10.4 Controller synthesis using discrete-time model Example: comparison of various controllers

Transcription

1 10. Digital 10.1 Basic principle of digital control 10.2 Digital PID controllers A 2DOF continuous-time PID controller Discretisation of PID controllers Implementation and tuning 10.3 Sampling of time-delay models State-space model with a time delay Second-order model with real poles and zero First-order model 10.4 ler synthesis using discrete-time model PID controller Time-delay compensation Dead-beat control Improving intersample behaviour 10.5 Example: comparison of various controllers KEH Dynamics and 10 1

2 10. Digital 10.1 Basic principle of digital control The block Sample receives the continuous-time signals yy(tt) and rr(tt), and discretises them to sequences of numbers, yy(tt ii ) and rr(tt ii ), defined at the time instants tt ii, ii = 1,2, In practice, this is an A/D converter. The block Hold receives the sequence of numbers uu(tt ii ), ii = 1,2,, from the control algorithm and sends out a piecewise continuous signal, uu tt = uu(tt ii ), tt ii tt < tt ii+1. In practice, this is a D/A converter. KEH Dynamics and 10 2

3 10. Digital 10.2 Digital PID controllers A 2DOF continuous-time PID controller Many variants of continuous-time PID controllers were introduced in Section 7.1. A flexible form of the PID controller is uu tt = KK c ee p tt + 1 TT i 0 tt ee ττ dττ + TTd dee f (tt) dtt + uu 0 (10.1) ee p (tt) bbbb(tt) yy(tt), ee(tt) rr(tt) yy(tt), ee f (tt) cccc(tt) yy f (tt) (10.2) TT f dyy f (tt) dtt + yy f tt = yy(tt) (10.3) Here bb and cc are setpoint weights yielding a 2DOF PID controller. The control law (10.1) can also be written uu tt = uu p tt + uu i tt + uu d tt + uu 0 (10.4) uu p tt = KK c ee p tt (10.5) uu i tt = KK c TT i 0 tt ee ττ dττ duu i (tt) dtt = KK c TT i ee(tt) (10.6) uu d tt = KK c TT d dee f (tt) dtt (10.7) KEH Dynamics and 10 3

4 10. Digital 10.2 Digital PID controllers Discretisation of PID controllers A discrete-time PID controller can be obtained by using a discrete-time approximation of the derivatives in (10.3), (10.6) and (10.7). There are many methods for discretization of a derivative. Forward-difference approximation According to the definition of a derivative, dxx tt xx tt+h xx tt = lim dtt h 0 h A natural approximation of the derivative is then dxx(tt) dtt xx tt+h xx(tt) h (10.8) (10.9) However, this is not a suitable approximation of the derivative term in (10.1) or (10.7), because it requires a future value at time tt + h to be known when the control signal uu(tt) is calculated. It is possible to avoid this causality problem by including a derivative filter. However, even then the derivative part becomes unstable if TT f h/2 (10.10) where h is the sampling interval. KEH Dynamics and 10 4

5 10.2 Digital PID controllers Discretisation of PID controllers Backward-difference approximation By a slight modification of (10.9), the derivative can be approximated by dxx(tt) dtt xx tt xx(tt h) This results in a stable controller for all non-negative values of TT d and TT f. The proportional part of the controller becomes (without approximation) where tt kk is the time instant of the kk th sample. Applying (10.11) to the integral part (10.6) gives h (10.11) uu p tt kk = KK c ee p tt kk (10.12) uu i tt kk uu i tt kk 1 h = KK c TT i ee(tt kk ) uu i tt kk = uu i tt kk 1 + hkk c TT i ee(tt kk ) (10.13) Because the setpoint rr(tt kk ) has not yet affected the integral at time tt kk it appears that a better approximation than hee tt kk = h rr tt kk yy tt kk for the increase of the integral between time instants tt kk 1 and tt kk would be h rr tt kk 1 yy tt kk. However, that would delay the effect of rr(tt kk ) by one sample, which does not happen in a continuous-time controller. Therefore, ee(tt kk ) is used in (10.13). KEH Dynamics and 10 5

