Ind. Eng. Chem. Res. 1993,32, 1102-1107
1102
Autotuning of Multiloop Proportional-Integra1 Controllers Using Relay Feedback Ai P. Loh,’ Chang C. Hang, Choon K. Quek, and Vinod U. Vasnani Department of Electrical Engineering, 10 Kent Ridge Crescent, Singapore 0511
In this paper, a simple method for autotuning multiloop PI controllers for multivariable systems is presented. The procedure is a combination of sequential loop closing and relay tuning often used for tuning single-loop PI/PID controllers. The proposed design offers three major advantages. Firstly, overall closed loop stability is guaranteed; secondly, very little knowledge of the process is required; thirdly, it is an autotuning procedure. Set point and load disturbance responses are improved by introducing set point weighting factors. Some simulation and comparative results are shown for distillation column models commonly used to demonstrate multiloop designs. 1. Introduction
The control of multivariable systems has always been a challenge to control system designers due to ita complex interactive nature. Many control schemes existing in the process industry today are made up of simple controllers acting in a multiloop fashion. Such controllers are typically proportional plus integral (PI)or proportional plus integral plus derivative (PID) structures, many of which are tuned largely on a single-loop basis. This procedure cannot guarantee stability when all the loops are closed simultaneously. The closing of one loop affects the dynamics of all the other loops and is precisely the cause for the loops going unstable when controllers designed individually are closed simultaneously during automatic operation. Stability can only be ensured by introducing appropriate detuning factors on the PUPID parameters (Luyben,1986). There are other techniques for multivariable control system design which ensures closed-loop stability by considering the multivariable system as an entity rather than as single loops acting in a multi-loop fashion. Such techniques as the characteristic locus and the inverse Nyquist array methods (Maciejowski, 1989) are computationally intensive and therefore less appealing to practicing engineers. Often, the controllersobtained are rather complex. Other methods, such as the linear quadratic Gaussian (LQG) (Anderson and Moore, 1989) and the internal model control (Morari and Garcia, 1982) approaches, are more amenable but they require full knowledge of the process. More recently, numerical design techniques have also been introduced, for example, dynamics matrix control (Cutler and Ramaker, 19801, which makes use of process knowledge and the desired performance trajectories to calculate the multivariable controller structure. These methods are mainly suitable for off-line design, and additional efforta are needed to introduce on-line autotuning or adaptive control. In this paper, we consider the design of multiloop PI/ PID controllers for multivariable systems. We are motivated by two factors. Firstly, the widespread use of PI/ PID structures in the process industry implies that they are well accepted and understood by process engineers. They are also known to be versatile for a wide range of processes. Secondly, the availability of methods to individually tune these controllers efficiently and without too much computational effort is most appealing. For example, the step response test used to tune PI controllers for first-order or overdamped systems is a quick means of achieving some reasonable design, although some fine tuning may need to be executed subsequently to obtain more acceptable responses. The relay autotuning method
(Astromand Hagglund, 1984)is another means to achieve quickdesigns. This latter method is particularly appealing because it does not require very much knowledge about the process. Only the sign of the steady-state gain is required to set the direction of switch of the relay. In order to retain the single-loop tuning structure, the design of the controller is achieved through sequential loop closing (Mayne, 1979)as follows. The first controller is constructed based on one subprocess while all the other subprocesses reamin on open loop. The next and the rest of the other loops are constructed one at a time but with all the previously designed loops closed. While this ensures stability, the problem of identifying the combined dynamics at each stage remains a computationally tedious one. The design method proposed in this paper combines the efficiency of relay autotuning and sequential loop closing. This has several advantages. Firstly, it guarantees closed-loop stability at each stage of the design. Secondly, the controllers are of PI/PID structures which have the merits outlined above. Thirdly, with sequential loop closing, the design procedure still retains the single-loop tuning structure which makes handling of the overall design easier. Finally, full knowledge of the process is unnecessary since the important design parameters are obtainable from the relay feedback. The following assumptions are made about the multivariable plant. Firstly, the manipulated and controlled variables have been paired using measures such as the relative gain array (Bristol, 1966). Any transfer function model in this paper is a reflection of this pairing. Secondly, the plant is multiloop gain stabiliiable and hence multiloop PI/PID stabilizable (Morari and Zafhiou, 1989). Thirdly, the plant is diagonally dominant with low pass characteristics. Some simulation results will be shown for distillation column models commonly used to demonstrate multiloop designs. Comparisons are also made with the “biggest log modulus tuning” method (Luyben, 1986),which designs using the Ziegler-Nichols tuning rules but with detuning factors introduced. 2. Review of Single Loop Relay Autotuning and the BLT Method 2.1. Relay Autotuning. In the relay tuning of singleinput single-output (SISO)systems, the process under relay feedback is set up as in Figure 1. Assume that the process is of low pass characteristics and that it is able to sustain oscillations when subjected to some appropriate feedback gain.
