Ind. Eng. Chem. Res. 1990,29, 1249-1253 Chem. Process Des. Dev. 1981,20, 147. Elaahi, A.; Luyben, W. L. Alternative Distillation C o n f i a t i o n s for Energy Conservation in Four-Component Separations. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 80. Elaahi, A.; Luyben, W. L. Control of an Energy-Efficient Complex Configuration of Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 368. Freshwater, D. C.; Ziogou, E. Reducing Energy Requirements in Unit Operations. Chem. Eng. Sci. 1976, 1 1 , 215. Frey, R. M.; Doherty, M. F.; Douglas, J. M.; Malone, M. F. Controlling Thermally Linked Distillation Columns. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 483. Lenoff, A. M.; Morari, M. Design of Resilient Processing Plants-I: Process Design Under Consideration of Dynamic Aspects. Chem. Eng. Sci. 1982, 37, 245. Levien, K. L.; Morari, M. Internal Model Control of Coupled Distillation Columns. AZChE J. 1987, 33 (l),83. Linhoff, B.; Dunford, H.; Smith, R. Heat Integration of Distillation Columns into Overall Processes. Chem. Eng. Sci. 1983,38,1175. Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Znd. Eng. Chem. Res. 1987, 26, 2490. Mosler, H. A. Control of Sidestream and Energy Conservation Distillation Towers. Proceedings of the AIChE Workshop on Industrial Process Control, Tampa, FL, 1979. Morari, M.; Faith, D. C. The Synthesis of Distillation Trains with Heat Integration. AIChE J. 1980,26, 916. Null, H. R. Heat-Pumps in Distillation. Chem. Eng. Prog. 1976, 73, 58. O’Brien, N. G.Reducing Column Steam Consumption. Chem. Eng. Prog. 1975, 72, 59. Ogunnaike, B. A.; Lemaire, J. P.; Morari, M.; Ray, W. H. Advanced
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Multivariable Control of a Pilot-Plant Distillation Column. AIChE J. 1983,29 (4),632. Petlyuk, F. B.; Platonov, V. M.; Slavinskii, D. M. Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Znt. Chem. Eng. 1965,5 (6),555. Rathore, R. N. S. Process Resequencing for Energy Conservation. Chem. Eng. Prog. 1982, 78, (12),75. Rathore, R. N. S.; Van Wormer, K. A.; Powers, G. J. Synthesis Strategies for Multicomponent Separation Systems With Energy Integration. AIChE J. 1974,20, 491,940. Rush, F. E. Energy-Saving Alternatives to Distillation. Chem. Ena. h o g . 1980, 76;-(7), 44.Stupin, W. J.; Lockhart, F. J. Thermally Coupled Distillation-A Case History. Chem. Ena. Proa. 1972.68 (10).71. Takama, N.; Kuriyama, T.;Niida, K.; Kinoshita, A.; Shiroko, K.; Umeda, T. Optimal Design of a Processing System. Chem. Eng. Prog. 1982, 78 (9),83. Tedder, D. W.; Rudd, D. F. Parametric Studies in Industrial Distillation. AZChE J. 1978, 24, 303. Tyreus, B. D.; Luyben, W. L. An Economic Study of Multi-Effect Distillation Systems. Hydrocarbon Proc. 1975, 54, 93. Tyreus, B. D.; Luyben, W. L. Dynamics and Control of Heat-Integrated Distillation Columns. Chem. Eng. Prog. 1976, 72, 59. Umeda, T.; Niida, K.; Shiroko, K. A Thermodynamic Approach to Heat Integration in Distillation Systems. AZChE J. 1979,25,423. Yu, C. C.; Luyben, W. L. Design of Multiloop SISO Controllers in Multivariable Processes. Ind. Eng. Chem. Process Des. Dev. 1986, 25,498.
