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20EI501-Process Control

Department: Electronics and Instrumentation Engineering

Batch/Year: 2021-25/III

Created by:

Ms. K.R. Chairma Lakshmi AP/EIE

Table of Contents

1. COURSE OBJECTIVES

2. PRE REQUISITES

4. SYLLABUS

SUBJECT CODE : 20EI501

SUBJECT NAME: PROCESS CONTROL

L T P C :3 2 0 4

UNIT I PROCESS MODELLING AND DYNAMICS

Need for process control – Mathematical Modeling of Processes: Level, Flow, Pressure and Thermal processes – Continuous and batch processes – Interacting and Non-Interacting system - Self regulation – Servo and regulatory operations – Lumped and Distributed parameter models – Heat exchanger – CSTR

– Linearization of nonlinear systems.

​

UNIT II FINAL CONTROL ELEMENTS

Actuators: Pneumatic and electric actuators – Control Valve Terminology - Characteristic of Control Valves: Inherent and Installed characteristics - Valve Positioner – Modeling of a Pneumatically Actuated Control Valve – Control Valve Sizing: ISA S 75.01 standard flow equations for sizing Control Valves – Cavitation and flashing – Control Valve selection

​

UNIT III CONTROL ACTIONS

Characteristic of ON-OFF, Proportional, Single speed floating, Integral and Derivative controllers P+I, P+D and P+I+D control modes – Practical forms of PID Controller – PID Implementation Issues: Bumpless, Auto/manual Mode transfer, Anti-reset windup Techniques – Direct/reverse action.

​

UNIT IV PID CONTROLLER TUNING

PID Controller Design Specifications: Criteria based on Time Response and Criteria based Frequency Response - PID Controller Tuning: Z-N and Cohen-Coon methods, Continuous cycling method and Damped oscillation method, optimization methods, Auto tuning – Cascade control – Feed-forward control

SYLLABUS

UNIT V MODEL BASED CONTROL SCHEMES

​

Smith Predictor Control Scheme - Internal Model Controller – IMC PID controller –

- Three- element Boiler drum level control - Introduction to Multi-loop Control

Schemes – Control Schemes for CSTR, and Heat Exchanger - P&ID diagram.

4. Course Outcomes

5. CO-PO/CO-PSO MAPPING

6. Lecture Plan-UNIT-III

7. ACTIVITY BASED LEARNING

​

1. Topic: You are a control engineer working for a company when your optimal controller breaks down. As a backup, you figure that by using coarse knowledge of a classical method, you may be able to sustain development of the product. After adjusting the gain to one set of data taken from a controller, you find that your ultimate gain is 4.3289.

​

From the adjusted plot below, determine the type of loop this graph represents;

then, please calculate K_c, T_i, and T_d for all three types of controllers.

Activity: Cogitate and connote

Students are made to analyze the response curve given, and using their feedback control system tuning knowledge, they can find out the appropriate values for the adjustable parameters of the controller.

2. Topic: Your partner finds another set of data after the controller breaks down and decides to use the Cohen-Coon method because of the slow response time for the system. Luckily the response curve was obtained earlier and is illustrated below. From this data he wanted to calculate K_c, T_i and T_d. Help him to determine these values. Note that the y-axis is percent change in the process variable.

​

Activity: Cogitate and connote

ACTIVITY BASED LEARNING

3. Topic: There are several ways to tune the parameters of a PID controller. They involve the following procedures. For each, name the procedure and explain how the given measured information is used to pick the parameters of the PID controller.

Case (a). The controller is set to P only, and the system is operated in "closed-loop", meaning that the controller is connected and working. The gain is tuned up until a resonance is obtained. The amplitude and frequency of that resonance is measured.

​

Case (b). The system is kept in "open-loop" mode, and a step-function change is manually made to the system (through a disturbance or through the controller itself). The resulting response of the system is recorded as a function of time.

Activity : Think and Act

Students can be asked to observe keenly the procedure of two cases and allowed to find the tuning method adopted before going into finding the parameters of the controller.

8. Lecture Notes-Unit-IV

Evaluation criteria

To achieve an optimum response for all the controlled process, selection of type of feedback controller(i-e P,PI,PID) and adjusting the parameters of the selected controller(i-e Kp, KI, KD) for a particular process is important.

​

Consider a general closed loop system as shown in fig.

When the load or setpoint changes, the response of the process deviates from the desired value and the controller tries to bring the output again close to the desired setpoint. The following graph shows the response of a controlled process to a unit step change in the load when different controllers are used.

The different controllers have different effect on the response of the controlled process. It clearly demands some performance criteria for the selection and tuning of controller.

The following the points are considered for the achieving the desired setpoint value in a closed loop system.

​

What type of feedback controller should be used to control the given process

How do we select the best values for the adjustable parameters of a feedback

controller.

What performance criteria should be used be used for the selection and tuning of controller.

There is variety of performance criteria used are such as Keep the maximum deviation (error) as small as possible. Achieve short settling time

Minimize the integral of errors until the process has settled to the desired setpoint.

Different performance criteria leads to different control design will be discussed in the following section.

​

Performance criteria

some basis for the comparison of

To start with performance criteria, establish alternative controller designs.

Consider the different feedback control systems producing the two closed loop response as in the below figure.

Response A has reached the desired level of operation faster than B.

If the criterion for the design of controller had been “return to the desired level of operation as soon as possible” then select the controller which gives the closed loop response of type A.

If the criterion had been “keep the maximum deviation as small as possible” or “return to the desired level of operation and stay close to it in the shortest time”, select the controller which gives the closed loop response of type B.

For the process control applications, any one of the following criteria is used Steady state performance criteria

Dynamic response performance criteria Simple performance criteria

​

Time integral performance criteria

​

I. Steady state Performance criteria

The principle steady state criterion usually is zero error at steady state. The proportional controller cannot achieve zero steady state error, while a PI controller can.

