3.1 Introduction
3.1 introduction
IdcntiUciilion ol'TlTO processes is more dilTieult due to the inleraetion between the loops.
For easier Held implementation, it is desirable to identify two SISO transfer function models of the flTO process. Many methods have been presented in the literature for the identification of TITO processes [45-58] by using a single relay or sequential operations of relays or by decentralized relay.
This chapter proposes new identillcation methods for TITO processes. The methods overcome some constraints of the conventional relay auto-tuning techniques available in the iileralLire. In section 3.2, we propose an on-line identification method where a pair of relay- controller (relay in parallel with a PID controller) is used for identillcation of two SISO second order plus delay process transfer function models oi the TITO process. A modified identillcation method is given in subsection 3.2.2 by introducing proportional controllers in the inner feedback loop. The proportional controller stabilizes the process and reduces the interaction between the loops. In section 3.3, we propose a new structure for the identillcation of a TITO process. In the structure, two preload relays are inserted in the error paths and two integral fillers are used in the feedback paths. The method ensures a smooth
limit cycle of the TffO process in the presence of measurement noise and loop interaction of various magnitudes. The concluding remarks to the methods are given in section 3.4.
3.2 Oii-linc IdentiUcation of TITO Processes 3.2.1 Identification wifii PII) Controllers
3.2.1.1 Identification Strnctnre
Pig. 3.1 shows the on-line identification structure for a TITO process. It comprises of an ideal relay in parallel with a PID controller in the error path of each loop. The proposed relay identification method induces the limit cycle outputs )'; and f, around the set-points li^ and /A, respectively. Based on the measurements of the limit cycle outputs, two SISO lumsler lunction models ol the process dynamics are identified. The relay heights /?, and /?,
■lie non-zero during identillcation and these values are set to zero after the relay test for
noimal closed loop operation. The PID controllers in the loop are able to reject the effect of static load disturbance during tbe identification. As shown in the figure, die static load
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3.2 On-line Identification of TITO Processes
dislurbaiiccs /., and and random mcasiircmcnl noises and A-/, appear at the inputs olTlie process G(.s) and output of the sensors, respectively.
Ri
Gdi (-v)
—
—
—►O.v:(.v)
/
—
-H -h:
i- iu. 3. 1 . Proposed on-line identification structure for a TITO process
3.2.1.2 Identification Procedure
The equivalent of the proposed identification structure (shown in Fig. 3.1) is redrawn in Fig.
3.2. It is apparent from Fig. 3.2 tiiat the controllers are in the inner feedback paths and help
in stabili/.inu the process during the identification. Attempt is made in this subsection toidentify two SISO second order plus time delay transfer function models of the TITO
process from the relay experiments. Let the model transfer matrix beG„As) - G„.^s) 0
0
(3.1)
which possesses the SISO transfer function models as given below
K c V / = 1,2 (.sT + 1)-
where , 7] and D, are the process model parameters to be estimated.
(3.2)
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Rt
Ri Gcll (s)
Li Ml
/'/h -I-/;,
A'y
R^ li: |—
—-h:
Gii{s) C,:(.s-) G:i (.v)
)'/
AA
G:: is)
}A
1
L:
G, I2 (x) < (
^ \
R^
M^
Fig. 3.2. Equivalent representation of the structure of Fig. 3.
C,(.v)
= a;',
c;„(.v) 0 0 o;,,(.v)
he the Pi!) eonlroller matrix where
I
V T s'' 2
+ • Vm-
i + AO V / = 1,2 (3.3)
In the above expression, /i is the derivative filter constant of the i''' loop and neglected in
the Ibllowiim for ease in analysis. Similarly, let the describing function matrix be
A' = A' 0
0 N
(3.4)
where /V, and A', are the gains of the ideal relays of loop 1 and 2, respectively. The estimation of process model parameters is carried out in the following manner.
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3.2 On-line Identification of TITO Processes
Estimation of Steady State Cilains ( K^)
Tlio philosophy of estimation of the steady stale gain for the SISO proeess described in
siihseelion 2.2.1.2 is extended for the TITO process. For t>2T.,„ when steady state limitcycle is obtained around the set-points, the average values of the process inputs
(/(,'^and and the process outputs (r,',,^and are measured. Then, a temporal
disturbanee is uiven to one of the set-points or reference inputs for a short duration and the
corresponding average values /h;,, .v^,, and are measured. If the limit cycles
occur around zero (r,(/) = 0), two temporal disturbances may be given to measure the a\ cra'jc values oi'the process inputs and outputs. Then, the steady state gains are estimated
by the relations
/
\
K. = /
(3.5)
("LyyL.. - "LyXLy )("Ly"i.yy ' "Ly"Ly )
where
Detailed derivation of the expressions given in (3.5) is presented in Appendix A.
Kstimafioii of Procc.ss Model Parameters
It is assumed that both the loops under limit cycle condition have the same frequencies. The
peak amplitudes A, and critical frequency co of the limit cycle outputs arc measured. The
describiim function of the ideal relays for each loop is given by
.V = V / -1,2 (3.6)
/r .1,
■fhe auto-tuning test results in stable limit cycle outputs for any 'h / h. * 0 when
= - f (3-7)
where
icy ) = [ / + )] ' identity matrix.
