Ind. Eng. Chem. Res. 1988,27, 969-973 Proceedings of the 19th International Symposium of The Combustion Institute, 1982, pp 1045-1065. Steinberger, R. L.; Treybal, R. E. “MassTransfer from a Solid Soluble Sphere to a Flowing Liquid Stream”. AZChE J. 1960,6, 227-232. Yang, W.-C.; Keairns, D. L. ‘Solid Entrainment Rate into Gas and
969
Gas-Solid, Two-PhaseJets in a Fluidized Bed”. Powder Technol. 1982,23,89-94.
Received f o r review May 18, 1987 Revised manuscript received January 20, 1988 Accepted February 22, 1988
Improved Multiloop Single- Input/Single - Output (SISO) Controllers for Multivariable Processes Thomas J. Monica,? Cheng-Ching Yu,’ and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
Three methods are proposed for improving the performance of multiloop single-input/single-output (SISO) controllers which have been tuned by a n approach illustrated in a n earlier paper. In t h a t paper, multiloop PI controllers were designed by detuning the controllers equally from SISO Ziegler-Nichols settings until a multivariable stability criterion was satisfied. I n the present paper, several methods are described for adding derivative action to the PI controllers. An empirical procedure is developed for weighting the detuning of the loops by the predicted final ITE of the process after load and set-point disturbances. The combination of weighted detuning and derivative action provides a significant performance improvement, with little added complexity. T h e results of both the earlier method and the enhancements described in this paper are compared t o Dynamic Matrix Control for several multivariable systems, from 2 X 2 u p t o 4 X 4.
In a recent paper, Luyben (1986) proposed a method for tuning P I SISO controllers for multivariable processes, which was called BLT tuning. The method consists of detuning all the controllers equally from Ziegler-Nichols settings until a safe margin of closed-loop stability is obtained, as indicated by a multivariable Nyquist plot. In the earlier work, good performance was obtained with very little expenditure of engineering or computing effort. However, comparisons of diagonal controllers with fully multivariable controllers such as DMC-which are demonstrated in this paper-indicate some incentive for developing improved versions of the diagonal controller. This paper is directed a t improving the diagonal BLTtuned controller performance so it approaches the fully multivariable controllers, without sacrificing the simplicity of the controller design and function. As stated in the earlier work, we do not claim that this method provides better control than full multivariable controllers. Our objective is to demonstrate a simple technique that provides reasonable performance for little engineering cost. In this work, we assume that selections of controlled variables, manipulated variables, and variable pairings have been made, for example by the method of Yu and Luyben (1986). We start with a matrix of transfer functions of order N X N . Simulations of three linear multivariable processes are presented. The performances of four multiloop SISO diagonal controllers are compared to the performance of a fully multivariable DMC controller for both load and set-point disturbances. The original BLT method is labeled BLT-1 in this paper. It uses only PI modes in the diagonal controllers and equal detuning of all loops. Two techniques were developed and combined that improve the performance of BLT control, while retaining its *Author to whom correspondence concerning this paper should be addressed. Present address: Department of Chemical Engineering, University of Houston, Houston, TX 77004. Present address: Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, R.O.C. 0888-5885/88/2627-0969$01.50/0
straightforward nature. In the first (BLT-2), derivative action is added to all controllers. A single F factor is used to adjust the integral time constant (q)and the controller gain (K,)of each loop, as in the BLT-1 method. Then, a separate F D is used to tune the derivative action (rD)of each loop in a similar manner. The second method (BLT-3) compensates for the fact that multiloop interactions are not normally symmetric. The detuning of each loop uses a weighting factor that depends on a ratio of a prediction of its integral total error (ITE) and the ITE of the other loops, for both load and set-point disturbances. The best results are obtained when these two modifications are applied in combination (BLT-4). This approach yields a decided improvement over BLT-1 tuning alone and provides a clear method for weighting the performance of each controlled variable according to it’s importance. As was true with the original BLT method, these techniques are limited to open-loop stable systems, and the question of integrity is only partially addressed: when all loops are on automatic, the system is stable; when any one loop is on automatic, the system is stable.
Discussion of BLT Procedures 1. BLT-1. We begin with a multiloop SISO system tuned by the BLT-1 method as described in the earlier work (Luyben, 1986). This relatively straightforward procedure, described below, was tested successfully in simulations on 10 multivariable process models. 1. Compute the Ziegler-Nichols tuning parameters of t,he diagonal elements of the process transfer function matrix, gii(s),as though the diagonal elements represented SISO systems. 2. Choose a detuning factor F. F should be greater than one. 3. Compute K , and T I J for each loop by
0 1988 American Chemical Society
970 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
fF='
Lc Max
..
