Analysis of Disturbance Observer Based Control forNonlinear Systems under Disturbances with

Bounded VariationWen-Hua CHEN and Lei GUO?

Department of Aeronautical & Automotive EngineeringLoughborough University, LE11 3TU, UK

Email:w.chen@lboro.ac.uk? Now with Research Institute of AutomationSoutheast University, Nanjing, 210096, China

Keywords: Disturbance attenuation; observers; nonlinearsystems; actuator nonlinearities; stability

Abstract— Control of nonlinear systems with unknown distur-bances is considered in this paper using disturbance observerbased control (DOBC) approach. The disturbances concernedare supposed to be time-varying but with bounded variation.For a nonlinear system with well-defined relative degree fromthe disturbances to the output, a nonlinear disturbance observeris presented to estimate the unknown disturbances and thenis integrated with a conventional controller using disturbanceobserver based techniques. Stability and performance analysisof the composite closed-loop system is provided using Lyapunovtheory and input-to-state stable concept.

I. I NTRODUCTION

Disturbances widely exist in engineering. This is one ofthe main reasons why feedback is so prevalence. Control ofnonlinear systems under disturbance has been an active re-search area in the last decades and several elegant approacheshave been presented. Basically, control methods addressingdisturbance attenuation can be classified according to theassumptions made on disturbances. The first approach is tomodel disturbances by a stochastic process, which leads tononlinear stochastic control theory [1]. The second approachis nonlinear H∞ control [16] where the energy of disturbancesis assumed to be bounded. The third approach is to deal withdisturbances generated by a neutral stable exogenous system,namely, nonlinear output regulator theory [13]. These direc-tions have been extensively investigated and many elaborateresults have been developed. However, in general, design ofcontrollers using these approaches involves the solution of aset of nonlinear partial differential equations (PDEs) or nonlin-ear partial differential inequalities (PDIs). Although significantprogress has been achieved in this area and research is stillgoing on, solution of those PDEs or PDIs, in general, can onlybe obtained for special cases or approximately. In additionto the above general approaches, there are also methods todeal with specific disturbances, for example, harmonics [4]. Bysufficiently using the information of the specific disturbances,elegant results can be achieved.

This paper addresses a class of disturbances with boundedvariation. This class of disturbances has been considered by

[3]. Compared with other modelling of disturbances wherethe information of the distribution, the stochastic properties orthe model of exogenous systems is required, less informationis required in this modelling. Actually, the only requirementis that the variation of the disturbance is bounded. Manyengineering disturbances can be approximately representedby this kind of disturbance after its high frequency part isfiltered by properly designed filters in the loop or the inertiaof systems.

A disturbance observer based control (DOBC) approachis adopted in this paper. In this approach the influence ofdisturbances is estimated by a disturbance observer and thencompensated for based on the estimate. DOBC has its roots inmany mechatronics applications in the last two decades [14],[9], [10], in particular for linear systems. Recently attempthas been made to establish theoretic justification of theseDOBC applications and extend DOBC from linear systemsto nonlinear systems. In the current DOBC framework [7],[6], similar to nonlinear regulation theory [13], it is assumedthat the disturbances are generated by an exogenous systemwith a known model. However, the stability results establishedunder this assumption may be invalid for engineering appli-cations since it is quite difficult to model the disturbancesas an output of an exogenous system for some engineeringproblems. Although simulation and experiment results havedemonstrated that the DOBC may work quite well even thoughthis assumption is not satisfied, it lacks theoretic justificationof these applications [7], [5]. It is obvious that this obstructsthe further development and applications of DOBC. To thisend, this paper intends to overcome this restriction by onlyassuming that the disturbances have bounded variation. It isshown that, under a properly designed nonlinear disturbanceobserver and a proper feedback, the tracking error or regulationerror caused by disturbances can be suppressed to an arbitrarilyspecified arrange.

II. PROBLEM FORMULATION

Consider a nonlinear system described as

x = f(x) + g1(x)u + g2(x)w (1)

Control 2004, University of Bath, UK, September 2004 ID-048

2

wherex ∈ Rn, u ∈ R andw ∈ R are the state vector, inputand disturbance, respectively.

