Module 6. Process control

Lesson 29

TYPES OF CONTROLLER MODES

29.1  Proportional Control Mode

In the proportional (throttling) mode, there is a continuous linear relation between value of the controlled variable and position of the final control element. In this control mode, the output of the controller is proportional to error e(t). The relation between the error e(t) and the controller output p is determined by a constant called proportional gain constant denoted as Kp. The output of the controller is a linear function of e(t).

            p(t) = Kpe(t) + p(0)                                       ……………………………………………………………. (1)

            Where:

                        Kp = Proportional gain constant

                        P(0) = Controller output with zero error or bias

The direct and reverse action is possible in the proportional controller mode. The error may be positive or negative because error (r-b) depending upon whether b is less or greater than the reference setpoint r(t).

If the controlled variable i.e. input to the controller increases, causing increase in the controller output, the action is called direct action. For example the output valve is to be controlled to maintain the liquid level in a tank. If the level increases, the valve should be opened more to maintain the level. On the other hand if the variable decreases, causing increase in the controller output, the action is called reverse action. Conversely, increase in the controlled variable, causing decrease in controller output is also a reverse action.

29.1.1  Characteristics of proportional mode

The various characteristics of the proportional mode are:

1.      When the error is zero, the controller output is constant equal to p0.

2.      If the error occurs, then for every 1 % of error the correction of Kp % is achieved. If error is positive, Kp % correction gets added to p0 and if error is negative, Kp % correction gets subtracted from p0.

3.      The band of error exists for which the output of the controller is between 0 to 100%.

4.      The gain Kp and error band PB are inversely proportional to each other.

29.1.2  Proportional gain

Proportional gain is the percentage change of the controller output relative to the percentage change in controller input. Gain, also called sensitivity, compares the ratio of amount of change in the final control element to amount of change in the controlled variable. Mathematically, gain and sensitivity are reciprocal to proportional band. The gain kP can be expressed as:

            KP = 100 / P

            Where

                        P = proportional band

The gain determines how fast the system responds. If the value is too large the system will be in danger to oscillate and/or become unstable. If the value is too small the system error or deviation from set point will be very large.

29.1.3  Proportional band

Proportional band, (also called throttling range), is the change in value of the controlled variable that causes full travel of the final control element. The  proportional  band  of  a particular  instrument  is  expressed  as  a  percent  of  full range. For example,  if  full range of  an  instrument  is  200oF  and  it  takes  a  50oF change  in  temperature to cause full valve  travel,  the  percent  proportional  band  is 50oF  in  200oF,  or  25%.

The proportional band P, express the value necessary for 100% controller output. If P = 0, the gain or action factor kP would be infinity - the control action would be ON/OFF.

29.1.4  Offset

Offset, also called droop, is deviation that remains after a process has stabilized. Offset is an inherent characteristic of the proportional mode of control.  

29.2  Integral Control Mode

With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset that remains when proportional control is used. In such a controller, the value of the controller output p(t) is changed at a rate which is proportional to the actuating error signal e(t). Mathematically it is expressed as:

           

Where Ki = constant relating error and rate

The constant Ki is also called integral constant. Integrating the above equation, the actual output at any time t can be obtained as

           

Where p(0) = controller output when integral action starts i.e. at t = 0.

29.2.1  Advantages

1.      Integral controllers tend to respond slowly at first, but over a period of time they tend to eliminate errors.

2.      The integral controller eliminates the steady-state error, but has the poor transient response and leads to instability.

29.2.2  Characteristics of integral mode

1.      If error is zero, the output remains at a fixed value equal to what it was, when the error become zero.

2.      If the error is not zero, then the output begins to increase or decrease, at a rate Ki % per second for every ± 1 % of error.

3.      The inverse of Ki is called integral time and denoted as Ti.

     

29.2.3  Application of P and I control modes

The comparison of proportional and integral mode behavior at the time of occurrence of an error signal is given in Table 29.1.

Table 29.1 Comparison of P and I controllers

Controller

Initial behaviour

Steady state behaviour

p

Acts immediately.

Action according to Kp.

