(10) PID Controller

1. (10) PID Controller

2. Continuous-time PID controller • Proportional–Integral–Derivative (PID) is the most widely used controller in process industry. • The output u(t)of 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. Note that system with only P-control has a steady-state error (Offset). • 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 and reduce overshoot.

4. Discrete PID Controller • To implement PID control using digital computer we convert the continuous-time equation: into discrete form by approximating integral and derivative using finite differences:

5. Discrete PID controller: position form • Using finite difference approximations, we can write • This is the positionform of discrete PID controller. Its drawback is that the need to store error values ek, k = 1 to nin order to calculate the controller output un.

6. Discrete PID controller: velocity form • From the position form: • Subtracting these two equations, we obtain the velocityform: • Here the current control signal un is an update of the previous value un-1. Only error values en, en-1, en-2in addition to un-1need to be stored in order to calculate un.

7. 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:

8. PID Controller Tuning • To obtain satisfactory response with PID control, the parameters {Kp, Td, Ti}should be tuned. • This can be done by trial and error(time-consuming and costly). • Many systematic methods for PID tuning such as classical Ziegler–Nichols (ZN) tuning rules. • Although ZN rules may exhibit large overshoot, they give educated guess for parameter values and provide a starting point for fine tuning until an acceptable result is obtained. • Discrete PID parameters K0, K1, and K2 can then be calculated from Kp, Td, Ti and the given sampling period T.

9. Ziegler Nichols’ first method • Start at steady state. • Apply a step input to the process (open-loop). • Record the process reaction curve (the step response). If the response is S-shaped, this method is applicable. • Fit a a first-order system plus a time delay (FOPDT) model to the process reaction curve. Where Lis the time delay, τ is the time constant and K is the dc gain.

10. Tuning rules With the aid of the following table, find the controller parameters (P, PI, or PID) corresponding to the FOPDT model obtained.

11. Fitting FOPDT model to process reaction curve Method 2: two-points method Method 1: Point of maximum slope The point of maximum slope may be very susceptible to noise!

12. Example The open-loop unit step response of a thermal system is shown below. Obtain the FOPDT model approximating this response. 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

13. According to ZN table, the settings of the continuous PID controller and the corresponding discrete-time PID controller parameters can be obtained as follows: • The transfer function of the controller is: