Ind. Eng. Chem. Process Des. Dev. 1981, 20, 188-196
188
Table I. Effect of Recycle on Product Quality (Operating Variables as Shown in Figures 21 and 22) total no. feed of volrecy- ume, cles mL 0
1 2 3 4 5 0 1 2 3 4
product product cuts, Ve/VT mL %
A Sodium Chloride-Glucose System 40 0.93-1.00 10 20.0 80 1.50-1.58 10 27.3 120 1.75-1.83 10 30.5 160 2.00-2.16 20 32.9 2.18-2.50 200 40 33.1 240 2.42-2.90 60 33.8
NaCl, %
0.45 0.07 0.01 0.002 0.001 0.0002
B. Sodium Chloride-Glycerol Systema 40 0.72-0.88 20 26.05 0.01 80 1.20-1.44 30 27.51 0.004 120 1.60-1.92 40 31.10 0.002 160 1.92-2.32 50 32.70 0.001 200 1.76-2.24 60 32.85 0.0003
Feed: NaCl= 2.86 wt %; glycerol = 33.75 wt %
volumes, low flow rates, higher temperatures, and small resin particles, all contribute to better separations. Present study indicates that the cross-linkage of the resin should be within the limit when the mass transfer characteristics change due to solute size. Within a range of 4 to 10% DVB content, this change was not observed. On the basis of Gaussian nature of the elution curves of the present study and others reported in literature, it has been found that
HETP = K(flow rate)0,5(feedv ~ l u m e ) ~ ~ ~ ~ ( p adiameter) rticle where the constant K is a function of temperature and the characteristics of the systems used. In the multipass experiments with a recycle procedure, industrial feasibility of the ion-exclusion process is indicated particularly in situations where the concentration of electrolyte to be removed is high enough to render conventional ion exchange uneconomical. Pilot plant studies followed by economic estimates are needed, if the ground work for industrial exploitation of ion exclusion is ready. Acknowledgment Thanks are due to M/S Hindustan Levers Ltd.,India, for providing the sample of crude glycerol for the recycle studies. Literature Cited Asher, D. R.; Simpson, D. W. J. my$.Chem. 1956, BO, 518. Gupta, A. K. M. Tech. Thesis, IIT, Kanpw, India, 1877. Martin, A. J. P.; Synge. R. L. M. J . Bkchem. 1941, 35, 1358. Mayer. S. W.; Tompklns, E. R. J. Am. Chem. Soc. 1947, 69, 2888. Rlelipp, 0. E.; Keller, H. W. J. Am. 011 Chem. Soc. 1956, 33 (3), 103. Setherland, D. N.; Mountfort. C. 8. Ind. Eng. Chem. pIocess Des. Dev. I S M , 8 . 75. Simpson, D. W.; Wheaton, R. M. Chem. Eng. R q . 1954, 50, 45. simpson, D. w.; Baumn, w. c. rnd. ~ n g chem. . 1954, 46, 1958. Singh, D. M. Tech. Thesis, IIT, Kanpw, India, 1978. Tayyabkhan, M. T.; Whlte, R. R. A I C M J . 1961, 7 , 672. Wheaton, R. M.; Baman, W. C. Ind. Eng. Chem. 1953, 45, 228. Wheaton, R. M.; Bauman, W. C. Ann. N. Y . Acad. Sol. 1953, 57, 159.
Received for review August 10, 1979 Accepted September 19, 1980
Frequency Domain Adaptive Controller Sandra L. Harris and Duncan A. Melllchamp' University of CalifornL, Santa Barbara, Santa Barbara, CallfornL 93 106
A frequency domain adaptive controller for application to the class of systems which includes single-hput/single-output linearizable deterministic processes in standard feedback loops was developed, tested, and evaluated. The algorithm modifies the controller parameters to match the shape of a desired closed-loop transfer function as the index of performance. A nonparametrlc frequency domain identification is followed by a simple optimization to determine new controller parameters and a tlme scale factor. This tlme scale factor is subsequently used to change the controller sampling time: a faster process time scale elicits a faster controller sampllng rate and vice versa. The adaptive algorithm was tested using off-line calculations, simulated systems (noise-free and noisecorrupted), and two bench-scale processes (liquid level and stirred tank). Standard controllers in all cases are digital PI or PID controllers. The technique does not require the control loop to be physically opened, and may be used on- or off-line for controller tuning.