Also for the proportional control, the steady state error(offset) tends to zero as Kp tends to infinity(Proportional gain-Kp)

II. Dynamic response performance criteria

The evaluation of the dynamic performance of a closed loop system is based on two types of commonly used criteria

Simple performance criteria: Criteria that uses only few points of the response. They are simpler but only approximate.

Time Integral performance criteria: Criteria that uses the entire closed loop response from time t=0 to t= very large value. These are more precise but more cumbersome to use.

​

1.Simple performance criteria

The simple performance criteria are based on some characteristic features of the closed loop response of a system. The most often quoted are overshoot, rise time, settling time, decay ratio and frequency of the transient.

Decay ratio= C/A Overshoot=A/B

Every one of the above mentioned characteristics of closed loop system may be used by the designer as the basic criterion for selecting the controller and the values of its adjusted parameters.

Every one of the above mentioned characteristics of closed loop system may be used by the designer as the basic criterion for selecting the controller and the values of its adjusted parameters.

The controller is designed in order to have minimum overshoot or minimum settling time and so on. One simple characteristic does not describe the desired dynamic response. Usually, we require more objectives to be satisfied (i-e minimum overshoot, minimum settling time etc )

Unfortunately, controller design based on multiple criteria lead to conflicting response characteristics. For example, if PID controller is used by decreasing the value of the overshoot (through a decrease in the value of gain Kp) which will increase the settling time. Such conflicts will always arise while using simple design criteria.

From all these performance criteria, the decay ratio has been the most popular by the practicing engineers. Specifically, experience has shown that a decay ratio C/A=1/4 is a reasonable settling time. This criterion is usually known as the one quarter decay ratio criterion.

​

One quarter decay ratio criterion

It is advantage of being readily measurable, as it is based on only two points on the step response. The measure of decay ratio is found by adjusting the control loop

until the deviation from the disturbance is such that each deviation peak is down by one quarter from the preceding peak. In this case the actual magnitude of the deviation is not included in the measure, nor the time between peak to peak. In this sense neither duration nor magnitude of deviation is directly involved in quarter amplitude criterion.

​

Case (i) In case of cyclic underdamped response, the most critical element is

sometimes a combination of duration and deviation, which must be minimized.

Above figure is a type of cyclic response. The system is adjusted to make each peak down by one quarter of the previous peak. Thus, if minimum deviation occurs at one loop setting and minimum duration at another, then neither is optimum.

One type of optimum measure of quality in these cases is to minimize the net area of the deviation as a function of time. Below Figure shows the minimum area criterion for cyclic response that adjusts the process control loop until the net area is

minimum. The sum of the shaded area can be expressed as

​

Where A= area of deviation b= measured value

A = _∫ r − b dt

r= setpoint value

ITAE is expressed by the following equation

α

2. Time integral performance criteria

The shape of the complete closed loop response from t=0 until steady state has been reached, could be used for the formation of a dynamic performance criterion. Unlike the simple performance criteria which use only isolated characteristics of the dynamic response (eg. Decay ratio, settling time) the criteria of this category are based on the entire response of the process. The integral criteria have the advantage of being more precise, i-e more than one combination of controller settings will usually give a ¼ decay ratio, but only one combination will minimize the respective integral criteria.

​

Integral of the square error(ISE)

​

ISE is expressed by the following equation

∫

2

ISE = e (

t)dt

0

Where e(t)= Ysp(t)-Y(t) is the deviation (error) of the response from the desired

setpoint.

ISE criteria gives more weight to large deviations. Integral of the absolute value of the error(IAE)

α

IAE is expressed by the following equation

IAE = _∫ e(t)dt

0

Integral of the time weighted Absolute error(ITAE)

α

ITAE = _∫ t e(t)dt

0

ITAE criteria weights deviations more heavily as time increases.

The problem of designing the ‘best’ controller can now be formulated as follows:

Select the type of controller and the values of its adjusted parameters in such a way as to minimize the ISE, IAE or ITAE of the system’s response.

The use of above three criteria depends on the characteristics of the system which has to control and some additional requirements impose on the controlled response of the process.

​

The following are some general guidelines

To strongly suppress large errors, ISE is better than IAE because the errors are squared and thus contribute more to the value of the integral.

For the suppression of small errors, IAE is better than ISE because when the small error is squared, it become even smaller.

To suppress large error that persist for longer times, the ITAE criterion will tune the controllers better because the presence of large ‘t’ amplifies the effect of even small errors in the value of the integral.

The shape of the expected closed loop response when the controller parameters are tuned using ISE, IAE and ITAE criteria

Two points to remember

​

Different criteria lead to different controller designs.

For the same time integral criterion, different input changes lead to different designs.

​

Selection of feedback controllers

The systematic way of selecting one of the popular feedback controllers for a given process can be stated in three steps:

​

Define an appropriate performance criterion i-e ISE, IAE or ITAE

Compute the value of the performance criterion using P or PI or PID controller with the best settings for the adjusted parameters Kp, KI, KD

​

Select the controller which gives the best value of the performance criteria.

​

Drawbacks

​

It is very tedious

It relies on models (transfer functions) for the process, sensor and final control element which not be known exactly.

It incorporates certain ambiguities as to which is the most appropriate criterion and what inputs to consider.

To select the most appropriate type of feedback controller, qualitative considerations of P,I,D is taken into account.

​

Proportional control

Accelerates the response of controlled process.

​

Produces an offset

Integral control

​

Eliminates any offset

​

Produces sluggish, long oscillating responses

If the gain Kp increases to produce faster response, the system becomes more oscillatory and may be led to instability.