0888-588519312632-1102$04.00/0 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1103
Remark 2: The ability of the detuned loop to reject disturbances is generally reduced, leading to poorer transients. 3. Proposed Tuning Method
Figure 1. Relay tuning for SISO system.
Figure 2. Multiloop control of a 2 X 2 system. Table I. Ziegler-Nichols Tuning Rules PI proportional gain k, = 0.45Ku integral time Ti = 0.8tu derivative time
PID k , = 0.6ku Ti = 0.5Tu Td 0.125Tu
If d is the relay amplitude and a is the amplitude of the first harmonic response at the output, assumingall higher harmonics have been effectively attenuated, then the relay is said to have a gain given by N ( a ) = 4d/7ra (1) This gain is the describing function (Cook, 1986), and it effectively plays the role of a gain-varying proportional controller. Under the feedback configuration in Figure 1, the gain k, (ultimate gain) will be such that the overall system output becomes self-oscillatory with period Tu (ultimate period). A PI or PID controller can then be tuned using Ziegler-Nichols rules (Ziegler and Nichols, 1942). These rules are given in Table I. The tuning is automated by initially closing the loop on a relay. After a few oscillation cycles, the controller parameters can be computed and the controller with these parameters may then be switched in to replace the relay. 2.2. The BLT Method. The BLT (biggest log modulus tuning) method was proposed by Luyben in 1986. It is an iterative method to tune a set of multiloop PI/PID controllers as shown in Figure 2. The Ziegler-Nichols settings for each individual loop are first calculated using the ultimate gains and periods of the diagonal elements in the transfer function model of the plant. These settings are then detuned by some factor Fwhich is usually between 2 and 5. A stability test is then performed based on the locus of W(s)= (-1 + det(1 + G(s) K ( s ) ) ) .F is further varied if some “distance criteria” based on W(s)is not satisfied. This is thus an off-line method that requires good knowledge of the process dyanmics. Remark 1: The detuning in the BLT method is based on one single parameter, F, primarily to achieve closedloop stability and better performance in terms of set point response and interactions. In principle, detuning with independent parameters may be considered with the added advantage of reducing interactions by different amounts of the other loops. However, the design becomes more complex.
