an adaptive robust nonlinear motion controller combined with disturbance observer-z9k

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454 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 2, MARCH 2010

An Adaptive Robust Nonlinear Motion ControllerCombined With Disturbance Observer

Zi-Jiang Yang, Seiichiro Hara, Shunshoku Kanae, Kiyoshi Wada, and Chun-Yi Su

AbstractParameter adaptation and disturbance observer(DOB) have been considered as two contrastively different ap-proaches to handle uncertainties in motion control problems. Thepurpose of this brief is to merge both techniques into one controldesign with theoretically guaranteed performance. It is shownthat the DOB compensates low-passed components of the lumpeduncertainties without the necessity of parameterization, whereasthe adaptive mechanism is only automatically activated in thecases where the fast-changing components of the uncertaintiesbeyond the pass-band of the DOB can be parameterized by un-known parameters. It is thus shown theoretically how the DOBand adaptive mechanism play in a cooperative way so that thecontroller is more effective than the individual ones. Simulationresults are provided to support the theoretical results.

Index TermsAdaptive control, disturbance observer (DOB),input-to-state-stability, motion control, robust control.

I. INTRODUCTION

IN THE motion control problems of nonlinear mechanical

systems, adaptive control, disturbance observer (DOB)

control, and sliding mode control (SMC) are typical but con-

trastively different approaches to handle the uncertainties

[1], [2].

So far, many papers have been published on the DOB-based

motion controllers [3], [4]. The key point of a DOB is to pass

the external disturbances and model mismatch, lumped as anerror term, through a low-pass filter and then to compensate

the external disturbance and model mismatch by the output of

the low-pass filter. Usually, the DOB-based motion controllers

are designed according to linear control theory. Most recently,

a novel robust nonlinear motion controller with the DOB has

been proposed, where the input-to-state stability (ISS) property

of the overall nonlinear control system is guaranteed [5]. Com-

pared to the adaptive control technique, the DOB-based con-

trol enjoys the advantages of simplicity, compensation ability of

unparameterizable uncertainties, reliable transient performance.

Manuscript received March 28, 2008; revised January 20, 2009. Manuscriptreceived in final form May 01, 2009. First published July 21, 2009; currentversion published February 24, 2010. Recommended by Associate EditorL.-A. Dessaint.

Z.-J.Yangis withthe Department of Intelligent Systems Engineering, Collegeof Engineering, Ibaraki University, Ibaraki 316-8511, Japan (e-mail: [email protected]).

S. Hara and K. Wada are with the Department of Electrical and ElectronicSystems Engineering, Graduate School of Information Science and ElectricalEngineering, Kyushu University, Fukuoka 812-8581, Japan.

S. Kanae is with the Department of Electrical, Electronics and Computer En-gineering, Fukui University of Technology, Fukui 910-8505, Japan.

C.-Y. Su is with the Department of Mechanical Engineering, Concordia Uni-versity, Montreal, Quebec H3G 1M8, Canada.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2009.2022549

However, the DOB-based controllers cannot sufficiently com-pensate some fast-changing uncertainties beyond the pass-band

of the DOB [1].

On the other hand, the adaptive control techniques have

also been widely used in motion control [1], [6][8]. If the

uncertainties such as friction and high-frequency ripples can

be parameterized, adaptive laws are adopted to achieve a small

tracking error after the transient phase. However, the adap-

tive control techniques may exhibit unsatisfactory transient

performance and cannot handle unparameterizable external

disturbances through parameter adaptation.

In contrast to the adaptive control or the DOB control, the

SMC drives the system states to a stable sliding surface andkeeps the states on it by a switching function. The SMC has a

simple structure and high robustness against the uncertainties.

Since the nonlinear system analysis methodologies of both

the adaptive control and SMC are well established, there have

been many works where these two techniques are combined to

achieve better performance [9]. Besides these, there also have

been some works that combine the DOB and SMC [10], [11],

where a linear nominal model is considered, and the modeling

error and external disturbances are lumped as a disturbance

term. We denote here this lumped disturbance term as . By

using the DOB, an estimate of , i.e., is obtained. And by

the DOB compensation, the lumped disturbance is reduced to, and a modest switching gain of the SMC is sufficient.

