Linear Control Systems Lectures #5 - PID Controller. Guillaume Drion Academic year

Size: px

Start display at page:

Download "Linear Control Systems Lectures #5 - PID Controller. Guillaume Drion Academic year"

Lee Long
5 years ago
Views:

1 Linear Control Systems Lectures #5 - PID Controller Guillaume Drion Academic year

2 Outline PID controller: general form Effects of the proportional, integral and derivative actions PID tuning Integrator windup and setpoint weighting 2

3 A simple controller to control complex systems: PID Closing the loop: the controller signal enters in the input Input SYSTEM Output CONTROLLER Classical controller: Proportional-Integral-Derivative (PID) where is an error measure between a reference and the output of the system. 3

4 The classical controller: PID controller PID stands for Proportional-Integral-Derivative. 4

5 The classical controller: PID controller Proportional term: considers the current value of the error. 5

6 The classical controller: PID controller Integral term: considers the past values of the error. 6

7 The classical controller: PID controller Derivative term: predicts the future values of the error. 7

8 The classical controller: PID controller Derivative term: predicts the future values of the error.!! Most PID controllers do not use derivative action!! 8

9 PID controller design: shaping the feedback gains Controller design: shaping the loop gains to improve the static and dynamic performances of the controller. 9

10 Two types of PID controllers Controller design: shaping the loop gains to improve the static and dynamic performances of the controller. In (a), P, I and D act on control error. In (b), I acts on control error, and P and D act on systems output. 10

11 Outline PID controller: general form Effects of the proportional, integral and derivative actions PID tuning Integrator windup and setpoint weighting 11

12 PID controller with error feedback The two forms encountered in control systems: T i = integral time constant, T d = derivative time constant. 12

13 PID controller design: pure proportional feedback Pure proportional feedback: steady-state error! Indeed: For a pure proportional feedback at steady-state, the error is given by The error goes to zero as k p goes to infinity, but increasing k p will eventually destabilize the closed-loop system (gain margin). 13

14 PID controller design: pure proportional feedback Pure proportional feedback: steady-state error! Indeed: For a pure proportional feedback at steady-state, the error is given by To avoid steady-state error, we can use a feedforward term: with u ff us called reset in the PID literature, and has to be adjusted manually. 14

15 PID controller design: derivative action Derivative action: predictive and anticipatory action. If k d is increased, system responses is damped. But derivative action amplifies high frequencies (hence reduces noise rejection). Derivative action should be used with a filter (= lead compensator): C d (s) = k ds 1+sT f 15

16 PID controller design: integral action Integral action: no steady-state error. 16

17 PID controller design 17

18 Outline PID controller: general form Effects of the proportional, integral and derivative actions PID tuning Integrator windup and setpoint weighting 18

19 Ziegler-Nichol s tuning Feedback gains are extracted from the dynamical response of the open-loop process. Two methods: a time-domain method and a frequency-domain method 19

20 Ziegler-Nichol s tuning - time-domain method Feedback gains are extracted from the step response of the process. a/ is an approximation of the time delay of the system. is the steepest slope of the step response. 20

21 Ziegler-Nichol s tuning - frequency-domain method Start with zero gain, and increase proportional gain until systems start to oscillate. k c = critical proportional gain, T c = period of oscillation,.! c = 2 T c 21

22 Ziegler-Nichol s tuning - Improvements Time-domain method: characterize the step response by K, and T in the model 22

23 Outline PID controller: general form Effects of the proportional, integral and derivative actions PID tuning Integrator windup and setpoint weighting 23

24 Integrator windup If the control variable saturates (i.e. reaches the actuator limits), there will be a residual error that will be continuously integrated by the controller. The integral term will build up, and eventually become very large. The control signal will then remain saturated even when the error changes, and it may take a long time before the integrator and the controller output come inside the saturation range. Integrator windup 24

25 Integrator windup 25

26 Anti-windup Anti-windup: avoiding error integration while in saturation 26

27 Integrator windup 27

28 Integrator windup If the control variable saturates (i.e. reaches the actuator limits), there will be a residual error that will be continuously integrated by the controller. The integral term will build up, and eventually become very large. 28

29 Setpoint weighting When there is an abrupt change in the reference, the proportional and derivative actions can become very big and lead to a large initial peak. To reduce this peak, we can only show a fraction of the reference to the proportional and derivative controllers: Setpoint (or reference) weight ( 2 [0, 1] ) Setpoint (or reference) weight ( 2 [0, 1] ) No setpoint weight on the integral action! This would lead to systematic steady-state error. 29

30 Setpoint weighting 30

31 The classical controller: PID controller PID stands for Proportional-Integral-Derivative. 31

MM7 Practical Issues Using PID Controllers

MM7 Practical Issues Using PID Controllers Readings: FC textbook: Section 4.2.7 Integrator Antiwindup p.196-200 Extra reading: Hou Ming s lecture notes p.60-69 Extra reading: M.J. Willis notes on PID controler