1. Lecture 9:
PID Controller

2. PID controller
The proportional–integral–derivative (PID), also called three-term, is the most widely used controller in process industry.
The output u(t) of the PID controller is the sum of three terms:
where
e(t) = r(t) − y(t), is the error (controller input)
r(t) is the reference input
y(t) is the plant output.
Ti is known as the integral time.
Td is known as the derivative time.

3. PID controller actions
Proportional: the error is multiplied by a gain. The higher is the gain, the faster is the response. However, very high gain may cause instability.
Integral: is used to remove steady-state error. However, integral action increases the overshoot and reduces system stability.
Derivative: is used to improve the transient response by reducing overshoot.

4. PID controller: transfer function
By taking Laplace transform of the equation:
we obtain the transfer function of the continuous-time PID controller:

5. Discrete PID Controller
To implement PID control using a digital computer we convert the following continuous-time equation into a discrete form:
To do this, a simple method is to approximate integral and derivative using finite differences:

6. Discrete PID controller: position form
Using finite difference approximations, we can write:
Using subscripts instead of arguments, then
This is called the position form of discrete PID controller. The drawback of this form is that in order to calculate the controller output un we need error values ek, k = 1 to n.

7. Discrete PID controller: velocity form
From the position form:
We can write
Subtracting these two equations, we obtain:
Here the current control signal un is an update of the previous value un-1. This is called the velocity form.

8. Transfer function of Discrete PID controller
The velocity form of discrete PID controller is:
Taking z-transform of both sides, we get the transfer function of discrete PID controller:

9. PID Tuning
The adjustment of controller parameters Kp, Td , and Ti in order to obtain a satisfactory response is called controller tuning.
The discrete PID parameters K0, K1, and K2 can be calculated from Kp, Td , Ti and the sampling period T.
There are many techniques for PID controller tuning. Here, we look at the classical Ziegler–Nichols (ZN) tuning rules.
Such rules will give a stable operation of the system. However, the resulting system may exhibit a large maximum overshoot in the step response, which is unacceptable. In such a case we need series of fine tunings until an acceptable result is obtained.
In fact, the Ziegler-Nichols tuning rules give an educated guess for the parameter values and provide a starting point for fine tuning, rather than giving the final settings for Kp, Td , and Ti in a single shot.

10. Ziegler-Nichols tuning rule
In order to use ZN rules, the process (while in open-loop) is subjected to a step input and the response curve (also called process reaction curve) is plotted. From this plot, the process is approximated by a first-order system plus a time delay (FOPDT):
Where
L is the time delay,
τ is the time constant and
K is the dc gain.

11. FOPDT: method 1
The point of maximum slope may be very susceptible to noise!

12. Alternative Method: the two-points method
0.63 
0.28 
t28%
t63%

13. Ziegler-Nichols tuning rule
According to ZN, the P, PI, and PID controller settings are:
Controller Kp Ti Td P ∞ PI
3.3L
PID 2L
0.5L

14. Example Solution: From the Figure: Δ = 40˚C, δ = 1 → K = 40,
The open-loop unit step response of a thermal system is shown. Obtain the transfer function of this system and use Ziegler–Nichols tuning rules to design a discrete-time PID controller (assume a sampling period of 1 sec):
Solution:
From the Figure:
Δ = 40˚C, δ = 1 → K = 40,
L = 5 sec,
τ = 20 sec.
Hence, the transfer function of the plant is

15. According to ZN table, the settings of PID controller are:
The discrete-time controller parameters are
The transfer function of the controller is thus:

16. Practical issues with PID control
In practice, some problems may occur with PID controllers. Here, we will discuss two problems:
Integrator windup: a problem associated with integral action and happens as a result of actuator saturation.
Derivative kick: a problem associated with derivative action and happens as a result of fast changes (such as step changes) in set point or reference inputs.

17. Integrator windup
It is often desirable to have short rise and settling times, and small overshoot (response in red).
With integrator windup, the rise time, overshoot and settling time become much larger (response in purple). The question is why?

18. Why large rise time?
In practice, a control action is limited by physical constraints. For example, the maximum output voltage from a device is limited.
Suppose that as a result of, say, large reference steps, large errors are generated, and hence, the controller output becomes larger than the limits of the actuator (actuator becomes saturated).
Then, large controller output can not be converted into large actuator signals. This makes the system slower than the ideal case (rise time becomes larger).

19. Why large overshoot & settling time?
Remember that: integral control is responsible for making the error zero at steady-state.
At steady-state, proportional controller output is zero while integral controller output is non-zero and driving the process.
In the ideal case, the value of integrator output is proportional to the area highlighted below.
As the error signal does not return to zero quickly, the integral term keeps adding up continuously and becomes quite large. Further buildup of the integral term while the controller is saturated is referred to as integral windup.

20. Why large overshoot & settling time?
Let us do the same thought experiment with the case of actuator saturation. Due to the increase in rise time, the integral of area (+) is much larger than the required integrator output at steady state. The enormous overshoot and large settling time are therefore absolute necessary to reduce the integrated error with negative error plane marked with area (-).
In summary, integrator windup occurs if the actuator is in saturation combined with integrating action in the controller.

21. Anti-windup
There are many anti-windup methods to solve the integrator windup problem.
For example, integral action can be stopped as long as the actuator is in saturation (conditional integration).
Of course, the rise time can not be reduced with anti-windup, but the overshoot and settling time can be much reduced.
Source: by Matthias Bauerdick

22. Derivative Kick
Another possible problem with PID is caused by the derivative action of the controller.
This may happen when the set-point r(t) changes sharply (e.g. a step change).
Under such a condition, the derivative term can give the controller output u(t) a kick (very high value), known as a derivative kick.
This is usually avoided in practice by taking the derivative of the controlled variable y(t), instead of the error e(t):