A Sliding Observer for Nonlinear Process

8/6/2019 A Sliding Observer for Nonlinear Process

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Pergamon Chemical Engineering Science, Vol. 52, No. 5, pp. 787-8 05, 1997Copyright 0 1997 Elsevm Science Ltd

PII: SOOO9-2509(96)00449-6 Pnnted in Great Britain. All rights reservedcm-2509197 $17.00 + 0.00

A sliding observer for nonlinear processcontrolGow-B in W ang, Sheng-Shiou Peng and Hsiao-Ping Huang*

Departm ent of Chem ical Engineering, N ational Taiw an University, Taipei, Taiwan, 10 617 ,R.O.C.

(Received 18 December 1995; in revised form 5 August 1996; accepted 12 September 1996)Abstract-Sliding observers are considered as nonlinear state estimators with goo d robustnessto bounded modeling errors. In this paper w e have developed sliding observers for processcontrol. The observer is hence designed so as to posses s invariant dynamic mo des which can beassigned independently to achieve the desired perform ance. Converg ence of the estimatingalgorithm is formu lated by using Lyapunov stability theorem s. Conditions for robustness tomodeling errors ar e derived by analyzing the norms of estimation errors. Fo r process control,servo-tracking and disturbance rejection for chemical reacto rs have been discussed by makinguse of this sliding observer. Simulation exam ples to demo nstrate the construction and perfor-mance of this proposed sliding observer for chemical process control are also presented. 0 1997Elsevier Science Ltd. All rights reservedKeyw ords: Sliding mode ; observer; observability; state estimation; nonlinear system.

1. INTRODUCTIONVariou s methods fo r designing nonlinear controllershave been reported in the literature. Unless the y areusing ad hoc designs, most of the nonlinear controllerssuch as those using feedba ck linearization (Hunt et a/.,1983; Kravaris and Chung, 1987), GM C (Lee andSullivan, 1988 ) and many others req uire the feedba ckof state variables to implement the control strategies.In practice, h owever, com plete state feedback is im-practical in most applications. Althoug h d irect inte-gration, i.e. open-loop observer, can be used to esti-mate the required state feedback and wo rks well fora few applications, a closed-loop observer with feed-back correction is still desirable, especially, when thesystem consists of modeling error(s) or pure inte-grator(s).

Althoug h the theories and applications for linearsystems are well developed, development of observersfor nonlinear systems still provide s an open area forresearc h. Till now, developm ent of observer s for non-linear systems has encountered many difficulties, suchas: requirement of extensive compu tational efforts,coupling with controllers, uncertainty in the perfor-mance or robustness, restrictive conditions to be satis-fied, etc. Som e of the major difficulties encountered inthe development of observers, w hich are reported inliterature, have been reviewed by Misawa and Hed-rick (1989).

*Corresponding author.

In the last decad e, attention has been focused onapplying the transform ed canonical form s for design(Bestle and Zeitz, 1 983; Keller, 1987; Kantor, 1989;Ding et al., 1990, etc.). For most of the nonlinearsystems, such a transformation can be definedthrough Lie derivatives of the output which is a func-tion of state variables (Gibon-Farg eot et al., 1994;Alvarez-Ramirez, 1995). Howev er, the resulting ca-nonical form s are not strictly linear and the trans-formations thus involve exogenou s inputs and a finitenumber of their derivatives. In case of simple system s,such difficulties involved in the transformations couldbe easily resolved. However, for systems having di-mensions greater than tw o, solution of the coordinatetransformations and existence of such transforma-tions appear to be the bottleneck.Recently, another type of observers that also usesliding mode have been reported. In the early work ofSlotine et al. (1987), the observer was constructed fora second-o rder nonlinear dynamic system involvingonly single measurem ent. The extensions of such ob-servers to nth-order and multi-output systems havealso been addr essed in the literature. Further develop-ment in this field of sliding obse rver was made byMisawa (1988). Canudas and Slotine (1991) havefurther applied such observers in robot manipulators.In all the studies men tioned above, a fram ewor ksimilar to a Luenberger observer was used by appen-ding a switching function with constant gains aspart of feedb ack correctio ns. To obtain such gains,different procedures have been proposed (Misawaand Hedrick, 1989). These procedures require certain

787


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78 8 Gow-Bin Wang et al.conditions to be satisfied or optimized with singularvalues which depend on the scaling factors. Th us, theprocedures developed for determining the switchinggains are complicated and the performance issue ofthe observers is also not addressed.

It is the purpose of the present paper to designsliding observers for control of nonlinear chemicalprocesses. In chemical process control, measurementsof indirect outputs are usually used to infer the keyoutputs which are rather difficult to be measuredon-line. The sliding observ er presented here is used toforce these indirect but measurable outputs to lie onspecified sliding surfa ces, so that the resulting esti-mated states can be used to implement the nonlinearfeedback control more efficiently. In order to designthe observer independently, time-varying gains for theswitching functions are used to keep the dynamicmodes of these estimated states invariant regardless ofthe position of the states. For formulating the nominalconvergence of the estimating algorithm, Lyapunovstability analysis has been used. Conditions for itsrobust stability are also derived herein. To demon-strate the potentia l use of this sliding ob server, rejec-tion of unknown disturbance by using a feedforward-like control is illustrated. Similar use for the model-based predictive control may be investigated in futureresearch. Simulations for the application of chemicalreactor control are given as illustrations for the poten-tial uses of such sliding observers.