The proposed tuning method is a two-step process in which the PI controller parameters are first determined, followed by a fine-tuning step. The first step performs sequential loop closing with relay autotuning similar to the SISO case. The second step improves the set point responses through set point weighting (Hang et al., 1991). The advantage of this over detuning will be discussed. 3.1. Sequential Loop Closing with Relay Autotuning. The main idea of sequential relay tuning is to tune the multivariable system loop by loop, closing each loop once it is tuned, until all the loops are tuned. To tune each loop, a relay feedback configuration is set up to determine the ultimate gain and frequency of the corresponding loop. The Ziegler-Nichols PI/PID settings are then computed on the basis of this information. For a 2 X 2 process, the tuning procedure is as follows. Either one of the diagonal elements (g~~(s) or gzz(s))in the transfer function model is first tuned using a relay. Subsequently, this loop is closed and a second relay tuner is placed on the other loop to tune the corresponding controller. Where the resultant transfer function at each stage (generally higher order than the original) isgain stabilizable and hence PI/PID stabilizable, there is no question about the stability of the overall design procedure. Gain stabilizability is a necessary requirement for the relay feedback to give any meaningful results since the related describing function essentially acts as a varying gain controller. Equivalently, if the relay feedback configuration is able to establish substained oscillations at every stage of loop closing, overall stability is assured. While there is specific analysis for stability, performance is harder to assess since there are usually time domain criteria associated with not only step responses but also interactions. The sequence of loop closing is important since it affects the amount of interaction entering all the previously tuned loops and therefore limits the quality of the set point responses for those loops. It is generally expected that when the slower loop is tuned first, the set point responses of the overall closed loop are inferior compared to when the faster loop is tuned first. This is because the slower loop is much more affected by interactions and tuning this first does not allow any interaction to be considered in the tuning. On the other hand, when the faster loop is tuned first, it has the advantage that it is less affected by any interactions. More importantly, it allows the slower loop to be tuned last and hence able to account for the interactions resulting from the closure of the faster loop. Even though the proposed sequence of tuning seems like the most logical scheme to reduce the effects of interactions, simulations have shown that in fact there is no real necessity to adhere to such a sequence. Instead, one could start with any loop and proceed to iterate over all the loops until a “stable” design has been obtained. Stability in this context refers to the convergence of all the controller parameters to approximately that of the design when the loops are tuned in descending order of the ultimate frequencies. This can be explained and illustrated by a 2 X 2 example as follows. Assume that the subprocesses gll(s) and g22(S) have different ultimate frequencies with gll(s) being the faster loop. Assume also that the slower gZz(S) loop has already been tuned with the controller &(s) obtained. With loop
1104 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993
I
0.25 03 0.1s 03
E 9.21 0
O.*
I 50
1 0 0 1 5 0 2 0 0 1 5 0 3 o Q 3 5 0 4 0 0
sa
1 o 0 1 5 0 2 0 0 1 5 0 3 W 3 5 0 a
t
4.Zl
0
I
9.d
o
50
io0
1%
h
MO
ZYI
MO
350
0
b
Figure 3. (a, top) Tuning transients in loop 1. (b, bottom) Tuning transienta in loop 2. 2 closed, the procedure calls for the tuning of the first loop which now has the combined dynamics as follows:
g12(s) g21(s) g22(s) &(s) (2) g22(s)[1+ g22(s) Kz(s)l At the ultimate frequency of gll(s),the second term in the above expression is negligible and hence the ultimate frequency of the combined dynamics remains approximately unchanged. Tuning of this loop and subsequent iteration on the other loop with this loop closed becomes similar to tuning the faster loop first. The above argument also applies to systems of higher dimensions particularly when all the ultimate frequencies are distinct and far apart. Otherwise, some iterations may be required. For cases where the resultant dynamics are more oscillatory (underdamped type), the PI/PID ZieglerNichols tuning formula as it stands is well-known not to cope very well. Typically, the step responses remain oscillatory with little chance of improvements except through excessive detuning or a complete revamp of the Ziegler-Nichols rules. We exclude such cases in our simulations even though the design technique still holds. 3.2. Set Point Weighting. The use of set point weighting to improve the set point response in SISO systems is described in Hang, et al. (1991). With set point weighting, the normal PI control law becomes g,,(s)
-
B is generally between 0 and 1. If the overshoot without set point weighting is large, then B should be chosen to be small. The effect of this is that the initial proportional kick introduced by the abrupt set point change is reduced according to the magnitude of the set point weights. This results in smaller overshoots in the set point response,
leading also to less interactions in the other loops. Its major advantage is that load disturbance response is not affected and transient characteristics such as settling time and damping are also not altered by its introduction. It also has the added advantage of simplicity compared to detuning since its value is generally limited to between 0 and 1 and is independent of the design. These can be shown as follows. For m X m multivariable systems, with set point weighting, the closed-loop transfer function relating the set points, R(s),and the plant outputs, Y(s),is given by where
B = dWP,,&--,B,) Since the closed-loop characteristic polynomial remains unchanged with or without the set point weights, we conclude that they do not affect closed-loop stability and may be introduced after the design. However, they do influence the response to set point changes in a manner similar to the SISO situation. Consider a 2 X 2 system with set point change only in loop 1. Initially, let the set point weights be zero. The corresponding outputs are G(S)
= &?:,,(SI
g(S)= g:,,(s)
R,(s)
R,(s) where g:ij(s) denotes the ijth elements in the closed-loop
Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 1105
a
I .4
-
1.2I -
”
a8
/(I&...
