ESTIMATION OF UNKNOWN DISTURBANCES IN NONLINEAR SYSTEMS

R. Sharma and M. Aldeen Dept. of Electrical and Electronic Engineering

The University of Melbourne Australia

ABSTRACT In this paper we extend the nonlinear unknown input observer design method of Koenig and Mammar [6] to a wider class of nonlinear systems where both input and output disturbances are present. The approach used in this paper is to decouple the disturbances from the rest of the system through a series of transformations on the both state and output equations. Once total disturbance decoupling is achieved, an appropriate observer for the disturbance free part is proposed and designed. The observer gain matrix is determined by solving an algebraic riccati equation. The designed observer is, subsequently, used for estimation of the unknown input and output disturbances. 1. INTRODUCTION.

Observer based techniques have been the subject of extensive research over the past four decades. For linear systems, the problem of observer design has reached maturity and various techniques have been proposed (e.g. Kudva et al [1], Guan and Saif [2], Hou and Muller [3]). For nonlinear systems, however, there still remain many open issues. This is particularly the case when such systems experience unknown disturbances in both the input and output, which are a source of erroneous information. Most of real life industrial systems, when examined in detail, are nonlinear, to some extent. For such systems, the issue of designing nonlinear observers in the presence of unknown input and output is of a practical importance. Fault detection and isolation (FDI) is but only one example of how important such observers can be when used for the purpose of process monitoring in presence of unknown disturbances. Thus, in the design of observers for fault detection purposes, the observers should be insensitive to unknown disturbances in the inputs and outputs and at the same time perform the task of monitoring and state estimating accurately. Whereas for linear systems, necessary and sufficient conditions for existence of unknown-input observers (UIO) have been fully established [1], little work has been reported in the field of nonlinear unknown input observer (NUIO) theory. It is for this reason that recent research efforts have been directed into addressing this deficiency.

One of the earliest studies in this field is the work of Wunnenberg [4]. In this approach, the nonlinearities acting on the system are assumed to be functions of the inputs and available measurable outputs. However, such assumption places restriction on its applicability. Seliger and Frank [5] proposed an alternative approach for a general class of nonlinear systems. The approach provides existence conditions and proposes a design method based on disturbance decoupling through state transformation. A draw back of this approach is that convergence of the error dynamics is not proven or discussed. More recently, [6] reported an unknown input reduced order observer for nonlinear systems for the purpose of FDI, which is also based upon complete disturbance decoupling. In another approach by Shields [7], an unknown input observer is designed for nonlinear descriptor systems which can be used in residual generation for fault detection. The advantage of this approach is that robustness to unknown inputs can be investigated without resorting to disturbance decoupling. None of the above mentioned approaches take into consideration the presence of plant output disturbances. The inclusion of plant output disturbances is an important practical problem as it provides an option for accounting for modeling uncertainties and could, therefore, enhance the robustness properties of the observation scheme significantly. Furthermore, the above approaches deal primarily with FDI and do not consider the problem of directly estimating the external unknown disturbances. In this article, we extend the unknown input nonlinear observer design methodology to the class of nonlinear systems which are affected by both unknown input and output disturbances. In addition, we use the designed nonlinear unknown input and output observer to estimate these unknown disturbances. An algebraic riccati equation (ARE) and a Lyapunov function are used for the purpose of gain computation and stability analysis. 2. SYSTEM DESCRIPTION AND ANALYSIS. The class of nonlinear systems considered in this paper is composed of a linear unforced part and a nonlinear state dependent controlled part and is described as follows:

),(+++= 1 uxgEdBuAxx� (1)

2+= DdCxy (2)

Control 2004, University of Bath, UK, September 2004 ID-017

where, nx ℜ∈ ; pu ℜ∈ ; my ℜ∈ ; 1 ;qd ∈ℜ

2qd ∈ℜ are the state, input, output, unknown input

and output disturbances, respectively, and matrices CEBA ,,, and D are constant and of appropriate

dimensions. In the ensuing analysis the following have been assumed to hold: A1. ),( uxg is lipschitz with lipschitz constant

.0γ That is,

21021 ),(),( zzuzguzg −≤− γ ., 21nzz ℜ∈∀

A2. .��

���

�+=�

�

���

� −D

Erankn

DC

EAsIRank

A3. ( ) ( ).0

CErankDrankD

CEDRank +=�

�

���

�

In order to obtain the disturbance free subsystem of (1)-(2), we define the following two transformations

��

���

�Φ= +E

T 11 and �

�

���

�Φ= +D

T 22 ,

where, ( )1

n q n− ×Φ ∈ℜ and ( )2

m q m− ×Φ ∈ℜ are the null

spaces of E and D , respectively, and +E and +D are pseudo inverses of E and D , respectively. Now, (1)-(2) can be re-written as

