Observers for bilinear systems with unknown inputs and application to superheater temperature control

Pergamon

PII:S0967-0661 (97)00029-4

Control Eng. Practice, Vol. 5, No. 4, pp. 493-506, 1997 Copyright @ 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0967-0661/97 $17.00 + 0.00

OBSERVERS FOR BILINEAR SYSTEMS WITH UNKNOWN INPUTS AND APPLICATION TO SUPERHEATER TEMPERATURE

CONTROL

Sang Hyuk Lee, Jaesop Kong and Jin H. Seo

School of Electrical Engineering, Seoul National University, Seoul, Korea

(Received April 1995; in final forra January 1997)

Abstract: This paper considers the problem of controlling the steam temperature of a superheater with a desuperheater. Since the metal temperature is usually not available for measurement, it is regarded here as an unknown input, and a bilinear unknown input observer is designed. A sufficient condition for the asymptotic stability of the proposed observer is derived, and the design of a state feedback controller is based on the proposed observer. Computer simulations show that the estimated value follows the superheater steam temperature under the variation of the external inputs, and that the outlet steam temperature is properly maintained, and a comparision with a PI controller is made. Copyright © I997 Elsevier Science Ltd

Keywords: Bilinear systems, unknown input, heat exchanger, temperature control, state feedback.

1. I N T R O D U C T I O N

This paper considers the problem of controlling the

steam temperature of a superheater using a

desuperheater. When the steam temperature is

controlled by state feedback, the relevant state of

the system is steam temperature. The system state

is largely influenced by the distribution of the

metal temperature. However, the metal temperature

is usually not available for measurement, and hence

is regarded as an unknown input, and an observer

for the system with an unknown input is proposed.

The problem of designing an observer for a

system with unmeasurable inputs has attracted

some attention in the literature. Hostetter and

Meditch (1973) proposed a method which assumes

some a priori knowledge of the disturbance. Kudva

et at (1980) gave necessary conditions for this

kind of observer to exist. In Bhattacharyya (1978),

a geometric approach has been proposed. Also,

Hara and Furuta (1976) and Funahashi (1979)

proposed methods which construct stable

minimal-order observers for bilinear systems.

The following section extends the theory of

observers for systems with unknown inputs

(Bhattacharyya, 1978, Kudva et al., 1980, Kurek,

1983, Yang and Wilde, 1988, Hostetter and Meditch,

1973). A sufficient condition for the asymptotic

stability of the proposed bilinear observer is

derived. The design procedure for the bilinear

observer is also discussed. In Section 3, a model of

a superheater is developed. To describe the

distributed parameter nature of the superheater, a

finite difference model is employed by dividing the

superheater into many segments. To satisfy the

condition for the construction of the unknown input

observer, an approximation scheme for the metal

temperature is proposed, considering the distribution

of the metal temperature. In Section 4, a state

feedback controller is constructed by applying the

493

494

optimal control theory. In Section 5, the simulation

studies are carried out and it is shown that the

designed observer is guaranteed to be

asymptotically stable at all operating points within

some bounds. Using computer simulations, it is

also shown that the estimated value follows the

steam temperature under the variations of the inlet

steam temperature, inlet steam mass flow rate and

flue gas temperature and that the outlet steam

temperature is properly maintained. The

performance of the proposed controller is compared

with that of a PI controller. Some conclusions

follow in Section 6.

Sang Hyuk Lee et al.

z(t) = [ Fo + i~E?i( t)FiJz( t) + [ Go + ~]_qj( t) Gj]

• u(t) + [ L 0 + ~]p,.(t)Li]w(t) (5)

Xe( t) = z( t) - Ew( t), (6)

where z ( t ) ~ R ' , x~(t)~R"; E~R'×m; Fo,

Fi~R*×*; Lo, L i a R ..... : Go, G j ~ R *~p.

Define the estimation error e(t) = xe(t) -- x(t).

Then, e(t) satisfies the following equation:

2. D E S I G N OF B I L I N E A R O B S E R V E R S

2.1 Observer for bilinear systems with unknown inputs

Consider a particular kind of bilinear systems with

unknown inputs:

x( t) = [ Ao + i~e ? i ( t) A i] x(t) + [ Bo + ~'].q~( t) B j] jE]

• u( t )+Dv(t ) (1)

w( t) = Cx( t), (2)

where x( t) ~ R* , u( "t) E R p, w( t) ~ R " and

v( t )ER ° are state, input, output and unknown

input, respectively; ~: ={1, "",/max}, )

: = { 1 , " ' , j ~ . x } , ] and ] denote sets; Ao, A,

E R "×', Bo, Bi ~ R ~×~, D ~ R "×°, C

R=×'; Pi(t) and qj(t) are input or time-varying

components lying within certain bounds, that is,

P7 <- pi( t) <- PT, i e I, (3)

q7 <- qj( t) <- e l , J E 2. (4)

Without loss of generality, it can be assumed that

D has full column rank and C has full row rank.

e( t) : xe ( t) - x( t)

: z(t) - Ew( t ) - X(t)

= [F0 + i~/)i(t)Fi]e(t)

+ [ Fo - ( EC + I)Ao

+ i~e?i( t)(Fi-- (EC+ 1)Ai)]x(t)

+ [ Go - (EC+ 1)t?o

+ j~3yqj( t)( G j - (EC+ 1)Bj)]u(t)

+ [ Lo + FoE + i~_3iPi( t)( L, + FiE) ]w( t)

- [ (EC+ I)D]v(t).

