Chapter 2 CONTROLLER PRINCIPLE

Introduction 

Controller principle have 4 type controller :
   -  On Off Controller ( Two position / Multiposition Mode )
   -  Analog Controller  ( P, I, D, PI, PD, PID )
   -  Digital Controller
   -  Fuzzy Controller

Controller modes refer to the methods to generate different types of control signals to final control element to control the process variable. Broad classifications of different controller modes used in process control are as follows:

     (1) Discontinuous Controller Modes
         - Two-position (ON/OFF) Mode
         - Multiposition Mode
         - Floating Control Mode: Single Speed and Multiple Speed

     (2) Continuous Controller Modes
        - Proportional Control Mode
        - Integral Control Mode
        - Derivative Control Mode

     (3) Composite Controller Modes
        - Proportional-Integral Control (PI Mode)
        - Proportional-Derivative Control (PD Mode)
        - Proportional-Integral-Derivative Control (PID or Three Mode Control)

2.2 Controller Modes

2.2.1 DISCONTINUOS CONTROL MODES

          Two position (ON/OFF) Mode
          The most elementary controller mode is the two-position or ON/OFF controller mode. It is the
          simplest, cheapest, and suffices when its disadvantages are tolerable. The most general form can be
          given by:
                                                                P = 0 % ep < 0 
                                                                      100 % ep > 0

          The relation shows that when the measured value is less than the setpoint (i.e. ep > 0), the

          controller output will be full (i.e. 100%), and when the measured value is more than the

           setpoint (i.e. ep < 0), the controller output will be zero (i.e. 0%).

           Neutral Zone : In practical implementation of the two-position controller, there is an

           overlap as ep increases through zero or decreases through zero. In this span, no change in

           the controller output occurs which is illustrated in Fig. 2.2.1.1

Fig. 2.2.1.1 Two-position controller action with neutral zone.

           It can be observed that, until an increasing error changes by Δep above zero, the controller output 

           will not change state. In decreasing, it must fall Δep below zero before the controller changes to 0%.              The range 2Δep is referred to as neutral zone or differential gap. Two-position controllers are                        purposely designed with neutral zone to prevent excessive cycling. The existence of such a neutral                  zone is an example of desirable hysteresis in a system.

           (Example 1)
           A liquid-level control system linearly converts a displacement of 2 to 3 m into a 4 to 20 mA control                signal. A relay serves as the two-position controller to open and close the inlet valve. The relay closes            at 12 mA and opens at 10 mA. Find (a) the relation between displacement level and current, and (b)            the neutral zone or displacement gap in meters.

           Solution

           Given data : Liquid-level range = 2 to 3 m i.e. Hmin = 2m & Hmax = 3m
                             Control signal range = 4 to 20 mA i.e. Imin = 4mA & Imax = 20mA

                             (a) Relation between displacement level (H) and current (I)
                             (b) Neutral zone (NZ) in meters.

             (a) The linear relationship between level and current is given by :
                                                         H = K I + Ho

                    The simultaneous equations for the above range are:
                                   For low range signal 2 = K x 4 + Ho
                                   For higher range signal 3 = K x 20 + Ho
                     Solving the above simultaneous equations we get:
                                    K = 0.0625 m/mA, & Ho = 1.75 m
              Therefore, the relation between displacement level (H) and current (I) is given by :
                                                                 H = 0.0625 I + 1.75
              (b) The relay closes at 12 mA, which is high level, HH
                                                      HH = 0.0625 x 12 + 1.75 = 2.5 m
                    The relay opens at 10 mA, which is low level, HL
                                                      HL = 0.0625 x 12 + 1.75 = 2.375 m
              Therefore, the neutral zone, NZ = (HH - HL) = (2.5 - 2.375) = 0.125 m

            (Example 2)
             As a water tank loses heat, the temperature drops by 2 K/min when a heater is on, the system gains              temperature at 4 K/min. A two-position controller has a 0.5 min control lag and a neutral zone of ±                4% of the setpoint about a setpoint of 323 K. Plot the heater temperature versus time. Find the                      oscillation period.

