Self-tuning IMC-PID control with interval gain and phase margins assignment

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 3, MAY 2001 535

Brief Papers_______________________________________________________________________________

Self-Tuning IMC-PID Control with Interval Gain and Phase Margins AssignmentW. K. Ho, T. H. Lee, H. P. Han, and Y. Hong

Abstract—The idea of pole-region assignment is extendedto interval gain and phase margin assignment. The internalmodel control proportional-integral-derivative IMC-PID designis examined from the frequency domain point of view. Equationsfor typical frequency domain specifications such as gain margin,phase margin and bandwidth are derived for the IMC-PID design.The gain and phase margins are monitored in real time and aself-tuning controller with interval gain and phase margin assign-ment is proposed. An implementation example in the laboratoryis also given.

Index Terms—Adaptive control, process control, process moni-toring, proportional control, robustness.

I. INTRODUCTION

CONTROL system design using pole-placement iswell-known [1]. It yields a unique solution for the

controller. However, a unique solution does not allow anyflexibility. Inevitably, there will be noise in the control system.In self-tuning control, the plant parameters obtained fromidentification varies from sample to sample and a pole place-ment design will give a time varying controller that variesfrom sample to sample because of noise. On the other hand,specifying a region for the closed-loop poles has the propertythat as long as the closed-loop poles are within the specifiedregion, there is no need to retune or vary the controller. Aself-tuning controller that only self-tunes when the closed-looppoles are outside the specified region can be implemented.The result is a piece-wise time invariant self-tuning controllerwith only occasional retuning. The implementation and sta-bility issues will become much simpler compared to that of acontinuously varying self-tuning controller. Therefore insteadof pole-placement where the closed-loop poles are placed atspecific locations, the idea has been extended to pole-regionassignment where the closed-loop poles are restricted to aregion. Such works are given in [2]–[6].

Pole-region assignment is also related to robust control. Inrobust control, a plant with uncertain parameters is given. Thelower and upper bounds of the parameters are also given. Acontroller may be found such that all the closed-loop polesare located in a specified region—a pole-region assignmentproblem—and remain in this region if the plant parametersremain within the upper and lower bounds [7], [8].

Manuscript received February 3, 1998. Manuscript received in final form Jan-uary 17, 2001. Recommended by Associate Editor D. Magee.

The authors are with the Department of Electrical Engineering, National Uni-versity of Singapore, Singapore 119260 (e-mail: [email protected]).

Publisher Item Identifier S 1063-6536(01)03368-1.

We have traditionally designed control systems using eithertime domain techniques to meet time-domain specification suchas closed-loop time constant, or frequency domain techniquesto meet frequency domain specifications such as gain and phasemargins and bandwidth [9]. Each approach has its advantages.Different techniques focus on different attributes of the controlsystem. For example, it is typical to measure the robustness ofthe system in the frequency domain using gain and phase mar-gins [10], [11]. This paper extends the idea of a pole-region as-signment for the closed-loop poles to an interval assignment forthe gain and phase margins in the frequency domain where aninterval, instead of specific values for the gain and phase mar-gins, is specified. The advantage of a piece-wise time invariantself-tuning controller is retained. The robust control interpre-tation is even more meaningful as gain and phase margins aretraditional indicators of stability robustness.

According to a survey [12] of the state of process controlsystems in 1989 conducted by the Japan Electric MeasuringInstrument Manufacturer’s Association, more than 90% of thecontrol loops were of the proportional-integral-derivative (PID)type. PID controllers are widely used and many formulas havebeen derived to tune the PID controller. Among them, the in-ternal model control (IMC) formula is well-known [13]–[18].The IMC-PID formula is attractive to industrial users becauseit has only one tuning parameter, , which is related to theclosed-loop time constant. This makes it easy and convenientto tune the PID controller to meet specified time domain perfor-mance.