/
/*
8
0
0 5
1
1 5
2
2 5
3
3 5
4
4 5
5
FD
Figure 1. Effect of F and FDfor the WB system.
where K,,, and r1J.mare the Ziegler-Nichols P I values. 4. Calculate the function W over the appropriate frequency range (i.e., near the (-1,O) point), where W = -1 f det[I G(iw)B(iw)] (3) 5 . Compute the function L,(iw) where
+
L,(iw) = 20 log I1
+"wl
(4)
6. Adjust F until the maximum L,"""= 2N. This method has been shown to produce stable multiloop SISO controllers with reasonable performance response. The determinant of eq 3 is the characteristic equation of the multivariable closed-loop system (MacFarlane, 1977) and rigorously determines closed-loop stability. For an openloop stable system, if the determinant Nyquist contour encircles the origin, the closed-loop system has a pole in the right-hand complex plane. The function W of eq 3 also rigorously indicates closed-loop stability, on the basis of whether it encircles the critical point (-1,Oi). It is important when applying this procedure to confirm the form of the Nyquist plot of the function W , because the log modulus function merely denotes nearness to the critical point. An unstable system could be tuned to a maximum log modulus of 2N because it encircles the (1,Oi) point but is not near it. 2. BLT-2. After tuning by BLT-1, derivative action can be incorporated through the steps that follow. 1. Choose a second detuning factor F D . F D should be greater than one. 2. Compute 7 D J, where (5)
and 7 ~ j . is z ~ the Ziegler-Nichols value for 7DJ. 4. Repeat steps 4 and 5 of the BLT-1 procedure. Change F D until L,"" is minimized, maintaining F D 1 1. The trivial case may result where LCmax is minimized for F D = m, Le., no derivative action. 5 . Reduce F , and repeat steps 3 , 4 , and 5 of the BLT-1 procedure until L,""" = 2 N . 6 . Repeat steps 4 and 5 above until no further reduction in F is possible. By this process we are shifting the phase angle of the closed-loop characteristic equation Nyquist contour by tuning the derivative terms so that they have the greatest effect near the closed-loop resonant frequency. To illustrate the BLT-2 procedure, consider the 2 X 2 Wood and Berry (WB) (1973) column example. BLT-1 tuning gives an original F value of 2.55 to achieve an LCm of 4. The BLT-2 tuning gives final values of F = 1.9 and F D = 1.9 as illustrated in Figure 1. The recursive BLT-2 procedure converged for all cases studied. The computing time required for the BLT-2 procedure is an order of magnitude greater than that of BLT-1 but is still only about 10 s on a Cyber 850 for a typical 4 X 4 system of transfer functions. The regulatory response of the BLT-2 controlled WB model is shown in Figure 2, along with BLT-1 and DMC.
30
L:
TIME MIN
Figure 2. WB load response controlled variables: BLT-1, BLT-2, DMC.
Table I. Load Transfer Functions WB
3.8 exp(-8s) gL(l) = 14.9s + 1
gL(2) =
4.9 exp(-3s) 13.2s 1
+
OR 0.14 exp(-lb) gL(l) = (19.2s 1)*
-11.54 exp(4.6~)
+
gL(3) =
7.01s + 1
0.53 exp(-10.5s) gL(2) =
6.9s
+1 A1
-0.86 exp(-6s) gL(l)
=
6.2s
+1
gL(3) =
-1.06 exp(-5s) gL(2) =
35s
1.2 exp(-9s) 24s +
-0.86 exp(4.25~) =
+
(4s+
1)2
Table 11. WB BLT-I, BLT-2, and DMC IAE Values for Load Disturbance BLT-1 BLT-2 DMC
4.67 2.55 19.90
61.5 42.3 32.6
The integrated absolute errors (IAEs) of these controllers are listed in Table 11. (Table I lists the load response transfer functions for the three systems discussed in this work. The process transfer functions for these systems were presented by Luyben (1986) and the original authors.) For this example, BLT-2 control displays a decided advantage over BLT-1 and appears to be at least as good as DMC. 3. BLT-3. In this section, a method is outlined for using individual weighting factors for the detuning of the SISO loops in a BLT system. The objective is to estimate the level of imbalance in the controller detuning and compensate for it. The ITE of each controlled variable in the system for set-point and load disturbances is estimated as suggested by Shinskey (1986). This measure only estimates the disturbance effect because the components of the response trajectory above and below the set point will partially cancel in the ITE computation. The s u m of these estimates is used to obtain a weighting factor for the detuning of each loop. Note that this procedure is dependent on variable scaling, but this can be used to relate performance to economic value. For any N X N set of diagonal PI controllers, the value of the manipulated variable of loop J is
Hereafter, we will assume mJ(0)= 0.