It is assumed that the disturbancew is unknown, time-varying but with bounded variation. That is,

|w| ≤ wd, (2)

wherewd is an unknown positive constant. Many engineeringdisturbances consist of both high frequency and low frequencyparts. After they are filtered by a properly designed filterin the loop or the inertia of dynamic systems, they can beapproximated by (2). Hence (2) represents a large variety ofdisturbances in engineering.

The objective of this paper is to design a nonlinear controllerusing the DOBC concept such that the influence of thedisturbance satisfying (2) can be attenuated to a specified level.Within this framework, a nonlinear disturbance observer isrequired to estimate the influence of the disturbance and thenit is integrated with a conventional controller. To establish ourtheoretic results, the following assumption is imposed in thispaper.

Assumption 1: There exists a smooth functionh(x) suchthat the relative degree fromw to h(x), ρ, is uniform well-defined for allx and t.

Remark 1:An obvious choice ofh(x) is the output of thedynamic system (1). However, if the relative degree from thedisturbance to the output is not uniform well defined, a smoothfunction h(x) should be found such that Assumption 1 issatisfied.

III. N ONLINEAR OBSERVER FORUNCERTAIN

DISTURBANCES

In this paper, a nonlinear disturbance observer to estimatethe disturbancew of system (1) is proposed as

{z = −l(x)g2(x)[z + p(x)]− l(x)[f(x) + g1(x)u]w = z + p(x)

(3)wherep(x) is a nonlinear function to be designed and relatesto the observer gain functionl(x) as

l(x) =∂p(x)∂x

. (4)

Let n0 denotesminx |Lg2Lρ−1f h(x)|. Assumption 1implies

thatLg2Lρ−1f h(x) 6= 0 for all x. Hencen0 is a positive scalar.

The performance of the nonlinear disturbance observer (3)is stated in Theorem 1 [8].

Theorem 1:Consider system (1) satisfying Assumption 1and subject to the disturbance (2). When the nonlinear observergain l(x) and the nonlinear functionp(x) are chosen as

l(x) = p0

∂Lρ−1f h(x)∂x

, (5)

andp(x) = p0L

ρ−1f h(x) (6)

respectively, there always exists a constantp0 such that the ob-server error dynamics is input-bounded-output-bounded stable.

Observererror dynamics

Systemdynamics

- e -

?w

w

Overall dynamics

- x

Fig. 1. Structure of the overall closed-loop system under DOBC

Furthermore, the steady state estimation error yielded by thedisturbance observer (3) is bounded by

∣∣∣∣wd

p0n0

∣∣∣∣ (7)

wherewd denotes the maximum variation rate of the distur-bance.

Remark 2: It should be noticed thatp0 should be chosen tobe positive whenn(x) > 0 and to be negative whenn(x) < 0.The convergence of the disturbance estimate can be specifiedby adjusting the design parameterp0.

IV. STABILITY ANALYSIS OF DOBC

In the DOBC scheme, the control consists of two parts,given by

u(t) = α(x) + β(x)w (8)

whereα(x) is a conventional control law designed under theassumption that there are no disturbances or the disturbancesare known, andw is the estimate yielded by the disturbanceobserver (3).

Substitution of the above control law into system (1) gives

x = f(x) + g1(x)α(x) + g1(x)β(x)w + g2(x)w= f(x) + g1(x)α(x) + (g1(x)β(x) + g2(x))w

−g1(x)β(x)e (9)

where the observer error is governed by the nonlinear dynam-ics

e = w − l(x)g2(x)e (10)

The overall closed-loop dynamics consist of the controlledsystem dynamics (9) and the observer error dynamics (10).The structure of the overall closed-loop system is shown inFigure 1.

Stability of system (1) under DOBC (8) is stated in Theorem2 and its proof is given in Appendix A.

Theorem 2:There exists a DOBC law (8) such that theclosed-loop system (9) is input-to-state stable in the sense thatwhen the disturbancew and its derivative are bounded, thesystem state and the observer error are bounded if the controllaw u = α(x) renders the closed-loop system input-to-statestable and the disturbance observer is given by (3) and satisfiesthe conditions in Theorem 1.