Introduction An adaptive controller seeks to maintain a defined set of system characteristics through self-modification over a wider range of external conditions than a standard controller could satisfactorily handle. Theoretical developments and applications of adaptive control have been reported by many authors. Advancements in computer technology during the past decade have encouraged the use of computers in process control, both in the design of control systems and in the use of on-line computers as real-time elements in control systems. These developments have increased the feasibility of applying adaptive control 0196-4305/a1/1 I 20-01a8$01.2510
techniques much beyond the situation that existed just a few years ago. Definition and Functions of Adaptive Control. A process may undergo dynamic changes severe enough to require some sort of compensating change in control strategy. An adaptive controller attempts to accomplish such a compensating change, preserving uniformity of overall system characteristics in the face of long and short-term process change. The adaptive algorithm generally performs three functions: identification, decision, and actuation. Identification consists of continuously (or periodically) measuring and determining the dynamic 0 1981 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981 189
0
CHANGED
A ADAPTED I
I
i
NEW CONTROLLER SETTlffiS (ACTUATIONI
~-----l
I
!&!
Y
-3
t
I INITIAL AND IP CURVE 2 PROCESS CHANGED, CONTROLLER UNADAPTED
-2
0
-I
I
REAL
Figure 2. Example case. Adaptation to time scale multiplied by 1.25; system’l,Table I.
by the index of performance) controller settings. The controller is then modified by making use of this information. When implemented using an on-line computer, all elements in the two control loops are internal to the adaptive contrdler computer program, with the exception of the process itself. Theoretical The Index of Performance. Central to any adaptive algorithm is the index of performance (IF’). This is a single number by which the quality of system performance is characerized. Selection of the IP strongly influences the three tasks of the adaptive controller. Controller modification depends on the measured IF’,optimal performance is defined in terms of the IP, and the choice of identification technique is guided by the IP. There are many possible indices of performance, and the ultimate goal of an adaptive controller, that of maintaining the system closed-loop response under possible adverse conditions, must be kept in mind. Hence a performance index particularly suited to chemical process control should be based on requiring the closed-loop response to exhibit certain desired properties. The desired characteristics might be expressed graphically, in the form of system equations, or in terms of the characteristics of a linear (for example, second-order) system. In the time domain these characteristics might include the rise time, settling time, percent of overshoot, etc.; or in the frequency domain, peak frequency, bandwidth, gain and phase margins, etc. However, it is more satisfying, as well as more complete in describing the system, to work with the overall response rather than some single system characteristic such as one of those mentioned above. The approach can be from either the time or frequency domain point of view. Although modern state space control and other time domain methods have enjoyed a certain popularity over the last two decades, recently there has been a trend to reexamine and extend classical, i.e., frequency domain, methods (Foss, 1973). These methods have several advantages including that they are conceptually simpler and are more familiar to the process control industries. The index of performance proposed for use and evaluated in this work is the “shape of the closed-loop transfer function in the complex plane” (see Figure 2). Such an IP has the property that it involves the total system response. Frequency appears only as a parameter in this representation; hence major time constant shifts which
190
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
occur naturally in flow processes (i.e., whenever flowrates change) can be accommodated in a manner similar to that used by Sutherland (1973),who proposed that the product of the process major time constant and the sampling frequency be maintained constant. In the present work, in addition to modifying the sampling rate (a shortened time scale, therefore faster system response, would elicit a faster controller sampling rate and vice versa), the controller time constants would be correspondingly shifted. This change, coupled with appropriate controller gain adjustments to balance process gain changes, would have the effect of potentially returning all parameters in the closed-loop transfer function to a form identical with that before process changes occurred (at least for cases of process gain and overall time scale changes, only). Thus the system characteristic equation is kept constant and the forward loop transfer function as well; hence matching the complex shape of the closed-loop transfer function would keep the closed-loop response constant. It has been demonstrated (Van den Bos, 1970)that, with some a priori knowledge, data points at as few as five properly chosen frequencies are sufficient to identify a process. Thus it should be sufficient for the adaptive controller to compare the measured (complex plane) transfer function with the desired shape at a similar number of points. The comparison function that was utilized as index of performance and minimized in this work is the “sum of the normalized distances between measured and desired points” over all tested frequencies in the complex plane NF
to step 3. (7) Implement the new controller in the actual system. (8) Initiate a new adaptive cycle, i.e. go to step 1.