6 Discretisation of PID controllers Backward difference It is convenient to derive the derivative part via the Laplace domain. Combination of (10.3) and (10.7) in the Laplace domain yields where EE d (ss) is the Laplace transform of TT f ssuu d ss + UU d ss = KK c TT d ssee d (ss) (10.14) The inverse Laplace transform of (10.14) gives TT f duu d (tt) dtt ee d tt cccc tt yy(tt) (10.15) + uu d tt = KK c TT d dee d (tt) dtt Application of (10.11) to both sides of (10.16) at time tt = tt kk now yields or after rearrangement TT f uu d tt kk uu d (tt kk 1 ) h + uu d tt kk = KK c TT d ee d tt kk ee d (tt kk 1 ) h (10.16) uu d tt kk = TT f TT f +h uu d tt kk 1 + KK ctt d TT f +h ee d tt kk ee d (tt kk 1 ) (10.17) The signals uu i (tt kk ) in (10.13) and uu d tt kk kk = 1, 2, and substituted into (10.4). in (10.17) are calculated recursively for KEH Dynamics and 10 6

7 10.2 Digital PID controllers Discretisation of PID controllers Tustin s bilinear approximation The forward- and backward-differences are asymmetrical in the sense that the difference xx tt xx(tt h) /h does not approximate the derivative in the middle of the range, i.e., at time tt 0.5h, but at tt h (forward) or tt (backward). A symmetrical approximation is obtained by 1 2 dxx(tt) dtt + dxx(tt h) dtt xx tt xx tt h h (10.18) In general, this approximation is better than the previous ones and for well chosen controller parameters, the resulting discrete-time PID controller approximates the continuous-time PID controller better than the previous ones. However, if the filtering time constant is so small that (10.10) applies, the PID controller will cause intersample ripple, also called ringing. As a result of this, the inputs and the outputs will oscillate between sampling points, which is a very undesirable feature. Thus, Tustin s approximation is not suitable (without some modification) if no derivative filter is used or if the sampling interval is too large. KEH Dynamics and 10 7

8 10.2 Digital PID controllers Discretisation of PID controllers Trapezoidal approximation In the forward- and backward-difference approximations, the signal to be approximated is assumed to be constant in the sampling interval. It appears better to assume that a changing signal changes linearly in the sampling interval. This assumption yields the trapezoidal approximation. The figure to the left illustrates the forward-difference, the backward-difference and the trapezoidal approximation of a signal yy. If the signal is a control error ee = rr yy, the signal and the approximation is discontinuous when a setpoint change is made, as shown to the right. KEH Dynamics and 10 8

9 Discretisation of PID controllers Trapezoidal approximation Using the trapezoidal approximation, (10.6) gives for the integral part uu i tt kk which can be rewritten as = hkk c TT i kk jj=1 0.5 ee(tt jj 1 ) + ee(tt jj ) (10.19) uu i tt kk = uu i tt kk 1 + hkk c 2TT i ee tt kk + ee(tt kk 1 ) (10.20) Also here, rr(tt kk ) has not affected the true integral at time tt kk, but use of rr tt kk 1 yy(tt kk ) instead of ee(tt kk ) would again delay the control effect of rr(tt kk ) by one sample, which does not happen in a continuous-time controller. Therefore, ee(tt kk ) is used in (10.20). This is also consistent with formal use of the trapezoidal approximation when the fact that rr(tt kk ) is constant in the sampling interval is neglected. Again, the derivative part is most easily derived via the Laplace domain. From (10.7) it follows that dee f /dtt is needed. Because the setpoint is constant in the sampling intervals, dee f dtt = dyy f dtt. From the Laplace transform of (10.3) it then follows that ssee f ss = ss EE TT f ss+1 d(ss) (10.21) KEH Dynamics and 10 9

