Feedback Control


IV.1 Introduction to Feedback Control 
Consider a chemical process plant shown in the Fig IV.1. It has an output  that needs to be maintained at a desired setpoint . The process is subjected to unavoidable disturbance (load) . One input is available in the process that can be manipulated in order to steer the process to its desired setpoint . Hence, in the event of process disturbance drifting the process output away from its desired setpoint, the objective of feedback control is to steer the process output at its setpoint by manipulating the process input. 
Fig. IV.1: A conceptual block diagram of feedback control loop of a chemical process 

A feedback control law consists of the following steps of action: 
•  Measurement of process output , such as pressure, temperature, flow, level, composition, using a suitable measuring device. 
•  Comparison of process output and its desired setpoint  and calculation of the deviation error 
•  Feeding the value of deviation error to the controller, which calculates the appropriate control action to be taken. 
•  Feeding the value of the control action to the final control element, that translates the control action into manipulated form of process input .
As the name suggests, an open loop process does not have any feedback loop mechanism. It consists of only the process along with its two inputs ( u and d) and one output (y). Input  is available for manual manipulation that is required for modeling exercise. On the other hand closed loop process has a feedback loop mechanism that consists of measuring device, controller and final control element. The process input is not available for manual manipulation, rather the controller (and the final control element) ensures automatic manipulation of . However, a setpoint  for the process output is available for manual manipulation in case the process objective demands so. 
Fig. IV.2 demonstrates some typical feedback control systems which are often encountered in a chemical process plant. This is an example of a distillation column. There are six control loops in this example, one each for pressure, temperature, flow, level, reflux and composition. The measurements are done through an appropriate measuring device cum transmitter (denoted by PT, TT, FT, LT, RT and CT). The measured output is compared with the respective setpoints (denoted by Psp , Tsp , Fsp , Lsp , Rsp and Csp) and errors are fed to the respective controllers (denoted by PC, TC, FC, LC , RC and CC) and control actions are subsequently initiated by their corresponding valves. 
Fig. IV.2: Schematic of a typical feedback control configuration of a distillation column

The feed is pumped into the column whose flow rate and composition are controlled. Feed mainly comes through a major supply line where the feed composition may vary due to various factors which are beyond the scope of the operator of the distillation column. An auxiliary supply line with guaranteed purity of material can be mixed with the main supply through a regulated dose in order to maintain the composition of the feed entering the column. A flow controller at the entry point of the column maintains the feed flow rate. The pressure of the top section of the column is maintained by regulating the flow of vapour going out of the column. The vapour is condensed and collected in a tank. The outlet temperature of distillate coming out the condenser (heat exchanger) is controlled by regulating the flow rate of cooling water used in the heat exchanger. The liquid level inside the storage tank in the downstream of condenser is controlled with a level controller by regulating the tank outlet. The L/D ratio of a distillation column is very important and it needs to be controlled by a ratio controller. The ratio transmitter is essentially a flow measuring device that sends the ratio of the two flow rates rather than the flow rates themselves. In all the above examples, only one type of final control element has been demonstrated viz . control valve. Nevertheless, there are various other final control elements available such as variable speed metering pump, heating regulators etc .
Note : The composition transmitter/controller is usually a high value delicate instrument which is not used in a rugged process plant environment. On-line gas chromatograph is an example of such sophisticated instrument. They are used only in a few critical operations where tighter regulation of composition is mandatory, such as automated drug delivery.
The above example indicates that the basic hardware components of a feedback control configuration are the following:
•  Process
•  Measuring devices
•  Transmission lines
•  Controllers
•  Final Control Elements
In the following sections, we shall discuss each and every hardware item in an elaborate manner.

IV.1.1 Process 
Any equipment that serves the targeted physical/chemical operation of the plant is termed as a process. Reactors, separators, exchangers, pressure vessels, tanks, etc. are examples of a process. Typically these processes are connected in a logical fashion and the output of one process becomes input to the other. Any disturbance/malfunction of one process may affect other processes in the downstream side (and upstream too, in case recycle streams are used). Detailed discussions on these processes are not within the scope of this course, however, the modeling techniques and related issues have already been discussed before. Process variables are primarily pressure, temperature, flow rate, level, composition, etc . From the process control perspective, it is crucial to study how the changes in one process variable affect the other, so that an educated measure of control action on one variable can be taken in order to maintain the other.
IV.1.2 Measuring Instruments or Sensors 
The success of any feedback control operation depends largely on accurate measurement of process variables through appropriate sensors. There are a large number of commercial sensors available in the market. They differ in their measuring principle(s) and/or their construction characteristics. Module VII accounts of a few of such measuring instruments. Details of such devices may be found in technical booklets dedicated for those individual items.
IV.1.3 Controllers 
A controller is basically a mathematical function block that reads the error between desired setpoint and the measured output and then computes the corrective action for the manipulated input that would steer process towards the desired setpoint. There are three basic types of feedback controllers which are widely used in the industry.
•  Proportional (P) controller
•  Proportional Integral (PI) controller
•  Proportional Integral Derivative (PID) controller
Let us study each one separately.