Step 1 consists of an identification of the closed-loop system by any suitable means. This approach, rather than identifying the process itself, conforms to the chosen IP, i.e., the shape of the closed-loop transfer function, which is the object of interest. Although the form of IP requires only data points representing the closed-loop transfer function, the uncontrolled process transfer function is required as an intermediate step when determining the IP for a “test” controller in the optimization sequence. Points representing the process transfer function, the result of the calculation in step 2, can be subject to large errors not only due to the problems of correlated noise, but because the sensitivity of G, to errors in GCL is high, especially at low frequencies
GCL aGP
- ‘2, -- G, dGCL 1 - GCLGc2Gf
1
(4)
For example, for a system including a PID controller [GC1 = Kc(l + 1/+) and Gc2 = (T$W + l ) ] and a low-pass filter, as frequency approaches zero, GcL, Gc2,and Gf all approach 1and the sensitivity approaches infinity. Thus small experimental errors in GCL would be magnified in GP. However, looking at the inverse operation, the closedloop transfer function is relatively insensitive to errors in G,; indeed the purpose of a feedback loop is to desensitize the system to process parameter fluctuations
i=l
where NF= number of frequencies (6 to 8), wi = weighting factor for calculation at frequency i, GI = value of desired GCL at frequency i, and GM = value of measured GCL at frequency i. A weighting factor may be included to allow those points which normally would be subject to greater noise problems to be weighted less in determining the comparison function, or those points located in a frequency region more dramatically affected by process changes could be weighted more heavily. This index of performance does not require a parametric identification of the system. Furthermore, only a nonparametric identification of the overall closedloop system is necessary, thus avoiding the problems of identifying the process itself while under closed-loop conditions. The Algorithm. Under the assumption of a deterministic single-input/single-outputlinearizable process in a feedback loop with parameters varying slowly compared to identification time, the following adaptive algorithm is proposed: (1)Identify the closed-loop system. (2) Obtain the process transfer function from the identified closedloop transfer function by mathematically “opening the loop” G, =
GCL/ Gc1 1 - GCLGc2Gf
(3) Choose new controller constants via a suitable search method. (4)Mathematically “close the loop” using G, from step 2 and the controller from step 3
GCL =
G,Gc* 1 + GpGclGc2Gf
(3)
(5) Calculate the IP. (6) Determine whether the optimization has reached a suitable stopping point. If not return
At low frequencies G, approaches the process gain, GC1 approaches infinity, and Gc2and Gf approach 1; thus the sensitivity approaches 0. It is claimed, a conjecture which cannot be directly proved but is subject to experimental test, that the errors introduced in mathematically opening the loop (step 2) are generally compensated by mathematically closing the loop (step 4). The process transfer function is used only as an intermediary in the closed-loop adaptation; hence any errors introduced will be subsequently removed and will not be critical. Off-line calculations demonstrate that this result is generally obtained. Step 3, the search for the updated controller, involves the controller parameters as independent variables. One characteristic of flow processes, particularly, is that a simple flowrate change often results in an equal shifting or scaling of the important process time constants. If the process time constants are shifted by a factor P, and P can be determined, and if the controller time constants are each multiplied by this factor, then the result will be the original closed-loop transfer function with every time constant multiplied by P. Furthermore, if the subsequent identification and control action utilizes a sampling time equal to the original sampling time multiplied by 8, the values of the transfer function at these new frequencies should equal the values obtained in the original identification at the original frequencies. Thus, the ideal adaptation to a time scale change in the process, /3, would be an equivalent scale factor change in the controller, as well, allowing the shape of GCL to remain constant despite process changes. If such an approach is to be used,the P must be allowed to change to reflect the new time scaling; i.e., the ideal closed-loop transfer function, GI, must be scaled in frequency by the factor 0.The scale factor is not independent
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 2, 1981
1Ql
Table I. Simulated Processes and Their PID Controllers” system
order
KP
7d
71
7 2
1 2
2DT
1 1
0.5
2 2
1 1
4
7 3
0.2
74
0.1
Kc 4.2 6
71
2 1.5
7D 0.5
0.35
Gains in volts/volt. Time constants in minutes.