10 Discretisation of PID controllers Trapezoidal approximation According to the trapezoidal approximation, ee d (tt) changes linearly between sampling points. The Laplace transform of these changes up to time tt = tt kk is (note that jj is an integer, not the imaginary unit j = 1) EE d ss = ee d(tt 0 ) ss + ee d tt 1 ee d (tt 0 ) hss 2 kk 1 + ee d(tt jj+1 ) 2ee d (tt jj )+ee d (tt jj 1 ) jj=1 e jjjjj (10.22) hss 2 Substitution into (10.21) and taking the inverse Laplace transform yields dee f (tt kk ) dtt = ee d(tt 0 ) e tt kk/tt f + ee d tt 1 ee d tt 0 TT f h (1 e (tt kk jjj)/tt f ) kk 1 + ee d(tt jj+1 ) 2ee d (tt jj )+ee d (tt jj 1 ) jj=1 (1 e (tt kk jjj)/tt f ) h Combination with the corresponding expression for dee f (tt kk 1 )/dtt yields dyy f (tt kk ) dtt = dyy f(tt kk 1 ) dtt e h/tt f + ee d tt kk ee d tt kk 1 h (1 e h/tt f) (10.23) Combination with (10.7) finally gives uu d tt kk = e h/tt fuu d tt kk 1 + KK ctt d (1 e h/tt f) h ee d tt kk ee d (tt kk 1 ) (10.24) KEH Dynamics and 10 10

11 10. Digital 10.2 Digital PID controllers Implementation and tuning Unification of methods The control laws according to the backward-difference and the trapezoidal approximation can both be expressed in the form uu i tt kk = uu i tt kk 1 + bb i1 ee tt kk + bb i2 ee tt kk 1 (10.25) uu d tt kk = aa d uu d tt kk 1 + bb d ee d tt kk ee d tt kk 1 (10.26) where uu i tt 0 = uu d tt 0 = 0 can be used as starting values for the iterative calculation. The control errors ee tt kk and ee d (tt kk ) are defined in (10.2b) and (10.15), respectively, and the parameters are given below. Method bb ii11 bb ii22 aa dd bb dd Backward Trapezoid KK c h TT i 0 KK c h 2TT i KK c h 2TT i TT f TT f + h e h/tt f KK c TT d h KK c TT d TT f + h 1 e h/tt f In general, the trapezoidal approximation is more accurate than the backwarddifference approximation, but both tend to be good choices. KEH Dynamics and 10 11

12 10.2 Digital PID controllers Implementation and tuning Preventing integrator windup If the control signal reaches a process constraint (e.g., a control valve fully open) when there is a control error, and the controller contains standard integral action, the absolute value of the integral will continue to increase as long as the sign of the control error is unchanged. This will result in integrator windup, which is illustrated in the figure. Here a setpoint change y is made at tt = 0 (upper). The control signal uu immediately reaches its maximum value 0.1 (middle). r The integral uu i continues to increase as long as yy < rr (lower). u The control signal will remain at the maximum value until the integral decreases sufficiently even though yy > rr. After this, u i the control signal might reach its minimum value and oscillations follow. KEH Dynamics and 10 12

13 Implementation and tuning Preventing integrator windup Integrator windup can be prevented in many ways. One way is to adjust the value of the integrator so that the calculated control signal with the adjusted integrator exactly reaches the constraint. This is achieved with uu i tt = KK c TT i 0 tt ee ττ dττ + 1 TT t 0 tt ees ττ dττ (10.27) where ee s tt min uu max, max uu min, uu tt uu(tt) (10.28) This function has the property that ee s = 0 if uu min uu uu max, otherwise ee s = uu max/min uu. TT t is the tracking time constant, which should be chosen as TT d < TT t < TT i. A reasonable value is TT t = TT i TT d. If TT d = 0, TT t = 0.1TT i can be used. The figure illustrates how this anti-windup works. (Note different time scale.) y u i r u KEH Dynamics and 10 13

14 Implementation and tuning Preventing integrator windup Discrete-time control For discrete-time control, (10.27) can be discretised similarly to (10.6). Because duu i (tt) the backward-difference approximation yields dtt = KK c TT i ee tt + 1 TT t ee s (tt) (10.30) uu i tt kk = uu i tt kk 1 + hkk c TT i ee tt kk + h TT t ee s (tt kk ) (10.31) and the trapezoidal approximation yields uu i tt kk = uu i tt kk 1 + hkk c 2TT i ee tt kk + ee(tt kk 1 ) + h 2TT t ee s tt kk + ee s (tt kk 1 ) (10.32) KEH Dynamics and 10 14

