state observer for a class of nonlinear systems and its application

5/22/2018 State Observer for a Class of Nonlinear Systems and Its Application

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Proceedings of the 2002 IEEE lntemationalConferenceonControlApplicationsSeptember 18-20,2002 Glasgow, Scotland,U.K.State observer for a class of nonlinear systems and its application

Xinkai Chen* and Gu isheng Zhai***Departm ent of Intelligent M echanics, Kinki University, Uchita, Naga-gun, Wakayama 649-6493, JapanE-mail: [email protected]; Fax: +8 1-736-77-4754**Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-85 10, JapanE-mail: [email protected];Fax: +8 1-73-457-8 187Abstract: In this paper, we consider the state observerproblem for a class of nonlinear systems which are usuallyencountered in the machine vision study. The formulationof the state observer is motivated by the sliding modemethods and adaptive control techniques. The proposedobserver is applied to the identification problems of themotion parameters and space position of a moving objec t byusing the perspective observation. Simulation results showthe proposed o bserv er is very effective.1. IntroductionPerspective observ ations arise naturally in the study ofmachine vision [1-81. The observation via a camera consistsof the perspective projection of poin ts in the 3-D scene ontothe image plane. Thus, points in 3-D space are observed upto a homogen eous line. In the study o f machine vision [3-71,we often encounter into the observation problem of a classof nonlinear system s which can be described by

(1)i 1 = @ (x,, ) X 2 +w, U)i ,= g ( x , x, , U)x1=A --,lnl E X , c Rhere >

x , = b2,,. . , x 2 , , TE X , c R ' and U E U c R k Then x n matrix @ (x, , U ) and the vector g ( x ,, , , U) arein general nonlinear func tions of their arguments. x1 is theoutput which is available. x2 is the unknown partial state.

For the case n 2n,, some formulations of the stateobservers are proposed in [3-71. However, for the casen < n 2 , o result has been reported about the state observer.And the parameter identification problem for a movingobject by using the perspective observation can beattributed to the case nl < n

In this paper, we consider the observer design problemfor the class of sy stems described in (1). For this purpose,we make the follow ing assumptions.( A l ) x, and U are available signals. X,, X , and Uare bounded sets.(A2) The functions @ , 0 , e , - and g , ., -) areknown

d@(xl are bounded for all

{

dt4 (x,, U ) andx 1 ~ X 1 c R n 1nd U E U C R ~ .

(A4) The function g ( x ,, , , ) is piecewise differentiablewith respect to x, , and the partial derivatives are bounded(at the undifferentiable points, we mean the left and rightside derivatives).(A5) There exist a positive constant p and a very smallpositive constant p such that

for all t 2 0.Under the above assumptions, we try to formulate thestate observer for the systems (1) in section 2, where themodified sliding mode method and the adaptive controltechniques are employed. In section 3, the proposed methodis ap plied to the identification of the motion param eters andthe space position of a moving object by using theperspective observation. Section 4 conc ludes this paper.

2. State observerDenote by x =[x,, lT t follows that for n = n +n 2 ,The d iscontinuous observer is constructed as

X E X C R and X = X , @ X , .[il@(x,, U ) i 2 +8(x,, U)+ w(t)

(3)

where the initial condition is determined asiz0s a vector with constant entries; I? is a positivedesig n parameter; is defined as

21(0) = X l ( 0 ) 2 2 2 (0) = 3220 9 (4)

q = min$ : >q-]and ~ \ ~ , ( t ) l ~ 2M } , ( 5 )and To = 0 ; > 0 is a large constant; w t) is definedas

.T

Si 0 are design param eters, iit) are defined as

0-7803-7386-3/02/ 17.000 002 IEEE 88



a re positive constants, (O) can be any positiveconstants for i = 1, - . e , n , .Remark I : q. defined in ( 5 ) are the discontinuous pointsof the system (3). By o bserving (3) and (5) , it can be easilyseen that 2, (t) is bounded by IF t)ll, M

Definee , = [ e , , , . . . , e , , p = x l - , e , = x 2 - i z . (8)

Now, combining (1) and (3) yieldsL. ( t )= e, -w(t) , e ,(0)= 0 9)

