1. LECTURE#11 PID CONTROL AUTOMATION & ROBOTICS
By: Engr. Irfan Ahmed Halepoto
Assistant Professor 1

2. Integral Term
While the proportional term considers the current size of e(t) only at the time of the controller calculation,
the integral term considers the history of the error,
how long and how far the measured process variable has been from the set point over time.
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error.
Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously.
The accumulated error is then multiplied by the integral gain and added to the controller output.

3. Role of Integral Mode
The role of integral control is to eliminate the "droop" seen in proportional control.
Integral is also called "reset control” as the set point is continuously reset as long as an error is present
Thus it automatically pushes the process value toward the desired set point once droop occurs.
Even when the set point is changed, the integral control will work to eliminate droop.
proportional and integral control

4. Integral Action: Reset Curve
Integral control is also referred to as reset control as the set point is continuously reset as long as an error is present
Integral adjustments that affect the output are labeled 3 ways:
Gain - expressed as a whole number
Reset - Expressed in repeats per minute
Integral Time - Expressed in minutes per reset

5. Integral Action-Reset Mode
Integral action drives the steady-state error towards ZERO but slows the response.
since the error must accumulate before a significant response is output from the controller.
Since an integrator introduces a system pole at the origin, an integrator can disturb loop stability.

6. Integral Function Controller output is proportional to error e.
Output of Integral control is constant only e=0, no steady state error.
Integral control is always slower than that of P control,
Reduce system stability (system pole at the origin).
Open loop gain is proportional to S0 , increase S0 reduce system stability.

7. Integral Control Mode
“The persistent mode”

8. Integral Term Error Calculation
The integral sum of error is computed as the shaded areas between the SP and PV traces.
Each box in the plot has an integral sum of 20 (2 high by 10 wide). If we count the number of boxes (including fractions of boxes) contained in the shaded areas, we can compute the integral sum of error.

9. Integral Term Error Calculation
So when the PV first crosses the set point at around t = 32, the integral sum has grown to about 135. We write the integral term of the PI controller as:

10. Integral Term Error Calculation
Note that the integral of each shaded portion has the same sign as the error.
Since the integral sum starts accumulating when the controller is first put in automatic, the total integral sum grows as long as e(t) is positive and shrinks when it is negative.
At time t = 60 min on the plots, the integral sum is 135 – 34 = 101.
The response is largely settled out at t = 90 min, and the integral sum is then 135 – = 108.

11. Integral Control Mode Response
In Integral Control Mode as long as an error is present, integral mode will continuously increment or decrement the controller’s output to reduce the error or tend the error toward zero.
If error is large, integral mode will increment or decrement the controller output fast, if the error is small, the changes will be slower.
For a given error, the speed of the integral action is set by the controller’s integral time setting (Ti).
A large value of Ti (long integral time) results in a slow integral action, while small value of Ti (short integral time) results in a fast integral action.
If the integral time is set too long, the controller will be sluggish, if it is set too short, the control lop will oscillate and become unstable.

12. Integral Control Mode-Units
Most controllers use integral time in minutes as the unit of measure for integral control, but some others use integral time in seconds, integral gain in repeats per minute or repeats per second.
Integral Time
Integral Gain
Minutes
Seconds
Rep / Min
Rep / Sec
0.05 3 20
0.333
0.1 6 10
0.167
0.2 12 5
0.0833
0.5 30 2
0.0333 1 60
0.0167
120
300
600
1200
 Units of the integral control mode

13. Proportional + Integral Controller
Because of the introduction of offset in a control process, proportional control alone is often used in conjunction with Integral control
Offset is the difference between set point and the measured value after corrective action has taken place

14. Proportional + Integral Controller
Commonly called the PI controller, its controller output is made up of the sum of the proportional and integral control actions
PI Controller Algorithm

15. PI Control Mode

16. PI Control Mode In PI control mode, the controller make the following:
Multiplies the Error by the Proportional Gain (Kp) and Added to the Integral error multiplied by Ki, to get the controller output.
The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller.
However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction).

17. PI Control Action P to improve response time and reject disturbance,
I to eliminate steady state error.

18. Proportional & Integral Gains
Comments:
Integral control has removed the steady-state error and improved the transient response, but it has also increased the system settling time.
Settling times can be lowered by increasing the gain.
This will increase the system bandwidth, but it will also decrease the stability margin
Step responses of the closed loop system for a = 3 and proportional gains of Kp = 25, 50, and 75. Where a = Ki / Kp.

19. Challenges of PI Control
There are challenges in employing the PI algorithm:
The two tuning parameters interact with each other and their influence must be balanced by the designer.  
The integral term tends to increase the oscillatory or rolling behavior of the process response.
Because the two tuning parameters interact with each other, it can be challenging to arrive at “best” tuning values.

20. Derivative Action
Derivative control is rarely used in controllers, because it is very sensitive to noise and it makes trial-and-error tuning more difficult.
Derivative action acts on the derivative or rate of change of the control error.
This provides a fast response, as opposed to the integral action, but cannot accommodate constant errors (i.e. the derivative of a constant, nonzero error is 0).
Derivatives have a phase of +90 degrees leading to an anticipatory or predictive response.
However, derivative control will produce large control signals in response to high frequency control errors such as set point changes (step command) and measurement noise.
In order to use derivative control the transfer functions must be proper. This often requires a pole to added to the controller.