2 . M A T H E M A T I C AL F O R M U L A T I O NConsider a general nonlinear system as

k = f(x, u)Y = h(x)

where x E R is a vector of state variables, u E R is theinput vector and y E RP is the output vector.

To construct the nonlinear observers for the systemgiven in eq. (1) it is essential to devise a correctingfunction @ such that, integration of the followingequation would p roduce estimates of x, i.e. 9:

4 = f(%, I) + qy - 9)(2)i = h(f).

3. SLIDING OBSERVER FOR SISO SYSTEMSIn the following text, for convenien ce, we shall use

x, instead of z, to denote the state in eq. (3). Considera SISO n onlinear system of the following form:

e = f(x, u)(6)

Y = Xl

As has been mentioned earlier, many nonlinear ob-servers have been reported in the literature. Althoughnot exhaustive, various methods of approach alongwith their disadvantages for the few reported nonlin-ear observers are summarized in Table 1. To over-come some major obstacles in the construction ofnonlinear observer for process control, a sliding ob-server is presented here.

where x = [x1 x2 ... x,IT is a state vector and u is anexternal input. The function f(x, u) in eq. (6) is subjectto a norm-bounded modeling error.

The sliding observer for this SISO system becomesii =f,(ir, u) + k, sgn(x, - ai)i2 =f2 (%, u) + k2 sgn(x, - a,)

(7 )

We will assume that the Jacobian matrix of h, i.e.J(h(x)), exists and is of full rank for all x E X, so that eq.(1) can now be transformed into the following system:

& =f,(%, u) + k,sgn(x, - x*i).Let Pi = xi - Ti, i = 1,2, .. , n; then the above equa-tion becomes

i = f*(z, u)(3 )y = cz

i?, = Afi + Sfi - kl sgn(Zi);, = Af2 + S f 2 k2 sgn(Z,)

where C = [I, 0] and X is an open subset of R. ;, = Afn + 6 f n k,sgn(x,)

Using eq. (3), z is partitioned into

so that an observer of the following form is con-structed:

i = P(&, u) + K(t)a (5 )where K is a time-varying gain matrix and u is givenas follows:

U=swh - 9wb2 - 9

i fb, - &I1nd the sign function , sgn(t), is defined as

i1 ift>O

sgn(t) = - 1 if t < 0.Further, let Z = z - P lie on sliding surfaces by

applying sliding conditions. The switching gainmatrix, K, is formulated in order to keep the dynamicpoles of Z* = z* - 4* invariant at desired constantvalues, which would lead to a good performance.Convergence of the estimating algorithm is for-mulated by using Lyapunov stability theorems andthe robustness to modeling errors are derived viaanalysis of norms. Incorporation of such a slidingobserver into a closed loop for process control isaddressed in the following.

(8 )


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A sliding observer for nonlinear process control 78 9where Afi =fi(x, U) -f;(%, u) and Sfi is the modelingerror due to structural deviation.

We can define a sliding function in terms of Z1 ass = 11

and a sliding condition for J, as either(9 )

or$s = -qsgn(s) (11)

whe re q is a positive constant. In order to drive Z1 toits sliding surfac e, dZ,/dt should have a sign oppo siteto that o f fl. To do so, we can assign th e value to kl as

kl >q+F. (12)Here , we assume that the dynamic uncertainty is ex-plicitly bounded, i.e.

IAfi + hfil G F (13)whe re F is a positive constant. Thus, inequality (12)can mak e the variant trajectories point towar ds thesurface s(t) = 0, where the tracking estimation error.C1 s zero.

Applying the concept of equivalent dynamics inaccorda nce with Filippov (Slotine and Li, 1991 ), theconvex combination of the dynamics on both sides ofthe surface s(t) leads to

i, = ;l(Af, + WI + k,) + (1 - y) (Afi + Sfi - k,).& = r(.G + ~~ + k2) + (1 - Y) (Afi + Sf2 - k,)

(14);, = 1/(4fn + 6f + k,) + (1 - y) (A& + S fn k,).

When k, is determined from eq. (12), we haveS = f, = 0, i.e.

dA f1 + W I + h) + (1 - Y) (Af i + Sfi - k,) = 0.Hence, the value of y is given as

h -Afi -6fiy= 2kl (15)Substituting eq. (15) into eq. (14) gives the reduced-order sliding observer dynamics in the form of

.& = Afz + Sfz - W kd (Afi + Sfd$3 = Af 3 + Sf3 - bdkd (Afl + Sfd

(16)% = Afn + 6fn - W k,) (Afl + Sfi).

By linearizing locally with respe ct to the point at %,eq. (16) can be treated as a state o bserver with the

following total differential form:k, = {.$;g}& + . . . + {?&%&}i

In the following, we first neglect the structural modelingerror ter ms. Linearization is made on the basis of eachpoint of the estimated states, instead of a fixed point ofequilibrium, hence, the deviations considered for lin-earization are Zis, rather than considering how far theyare away from the 6xed point. C onsequently, this wouldbe less restrictive comp ared to the extended linearizationmethod of Baumann and Rugh (1986).