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 971
At steady state, as t goes to infinity, eJ goes to zero. Thus, lim [mJ(t)] = K-Some, cJ
dt
(7)
71J
t-m
So the final ITE of any controlled variable can be computed from
We can easily compute mJ(a) from the process steady-state gains as follows:
m ( a ) = G-'(O)x(m) - G-'(O)GL(O)d(m)
(9)
To compute the ITEs for a step change in xKwt, where xWt = [O,...,0,1,0 ,...,01, the steady-state value of m which drives x (a)to xBetis determined by
m(m) = G-l(O)x(m)
(10)
where x(m)
= [O,...,0,1,0,...,01
[gik(O),&O)
,...
a5
30
53
TIME M N
Figure 3. WB load response controlled variables: BLT-1, BLT-3, DMC. Table 111. WB IAE Values for BLT-1, BLT-3, and DMC Load Disturbances
IAE( x 1)
IAE(x2)
4.67 9.18 19.90
61.5 28.1 32.6
BLT-1 BLT-3 DMC
it is empirically logical to weigh both the load and set-point disturbances in the SJ function. The largest SJ is selected as a normalization factor:
(12)
= max SJ (17) We wish to adjust the loop tuning so that the functions of the absolute ITEs of each loop are equal. To do so, we calculate individual detuning factors for each loop, Fj:
where g;i(O) is the iKth element of G-'(O). m (m) for a unit step load disturbance is
m(m) = G-l(O)G,(O)
5
(11)
So for a unit step in xKset, m(m) =
C
(13)
SMM
FJ = F(SMAX/SJ)~'~
or
The P I controller parameters are computed as
N
mi(..) =
(18)
[gTJ(o)gLJ(0)l
J= 1
(14)
To obtain the ITE for controlled variable J for either case, we employ eq 8:
At this point we can use the ITEs to generate a measure of the error in each controlled variable of the SISO system based on either set-point changes, load disturbances, or any arbitrary combination thereof. We use this information to determine different detuning factors for each of the SISO loops to compensate for the unequal response of the variables. Several ways to weigh the ITE were investigated (Monica, 1987). The best procedure found was to compute a function of the absolute values of the ITEs for a disturbance in each set point and for a load disturbance as follows: N
SJ = C Labs (ITEJ-sET~) / N 1 + abs (ITEJ-LoAD)
Kcj=
K c J-ZN
FJ
(20) 71 J = 71J-ZNFJ Consider the WB example. For this 2 X 2 system, gll(0) = 12.99, g12(0) = -18.9, g21(0) = 6.6, and gZ(0) = -19.4. ALSO g;:(O) = 0.155, g&O) = -0.151, gii(0) = 0.053, and gii(0) = -0.103. The Ziegler-Nichols PI tuning constants are Kcl.m = 0.96, 7I1-m = 3.25, Kc%, = -0.19, and 7B-m = 9.20. The ITEs of the controlled variables for a unit step change in xFt are 0.532 and -2.57 and for a unit step change in xZset are -0.519 and 5.04. The ITEs of the controlled variables for a unit load disturbance are -0.519 and 14.96. The S vector that results is
S = [1.04,18.77IT Therefore, the individual detuning factors are Fl = F(4.25), F2= F(l.OO). Note that the F factor for BLT-1 is 2.55 (see Table V). The BLT-3 settings produce a tighter x 2 loop
and a looser x1 loop. Figure 3 depicts the regulatory response of the WB model under BLT-1, BLT-3, and DMC control. Table I11 SJ = abs (71j / K CJ ) x lists the corresponding IAE data. BLT-3 is better than N BLT-1 and DMC for this case. [abs (mJ-SETi(m)/N)l + abs ( ~ J - L o A D ( ~ ) both )) i=l 4. BLT-4. As a last approach, a combination of BLT-2 and BLT-3 was explored. First the BLT-3 procedure was SJ = N used to get individual weighting factors for each loop. abs ( 7 1 j / K ~ j ) C { a b[g>XO)/Nl s + abs [g>j(O)g~i(O)l) Then the BLT-2 procedure was used with individual FD is1 factors for each loop: (16) FDJ = FD(SMAX/SJ)~'~ (21) Note the division of the set-point ITEs by order of the For the WB system, the results of BLT-4 tuning are F = system, N . This forces the S function to depend equally 1.18,F1 = 5.02, F2 = 1.18, F D = 1 . 5 5 , F ~= ~6.59, andFD2 on the N possible set-point disturbances and the load = 1.55. Figure 4 compares BLT-4 results for the WB disturbance. The SJ)s are computed in this way because i=l
