 3.2: An observer in the form of (6) can be

designed to provide an exponentially convergent state

estimate for the general periodic disturbance model (4) if

the Lipschitz constant � for h1 satisfies �5 1ffiffi2p .

Proof: Our proof is constructive. Let

P ¼

p �1

4�2!

�1

4�2!p

2664

3775,

where p4 14�2!

. It is easy to see that P is positive

definite. A direct evaluation gives

PAþ ATP�HTH

�2¼ �

1

2�2I:

Therefore the second condition in (7) is satisfied.

Following the first condition specified in (7), we set

L ¼

4!ð4p�2!Þ

ð4p�2!Þ2 � 1

4!

ð4p�2!Þ2 � 1

26664

37775: ð9Þ

The rest part of the proof can be completed by

invoking Lemma 3.1.

4. Disturbance rejection

In this section, we apply the technique developed in

the previous section to the internal model design

for the asymptotic rejection of general periodic

disturbances for the nonlinear systems in the

form (1).We introduce two assumptions about the system.

Assumption 1: The feedforward term �(v) can be

modelled as the output of a system in the format

shown in (4) and the Lipschitz constant of the output

nonlinear function � satisfies �5 1ffiffi2p .

Assumption 2: The subsystem

_z ¼ f ðz, v, yÞ

is Input-to-State Stable (ISS) with state z and input y,

characterised by ISS pair (�, �). Furthermore, �(s)¼O(a2(s)) as s! 0.

Before introducing the control design, we need to

examine the stability issues of the z-subsystem, and

hence introduce a number functions that are needed

later for the control design and stability analysis of

the entire system. In the following analysis, we use the

technique of changing supply functions shown in

Sontag and Teel (1995) and Isidori (1999). From

Assumption 2, there exists a Lyapunov function Vz(z)

that satisfies that

�1ðkzkÞ � VzðzÞ � �2ðkzkÞ,

_VzðzÞ � ��ðkzkÞ þ �ðj yjÞ,

(ð10Þ

where �, �1 and �2 are class K1 functions, and � is a

class K function. Let be a K1 function such

that (kzk)� a2(z) and (s)¼O(a2(s)) as s! 0.

Since (s)¼O(a2(s))¼O(�(s)) as s! 0, there exists a

smooth nondecreasing (SN ) function ~q such that,

8r2Rþ,

1

2~qðrÞ�ðrÞ � ðrÞ:

Let us define two functions

qðrÞ :¼ ~qð��11 ðrÞÞ,

ðrÞ :¼

Z r

0

qðtÞdt:

Define

~VðzÞ :¼ ðVðzÞÞ,

and we obtain

_~VðzÞ � �qðVðzÞÞ�ðzÞ þ qðVðzÞÞ�ðj yjÞ

� �1

2qðVðzÞÞ�ðzÞ þ qð�ðj yjÞÞ�ðj yjÞ

� �1

2qð�1ðkzkÞÞ�ðzÞ þ qð�ðj yjÞÞ�ðj yjÞ

¼ �1

2~qðkzkÞ�ðzÞ þ qð�ðj yjÞÞ�ðj yjÞ,

where � is defined as �(r) :¼ �2(��1(2�(r))) for r2R

þ.

Let us define a smooth function �� such that ��ðrÞ �qð�ðjrjÞÞ�ðjrjÞ for r2R and ��ð0Þ ¼ 0, and then we have

_~VðzÞ ¼ �ðkzkÞ þ ��ð yÞ: ð11Þ

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Based on observer design presented in the previous

section, we design the following internal model

_� ¼ A�þ b�1L 0ð yÞ þ Lu� b�1ALy

� Lhð�� b�1LyÞ, ð12Þ

where L is designed as in the previous section in (9).The control input is then designed as

u ¼ �b�1 0ð yÞ þ k0yþ k1yþ k2��ð yÞ

yþ k3 � ð yÞ

� �þ hð�� b�1LyÞ, ð13Þ

where k0 is a positive real constant, k1 ¼

��1b2ð� þ kHkÞ2 þ 34, k2¼ 4��1jjb�1PLk2þ 2 and k3 ¼

4��1jjb�1PLk2 þ 12.