​

Derivative control

​

Anticipates future errors and introduces appropriate action

​

Introduces a stabilizing effect on the closed loop response of the process.

It is clear from the above points that a three mode PID controller should be the best. It offers the highest flexibility to achieve the desired control response by having three adjustable parameters. It introduces a more complex tuning problem since three parameters to be adjusted.

​

The following rules are adopted to select the most suitable controller.

Simple proportional controller can be used if small offset is acceptable. If the process has an interacting action(1/s term in the transfer function), proportional control does not exhibit offset. Therefore for gas pressure or liquid level control it is used.

A PI controller should be used when P control alone cannot provide small steady state errors. For flow control PI is used. The response of flow control is fast. The speed of the closed loop system remains satisfactory despite the slow down caused by the integral control mode.

PID controller is used to increase speed of closed loop response and retain robustness. For a multicapacity process whose response is very sluggish, addition of PI makes more sluggish. In such case, addition of derivative control with the stabilizing effect allows the use of higher gains which produce faster response without excessive oscillations. For temperature and composition control it is used.

Controller Tuning

The selection of best values for the adjustable parameters of a feedback controller is known as controller tuning problem.

​

There are three general approaches used for the tuning of controllers

Use simple criteria such as one quarter decay ratio, minimum settling time, minimum largest error and so on. Such an approach is simple and easily implementable on an actual process. Usually, it provides multiple solutions. Additional specifications on the closed loop performance will then be needed to break the multiplicity and select a single set of values for the adjusted parameters.

Use time integral performance criteria such as ISE,IAE or ITAE. This approach is rather cumbersome and relies heavily on the mathematical model(transfer function) of the process. Applied experimentally on an actual process, it is time consuming.

​

Use semi empirical rules which have been proven in practice.

​

Empirical Tuning Methods

Process reaction curve method Ultimate cycling method

Damped oscillation method

​

Process reaction curve method (Cohen and coon Method)

The most popular of the empirical tuning methods known as the process reaction

curve method developed by Cohen and coon.

The basic approach is to open the process control loop so that no control action (feedback) occurs. This is done by disconnecting the controller output from the final control element. All the process parameters are held at their nominal values. This method can be used only for systems with self regulation.

This is also called as open loop Transient response method.

Consider the control system below which has been opened by disconnecting the controller from the final control element.

Introduce a step change of magnitude A in the variable ‘C’ which actuates the final control element. In the case of a valve, C is the stem position. Record the value of the output with respect to time. The curve Ym(t) is called the process reaction curve.

​

Between Y_mand C, the following transfer function is obtained

The above equation shows that the process reaction curve is affected not only by the dynamics of the main process but also by the dynamics of the measuring sensor and final control element.

Cohen and coon observed that the response of most processing units to an input change had a sigmoidal shape, which can be approximated by the response of a first order system with dead time.

C(s)

Y (s)

m

= G_f (s)G_p (s)G_m (s) (1)

PRC

G (s)=

Where s is the slope of the sigmoidal response at the point of inflection

​

t_d= time elapsed until the system responded.

C(s)

Y (s)

m

=

G_PRC (s)=

_Ke −t_ds

Which has three parameters

Static gain K Dead time td Time constant τ

It is easy to estimate the values of three parameters _{K =} Output(at steady state) ₌ B

Input(at steady state) A

s

τ = ^B

Cohen and coon used the approximate model of equation (2) and estimated the values of the parameters K, td and τ as indicated above.

Then they derived expressions for the best controller settings using load changes and various performance criteria such as

​

One quarter decay ratio

​

Minimum offset

​

Minimum integral square error(ISE)

The results of their analyse are summarized below

​

Proportional controller

Corrections to Kp are sometimes used to obtain quarter amplitude criterion of the response

d

p

τ Kt

Proportional gain = K =

τ ⎡ ^t_d ⎤

d ^⎣

Proportional Integral controller

⎦

K_p = _{Kt ⎢}1 + _{3τ ⎥}

⎦

τ ⎡ ^t_d ⎤

d ^⎣

K_p = _{Kt ⎢}0.9 + _{12τ ⎥}

τ

​

Proportional Integral Derivative controller

20t

9 +

d

30 + 3t_d

T_I = t_d

⎦

4τ ^⎥

⎢ ₃

K =

τ ⎡ 4 ₊ ^t_d ⎤

Kt

d ^⎣

p

τ

I

8t

13 + ^d

32 + ^6td

^{T = t} d ^τ

τ

2t

4

11 + ^d

T_D = t_d

Remarks

The controller settings given above are based on the assumption that the first order plus dead time system is a good approximation for the sigmoidal response of the open loop real process.

If the approximation is poor, then cohen and coon settings should be viewed as first guesses needing certain online correction.

It is noticed that all physical processes encountered in a chemical plant are simple first order or multicapacity processes whose response has the general overdamped shape. The oscillatory underdamped behaviour is produced mainly by the presence of feedback controllers. Therefore when the loop is open the response takes the sigmoidal shape of a overdamped system.

The proportional gain of PI controller is lower than that of P controller. This is due to the fact that integral control mode makes the system more sensitive and thus the gain value needs to be more conservative.

The stabilizing effect of derivative control mode allows the use of higher gain in the PID controller.

​

Ziegler Nichols Method (Ultimate cycle Method)

The process reaction curve was originally developed by Ziegler Nichols. The corrections were developed by cohen and coon (when the quarter amplitude criterion is indicated) which become popular later on.

Ziegler- Nichols have developed another method of controller setting assignment that has come to be associated with their name. This technique also called the ultimate cycling method.

It is based on adjusting a closed loop until steady oscillations occur. Controller settings are based on the conditions that generate the cycling. This method is based on frequency response analysis.