.5(1
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(J('K, )(A', + G(/Vo .,)) = -1 V /■ = 1,2 Using (3.2), (3.3) and (3.6) in (3.8), \vc obtain
—' Kc t(ci. +a, ) = -1 (I- ./>'!,'/:)■
where 4/7
1 T• h, \
(3.8)
(3.9)
n A
+ A". + r.'• /
= A . OJ.T' -
j
(3.10)
(3.11) F.ciiiating the magnitudes of both sides of (3.9), one obtains
(o T - ;r + tan
+ "i - 1 (3.12)
Similarly, equating the phase angles of both sides or(3.9) gives
2tan '(ri7,.,.7;) (3.13)
1 Then, (3.12-3.13) gives the e.xpressions for the transfer function model parameters as
(3. 14)
^7, ^
y - V ~ '
(i\ ,.
n + tan D =
/ A
C7,
^ "l/ J
■ 2 l»n"' r)
(0. (3.15)
Two stages of relay tests are performed to estimate the process model parameters. In the first stage of relay test, the initial controller parameters for each loop are set in a similar manner as given in the subsection 2.2. 1.2. Once stable limit cycle outputs are obtained, two
lemporal .Sl.SO models of the flTO process are identilled using the limit cycle parameters and the expressions given in (3.5) and (3.14-3.15). Thereafter, the controller parameters are updated using (4.44) and (4.58-4.59) of chapter 4 for speciHed values of loop gain margins ( .i,'™ ) and phase margins ( ), where the subscript / = 1, 2. The second stage of relay test is
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3.2 On-line Identification ofTlTO Processes
pcrloniicd using llic abine conlrollcr sellings to obuiin the final SISO iransfer funetion process models of the TITO process.
3.2.1.3 Load Disliirhance Rejection
The output of a TITO process for a step load disturbance is
r(.v) ^ [/ + .V(7(.v) + G.,(7(.v)] ' (;(.s)Z.(.v) (3.16)
The presence of the controllers in the loops during identification results in zero steady state
value of the outputs to the step load disturbance inputs implying symmetrical steady state limit cycle outputs around the set-points. This ensures accurate estimates of the limit cycle
parameters under static load disturbances.
3.2.1.4 iMcasiircmcnt Noise
To measure the amplitudes and the loop frequency accurately, the smooth simials from the noisy initputs are recovered using the technique discussed in subsection 2.2.1.4. The best fit outputs for the ri'fO process can be e.xprcssed as
fi ^ S
"■ v-os(A7/;/)+ sinCA'//;/) (3.17)
k II
where, i''(/) recovered output for the /" (/ = 1,2) loop, (A, = 0,1,...8) and
v:t (A, = 0,1,...8) are the Fourier coefficients and o) is the frequency of the recovered
outputs.
3.2.1.5 Simulation Results
do illustrate the methodology discussed in the preceding subsections, two examples are
presented here. One more example is given in appendix B.l for further illustration. The
auto-tuning test starts with the choice of /f,', = 0. 1 , T,] = 1 and The final model of the process is obtained using the procedure given in the subsection 3.2. 1 .2, The effect of measurement noise on the process model parameters is investigated in simulations by
introducing normally distributed random additive noise with zero mean and varying ai iaiice. 3/^(0, ), iii the outputs duiing the relay tests thus making the oscillations noisy.
I he relay test is performed with /;. = /?, = | .
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I^MinipIc 3.1
C'oiisiclci" ii rrrO process [62] wilh (he iranslcr nialrix
"(i.5.v + l) l).2(0.75.v + i)
I
(''(.s) = cl(.'n(s]
().6(().75.v + l) 0.8(1.2v + l) ^
where ck'n(s) = (I + .v)(l + 2.v)"(l +0.5.v). The gain margin of 2, phase margin or45" and the derivative lliler eonslanl of ().()! are used for both the loops to design the PID eontrollers for the seeond stage of relay test, 'fhe process model parameters along with the controller parameters are given in Table 3.1(a). Similar relay tests are performed wilh additive noise ol \arioiis tr;, such that the SNR varies Irom 10 dB to 30 dB. The estimated parameters (7], 73, /J>| and D,) with different noise levels and their respective errors are given in Table 3.1 (b). The recovered and noisy outputs signals wilh SNR = 20 dB are shown in Fig. 3.3.
From the results, it is seen that the process is identified very well by the proposed method although it is subjected to high level of noise. The largest errors in the estimated parameters
7] and D, are 0.18 'f(i and 0.78 'fo, respectively which are within the aceepiable limit.
fable 3. 1 (a) Controller and process model parameters for example 3.1
Loop 1 Loop 2
Stage 1
A^;, =0.1, r;, = 1,7;;, =0
a;, = 0.1,7;', = 1.7;;_, = o
A, = 0.849, 7] = 1.6698, D, = 0.4370 A, = 0.678, r, = 1.5343, D, = 0.4272 Stage 2
a;, = 3.5298, T;, = 1.671 1, 7;;, = l .6698