E-6-6
EVAPORATOR CONTROL DESIGN : A QUANTITATIVE FEEDBACK THEORY
APPROACH
Rahul Kundergi and P.S.V.Nataraj
Systems and Control Croup
Department of Electrical Engg.
Indian Institute of Technology, Bombay 400076 India
Email: [email protected]
ABSTRACT:
In this paper, a robust multivariable
2.2 DERIVATION OF THE SINGLE-LOOP STRUCTURES
The set of equivalent singleloop StNctureS are derived. as given in Fig2. From
above Fs),the 'plants' in the rows of Fig 2 respectively are:
kle- I f
e-0.027s
3
'1 I = (00032r+ 1)(0.0687s+ I ) '22 =
f w matrix
~
degree of freedom feedback
structure is designed for an evaporator example. The evaporator is described by
a 2x2 transfer matrix having uncertain time-delays and gains. The desired
trachng properties of the closed loop system are a priori given. and these are to
be achieved despite the large parametric uncertdintj.1 The third MIMO
quantitative feedback technique (QFT) of Horowitz is used for the design. The
obtained results are verified in both frequency and time domains through
simulations. and found to be acceptable over the range of uncemnty
considered.
'
Mth the Lsturbances d l 1,..d22as
-k,k,(O 0042r + 1)(0.142+ l)e(-'I+OO'S'J
2(00032s+ 1)(0.0687s+ 1)(0.16%+ 1)
1. INTRODUCTION:
The traditional process of evaporation has been very popular in the chemical
industry. e.g in desalination plants and sugar industry. The process is
characterized by very complex dynamic behavior : time-delays. asymmetry in
output responses due to non-linearity. timevarying nature due to drifting of
heat-transfer coe5cients. etc. As such . it is a complex task to obtain even a
satisfactory model of the single-effa plant.: it is more so for multiple-effect
evaporator arrangements that are heavily used in sea water plants for
economizing on energy consumption. The problem is aggravated due to large
ignorance of the values of key model parameters. For example, most of the
heat transfer data reported in the literature were obtained with water or very
dilute solutions: the heat t d e r mefficients in actual practice therefore
usually varies widely from these estimates Hence. in order to effectively
control such plants. the control designer needs to come up with a feedback
system that is robust with respect to the parametric uncertainties i.e he needs
to synthesize a feedback system that achieves the desired tracking and
regulation properties, and is stable over the entire range of possible parameter
variations.
c
6%; ;j
I)
.S---+I)
0.75
Finally. the bounds Flu. Fll on the prefilter magnitude are derived @'ko
and HouDis. 1988) and a urefilter
s
9
(-
0.67
+ I)(-
2.5* ')
is synthesized using stmght line approximations on a Bode plot
Channel 2 : The design frequencies are RE (0 5 , I , 5. IO) The shifted
bounds B2(o) on Lm20(10) are &splayed in Fig 4 A controller gz(s) is
synthesized by hand so that the resulting L d o o o ) obeys the correspondng
B2(u)
2. DESIGN PROBLEM:
2.1 PROBLEM STATEMENT:
The apple juice evaporator example considered in this work is similar to that
demibed in (Figueroa et al., 1991). This is a triple-effect evaporator with
counter-current pre-heating and parameters tuned to fit real &ta. The
Controlled Mriables are : the concentration (y1) and level (y2) both in the third
effect. %vhik the manipulated variables are inlet steam pressure
and
Output flow (9).
The 2 x 2 plant transfer matrix is
i
+
g,(s)=?
In this paper. w apply the third multi-input multisutput (MIMO)quantitative
feedback theory (QFT') technique of Horowitz (1979) to design a robust
feedback system for a an industrial-wale triple-effect (apple juice) evaporator
example . The plant transfer function is a multivariable one with 2 inputs and
2 outputs. and has significant t i m d l a y s in all elements except one . For the
p u r p ~ e sof design. we shall assume that there is large uncertainty ( i IO %) in
four key model parameters: three gains and one-time-delay. A similar example
but with different uncertainty and performance requirements has been solved
wing the structured singular-value approach by Figueraa et al. (1991).
i
.