For the stability of the closed-loop system, we havethe following theorem.

Theorem 4.1: For a system in the form shown in (1), the

output feedback control design with the internal model

(12) and the control input (13) ensures the boundedness ofall the variables of the closed-loop system, and the

asymptotic convergence to zero of the state variables z

and y and the estimation error (w� �þ b�1Ly).

Proof: Let

¼ w� �þ b�1Ly:

It can be obtained from (12) that

_ ¼ ðA� LHÞ þ b�1Lðh1ðwÞ � h1ðw� ÞÞ

þ b�1LaðzÞ þ b�1L ð y, vÞ:

Let Vw¼ TP . We obtain

_Vwð Þ � ��k k2 þ 2j Tb�1PLaðzÞj þ 2j Tb�1PL ð y, vÞj

� �1

2�k k2 þ 2��1jjb�1PLk2ða2ðzÞ þ j ð y, vÞj2Þ

� �1

2�k k2 þ ðk2 � 2ÞðkzkÞ þ k3 �

1

2

� �y � ð yÞ,

ð14Þ

where � ¼ 12�2� 1.

Based on the control input (13), we have

_y ¼ �k0y� k1y� k2��ð yÞ

y� k3 � ð yÞ þ aðzÞ þ ð y, vÞ

þ bðhðw� Þ � hðwÞÞ:

LetVy ¼12 y

2. It follows from the previous equation that

_Vy¼�ðk0þk1Þy2�k2 ��ðyÞ�k3y � ðyÞþyaðzÞ

þy ðy,vÞþybðhðw� Þ�hðwÞÞ

��k0y2�k2 ��ðyÞ� k3�

1

2

� �y � ðyÞþðkzkÞþ

1

4�k k2:

ð15Þ

Let us define a Lyapunov function candidate forthe entire closed-loop system as

V ¼ Vy þ Vw þ k2 ~Vz:

Following the results shown in (10), (14) and (15),we have

_V � �k0y2 �

1

4�k k2 � ðkzkÞ:

Therefore we can conclude that closed-loop system isasymptotically stable with respect to the state variablesy, z and the estimation error .

5. Examples

Several types of disturbance rejection and outputregulation problems can be converted to the form (1).In this section, we show two examples. Example 1 dealswith rejection of general periodic disturbances, andExample 2 demonstrates how the proposed methodcan be used for output regulation.

5.1 Example 1

Consider

_x1 ¼ x2 þ �1ðx1Þ þ b1u,

_x2 ¼ �2ðx1Þ þ �ðwÞ þ b2u,

_w ¼ Aw

y ¼ x1,

8>>><>>>:

ð16Þ

where y2R is the measurement output, �i: R!R, fori¼ 1, 2, are continuous nonlinear functions, �: R

2!R

is a nonlinear function which produces a periodicdisturbance from the exosystem state w, and b1 and b2are known constants with the same sign, which ensuresthe stability of the zero dynamics. The control objec-tive is to design an output feedback control input toensure the overall stability of the entire system, and theasymptotic convergence to zero of the measurementoutput. The system shown in (16) is not in the form of(1) and the disturbance is not matched. We will showthat the problem can be transformed to the problemconsidered in the previous section.