��

���

�+��

���

�+�

�

���

�+�

�

���

�=�

�

���

�

0),(

0 2

1 uxg

Dd

Edu

Bx

C

A

y

x� (3a)

Multiplying (3a) with ,0

0

2

1��

���

�

T

Tyields

��

���

�+��

���

�+�

�

���

�+�

�

���

�=�

�

���

�

0),(

01

22

111

2

1

2

1 uxgT

DdT

EdTu

BTx

CT

AT

yT

xT �

(3b) Defining,

��

���

�=

2

11 x

xxT , (4)

equation (3b) is equivalent to

��

++++=Φ+++=

+ ),(),(

122221212

112121111

uxgEduBxAxAx

uxguBxAxAx�

�

(5)

and

��

++=+=

22221212

2121111

dxCxCy

xCxCy (6)

where, =−111 ATT �

�

���

�

2221

1211

AA

AA; BT1 = �

�

���

�

2

1

B

B; ;

01 �

�

���

�=

IET

;2

12 �

�

���

�=

y

yyT ;

02 �

�

���

�=

IDT ;

2221

1211112 �

�

���

�=−

CC

CCCTT

The disturbance free subsystem can now be expressed as

),(112121111 uxguBxAxAx Φ+++=� (7)

2121111 xCxCy += (8)

In order to eliminate 2x from the above subsystem, we

assume 12C to be of full column rank and define the following transformation,

��

���

�Φ= +

12

33 C

T (9)

where, ( )3

q m q× −Φ ∈ ℜ is the null space of 12C and +12C is

the pseudo inverse of 12C . Application of 3T on (8), gives

1111 xCy = (10a)

)( 1111122 xCyCx −= + (10b)

where, ;11311 CC Φ= ;131 yy Φ= Substituting (10b) in (7), the final disturbance free system is expressed as

),(111121211 uxguByCAxAx Φ+++= +� (11)

1111 xCy = (12)

where, ( )11121211 CCAAA +−= ; For the system of equations (11) and (12), we propose the following reduced order observer

),ˆ(ˆˆ1111 uxgJuGyxFx Φ+++=�

(13)

11ˆˆ NyxMx += (14)

Theorem1. If assumptions A1, A2 and A3 hold and (i) 12C is of full column rank.

(ii) Pair ( )11, CA is detectable. then, (13)-(14) can act as an unknown input nonlinear observer for system (1)-(2), with

;11CKAF −= ;31212 Φ+= + KCAG ;11 BBJ Φ==

;11121 CCEM T +−Φ= ;= +12CEN

where, K is an observer gain matrix whose computation may be based upon the solution of an ARE.

Proof. Detectability of the pair ( )11, CA guarantees the existence of the following observer for the disturbance free subsystem (11)-(12),

)ˆ(),ˆ(ˆˆ1111111121211 xCyKuxguByCAxAx −+Φ+++= +�

(15) where K is the observer gain matrix. Comparing (13) and (15) gives the desired values for

GF , and .J Also, from (4) we have

Control 2004, University of Bath, UK, September 2004 ID-017

[ ] ��

���

�Φ=

2

11 x

xEx T (16)

By substituting (10b) in (16), we obtain [ ] 112111121 yCExCCEx T ++ +−Φ= (17)

[ ] 112111121ˆˆ yCExCCEx T ++ +−Φ= (18)

Theorem2. The following statements are equivalent.

(i) The matrix 12C is of full column rank and

the pair ( )11, CA is detectable.

(ii) ��

���

�+=�

�

���

� −D

Erankn

DC

EAsIrank ∀ complex .s

Proof. The proof of this theorem can be found in Hou and Muller [9].

3. GAIN COMPUTATION. Consider the following ARE.

01

111120 =++�

�

��

� −++ IQPCCIPAPPA TT εε

γ

(19) where, .0>ε It should be noted that this form of ARE leads to the provision of a more general set of design parameters than in [6]. Hence, a more appropriate gain matrix can subsequently be obtained.

Let, TCPK 1121ε

= and ( ) ,12

01−

= γPP 1P is symmetric

positive definite. Using these definitions in (19) gives

( ) ( ) ( ) 01120111111 =+++−+− IPIQPPCKACKAP

Tεγ

(20) The ARE (20) contains three design parameters ( )0,, γε Q , which provide a greater design freedom than in [6]. By appropriate choice of these parameters, a suitable observer gain matrix, ,K can be obtained. Detailed analysis and algorithm for determination of P and K using this technique can be found in Raghavan and Hedrick [10]. 4. STABILITY ANALYSIS. Let,

xxe ˆ−= (21)

and 111 x̂xe −= (22)

then ( ) geCKAe ~11111 Φ+−=�

where, ).,ˆ(),(~ uxguxgg −= Let us introduce the following Lyapunov function 111 ePeV T= This implies