Theorem 2 establishes the stability of the closed-loop sys-tem under the DOBC scheme.β(x) in the the control law

Control 2004, University of Bath, UK, September 2004 ID-048

Observererror dynamics

Systemdynamics

- e -w

Overall dynamics

- x

Fig. 2. Structure of the overall closed-loop system under DOBC for systemssatisfying matching condition

can be constructed following the procedure in Reference [12,p.27-28]. However, for a class of nonlinear systems wherethe disturbance satisfies so-called matching condition [2],more explicit results will be established in the following. Adisturbance satisfies matching condition if it can be directlyreached by system inputs. This kind of nonlinear system hasbeen investigated by many nonlinear control methods. In thefollowing it is assumed that there exists a nonlinear functiong0(x) such that

g1(x)g0(x) = g2(x) (11)

This condition can also be extended to the nonlinear systemsthat can be transformed to satisfy it by a diffeomorphism [11].

When the nonlinear system (1) satisfies the matching con-dition (11) and the DOBC law (8) is applied, it follows from(9) that the controlled system dynamics are described by

x = f(x) + g1(x)α(x)+(g1(x)β(x) + g2(x))w − g1(x)β(x)e

= f(x) + g1(x)α(x) + g2(x)e (12)

The structure of the overall closed-loop system consistingof the observer error dynamics (10) and the above controlleddynamics, (12), is shown in Figure 2. The stability of theDOBC for this class of nonlinear systems is established inTheorem 3 and its proof can be found in Appendix B.

Theorem 3:: Consider a nonlinear system satisfying As-sumption 1 and matching condition (11). Suppose that thecontrol law u = α(x) renders the closed-loop system input-to-state stable. Then the closed-loop system under the DOBC(8), wherew is given by the nonlinear disturbance observer(3) with nonlinear gain (5),(6) and a properly chosenp0, andβ(x) is chosen as−g0(x), is input-to-state stable in the sensethat when the variation rate of the disturbance is bounded, theobserver error and the system state are bounded.

Remark 3:Theorem 3 points out that when the matchingcondition is satisfied, the bounds of the state and the observererror under the DOBC scheme only depend on the variationrate of the disturbances, which is different from Theorem 2where the bounds of the state and the observer error dependon both the amplitude of the disturbance and its derivative.Inparticular, when the disturbance is constant in a period of time,zero steady state error can be achieved.

V. SIMULATION EXAMPLE

An electro-mechanical system with a nonlinear spring canbe modelled as a controlled Duffing’s equation, described by

[15]mx + cx + f1x + f3x

3 = ktu + b (13)

where x, u and b are displacement, controller force anddisturbance respectively.m, c and kt are the mass, dampingand the torque constant respectively. The characteristics of thenonlinear spring are represented by the parametersf1 andf3.b is the external disturbance and it is unknown load in thiscase. The output is the displacement

y = x

For this simple nonlinear systems, a number of nonlinearcontrol methods can be applied and the feedback linearisationtechnique is adopted in this paper which is given by

u(t) = m (k1(yd − y)− k0x) /kt ++(cx + f1y + f3y

3)/kt (14)

. With properly chosen control gaink0 andk1, good trackingperformance can be achieved.

However the feedback lineairsation control law designedabove may have quite poor disturbance attenuation ability.Consider that unknown variable load is applied on this electro-mechanical system. For example, when the load time historygiven in Fig. 3 is applied, the position error under the designnonlinear control law is shown in Figure 2 by the dashed line.The parameters for the simulation has the following values:m = 1 kg, c = 5 Ns/m, f1 = 100 N/m, f3 = 500, 000 N/m3,kt=1N/v, k0 = 16N/m/s, k1 = 99N/m. With the increase ofthe load, the position error significantly increases.

Then a nonlinear disturbance observer is designed andintegrated with the above designed controller. As shown by thesolid line in Fig 4, the position error is significantly reduced.Further more the magnitude of the position error does notdepends on the magnitude of the load but the variation rateof the load. For the period of from 10 second to 20 second,the load is quite large but since it remains a constant value,the position error gradually reduces to zero. This confirms thetheoretic result presented in Theorem 3. To further investigatethe performance of the proposed approach, random noise isadded in the unknown load. It is found that the controllermaintains good performance.

VI. CONCLUSION

A DOBC approach to the control problem of nonlinearsystems with a large variety of disturbances was presented inthis paper. The uncertain disturbances could be time-varyingand no explicit modelling of the disturbances is required.This work significantly extends the applicability of the DOBCapproaches for nonlinear systems and provides theoretic jus-tification for some DOBC applications where the assumptionthat the disturbance is generated by a known model is notsatisfied.