Let

�z ¼ x2 �b2b1

x1:

In the coordinates ð y, �zÞ, we have

_y ¼ �zþb2b1

yþ �1ð yÞ þ b1u,

_�z ¼ �b2b1

�zþ �2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

yþ �ðwÞ:

8>>><>>>:

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Consider

_�z ¼ �b2b1�z þ �ðwÞ:

As shown in Ding (2008b), there exists a steady state

solution, and furthermore, we can express the solution

as a nonlinear function of w, denoted by �z(w). Let usintroduce another state transformation with

z ¼ �z� �zðwÞ. We then have

_y ¼ zþb2b1

yþ �1ð yÞ þ b1ðuþ b�11 �zðwÞÞ,

_z ¼ �b2b1

zþ �2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y:

8>>><>>>:

ð17Þ

Comparing (17) with (1), we have

aðzÞ ¼ z,

ð yÞ ¼b2b1

yþ �1ð yÞ,

b ¼ b1,

hðwÞ ¼ �b�11 �zðwÞ,

f ðz, v, yÞ ¼ �2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y:

From a(z)¼ z, we can set (kzk)¼kzk2¼ z2.It can be shown that Assumption 2 is satisfied by

(17). Indeed, let VðzÞ ¼ 12 z

2, and we have

_Vz ¼ �b2b1

z2 þ z �2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y

!,

� �1

2

b2b1

z2 þ1

2

b1b2

�2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y

!2

:

Let

~Vz ¼ 2b1b2

Vz,

and finally we have

_~Vz � �ðjzjÞ þb1b2

� �2

�2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y

!2

:

ð18Þ

It can be seen that there exists a class K function �(jyj)to dominate the second term at the left-hand side of

(18), and the z-subsystem is ISS. For the control

design, we can consider

��ð yÞ ¼b1b2

� �2

�2ð yÞ �b2b1�1ð yÞ �

b2b1

� �2

y

!2

:

The rest part of the control design follows the stepsshown in Section 4.

For the simulation study, we set the periodicdisturbance as a square wave. For convenience, weabuse the notations of �(w(t)) and h(w(t)) as �(t) andh(t). For � with t in one period, we have

� ¼d, 0 � t5 T

2 ;

�d,T

2� t5T,

8<: ð19Þ

where d is an unknown positive constant, denoting theamplitude. We obtain

h ¼ d �hðtÞ,

where

�hðtÞ ¼

�1

b21� e

�b2b1t

� �

þ1

b2e�

b2b1ttanh

T

4

b2b1

� �, 0 � t5

T

2;

1

b21þ e

�b2b1t� 2e

b2b1ðT2�tÞ

� �

þ1

b2e�

b2b1ttanh

T

4

b2b1

� �,

T

2� t5T:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð20Þ

Eventually, we have the matched periodic disturbanceh(w) given by

hðwÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw21 þ w2

2

q�h arctan

w2

w1

� �� �:

Note thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw21 þ w2

2

qdecides the amplitude, which can

be determined by the initial state of w.In the simulation study, we set T¼ 1, d¼ 10,

�1¼ y3, �2¼ y2, b1¼ b2¼ 1. The simulation results areshown in Figures 1–4. It can be seen from Figure 1 thatthe measurement output converges to zero and thecontrol input converges to a periodic function. In fact,the control input converges to h(w), as shown inFigure 2. As for the internal model and state estima-tion, it is clear from Figure 3 that the estimatedequivalent input disturbance converges to h(w), and �converges to w.

5.2 Example 2

In this example, we briefly show that an outputregulation problem can also be converted to the formin (1). Consider

_x1 ¼ x2 þ ðey � 1Þ þ u,

_x2 ¼ ðey � 1Þ þ 2w1 þ u,

_w ¼ Aw,

y ¼ x1 � w1,

8>>><>>>:

ð21Þ

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where y2R is the measurement output and

w1¼ [1 0]w. In this example, the measured output

contains the unknown disturbance, unlike Example 1.

The control objective remains the same, to design an

output feedback control law to ensure the overall

stability of the system and the convergence to zero of

the measured output. The key step in the controldesign is to show that the system shown in (21) can be

converted to the form as shown in (1).Let

�z ¼1

1þ !2½1 �!�w,

0 5 10 15−0.05

0

0.05

0.1

0.15

0.2

time (s)

y

0 5 10 15−20

−15

−10

−5

0

5

time (s)

u

Figure 1. The system input and output of Example 1.