Le tk,

k, = k,[!Xwhere f, E R- andH(k)4

af2 kz afi- ---ax, kt ax,af3 k afl- ---ax2 h ax2

afn km afi- _--ax, k, ax,

E R(-1)X(-1)

*2_1 -1H(B) .X (18)7af2 kz afl. ----

ax, h ax,af3 ks afl. . . --- -ax, h ax,

afn kn afl----ax, h ax, 1

(19)To keep the eigenvalues of the observer invariant, theswitching gains, k,A [kz k3 ... kJT, can be directlycalculated by a specific formula depicted below. Fromeq. (19), it becomes

H(k)- af2 af2

ax, . dx,. :

ai i-ax, . ax, I

p v,f, - BV,,flJAA, - /?c,. (20)


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It is clear that by using th e sliding condition for the It has been shown that the switching gains,state corresponding to the measured output, state ki, 2 < i < n, are calculated according to eq. (22),xi is forced to lie on a sliding surface. A s a result, the hence enabling matrix H(% ) to have specific eigen-reduced -order observer has invariant dynam ic poles values in the LHP. There fore, the stability of theand is guarantee d to converge a symptotically. On the reduced -order observer for nominal case, i.e. 6f = 0 ,other hand, it should be noted that this approach can be guaranteed.would inevitably introduce tracking error. Hence, The issue of robustness of the observer concernsa trade-off between tracking accuracy and control

whether or not x^,diverges in the presence of modelingpower has to be achieved by suitably choosing the errors. From eqs (17) and (25), it can be defined thatboundary layer thickness 4 (Slotine and Sastry, 1983) . i, = H(f)%, + 6,

4. NOMINAL CONVERG ENCE AND ROBUSTNES S Further, the solution for .C7,an be expressed asSubstituting xi, corresponding to the measu red soutput, onto a defined sliding surface , we further show r7, = eH$(0) + eHctmr)S, r. (26)that even for the remaining states the observer is 0

stable and is robust to modeling errors. If il%,(O)/I< a and there exist b, e and N such thatLet the Lyapunov function I/ be of the form IIa f ll G @ + N e - M l + IIk, II/ k,) (27)V(1) = !{2: + 2; + .. + n;}. (24) then we haveThen we have b N --E*

ripi) = II .il + 5z2.k2 + ... +x,.&. 11 , /I < ae- + - + - eAn 1, - E (28)


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A sliding observer for nonlinear process control 793where -A,,, (i., > 0) represents the greatest eigen-value of H(f), I/?&11s the vector norm of 8, and a, b,N and E are all positive constants. The derivation ofeq. (28) is given in the Appendix. It may hence beconcluded that

(i) iii,/1 would remain bounded, if (/Sf(l Q(h + Ne-)/(l + likJ/kl), b > 0 and 0 < F:< i,.(ii) l/%,(1 0. if h = 0, 116f(l (Ne-Ml + llk,ll!k,)

and 0 < c < i,,.In other words, the estimation errors of the reduced-order observer rem ain bounded if //6f /I is bounded asgiven in eq. (27).5. S L I D I N G O B S E R V E R S F O R M U L T I - O U T P U T S Y S T E M S

For multi-output systems, the construction of slid-ing observer can be derived directly from the SISOsystem by approximately the same way as describedabove. Nevertheless, the method of finding switchinggain matrix K would require more sophisticatedmathematical manipulations. For process control,however, we are more interested in formulating theproblem as follows.

Instead of using a full matrix, we assign a blockdiagonal form to K in eq. (5). In other words, we canrenumber the system and divide the system into sev-eral subsystems. Within each subsystem, the stateestimations are corrected by a sign function based onthe same ou tput. The key issue would be the assign-ment of each pivot switching gain that would keep theoutput on its sliding surface, and the reduced-ordersystem to have invariant dynamic modes. We willillustrate these procedures by using a system whichhas two outputs. First, according to the system

k = f(x, u)y = cx

we can renumber the states such that

Partition the vector x into x 0 and xb, so that? = [a, b* z&Ji* = [,?, &+I .


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19 4 Gow-Bin Wang et al.

Fig. 1. The serial CSTR system.

In other words, the computations for k: and kg cannow be decoupled. Each of the k: and k f can becalculated individually by using eq. (22) for each of thesubsystem s. In the following, we will illustrate thisproposed sliding observer with an example of estima-tion of a chemical reaction system.

Example (State estimationfor a serial chemical re-action system): Figure 1 show s the serial continuous-stirred tank reactor (CSTR) system, belonging to themulti-output case and studied by Henson and Sebor g(1990). The dynamic behavior of this system can begoverned by the following equations:Cl = +(C, - Cl) - koCl ex p

1 ( >- g

1

T~=+(T,- T1)+ (- ~WOCl exp E( >

--1 PC, RT

+ O. O l u [ l -exp(-%)1(7,1-x4

it 3 = x1 - x3 - kOx,exp

It is apparen t that this reacting system has twomeasurab le outputs, i.e. it can be split into tw o sub-systems. Therefore, to estimate the concentrationsin the reactor, one can design the sliding observer asfollows:

+ k1 sgn(Z,/$)

C?,= $(Cl - C,) - koCz ex p (37)

The nominal values of the param eters of this serialCST R system are given in Table 2. The state variablesx, system outpu t y and manipulated input u are de-fined as follows:

x&CC1 T1 C2 T21T, yp[T1 T2 T ) upq,.Hence, the dynamic model of the serial CSTR indimensionless form gives

i1 = 1 -x1 -koxlexp

4, = T, - XI2 + k(-AH),exp ( >- -$+o.olu[l -:+Jii:- a,)+ k2sgnGdd4

+ k3 sgn(W4)(39)

where &g x4 - &. The switching gains kz and k4 ar edetermined according to eq. (12). The other tw oswitching gains can be directly calculated accordingto eq. (22).