The lumped uncertainty , however, may include signal de-

pendent uncertainties. Moreover, the DOB itself introduces

an extra loop into the system, and hence we cannot easily say

and are bounded prior to control design. In this context,

it is considered that there still exist possibilities to revise this

approach. As a demerit of the SMC, in the neighborhood of the

sliding surface, the switching function may chatter due to noise

or finite sampling frequency.

The combination of the DOB and adaptive control, however,

has not been well studied, due to the fact that the stability of

the DOB-based controllers for nonlinear systems has not beenwidely studied in the literature whereas the adaptive control

systems usually require strict Lyapunov-like stability analysis.

Therefore, it is of not only theoretical but also practical interests

to investigate what performance can be achieved by exploiting

the advantages of the two different techniques.

In this brief, we propose a motion controller that merges both

advantages into one control design so that it can handle broader

classes of uncertainties with theoretically guaranteed perfor-

mance. It is shown that the DOB compensates low-passed com-

ponents of the lumped uncertainties, whereas the computation-

ally demanding adaptive mechanism is only automatically ac-

tivated in the cases where the fast-changing components of the

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YANG et al.: AN ADAPTIVE ROBUST NONLINEAR MOTION CONTROLLER COMBINED WITH DISTURBANCE OBSERVER 457

Although is included in , since is strictly proper so

that the phase of is delayed, is implementable.

To let the adaptive parameters stay in the prescribed range,

we adopt the adaptive laws with projection [12], shown in (26)

at the bottom of the page, where , , and

is the th element of defined in (4), shown in

(27) at the bottom of the page. where , ,and is the th element of defined in (4).

It can be verified that the adaptive laws satisfy

(28)

Remark 2: Since the complementary filter is a high-

pass filter, in the case of slowly-changing signals, the amplitudes

of and are trivial so

that the computationally demanding adaptive laws become to

be lazy or unnecessary. On the other hand, when the pass-

band of the DOB is less broad, the adaptive laws work more

significantly. Therefore, we can conclude that the DOB and the

adaptive laws play in a complementary and cooperative way to

account for uncertainties.

IV. STABILITY ANALYSIS

In this section, we first show the ISS property of each

subsystem that specifies the boundedness and transient perfor-

mance. Then we will show our main theoretical contribution

in this brief, i.e., how the adaptive laws and DOB play in a

cooperative way; what control performance can be achieved

if these two different techniques are merged into one control

design. Finally, we summarize the theoretical results for the

overall error system.

A. ISS Property Analysis of Each Subsystem

Applying to subsystem (16), we have

(29)

and the state-space form

(30)

where

(31)

The ISS property is described in the following lemma [15].

Lemma 1: If the virtual input is applied to the subsystem

, and if is made uniformly bounded at the next step, then

the subsystem is ISS, i.e., for

To establish the ISS property of the subsystem , we first

rewrite (24) by using (22)

(32)

Then we have by using (22), (23), and (32)

(33)

where

(34)

(35)

According to Assumptions 15, it is obvious that each term in

the numerator of is counteracted by a nonlinear damping

for

for

otherwise

(26)

for

forotherwise

(27)



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Fig. 1. Reference position y and its velocity _y , and the unmodelable externaldisturbance f = b .

A. Description of the System Model

The system model in [7] is described as follows:

(53)

where is the control voltage, is the position, is the

velocity, , and [7].

The periodic ripple disturbance is given as

(54)

where , , . It is assumed

here that only the magnetic pitch is known.

The friction model is given as

(55)

where , , , .

The reference trajectory and its derivative are shown in

Fig. 1. To verify the robustness against the unparameterizedexternal disturbance, we introduce the external disturbance

as shown in Fig. 1. The controller is implemented at a

sampling period of ms . To verify the noise effects, a

uniformly distributed noise between m and m

is added to the measurement of . The measurement of is

obtained by pseudodifferentiation.

B. Results of the Adaptive Sliding Mode Controller

The adaptive sliding mode controller is given as follows [7].

1) Define the tracking errors

Fig. 2. Results of the adaptive sliding mode controller.

2) Define the PID sliding surface

3) Introduce the dead-zone to the sliding surface

4) Construct an adaptive sliding mode controller

Notice that and have been lumped as one term

such that .