Unlike the process reaction curve method which uses data from the open loop response of a system, the Ziegler Nichols tuning technique is a closed loop procedure.

​

The following steps are followed in Ziegler Nichols method

​

Bring the system to the desired operational level(design conditions).

Using Proportional control only and with the feedback loop closed, introduce a setpoint change and vary the proportional gain until the system oscillates continuously. The frequency of continuous oscillation is the cross over frequency ‘ωco’

Let ‘M’ be the amplitude ratio of the system’s response at the cross over frequency.

​

Compute the following two quantities

Using the values of Ku and Pu Ziegler Nichols recommended the following settings for feedback controllers.

u

₌ 1

M

Ultimate gain = K

ω

co

u

Ultimate period of sustained cycling = P = ^2π min/cycle

The settings above reveal the rationale of the Ziegler Nichols methodology

​

For Proportional control alone, use a gain margin equal to 2.

For PI control use a lower proportional gain because the presence of integral control mode introduces additional phase lag in all frequencies, with destabilizing effects on the system. Therefore, lowerK maintains approximately the same gain margin.

The presence of the derivative control mode introduces phase lead with strong stabilization effects in the closed loop response. Consequently, the proportional gain K, for a PID controller can be increased without threatening the stability of the system.

​

Damped Oscillation Method

In many plants, sustained oscillations for testing purpose are not allowable and the ultimate frequency method as discussed under Ziegler Nichols method cannot be used to secure the optimum settings (better quality output with minimum cost)

The following modification is easy to follow and perhaps more accurate than the ultimate method. By using only proportional action and starting with a low gain, the gain is adjusted until the transient response of the closed loop shows a decay ratio of ¼

​

The reset time and derivative time are based on the period of oscillation, P, which is

always greater than the ultimate period P_u.

​

For PID control

​

T_D = P/6 T_I= P/1.5

With the derivate and integral times at the above values, the gain for ¼ decay ratio is gain established by transient response tests

Determination of optimum settings for mathematically described processes using time response

Using time response methods, PID parameters such as proportional gain, integral time and derivative time are calculated. The Ziegler Nichols closed loop tuning formula is used.

​

Methods for Time domain Analysis

The two popular methods are Root locus Technique Routh array Method.

I. Root locus Technique

It is the graphical method for sketching the locus of the roots in the S plane as the parameter varies.

​

The path taken by the roots of characteristic equation when open loop gain K is

varied from 0 to infinity are called root locus.

Procedure to construct the root locus and finding the PID parameters Location of poles and zeroes

Draw the real and imaginary axis. The poles are marked as X and zero as o. The number of root locus is equal to number of poles of open loop transfer function. Origin of root locus is pole and ends at zero or infinity.

​

Root locus on real axis

To determine part of root locus on real axis, take a test point in real axis. If total no of poles and zeroes on right is odd then the test point lies on the root locus. If even, then test point does not lies on root locus.

Angle of asymptotes and centroid If n is the No of poles

​

M is the No of finite zeroes

n-m branches will terminate at zeroes at infinity. These n-m root locus branches will go along an asymptote path and meets the asymptote at infinity.

​

Hence no of asymptotes is equal to number of root locus branches going to infinity

Breakaway and Break in Points

The break away and break in points either lie on real axis or exist as complex conjugate pairs. If there is a root locus on real axis between 2 poles then there exist break away point (2 zeroes then break in points)

​

Characteristic equation B(s) + KA(s) = 0

​

K= -B(s)/A(s) , dK/ds=0

​

The roots provide break away and break in if K is real and positive.

​

Angle of Departure and Arrival

=180-[θ₁+θ₅+ θ₂]+[ θ₃+ θ₄]

Angle of asymptotes = ^{± 180(2q + 1)} q = 0,1,2,....

n − m _{Centroid =} Sum of Poles − Sum

Point of intersection of root locus using imaginary axis

The point where the root locus intersects on the imaginary axis can be obtained by anyone of the method

Routh array method Trial and error method

Put s=jω in the characteristic equation. And separate real and imaginary parts are equated to zero. By this method the value of K and ω are calculated.

Obtain the value of ultimate gain and ultimate period Ultimate gain= Ku= K

​

Ultimate Period=Pu = 2π/ω

K is the system stability and ω is the sustained oscillation frequency Formula to find PID settings using Ziegler Nichols method

K_p= 0.6K_u Ti = P_u/2 T_D = P_u/8

II. Routh array Method

The routh array method is an analytical method for finding the tuning parameters of the controller.

​

The characteristic equation of the system is given by 1+G(s)H(s)=0

​

Find the value of K and ω by routh array.

Obtain the value of ultimate gain and ultimate period

Ultimate gain= Ku= K Ultimate Period=Pu = 2π/ω Formula to find PID settings using Ziegler Nichols method

K_p= 0.6Ku T_i = Pu/2 T_D = Pu/8

Determination of optimum settings for mathematically described processes using frequency response

Using frequency response methods, PID parameters such as proportional gain, integral time and derivative time are calculated. The Ziegler Nichols closed loop tuning formula is used.

​

Methods for frequency domain Analysis

​

The popular methods are

Bode plot Polar plot

Nichol’s chart Nyquist plot.

Procedure to construct the Bode plot and finding the PID parameters Consider the transfer function of open loop system as

Find the corner frequencies ω_c1= 1/T₁ and ω_c2= 1/T₂ and form the tabulation relating factor term, corner frequency, slope, change in slope.

Select the lower frequency which is less than ω_c1 and higher frequency which is greater than corner frequencies.

For the given transfer function, find phase angle and substitute different values of ω and form phase angle plot tabulation.