(c
972 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 x1
-3& .-,-
x1
1
xi
BLTl
BLTl
BLT3
BLT4
MVK:
-/-------DIx
BLTl
x2
f
BLT3
BLT4
twc 30
5
45
EO
TIME [MINI
x3
Figure 4. WB load response manipulated variables: BLT-1, BLT-2, BLT-3, BLT-4, DMC. M1
M1
:'&
'
BLTl
ELT4
BLTl
''iz-_
..- '
,
m
,
30
45
5
BLT2
-:ME (MIN
BLT3
Figure 6. OR load response controlled variables: BLT-1, BLT-4, DMC.
35
M1
:y-----
-.
x1
"_.
X' X'
-
---,
BLT1 BLT4
M c--c.p-
x2'-
BLTl
'J
X2 i .
'1
hJ2
BLT4
--I--
FA2 - / *
,
I
3c
1
A5
x2
x3 53
- M E UIN
x3
G 7
BLT4
-
m
Lu-
BLTl
e.
_ _\/L---------------
BLT4
3-
Figure 5. WB load response controlled variables: BLT-1, BLT-4, DMC.
column with BLT-1 and DMC. x4
5. DMC. In this work the SISO BLT controllers are compared to the multivariable DMC controller (Cutler, 1983). Implementation of DMC requires first obtaining a multivariable step response coefficient matrix and then selecting a prediction horizon (NP), a manipulation horizon (NM), and a tuning factor (f) (Yu, 1986). In this work, the values of NP and NM were chosen as 15 and 40 min, respectively, values which are large enough to account for the given process time constants. In each case, the value off was tuned so that the DMC manipulative action was comparable to that of the BLT controllers (Le., the maximum magnitudes of the changes in the manipulated variables were kept approximately the same for all controllers). Figure 5 shows the manipulated variable dynamics for the WB example for BLT-1, BLT-2, BLT-3, BLT-4, and DMC.
Results and Conclusions The performances of BLT-4 and DMC were compared for three cases: the 2 X 2 WB system, the 3 X 3 OR system (Ogunnaike et al., 1983),and the 4 X 4 A1 system (Alatiqi, 1985). Comparative load rejection dynamics are given in Figures 4,6, and 7. IAE data from these runs are listed in Table IV. The controller parameters for these systems are listed in Table V. In the three cases studied, BLT-4 is obviously superior to BLT-1 control, and DMC and BLT-4 provide comparable load disturbance response. Three methods are presented for improved tuning of PID controllers for MIMO systems. In particular, one
x4
53p: " I / -
BLT4
i-
I BC
I
I
120
180
,
mJc
240
TIME M l h
Figure 7. A1 load response controlled variables: BLT-1, BLT-4, DMC. Table IV. WB, OR, and A1 IAE Values for BLT-1, BLT-4, and DMC Load Disturbances
BLT-1 BLT-4 DMC
4.67 7.22 19.90
BLT-1 BLT-4 DMC
0.84 0.72 1.73
61.5 19.0 32.6
OR Load Disturbance
BLT-1 BLT-4 DMC
7.21 4.73 3.23
65.40 24.30 28.20
A1 Load Disturbance 56.60 2.08 29.70 13.30 17.30 15.30 15.60 11.10 4.72
37.60 11.00 5.70
method, called BLT-4 in this work, is recommended for multivariable SISO PID tuning. This systematic procedure, based on Luyben's BLT-1 controller, is easily applied and includes a clear mechanism for weighting the value of the controlled Variables. The results are stable, provide an improvement over BLT-1 controller, and compare favorably with an implementation of DMC on the same
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 973 Table V. Control Parameters for BLT and DMC WB OR A1 ZN P I Settings K, 0.96, -0.19 3.24, -0.63, 5.66 5.13, 6.62, 2.66, 4.55 TI 3.25, 9.20 7.62, 8.36, 3.08 32.1, 3.33, 3.29, 12.3 TD
0.488, 1.38
ZN T D Settings 1.14, 1.25, 0.46 4.82, 0.5, 0.494, 1.845
F = 2.55
BLT-1 F = 2.15
F = 1.18 F1 = 5.02 Fz 1.18
BLT-4 F = 1.16 F1 = 2.67 F2 1.32 F3 = 1.16
FD = 1.55 FD1 = 3.57 F D =~ 1.77 FD, = 1.55
f NP NM
10.0 40 15
F = 2.25 F = 1.0 F1 = 1.0 F2 = 6.87 F3 = 1.56 F4 = 1.17 FD = 1.90 FD1 = 1.90 F D =~ 13.05 FD3 = 2.96 FD, = 2.22
DMC Settings 0.1 0.1 40 40 15 15
systems. The method was applied successfully to three systems from the literature.