( ) ( ) 11111111 ePCKACKAPeVTT

��

���

� −+−=� gPeT ~2 111 Φ+

or ( )[ ] gPeeIPIQPeV TT ~2 111111

201 Φ+++−= εγ�

≤ ( )[ ] ePeeIPIQP 01112

11120 2 γεγ Φ+++−

From equations (17), (18) and (21) we obtain xxe ˆ−= = ( ) .ˆ

111 eMxxM =− Thus, we have

( )[ ] 21011

2111

20 2 eMPeIPIQPV γεγ Φ+++−≤�

( )( ) ( )[ ]MPIPIQPe 11min011max

20

21 2 Φ+++−≤ σγεσγ

where, maxσ and minσ represent maximum and minimum singular values, respectively. Thus, the system is asymptotically stable if

( )( )( )IPIQP

MP++

Φ>

11max

11min0

2εσ

σγ (23)

By suitable choice of design parameters ( )0,, γε Q , (23) can be satisfied and asymptotic stability of the error dynamics can be guaranteed. 5. ESTIMATION OF UNKNOWN DISTURBANCES. In [6], the estimated states are used for residual generation. Here, we utilize the state estimates to reconstruct the unknown disturbances associated with both inputs and outputs. Although the theory presented in this paper is used for estimation of unknown disturbances, it can equally be used for fault detection in presence of disturbances in nonlinear systems. A separate paper is under preparation in this regard. (1) Input disturbances:

From (5), we obtain input disturbances as

),(222212121 uxgEuBxAxAxd +−−−−= � Substituting for 2x from (10b) and rearranging, we obtain

),(541322211 uxgHuHxHyHyHd iiiii +++Φ+Φ= � where,

+= 121 CH i ;

( )+++ −−= 1222121211122 CACACCH i ;

���

��� +−−= ++

1112222111123 CCAAACCH i ;

[ ]2111124 BBCCH i −−= + ;

[ ]++ −Φ−= ECCH i 111125 ;

Control 2004, University of Bath, UK, September 2004 ID-017

Therefore, an estimate of input disturbances is given by

),ˆ(ˆˆ541322211 uxgHuHxHyHyHd iiiii +++Φ+Φ= �

(24) Remark: It can be shown that (24) may be also be expressed as

),ˆ(ˆˆ541312111 uxgHuHxHyHyHd iiiii ++++= �

It may be noted here that, since, 1y is disturbance free, the existence of its derivative is not affected by the unknown behavior of output disturbances, and thus estimate of the input disturbance is independent of the output disturbance. (2) Output disturbances: Using (6), the output disturbances can be written as

22212122 xCxCyd −−= On substituting 2x from (10b) and rearranging, the above equation becomes

1212 xHyHd oo += (25) where,

[ ]++ +Φ−= DCCH o 212221 ;

[ ]111222212 CCCCH o++−= ;

Thus, the output disturbances can now be estimated as

1212ˆˆ xHyHd oo += (26)

The estimates 1x̂ and x̂ can be obtained from the nonlinear unknown input observer defined by system (13)-(14). 6. NUMERICAL EXAMPLE In order to illustrate the methodology described in sections 2-5, we consider the example of a single link flexible joint robot. The system description for this system is given by the following set of algebraic equations [10],

( )

( ) ( )

1

1 1

1 11 1

sin

m m

Rm m m

m m m m

m m

K KBku f

J J J J

k mghJ J

τ τ

θ ω

ω θ θ ω

θ ω

ω θ θ θ

=

= − − + −

=

= − − −

�

�

�

�

where, mθ and mω are, respectively, the position and angular velocity of the motor and 1θ and 1ω represent those of the link. u and f denote the known input and the fault signal, respectively. Using the values of the parameters as in [10], a state space description for the system of the following form is obtained:

3

00 1 0 0 0 0048.7 12.4 48.7 0 21.6 21.600 0 0 1 0 0

66.4sin19.4 0 19.4 0 0 0

1 0 0 0 10 1 0 0 20 0 1 0 1

out

x x u f

x

y x d

� �� � � � � �� �� � � � � �− − − � �� � � � � �= + + +� �� � � � � �� �� � � � � �−−� � � � � � � �� � � � � � � �

−� � � �� � � �= +� � � �� � � �� � � �

�

where, 1 2 3 4 1 1 .T T

m mx x x x x θ ω θ ω= =� � � �� � � � This example satisfies conditions A1-A3. For the fulfillment of condition A1, we choose 0 1.γ = In the following, we will estimate the unknown input (fault) and output disturbances, f and outd . Using,