Since e(t) is required to converge to zero

irrespective of v(t), E is chosen to satisfy the

relation:

(EC + I ) D = 0. (7)

Let P : = E C + I , L o ' = Lo + F o E and L,

: = L , + F I E . Then, with a choice of E

satisfying (7), e(t) satisfies

e(t) = [Fo + i~/)i(t)Fi]e(t)

+ [ F o - P A o + LoC

+ i~E / ) i ( t ) (F i -PA i+ LiC)]x(t)

+ [ Go - PBo + ~ j ( t ) ( Gj - PBj)]u(t).

For the system (1) and (2), an observer which

reconstructs the state x(t) without the knowledge

of the unknown input v(¢) is constructed using

measurement w(t) and input u(t) as follows:

If matrices F0, F,, Go, G, Lo and L, are

constructed such that

Fo - PAo + (Lo + FoE)C = 0, (8)

Observers for Bilinear Systems with Unknown Inputs

Fi - P A i + (L i + F i E ) C = O, i ~ 1, (9)

Go - P Bo = 0, (10)

G) - P Bj = O, j ~ ], (11)

then for all Pi( t) and qi( t), e( t) satisfies

e(t) = [F0 + ~ i ( t) Fi] e( t). (12)

It follows from (12) that if E, F0, F,, Go, G,

Lo and L, are constructed so that (7), (8), (9),

(10) and (11) are satisfiedand F 0 + i ~ i ( t ) F z is

stable for all Pi(t) satisfying (3), then Xe(t)

converges to x(t) irrespective of v(t).

495

inverse (CD) +. Under this condition, the general

solution to (7) is given by

E = - D ( C D ) + + K(Im - CD(CD)+), (14)

where K ~ R "×m is an arbitrary matrix.

Let R: = I - D( CD) + C. Then, it "can be seen

that, with a particular choice E = - D(CD) + (7)

is satisfied and P becomes R. Assume that

(C, RA 0) is detectable, which implies that there

exists an L0 such that R A 0 - LoC is stable.

With this choice of L0, F0 and L 0 are

constructed as follows:

F0 = RAo - Lo C, (15)

2.2 Construction of bilinear observers

From the results in Section 2.1, a bilinear observer

can be constructed as follows: first, E is chosen

to satisfy (7) and F 0 and L 0 are constructed so

that (8) is satisfied while F0 remains stable. Next,

F,, L,, i ~ I, Go and Gj, j ~ ] , are

constructed so that (9), (10) and (11) are satisfied.

If F0 and Fi, i ~ 1, have been chosen so that

F 0 + i~_al~)i(t) F, is stable for all I)i(t) satisfying

(3), then the observer asymptotically estimates the

states. To guarantee

F0 + / % ~ i ( t ) Fz for chosen

~ i(t) Fi is regarded as a

condition on Pi(t)'s is proposed

stability of F0 + ~ i ( t ) Fi .

the stability of

Fo and Fi, iE 1,

perturbation and a

to ensure the

In the following, it is assumed that

rank( CD) = q, q <- m (13)

which is necessary and sufficient for the existence

of E satisfying (7). Under the above condition,

matrix CD has full column rank and has left

Lo = Lo - FoE. (16)

With this construction of Fo and Lo, (8) is

satisfied and Fo is stable

The above results can be summarized as a theorem.

Theorem 1: Assume that rank(CD) = q <-m.

If (C, RA0) is detectable, then there exist E ,

F0 and L0 such that (7) and (8) are satisfied and

F0 is stable.

Once P is chosen, (9), (10) and (11) can be

satisfied by cons~ucting matrices Fi, Li , Go

and Gi as follows:

Fi = P A i , i E I, (17)

Li = - Fi E, i ~ 1, (18)

Go = P Bo, (19)

Gj = P B j , j ~ ). (20)

The next theorem reformulates the assertion in

Theorem 1 in terms of the invariant zeroes of the

496

system ( C , A0, D) which stands for


obtained:

x( t ) = A o x ( t) + D v

w(t) = c x ( t ) .

T h e o r e m 2: Assume that rank( CD) = q <- m. If

all the invariant zeroes of the system ( C , A o , D)

have negative real parts, there exist E, Fo and

Lo such that (7) and (8) are satisfied and Fo is

stable.

Proof: See Appendix,

Next, W: --- 19(CD) + C is defined. Then,

R W = [ I - D( CD) + C]D( CD) + C = O,

which implies the relations:

R 2 = R ( I - W) = R ,

W 2 = ( I - R ) W = W.

Therefore, R and W are projection operators on

R" such that

I = R + W .

Hence, a direct sum decomposition of R ~ results:

R ~ = I m R ~ ImD.

For later use, the above results are summarized as

a lemma.

Lerrm~ 1: Assume that rank( CD) = m. Then,

the state space has a direct sum decomposition:

R ' = I m R (~ I m W

= I m R @ I m D .