The contributions and limitations of this paper are as follows.The IMC-PID design is examined from the frequency domainpoint of view. Equations for typical frequency domain specifica-tions such as gain and phase margins and bandwidth are derivedfor the IMC-PID design. Equations for real-time monitoring ofthe gain and phase margins of a PID control system are alsoderived. Using the equations derived, a self-tuning IMC-PIDcontroller with interval gain and phase margin specifications isproposed. A laboratory implementation example is also given.This paper considers the commonly used controller and plantmodel in process control—the PID controller and the stablefirst-order plus dead-time plant model—because of their wide-spread usage. Theoretically, and (the gain and phasemargin, respectively) only define two points and cannot guar-antee that all parts of the Nyquist curve do not encircle the point

as required by the Nyquist stability theorem [19],[20]. However, in practice, the gain and phase margin designtechnique is well accepted [1], [8]–[10], [19], [20]. Examplesare given for an unstable plant in [21] to show that gain and

1063–6536/01$10.00 © 2001 IEEE

536 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 3, MAY 2001

phase margins are not good indicators of stability robustness.The design of controller for the unstable plant should be con-sidered carefully and separately from the class of common plantmodels and controllers. The results in this paper do not apply tothe class of unstable plants.

II. FREQUENCY DOMAIN CHARACTERIZATION OF THE

IMC-PID CONTROL SYSTEM

In this section, the IMC-PID design is examined from the fre-quency domain point of view. Equations for typical frequencydomain specifications such as gain and phase margins and band-width are derived for the IMC-PID design.

A. The IMC-PID Design

The IMC-PID controller has been discussed elsewhere. In thissection, only the equations necessary for the derivation of itsfrequency properties such as gain and phase margins and band-width are given.

The PID controller is given as

(1)

The IMC-PID formula is given as [15]

(2)

(3)

(4)

for the first-order plus dead-time process model

(5)

where the tuning parameter, , is related to the closed-looptime constant as follows. If we approximate the deadtime in theprocess model of (5) with the Pade approximation [20]

then using (2)–(4), the closed-loop transfer function with thePID controller is approximately given as

(6)

B. Gain Margin, Phase Margin and Bandwidth

Analytical expressions for the gain and phase margins andthe bandwidth can be obtained as follows. Denote the gain andphase margins by and , respectively. From the basic def-initions of gain and phase margin, the following set of equationsis obtained:

(7)

(8)

(9)

(10)

where the phase margin is given by (7) and (8), and the gainmargin by (9) and (10). The frequency,, at which the Nyquistcurve has a magnitude of 1 (0 dB) may be considered as thebandwidth of the system.

Substituting (1) and (5) into (7)–(10) gives

(11)

(12)

(13)

(14)

In general, (11)–(14) can be solved numerically but not analyti-cally because of the function (we will show that theseequations can be solved approximately when needed in Sec-tion III). However, for the IMC-PID design, the equations canbe solved analytically and its frequency domain characteristicscan be obtained explicitly. Substituting (2)–(4) into (11)–(14)give

(15)

(16)

(17)

(18)

Solving (18) gives a constant and for convenience is denoted as

(19)

C. Interpretations of the Equations

Equations (15)–(18) give explicit analytical relations be-tween the tuning parameter, , gain margin, , phasemargin, , and bandwidth, . From the equations, inter-pretations can be made that are necessary for the design ofthe self-tuning IMC-PID control with interval gain and phasemargins.

1) The gain and phase margins for the IMC-PID design arerelated. Substituting (16), (17), and (19) into (15) gives

(20)

Equation (20) gives the achievable gain and phase mar-gins and Fig. 1 shows the curve for the above relationship.For the IMC-PID design, only gain and phase margincombinations along the curve can be obtained. Combina-tions of gain and phase margins outside the curve are notachievable.


Fig. 1. Gain and phase margins for IMC-PID design.

2) Using (17) and (19), the tuning parameter,, for theIMC-PID controller can be related to the gain margin,

, by

(21)

The curve in Fig. 1 is calibrated in terms of which isrelated to the closed-loop time constant as shown in Equa-tion (6). The curve in Fig. 1 clearly shows the tradeoff be-tween performance and robustness for the IMC-PID de-sign. Specifying a small (fast closed-loop time con-stant) gives small gain and phase margins and specifyinga large (slow closed-loop time constant) gives largegain and phase margins. From a control design point ofview, Fig. 1 enables the user to see this tradeoff and thenselects the operating point on the curve that will meet hisrequirements of performance (closed-loop time constant)and robustness (gain and phase margin).