ACKNOWLEDGEMENT

The author would like to thank financial support for thiswork from UK Engineering and Physical Science ResearchCouncil under the grant GR/N31580

Control 2004, University of Bath, UK, September 2004 ID-048

4

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40External load time history

Time (sec)

Load

(N

)

Fig. 3. Time history of unknown load

0 5 10 15 20 25 30−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05Performance with and without disturbance observers

Time (sec)

Pos

ition

err

or (

m)

Without disturbance observerWith disturbance observer

Fig. 4. Position error comparison with and without disturbance observer

REFERENCES

[1] K. J. Astrom. Introduction to Stochastic Control Theory. AcademicPress, New York, 1970.

[2] B.R. Barmish and G. Leitmann. On ultimate boundedness controlof uncertain systems in the absence of matching condition.IEEETransactions on Automatic Control, 27:153–158, 1982.

[3] G. Bartolini, A. Ferrara, and A. A. Stotsky. Robustness and performanceof an indirect adaptive control scheme in presence of bounded distur-bances.IEEE Transctions on Automatic Control, 44:789–793, 1999.

[4] D.S. Bayard. A general theory of linear time-invariant adaptivefeedforward systems with harmonic regressors.IEEE Transactions onAutomatic Control, 45(11):1983–1996, 2000.

[5] W.-H. Chen. Nonlinear PID predictive control of two-link roboticmanipulators. InProceedings of 6th IFAC Symposium on Robot Control,pages 217–222, Vienna, Austria, 2000.

[6] W.-H. Chen. A nonlinear harmonic observer for nonlinear systems.ASME Journal of Dynamic Systems, Measurement and Control, 125(1),2003.

[7] W.-H. Chen, D. J. Ballance, P. J. Gawthrop, and J. O’Reilly. Anonlinear disturbance observer for two-link robotic manipulators.IEEETransactions on Industrial Electronics, 47(4):932–938, August 2000.

[8] W.-H. Chen and L.Guo. Control of nonlinear systems with unknownactuator nonlinearities. InSubmitted to IFAC NOCLOS, Stuggart,German, 2004.

[9] Y.H. Huang and W. Messner. A novel disturbance observer designfor magnetic hard drive servo system with rotary actuator.IEEETransactions on Magnetics, 4(Pt.1):1892–1894, 1998.

[10] J. Ishikawa and M. Tomizuka. Pivot friction compensation using anaccelerometer and a disturbance observer for hard disk.IEEE/ASMETransactions on Mechatronics, 3(3):194–201, 1998.

[11] A. Isidori. Nonlinear Control Systems: An Introduction. 3rd Ed.Springer-Verlag, New York, 1995.

[12] A. Isidori. Nonlinear Control Systems: II. Springer-Verlag: London,1999.

[13] A. Isidori and C. J. Byrnes. Output regulation of nonlinear systems.IEEE Transactions on Automatic Control, 35:131–140, 1990.

[14] C.J. Kemf and S. Kobayashi. Disturbance observer and feedforwarddesign for a high-speed direct-drive positioning table.IEEE Transactionson Control Systems Technology, 7(5):513–526, 1999.

[15] E. A. Misawa. Discrete-time sliding mode control for nonlinear systemswith unmatched uncertainties and uncertain control vectors.ASMEJournal of Dynamic Systems, Measurement, and Control, 119(3):503–512, 1997.

[16] A. J. van der Schaft. L2-gain analysis of nonlinear systems and nonlinearstate feedback H∞ control. IEEE Trans. Automat. Control, 37:770–784,1992.

APPENDIX A: PROOF OFTHEOREM 2

When the control lawu = α(x) is applied to system (1),the closed-loop system is given by

x = f(x) + g1(x)α(x) + g2(x)w (15)

Condition (1) in Theorem 2 implies that there exists anLyapunov functionV (x) such that

∂V

∂x(f(x) + g1(x)α(x) + g2(x)w) ≤ −γ1(‖x‖) (16)

when ‖x‖ ≥ c1(‖w‖) whereγ1(·) and c1(·) are a classK∞andK function respectively [12, p.7].