For an important special case where

rank( CD) = q = m, the following theorem is

obtained using the above direct sum decomposition

of R ~.

T h e o r e m 3." Assume that rank( CD) = q = m.

(C, RA 0) is detectable if and only if the stable

subspace of RA0 contains Im R.

Proof: See Appendix

From Theorems 1 and 3, the following corollary

results.

Corollary 1: Assume that rank( CD) = q = m.

If the stable subspace of R.A0 contains ImR,

there exist E, F0 and L 0 such that (7) and (8)

are satisfied and F0 is stable.

R n = I m R @ I m W .

From the definition of W, it can be seen that

Im W c Im D.

Since WD = D(CD)+CD = D, it can be seen that

Im WD ImD.

Suppose that (12) holds. In the following, a~x(M)

and Crmin(M) denote the maximum and minimum

singular values of the matrix M, respectively. In

the next theorem, i ~ i ( t ) f ~ is regarded as a

perturbation term, and a condition is derived on

Pi(t), i ~ ~, which guarantees the stability of

F0 + Y[Pi(t) F,.

Hence, it follows that

Im W = I m D ,

and another direct sum decomposition of R" is

T h e o r e m 4." Let Fo be stable, M = M r be

positive definite, and H = H T be a solution of the

Lyapunov equation:

Then,

satisfying


F~H + HFo + M = O. (21) 3. S U P E R H E A T E R M O D E L L I N G A N D

O B S E R V E R D E S I G N

Fo + . ~ i ( t ) F , is stable for all Pi(t) List of Symbols

°2~ ( M ) Vt . (22) '(t)]2( i~o2,~, ( F'[ H + HF i) '

T., = metal temperature (]2)

T = steam temperature (]2)

7", = inlet steam temperature (]2)

Proof: See Appendix.

The above results for the existence of the observer

are summarized in the following corollaries.

Corollary 2: Assume that rank( CD) = q <- m

and that all the invariant zeroes of the system

( C , A o , D ) have negative real parts. Then there

exists an observer (5) and (6) if (22) is satisfied

for some Fo, F i, L i, i E~l and Go and G,

j E )', satisfying (15)-(20), while F 0 remains

stable.

Corollary 3: Assume that rank( CD) = q = m

and that the stable subspace of R A 0 contains

I m R . Then there exists an observer (5) and (6)

if (22) is satisfied for some F0, F,, L,, i ~ 1

and Go and G , j ~ ~, satisfying (15)-(20), while

F0 remains stable.

The following remark illustrates how to construct

stable F0 using the free parameter K in (14),

Remark 1: If rank(CD) = q( m, the free

parameter K of the general solution to (14) can

be utilized to construct F0 and L 0 such that

(8) is satisfied and F0 is stable. In this case,

detectability of (C, PAo) is required, which is

weaker than detectability of (C, RAo). However,

detectability of (C, PA o) is more difficult to

check than that of (C, RA0). There is no known

systematic method to make (C, PA0) detectable

by choosing a suitable parameter K when

( C, R A o) is not detectable.

497

To = outlet steam temperature (]2)

Td = spray water temperature (]2)

Hi = inlet steam enthalpy (kcal]/eg)

Ho = outlet steam enthalpy (kcal]kg)

wi = inlet steam mass flow rate (kg[s)

wo = outlet steam mass flow rate (kg]s)

wa = spray water mass rate (kg] s)

Q ~ = heat input rate from flue gas

( kca# s)

Q.~ = heat input rate from metal

( kcal/ s)

V; = steam volume ( m 3)

Vs = volume of each segment ( m a)

p = steam density (kg/mZ).

Ca = superheated steam heat

( kcal/ kg ]2 )

to metal

to steam

capacitance

( kcai/ kg ]2 )

capacitance

Ct~ = spray water heat capacitance

Cm = superheater tube heat

( kcal] kg ]2 )

a.~ = heat Ixansfer rate from metal to steam

( kcal/ m2 s ]2 )

a ~ = heat transfer rate from gas to metal

(kcal]m 2 s]2 )

M,, = mass of superheater tube (kg)

Sl = external heating surface from gas to metal

(m s)

$2 = internal heating surface from metal to

steam (m s)

In the operation of a power plant superheater,

exacting demands are made on the steam

teml~erature maintenance at the outlet. For

498

temperature control at the outlet of a superheater,

the relevant system state is the temperature

pattern along the superheater tube. This is

described by a distributed-parameter system, which

involves an infinite number of state variables. To

derive a simplified model for control purposes, the

superheater is divided into segments, and a lumped

model is derived, which represents a finite number

of intermediate temperatures. The results are

shown in Subsection 3.1.

In the interest of simplicity in practical

implementation, the observer is constructed based

on the lumped model with fewer segments than the

superheater model described above. It is also

illustrated how to approximate the unknown inputs

to satisfy the conditions for the construction of the

observer proposed in Subsection 3.2.

3.1 Superheater modelling

To describe the distributed parameter nature of

the superheater accurately, the superheater is

divided into many segments. Each segment is

taken as a control volume to be approximated as a

simplified single capacitance. Using the control

volume approach, a lumped model for each segment

is derived as shown in Fig. 1. The steam and the

flue gas are separated by a metal tube, which

forms a heat-exchange surface.


