Distributed Real-Time Control Systems

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1 Distributed Real-Time Control Systems Chapter 9 Discrete PID Control 1

2 Computer Control 2

3 Approximation of Continuous Time Controllers Design Strategy: Design a continuous time controller C c (s) and then compute a discrete time approximation C d (z). Advantage: Use already available knowledge in continuous time controller design. Computer clock A/D e(kt) Control Algorithm u(kt) D/A» E(s) C c (s) U(s) Problem: Determine C d (z), such that: A/D + C d (z) + D/A» C c (s) 3

4 Consider: Example Us () a C c Hs (s) () = U(s) u(t) + au(t) = ay(t) E(s) Ys () = a s + s a+ a Writing in the integral form: t u() t = [- au( t) + ay( t)] dt Û ò o kt -T u( kt ) = [- au( t) + ay( t)] dt + [- au( t) + ay( t)] dt Û ò o T ò kt-t u( kt ) = u( kt -1) + area of au(t)+ay(t) in the interval kt-1 < t < kt 4

5 Different Approximations kt-t kt Forward approximation kt-t kt Backward approximation. kt-t kt Trapezoidal approximation. 5

6 Forward Approximation kt-t kt Forward approximation ukt ( ) = ukt ( - T) + T[ -aukt ( - T) + aykt ( -T)] = (1-aT) u( kt - T) + aty( kt -T) H F -1 atz a () z = = (1-aT) z ( z - 1)/ T + a 6

7 Backward Approximation kt-t kt Backward approximation. u( kt ) = u( kt - T ) + T[ - au( kt ) + ay( kt )] u( kt -T ) at = + ykt ( ) (1 + at) (1 + at) H F at 1 atz a ( z) = = = at 1 - z /(1 + at) z(1 + at) -1 ( z - 1) / Tz + a 7

8 Trapezoidal Approximation kt-t kt Trapezoidal approximation. T u( kt ) = u( kt - T ) + [-au( kt - T ) + ay( kt -T )- au( kt ) + ay( kt)] at /2 at /2 = ukt ( - T) + [ ykt ( - T) + ykt ( )] 1 + at / 2 (1 + at / 2) H F at( z + 1) a ( z) = = (2 + at) z + at -2 (2 / T)[( z - 1) /( z + 1)] + a 8

9 Approximation Methods Euler s Method or Forward Difference: d x( t+ T) -x( t) xt () = dt T Transformation s = z -1 T Backward Difference: d x() t -x( t-t) xt () = dt T Transformation s = z -1 zt Trapezoidal, Tustin s Method or Bilinear: d x t + x(t T) dt 2 = x t x(t T) T Transformation s = 2 z -1 Tz+ 1 9

10 Stability Regions Mapping Mapping of the left complex semi-plane (in light blue) s = z -1 T s = z -1 zt s = 2 z -1 Tz+ 1 Im Im Im 1 Re 1 Re 1 Re Forward difference Backward difference Tustin 10

11 Frequency Interpretation H( z) H( z) H( z) 1 1 Gs () = = 3 2 s + 1.5s + s+ 0.5 ( s+ 1)( s i)( s i) Forward Backward Tustin = = = z + ( T) z + (6-6T + 2 T ) z+ ( T - 2T + 3T - 2) ( T + 2T + 3T + 2) z + (-2T - 6T - 6) z + (3T + 6) z ( T + 4T + 12T T 2T z Discretizations T ( z+ 1) ) z + (3T + 4T -12T - 48) z + (3T - 4T -12T + 48) z+ ( T - 4T + 12T -16) 50 T=0.5 s, W Nyquist =2p rad/s 50 T=1.0 s, W Nyquist = p rad/s Mag [db] Freq [rad/s] Freq [rad/s] Fase [grau] fase [grau] Freq [rad/s] Freq [rad/s] Legend: Forward, Backward, Tustin, Continuous. 11

12 Frequency Mapping Tustin s method provides a map of the s-plane imaginary axis (continuous frequencies) to the unit circumference (discrete frequencies) s = jw c z = exp(jw d T) jw c e jw dt -jw c e- jw dt This map is not linear. It compresses the high frequencies.! c = 2 T tan(! dt 2 ) 12

13 Frequency Pre-warping We can correct the transformation at a particular point in frequency w 1 : s' = w1 tan( w T Example for T= 0.01 sec, w 1 = 60p. 1 T z / 2) 2 z Default frequency mapping 60 Pre-warp for fc = 30Hz f cont (Hz) 40 f cont (Hz) f disc (Hz) f disc (Hz) 13