Offset error always present.

Larger the Kp, smaller the error.

I

Acts slowly.

It is the time integral of the error signal.

Error signal always become zero.

 

It can be seen that proportional mode is more favorable at the start while the integral is better for steady state response. In pure integral mode, error can oscillate about zero and can be cyclic. Hence in practice, integral mode is never used alone but combined with the proportional mode, to harness the merits of both modes. 

29.4  Derivative Control Mode

In this mode, the output of the controller depends on the rate of change of error. Hence, it is also called rate action mode or anticipatory action mode. The mathematical equation for the mode is:

           

            Where Kd = Derivative gain constant

The derivative gain constant indicates by how much % the controller output must change for every % per second rate of change of the error. Generally Kd is expressed in minutes. The important feature of this type of control mode is that for a given rate of change or error signal, there is a unique value of the controller output.

The advantage of the derivative control action is that it responds to the rate of change of error and can produce the significant correction before the magnitude of the actuating error becomes too large. Derivative control thus anticipates the actuating error, initiates an early corrective action and tends to increase stability of the system, improving the transient response. The derivative or differential controller is never used alone because when error is zero or constant, the controller has either no output or the nominal output for zero error.

29.4.1  Characteristics of derivative control mode

For a given rate of change of error signal, there is a unique value of the controller output. When the error is zero, the controller output is zero. When the error is constant i.e. rate of change of error is zero, the controller output is zero. When the error is changing, the controller output changes by Kd % for even 1 % per second rate of change of error.

When the error is zero or a constant, the derivative controller output is zero. Hence, it is never used alone. Its gain should be small because faster rate of change of error can cause very large sudden change of controller output. This may lead to instability of the system.

29.4.2  Advantages

1.      With sudden changes in the system the derivative controller will compensate the output fast.

2.      The long term effects the controller allows huge steady state errors.

3.      A derivative controller will in general have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response.

29.4.3  Drawbacks of derivative action

1.      The output of controller is zero at constant error condition.

2.      It will amplify the noise present in the error signal.

29.5  Composite Control Modes

Due to offset error, proportional mode is not used alone. Similarly, integral and derivative modes are not used individually in practice. Thus, to take the advantages of various modes together, the composite control modes are used. Composite modes of controller operation combine advantages of each ‘pure’ mode. The various composite control modes are:

1.      Proportional + Integral mode (PI)

2.      Proportional + Derivative Mode (PD)

3.      Proportional + Integral + Derivative Mode (PID)

29.6  Proportional-Integral Control

This is a control mode that results from a combination of proportional mode and integral mode. The analytical expression for this is:

            p(t) = Kp e(t) + Kp Ki ∫ e(t) dt + p(0)

The main advantage of this composite control mode is that the one-to-one correspondence of proportional mode is available and integral mode eliminates the inherent offset.

29.7  Proportional-Derivative Control Mode

This involves the series or cascade combination of proportional and derivative modes. The controller output could be expressed as:

           

This system cannot eliminate the offset of proportional controllers. However, it can handle fast process load changes as long as the offset error is acceptable.

29.8  Proportional Integral Derivative Controller (Three Mode Controllers)

The three mode controller uses proportional, integral and derivative (PID) action and is the most versatile of all controller actions. The proportional part of this controller multiplies the error by a constant. The integral part integrates the error. Finally, the derivative part differentiates the error. The functions of the individual proportional, integral and derivative controllers complement each other. If they are combined it is possible to make a system that responds quickly to changes (derivative), tracks required positions (proportional), and reduces steady state errors (integral). The output of the controller is the sum of the previous three signals as given in the following equation:

           

Where Kp, Ki and Kd are the proportional, integral and derivative gains respectively.

The proportional, integral and derivative terms must be individually adjusted or ‘tuned’ to a particular system.

29.8.1  Advantages

1.      This mode eliminates the offset of proportional mode. 

2.      It provides the most accurate and stable control of the three controller types.

3.      It is recommended in systems where compensation is required for frequent changes in load, set point, and available energy.

4.      It can help achieve the fastest response time and smallest overshoot.