            Solution

            Given data : Temperature drops = 2 K/min
                               Temperature rises = 4 K/min
                               Control Lag = 0.5 min
                               Neutral zone = ± 4%
                               Setpoint = 323 K
            ± 4% of 323 = 13 K. Therefore, the temperature will vary from 310 to 336 K (without considering               the lag)
            Initially we start at setpoint value. The temperature will drop linearly, which can be expressed by
                                                                   T1(t) = T(ts) – 2 (t – ts)
           where ts = time at which we start the observation
           T(ts) = temperature when we start observation i.e. 323.
           The temperature will drop till - 4% of setpoint (323K), which is 310 K.
           Time taken by the system to drop temperature value 310 K is 310 = 323 – 2 (t – 0) t = 6.5 min
           Undershoot due to control lag = (control lag) x (drop rate) = 0.5 min x 2 K/min = 1 K
           Due control lag temperature will reach 309 instead of 310 K. From this point the temperature will rise            at 4 K/min linearly till +4% of setpoint i.e. 336
           K, which can be expressed by :
                                                                         T2(t) = T(th) + 2 (t – th)

          where th = time at which heater goes on
                    T(th) = temperature at which heater goes on
                                                                 336 = (310-1) + 4 [t – (6.5 +0.5)]
                                                                   t = 13.75 min
          Overshoot due to control lag = (control lag) x (rise rate) = 0.5 min x 4 K/min = 2 K
          Due control lag temperature will reach 338 instead of 336 K.
          The oscillation period is = 13.75 + 0.5 +0.5 + 6.5 = 21.25 ≈ 21.5 min
          The system response is plotted as shown in Fig. 2.2.1.2 with undershoot and overshoot values.
       

Fig. 2.2.1.2 Plot of heater temperature versus time for Example 2

2.2.2 Multiposition Mode

         It is the logical extension of two-position control mode to provide several intermediate settings of the              controller output. This discontinuous control mode is used in an attempt to reduce the cycling                        behaviour and overshoot and undershoot inherent in the two-position mode. This control mode can be          preferred whenever the performance of two-position control mode is not satisfactory. The general                form of multiposition mode is represented by
                                                      p = pi e p > ei i =1, 2,...., n
         As the error exceeds certain set limits ± ei, the controller output is adjusted to present values pi.

         Three-position Control Mode : One of the best example for multiposition control mode is three                  position control mode, which can be expressed in the following analytical form:
                                                                   P =  100% ep > +e1
                                                                      =   50% -e1 < ep < +e1
                                                                      =   0% ep < - e1
        As long as the error is between +e1 and -e1 of the setpoint, the controller stays at some nominal setting         indicated by a controller output of 50%. If the error exceeds the set point by +e1 or more, then the               output is increased to 100%. If it is less than the setpoint by -e1 or more, the controller output is                   reduced to zero. Figure 2.2.2.1 illustrates three-position mode graphically.

Fig. 2.2.2.1 Three-position controller action

         The three-position control mode usually requires a more complicated final control element, because it            must have more than two settings. Fig. 2.2.2.2 shows the relationship between the error and controller          output for a three-position control. The finite time required for final control element to change from one          position to another is also shown. The graph shows the overshoot and undershoots of error around the          upper and lower setpoints. This is due to both the process lag time and controller lag time, indicated by          the finite time required for control element to reach new setting.

        Fig.

Fig. 2.2.2.2 Relationship between error and three-position controller action, including the

effects of lag.

2.2.3 CONTINUOS CONTROL MODES

            In continuous controller modes the controller output changes smoothly in response to the 

         error or rate of change of error. These modes are an extension of discontinuous controller  

         modes. In most of the industrial processes one or combination of continuous controllers 

         are preferred.

         Analog Controller ( P, I, D, PI, PD, PID )

The Control Units are in general build on the control principles

• proportional controller
• integral controller
• derivative controller

Proportional Controller (P-Controller)

• a linear relationship exits between the controller output and the error
• range of error to cover the 0% to 100%  controller output is called proportional band
• 
  
  
  p=Kp Ep + Po   (2.0)
  

where Kp = proportional gain (% per %) 

p0 = controller output with no error or zero error (%)

The equation (2.0) represents reverse action, because the term KpEp will be subtracted 

from Po whenever the measured value increases the above setpoint which leads negative 

error. The equation for the direct action can be given by putting the negative sign in front 

of correction term i.e. - KpEp. A plot of the proportional mode output vs. error for 

equation (2.0) is shown in Fig.2.0. 

In Fig.2.0, Po has been set to 50% and two different gains have been used. It can be 

observed that proportional band is dependent on the gain. A high gain (G1) leads to large 

or fast response, but narrow band of errors within which output is not saturated. On the 

other side a low gain (G2) leads to small or slow response, but wide band of errors within 

which output is not saturated. In general, the proportional band is defined by the 

equation:  

PB = 100/Kp

The summary of characteristics of proportional control mode are as follows: 

1. If error is zero, output is constant and equal to p0.

2. If there is error, for every 1% error, a correction of Kp percent is added or 

subtracted from p0, depending on sign of error. 

3. There is a band of errors about zero magnitude PB within which the output is not 

saturated at 0% or 100%.