Received for review November 16, 1989 Accepted February 26, 1990
A Regulating and Tracking PID Controller Ricardo J. Mantz* and Eugenio J. Tacconi LEICZ,t Departmento de Electrotecnia, Facultad de Ingenieria, UNLP, CC 91, 1900 La Plata, Argentina
A simple and robust method to design a modified PID controller with good regulating and tracking characteristics is presented. Only two limit cycle measurements of the plant under relay control are required to design the modified PID controller. The design method has shown to work well in a number of simulations of industrial applications. One example is presented. 1. Introduction
The use of PID controllers as industrial regulators is widespread. Engineers and process operators are familiar with them and often have a good feeling toward the effects of their different parameters. However, PID controllers have restrictions because they only have three tuning gains. For instance, they cannot be designed so as to have, simultaneously, an appropriate reduction of disturbances and good transient behavior toward changes in the reference signal. Usually they are adjusted in order to have good disturbance rejection, and frequently a poorly damped pair of complex conjugate poles results, causing undesirable overshoots within the tracking step responses. In recent papers (Gawthrop, 1986,Eitelberg, 1987; Hippe et al., 1987; Mantz and Tacconi, 1989, a slight change in a classical PID structure is analyzed. It consists of weighting the reference signal for each type of action (proportional, integral, and derivative) in a different way (Figure 1). Thus, the input-output transfer function is modified without changing the disturbance rejection characteristic. From now on, Fi = 1is chosen so that the controlled magnitude equals the reference at steady state. t Laboratorio d e Electrdnica Industrial C o n t r o l e Instrumentacidn.
0
The corresponding tracking transfer function of the closed-loop system is -~ -( s -) PID’(s) Gp(s) (1) R(s) 1 + PID(s) Gp(s) where
PID(s) and PID’(s) are the transfer functions of the classical PID and the modified PID controller, respectively. Equation 1 can also be written as Y(s) PID’(s) P U S ) Gp(s) -=(4) R ( s ) PID(s) 1 + PID(s) G,(s) From this expression, the input-output transfer function of the system can be considered as two blocks in cascade connection: one represents the closed-loop system compensated by a classical PID controller, and the other one perfectly cancels the two fixed zeros introduced by this classical PID and locates two other zeros somewhere in the s plane. 1990 American Chemical Society
1250 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 IlmG
I
Figure 1. Modified PID control structure.
When the classical PID controller is adjusted to have a closed-loop system with good disturbance rejection, an underdamped behavior with two complex conjugated dominant poles frequently results, causing undesirable overshoots within the tracking step responses (Hippe et al., 1987; Takahashi et al., 1972). The basic idea is to adjust the Fp.and Fd values in such a way that the zeros of the tracking transfer function, introduced by the modified PID, cancel the annoying effect of the complex closed-loop dominant poles. In this way, it is possible to improve the tracking behavior without modifying the regulating characteristics. In fact, the classical PID is usually designed with TiZ 4Tdhaving its zeros placed on the real negative axis. As these zeros attract the closed-loop poles, if the closed-loop gain is high enough, the system compensated by a classical PID will have one or two additional poles placed close to them. The effect of the complex conjugate dominant poles of the tracking transfer function can be cancelled by the proper Fpand Fdcoefficients. Then, the tracking response might be strongly influenced by the closed-loop poles placed near the zeros of the classical PID. Thus, the overshoot of the response to a reference step would be considerablyreduced or eliminated. A method to design a modified PID controller, based on the knowledge of critical gain and frequency, is carried out in Mantz and Tacconi (1989) by making assumptions that are only valid for minimum-phase systems. However, this is a serious limitation, and the above method cannot be employed in most thermic or chemical processes because of the important role played in these systems by the process time delay. The design method presented in this work combines different strategies in order to obtain the following characteristics: (i) it does not have to be limited to minimum-phase systems; (ii) it only has to require measurements of the plant transfer function, which must be carried out under plant safety conditions; (iii) the number of measurements has to be reduced to the minimum; (iv) it has to be suitable for automatic tuning of PID controllers as described in Astrom (1982) and Astrom and Hagglund (1984). 2. Proposed Method 2.1. Principles of the Proposed Method. In order to assign the Fpand Fd values, it is necessary to estimate the position of the complex conjugate dominant poles of the closed-loop system compensated by a classical PID controller. A simple method to estimate the closed-loop dominant poles from the knowledge of only two points on the Nyquist curve of the open-loop transfer function has been recently discussed by Hagglund and Astrom (1984,1985). In this method, the open-loop transfer function, G(s), is viewed as a map from the s plane to the G plane (Figure 2). If C represents the location of a closed-loop dominant pole in the s plane, this point is mapped by the critical point C’ = -1 in the G plane. Now consider the conformal map A’B’C’of the triangle ABC in the s plane. If AB%’
a
b
Figure 2. Conformal mapping from the s plane to the G plane.