In semilog sheet, plot magnitude and phase angle plot

Magnitude plot: Magnitude versus logω Phase angle plot: Phase angle versus logω

K

s(1 + sT₁ )(1 + sT₂ )

G(s) =

⎝ c1 ⎠

^⎛ ω_c2 ^⎞

Gain at ω_c2 = Change in slope from ω_c2 to ω_c1 × log^⎜ _ω ^⎟ + Gain at ω_c1

From the Bode plot find gain margin(G.M), Phase Margin (PM), gain cross over frequency (ω_gc) and phase cross over frequency(ω_pc)

Procedure to find ultimate gain and ultimate period Ultimate gain= Ku= 10GM/20

​

Ultimate Period=Pu = 2π/ωpc

PID parameters using ultimate cycling method

​

K_p= 0.6Ku T_i = Pu/2 T_D = Pu/8

​

Auto tuning

Many process control systems have an automatic tuning (auto tune) feature. The operator can simply push the auto tune button and the controller tuning will automatically carried out. When the operator feels that the current set of tuning parameters is not performing well then he will put the system in auto tune mode.

Astrom and Hagglund proposed a powerful auto tune variation (ATV) method for obtaining the ultimate gain and ultimate period. The method consists of adding a relay where the error signal is input to the relay. The relay toggles the process input between +h% to –h% on detecting a zero crossing.

The action of the relay causes the process input to toggle around the steady state by ±h% for every zero crossing in the error signal corresponding to the output crossing the set point. Sustained oscillations result and the system ends up in a limit cycle as above ( output of process marked as ‘y’ in Y axis)

The period of oscillations is the ultimate period Pu. The amplitude ‘A’ of the output oscillations gives the ultimate gain Ku as Ku = 4h/Aπ

The auto tune variation method has advantages over open loop step methods. The method automatically finds the critical frequency(or period) of the process. Also large deviations away from the steady state are avoided as this is a closed loop test. Finally, the amplitude at the critical frequency (ultimate period) is obtained so that the identification procedure is more accurate than step/pulse test.

​

Cascade control

In cascade control configuration, one manipulated variable and more than one measurement. In this scheme there will be two controllers namely primary and secondary controller.

The output of the primary controller is used to adjust the setpoint of a secondary controller, which in turn sends a signal to the final control element. The process output is feedback to the primary controller, and a signal from an intermediate stage of the process is feedback to the secondary controller.

The outer loop (primary controller) controller output is the set point of the inner loop (secondary controller). Thus, if the outer loop controlled variable changes, the error signal or input setpoint of the inner loop changes. Even though the measured value of the inner loop has not changed, the inner loop

experiences an error signal and thus new output by virtue of its setpoint change.

Cascade control provides better control of the outer loop variable than through single variable system. The disturbances arising within the secondary loop are corrected by the secondary controller before they can affect the value of the primary controller output. This benefit has led to the extensive use of cascade control in chemical processes.

Consider a process consisting of two parts: Process I and Process II. Process I (Primary) has an output variable which we want to control. Process II (secondary) has an output that we are not interested in controlling but which affects the output we want to control.

Fig: Open loop Process

Fig: Conventional feedback system

Fig: Cascade control

​

The two advantages of cascade control

Disturbance affecting the secondary variable can be corrected by the secondary controller before a pronounced influence is felt by the primary variable.

​

Closing the control loop around the secondary part of the process reduces the

phase lag seen by the primary controller, resulting in increased speed of response.

​

Closed loop behaviour of cascade control system

The closed loop response of the primary loop is influenced by the dynamics of the secondary loop, whose open loop transfer function is

​

^Gsecondary ^{= G}cII ^GpII ^{= 0}

The stability of the secondary loop is determined by the roots of its characteristic

equation

​

1+ G_cII G_pII = 0

​

For the primary loop overall transfer function is

The characteristic equation whose roots determine the stability of primary loop is

Selection and Tuning of cascade control

The two controllers of a cascade control system are standard feedback controllers(i-e P, PI,PID). Generally, a proportional controller is used for the secondary loop, although a PI controller with small integral action is not unusual. Any offset caused by P control in the secondary loop is not important since we are not interested in controlling the output of the secondary process.

The dynamics of the secondary loop are much faster than those of the primary loop. Consequently, the phase lag of the closed secondary loop will be less than that of the primary loop.

This feature leads to the following important result, which constitutes the rationale behind the use of cascade control.

The cross over frequency for the secondary loop is higher than that for the primary loop. This allows us to use higher gains in the secondary controller in order to regulate more effectively the effect of a disturbance occurring in the secondary loop without endangering the stability of the system.

​

The tuning of two controllers of a cascade system proceeds in two steps:

First, determine the settings for the secondary controller using one of the methods (cohen coon, Ziegler Nichols or other employing time integral criteria or phase and gain margin considerations)

^GcII^GpII

⎞

⎝

cII pII _⎠

⎛

G

^GPrimary ^{= G}cI ^⎜ _{1 + G}

^{⎟ G}pI

cII pII

^GcII^GpII

⎟

⎠

⎜

⎝

⎛ ⎞

1 + G G

1 + G_cI ^{⎜ ⎟} G_pI = 0

Second, from the bode plot of the overall system determine the cross over frequency using the settings for the secondary loop we found above. Then using the frequency response techniques, chose the settings for the primary controller(using bode plots)

​

Applications

​

Jacketed CSTR: Consider the CSTR, where the reaction is exothermic and the heat

generated is removed by the coolant, which flows in the jacket around the tank.

The control objective is to keep the temperature of the reacting mixture T, constant at a desired value.

Disturbances: Feed Temperature Ti and coolant Temperature Tc Manipulated variable: Coolant flow rate Fc

Simple feedback control: T will respond much faster to changes in T_i than to changes in T_c. Therefore the simple feedback control will be effective in compensating for changes in T_i and less effective in compensating for changes in T_c.