3) Instead of using as the tuning parameter, the IMC-PIDcontroller can also be designed from the gain margin spec-ification. Substituting (21) into (2) gives the IMC-PIDformula for in terms of

(22)

Note that the IMC-PID formula for (3) and (4) aregiven in terms of only the process parameters,and ,and are independent of the specificationsand . Todesign from a given phase margin specification, one candetermine from Fig. 1 or, equivalently, numericallysolve for from (20). Then, can be computed from(22).

4) Instead of using as the tuning parameter, the IMC-PIDcontroller can also be designed using bandwidth,, as

the tuning parameter. From (16), the relation betweenand is given as

This approach can be useful for a noisy system.5) The commonly used first-order plus dead-time process

model is considered in this paper. If this is a poor modelof the real process, then a larger gain and phase marginmay have to be specified.

The observations mentioned above form the basis of the self-tuning IMC-PID control described in the next section.

III. SELF-TUNING IMC-PID CONTROL WITH INTERVAL GAIN

AND PHASE MARGINS ASSIGNMENT

The proposed algorithm is as follows. An interval of the curvein Fig. 1 is specified. A nominal gain and phase margin pair onthe interval is also specified. The nominal gain margin (phasemargin) can be simply chosen as the average value of the gainmargins (phase margins) at the end points of the interval andonce the nominal gain margin (phase margin) is chosen, thenominal phase margin (gain margin) is given. During opera-tion, the system gain and phase margin are monitored in realtime. When the gain or phase margin falls outside the specifiedinterval, the controller self-tunes to give the nominal gain andphase margin. Otherwise, the controller is fixed.

The gain margin, , and phase margin, , of the systemcan be monitored using (11)–(14). To do so, we have to solvefor in (12) and in (14). From (12), is given by

(23)

where

Knowing , the phase margin, , can be obtained from (11).To monitor the gain margin, , we have to solve for in

(14). Summing the three functions in (14) gives

for (24)

and

for (25)

where

and

Equations (24) and (25) can be solved for numerically butnot analytically due to the presence of the function. Forreal-time monitoring of the gain and phase margins it is neces-sary to have an analytical solution. An approximate analyticalsolution can be obtained for using a least squaresapproximation. Using a linear model, , the ob-jective function is given as


Fig. 2. Approximation of thearctan function (approximation: dashed line; exact: solid line).

The solution for to give the minimum can be obtained from

giving

Fig. 2 shows the approximation.The identity

can be used for and . The approximation for thefunction is therefore given as

(26)

Using (26), (24) and (25) can now be expressed as

(27)

where the coefficients to are given in Table I for the threecases in (26).

Equation (27) is either a third- or fourth-order equation andthere are analytical solutions given in mathematics handbooks[22]. Equation (27) has to be solved for four different sets ofcoefficients ( to ) as listed in Table I giving a number ofsolutions for . Considering that is necessarily positive real,

TABLE ICOEFFICIENTS OF(27)

NOTE THAT � = �T T ; � = �T + �T � T T ; � = T + T � �

the valid solution can be obtained by substituting each positivereal solution for into the residue function [from (14)]

(28)


Fig. 3. The gain and phase margin of the system can be estimated by interpolating between identified points on the Bode Plot.

Fig. 4. Pairs of band-pass filters and recursive least-square estimators can beused to identify points on the Bode Plot.

and choosing the value that gives the smallest. Knowing ,the gain margin, , can be obtained from (13). An example inthe next Section will clarify matters.

An alternative method of estimating the gain margin, ,and phase margin directly from the Bode Plot can be im-plemented. The details are given in [23]. Only the key ideas aregiven here. Points on the Bode Plot (see Fig. 3) can be estimatedusing pairs of band-pass filters and recursive least-square esti-mators as shown in Fig. 4. Once a number of points on the BodePlot are known, and can be obtained from interpolation.In practice, the two methods of estimating the gain margin,,and phase margin given in this paper can be used to validateeach other by comparing the results before using the new esti-mates.

Fig. 5. Coupled-tanks system.

IV. I MPLEMENTATION EXAMPLE

The self-tuning IMC-PID controller was tested in the labora-tory on a coupled-tank as shown in Fig. 5. The controlled vari-able is the liquid level, , in Tank 2.