The mass conservation law is also applied to the

steam flow to obtain

VK-•dt = w i - wo. (25)

Recall that if the velocity of a compressible flow is

sufficiently slower than the speed of sound, the

flow may be approximated as an incompressible

one (McCormack and Crane, 1973, Daugherty et al.,

1985), and consequently d p / d t = O. Since the

velocity of the steam in the superheater is

considerably slower than the sound in the power

plant boiler, the approximation can be adopted, and

as a consequence, the density of the steam can be

excluded from the states. Then, (25) is reduced to

wi = wo. Assuming that the pressure inside the

tube is constant, the enthalpy of the steam

satisfies the relation d H = CpdT, where Cp is

the constant-pressure specific heat. Assuming

convection is the exclusive heat txansfer mode for

the superheater, the heat transfer Q,~ and Qg,~

are expressed in terms of the heat transfer rates

a ~ and a,~ and heating surface S:

Q,,~ = a,,~S2(Tm- 7), (26)

Qg,,, = a ~ S l ( T , - T.,). (27)

metal Tm

Hi, Ti, w i Ho, To, Wo

flue gas direction =~ ~ Qgm

Fig. 1. Control volume of superheater.

It is also assumed that the heat transfer rates a~m

and a~s are constants.

There exist several possibilities for flow

arrangement in heat exchangers. The principal ones

can be summarized as:

Parallel flow: The hot and cold fluids enter at the

same end of the heat exchanger, flow in the same

direction, and leave together at the other end.

Assuming that the system is lumped, the energy

conservation law is applied to the control volume

to obtain

Counter flow: The hot and cold fluids enter at the

opposite ends of the heat exchanger and flow in

opposite directions.

( VfpHo) = w iHi - - woHo+ Q,~, (23) Cross flow: In the cross-flow arrangement, the

hot and cold fluids flow in directions depending on

the design.

-ff•(MmCmTr,) = Q, ,~-Q~s. (24) The temperature profiles for the parallel-flow and

Observers for Bilinear Systems with Unknown Inputs 499

counter-flow arrangements are shown in Fig. 2. + amsS2(z I --Xl)"4-Ct, Ziwi+ Ct, dTdw d (30)

Temperature Tern

0 L

Fig. 2. (a) Parallel flow.

~rature

metal

0 L (b) Counter flow.

d z l = a , ,~Sl ( T z - z l ) - a ~ S 2 ( z l - x l ) M ~ C . d t

(31)

In the kth segment, k = 2,-- ' , n, (28) and (29)

yield the state equations:

VsloCp ~-~ k : Cp( wi'J- Wd)(X k_ 1 -- Xk)

+ a~,~S2(zk-- x~), (32) Using (26) and (27), (23) and (24) become

dT VhoCp----d- i- ~- CpwiTi - CpwoZo

+ a~S2( T i n - 73, (28)

d'zk -- ct~Sl( Tgk- zk) -- a~S2(zk-- xk). M , C , dt

(33) (Lee et. aL, 1994).

dTm MmCm dt - as~Sx( T g - Tm)

- a,~S2( T i n - 73. (29)

Now, the superheater is divided into n segments

as shown in Fig. 3. (28) and (29) are applied to

each segment to set up the state equations with

X = [X l . X2, "",Xn]T: = [T1, 7"2, "", T. ] z,

z = [ z l , z2, " " , z~] r : = [Tin1, T,,~, "", T , j 7

Wd, Ta, Ct~

desul~rheater

L----it. ..,4~ ~

T1 T2 7"3 ........ T,,-x T~ wi, 7", Wo, To

T~l T,~ T~3 ........ Tmn-I T ~

T T T T T Tgl Ts2 Ts~ T ~ - i Ts~

32 Observer design

To control the outlet steam temperature of a

superheater via a state feedback controller, it is

required that the steam and metal temperature

along the superheater tube be available. In a

superheater, the heat input from the external flue

gas is usually not available for measurement.

Hence, the consmlction of an unknown input

observer should be considered. However, in a

power plant superheater, the measurements for

control purposes are quite restricted, and only inlet

and outlet steam temperatures are usually available

for measurement. If the measurement of metal

temperature is available, it is possible to construct

an observer to estimate the steam temperature

without knowledge of the external heat input.

However, with the restfictegl measurement of the

inlet and outlet steam temperatures, it can be

shown that the conditions for the existence of an

observer proposed in Section 2 are not satisfied,

and hence such an observer cannot be constructed.

Fig. 3. Partition of a superheater

In the first segment, the desuperheater is included

and (28) and (29) are modified as follows:

V~oC~ dx----A-1 - Cp( wi + w~)xl dt

On the other hand, it can be seen that the

conditions for the existence of an observer

proposed in Section 2 are satisfied for an

appropriate reduced model derived from the model

developed in Subsection 3.1. In this reduced model,

the system state is composed of steam

temperatures and the metal temperatures are

regarded as unknown inputs.