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796 Gow-Bin Wang et al.

:lik_-_::r

0 1 2 3 4 5 6 7 8 9 10Time (min)

~~~~

t@-y+yy

0 1 2 3 4 5 6 7 8 9 10Time (min)Fig. 2. Estimation results of the serial CSTR system under sliding observer w ith observer poles

Pm = Pm = - 5.

As the vector u does not explicitly appear in the above governed by the following equations:formulation, to construct an observer fo r closed-loopsystem, it would be required to examine the observ- % = f(x, u)ability condition over a given domain of x only. P = f(2, u) + cD(y j ) (47)

State afine systems: For some chemical reaction y = cxsystems (Gibon-Fargeot et a l . , 1994), the dynamic where @ designates a correcting function which isequations can be described by the following equa- used in the observer. If we define the difference be-tions: tween the true states and the estimated states asg = W(r)* YW + b(e), Y(t)) (46) 2=x-fy = cx. then the above governing equations becom e

The observability, for this kind o f systems, would ir = f(x, u)depend on whether or not the (A, c) pair is observable. (48)

The other criterion to be considered while using the P = f(x, u) - f($ u) - (D& i).observer for nonlinear proc ess control is whe ther in- In linear systems, it is easy to show that 12 nd x havecorporation of the proposed sliding observer into independent dynamic modes, and the dynamic modesa control system would cause only stability problem . of the overall system would be a combination of bothTo answer this question, we shall consider a general sets. Thus, incorporating a stable observer intoobserver of eq. (2). The closed-loop system will then be a stable state feedback control system would result in


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A sliding observer for nonlinear process control 191

Fig. 3. Estimation results of the serial CSTR system under sliding observer with observer polesP O1 = P O2 = - 2 .

the formulation of a stable closed-loop system. In Moreover, within the neighborhood of some equilib-general, it is difficult to reach such a conclusion for rium point x0, we can further derive thatnonlinear systems. However, it is essential to knowthat, if the estimated states converge asymptotically totheir true values with bounded transient errors, then i=f(x.u(xo))+(,f+~~)~~=~(x-x, lwhether a stable state feedback system remains stableafter incorporating the proposed sliding observer. + W(xo,2)(x - x0) + o(i? (51)

According to the result given in the previous sec-tion, eq. (48) for the system that uses the proposed ^sliding observer now becomes =Z V,f,, .\,] +EeSui?x \_1 + W(x,, 2) > (x -- xg)

(49) where(50) W(xo> ~,~V, (Q(xF) = Vxq, Iyz x0% + V,q , Ix = x, .G

It can be obviously seen from eq. (50) that g7, isindependent of x. On the other hand, eq. (49) can be + + vxqI.=.,~linearized with respect to 2 at the point of i = 0, i.e.P = x, as and

i = f(x, u (x - 2)) = f(x, u (x)) + E ; _= 1 + o(2).x cl Q(x)+; _~ = cw) 9b4 . . . m )i . (52)I / X-0


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798 Gow-Bin Wang et al.Le t

A(x,,1)~V,fJ.=., +:g _ + W(x,, 3x - 7.(53)

p 4(x,) + W(x0, 5).where

Al(xa, 0) = ,4(x,).The local stability of this system can be guaranteed

by studying the eigenvalues of the following linearizedmatrix at the equilibrium point x0:

A(x,, %) = ,4(x,) + W(xo, 2).It is known that, with out loss of generality, one can setx0 as 0. Thus, we have

A(0,2) = A(0) + W(0, 2).Notice that the ideal system, whose all states areaccessible, is stable such that all the eigenvalues ofA(0) lie in the LHP. The effect of the observer on itsstability is limited to the extra term, W(O ,r?). Thestability o f this observer- based closed-loop systemshould be discussed in light of the following theorem .

Theorem (P erturbation theorem for t he eigenvalue,Stewart and Sun, 1990): Let 3, be a simple eigenvalue ofthe matrix A, with right and left eigenvectors v and w,and let A = A + E be a perturbation of A. Then thereexists a unique eigenvalue 1 of .& such that

X=n+ oHEvgHy + o(llEl12)

Based on the above theorem , we haveX[A] = )*[A] -z + 4l Wl12). (55)

We can therefore conclude that ifoHWvmax 7 < I~il Viil IV

the incorporation of this sliding observer would notaffect its nominal stability which is derived from theideal system. As can be seen from eq. (51), as f ap-proaches zero as time passes, the observer based sys-tem will convert to an ideal system.