5) Construct the adaptive laws

where .

In [7], the term included in the friction

term is not estimated by parameter adaptation, but is counter-

acted by the sliding mode control term.

The results are shown in Fig. 2. It can be verified that the re-

sults reflect the theoretical conclusion in [7] quite clearly: the

sliding surface signal does converge to the dead zone shown

by the dashed lines. However, we have found that it is difficult

to obtain acceptable results when the dead zone width is fur-

ther reduced. As well known, the sliding mode type controller

may suffer from the chattering problem due to the switching ac-

tion. It is uneasy to reduce the chattering without degradation ofprecision.



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YANG et al.: AN ADAPTIVE ROBUST NONLINEAR MOTION CONTROLLER COMBINED WITH DISTURBANCE OBSERVER 461

C. Results of the Proposed Controller

We restate the plant model (53) according to the problem set-

ting (1), where

(56)

where is the periodic position de-

pendent ripple disturbance modeled in (54). Since the term

is not estimated by the adaptive

laws in [7], to perform a fair comparative study, we also do not

estimate this term and we handle this term by the nonlinear

damping terms and the DOB.

The nominal functions are given as

(57)

where denotes the nominal value of .The known nonlinear bounding functions required in As-

sumption 4 are given as

(58)

in (5) is modeled by a single parameter as

.

In [7], it is assumed that the form of the ripple disturbance

is known. In contrast, in our method, the ripple disturbance is

approximated by a periodic RBF network of period . Define

the truncated variable as

(59)

where is a function that rounds the entry to the greatest

integer less than it. Then is modeled as

(60)

where is a periodic RBF network

(61)

where , and

are Gaussian basisfunctions whose centers are equidistantly located in

. The essential width of the basis functions

is chosen as . The nonlinear

function is thus modeled as

(62)

To summarize, since the nominal function is chosen as

, we can model in (5) as

(63)

Fig. 3. Results of the proposed controller ( = 0 : 0 1 ) .

The bounds of the unknown parameters are given as follows:

(64)

The time-constant of , control gains and adaptive gains

are given as

(65)

Notice that different values of are used for comparative inves-

tigations of the cooperative performance of the DOB and adap-

tive laws.

The results are shown in Figs. 35, where from the top to the

bottom are respectively the position error , velocity error ,

control input and the estimate of . Due to the space limita-

tion, the behaviour of the other parameter estimates is omitted.

We can find that when is sufficiently small, the proposed con-

troller brings significant improvement to suppress the error am-

plitudes. Also, we can find that due to the cooperative effectsof the DOB and adaptive laws, the control performance is not

sensitive to the value of .

Furthermore, when the value of is smaller, the adaptive pa-

rameter converges more slowly. This is because that when

is smaller and thus the pass-band of is broader, the

high-pass filter cuts off more signal com-

ponents so that the adaptive law (27) works with less signal in-

formation. We have also verified that if either of the DOB or the

adaptive mechanism is removed, the results are not acceptable.

The results, however, are omitted due to the limitation of paper

length as a brief.

Also, comparing the results of Figs. 35 with those of Fig. 2,we can conclude that at least for the example under study, the



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Fig. 4. Results of the proposed controller ( = 0 : 0 3 ) .

Fig. 5. Results of the proposed controller ( = 0 : 1 ) .

proposed controller does not suffer from the chattering problem

and thus achieves smaller position-tracking error. However, due

to the use of the DOB, we have to calculate some signals that

are filtered by the low-pass filter . Therefore, our method

requires more computing burden.Finally, we can conclude that all of these results exactly re-

flect the theoretic analyses.

VII. CONCLUSION

In this brief, an adaptive robust nonlinear motion controller

combined with the DOB for positioning control of a nonlinear

SISO mechanical system has been proposed. Stability analysis

was performed as well. Extensive simulation studies were car-

ried out to support the theoretical analysis. Our major contribu-

tion is to incorporate the DOB technique and adaptive techniquewhich have been considered as two contrastively different ap-

proaches in the literature. Moreover, it has been shown theoret-

ically how the DOB and adaptive mechanism play in a coopera-

tive way so that the proposed controller is able to handle broader

classes of uncertainties than the individual adaptive controllers

or DOB-based controllers.

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