can be approximated by a triangle, the real and imaginary parts of the complex dominant poles are given by ..
od =02 (5) These equations have been obtained in Hagglund and Astrom (1985) by choosing a frequency, w2, so that the A%’ line interests the Nyquist curve orthogonally. Thus, the vectors 1 + G ( j w 2 ) and G(jw2)- G(jw,) can be considered orthogonal between them if w1 is near wz. In order to apply this method to estimate the dominant poles, points A’ and B’ have to be determined experimentally. In Astrom (1982), Astrom and Hagglund (1984), and Hagglund (1981), a simple and robust technique to estimate the points on the Nyquist curve, G( j w ) , has been described. In this technique, by performing closed-loop measurements with a hysteresis relay introduced in the loop, the real and imaginary parts of point Pi(ci,di)on the open-loop Nyquist curve are determined by
where M represents the relay amplitude, Hi the hysteresis width, and Bithe limit cycle fundamental-harmonic amplitude. This technique has been chosen here because it is easy to be implemented and because, by modifying the relay amplitude, the limit cycle amplitude of the plant output can be controlled by performing the measurements under plant safety conditions. Combining this measuring technique with eq 5, the estimation of the dominant poles of the closed-loop system can be performed. This measurement technique is particularly attractive because by combining the automatic determination of two points on the Nyquist curve with some design method based on dominant poles, an autotuning method can be obtained (Hagglund and Astrom, 1985). If the value of the relay hysteresis width is arbitrarily chosen, the frequency corresponding to point A’(c2,d2) (Figure 2) will not be generally close to the damped natural frequency of the closed-loop dominant poles. Thus, the A’C’line will not intersect the Nyquist curve orthogonally. One drawback of the above estimation method is that usually more than two measurements of the total open-loop transfer function have to be perform to satisfy the orthogonality condition. This problem has been solved here by extending the estimation method to any pair of points on the Nyquist diagram. In this case, the following equations result:
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1251 The approximate expressions to estimate the closed-loop dominant poles presented in Hagglund and Astrom (1984) and based on amplitude and phase margins are a special case of eq 7. In that case, w2 and w1 are the frequencies where the Nyquist curve intersects the negative real axis and the unit circle, respectively. Equations 7 can be used with any pair of points on the Nyquist curve, but when the distance among points A‘, B’, and C’ is reduced, the error related to the triangle approximation is improved. The reduction of the distance between points A ’and B ‘also increases the sensitivity of the estimated values of u and wd to measurement errors. Once the real and imaginary parts of the closed-loop complex dominant poles have been estimated, the Fpand F d coefficients of the modified PID’(s) controller (eq 3) can be calculated in order to place its zeros on the estimated closed-loop dominant poles. From eq 3,
By assigning the Fpand Fdvalues in this way, when a change in the system reference occurs, the influence of the complex conjugate dominant poles on the output response is substantially reduced. In order to simplify the mathematical treatment, ideal PID structures have been considered for both classical and modified PID controllers (eqs 2 and 3). In a practical PID controller, one or more additional poles are always included to reduce the derivative kick at the controller output. These poles, usually placed in the high-frequency band, do not modify substantially the plant output. Thus, to design the modify PID controller, the above equations can still be applied without any corrections. 2.2. Practical Considerations. The design of the modified PID controller following the procedure indicated in the previous section requires several measurements. From a practical point of view, it is important to reduce these measurements to a minimum number. The estimation of the closed-loop dominant poles (eq 5 or 7) requires the knowledge of two points of the total open-loop frequency response, G ( j w ) , given by the PID controller and the plant, G,, in cascade connection
G( jo) = PID(j w ) G,( j w )
(9)
These two points can be determined directly by the method described above. For its application, it is obviously required that the a priori design and inclusion be used in the loop of the PID controller. A t least one measurement of the plant G,(jw) is then required to design the PID controller, and a minimum of two other measurements of the total open-loop transfer function will be necessary in order to estimate the closed-loop poles. Besides, if the parameters of the PID controller are modified, new measurements have to be performed. A more interesting possibility suggested here is to apply the measuring method to determine, under plant safety conditions, two points ( a and b ) on the Nyquist curve of the plant transfer function G (Figure 3). These two points can be used to design a ciassical PID controller. Once the parameters of the PID controller are determined, points a and b can be used to calculate the corresponding points A ’ and B’ of the total open-loop frequency response C ( j w ) (Figure 3). Then, eq 7 give an estimation of the closed-loop dominant poles, and the modified PID controller is designed by applying eq 8.