500

To derive a model for the construction of the

observer, the superheater is divided into v

segments, and equations analogous to (30)-(33) are

derived. Regarding the metal temperatures as

unknown inputs, the equations analogous to (31)

and (33) are discarded, and the s ta te-space model

for an observer consists of the equations analogous

to (30) and (32) while taking z~, k = 1 , - " , u, to

be unknown inputs T ,~ , k = 1, "", u. After these

processes,


B01 = [ bl 0 "'" 0] z, B12 : [ b2 0 "" 0] :r

dx],, : _ Cp( wi-t- Wd)X 1 V, oCp dt

+ ~'msS2( Trot - x ~ ) + CpTiwi-l- CmTewd, (34)

and in the k th segment, k = 2, . . ' , v,

a.~S2 1 Ct~Td where al VspCp ' a2 = Vsp ' bx = V~oCp '

b2 = a2.

It is assumed that the metal temperature

distribution varies smoothly, and that a

measurement point at the inlet is located

sufficiently close to the heated section, and

consequently the temperature measurement at the

inlet of the superheater contains information about

the metal temperature in the first segment. In

cases where the outlet steam temperature is also

available for measurement, the measurement

equation can be taken as follows:

dxk _ Cp( w i + w e ) ( x k - t - x , ) V~pCp dt [100

w( t )= Cx( t ) = 0 0 0 ... 0 (37)

+ a.~S2( T . ~ - xk). (35)

Regarding metal temperatures T.a, k = 1, "", u,

as unknown inputs v ( t ) in (34) and (35), (34) and

(35) can be written in the following form:

Jc(t) = [ Ao + Pl( t)Aa + P2(t)A2]x(t)

+ [Bo + ql (OBt]u( t ) + Dv( t ) , ( 3 6 )

The measurements w(t) satisfy w(¢) E / ~ .

However, it follows from (13) that the number of

unknown inputs must be less than or equal to the

number of measurements to construct an unknown

input observer. Therefore, a sufficient condition in

Section 2 for the existence of an unknown input

observer cannot be satisfied unless the steam

temperature measurements are taken at more points

than there are unknown inputs in the superheater.

m

: = A x ( t ) + Bu( t ) + D r ( t ) ,

where

i0l(t) = W,,

P2(t) = wa,

-v(t) = [Tml T , e " " • T,,~]T,

qt( t ) = Ti , u ( t ) = [ w # wi] r

The matrices are given by

Ao = d i a ~ ax, at . . . . . . . al ] ,

[a20 ......... !l a 2 - - a 2 . . . . . . . . .

A I = A ~ = 0 az - a2 . . . . . . ,

0 "'" 0 a2 - a2 0 . . . . . . . . . a2 - a2J

However, in the superheater, the metal temperature

distribution generally varies smoothly with distance

along the superheater and takes a special form

along the superheater. Therefore, the metal

temperature distribution v ( t ) can be approximated

as a special function with a small number of

parameters. In this paper, the metal temperature

distribution v ( t ) is expressed as a second-degree

polynomial in the distance l from the cold steam

inlet end. In the case where the temperature

measurements are taken at more points in the

superheater, more precise approximation schemes

can be adopted. Then, for the superheater of total

length L, the metal temperature Tin(l, t) at the

distance l and time t can be written as

[ ]' [ ] , - - B o = 0 0 " " 0 , B l = 0 0 r 0 , D = - A o ,

Tm(l , t ) = v l ( t ) ( l - L ) 2 + v2(t), (38)

where vl(t) and v2(t) are unknown functions of


time. If d l = L/v, where v is the number of

segments, the metal temperature T ~ ( / , t) in the

kth segment can be approximated as

T~(I , t) = vl(t)[ ( k - O . 5 ) d l - L ] 2 + v2(t) (39)

k = 1 ,2 , . - - , v.

501

(IW i - Winl2+lWd - Wdn[ 2) <: X = Co~t . (42)

Section 5 shows that with an actual superheater

specification, the above sufficient condition for

asymptotic stability of the bilinear observer can be

satisfied,

Then, v ( t ) in (36) can be written as

7(O= (0 .SzJ l - - L) 2 1 ]

(1 .5:dl - L) z 1: v(t),

[ ( v -- 0 . 5 ) d l - - L] 2 1

(40)

where v ( t ) = [ v l ( t ) v2( t ) ] r is treated as an

unknown function in (1). Therefore the matrix D

in (1) can be obtained from D v(t) = Dr(t):

D =

( 0 . 5 a l l - L)2d d ] ( 1 . 5 z l l - L)2 d d ]

[ ( v -- 0 .5)a l l - - L]2d d

where d = - al.

Then, the matrix CD in (13) is given by

[ (0"5z l l -L)2d d] (41) CD = [ ( v - 0 . 5 ) a l l - L]2d '

and it can be checked that rank(CD) = 2. Hence,

if v( t )=[vl( t ) v2(t)] r is instead regarded as

an unknown input in (1), then the necessary

condition in (13) is satisfied.