IV.1.3.1 Proportional Controllers 
The actuating output of a controller is proportional to the error between the setpoint and process output. Higher the error, higher will be the control action. The control law is given as:
(IV.1) 
where  is called the gain of the controller and  is the bias signal. When error signal  is zero (i.e., the process output reaches its desired setpoint), the control signal  stabilizes at its bias value . The deviation form of actuating signal is
(IV.2) 
Hence the transfer function of the proportional controller is
(IV.3) 
In industrial lingo, the proportional controller is also termed as “Gain” controller. Equivalent representation of proportional gain is proportional band . It is the amount of change in error that will cause the control action to go from full OFF to full ON. The amount of change in error is calculated as a percentage of full-scale error,
(IV.4) 
e.g. , consider a level controller acting on a tank where we measure the level from bottom to top as 0 to 100%. A control valve on the outlet of the tank maintains the level in the tank. The PB is defined as the range of level over which the control valve will go from fully closed to fully open . Let us take the example of a tank whose maximum level is 5 m. Suppose we decide that if the tank level should fall to 20% (1 m) we want the control valve fully closed (0% open) and if the tank level rises to 60% (3 m) we want the control valve to be fully open (100% open). If the tank level is between 20% and 60% we want the control valve to be open in pro rata basis. This controller would have a PB of 40% (60% - 20%). So if the tank level were to rise to 2.5 m or 50% of the full tank (75% up the PB), the control valve should be set to 75% open).

Fig. IV.3: Proportional Band of Feedback Controller 
Usually, the workable range of PB is
(IV.5) 
The condition of  is a bit tricky to visualize. Since the level of the tank can only go up to 100%, a PB of (say) 250% causes the valve to move only through a portion of its available range as the tank goes from empty to 100% full. Let's assume that we set the lower end of the proportional band at -75% and the upper end at 175%. That gives a PB of 250%. Physically it means that the level would have to fall way below the bottom of the tank to fully close the valve and way above to fully open the valve. Since the level of the tank cannot really go that far, it means that the valve will never fully close or fully open. Tuning is the process of finding the gain or PB that provides the optimal response of control valve for controlling the process.

IV.1.3.2 Proportional Integral Controllers 
The actuating output of a PI controller is given as:
(IV.6) 
where  is the integral time constant (or the reset time) in minutes. The PI controller not only actuates on the basis of current error, but it also accounts for the history of all the past errors that has been encountered since the control action has started. From eq.(IV.6) the transfer function of the PI controller is
(IV.7) 
In industrial lingo, the PI Controller is also termed as “Gain-Reset” controller. The reset time is a design parameter which usually varies within the range 
At this point, it is worth explaining the significance of the term reset . Suppose the error between desired setpoint and process output changes by a constant step of magnitude . The effect of integral term of eqn. (IV.6) after every minutes is given as
(IV.8) 
In other words, the integral action repeats the response of the proportional action every minutes and “resets” itself for an integral action. Sometimes the controllers are calibrated in terms of reciprocal of reset time, (repeats per minute). This is known as reset rate .

The reset term causes the control action changing as long as there exists a non-zero error in the system. Often this error cannot be eliminated quickly and given enough time, they produce larger values for integral terms. Such situation is often observed when the system undergoes a large change in setpoint (say a positive change) and the integral term accumulates a significant error during the rise. This condition is termed as Integral Windup . The control action in turn keeps on increasing until it reaches the control valve saturation ( i.e. control valve fully open or fully closed). Even if the error changes its sign (as the process output overshoots the desired setpoint), this accumulated error has to unwind completely before control action is reversed. Various measures can be taken to address the issue of integral windup such as:
•  Re-initializing the integral action to a desired value
•  Increasing the setpoint in a suitable ramp (rather than a single step jump)
•  Disabling the integral action until the process output enters the controllable region
•  Preventing the integral term from accumulating above or below pre-determined bounds
IV.1.3.3 Proportional Integral Differential Controllers 
The actuating output of a PID controller is given as:
(IV.9) 
where  is the derivative time constant (or the preact time) in minutes. The PID controller not only actuates on the basis of current and past errors but it also anticipates the error in immediate future and applies an additional control action which is proportional to the current rate of change of error. Hence the transfer function of the PID controller is
(IV.10) 
In industrial lingo, the PID Controller is also termed as “Gain-Reset-Preact” controller.
The major drawback of a PID controller is that for a noisy response in a process, the controller can erroneously actuate a high derivative control action.