11

1 0 0 00 0 1 0

;0 0 0 10 0.0463 0 0

TE+

� �� �� � � �= =� � � �� �� �

−� �� �

22

0.8165 0.5266 0.23670.4082 0.2367 0.8816 ;0.1667 0.3333 0.1667

TD+

−� �Φ� � � �= = −� � � �� � � �−� �

33

12

0.4100 0.91210.0731 0.0329

TC +

Φ� � � �= =� � � �−� �� �

,

the disturbance free subsystem is obtained as

1 1 1 1

1 1 0 1.5798 0.7101 00 0 1 0 0 0 ( , )

19.4 19.4 0 0 0 0x x y u g x u

− −� � � � � �� � � � � �= + + + Φ� � � � � �� � � � � �−� � � � � �

�

1 10.7071 0.7071 0y x= � �� �

For the computation of gain K, choosing

3Q I= and 1ε = satisfy inequality (23) with 00.0095 γ< . Thus, the observer described by (13)-(14) has the following design parameters

1.3712 0.6288 00.405 0.405 1 ;

19.3161 19.4839 0F

−� �� �= − −� �� �−� �

1.795 0.23130.2348 0.5224 ;0.0486 0.1082

G

−� �� �= � �� �� �

00 ;0

J� �� �= � �� �� �

0.52490.5728 ;0.1186

K� �� �= � �� �� �

1 0 01 1 0

;0 1 00 0 1

M

� �� �−� �=� �� �� �� �

0 01.5798 0.7101

0 00 0

N

� �� �−� �=� �� �� �� �

.

Coefficients corresponding to input disturbance estimation equation (24) are as follows:

1 0.0731 0.0329iH = −� �� � ;

2 0.8338 0.3748iH = −� �� � ;

3 1.7269 1.7269 0.0463iH = − −� �� � ;

4 1;iH =

5 0.0463 0.0463 0.0463 0iH = −� �� � .

Similarly, coefficients corresponding to output disturbance estimation equation are as follows:

1 0.5 0 0.5oH = −� �� � ; 2 0.5 0.5 0oH = −� �� � .

Control 2004, University of Bath, UK, September 2004 ID-017

7. SIMULATION RESULTS. The properties of the proposed unknown nonlinear observer are now demonstrated through the following simulation study. We first introduce a unit step in u at time t=15s. Then we simulate a gradual loss of motor excitation signal starting at t=25s and finishing at t=30s at which time the excitation signal is totally lost. We also assume that the measurement is contaminated by a random noise signal. The aim of this study is to estimate both the loss of the excitation signal, f, and the output noise signal, outd . Figure 1 shows the actual and estimated fault signals. From this figure, it is clear that estimation of the fault signal is insensitive to any other inputs. The initial oscillations are due to the different plant and observer initial conditions, which can be eliminated if the observer is always turned on. Figures 2 show the estimated random output noise signal. Figures 3 and 4 demonstrate the actual and estimated responses of the motor and link speeds, respectively, to the step change in u at t=15s and also to the gradual loss of the excitation signal starting at t=25s. Thus, it is clear from this study that the approach presented in this paper is able to asymptotically estimate any unknown input and/or fault signal in the state or output.

Fig(1): Estimation of fault signal, f

Fig(2): Estimation of unknown output disturbance

8. CONCLUSION An observer for a class of nonlinear systems with unknown input and output disturbances has been presented in this paper. The designed observer is used for the estimation of the unknown disturbances. A more

general ARE, which allows for more design freedom than in existing publications, is proposed for the computation of the observer gain matrix. A stability condition is derived from a Lyapunov function. Finally, the effectiveness of the technique is illustrated with the help of a numerical example.

Fig(3): Responses of actual and estimated motor speeds

Fig(4): Responses of actual and estimated link speeds

References. 1. Kudva, P., Viswanadham, N. and Ramakrishna,

A., 1980, IEEE Trans. Automat. Contr.,25, 113-115.

2. Guan, Y. and Saif, M., 1991, IEEE Trans. Automat. Contr., 36, 632-635.

3. Hou, M. and Muller, P.C., 1992, IEEE Trans. Automat. Contr., 37, 871-874.

4. Wunnenberg, J., 1990, VDI-Verlag, Reihe 8, Nr.222., Chapter 4, 64-72.

5. Seliger, R. and Frank, P.M., 1991, Proceedings of the 30th IEEE conference on Decision and Control, 2248-2253.

6. Koenig, D. and Mammar, S., 2001, Proceedings of the American Control Conference, 2143-2147.

7. Shields DN, 1997, International Journal of Control, 67, no.2, 153-168.

8. Koenig D. and Patton R.J., 1999, 14th World Congress IFAC, Beijing P.R. China, July 5-9, 1999.

9. Hou, M. and Muller, P.C., 1994, International Journal of Control, 60, 827-846.

10. Raghavan, S. and Hedrick J.K., 1994, International Journal of Control, 59, 515-528.

Control 2004, University of Bath, UK, September 2004 ID-017