Let win and wd,~ denote the nominal values of

wi and wd, respectively. If

Ao = Ao + w~A1 + wdnA2,

p l ( t ) = w i - win,

the state equation of the form (1) is obtained. With

the superheater model thus derived, the condition in

(22) can be given as follows:

4. S T E A M T E M P E R A T U R E C O N T R O L

A state feedback controller is constructed based on

the proposed observer. The output equation is

given as:

w

y ( t ) = C x ( t ) = [ O 0 0 . . . 0 1]x(t). (43)

vn denotes the known nominal value of the

unknown input v. Since the steam mass flow rate

is measurable, it can be treated as a known

disturbance. Denoting by w a. the nominal value of

Wd, UO is defined as u0: = w d - - w a n . If the

system is discretized without the unknown input

D ( v - v n ) , it follows that

x(k+ 1) = [ A ~ + (uo(k) + wd,,)A~+ wi(k)A~]x(k)

+ B'~l(Uo(k) + Wan) + T~'~2wi(k) + Davn. (44)

Denote the desired steam temperature by T , and

take a cost function f l with the one step ahead

as the terminal horizon (Goodwin and Sin, 1984):

f = 1/2[ (Cx(k+ 1) - Tr) r - ~ - C x ( k + 1) - Tr)

+ (-Cx(k) - T,.) ~ C x ( k ) - Tr) + uD(k)2R] (~5)

m

where R is an input weighting matrix, P and

Q are state weighting maWices. Then the total

mass flow rate wd(k) of the spray is given in

terms of Uo(k), which minimizes f l and the

nominal value Wdn :

wd(k)= uo(k)+ wd. = [ R + (Ad x(k) + Bdl) T"-C r

• P C(A~x(k) + Bdl)] -1 [ (A¢x(k) + Bdl) T

- C ~ Tr- -C(A~+ wa~A~ + wi (k )d~ x(k)

--'-CT iS~2w i( k) --'C Bgl w d. - "C Dd v .] ]

502

-b wan

(Goodwin and Sin, 1984).


(46) bl = 0 . 2 4 , d = 0 . 4 9 ,

wi. ( 420 (kg/s), w, <50 (kg/s).

5. S I M U L A T I O N

Computer simulations were performed to verify the

performance of the proposed controller, and a

comparison with a PI controller is made. It is

shown that the estimated value follows the steam

temperature under the variation of inlet steam

temperature and flue gas temperature, and that the

proposed controller maintains the outlet steam

temperature properly. In this case, the observer is

constructed using only measurement w(t) and

input u(t) without knowledge of the unknown

input v(t). An observer is constructed when the

flue gas flows in parallel with the steam inside,

with the external inputs varying. The external

inputs are inlet steam temperature and flue gas

temperature.

To describe the distr ibuted-parameter nature of the

superheater, the superheater is divided into 20

segments. Therefore, (30), (31), (32) and (33) are

simulated with n = 20. So, the superheater can be

approximated as a system with states consisting of

20 steam temperatures, x i, i = 1 . . . . . . 20, and 20

metal temperatures, zi , i = 1 . . . . . . 20.

In the interest of simplicity in implementation, the

state model for the observer is derived by

decomposing the superheater into 5 segments, and

hence (34) and (35) are simulated with ~ = 5. The

metal temperature Tin(l, t) as a function of the

distance l from the cold steam inlet end and time

t is approximated as a second-degree polynomial

as in (38). As shown in (34) and (35), the metal

temperatures T,~, k = l , " ' , 5 , are regarded as

unknown inputs. In equation (36), the unknown

input v ( t ) represents T,~, k = 1 , ' " , 5 , and T ~ ,

k = 1 , " ' , 5 are obtained as in (39).

The following values obtained from an actual

superheater specification were used in the

simulation:

L = 3 2 . 8 ( m ) , a l = - 0 . 4 9 , a2 = 2 .08×10 -3,

With win = 280 (kg/s) and Wdn = 2 0 (kg/s), it

can be checked that there is no invariant zero of

the system ( C , A , D) in (36). Thus a stable

matrix F 0 can be constructed. Applying Theorem

4 with Q = 2 I and ~ : 1 r e su l t s i n a bound:

( [ Wi - -Win [ 2 + W d -- w ~ z) ~ 74,325.

In equation (45), the input weighting matrix R is

taken to be 1, and the state weighting matrices

P and Q are taken to be 51. Those weighting

matrices are used in the state feedback controller

(46) with the est imated state xe(k) replacing the

state x(k). In the simulation, the discretization

interval is 0.1 second. The initial values of the

states xi, i = 1 . . . . . . 20, in (30)-(33) vary linearly

from xl = 410 (°C) to x20 = 550 (°C), and the

states zi , i = 1 . . . . . . 20 also vary linearly from

Zl = 550 (°C) to z20 = 560 (°C). Initial

observer states are given the values of 415, 454,

479, 508 and 550 (°C). The initial flue gas

temperature profile for the parallel flow

arrangement is illustrated in Fig. 4, where the flue

gas temperatures at l---- 0 and l = L are 850 (°C)

and 800 (°C), respectively. The initial flue gas

temperature profile is approximated as a

second-degree polynomial. In the simulation, flue

gas temperatures at l = 0 and l = L vary in time

independently, and inbetween the temperature

distribution changes smoothly according to the

variation of the flue gas temperatures at l = 0 and

l= L. Fig. 5 illustrates the flue gas temperature

variations at l = 0 and I = L , where the upper

part represents the temperature variation at l = 0

and the lower part represents the temperature

variation at I = L . The inlet steam temperature

variation is a known external input. Fig. 6 shows

the inlet steam temperature variation. As shown in

Fig. 6, the inlet steam temperature changes

abruptly at 300 (s) and 600 (s). Variation of the

inlet steam mass flow rate is shown in Fig. 7. The

inlet steam mass flow rate decreases linearly from

500 (s) to 600 (s).