2.3 DESIGN EXECUTION:
At the outset. since a basically non-interacting system is desired. it is best to set
fI2(s)=f2l(s)=O. The obligations on LI and L2 are known by refemng to Table
1 We consider the first row of Table 1 in the first step of the design
Channel 1: The deslgn frequencies are RE (0 05.0.1.0.5.1 0.1.5.2.5.8.10)
The templates of q1 lare computed at o E R d . and used to derive the bounds on
LloCjo). as described in Horowitz's paper. Due to the NMP character of 411.
the bounds are shifted to the right in Nichols chart. The shifted bounds El(")
are displayed in Fig. 3 A controller gl(s) is designed so that the resulting
L,~oCjo) satidies the corresponding Bl(u) :
Lastly.
hand a prefilter f22(s) satisfylng its bounds FzU(s).F21(s) is designed by
I
k,e-'is
--,-
P ( s ) = (0.0032s+ IXO.0687~+ I)
k2e4.0szr
I (0.0042s+ IXO.1422~+ I )
(0.116% + 1)
Suppose there is the followng considerable uncewnty in the plant parameters
kl.tl.kZ.andk3
klE [0 58.1 OS]. k2 E [-02.-0 261.
k3 E [-I 86.-2 0461.11 E [0 069.0 0851
The above gven parametnc uncemnty generates the set of plants P The
nominal values are taken as
k l = O 8 3 . t l = O O 7 7 . k 2 = - 0 2 3 , k 3 = - l 953
Further suppose that. despite the parametnc uncemnty. the time doman
uaclung specifications gven in Table 2 are to be acheved The equivalent
frequency domam specs can be generated by using appropnate transfer functlon
models
2.4 DESIGN VERIFICATION:
Using the above obmned controller and prefilter matrices. the MIMO structure
of Fig 1 w simulated in the frequency and time domans The time doman
results for a unrt step in the setpoint are shown in F i g 5 and 6 The Figures
show that the all the time d o m n specs are satisfactonly met. over P
3. CONCLUSIONS:
Using the third MIMO QFT' technique. h r l y simple controller and prefilter
matnces have been designed Through simulations, these have been found
effective in achiewng the gven specs. despite the large uncemnt). in four
model parameters, including a time-delay It may be noted that in the present
work. sophisticated techniques such as design optimmtion and channel tradeoffs have no: been attempted These may be pursued in future applications
1473
0-7803-1872-2/94/$4.000 1994 IEEE
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on February 12, 2009 at 05:03 from IEEE Xplore. Restrictions apply.
REFERENCES:
Horowitz. I., 1979. Quantitative synthesis of uncerrain multiple input-outplt
feedback systems.lnt. J . (bnrrol. 30, 81-106.
Horowitz. I and Sidi. M . 1972. Synthesis of feedback systems with large plant
ignorance for prescribed time-domain tolerances. lnf. J. Conrrol. 16, 287-309.
Figueroa J.L.. Agamennoni. O.E.. m e s . A.C.. and Romagnoli. J. A.. 1991,
Robust multivariable controller design m a h o d o l w spbiliry and performance
requirements. Chem. Engng. Sci.. 46, 1299-1310.
Table 1 :
IM",
Specification types to b e satisfied b y Li
Fqg 5 ( b ) : Closed loop stcp r r s e
Channel I
of
y,
to
* , z 0.ogs.
YO
1,,",,I :::
2 -... A,, spec
A-
lo
-170
Chmnel 2-
TO"..
Fbg S l a 1 : C i o s r a
09
1000
s t e p rrsponsc
y , t o r, ;t
,
z 0.069
Table 2:
Tracking performance specs.
Unit step in rl :
y, Upper bound : Overshoot is 16.3%. t, 4 . 5 .
1
Fig 1 ' The two maQu-degree-of freedom structure
Fig 2 The four equivalent ~rgle-loopShlldures to fig 1
1474
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r,
,
t , :0