*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 K_{p}.
The output of the controller is a linear function of e(t).

p(t) = K_{p}e(t)
+ p(0)_{
. }(1)

Where:

K_{p
}= 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 p_{0.}

2.
If the error occurs, then for every 1 % of
error the correction of K_{p} % is achieved. If error is positive, K_{p
}% correction gets added to p_{0} and if error is negative, K_{p
}% correction gets subtracted from p_{0}.

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

4. The gain K_{p} 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 *k_{P}*
can be expressed as:

K** _{P}** = 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 200^{o}F and it
takes a 50^{o}F change in
temperature to cause full valve travel, the
percent proportional band is 50^{o}F in
200^{o}F, or 25%.

The proportional
band *P,* express the value necessary for *100%* controller output.
If *P = 0*, the gain or action factor *k_{P}*

**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 K_{i }= constant relating error
and rate

The constant K_{i}
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 K_{d }% 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 K_{p},
K_{i }and K_{d }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.