Fig: CSTR - Conventional feedback

Cascade control: We can improve the response of simple feedback control to changes in the coolant temperature by measuring T_c and taking control action before the effect has been felt by the reacting mixture. Thus if T_c goes up, increase the flow rate of coolant to remove same amount of heat. Decrease F_c when T_c decreases.

Fig: CSTR – Cascade control

There are two different control loops using two different measurements, T and T_c. But sharing common manipulated variable F_c. The loop that measures T (controlled variables) is the dominant or primary or master control loop and uses a setpoint supplied by the operator whereas the loop that measures T_c uses the output of the primary controller as its setpoint and called as the secondary or slave loop.

​

Feed forward control

Introduction

Feedback control loops can never achieve perfect control of a chemical process i-e keep the output of the process continuously at the desired set point value in the presence of load or set point changes. The reason is simple.

A feedback controller reacts only after it has detected the deviation in the value of the output from the desired set point.

A feed forward configuration measures the disturbance(load) directly and takes the control action to eliminate its impact on the process output. Therefore, feedforward controllers have the theoretical potential for perfect control.

But, as it is difficult to measure all possible disturbance variables and to predict their effect quantitatively, feed forward control is generally used along with feedback control. In most cases, a combination of feed forward and feedback techniques can correct process deviations in the shortest time. Feed forward loops usually corrected by feedback trimming.

The feed forward system is more costly and requires more engineering effort than feedback system.

​

Structure of feed forward control scheme

In feed forward control scheme, it measures the disturbance directly and then it anticipates the effect that it will have on the process output. Subsequently, it changes the manipulated variable by such an amount so as to eliminate completely the impact of the disturbance on the process output (controlled variable). Control action starts immediately after a change in the disturbance has been detected.

It is clear that feedback acts after the fact in a compensatory manner, whereas feedforward acts beforehand in an anticipatory manner.

Design of feed forward controller

The design of feed forward controller arises directly from the model of a process.

​

Consider the block diagram of an uncontrolled process.

Fig: Process Block diagram

​

The process output is given by

(1)

Y(s) = G_p(s) m(s) + G_d(s) d(s)

Let Y_sp(s) be the desired set point for the process output

Y(s) = Y_sp(s) sub in eqn(1)

​

Y_sp(s) = G_p(s) m(s) + G_d(s) d(s)

​

Y_sp(s) - G_d(s) d(s) = G_p(s) m(s)

Equation(2) determines the form that the feed forward control system should have and shown in below diagram. It also determines the two transfer functions, G_c and G_sp, which complete the design of the control mechanism.

p

⎦

( )^⎤ ( )

^⎢ G (s)

⎣ d

⎡^{Y (s)}

G (s) m(s) = ^sp − d s _⎥ G s

d

G (s)

⎣ ^d

_⎦ G _p (s)

⎡^Y_sp ^(s)

^⎤ G (s)

m(s) = _⎢

− d(s)_⎥ ^d

(2)

Fig: Block diagram of feedforward loop

G_c(s) = G_d(s) / G_p(s) = Transfer function of the feed forward controller (3)

​

G_sp(s) =1 / G_d(s) = Transfer function of the set point element. (4)

From eqns (3) and (4) it is clear that a feed forward controller cannot be a conventional feedback controller(P,Pi,PID). Instead , it should be viewed as a special purpose computing machine. This is the reason it is sometimes referred to as a feed forward computer.

The design equations (3) and (4) demonstrate that feed forward control depends heavily on a good knowledge of the process model (G_p, G_d). Perfect control necessitates perfect knowledge of G_p and G_d, which is not practically possible. This is the main drawback of feedforward control.

The inclusion of sensors that measures the disturbance and final control element alters the design of the transfer functions G_c(s) and G_p(s).

To analyse it, the measuring device and final control element are included in the block diagram.

Fig: Block diagram of feedforward loop with measuring element

​

Y = G_pG_f G_c [G_spY_sp-G_md] + G_dd

​

= G_pG_f G_c G_spY_sp- G_pG_f G_c G_md + G_dd

​

Y = G_pG_f G_c G_spY_sp +[ G_d - G_pG_f G_c G_m] d (5)

The design transfer functions G_c and G_sp can now be identified by the following requirements.

​

Disturbance rejection

The controller should be capable of completely eliminating the impact of a disturbance change on the process output. This implies that the coefficient of d in eqn(5) should be zero.

G_d - G_pG_f G_c G_m = 0 G_d = G_pG_f G_c G_m G_c = G_d / G_pG_f G_m

Set point tracking

The control mechanism should be capable of making the process output track exactly any changes in the set point (i-e Keep Y = Y_sp). This implies that the coefficient of Y_sp in eqn(5) should be equal to 1.

​

G_pG_f G_c G_sp = 1

Practical aspects on the Design of feed forward controllers

Assume G_m= G_f = 1. The two process transfer functions G_p(s) and G_d(s) has two elements

​

• The static element, which corresponds to the static gain
• The purely dynamic element, which is function of s. Thus G_p(s) = K_pG_p’(s) G_d(s) = K_dG_{d ’}(s)

​

Design of steady state feedforward controllers

At steady state, the static elements of the process transfer functions only retained.

​

Thus G_p = K_p G_d = K_d

G G G

G G

d

p f

⎥

G ⎤

Gsp = 1

⎢

^⎢_⎣ p f m ^⎥_⎦

⎡

^Gd ^Gsp _{= 1} G_m

d

sp

G

G

₌ G_m

sp

p d

c

K

K

Thus design transfer functions G

= ^Kd G = ¹

Design of simple dynamic feed forward controllers

Instead of using the exact transfer functions G_p(s) and G_d(s), it is possible to use approximations to them. Improved results are obtained over steady state feed forward controller.