1
”
I
P
Figure 3. Open-loop transfer function and plant Nyquist curves.
It has been mentioned before that the precision of the estimation method depends on the positions of points A’ and B’. Thus, in order to limit the number of measurements to a minimum of two, it is necessary to have a good feeling about how to choose the relay hysteresis width. A relationship and discussion between the procedure described here and that described in Mantz and Tacconi (1989) is made in the Appendix. From this discussion, the following procedure is recommended: A relay without hysteresis is used in the first measurement to determine the critical gain (k,) and frequency (w,) (u(-l/k,,O)). For the second measurement, a hysteresis width (H2)larger or equal to 0.188M/kc is suggested. Thus, the corresponding frequency (wl) and the point coordinates (b(al,&))are determined. In many design methods, the knowledge of points a(l/k,,O) and b(a,,&) is enough to design the classical PID regulator. Then, the corresponding A’ and B’ points of the total open-loop transfer function are calculated. With these points, the position of the closed-loop complex dominant poles is estimated and the design of the modified PID controller is performed. 3. Example As an application example, the same non-minimumphase plant considered in Hagglund and Astrom (1984) is analyzed by computer simulation. Its transfer function is given by
G,(s)
1.65e-lZS 20s 1
-
+
A first measurement, using a relay without hysteresis allowed to determine the critical point u(-l/k,,O), was w, = 0.157 rad/s, l / k c = 0.5 (11) where k, and w, are the critical gain and frequency, respectively. A second measurement, b(al,&),has been performed by using a relay hysteresis width given by H2 = 0.8M/k,. For the point b(al,Pl), the following values have been obtained: u1 = 0.12 rad/s, a1 = -0.55 rad/s, p1 = -0.31 rad/s (12) Assuming that the classical PID controller is designed by Ziegler and Nichols rules, K, = O.6kc = 1.2, Ti = A / W , = 20 S, T,j = ( T / ~ ) w , = 5 s (13) In the PID’(s) derivative action, an additional pole placed at 1.66 Hz has been included. This classical PID controller gives an underdamped behavior response that is normally accepted as a regulating controller because it gives a good rate of change of the controlled variable. Nevertheless, due to the overshoot magnitude, this underdamped behavior cannot be accepted for large changes in the reference signal.
1252 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
-2
I
I
I_.-
-1
I
I
~
..
~
I
I
I
0
1
.
I ._.I .I
1
]...I
100
I ~
200 t (sed
Figure 4. Set-point and step load disturbance responses. (a) Classical PID (b) modified PID.
Since the Ziegler and Nichols method has been used, point A'(c2,d2)of the total open-loop transfer function G(s) is fixed, independent of the plant, and is given by (see Appendix) c2 = -0.6 rad/s, d2 = -0.28 rad/s (14) Point B'(c,,d,) will be given by w1 = 0.12 rad/s, c1 = -0.59 rad/s, dl = -0.5 rad/s (15)
From eq 7, the closed-loop dominant pole estimation is a = -0.07 rad/s, a d = 0.21 rad/s (16) Then the Fpand Fd coefficients are from eq 8 Fp = 0.14, Fd = 0.2
(17)
Figure 4 shows set-point and step load disturbance responses of the corresponding classical control loop with Fp= Fd= 1 (curve a), and with Fpand Fdvalues given by eq 17 (curve b). For the modified PID controller, a step response without overshoot is obtained, but although the rise time is longer, the settling time has been slightly improved. Figure 4 also shows that the modified PID controller presents the same load disturbance response, near the "quarter decay", to that corresponding to the Ziegler and Nichols classical PID. The manipulated variables for both classical and modified PID controllers are shown in Figure 5. When the modified PID controller is used, the derivative kick amplitude is highly reduced. As the step response has a smaller rate of change, the system will have the advantage of requiring a smaller amount of power. I t is important to point out that the modification introduced by Fpand Fddoes not affect the stability and can be used in processes with greater dead times and any kind of plant for which a classical PID controller is usually applied. The stability depends only on the tune of the classical PID controller, and the proposed method is independent of the rules used to tune it, although the proposed method will, of course, inherit the limitation of the PID algorithms. For simplicity, the method has been developed considering an analog controller, but the extension to a digital controller is immediate. The implementation of the modified PID controller does not require more complexity than that involved in usual analog or digital PID controllers. A digital PID controller, being suitable for automatic tunning as described in Astrom (1982) and Astrom
i
l
i
I
I
1
0 50 t (sed Figure 5. Manipulated variable for a unit reference step. (a)
Classical PID; (b) modified PID.