of the changes determined in the optimization sequence for the controller time constants; in fact, those changes should directly determine 0.In this work it is proposed that the scale factor be calculated as the geometric mean of the changes in the updated controller time constants; , T,, then Le., if the controller time constants are T ~T, ~...,
p = i=l b(
2)“
where subscript u denotes “updated” and o denotes “original”. Hence, at step 5 in the adaptive algorithm, the index of performance is calculated using 0defined by eq 6. Implementation of this feature has one interesting outcome: viz., not only does the controller adapt to changes in the process, but also the IP adapts to changes in process “time scale”. Obviously process gain changes can be accommodated exactly by reciprocal changes in the gain of the forward controller element, GC1.Hence simpler process changes involving gain changes, time scale change, or a combination of the two, can be handled exactly and in a predictable way. More complicated process changes potentially can be compensated for satisfactorily by more complicated and nonpredictable controller parameter changes. The study of this adaptive controller and an experimental evaluation of some of its properties are the subjects of the remainder of this paper. Implementation The adaptive algorithm was implemented and tested using the facilities of the U.C. Santa Barbara Real-Time Computing Laboratory described elsewhere (Harris and Mellichamp, 1980). To begin with, an appropriate form of controller must be chosen for experimental tests. In this work the assumption has been made that a proportional-integral-derivative (PID) controller is adequate. This form, and its simpler analogs (PI, PD), are frequently used in industry. Bristol (1977) has stated “...there is no proposed digital controller better than the PID controller for the general purpose single-loop control function”. Shinnar (1976) states that a simple PID controller outperforms most of the optimal controllers published. Thus the choice of this family of controllers for use in an adaptive controller seems quite general and, hence, logical; however, the adaptive algorithm is by no means limited in the choice of controller. For a PID controller the number of independent variables which must be specified is three (Kc,TI, T D ) , and
p=
(E)”’
(7)
For P I and PD variants, p is simply set equal to a ratio of updated to original value of T~ or 7 D as appropriate. The remaining element which was used in the process loop was a double exponential filter included before the derivative portion of the controller to smooth the data and prevent noise from causing controller saturation. The filter transfer function is
where 7f = the filter time constant, which is much smaller than the process or controller time constants. The procedure selected to carry out the identification of the closed-loop system (step 1of the algorithm) was the same one described by authors previously; i.e., a multifrequency binary sequence designed by Harris (1978) to contain eight frequencies of interest was used to perturb the setpoint of the controller. (The choice of a piecewise continuous setpoint perturbation requires placing the controller derivative mode in the feedback path, i.e., as Gat to prevent controller output saturation a t the instant of setpoint changes. The use of derivative mode on ‘‘process output only” is a common approach with industrial systems. The proportional and integral modes made up the forward loop, Gc1J The closed-loop transform of the process was obtained through a Fourier transform of the resulting small output deviation. Details are given by Harris (1978). Since only numerical values of the objective function are available, the optimization itself (steps 3-6) must utilize a direct search method. The complex technique of Box (1965) was used, although several other direct search methods yielded equivalent results in preliminary testa. The search was stopped when parameter changes gave no significant improvement in IP value (i.e.,