With the specified initial conditions and temperature (~) g o o

variations, it has been checked how the observer

estimates the superheater steam temperature. The

first and the fourth states in the observer

geometrically correspond to the fourth and the

sixteenth steam states of the superheater model

partitioned into 20 segments. They are compared in

Figs 8 and 9. Since the initial observer values are

arbitrarily given, there is some deviation of the

estimate from the true value in the first 200 (s)

even though it is not clearly noticeable in Figs 8 (~)

and 9 due to time and magnitude scales. In Figs 8

and 9, it can be seen that the estimate follows the ... ~ k i . ~ t L ~ d ;*ta~

true value very closely, even though a small offset . - ' ~]~11~ ~'~lll]~ can be noticed, which results from the crude ...l "'--i',hhl~tlj,,.i.kLjibt. dividing scheme for the observer model (34) and . . [ l ~ W , , q ~ , , t r l . ~ r

/ l (35) and the approximation of the metal . ,

temperature (38). Fig. 10 shows the spray mass . . . . . . . . .

rate. It can be seen that the spray mass rate

changes against the variation of the inlet steam

temperature. The outlet steam temperature is

shown in Fig. 11. The abrupt changes of the inlet (~)

steam temperature at 300 (s) and 600 (s) cause

the outlet steam temperature to change after 5-6

(s) by about one-third of the inlet steam E ~

temperature change. Also, the smooth decrease of . .

the inlet steam mass flow rate from 500 (s) ""

causes the outlet steam temperature to change , ,

after 5-6 (s) by about a 2 (°C) increase of the

outlet steam temperature. The time delay of 5-6

(s) is not particularly noticeable in Fig. 11

because of the time scale, but has been confirmed (~,ls) using separate data. As shown in Fig. 11, the "'0

superheater outlet temperature is properly 4 ~

maintained at 540 (°C) under the changes of inlet

steam temperature at 300 (s) and 600 (s), inlet s ~

mass flow rate from 500 (s) to 600 (s) and flue

gas temperature. Next, a PI controller is applied to . .

the steam temperature control under the same

simulation environment, and its performance is

compared with that of the proposed state feedback

controller. The integral gain and the proportional (~) s ~

gain are given the values of 1 and 5 after ,,,

extensive tuning. The outlet steam temperature "'* s ~

which is controlled by the PI controller is shown - -

in Fig. 12. In Fig. 12, there is a rapid fluctuation ,..

of the outlet steam temperature due to the "'* 4 ~

variations of the flue gas temperature. From Figs --.

11 and 12, it can be concluded that the proposed "'°~ '"* "~* "'*

state feedback control method based on the

unknown input observer provides better Fig.

performance than the PI controller in maintaining

the outlet steam temperature.

8 5 o ~ 800

75O

l=O

503

l=L Fig. 4. Distribution of initial flue gas temperature.

Fig. 5. Flue gas temperature at

Fig. 6. Inlet steam temperature.

\

T o o ~ o • o o

T i m e ( s )

l ~ 0 and l = L .

7OO W O e o o

T i m e ( s )

i a t 2 ~ • t o 4o1~ l o t . ~ T o o ~ o e ~ o

T i m e ( s )

Fig. 7. Inlet steam mass flow rate.

8. Steam temperature (dashed

estimated value (solid line).

T i m e ( s )

line) and

504 Sang Hyuk Lee et al.

(~:)

~ 4 6

m 4 0

6 N

m m

m z a

S $ 1

6 1 0 o o 2 ~ I o o 4 ~ I o a ° ~ 7 o o m o I O 0

Time(s)

Fig. 9. Steam temperature (dashed line) and

estimated value (solid line).

( ~¢/ s)

7 0

g o

, o

o

Time(s)

Fig. 10. Spray mass rate.

('c)

, g . .~ , .g° .M .~'. .M ,,;o ~;° .~°

Time(s)

Fig. 11. Outlet steam temperature(controlled by

state feedback controller).

For temperature control at the outlet of a

superheater, the relevant system state is the

temperature pattern along the superheater tube,

which is not usually available for measurement.

This paper proposes to use the theory of unknown

input observers in estimating the steam

temperature without the measurement of the metal

temperature distribution. An observer has been

constructed by regarding the metal temperature

distribution as an unknown input and

approximating it as a second-degree polynomial. A

closer approximation can be used if more

measurements are available. A cost function with

the one step ahead as the terminal horizon is

chosen, and a state feedback controller is

constructed based on the proposed observer.

Using a computer simulation, it has been shown

that the proposed observer closely estimates the

states of the system with unknown inputs under

the variation of inlet steam temperature and flue

gas temperature. Also, the superheater outlet

temperature is properly maintained under the

variation of inlet steam temperature and flue gas

temperature. By comparisons with a PI controller,

the performance of the proposed controller has

been verified. With slight modifications, the same

procedures can be adapted to the control of various

heat exchangers.