and Hagglund (1984), is more attractive. 4. Conclusions A modified PID controller structure is attractive because the tracking behavior can be improved without modifying the regulating characteristics. In this work, different and useful strategies are combined in order to obtain a simple method to design a modified PID controller. The design only requires two measurements of the plant which are carried out under safety conditions, and its applicability is not restricted to minimum-phase systems. The method has been satisfactorily verified for a variety of plant transfer functions, and the tracking behavior has always been improved. One example is presented.
Acknowledgment We thank the anonymous referees for very helpful comments and suggestions. This work was supported by CIC Pcia. B.A. and CONICET, and Mantz and Tacconi are members of CIC Pcia. B.A. and CONICET, respectively. Appendix. Choice of the Relay Hysteresis Width The designing method of the modified PID controller has been outlined in section 2.1. The procedure described in that section requires at least one plant measurement to design the classical PID. Two other measurements of the total open-loop transfer function to estimate the closed-loop dominant poles are required. Besides, the errors in the dominant pole estimation depend on the position of points A'and B'(Figure 3). Thus, depending on the precision required, additional measurement could be necessary. Fortunately, suitable points to design the PID controller are also suitable to estimate the closed-loop dominant poles. This is due to the fact that in both cases the plant transient behavior is the required information. Thus, the measuring points have to be chosen in order to characterize as best as possible the plant transient behavior. The two points of the frequency response more largely used in classical literature to determine the relative stability and to characterize the closed-loop transient behavior are those corresponding to the amplitude and phase margins. The point that determines the amplitude margin is the intersection between the plant Nyquist curve and the negative real axis and is called the critical point. In many regulating process control problems, the classical PID controller can be designed by using only the information corresponding to this point (critical frequency (oc) and gain ( k c ) ) .
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1253 For the particular case of a minimum-phase plant controlled by a Ziegler and Nichols PID regulator, the classical and modified PID controllers are designed in Mantz and Tacconi (1989) in terms of the critical gain and frequency. The procedure employed in that reference to estimate, in a rough way, the location of the closed-loop dominant poles is related with Bode theorems, and it makes use of the following suppositions: (i) the total open-loop transfer function gain at the critical frequency varies 34.4 dB/ decade, approximately; (ii) the closed-loop system has a pair of complex dominant poles with a natural frequency (w,) near 0.790,; (iii) the closed-loop system response is near the one called “quarter decay” which corresponds to [ = c/w, = 0.21 if it is associated with a second-order system (Takahashi et al., 1972). The Fp and Fd coefficients obtained from the above suppositions are constant, are independent of the plant, and are given by Fp = 0.17, Fd = 0.654 (18) This fair and rather surprising result can be better understood in the context of the present work. A first measurement, using the technique described in section 2, is performed in order to determine the critical point a(l/hc,O). In this case the plant is connected in a feedback loop with a relay without hysteresis to measure the amplitude (B,) and frequency (w,) of the limit cycle fundamental harmonic. If M is the relay amplitude, the critical gain is approximately given by (eq 6) 4M k, = 7 (19)
-
rnl