When the disturbance is estimated by the disturbance ob-server (3) and the DOBC law (8) is applied, as shown inFigure 1, the closed-loop system is described by (9). It can beshown that if‖x‖ ≥ c1(‖w‖), the derivative of the Lyapunovfunction V (x) along the system’s trajectory satisfies

φ =∂V (x)

∂x(f(x) + g1(x)α(x)

+(g1(x)(x)β(x) + g2(x))w − g1(x)β(x)e

≤ −γ1(‖x‖) +∂V (x)

∂xg1(x)β(x)w (17)

Similar to the proof in [12, p.27], it can be concluded thatthere exists aβ(x) such that

φ ≤ −12γ1(‖x‖) (18)

when ‖x‖ ≥ c2(‖w‖) and ‖x‖ ≥ c1(‖w‖) where c2(·) is aclassK function. It follows from (10) that these two conditionsare implied by

‖x‖ ≥ c3

(∥∥∥∥ew

∥∥∥∥)

(19)

wherec3(·) is also a classK function.When an Lyapunov function candidate for the overall

closed-loop system is chosen as

Va(x, e) = V (x) +12e2, (20)

its derivative along the trajectory of the closed-loop system inFigure 1 is yielded by

Va(x, e) =∂V

∂xx + e1e1

= φ− p0n0e1 + e1w (21)

Control 2004, University of Bath, UK, September 2004 ID-048

It follows from (18) that

Va(x, e) ≤ −12γ1(‖x‖)− p0n0e1 + e1w (22)

if condition (19) is met.Furthermore, it can be shown that

Va(x, e) ≤ −12γ1(‖x‖) (23)

if

‖e‖ ≥∣∣∣∣

w

p0n0

∣∣∣∣ (24)

Combining condition (24) with (19) obtains

‖x‖ ≥ c3

(∥∥∥∥ew

∥∥∥∥)

≥ c4

(∥∥∥∥ww

∥∥∥∥)

(25)

where c4(·) is also a classK function. Hence it has beenproved that (23) is met if(x, e) satisfies condition (24) and(25). This implies that the observer error and the system stateof the closed-loop system under boundedw andw in Figure1 are bounded. Therefore the closed-loop system under theDOBC is input-to-state stable.

APPENDIX B: PROOF OFTHEOREM 3

Suppose that there existsu = α(x) such that system (1) isinput-to-state stable. Similar to the proof of Theorem 2, thisimplies that there exists an Lyapunov functionV (x) such that

V (x) =∂V (x)

∂x(f(x) + g1(x)α(x)) +

∂V (x)∂x

g2(x)w

≤ −γ1(‖x‖) +∂V (x)

∂xg2(x)w (26)

Furthermore there exist a classK∞ functionγ2(·) and a classK function c1(·) such that

V (x) ≤ −γ2(‖x‖) (27)

if‖x‖ ≥ c1(‖w‖). (28)

Define Va(x, e) = V (x) + 12e2 as an Lyapunov candidate

for the overall closed-loop system in Figure 2. Differentiationof the Lyapunov function along the trajectory of the overallclosed-loop system consisting of the observer error dynamics(10) and the controlled system dynamics (12) yields

Va(x, e) =∂V (x)

∂xx + ee

=∂V (x)

∂x(f(x) + g1(x)α(x))

+∂V (x)

∂xg2(x)e + e(−p0n(x)e + w)

(29)

Similar to the proof of Theorem 1, without loss of generality,it is assumed thatn(x) > 0 for all x and t. Using the resultsin (26), (27) and (28), one can conclude that

Va(x, e) ≤ −γ1(‖x‖) +∂V (x)

∂xg2(x)e + e(−p0n(x)e + w)

≤ −γ2(‖x‖)− p0n0e2 + ew

≤ −γ2(‖x‖)− p0n0e2 + ew (30)

if‖x‖ ≥ c1(‖e‖) (31)

Furthermore, it follows from (30) that

Va(x, e) ≤ −γ3(‖x‖) (32)

if

‖e‖ >

∣∣∣∣w

pono

∣∣∣∣ (33)

whereγ3(·) is a classK∞ function.Combining (31) and (33) shows that (32) is met if

‖x‖ ≥ c1(‖e‖) > c1

( ‖w‖pono

)(34)

ande satisfies (33).Hence the state and the observer error of the closed-loop

system under the DOBC are bounded if the variation of thedisturbance is bounded. QED.

Control 2004, University of Bath, UK, September 2004 ID-048