IV.1.4 Transmission Lines 
Measurement and/or control signals are carried through various transmission lines. Various process piping, connection and transmission lines, as per the standard set by International Society of Automation (ISA), are listed in the following figure.
Fig. IV.4: Representation of process piping, connection and transmission lines 

A heavy solid line represents piping,a thin solid line represents process connections to instruments, a dashed line represents electrical signals (e.g., 4–20 mA connections), a slashed line represents pneumatic signal tubes, a line with circles on it represents data links. Other connection symbols include capillary tubing for filled systems, (e.g., remote diaphragm seals), hydraulic signal lines, and guided/unguided electromagnetic or sonic signals. Electric/electromagnetic signals are instantaneous. Unless the process changes very fast and/or the transmission lines are too long, the dynamic behaviors of electric/electromagnetic transmissions are also usually ignored.

IV.1.5 Final Control Element 
The hardware component of a control loop that resides between the process and the controller and implements the control action is called the Final Control Element (FCE). It receives the control signal from the controller and regulates the value of the manipulated variable accordingly. An ideal FCE should be an instantaneously operating hardware that does not induce any time lag between process and controller. However, in reality it is impossible to find such an instantaneously operating FCE. Nevertheless any FCE, whose time constant is very small, is considered to be a good hardware for a control system.
Various types of FCEs are available in process industries that are widely used in control applications. They can be largely classified based on their energy source, viz . Pneumatic, Hydraulic and Electric, etc .
A few of them are discussed in Module VII.
IV.2 Dynamic behaviour of feedback controller 
Consider the generalized closed loop process given in the Fig IV.5.
Fig. IV.5: Generalized block diagram of feedback control loop of a chemical process

If we assume that the transmission line does not affect the signal flow, dynamics of the transmission lines can be completely ignored. Hence the following sub-processes will constitute the overall dynamics of the process:
Process 
(IV.11a) 
Measurement 
(IV.11b) 
Controller 
(IV.11c) 
  
(IV.11d) 
Final Control Element 
(IV.11e) 
Hence using eqs. (IV.11 b-e) on eq. (IV.11a), following algebraic manipulation will be useful.
(IV.12) 
Rearranging the above we obtain,
(IV.13) 
The eq. (IV.13) represents the closed loop response of the process. The pictorial representation is given in the Fig IV.6
Fig. IV.6: Simplified block diagram of feedback control loop of a chemical process

Where
(IV.14) 
(IV.15) 
 and  are the closed loop transfer functions (CLTF) of the process. maps the effect of change of setpoint on the process output whereas  maps the effect of change of load (disturbance) on the process output.
Two types of control problems are encountered with the feedback control systems, viz . servo and regulatory.
When setpoint of a process undergoes a change while the disturbance affecting the process remains constant, i.e. , the objective of the control system would be to steer the output as close as the setpoint trajectory. In such situation,
(IV.16) 
However, when the setpoint remains constant, i.e. , while the disturbance forces the process output to move out of the track of the setpoint, the objective of the control system would be to reject the effect of disturbance as soon as possible and steer the output back to the setpoint trajectory. In such situation,
(IV.17) 
Note that the CLTFs depend not only on the process transfer functions, but also on the transfer functions of measuring element, controller and final control element.
To expedite the construction of overall closed loop transfer function of any feedback loop, following rules may be applied:
1.  The denominator of the overall transfer function is “the product of all transfer functions in the feedback loop PLUS one” i.e ., 
2.  The numerator of the overall transfer function is “the product of all transfer functions in the forward path between setpoint and the controlled output (for servo problem) or the load and the controlled output (for regulatory problem)” i.e .,  or 




IV.2.1 Closed loop response of liquid level in a storage tank: A case study 
Fig IV.7 represents the closed loop block diagram of a storage tank whose liquid level needs to be controlled at a predefined setpoint. 
Fig. IV.7: Schematic of closed loop block diagram of storage tank 

We analyze various components of the above, 
Process : The process has two inputs and one output. Input  can be manipulated whereas input  is the source of disturbance. Output  varies proportionally with the square root of the height of the liquid in the tank as, The material balance around the tank gives the following model, 