Although the stability of such an observer-b asedsystem cannot be said to be totally independent of theobserver, th e stability condition can be achieved m oreeasily since this sliding observer provides the guaran-teed convergence as in the previous analysis. A s a re-sult, if the control using full state fee dback is globallystable w ithin a defined domain of x, it should bepossible to incorpora te a carefully designed observ erwhich will not cause a stability problem . Conse-quently, design of the state feedback c ontrol and theobserver can be impleme nted separately. In other

word s, one can design a stable control system byassuming that all state variables are fed back. A slid-ing observer as depicted above can then be construc-ted. Following this guideline, we illustrate in the fol-lowing sub sections the use of such a sliding observe rfor controlling chemical proc esses.6.1. Setpoint tracking of chemical reactor control

It is known that all control systems are nonlinear toa certain extent. H ence, the developm ent and applica-tion of nonlinear control algorithm s are attractinggreat attention in the recent years. Here, we utilize theGLC structure to deal with the servo control problemof an isothermal CSTR. The GLC approach was firstproposed by Kravaris and Chung (1987) or obtainingthe linear relationship between the transform ed inputsand proce ss outputs for nonlinear systems (Bequ ette,1991). Therefore, the linear control theory, which iswell-develop ed for the linear systems, can be utilizedto com plete th e controller design.For a minimum phase nonlinear system with rela-tive order r, the state feedback control law

u= v - Ljh(x)L,Lj- h(x)can directly transform the original nonlinear systeminto

y = a. (58)Furtherm ore, the new manipulated input v is set to be

r = - #gJ- 1) _ tj_ry-2- - 02Y - Qlb - Ys,). (59)

The closed-loop transfer function (CLTF ) of the sys-tem can then be obtained by combining eqs (58) and(59) as

Y(S) Ql-=ys,(s) s + t&s- + + e,s+ e1 (60)It may be mentioned here that m ost of the nonlinearcontrol tech niques utilize state feedba ck compensa torlaws. Hence, the employment of state observersbecom es n ecessary. In the following, we solve theestimation and control prob lems of the chemical rea c-tion system by applying the sliding observer de-veloped above.

Let us consider a well-mixed CSTR with isothermalreaction as

A=B+C.By denoting the concentrations of species A, B andC as xi, x2 and x3 , respectively, material balances forthis CSTR are described by the following dimension-less equations:

iii = 1 - xi - &X i + D,4& = Dlxl - x2 - D2xf - D 3x: + ui3 = D3x: - x3

y = h(x) = x3.(61)


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A sliding observer for nonlinear process control 19 9Since the system has relative order r = 2, according to in form ofeq s (58) and (59), we have P, = 1 - x^l - DlzZl + D& + k l sat&/+)

j;=v= - &J; - Hl(Y - Y J ( 6 2 ) $ 2 = DI P 1 - 2 2 - D& - D& + u + k2 sat(Z,/4)Then the correspond ing nonlinear control law de-rived from eq. (57) gives 23 = D3x*$ - g3 + k3 sat&/$) (65)

U= - 2D,~,(D,P, - i2 - D$;) + (D$; - y) - fQD $; - y) - Bl(y - y,,)2D,.u*, (63)

where f31 and e2 are tuning parameters of the GLC where & = x 3 - 2, and the switching gains are deter-appro ach. The overall closed-loop transfer function is mined from eqs (12) and (22).thus obtained from eq. (60) as During the simulations, we choose the processY(S) 01 parameters as D, = 3, D2 = 0.5 and D 3 = 1 and set-=Y,&) sz + 02s + 81 (64) the initial co nditions of the true states x(O) and theestimated states $0) as CO.3560.921 0.8481T andFurthermore, the estimated values of the unmeasur- [0.5 1.0 0.8481T, respectively. Two tuning parametersable states, $I and & , in the nonlinear control law, i.e. of the GLC approach are placed at @I = 4 and e2 = 4 .

eq. (63), are provided by the proposed sliding observer In this study, the control objective is to mak e the

0.10.05

051 -0.05

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I I I I I I I I I..-I_,

... po1 = po2 = -4.0 -C....A JJO] = po2 = -1.5 .. ---_

I I I I I 1 I I I

0 1 2 3 4 5 6 7 8 9 10Dimensionless Time

-fJ1 = po2 = -4.0 -po1 = po2 = -1.5 -

x2 -0.1 --0.12 I .:-0.14 2, :-0.16 - .,,_ -0.18 I I I I I I I I

0 1 2 3 4 5 6 7 8 9 10Dimensionless Timev V ._ ._ 1 I I_-.._-__. po1 = po2 = -4.0 -po1 = po2 = -1.5 ... -._

zc z-0.0015 ; ;

-0.00 2 -: ,--- .-_-0.0025 - I I I I I I I I

0 1 2 3 4 5 6 7 8 9 10Dimensionless TimeFig. 4. Results of estimation errors for the isothermal CST R system und er sliding observer w ith observer

poles pal = poz = - 4 and - 1.5.