+-
.
-
1
l i . 0.8
I
C
i
h
&lm
Figurn 4. (a) Response for set point at loop 1. (b)Response for set point at loop 2. (c) Response for set point at loop 1. (d) Response for set point at loop 2. Legend (-1 proposed design; (- - -) BLT design.
transfer matrix, G,(s), with @ = 0. With nonzero set point weights, the responses become
q ( s ) = g:&)(B,Ti,s Y&s) = g:&)(B,Ti,s
+ 1)R,(s) + 1)R,(s)
4. Case Studies
Assuming zero initial conditions, the corresponding time domain signals are
+ Y!(t) = @,TilY;(t) + Y i ( t )
rejection properties. Their use is simpler and gives a more direct control on the level of interactions compared to detuning.
y g t ) = @,Tig!(t)
(4)
y;(t)
(5)
In general, from (4) and (51, the amplitudes of signals for nonzero weights will be inferior compared to that of zero weights, in the sense that overshoots and undershoots will be larger, since the weights and integral times are positive. More importantly, the steady state in each loop always occur at the same time regardless of the value of @. When the response for a particular @ has settled after time t = t l , the response for any @ will also settle exactly after t l . The weights do not alter the transient conditions very much except for magnitudes of signals which increase with the weights. Settling times remain the same. Hence, for a conservative design in terms of overshoots and interactions, zero set point weighting is recommended. Any other @ will introduce larger overshoots/undershoots and interactions. The response to disturbances D(s) at the input to the plant is given by
Y(S)= (I + G ( S ) K ( S ) ) - ~ GD((~S)) This suggests that the disturbance rejection properties are not dependent on the value of the set point weighting. In summary, set point weights are used to reduce overshootsand interactionswithout sacrificingdisturbance
Example 1: Consider an eight tray + reboiler distillation column separating methanol and water with the following transfer function model (WB) (Luyben, 1986):
The manipulated variables are reflux and steam flow while the controlled variables are the distillate and bottoms compositions (qd and Qb, respectively). Loop 1 has an ultimate frequency of 1.61 rad/s while loop 2’s is 0.56 rad/ s. Hence, loop 1was tuned first followed by loop 2 with loop 1 closed. The detailed tuning sequence is shown in Table I1 while the tuning transients are given in Figure 3. The corresponding PI settings and set point weighting are in Table 111. Results are given to compare the design with the BLT method. The set point and load distrubances are shown in Figure 4 for independent set point changes. Load disturbances are assumed to occur at the inputs of the process. Example 2: Luyben (1986) also studied a 24-tray tower separating methanol and water with the following transfer function model (VL) for controlling the temperature on the 4th (stripping section) and 17th (rectifying section) trays.
1106 Ind. Eng. Chem. Res., Vol. 32,No. 6, 1993
I lir
b
Figure 6. (a) Response for set point at loop 1. (b) Response for set point at loop 2. (c) Response for set point at loop 1. (d) Response for set point at loop 2.