Where, is the constant of proportion. Linearization of the model around its nominal operating point , expressing it in its deviation form and converting it in Laplace domain yields the following open loop transfer function: 
(IV.19) 
Incidentally, in this example . Needless to mention, this is a trivial case and should not be generalized. 
Measuring device : Differential Pressure (DP) cell is a measuring device that measures differential pressure (ΔP) between two ends. output of DP cell maps a second order behavior with the differential pressure. In this example, DP cell measures the height of liquid level in the tank by comparing pressure exerted by liquid at one end with that of atmospheric pressure at the other. The pressure exerted by the liquid on the DP cell is linearly proportional to height of the liquid. Hence, 
(IV.20) 
where measured value of the height of the liquid, actual height of the liquid, constants. 
Controller : As is the setpoint, the error can be expressed as 
(IV.21) 
Let us assume that we use a PI controller, the control action can be expressed as 


Control Valve : Let us assume that the control valve for this system follows a first order dynamics, hence
(IV.23) 
The following figure is a representation of closed loop control configuration of liquid level in a tank.
Fig. IV.8: Schematic of closed loop control configuration of liquid level in a storage tank 

IV.2.2 Effect of P controller on a process 
Without losing the generality, let us assume for the following derivation. Let us concentrate on servo problem only. Regulatory problem can be solved in an analogous manner. We shall find closed loop response of first order and second order processes under the influence of a P controller.

The uncontrolled responses of the processes are
(IV.24) 
(IV.25) 
The controlled responses of the processes are
(IV.26) 
(IV.27) 
From the above equations, following conclusions can be arrived:
1.  The order of the process does not change with a P controller.
2.  Static gain of the processes are reduced.
3.  The time constant of the first order process is reduced, i.e. the closed loop response is faster. The natural period of oscillation and damping factor of the second order process are also reduced, i.e., an overdamped process may become underdamped with P controller.

IV.2.3 Offset in the output due to a P controller 
The ultimate closed loop response at  under P controller never reaches its desired setpoint. There always remains a discrepancy between the final value of response and the setpoint. This discrepancy is called the offset . Following derivation gives an account of the offset shown by first order and second order processes.
The final values of the processes on unit step change in setpoint are
(IV.28) 
(IV.29) 
Hence,
(IV.30) 
On choosing , a zero offset can theoretically be obtained. However, such a higher value of Kc is neither practical nor possible. Note that, processes having integrator term  in their transfer functions do not exhibit offsets.
The following figures show the offsets exhibited by a first and second order processes. In both the cases, offset is 

(a) First order process 
(b) second order process 
Fig. IV.9: Offsets exhibited by first order and second order processes
Note that both natural period of oscillation and damping factor decreased in closed loop response;more the value of controller gain less are the values of natural period of oscillation and damping factor. As a result the closed loop response becomes more oscillatory than its open loop counterpart.
IV.2.4 Effect of integral control action on a process 
We shall find closed loop response of first order process under the influence of an “integral only” controller.
(IV.31) 
It indicates that the order of the dynamics for the closed loop response is increased when integral control action is used. As a result, the response becomes sluggish.
Following derivation gives an account of the offset shown by first order closed loop process with a PI controller. The final value of the process on unit step change in setpoint is


The offset = 1 - 1 = 0, indicates that the integral control action eliminates any offset. It also indicates that the increase in controller gain  and decrease in reset value  reduce the value of natural period of oscillation and damping factor of closed loop response. Hence, we can improve the speed of the closed loop response at the expense of higher deviations and longer oscillations. In other words, high  and low  makes the closed loop response more sensitive and proportional action will be stronger. On the other hand, low  and high  makes system sluggish but integral action would be stronger (no offset, no overshoot).
IV.2.5 Effect of derivative control action on a process 
We shall find closed loop response of first order and second order processes under the influence of a “derivative only” controller.
(IV.33) 
(IV.34) 
It indicates that the derivative control action does not change the order of the response. It also indicates that time constant of the first order process is increased by a positive quantity of i.e ., the derivative control action makes a process sluggish. On the other hand, natural period of oscillation of second order process remains same. Damping factor increases with increase in and/or  Increase in damping factor means less sensitivity of process response and more robustness in the process behavior.