21. Derivative Mode
For rapid load changes, the derivative mode is typically used to prevent oscillation in a process system
The derivative mode responds to the rate of change of the error signal rather than its amplitude
Derivative mode is never used by itself, but in combination with other modes
Derivative action cannot remove offset

22. Role of Derivative
The role of derivative control is to reduce or eliminate overshoot and undershoot.
Derivative control, also called "rate", measures the rate of parameter change in the process value.
For Example:
If the temperature rises too fast, it will switch the heater off to prevent overshoot.
If the temperature is falling too quick, more power will be provided to the heater to reduce undershoot.
Derivative action anticipates overshoot and undershoot and makes adjustments to the power given to the heater to prevent them.

23. Derivative Control

24. Derivative Function or
Output of controller is proportional to the differential of error or
Prediction:
Adjusting the output according to speed of error.
D function must be formed to PD or PID controllers.
Controller takes no action if rate of change very small, accumulate error.

25. Derivative Control Mode
“The predictive mode”

26. Derivative Control Mode
We do not achieve zero offset; do not return to set point!

27. Derivative Control Mode Response
Derivative control mode is sometimes called Rate, as it produces an output based on the rate of change of the error.
Derivative mode produces more control action if the error changes at a faster rate.
If there is no change in the error, the derivative action is zero.
Derivative mode has an adjustable setting called Derivative Time (Td).
The larger the derivative time setting, the more derivative action is produced.
If the derivative time is set too long, oscillations will occur and the control loop will run unstable.
Derivative control mode can make a control loop respond a little faster than with PI control alone.

28. Proportional + Derivative Controller

29. Proportional + Derivative Controller
In Proportional Derivative mode, the controller make the following:
Multiplies the Error by the Proportional Gain (Kp) and Added to the Derivative error multiplied by Kd, to get the controller output.
The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint.
Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the stability.
However, differentiation of a signal amplifies noise and thus this is highly sensitive to noise in the error term, and can cause a process to become unstable.

30. Proportional and Derivative Gains
Comments:
The PD controller has decreased the system settling time considerably; however, to control the steady-state error, the derivative gain KD must be high.
This will decrease the response times and increase the bandwidth of the system and may make it susceptible to noise.
Step responses of the closed loop system for a =10 and derivative gains of KD = 10, 27, 50, and 75. Where a = Kp / Kd.

31. Proportional + Integral + Derivative Controller
Commonly called the PID controller, its controller output is made up of the sum of the proportional, integral, and derivative control actions .
Standard (Noninteractive) PID controller algorithm.

32. PID Controller
In Proportional Derivative Integral mode, the controller make the following:
Multiplies the Error by the Proportional Gain (Kp), Added to the Derivative error multiplied by Kd and Added to the Integral error multiplied by Ki, to get the controller output.

33. PID controller’s response
PID control provides more control action sooner than what is possible with P or PI control. This reduces the effect of a disturbance, and shortens the time it takes for the level to return to its set point.
P, PI, and PID controllers’ response to a disturbance

34. PD and PID Control Properties
Steady state, de/dt=0, PD control has steady state error.
D function reduces oscillation, increases system stability.
Adding D increase open loop gain, increase response speed.
Sensitive to disturbance.

35. Proportional, Integral and Derivative Gains
Comments:
Using both integral and derivative control (PID) has removed steady-state error and decreased system settling times while maintaining a reasonable transient response.
Step responses of the closed-loop system for a =15, b = 50, and derivative gains of KD = 5, 10, and 15. Where a = Kp / Kd and b = Ki / Kd

36. PID CONTROLLER…..

37. PI Summery Characteristics
Compensator increases the system type by one, which helps with error control.
Increases phase-lag at low frequencies.
Generally, increases damping, rise times, and settling times and reduces overshoot.
Decreases bandwidth.
Not sensitive to high frequency noise.
Acts as a low-pass filter.

38. PD Summery Characteristics
Compensator is anticipatory; it responds to the error and its derivative.
Phase lead is provided starting one decade below the zero.
Generally, increases damping and reduces %OS.
Generally, reduces rise and settling times.
Increases bandwidth.
Increases phase and gain margins.
May render a system susceptible to high frequency noise.
Acts as a high-pass filter.

39. PID Summery Characteristics
Combined effects of PI and PD compensation.
Cascade of a PI and PD compensator.

40. PID Limitation
A Proportional+Integral controller is optimal for a first order linear process without time-delays.
Similarly, the PID controller is optimal for a second order linear process without time-delays.
In practice, process characteristics are nonlinear and can change with time.
Thus the linear model used for initial controller design may not be applicable when process conditions change or when the process is operated at another region.

41. PID Limitation------ SOLUTION
One solution is to have a series of stored controller settings, each pertinent to a specific operating zone. Once it is detected that the operating regime has changed, the appropriate settings are switched in.
This strategy (called parameter- or gain-scheduled control) is sometimes used in applications where the operating regions are changed according to a preset and constant pattern.
In applications to continuous systems, however, the technique is not so effective.
Any other option……………………………..
A more elegant technique is to implement the controller within an adaptive framework. Here the parameters of a linear model are updated regularly to reflect current process characteristics.
The settings of the controller can be updated continuously according to changes in process characteristics.
Such devices are therefore called auto-tuning/adaptive/self-tuning controllers.