First of all, for the system (9), et us confirm theboundedness of $e2.By Remark 1 and assumption (Al) , itcan be easily seen that e 2 ( t ) is uniformly bounded.Further, by applying assumption (A3), the uniformboundedness of e, is obvious, i.e. there exist A, > 0such thatwhere ( ez), denotes the i-th entry of @e,

i , ?)= g ( x l , x z , ) - g ( x , , i , , )-r@ (t)w(t)I *, ) I Ai (10)

el,leli + SiRemark 2 : For i = 1, ..-, , the terms 1 t ) - ,which are motivated by the sliding mode method [ lo], are

introduced to compensate the bounded items (@ez, and toassure that leli(t)l and leli(t)l are very small. Unlike thetraditional sliding mode method, the upper bou nds A in(10) need not be known n our method. The upper boundsA, are adaptively updated by (7). Further, thediscontinuous form sign(el, t ) ) in the traditional slidingmode m ethod is modified asA

eli + SiIn order to derive the main result, the next lemma isintroduced for the co nstructed system (3).Lemma 1:About the constructed system (3), the following

results hold for' i = l , . . - , n , .(1). leli t)l and iit) are uniformly bounded.(2). There exist t i >0 and functions ,(U) >0 with theproperty l i m q (U)= 0 such that leli t)l I 26, andle,, t)l , (6,) for all t > t: .Proof: The proof is composed of the following two steps.Step 1 For i = l,..., n the boundedness of leli t ) l andi t) can be proved by differentiating

U 4 0

V, ( t )= (e,, t))' +Step 2 The second result can be proved by *differentiating(e,,(t))' based on the equation

for t > ti.Remark 3: In the constructed system (3), ( f ) is theauxiliary output. The dynamics of ,?,(t) is introduced inorder to estimate the unknown inner product e,. Lemma

tells us that its corresponding estimate is w ( t ) (see thefirst equation in (9)) (see also Remark 2). This estimate isemployed in the second equation in (3) to force i 2 t ) tobe very close to x, (t) as is very large.

The next theorem giv es the main result of this paper.Theorem 1. e , @ ) is uniformly bounded and decreasesexponentially. Further, there exist T 2 and a functionE @ , , Vq 1> 0 with the property2 v , olim ~ ( v , ,. e , v n l ) 0 such that=I

/le, t)ll, I E@, 9 * . 9 6 , ) (13)for all t 1T , i.e., & ( t ) generated in (3) is theapproximate estimate of x 2 ( t ) for sufficient large t bychoosing very small parameters 6i i = y - - - , n l ) .Proof: By the assumptions (A4), it can be easily seen thatthere exists a norm bound ed matrix S2(8,6,y, t) such thatwhere the mean-value theorem is employed .g ( x 1 , x 2 , ) - g ( x , , & , U) = Q ( x , , x , , 2 , , u ) e 2 ( t ) , (14)

K ( t ) = -r@t)el ( t ) .Define (1 5 )From Lemma 1, it can be seen that there exists a function4 v 1 Vnl 1> 0 with the propertylim d ( v , , - . , v n , ) 0 such that

5 V + O=I

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follows that q-q- is greater than a positive constant,say x (As p is very small, we can conclude thatx >> p , for all i. On every such a time interval, thevariable lie t)ll, decreases exponentially (see (28)) and, at

, the estimation error e , @ ) satisfies[le2(q+ 0111 < Ile2 T - 0)11 Thus, ]le2t)ll, decreasesexponentially until it becomes very small. Therefore,Ile, (t)ll, is bounded and (13) is proved.Remark 4 : The convergence of the observer is guaranteedby the assumption (A4), which is the so-called persistentlyexisting (PE) condition in the adaptive control literature [ 9 ] .We also call it the convergence condition of the obse rverin the following of this paper.3. Application to machine visionConsider the movement of the objec t described by

dtwhere ~ ( t )[xl, x,, x 3 r is the position in the space;mi (t) and b,(t) i = 1,2,3) are the mo tion parameters.