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800 Gow-Bin Wang et al.dimensionless concentration x3 track its setpointy,, = 0.75 as soon as possible. Let the switching gaink3 = 1.5 and the boundary layer thickness C$= 0.01.From the results of estimation errors show n in Fig. 4,it is clear that the unmeasurable concentrations canbe effectively e stimated by the proposed sliding ob-server. The servo control responses shown in Fig. 5demonstrate the tracking performance of this slidingobserver. Furthermore, it may be noted that the as-signed poles of the reduced-order sliding observer doinfluence the convergence rate for state estimation.

Moreover, when the modeling errors occur due toparametric uncertainty, the proposed sliding observercan still perform well by setting this parameter as anunknown state variable.6.2. Dist urbance rejectionfor chemical reactor control

The presence of unknown disturbance in the pro-cess would prevent fi from converging to zero. Asa result, the estimated states deviate from their truevalues which, in turn, would degrade the control per-formance. One way to reject such unknown distur-bance is to include it in the observer as an extra statevariable. For constant disturbance, this extra stateserves as an integrator, i.e. for SISO systems, we have

states, xZr . . ..x.+,, including the unkno wn distur-bance.

The matrix that corresponds to H(%) in eq. (19)would now become~*(%,.-&+I) =

V&f, Bcvx,fll V++,f, Bcvx+,fi1- * CVJI1 - IcIcvx+,fil 1 . (67)

The existence of parameter sets of p and $ of sucha system w ould depend on the observability pair(A*, c*) of the following nature:

A* =[

Vx,f, VX r0 xcn- 1) 01X1 1 (68)

c*= CVx,h vr+,.fil.If the pair (VJ l, V,,f,) is observable, a necessary andsufficient cond ition for this augmented system to beobservable is

k =f(x ,x ,+1 = 4 4 , ~ , + , = O , .v=x1 (66) Equation (69) is a result of applying the observabilitywhere d is an unknown disturbance. A sliding ob- condition (Morari and Stephanopoulos, 1980 ) toserver can be constructed to estimate the unmeasured the (A*, c*) pair. For multi-output systems, the

0.9

0.85XR

0.8

:- I I I I I I I. ... .,, po1 = po2 = -4.0 - -po1 = po2 = -1.5

. . ._..._.._

0 1 2 3 4 5 6 7 8 9 10Dimensionless TimeI I I I 1 I I I I

1.4 _ ,?...,, po1 = po2 = -4.0 -po1 =po2 = -1.5 1.2 - : ....(

u

I I I I I I I I I I I0 1 2 3 4 5 6 7 8 9 10

Dimensionless TimeFig. 5. Servo control responses of the isothermal CSTR system under sliding observer with observer poles

pOl = p02 = -4 and - 1.5.


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A sliding observer for nonlinear proce ss control 80 1number of unknown disturbances to be included in anobserver is at the most equal to the number of systemoutputs.

In the following ex ample , w e consider a well-mixedCSTR with first-order, irreversible, exo thermic reac-tion. Material and energy balances fo r such exothe r-mic CSTR are described by

dC q- = v (C, - C) - k&expdt

+~(Tc-nwhere C and T represent the reactant exit concentrationand reactor temperature, respectively. Let the four im-

portant dimensionless parameters be denoted as

D = koVemDbP q D 7 = (- AfW,oD,PC,Tf

(71)

The correspond ing dimensionless variables are de-fined as

Cx1 =-) T - T,CfO x2=- 4,T, dl cc,, +CfOTc - T, (72)u = ~ D4D5,Tf y = c, J,,, = T

where d, represents the unknown disturbance andCfo is the nominal value of the feed composition.After introducing the dimensionless quantities given

0.025 , I0.02

0.0150.01

0.0050

po1 = po2 = -4.0 -po1 = po2 = -1.5 -.open-loop observer -

k........ --..I__ I I I I I0 1 2 3 4 5 6 7 8 9 10Dimensionless Time0.03 , I I I 1 I I I r 1 Ipal = pox = -4.0 -PO] =po2 = -1.5 ..

open-loop observer -

-0.03 1 I I I I I I I I I I0 1 2 3 4 5 6 7 8 9 10

Dimensionless TimeI I I I I 1 I I I

po1 = po2 = -4.0 -po1 = po2 = -1.5 open-loop observer -

...X.L -k.I -$--.__J I I I I I0 1 2 3 4 5 6 7 8 9 10

Dimensionless TimeFig. 6. Results of estimation errors for the exothermic CSTR system under sliding observer with observer

poles pal = po2 = - 4 and - 1.5 and under the open-loop observer.


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80 2 Gow-Bin Wang et al.

P C ? 1 = PO2 = -4.u -PO ] =po2= -1.5 -open-loop observer -

0 1 2 3 4 5 6 7 8 9 10Dimensionless Time-0.93 I I I I I I I I I

po1 = po2 = -4.0 -po1 = PO.2 = -1.5 -open-loop observer -

-1.03 L I I I I I I I I I0 1 2 3 4 5 6 7 8 9 10Dimensionless TimeFig. 7. Results of disturbance rejection of the exothermic CST R system und er sliding observer with

observer poles po, = poz = - 4 and - 1.5 and under the open-loop observer.

above, the resulting normalized model is governed as where y, is the system m easurement andf * ( x ) = x 3 - x 1 - D, s ~ w( , + z 2 , D, )

Y = Xl> y m = x 2 .In this case, the unknown disturbance dl is regardedas another state variable. According to eq. (73), wehave

1, =x3--, -D6s,exp(l~,D,)efl(x)