0 5 10 15−20

−15

−10

−5

0

5

time (s)

u an

d h

uh

Figure 2. Control input and the equivalent input disturbance of Example 1.

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and it is easy to check that �z satisfies

_�z ¼ ��z þ ½1 0�w:

Let z¼ x2��z� x1. We obtain

_y ¼ zþ yþ ew1ðey � 1Þ þ ðu� hðwÞÞ,

_z ¼ �z� yþ 2w1,

�

0 5 10 15−20

−10

0

10

20

30

40

time (s)

w a

nd e

stim

ates

w1

w2

1

2

h

h

Figure 4. The exosystem states and the internal model states of Example 1.

0 5 10 15−4

−2

0

2

4

6

8

time (s)

h an

d es

timat

e

hestimate of h

Figure 3. The equivalent input disturbance and its estimate of Example 1.

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where hðwÞ ¼ 2þ!2

1þ!2 ½�1 !�w� ðew1 � 1Þ. It can be seen

that we have transformed the system to the format as

shown in (1) with ð y, vÞ ¼ ew1 ðey � 1Þ.To make H¼ [1 0], we introduce a state transform

for the disturbance model as

� ¼2þ !2

1þ !2

�1 !

�! �1

" #w:

It can be easily checked that _� ¼ A�. The inverse

transformation is given as

w ¼1

2þ !2

�1 �!

! �1

� ��:

With � as the disturbance state, we can write the

transformed system as

_y ¼ zþ yþ e1

2þ!2½�1 �!��

ðey � 1Þ þ ðu� hð�ÞÞ,

_z ¼ �z� yþ2

2þ !2½�1 �!��,

_� ¼ A�,

8>>>><>>>>:

ð22Þ

where hð�Þ ¼ �1 � ðe1

2þ!2½�1 �!��

� 1Þ. Note that ey�1y is a

continuous function, and we can take � ð yÞ ¼ d0yðey�1y Þ

2

where is d0 is a positive real constant depending on the

frequency and the knowledge of an upper limit of the

disturbance amplitude. The control design presented in

the previous section can then be applied to (22).Simulation studies were carried out with the resultsshown in Figures 5–8.

6. Conclusions

In this article, a new control method has beenproposed for the asymptotic rejection of generalperiodic disturbances in a class of nonlinear dynamicsystems. In the proposed design, general periodicdisturbances are modelled as nonlinear outputs of asecond-order linear system with the phase and ampli-tude information to be determined by the statevariables. By using this formulation, an internalmodel is designed based on observer design for systemswith nonlinear output functions. The proposed methodrelaxes certain restrictions on the wave profiles in theprevious results, due to the new estimation method forphase and amplitude information. The ISS character-isation of the internal dynamics of the system extendsthe asymptotic rejection of general periodic distur-bances to a class of nonlinear dynamic systems withnonlinear zero dynamics. The proposed design ofnonlinear Lipschitz internal models may also beapplied to the general periodic disturbance rejectionof other nonlinear systems with different control designtechniques such as the systems with nonminimum

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.05

0

0.05

0.1

0.15

0.2

time (s)

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−300

−250

−200

−150

−100

−50

0

50

time (s)

u

Figure 5. The system input and output of Example 2.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−25

−20

−15

−10

−5

0

5

10

15

20

time (s)

h an

d es

timat

e

hestimate of h

Figure 7. The equivalent input disturbance and its estimate of Example 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−300

−250

−200

−150

−100

−50

0

50

time (s)

u an

d h

uh

Figure 6. Control input and the equivalent input disturbance of Example 2.

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phase Ding (2005) and the systems with the non-linearity of unmeasured state variables Ding (2009b).

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−30

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−10

0

10

20

30

40

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µ a

nd e

stim

ates

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µ2

1

2

h

h

Figure 8. The exosystem states and the internal model states of Example 2.

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