It is supposed that the observed position in one imageplane is defined by

If it is necessary, it is supposed that the ob served position inanother image plane is defined byr

where m and n are constants.The perspective obser vatio ns are defined in (30) and(30). The combination of the observations in (30) togetherwith (30) is called stereo vision.LetW)= [u , , ,, w3fT, W ) = b 1 , b, , b3fT (31)

In the machine vision research, we make the followingassumptions.(MI) . Th e motion parameters w, t) and bi(t) i = 1,2,3)are bounded.(M2). x 3 ( t ) meets the condition x3(t) > 77 > 0 , where77 is a constant.(M3). m and n are known constants(M4). y ( t ) and y*( t ) arebounded.Remark 5: It is easy to see that assumptions (A2) and (A4)are reasonable by refemn g to the practical systems.

90



Remark 6: The considered motion described in (34) can(38)uch more complicated than the motions in [1-81.

x ( t ) and the motion parameters @ , ( t ) if ~ 3 ( t )s available. SO, f . ~ 3 ( t ) an be estimated, theposition ~ ( t )an also be estimated.bi (t) i = 1,2,3) by using the perspective observations. In the following, we try to estimate y3( t ) and the

can be calculated ascover a large class of practical perspective systems. It is l ( t ) Y2 (0 1x 1 t )= x 2( t )= x 3 t )=30 Y3 (0 Y3 ( t )The purpose of this research is to estimate the position

3.1 PreliminariesDefine motion parameters 0 ( t ) and b i ( t ) i = 1,2,3)3.2 Estimation of O t )

Based on the equation (37), an observer of O t ) can beconstructed by using the result of Theorem 1.To assure the convergence of the observer, the followingassumption is needed.(M6) There exist a positive constant 77 and a very smallpositive constant p such thatpPTW W T 171 (39)for all t 2 0 .Remark 7 In the estimation of O ( t ) , only the observationobtained by one camera is needed.3.3 Estimation of y 3 t )

(35) Even though the vector O (t ) can be estimated, it can beIt should be noted that cp(t) is available. seen that only the product [bl b2 b3b3 can beIn order to estimate the motion parameters @,( t ) and

b, (t) i = 1,2,3) we make the following additionalassumptions.(M5) The dynamics of q r ) and b,(t)(i= 1,2,3) areknown as

(36)Further, the functions q 6, t ) and p(z9, t )b arepiecewise differentiable with respect to 6 and the partialderivatives are bounded (at the undifferentiable points, wemean the left and right side derivatives).By using (33)-(36), system (33) gives

h t)= 6, t ) d ( t ) = p ( 6 , ) b .

(37)

It is obvious that the position of the object in the space

estimated. The remaining task is to estimate y 3 and[b , b , b 3 r In this section, y 3 is estimated byappealing to the stereo vision.

Now, by differentiating y*( t ) defined in (30'), it yields

+ nu, 0 2Y ; + n@,Y,.b3

Thus, based on (40), the observer of y 3 can beconstructed by applying the result of Theorem 1, where theestimation of O t ) obtained in section 3.2 is needed.

In order to assure the convergence of the observer, thefollowing assumption is needed.(M7) There exist a positive constant 77 and a very smallpositive constant p' such thatp .q *U; + YY;P +(-- +*Y: n w ; P b

2777 (41)for all t 2 0 .

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3.4 Estimation of [b, b, b3pNow, based on the results in Se ctions 3.2 and 3.3, theobserver of b can be constructed based on the followingdynamicsW=b 0 - Y 1 l y , . b + L , 1+v: Y l y Z h )~ , ( t > = b - Y 2 b d + [ - Y 1 Y l Y 2 l + r : b ( 0 ( 4 2 )d ( t )= p(o, ) b

It can be easily seen that, if the assumption (39) holds,where the first two equation in (42) is the rewritten form ofthe first two equa tion in (33).then the convergence condition for the observer of (42) isalso satisfied.4. Example and simulation resultsIn this section, we present the simulation results for anexample. The simulation is done by Matlab. The samplingperiod is chosen as 0.05. Suppose the measured image datais corrupted by 1% random noise.Con sider the movement of the object described by

dtwith [x,(O) x2(0) x3(0)F = [2 2 4r.