We may now apply the GLC structure to solve theconcentration control problem of the exothermicCSTR. The relative order r of this CSTR is equal to 2;thus, let

ji=v= - @2 J ; - B,(Y - Ysp). (76)Then the corresponding nonlinear control law can bedirectly derived from eqs (74) and (76):

u= (3, 1 - D6evCy,lU + ~,lD,)l~f,(~) - {D64U + ym/D4 F2wCy,JU + ~,dJD,)l)f~(x*) @ A4 Y,,)D,W + y,/D,)~ZewCy,lU~,lD,)l (77)i2=D6D,x,exp(l+~2,D,)-(l+D,)xl+uhere O1 and B2 are tuning parameters of the GLCapproach. The overall closed-loop transfer function

(74) can be obtained directly from eq. (76) as&f,(x) + 9 2 w Y(S) 4-zzij = 0, (78)y = Xl, Y, = x 2 . Y&l s2 + e2s 4 81


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A sliding observer for nonlinear proce ss control 80 3Furthermore, the estimated values of the unmeasur- no requirement of canonical transform ation, achieve-able states, i1 and &, in the nonlinear control law, ment of desired perfo rman ce by allocating the ob-given by eq. (77), are provided by the prop osed sliding server poles, knowledge of convergence and robust-observer in form of ness of the estimation to the designer, etc. Potential

(1 +$o,)uses of this sliding observer towa rds servo-tracking

4, = ?3 - ?I - DbPl ex p + kl sat &/41 and disturbance rejection for proce ss control are dis-cussed. Estimation of unmeasurable states and con-trol of chemica l re actor are illustrated.&=D,W,exp(l +tl,D,)-(l +D5)i2 Acknowledgement

(79) Financial suppor t from the National Science Council of+ u + k2 sat&/$) the Republic of China(NSC-84-2214-E002-036)s gratefullyacknowledged.

j3 = + k3sat(ZZ/4).where Z2 = x2 - z?~ nd the switching gains are deter-mined from eqs (12) and (22).

For comparisons, an open-loop observer is used toestimate the unmeasurable states by directly integrat-ing the following differential equations:

For the exothermic CSTR, the four process para-meters in eq. (71) are chosen as D4 = 5, D5 = 0.5,D6 = 1 and D7 = 2. Initial conditions for the truestates x(0) and the estimated states a(O) are set asCO.50 1.051T and CO.50 l.OIT, respectively. It shouldbe noted that, in this exam ple, th e control objective isto mak e the dimensionless concentration x1 remainon its setpoint ysp = 0.5 in the face of the unknowndisturbance. By applying the GLC approach, the twotuning parameters are set as o1 = 4 and e2 = 4. Fur-ther, the switching gain k2 and the boundary layerthickness 4 are set to be 1.5 and 0.0 1, respectively.Figure 6 reveals the existence of offset for estimationof the states for load change since the open-loopobserver cannot provide correct estimated states. Onthe other hand , the response results show n in Fig. 7demonstrate good robustness features of the reduced-order sliding ob server in face of existence of an un-known disturbance.

7. CONCLUDING REMARKSA sliding observ er, w hich beha ves like a reduced-

order observer, is presented. This observer has beenshown to overcome some major difficulties involvedin constructing nonlinear observer s for state estima-tions, especially for nonlinear proc ess control. Toachieve this, the switching gains in the observer aremade to be time-varying. Convergence of the estima-tion is analyzed by using Lyapunov stability the-orem s. Robustness conditions, which would guaran-tee the observer to have a bounded error norm whenfacing modeling error, ar e also derived. Th e advant-ages of this prop osed sliding observer include: simpleand less restrictive design and construction, no needof extensive comp utations during its implementation,

RRRR,

sa tsgnTuuVVW

a, bAI, A2r -41A,A(x,, 3CPCl.Cc,dDI-D7Ef, gFA fhH(k)AHJkoKLMNPO19 O2Pl

NOTATIONpositive constantsheat transfer areamatrix in eq. (20) or eq. (32)matrix in eq. (53)heat capacityvector in eq. (20)concentrationmatrix in eq. (32)disturbanceprocess parameters of the reaction sys-temactivation energyvectors of nonlinear functionspositive constantdefined as f(x, u) - f(f, u)output functionsmatrix in eq. (19)heat of reactionJacobian matrixspecific reaction rate constanttime-varying gain matrixLie operatoroperator defined in eq. (42)positive constantpoles of the reduce d-order sliding ob-serverinverse of the transpose of the observabil-ity matrix of [A,, c,] pairfeed flow ratematrix in eq. (52)relative orderideal gas constantreal scalar fieldn-dimensional real vector fieldlower triangular Toeplitz matrix withfirst column [l a2. an_2 a,_,]sliding surfacesaturation functionsign functiontimetemperatureinput vectoroverall heat transfer coefficienttransformed control variablereactor volumeLyapunov function candidate