In order to compare the method with that proposed by Mantz and Tacconi (1989),the regulator PID parameters are determined by applying the Ziegler and Nichols rules. Once the PID controller is designed, from eq 2 and 9, point A’(c2,d2)= G(jwc) of the total open-loop frequency response can be calculated. Actually, for Ziegler and Nichols PID design, point A’is fixed, is independent of the plant, and is given by c2 = -0.6 rad/s, d2 = -0.28 rad/s (14) The estimation of the closed-loop complex dominant poles (eq 7) requires the knowledge of another point G(jw,) = B’(cl,dl). It is possible to determine a fixed point on the G plane such that, if this point corresponds to G(j w l ) = B’( jwl), the suppositions made in the above reference will be approximately verified. The frequency and coordinates of this point are w1 = 0.77wc, c1 = -0.997 rad/s, d, = -0.283 rad/s (20) When these two points, A’(c2,d2)and B’(cl,dl),are used in eq 7, the real and imaginary parts of the estimated complex dominant poles are (r E -0.162~,, wd 0.77~~ (21) With these values, the same modified PID’ controller as in the previous reference is obtained (eq 18),and the corresponding suppositions are closely verified 20(10g IB’I - log IA’I) = -34.4 dB/decade (i) log w1 - log w2 (ii) = 0.206 (iii) w, = 0 . 7 8 7 ~ ~
Thus, assuming the suppositions made in that reference, the method presented here gives the same results and only requires knowledge of the critical point. One advantage of the present method is that a second measurement can be performed in order to check the above suppositions. The second point used here (B’(cl,dl))was given by eq 19, and as the PID controller is already known, the Nyquist point of the plant transfer function b(cul’,Pl’) corresponding to B’(cl,dl) is given by w,’ = O.77wc, al’= -1.69/kc rad/s,
P1’ = -0.148/kC rad/s (23) From eq 6 and 23, the required hysteresis width to perform the second measurement can be determined:
H2 = 0.188M/kC
(24)
For processes with large dead times, points A’ and B’ obtained by using this hysteresis width are usually close together, increasing the measurement errors substantially. Thus, for these kinds of processes, it is advisable to use a hysteresis width larger than the one given by eq 24. The second recommended measurement (b(cul,Pl)) is performed by connecting the plant in a feedback loop with a relay with the chosen hysteresis width. Then, the amplitude (B,) and frequency (wl) of the limit cycle fundamental harmonic are measured and the coordinates of point b are calculated by using eq 6. It is important to point out that, although some approximations are involved, the method works properly because it is enough for the complex conjugate zeros to be assigned near the closed-loop dominant poles to reduce their effects substantially. Literature Cited Astrom, K. J. Ziegler-Nichols Auto-Tuners. Internal Report LUTDF2/(TFRT-3167)/01-025/(1982); Department of Automatic Control, Lund Institute of Technology: May 1982. Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatics 1984, 20, 645-651. Eitelberg, E. A. regulating and tracking PI(D) controller. Znt. J. Control 1987,45, 91-95. Gawthrop, P. Self tuning PID controller: Algorithms and Implementation. IEEE Trans. Autom. Control 1986, AC-3, 201-209. Hagglund, T. A. PID tuner based on phase margin specification. Internal Report LUTDF2/ (TFRT-7224)/ 1-020/(1981); Department of Automatic Control, Lund Institute of Technology: Sept 1981. Hagglund, T.; Astrom, K. J. A new method for design of PID reguDelators. Internal Report LUTDF2/TFRT-7273/1-032/(1984); partment of Automatic Control, Lund Institute of Technology: July 1984. Hagglund, T.; Astrom, K. J. Automatic tuning of PID controllers based on dominant pole design. Presented at the Workshop on Adaptive Control of Chemical Processes; IFAC: Frankfurt, 1985. Hippe, P.; Wurmthaler, Ch.; Dittrich, F. Comments on “A regulating and tracking PI(D) controller”. Int. J. Control 1987, 46, 1851-1856. Mantz, R. J.; Tacconi, E. J. Complementary rules to Ziegler and Nichols ones for a regulating and tracking controller. Int. J. Control 1989,49, 1465-1471. Takahashi, Y.;Rabins, M.; Auslander, D. Scalar input-output linear systems and feedback control. In Control and Dynamic System; Addison-Wesley: Reading, MA, 1972.
Received for review October 20, 1989 Revised manuscript received March 2, 1990 Accepted March 14, 1990