Any real-time identification technique that can estimate thesimple first-order plus dead-time process model of (5) can beused. In this implementation the continuous time least squareestimator given in [24] was used. Noise, , in the systemwas first estimated. It was taken to be the difference betweenthe maximum and minimum values of the steady-state processoutput when the control signal is held constant [25].

The time when a set-point change is made to the time whenis estimated as , the dead-time. Consider the

first-order model of (5) (without dead-time):


Fig. 6. Real-time experimental results.

where . The gain, , and time constant,, isestimated as follows. Introduce the low-pass filter

The transformed model whereis replaced by is given as

(29)

where

(30)

(31)

Sampling of all variables in (29) and application of the standardrecursive least squares estimation algorithm to estimateisobviously possible

where is the covariance matrix and is the forgettingfactor. An estimate of and is obtained from and (30)and (31). In this implementation, the sampling interval is chosenas 5 s and and are chosen as 50 s and 0.99, respectively.Once the model of (5) is estimated, the phase margin of thesystem was monitored on-line using (11) and (23), and the gainmargin, (13), (27).

With the help of Fig. 1, the intervals for gain and phase marginwere specified as 1.5 to 2.5 and 52to 71 , respectively. Thenominal gain margin is chosen as 2, the mean of 1.5 and 2.5.From (20) and (21), the nominal phase margin is then given as65 and . When the system gain or phase marginfell outside the interval, the IMC-PID controller will self-tune


to give , and using (2)–(4) and(21). The phase margin of the system was monitored on-lineusing (11) and (23), and the gain margin using (13) and (27).The results are given in Fig. 6. Notice that the gain margin curvestayed within the specified interval of 1.5 to 2.5 from 12:30pmto 12:44pm and the IMC-PID controller was fixed. At 12:44pm,a valve in the coupled-tank was manually tightened to change itsdynamics. Notice that the gain margin curve went beyond 2.5 atabout 12:46pm and 12:48pm. On both occasions the IMC-PIDcontroller self-tuned and the gain and phase margin were resetto 2 and 65, respectively. From 12:48pm to 1:08pm, the gainand phase margin stayed with the specified interval of 1.5 to 2.5and 52 to 71 , respectively, and no retuning was necessary. At1:08pm, a valve was manually opened to change the dynamics ofthe coupled tank. At about 1:10pm and 1:13pm the gain marginwent outside the specified interval and on both occasions theIMC-PID controller was self tuned to reset the gain and phasemargin to 2 and 65, respectively. Throughout the experiment,the phase margin curve was within the specified interval andtherefore did not trigger any retuning. The process model iden-tified at 1:30pm was

and the PID controller was given as

giving the gain margin and phase marginas shown in Fig. 6.

Notice that in the derivation of (11), (13), (23), and (27) wehave not restricted them to the IMC-PID design. They can beused to monitor the gain and phase margin of any system witha PID controller and a first-order plus dead-time process modelwhich is widely in the process control industry. Typically, timedomain information such as the set-point, the controller outputand the process output signals are presented on-line in con-trol systems for the monitoring of performance and robustness.These signals are typically plotted on terminals for inspectionby operators. It is highly desirable to supplement these plots oftime domain signals with a plot of the gain and phase marginas was done in Fig. 6. Gain and phase margin are tradition fre-quency domain measures of stability robustness. It is importantto know the stability margins as the process model used is oftenan approximation of the system dynamics.

V. CONCLUSION

The IMC-PID design is examined from the frequencydomain point of view. Equations for designing the IMC-PIDcontroller from typical frequency domain specifications such asgain margin, phase margin or bandwidth are derived. Equationsfor real-time monitoring of the gain and phase margin ofa PID control system are also derived. Using the equationsderived, a self-tuning IMC-PID controller with interval gainand phase margin specifications is proposed. The advantage isthat a piece-wise time invariant self-tuning controller with only

occasional retuning and the implementation and stability issuewill become much simpler compared to that of a continuouslyvarying self-tuning controller. A laboratory implementationexample is also given. The controller only retuned when theplant dynamics was changed. Otherwise, the controller wasfixed.

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Self-tuning IMC-PID control with interval gain and phase margins assignment