Offset: An important characteristic of the proportional control mode is that it produces a 

permanent residual error in the operating point of the controlled variable when a load 

change occurs and is referred to as offset. It can be minimized by larger constant Kp

which also reduces the proportional band. Figure 2.1 shows the occurrence of offset in 

proportional control mode.

- image

Consider a system under nominal load with the controller output at 50% and error zero as 

shown in Fig.2.1. If a transient error occurs, the system responds by changing controller 

output in correspondence with the transient to effect a return-to-zero error.

Problem 6 

For a proportional controller, the controlled variable is a process temperature with a range 

of 50 to 130 o

C and a setpoint of 73.5 o

C. Under nominal conditions, the setpoint is 

maintained with an output of 50%. Find the proportional offset resulting from a load 

change that requires a 55% output if the proportional gain is (a) 0.1 (b) 0.7 (c) 2.0 and (d) 

5.0. 

Solution: 

 Given data: Temperature Range = 50 to 130 oC 

 Setpoint (Sp) = 73.5 oC 

 Po = 50% 

 P = 55% 

 ep = ? 

 Offset error = ? for Kp=0.1, 0.7, 2.0 & 5.0 

For proportional controller: P = Kp ep + Po 

 ep = [p-Po] / Kp = [55 – 50] / Kp = 5 / Kp %

(a) when Kp = 0.1 Offset error, ep = 5/0.1 = 50% 

(b) when Kp = 0.7 Offset error, ep = 5/0.7 = 7.1% 

(c) when Kp = 2.0 Offset error, ep = 5/2.0 = 2.5% 

(d) when Kp = 5.0 Offset error, ep = 5/5.0 = 1% 

[It can be observed from the results that as proportional gain Kp increases the offset error 

decreases.] 

 Integral Control Mode 

• eliminates the offset error problem by allowing the controller to 
• adapt to changing external conditions by changing the zero-error output. 
• Integral action is provided by summing the error over time, multiplying that sum by a gain, and adding the result to the present controller output
•  error becomes positive or negative for an extended period of time, the integral action will begin to accumulate and make changes to the controller output
• 
  
  
  
  

Introduction to Control System

Introduction to Control System

1.1 Definitions

   System : A system is a combination or an arrangement of different physical components which act together as an entire unit to achieve certain objective.

Control system : To control means to regulate, to direct or to command. Hence a control system is an arrangement of different physical elements connected in such a manner so as to regulate, direct or command itself or some other system.

In other meaning that we can describe control system is a group of components that maintains a desired result by manipulating the value of another variable in the system. an example that control system is use such as in electrical control. pneumatic control, hydraulic control.

1.1.1 Description of electrical, pneumatic and hydraulic control

Electrical control

electrical control is a control system of an electrical current that's uses either direct current or alternating current as a source of supply. Electrical control has two fundamental laws of current law and voltage law kirchoff. research, for electrical control system involving resistor, capacitor, inductor and amp.  

 

Basic block diagram of an electrical control system manually.

Basic block diagram of an electrical control system automatically by PLC

Explanation about current law and voltage law kirchoff

• Kirchoff current law (the law of the node) states that the algebraic sum of all current entering and leaving a node is ZERO.
• Kirchoff voltage law (loop law) states at any instant the algebraic sum of the voltages around any loop in a circuit is ZERO.

Example :

• Circuit L-R-C

• Complex Impedence

 

 
Pneumatic control
Pneumatic control is the most efficient way to transmit signals and power, whether liquid or gas fluid. Pneumatic control also is a system that uses compressed air to generate power / energy to do the work. The use of pneumatic system :

• Industrial process (the food, petrochemical and industry the use of robotics)
• Speed and position control systems
• Brake systems, horns and bumps.
• Water spraying systems, elevators and doors automatically.
  
  

This system can be controlled manually and automatically.

• Manually Controlled

• Automatically Controlled

 4 elements of pneumatic control systems :

• Compressed air supply.
• Control valves.
• Connecting tube.
• Transducers.
  

Hydraulic control
Hydraulic control is a system that uses fluid to force / energy to do a job. Used in the automobile industry such as power systems, brake system, cranes, jack cars, satellites and so on. The fluid commonly used in oil. 3 elements hydraulic control system of :

• Cylinders
• Hydraulic fluid supply
• Control valves

 

                                              Block Diagram of an Hydraulic Control

1.2 Identify general terms used in process control
 
This part, we focus about general terms used in process control.

• Input : The signal is fed to the system and is known also as a reference or set point.

• Process : An on going process that consists of controlled movement systematically to produce certain results.

• Feedback : A device used to measure the output signal.