​

Consider G_p(s) and G_d(s) are approximated by first order lags

Examples

​

Stirred tank heater.

Fig: Example for feedforward control-Stirred Tank heater

The control objective is to keep the temperature of the liquid in the tank at a desired value despite any changes in the temperature of the inlet stream. Feed forward loop measures the temperature of the inlet stream (disturbance) and adjusts appropriately the steam pressure (manipulated variable). Thus if the inlet temperature decreases, the steam pressure is increased and if the inlet temperature increases, the steam pressure is decreased.

1

G (s)

p

d

G (s)

=

βs + 1 αs + 1

βs + 1

1

αs + 1

=

G_c (s)=

G (s)

1

d

βs+1 introduces phase lead and 1/αs+1 introduces phase lag. α, β are adjustable

parameters.

sp

= αs + 1

G (s)=

Heat Exchanger

Fig: Example for feedforward control - Heat Exchanger

​

​

The objective is to keep the exit temperature of the liquid constant by manipulating the steam pressure. There are two principal disturbances that are measured for feed forward control which are liquid flow rate and liquid inlet temperature.

​

Boiler drum level control

Fig: Example for feedforward control – Boiler drum

The objective is to keep the liquid level in the drum constant. The two disturbances are the steam flow from the boiler, which is dictated by varying demand elsewhere in the plant and the feedwater flow is the manipulated variable

Distillation column

Fig: Example for feedforward control – Distillation column

The two disturbances are feed flow rate and composition. The manipulated variables are steam pressure in the reboiler and reflux ratio. The composition of overhead or bottom product is the control objective.

​

Continuous stirred tank reactor

Fig: Example for feedforward control – CSTR

​

The two disturbances: inlet concentration and temperature

The two manipulated variables: Product withdrawal flow rate and the coolant flow rate.

​

Two objectives: To maintain constant temperature and composition

Advantages and disadvantages of feed forward and feedback control

Feed Forward

​

Advantages

Acts before the effect of a disturbance has been felt by the system Good for slow systems (multicapacity) or with significant dead time.

It does not introduce instability in the closed loop response.

​

Disadvantages

Requires identification of all possible disturbances and their direct measurement.

​

Cannot cope with unmeasured disturbances

​

Sensitive to process parameter variations

Feedback

​

Advantages

It does not requires identification and measurement of any disturbances Insensitive to parameter changes

​

Insensitive to modeling errors

​

Disadvantages

It waits until the effect of a disturbance has been felt by the system before control action is taken

It is unsatisfactory for slow processes or with significant dead time.. It may create instability in the closed loop response.

PROBLEMS

​

1. For the control system shown below in fig, determine controller settings for a PI controller using Z-N method and C-C method.

𝑆+1

−𝑠

Process Transfer Fuction𝐺 𝑠 = ^𝑒 ---(1)

The general first order equation with delay time is expressed as

𝐶(𝑠) ₌ 𝐾

𝑅(𝑠) 𝜁𝑆 + 1

_𝑒−𝑡_𝑑𝑠

Comparing eq(1) &eq(2)

K=1 (Steady state gain)

​

t_d=1 (Time delay)

​

T=1 (Time constant)

​

C-C Method:

𝐶

𝐾 =

𝑇

𝐾_𝑃𝑇_𝑑

12𝑇 1

^{0.9 + 𝑇𝑑} = ¹ 0.9 + ¹

12

= 0.983

30 + 3𝑇𝑑/𝑇 30 + 3

^{𝜁𝐼 = 𝑇𝑑} 9 + 20𝑇𝑑/𝑇 ⁼ 9 + 20 ^{= 1.14}

Z-N Method:

​

Obtain the cross-over frequency by applying the Bode-stability criterion.

-1

PROBLEMS

K_C = 0.45K_CU =0.45 X 2.26 = 1.02

𝐼

𝑢

𝑟 = =

𝑃 3.09

1.2 1.2

= 2.58

2. Using Z-N rules, determine K_C and 𝜁_𝐼 for the control system shown in the figure.

-180°=-tan^-1(ω) – (180/𝜋)(1.02)ω

​

ω_co = 2rad/min

​

The Amplitude ratio (AR) at the crossover frequency for the open-loop can be written as

√(1 + 𝜔²)

𝐴𝑅 =

=

1

𝐾_𝑐 2.24

𝐼

𝑃

𝑢

𝑟 = =

The amplitude ratio for a transport lag is 1. According to the bode-criterion, the AR is 1.0 at the cross over frequency. So, substituting AR=1 in the above equation,

𝐾_𝑐𝑢 = 2.24

K_C = 0.45K_CU =0.45 X 2.24 = 1.01

​

2𝜋

𝜔

𝑐 𝑜

1.2 1.2

= 2.62𝑚𝑖𝑛

3. In the application of Z-N method the process begins oscillation with 30% proportional band in an 11.5 min period. Find i) the nominal 3 mode controller settings ii) settings to give quarter amplitude response.

100 100

𝐾_𝑢 = _𝑃𝐵 = ₃₀ = 3.33

​

Also given 𝑃_𝑢= 11.5𝑚𝑖𝑛

​

Proportional gain

𝐾_𝑢 3.33

^{𝐾𝑃 =} 1.7 ⁼ 1.7 ^{= 2}

PROBLEMS

𝑃_𝑢 11.5

𝑟_𝐷 = 𝑇_𝐷 = ₆ = ₆

= 1.92𝑚𝑖𝑛

4. A step function disturbance is 4.7% in the controlled variable if a quarter amplitude

criterions is used. Find the error.

The amplitude of each peak must be ¼ of previous peak.