Nomenclature B(iw) = controller transfer function or transfer function matrix 4 s ) = load disturbance
eJ = feedback error in loop J
f = DMC tuning factor F = BLT proportional and integral detuning constant FJ = BLT detuning constant of loop J FD
= the BLT derivative detuning constant
FDJ= BLT derivative detuning constant of loop J giJ(s) = the iJth element of G(s) gL&) = the J t h element of GL(s) G(s) = process transfer function or transfer function matrix GL(s) = load transfer function or transfer function matrix I = identity matrix ITEJ = the integral total error of the J t h controlled variable Kd = proportional gain of loop J KdVZN= PI Ziegler-Nichols gain of loop J L, = closed-loop log modulus L,”” = maximum closed-loop log modulus mJ(t)= J t h manipulated variable as a function of time m ( t ) = manipulated variable vector as a function of time m (s) = manipulated variable vector N = system order
NM = DMC manipulation horizon NP = DMC prediction horizon s = Laplace operator SJ = function of the predicted ITEs of loop J S M A X = maximum SJ S = SJ vector t = time W = function of the return difference x ( t ) = controlled variable vector as a function of t x (s) = controlled variable vector x&) = controlled variable set-point vector Greek Symbols w = frequency
integral time constant of loop J Ziegler-Nichols PI integral time constant of element g d s ) of G(s) T D J = derivative time constant of loop J T D J.ZN = Ziegler-Nichols PID derivative time constant of element gJ&) of G(s) T ~ J=
711.2~=
Literature Cited Alatiqi, I. “Composition Control of Distillation Systems Separating Ternary Mixtures with Small Intermediate Feed Concentrations”. Ph.D. Dissertation, Lehigh University, Bethlehem PA, 1985. Culter, C. R., ‘Dynamic Matrix Control-an Optimal Multivariable Control Algorithm with Constraints”. Ph.D. Dissertation, University of Houston, 1983. Luyben, W. L., ”A Simple Method for Tuning SISO Controllers in Multivariable Systems”. Znd. Eng. Chem. Process Des. Dev. 1986, 25, 654. MacFarlane, A. G. J. ”Return-difference and Return-ratio Matrices and Their Use in Analysis and Design of Multivariable Feedback Control Systems”. Znt. J. Control 1977, 25, 81. Monica, T. J. “An Exploration of Solutions to the Multivariable Control Problem”. M.S. Dissertation, Lehigh University, Bethlehem PA, 1987. Ogunnaike, B. A,; Lemaire, J. P.; Morari, M.; Ray, W. H. ‘Advanced Multivariable Control of a Pilot-plant Distillation Column”. MChE J . 1983,29,632. Shinskey, F. G. “Computer-Aided Design Procedures”. Presented at the Lehigh University Distillation Control Short Course, Bethlehem, PA, May 1986. Wood, R. K.; Berry, M. W. ‘Terminal Composition Control of a Binary Distillation Column”. Chem. Eng. Sci. 1973, 28, 1707. Yu, C. C.; Luyben, W. L. “Design of Multi-loop SISO Controllers in Multivariable Processes”. Znd. Eng. Chem. Process Des. Dev. 1986, 25, 498. Yu, C. C. “Control a Multicomponent Distillation Column Using Composition Estimation”. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1986. Received for review August 20, 1987 Revised manuscript received February 8, 1988 Accepted February 22, 1988