Table 11. Steps in Tuning Sequence s t e ~time ~ events 1 0 loop 1 is tuned with loop 2 open 2 15 loop 2 tuned with loop 1 controlled by the PI controller designed in step 1 3 60 loop 2 controller obtained; loop is closed and plant is allowed to settle 4 130 both loops closed; set point change in loop 1 5 270 both loops remain closed; set point change in loop 2 Table 111. Controller Parameters
WB tuning sequence
VL
KP Ti
1,2 291 0.868,4).0868 -1.353,3.36 3.246,10.4 3,1.33
B
090
0,o
ww 192 48.1,-25.4 18.99,26.3 090
The ultimate frequency of loops 1and 2 are 1.66 and 4.55 radls, respectively. Loop 2 was hence tuned first and the results of the tuning are given in Table 111. The set point and load disturbances are given in Figure 5. Example 3: Another distillation column model (WW) obtained from Luyben (1986) has the following transfer function:
Loop 1was tuned first. The PI controller parameters are
also given in Table 111. Figure 6 shows the set point and load disturbances. 5. Discussion
The simulation results show that the proposed design (solid lines) with zero set point weights compare favorably with the BLT method (dashed lines). The settling times for set point responses are significantly better for the WB and WW models while for VL they are comparable. The level of the interactions in most cases was significantly lower than that achieved by the BLT method despite detuning in the latter. As an example, in Figure 4b, the interaction in loop 1 due to set point change in loop 2 is much larger in the BLT method. In Figure 4c, the maximum amplitudes are comparable but the settling time is significantly slower in the BLT method. This points out the general problem associated with detuning. The relationships between detuning factors and their performances are not easily determined particularly when many factors are invoked. With a single detuning factor, as in the case of the BLT method, performance in terms of settling times will generally deteriorate while the level of interactions will reduce due to the reduction in loop gains. In fact, the transient conditions may alter significantly compared to that without detuning. This is in contrast to set point weighting where it was shown that settling times remain as well as can be expected with the Ziegler-Nichols rules without detuning. Overshoots and interactions are reduced. In some cases, the reduction is linear with respect to the set point weights. Detuning also affects input load disturbance rejection properties in the same manner as set point responses. In all the cases, responses from load disturbances settle relatively slower in the BLT method. With set point
Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 1107
a
“/{
1
d
i
I
Fieure 6. (a) Response for set point at loop 1. (b) Response for set point at loop 2. (c) Reaponae for set point at loop 1. (d) Responae for set point at loop 2.
weighting, load disturbance characteristics are as well as can be expected without the weightings because loop gains are not artificially reduced as in detuning. 6. Conclusion A method to autotune multiloop PI controllershas been proposed. The method effectively extends the relay autotuning technique of the SISO case to that for multivariable systems. It offers several advantages. Firstly, it is conceptually simple as it retains the singleloop tuning structure. Secondly, stability is ensured at every stage of the design through sequential loop closing. Thirdly, very little knowledge of the plant is required. Simulations on distillation column models have demonstrated the effectiveness of the design method. The procedure is more efficient when the fastest loop is tuned first since it is least likely to be affected by interactions. For “larger”systems,the order of tuning should correspond to the order of decreasing speeds as determined by their ultimate frequencies. For plants where the ultimate frequencies of the diagonal elements are similar, the sequence of tuning is not critical. However,some iteration may be expected in order to obtain the “best” design.
Acknowledgment V.U.V. is grateful to Singapore Technologies and the National University of Singapore for financial support.
Literature Cited Anderson,B. D. 0.;Moore, J. B. Optimal Linear Quadratic Methods; Prentice Hall, Englewood Cliffs, NJ, 1989. Aetrom, J. J.; Hagglund, T. Automatic Tuning of Simple Regulators WithSpecificationson Phaaeand AmplitudeMargina. Automatica 1984,20,645-651. Briatol, E. On a New Meaaure of Interaction for Multivariable Procesa Control. IEEE Trans. Autom. Control. 1966,AC-11, 133-134. Cook, P. Non-Linear Dynamical System; Prentice Halk Englewood Cliffs, NJ, 1986. Cutler, C. R.; Ramaker, B. L. Dynamic Matrix Control. Proceedings of the Joint Automatic Control Conference; 1980. Hang, C. C.; Aatrom, K. J.; Ho,W.K. Refinements of the Ziegler Nichols Tuning Formula. IEE Proc. Part D,1991,138(March), 111-118. Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Syatema. Znd. Eng. Chem. Process Des. Dev. 1986, 25,654-660.
Maciejowaki, J. M. Multivariable Feedback Design; Addison Wesley: Reading, MA, 1989. Mayne, D. Q.Sequential Design of Linear Multivariable Syatem. ZEE Proc. Part D 126 (June), 1979,559-563. Morari, M.; Garcia, C. E. Internal Model Control. Ind. Eng. Chem. Process Des. Dev. 1982,21. Morari, M.; zafiriou, E. Robust Process Control; Prentice HaU Englewood Cliffs, NJ, 1989. Ziegler, J. G.; Nichols, N. B. Optimum Settinga for Automatic Controllers. Trans. ASME 1942,65,433-444. Received for review November 13,1992 Accepted February 22, 1993