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80 4 Gow-Bin Wang et al.W(x,, 2) defined as V,(Q(x) %) Kailath, T. (1980 ) Linear Systems. Prentice-Hall,X state vector Englewood Cliffs, NJ, U.S.A.Y output vector Kantor, J. C. (1989 ) A finite dimensional nonlinearYm measured output observer for an exothermic stirred-tank reactor.2 state vector Chem. Engng sci. 44, 1503-1510.Keller, H. (1987)Non-linear observer design by trans-Greek letters

formation into a generalized observer canonicalform. Int. J. Control 46, 1915-1930.a 4 [XI c!s cr]

B vector defined in eq. (20) Kravaris, C. and Chung, C. B. (1987 ) Nonlinear statefeedback synthesis by global input/output lineariz-Y coefficient in eq. (15 ) ation. A.1.Ch.E. J. 33, 592-6 03.b f modeling error due to structural devi- Lee, P. L. and Sullivan, G. R. (1988 ) Generication model control (GMC). Comput. Chem. Engng 12 ,& vector defined in eq. (25) 573-580.1 positive constant Lewis, F. L. (1992 ) Applied Optimal Control and Es-;

positive constant timation. Prentice-Hall, Englewood Cliffs, NJ,tuning parameters of the GLC approach U.S.A.eigenvalues Meditch, J. S. and H ostetter, G. H. (1974 ) ObserversA for systems with unknown and inaccessible inputs.

V vector Int. J. Control 19, 473-480.P density Misawa, E. A. (1988 ) Nonlinear state estimation usingd vector defined in eq. (5) sliding observers. P h.D. thesis, Massa chusetts Insti-@ correcting function in eqs (2) and (47) tute of Technology, Cambridge, U.S.A.;

boundary layer thickness Misawa, E. A. and Hedrick, J. K. (1989 ) Nonlinearpositive constant observers - a state-of-the-art survey. ASME J.

w vector Dyn. System Measurement Control 111, 344-352.Morari, M. and Stephanopoulos, G. (1980 ) Part II:Subscripts structural aspects a nd the synthesis of alter-r reduced-order system native feasible control schemes. A.I.Ch.E. J. 26 ,232-246.Superscripts* estimated value

deviation value

REFERENCESAlvarez-Ramirez, J. (1995 ) Observers for a class of

continuous tank reactors via temperature measure-ment. Chem. Engng Sci. 50, 1393-1399.

Baumann, W. T. and Rugh, W. J. (1986) Feedbackcontrol of nonlinear systems by extended lineariz-ation. IEE E Trans. Automat. Control AC-31,40-46.Bequette, B. W. (1991 ) Nonlinear control of chemicalprocesses: a review. Ind. Engng Chem. Res. 30 ,1391-1413.Bestle, D. and Zeitz, M. (198 3) Canonical form ob-server design for non-linear time-variable systems.Int. J. Control 38, 419-431.Canudas de Wit, C. and S lotine, J.-J. E. (1991 ) Slidingobserver for robot m anipulators. Automatica 27,859-864.

Chen, C. T. (1984) Linear S ystem Theory and Design.

Slotine, J.-J. E., Hedrick, J. K. and Misawa, E. A.(1987 ) On sliding observers for nonlinear systems.ASM E J. Dyn. System Measurement Control 109 ,245-252.Slotine, J.-J. E. and Li, W. (199 1) Applied Nonlinear

Control. Prentice-Hall, Inc., Englewood Cliffs, NJ,U.S.A.Slotine, J.-J. E. and Sastry, S. S. (1983 ) Tracking con-

trol of nonlinear systems using sliding surfaces withapplication to robot manipulators. Int. J. Control38,465-492.Stewart, G. W. and Sun, J. G. (1993 ) Matrix Perturba-tion Theory. Academic Press, San Diego, U.S.A.Vidyasagar, M. (199 3) Nonlinear Systems Analysis.

Prentice-Hall, Englewood Cliffs, NJ, U.S.A.APPENDIX A: DERIVATIONS FOR EQ. (28)

From eq. (17), we havek, = Hi, + 6,. (Al)

Thus, we can solve thati, = e HV, ( 0 ) + eHcrmr), dz. WI

Holt, Rinehart and Winston, New Y&k , U.S.A.Ding, X., Frank, P. M. and G uo, L. (199 0) Nonlinear Le t 11 ,(O)/< a; we haveobserver design via an extended observer canonicalform. Systems Control Lett. 15, 313-322. ll?,l/ < a/leH/l + rIJeH(r-rJIIIS,lIdrGibon-Fargeot, A. M., Hamm ouri, H. and Celle, F. s 0(1994 ) Nonlinear observers for chemical reactors.Chem. Engng Sci. 49, 2287-2300. c ae -&,f + I em& /a,,1& (A3)Henson, M. A. and Seborg, D. E. (I 990) Input-output 0linearization of general nonlinear processes.A.I.Ch.E. J. 36, 1753 -1757 . where - 1, (1, > 0) is the greatest eigenvalue of H(t). From

Hunt. L. R.. Su. R. and Mever. G. (1983) Globaleqs (20), (22) and (25), it is known that

,, \ Itransformations of nonlinear systems. IEEE Trans.Automat. Control AC-28, 24-31. (A4)


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A sliding observer for nonlinear process control 80 5Then it can be derived that

116,II IlSfJ +F Ilkrll < 116f/JTherefore, if there exist h: N and E such that

=ae-m+JL(l -e-my+%i. (A5) [, _ e-m-l]IIWI G (b + Ne-VI + lIk,ll/%)

where h, N and E are all positive constants and E < i,,, weshall obtainh N< ue- I + _ +_e-i. --E (A@4n m

Hence, it is proved that /Ii, 11s bounded.I(%,I1< ae-m + e-m s e- [h + Ne-&*I dr0

A Sliding Observer for Nonlinear Process

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