14 Discretization of the PID controller Forward: s =,-. / Backward: s =,-.,/ Tustin: s = 0 /,-.,1. Integral Term I(z) E(z) = T z 1 I(z) E(z) = zt z 1 I(z) T z + 1 = E(z) 2(z 1) I(s) E(s) = 1 s i k = i k 1 + Te(k 1) 1 sample delay i k = i k 1 + Te(k) No delay i k = i k 1 + T 2 e k + e(k 1) No delay. Better noise attenuation (zero at -1) Derivative Term D(z) Y(z) = a z + 1 z 1 + at K < D(z) Y(z) = a z + 1 z at K < D(z) Y(z) = a z + 1 z 1 + at + at 1 2K < 2K < D(s) Y(s) = K <as K < s + a pole: p = A lim p A F pole: p A lim p = A F pole: p = A-?/ A 1?/ lim p = A F May become unstable Always stable Always stable but high frequency noise amplification (pole tends to -1) 14

15 Discrete PID Control Sampling time = T. Discrete time = k. ( bu ( kt) y( )) P( kt) = K kt p c - I( kt) = I( kt -T ) + K p æ e( kt) + e( kt -T ) KiTç è 2 ö ø Tustin K K K a d p d D( kh) = D( kt -T ) - -T K + at K + at d d ( y( kt) - y( kt )) Backward The backward difference method is chose because the pole goes to 0 when K d is zero which represent a pure delay. For the Tustin s method, the pole goes to -1, which amplifies high frequency noise. 15

16 Digital Implementation K1 = Kp*b K2 = Kp*Ki*T/2 K3 = Kd/(Kd+a*T) K4 = Kp*Kd*a/(Kd+a*T) y_ant = 0; i_ant = 0; d_ant = 0; e_ant = 0; Loop wait for next sampling time y = read A/D e = ref y p = K1*ref Kp*y i = i_ant + K2*(e + e_ant) d = K3*d_ant K4*(y - y_ant) u = p+i+d y_ant = y i_ant = i d_ant = d e_ant = e write u to D/A 16

17 Other Common Improvements Substitute K p e by K p e e to reduce the rise time. Dead zone to avoid switching Mean or Median filter for noise reduction. Derivative sample period. Integrator anti-windup. 17

18 Homework Write the pseudo code of the discrete PID with anti-windup. 18

19 Analysis of the ZOH Impulse Response 19

20 Analysis of the ZOH Frequency Response H zoh (j!) =Te j! T 2 sinc! T 2 At low frequency, a ZOH can be approximated by an additional phase delay: wt df = - 2 Influences negativelly the phase margin. 20

21 Choice of the Sampling Period Intuition may suggest the faster, the better, but: Uses more computational resources/energy Increases noise sensitivity (lower signalto noise ratio) T should be chosen as a function of the desired closed loop bandwidth f c. Rule of thumb: T f c < 0.05 i.e. f s > 20 f c With this condition, at f c the ZOH has Gain > 99% and Phase lag < 10 deg. 21

22 Homework It is intended that a closed loop control system attenuates an external disturbance in less than 1 second. 1. What closed-loop bandwidth should you choose for your design? 2. For that bandwidth, which is an adequate sampling frequency? Note: In a system without significant overshoot: the bandwidth of a system (in rad/s) is approximately the value of the slowest pole the time constant is the inverse of the pole (in rad/s). the settling time is approximately 3 times the time constant. 22

23 Quantization Clock A/D Control Algorithm D/A Process 23

24 Quantization Problems Quantization at A/D: Determines which values can actually be measured at the sensor. Setting desired values different from the ones that can be measured will generate feedback errors in steady state and led to output jitter. Quantization at D/A: Determines which output values can actually be achieved by the system. Trying to compensate errors that cannot be cancelled by the actuator may also lead to jitter effects. Quantization at controller. Quantization errors at the coefficients of the controller and at the internal variables representations may lead to controller performance issues. Most notably, can destabilize a systems. 24

25 Truncation: e = x-xq <q Quantization Models x q e q double x; int xq; xq=(int)x; x 3q x q 2q Rounding: e= x-xq <0.5q double x; int xq; xq=(int)(x+0.5); x q q/ 2 Whenever possible, rounding should be used, because the average error is smaller. Note : this result is independent of the representation. 3q/2 x 5q/2 e q/2 x 25

26 Mitigating Quantization Errors Use a deadzone on the error to prevent controller actions to quantization noise. Increase resolution of your A/D, D/A Low pass filter the input (averaging) and represent with floating point. Use output dithering. 26

27 Example: PI controller T=0.02s Set point = 15 27

28 Example: A/D Quantization Quantization step = 2 28

29 Example: D/A Quantization Set point = 10 Quantization step = 1 29

30 Workaround: Error Deadzone Set point = 17.5 Deadband: [ ] 30

31 Deadzone PSEUDO CODE y function y = deadzone(x, xmin, xmax) { if ( x >= xmax ) y = x-xmax; else if (x <= xmin) y = x + xmin; else y = 0; } slope=1 slope=1 x 31

32 Example: Output Noise 32

33 Increase deadband Deadband: [ ] 33

34 Deadzone Tradeoff Pros: Reduces noise and quantization error effects in steady state. Cons: Increase static error. 34

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