• Controller : A device that controls the process of the system and respond to the error signal to reduce the error to produce the required output.

• Output : Usually consists of variables such as temperature, pressure, velocity, etc.

• Error signal : The difference between the input signal and the feedback signal/output.

Example :

1.3 Basics process of control system


1.3.1 Temperature control  

          example: heating electrical system ( dry paint)


              


The system is start from set point (sp) . the sp is desired value or desired temperature. E.g if we wanted the desired temperature of the heating system to be 280 celcius, the sp would be adjusted to 280 celcius.

Next part of the control system that we will be examine is the process variable (PV). Which is the signal that comes from signal. Eg in the heating system, the pv is the signal from thermocouple which is the sensor for this system. The PV also called as feedback signal and it's present value or actual value of the temperature at the instant the sensor reading takes place. Since the sensor continually reading, PV also continually indicate in change temperature.

Summing junction is the place in control system where the sp is compared to the pv. that means if the sp is 280 celcius and the pv signal indicates the actual temperature is 270 celcius , the difference is 10 celcius. the summing junction is identified as sigma.

the difference between sp and pv is called error. The error can be positive value if  the sp is larger than the pv, or it can be negative value if the sp is smaller than the pv.

the controller will use gain, reset and rate to adjust the output signal in responses to the amount of error. gain, reset and rate are also called proportional, intergal and derivate (PID). the PID value can be adjusted to change the speed of responses for the system. E.g gain value, can be used to make the output change the temperature at a given rate of 1 celcius per minute, or it can be set to provide a change of 3 celcius per minute.



1.3.2 Pneumatic controller

 fig 1.3.2 

In automated industrial processes, it is always essential to keep the process variables such as temperature, flow rate, system pressure, fluid level, etc. at the desired value for safety and economical operation. Consider an example where the flow of water through a pipe has to be kept constant at some predetermined value (Fig. 1.3.2). Let the value of flow to be measured is ‘V' (process variable PV). This flow rate is compared with the required flow value say ‘V 1 ' (set point SP). The difference between these two values is the error which is sent to the controller. If any error exists, the controller adjusts the drive signal to the actuator, informing it to move the valve to give the required flow (zero error). This type of control system is called closed loop control system. It mainly includes a controller, actuator and a measuring device.

The control can be achieved by using control electronics or by pneumatic process control. The pneumatic systems are quite popular because they are safe. In the process industries like refinery and chemical plants, the atmosphere is explosive. Application of electronics based systems may be dangerous in such cases. Since the pneumatic systems use air, there are very scant chances of any fire hazards. Even though electrical actuators are available, but most of the valves employed are driven by pneumatic signals.

1.4 Open-loop and Closed-loop

1.4.1 Open-Loop system

A system in which output is dependent on input but controlling action or input is totally  independent of the output charges in output of the system. refer in figure 1.4 a.

 figure 1.4 a

The good example of an open-loop system is an electric switch. This is because output is light and switch is controller of lamp. Any change in light has no effect on the on-off position of the switch, i.e its controlling action. Some other example are traffic signal, automatic toaster system etc.

1.4.1.1 The advantanges and disadvantanges of open-loop system

Advantages	Disadvantages
1)such system simple in contruction	1) such system are inaccurate & unreliable because accuracy of such system are totally dependent on the accurate precalibration of the controller.
2) very much convenient when output is difficult to measure	2) such systems give inaccurate results if there are variations in the external environment.
3) such system are easy from maintenance point of view.	3) similarly they cannot sense internal disturbances in the system after the controller stages.
4) generally these are not troubled with the problems of stability.	4) to maintain the quality & accuracy, realibration of the controller is necessary time to time.

1.4.2 Closed-loop system

A system in which the controlling action or input is somehow dependent on the output or changes in output is called closed loop system. (for dependence of input on the output, such system uses the feedback property. refer figure 1.4 b for block diagram.

figure 1.4 b

1.4.2.1 The advantanges ang disadvantages of closed-loop system

Advantages	Disadvantages
1)accuracy of such system is always very high because controller modifies and manipulates the actuating signal such that error in the system will be zero.	1)such systems are complicated and time consuming from design point of view and hence costlier.
2) such system senses environmental changes, as well as internal disturbances and accordingly modifies the error.	2) dueto feedback, system tries to correct the error time from time. Tendency to overcorrect the error may cause oscillattions without bound in the system. Hence system has to be designed taking into consideration problems of instability due to feedback. The stability problems are severe and must be taken care of while designing the system.
3) in such system, there is reduced effect of nonlinearities and distortions.
4)bandwidth of such system i.e. operating frequency zone for such system is very high.