​

Given the amplitude of first peak is 4.7%

​

a₁=4.7%

Amplitude of peak a₂ = a₁ / 4 =4.7/4 = 1.18%

​

Amplitude of third peak, a₃ = a₂ / 4 = 1.18/4 =0.3%

5. Find the optimum controller setting parameters using P & PI controller for given transfer function 1/(4s+1) with transportation lag of 0.5sec.

−0.5𝑠

Given the transfer function is ^𝑒

4𝑆+1

When we compare the above transfer function with first order transfer function with delay

PROBLEMS

By Process reaction curve method,

​

For Proportional,

1 𝑟

𝐾_𝑐 = _{𝐾 𝑡𝑑}

𝑡_𝑑 1 + _3𝑟

𝑐

1 4

𝐾 =

1 0.5

0.5

1 +

3𝑋4

= 8.33

For Proportional + Integral,

1 𝑟

𝐾_𝑐 = _{𝐾 𝑡𝑑}

𝑡_𝑑

0.9 + _12𝑟

1 4

^{𝐾𝑐 =} 1 0.5

0.5

0.9 + _12𝑋4

= 7.28

𝑟_𝐼 = 𝑡_𝑑

30 + ^3𝑡𝑑

𝑟

_{9 +} 20𝑡_𝑑

𝑟

𝑟_𝐼 = 0.5

_{30 +} 3𝑋0.5

4

_{9 +} 20𝑋0.5

4

= 1.320

10. ASSIGNMENTS

11. PART-A

12. PART-B

13. SUPPORTIVE ONLINE CERTIFICATION COURSES

• Online Course: SWAYAM

​

Course Name: Chemical Process control

Course Instructor: Prof. Sujit Jogwar, IIT Bombay Duration: 8 weeks

​

AICTE approved FDP course

​

• Online course:Coursera

Course Name: Sensor Manufacturing and Process Control

​

University: University of Colorado Boulder

Course Instructor: James Zweighaft, Jay Mendelson Duration: 5 weeks

​

• Online course:Udemy

​

Course Name: Introduction to process control and Instrumentation

Course Instructor: WR Training, Petroleum Petrochemical & Chemical Engineering

​

Duration: 10 sections,75 lectures

14. REAL TIME APPLICATIONS IN DAY TO DAY LIFE AND TO INDUSTRY

Modeling and control of a real time shell and tube heat exchanger, Resource- Efficient Technologies Volume 3, Issue 1, March 2017, Pages 124-132

Tuning and Retuning of PID Controller for Unstable Systems Using Evolutionary Algorithm, Research Article | Open Access Volume 2012 |Article ID 693545 | https://doi.org/10.5402/2012/693545

15. CONTENT BEYOND SYLLABUS

Tyreus – Luyben Method:

The Tyreus-Luyben [10] procedure is quite similar to the Ziegler–Nichols method but the final controller settings are different. Also this method only proposes settings for PI and PID controllers. These settings that are based on ultimate gain and period are given in Table

The C-H-R Method:

This method that has proposed by Chien, Hrones and Reswich [1] is a modification of open loop Ziegler and Nichols method. They proposed to use “quickest response without overshoot” or “quickest response with 20% overshoot” as design criterion. They also made the important observation that tuning for set point responses and load disturbance responses are different.

To tune the controller according to the C- H-R method the parameters of first order plus dead time model are determined in the same manner of the Z-N method. The controller parameters can then be determined from the below Tables.. The tuning rules based on the 20% overshoot design criterion are quite similar to the Z-N method. However when the 0% overshoot criteria is used, the gain and the derivative time are smaller and the integral time is larger. This means that the proportional action and the integral action, as well as the derivative action, are smaller.

Tuning relations for C-H-R method. Load rejection

Tuning relations for C-H-R method. Set point tracking

16. ASSESSMENT SCHEDULE (PROPOSED DATE & ACTUAL DATE)

17. Prescribed Text Books & Reference Books

​

TEXT BOOKS:

1. Seborg, D.E., Edgar, T.F. and Mellichamp, D.A., “Process Dynamics and Control”, WileyJohn and Sons, 2nd Edition, 2003.
2. Bequette, B.W., “Process Control Modeling, Design and Simulation”, Prentice Hall of India, 2004.
3. Stephanopoulos, G., “Chemical Process Control - An Introduction to Theory and Practice”, Prentice Hall of India, 2005.

​

REFERENCES:

1. Coughanowr, D.R., “Process SystemsAnalysis and Control”, McGraw - Hill International Edition,2004.
2. Curtis D. Johnson, “Process Control Instrumentation Technology”, 8th Edition, Pearson, 2006.
3. Considine, D.M., Process Instruments and Controls Handbook, Second Edition, McGraw, 1999.
4. Bela.G.Liptak., “Process Control and Optimization”., Instrument Engineers’ Handbook., volume 2, CRC press and ISA, 2005.
5. Ramesh C. Panda., T.Thyagarajan., “An Introduction to Process Modelling Identification and Control for Engineers” Narosa Publishing house Pvt. Ltd, 2017.

18. MINI PROJECT SUGGESTIONS

​

​

Project 1:

Using Process reaction curve method, determine the closed loop response of a second order system using PID controller in Matlab.

​

Project 2:

​

Using Process reaction curve method, determine the closed loop

response of a second order system using PI controller in Matlab.

​

Project 3:

Using ultimate cycling method, determine the closed loop response of a second order system using PID controller in Matlab.

​

Project 4:

Using ultimate cycling method, determine the closed loop response of a

second order system using PI controller in Matlab.

​

Project 5:

Determine the closed loop response of a second order system satisfying IAE criteria in Matlab. Use PI controller

​

Project 6:

Determine the closed loop response of a second order system satisfying ISE criteria in Matlab. Use PI controller

Thank you

Disclaimer:

​

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