A C K N O W L E D G E M E N T S

(v.) s ~

m 4 0 -

a ~

6 , o i o o ~ s a o

Time(s)

Fig. 12. Outlet steam temperature(controlled by PI

controller)

The authors would like to thank the referees for

making many valuable suggestions to improve the

paper.

This work has been supported in part by Electric

Engineering & Science Research Institute under

Grant 94-02 which is funded by Korea Electric

Power Company, and by Engineering Research

Center for Advanced Control and Instruction

under Grant 96-25.

6. CONCLUSIONS

An observer for billnear systems with unknown

inputs is proposed and the design procedure is

discussed. A sufficient condition guaranteeing

asymptotic stability of the proposed observer is

derived based on the Lyapunov stability theorem

and characterized in terms of invariant zeroes.

REFERENCES

Bhattacharyya, S.P. (1978). Observer Design for

Linear Systems with Unknown Inputs, I E E E

Trans. Auto. Control, AC-23, pp. 483-484.

Daugherty, R.L., J.B. Franzini and E.J. Finnemore

(1985). Fluid Mechanics with Engineering

Applications, McGraw-Hill.


Funahashi, Y. (1979). Stable State Estimator for

Bilinear Systems, Int. f Control, Vol.29,

No.2, pp. 181-188.

Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering Prediction and Control, Prentice-Hall.

Hara, S. and K. Furuta (1976). Minimal Order

State Observers for Bilinear Systems, Int. ]. Control, Vol.24, No.5, pp. 705-718.

Hostetter, G. and J.S. Meditch (1973). Observing

Systems with Unmeasurable Inputs, IEEE Trans. Auto. Control, AC-18, pp. 307-308.

Kudva, P., N. Viswanadnam and A. Ramakrishna

(1980). Observers for Linear Systems with

Unknown Inputs, IEEE Trans. Auto. Control, AC-25, pp. 113-115.

Kurek, J.E. (1983). The State Vector Reconstruction

for Linear Systems with Unknown Inputs,

IEEE Trans. Auto. Control, AC-28, pp.

1120-1122.

Lee, S.H., J.S. Kong and J.H. Seo (1994).

Superheater Steam Temperature Control

Based on Observer for Systems with

Unknown Inputs, Proceedings of the 1st Asian Control Cor~erence, pp. 1017-1020.

McCormack, D.D. and L. Crane (1973). Physical Dynamics, Academic Press, New York.

Yang, F. and R.W. Wilde (1988). Observers for

Linear Systems with Unknown Inputs, IEEE Trans. Auto. Control, AC-33, pp. 677-681.

r a n k [ M . c A o D] < 0 n + q .

505

Since q K m, tl is an invariant zero of the

system ( C , A0, D). Hence, if all the invariant

zeroes of the system ( C , A0, D) have negative

real parts, then (C, RA0) is detectable and the

assertion follows from Theorem 1.

Proof of Theorem 3

Since rankCD= q = m, ( CD) +

(CD) -1, and

CR= C ( I - D( CD)-I c) = 0 .

becomes

Hence, the observability matrix of (C, RA0)

becomes

i o,1:l 1 ' o

wo=

which implies that

Ker W~ = Ker C.

A P P E N D I X :

Proofs of Theorems 2, 3, 4.

On the other hand, from (22),

Ker C D I m R .

Proof of Theorem 2

Suppose that a complex number 2 corresponds to

an unobservable mode of (C, RA0). Then, there

exists a nonzero vector x ~ R" such that

. 0]z -_ 0

or equivalently,

3 I . - Ao D x = 0 ,

which implies that

Since q = m, d im(Ker C) = n - m and

d im( ImD) = m. Thus it follows from the direct

sum decomposition of R n that

dim ( Im R) = n - m, which implies that

Ker C = Im R.

Therefore

Ker Wo -- Im R,

that is, the unobservable subspace of (C, RA 0) i s

equal to Ira R . Hence (C, RA0) is detectable if

and only if the stable subspace of RA0 contains

ImR.

506

Proof of Theorem 4

Since F0 is stable, H is positive definite. A

Lyapunov function V(e) = e r H e is introduced.

By the Lyapunov stability theorem, e(t) is

asymptotically stable, if there exists ~ > 0 so that

~e) < -e l le l l~ , Vt. (A1)

It will be demonstrated that (22) implies (A1). Let

Mi: = F T H + HFi. Then the time derivative of

V(e) is

Sang Hyuk Lee e t al.

( g l P,(t) I z)m( go2m~ <M,)),/2

< a"a.(M)- ~, Vt.

A~lying the following inequality

( ~,~.yl p,<t) I =)'/= ( ,~oL.(O;)) '/=, (A2)

it follows that

a ~ ( ~ , ( t ) M i ) < a"a,(M) - e, V t . (A3)

f/(e) = eT[ ( F [ H + HFo)

+ i~ i ( t ) (F i rH+ gFi)]e

= - erMe + eT[ i.~i(t) Mile.

Since

It follows from (20) that there exists a positive

number e, e ( a.an(M), so that

a"an(M) Ile[l~ < eTMe < a.~llell~,

(A3) implies (A1), which completes the proof.

Documents

Observers for bilinear systems with unknown inputs and application to superheater temperature control