Figure1 shows the motion in the space. Figure 2 showsthe observed image data by Camera 1. Strictly speaking, itshould not be noted by y, and y , as coordinate axes inFigure 2 since the definition of y1 and y2 in (30) isslightly different from the observed data due to theexistence of random noise. The image data observed byCamera 2 is similar as that shown in Figure 2. Thesimulation results are shown in Figures 3-9. Figures 3-5show the difference between w t ) and its estimate h t)Figure 6shows the difference between y3 and its estimatej 3 Figures 7-9 show the difference between b( t ) and itsestimate b ( t ) . It can be seen that very good estimates areobtained.

5-10 -10

Fig. 1 The motion in the space.

2 -1 -0 -

-1 -

--d5 -2 -1 '5 -; - 0 5 O 05Y, ( t ) lFig. 2 The observed image data by C amera 1.

1 51

0.50

-0 50 5 10 t 15Fig. 3 The difference between o, nd its estimate hl

5 10 t 15Fig. 4 T he difference between w, and its estimate 6 .420

-2

-6-a I5 10 t 15

Fig. 5 The difference between 0, and its estimate h3.5 ConclusionsIn this paper, we consid ered the observation problem fora class of nonlinear systems which are often encountered inthe study of machine vision. The proposed formulation is

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motivated by sliding mode method and adaptive controltechniques. The proposed algorithm is very simp le and easyto be implemented.The formulated observer is applied to the machine visionproblems, where the motion parameters and the position ofa moving object in the space are identified by using theperspective observation. The parameters in w ( t ) can beidentified by using only one camera. However, the stereovision is necessary in order to identify the variable y 3 ( t ) .The identification of b ( t ) is performed based on theidentifications of @ ( t ) and y 3 ( t ) Simulation resultsshow the proposed o bserver is very effective.

15I Il o t

I0 5 10 t 15101Fig. 6 The difference between y 3 and its estimate j 3

IOl

5 10 t 15Fig. 7 The difference between b, and its estimate b,

151050-5

-10-15-20

I5 10 t 1530;Fig. 9 The difference between b3 and its estimate b3 .

References[ l ] N. Ayache, ArtiJcial Ksion fo r Mobile Robots, StereoVision and Multisensory Perception (MIT Press,Cambridge, MA, 1991).[2] M. Boutayeb, H. Rafaralahy and M. Darouach,Convergence analysis of the Extended Kalman Filterused as an observer for nonlinear deterministicdiscrete-time systems, IEEE Trans. on AutomaticControl, Vol. 42, pp. 581-586, 1997.[3] X. Chen and H. Kano, Observation for the perspectivesystem, Proc. European ControlConference, pp.

[4] X. Chen and H. Kano, A new state observer forperspective systems, IEEE Trans. on Automatic Control,[5] B. K. Ghosh, H. Inaba and S. Takahashi, Identificationof Ricatti dynamics under perspective and orthographicobservationsyYyEEE Trans. on Automatic Control, Vol[6] M. Jankovic and B. K. Gh osh, Visually guided rangingfrom observation of points, lines and curves via anidentifier based nonlinear observer, Systems & ControlLetters, Vol. 25, pp. 63-73, 1995.[7] E.P. Loucks, A perspective System Approach to Motionand Shape Estimation in Machine Vision, Ph.D Thesis,Washington University, 1994.[8] L. Matthies, T. Kanade, and R. Szeliski, Kalmanfilter-based algorithm for estimating depth from imagesequence, International Journal of Computer Vision,Vol.[9] S. Satry and M. Bodson, Adaptive Control, Stabiliw,Convergence, and Robustness (Prentice Hall, EnglewoodCliffs, New Jersey, 1989).[ lo] V. I. Utkin, Sliding modes in Control Optimization(New York Springer-Verlag, 1992).

697-702,2001.

Vol. 47, No.4, pp. 658-663,2002.

45, NO. , pp. 1267-1278,2000.

3, pp. 209-236, 1989.

Fig. 8 The differenc e between b, and its estimate b,

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state observer for a class of nonlinear systems and its application

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