IV.2.6 Insight into the cause of offset 
The existence of offset in a proportionally controlled process indicates that the input manipulation has reached some sort of saturation before the controlled output could even reach the setpoint. As a result, manipulated input no longer changes even if the error between setpoint and controlled output is non-zero. The above fact can be demonstrated mathematically as follows.
The equations of P and PI controllers are:
(IV.35) 
(IV.36) 
The gradient of the dynamics of above control actions can be derived by simple differentiation,
(IV.37) 
(IV.38) 
The above equations show that in case of P controller gradient of control action becomes zero if there is no change in the value of error. In other words, when the error dynamics reaches a saturation point, the control action reaches saturation too. Mere existence of non-zero error does not yield any change in control action. On the other hand, gradient of control action in a PI controller not only depends upon the gradient of error dynamics but also upon the absolute value of the error at that moment of time. Hence, until the error and its gradient both are zero, the gradient of control action will not be zero. This fact ensures offset-free control action by PI controller.

IV.3 Stability of a closed loop process 
The stability of a process is realized by the location of its poles. Existence of a positive pole leads to the instability of the process. One of the main purpose of the controller is to ensure stable closed loop process for the otherwise unstable open loop process.
IV.3.1 Stabilization of unstable system with a P controller 
Consider an unstable process with transfer function
(IV.39) 
Let us assume . The closed loop transfer function will be
(IV.40) 
It is evident from the above equation that one needs to set  in order to make the closed loop transfer function to be stable.

IV.3.2 Destabilization of a stable system with a PI controller 
Consider a stable process with transfer function


Let us assume  and let us take a PI controller with gain 50 and reset 0.2, then the closed loop transfer function will be
(IV.42) 
The roots of the denominator of the closed loop transfer function are . The closed loop system has complex poles with positive real parts. Hence the open loop stable system becomes unstable due to inappropriate choice of controller parameters. On the other hand, if we choose a PI controller with gain 10 and reset 0.2, then the closed loop transfer function will be
(IV.43) 
The roots of the denominator of the closed loop transfer function are . The closed loop system has complex poles with negative real parts. Hence the system remains stable.
Following observations are made from the above two exercises:
•  Stability characteristics are determined by the poles of 
•  An otherwise stable system can be destabilized by wrong choice of controller parameters

IV.3.3 Routh Hurwitz Criterion for Stability 
The Routh Hurwitz criterion does not calculate the actual values of closed loop poles, rather it calculates whether any of the poles is on the left hand side of the imaginary axis of complex plane. In other words, the Routh Hurwitz criterion find whether a closed loop transfer function is stable. In this process, it also finds a limiting condition for controller parameters which would ensure stability for closed loop system. The denominator of a closed loop process transfer function is termed as characteristic equation . The Routh Hurwitz criterion works on the characteristic equation as follows:
Step 1.  Expand the characteristic equation into a polynomial
(IV.44) 
Step 2.  If  is negative, multiply the whole polynomial with 
Step 3.  If any of the coefficients is negative, then there exists at least one unstable pole in the closed loop system. No further analysis is required.
Step 4.  If all the coefficients  are positive, then form the following table, called Routh Array .  
Table IV.1: Routh Array 

Step 5.   Examine the first column, 
  • a.  A system is stable if all the above elements in the first column of the Routh Array are positive 
  • b.  If any one of the elements of first column is negative, we have at least one pole at the right hand side of the imaginary axis and the closed loop system is unstable 
    c.  The number of times the sign changes (from positive to negative to positive again and so on) indicates the number of unstable poles present in the closed loop system 
The last example (Eq. IV.42) can be revisited with Routh Hurwitz criterion. The polynomial coefficients, can be written in a Routh Array,

                                                     Table IV.2: Routh Array of eq. (IV.42) 

The first column of the Routh array is . Hence the system has unstable pole(s). The sign is changed twice in the column (positive to negative and then negative to positive). Hence the closed loop system has two unstable poles 
The Routh Array can also be used for finding the stability limits of the controller parameters. 

The Routh Array is 
                                                    Table IV.3: Routh Array of eq. (IV.45)
Only the third element of the first column is capable of being negative due to inappropriate choice of controller parameters. The stability of the closed loop system will be ensured if 

For example at , then the condition of stability is ensured if 

IV.3.4 Root Locus analysis 
The root locus analysis is an useful tool for determining the stability characteristics through graphical means. Root loci are plotted in complex plane as the gain varies from zero to infinity.

Fig. IV.10: Root Locus diagram 

The Fig. IV.10 demonstrates the root locus diagram of the said example. The loci of the complex pair of poles cross the imaginary axis at 

Let us take a second example where transfer function of a process is 
(IV.46) 
The characteristic equation is 
(IV.47) 
The following table shows the values of roots of the characteristic equation as  changes from 0 to .


(IV.41) 


(IV.22) 

(IV.18) 

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