15 10.2 Digital PID controllers Implementation and tuning Incremental control algorithms The control algorithms where the control signal is calculated by (10.4) are called positional algorithms. Sometimes it is more convenient to calculate the control signal as an addition to its previous value, i.e., uu tt kk = uu tt kk 1 + uu p tt kk + uu i tt kk + uu d tt kk (10.33) where uu tt kk uu tt kk uu(tt kk 1 ) (10.34) Such an algorithm is called an incremental algorithm. It should only be used with integral action. The given algorithms for the integral part are essentially in incremental form, but the other parts have to be determined by (10.34). The result is uu p tt kk = KK c ee p tt kk ee p (tt kk 1 ) (10.35) uu i tt kk = bb i1 ee tt kk + bb i2 ee tt kk 1 (10.36) uu d tt kk = aa d uu d tt kk 1 +bb d ee d tt kk 2ee d tt kk 1 + ee d tt kk 2 (10.37) If anti-windup is desired, (10.36) should be modified as (10.31) or (10.32). KEH Dynamics and 10 15

16 10.2 Digital PID controllers Implementation and tuning Choice of sampling interval There are no firm rules on how to choose the sampling interval. A long sampling interval (low sampling rate) has many drawbacks such as longer apparent time delay h/2 the aliasing effect no frequency above the Nyquist frequency ππ/h can be detected An unnecessarily short sampling interval (high sampling rate) means more measurements, calculations and control actions with almost no new information at sampling instants. The following are some guidelines on the choice of sampling rate: Use 10 times the rate suggested by Shannon s sampling theorem. Use 4 10 samples per rise time of the closed-loop system. Use samples per period of oscillation of the closed-loop system. The sampling frequency should be times the bandwidth. Choose ωω g h = , where ωω g is the gain crossover frequency, i.e., the frequency where the amplitude ratio of the open-loop system is 1. Note that we have not defined all above terms in this course. KEH Dynamics and 10 16

17 10.2 Digital PID controllers Implementation and tuning Tuning Because the controller parameters in (10.12), (10.25) and (10.26) are functions of the parameters of a continuous-time PID controller, any tuning method for continuous-time PID controllers can be used. Because the sampling increases the average time delay by half a sampling interval, it is advisable to increase the time delay of the system by h/2 before calculating the continuous-time controller parameters. In the derivation of (10.17) and (10.24), it was not taken into account that the setpoint changes stepwise, and only at sampling instants. This kind of setpoint changes can be handled exactly. The correct handling is achieved if the derivative setpoint weight cc in the continuous case is changed to cc = hcc TT f 1 e h/tt f cc = (TT f+h) cc TT f for the trapezoidal approximation for the backward-difference approximation (this is approximate) KEH Dynamics and 10 17

18 10.2 Digital PID controllers Implementation and tuning Implementation in SIMULINK Mathwork s SIMULINK software provides two discrete PID controller blocks, one of which is a 2DOF controller. The type of discretization can be chosen, e.g., backward Euler or trapezoidal. The controller parameters are PP = KK c II = KK c TT i DD = KK c TT d TT s = h (not adjustable here) If the trapezoid approximation is used both for the integrator and the filter 1 e h/tt f NN = 2 h 1+e h/tt f If the backward approximation is used NN = 1/TT f SIMULINK s derivative implementation requires NN > 0! KEH Dynamics and 10 18

19 10. Digital 10.3 Sampling of time-delay models State-space model with a time delay Sampling A continuous-time system with the same time delay θθ for all inputs has the state-space representation xx tt = AAAA tt + BBBB(tt θθ) (10.38) If the system is sampled with the sampling interval h, the solution is xx tt kk+1 = e AAh xx tt kk + e AAtt kk+11 ttkk tt kk+1 e AAττ BBBB(ττ θθ)dττ (10.39) where tt kk and tt kk+1 are two adjacent sampling points, tt kk+1 = tt kk + h. Assume that the time delay is smaller than the sampling interval, i.e., 0 θθ < h. If the input uu(tt) is constant in the sampling interval tt kk tt < tt kk+1, uu(tt θθ) is not constant over the same interval if θθ > 0; the value of uu(tt θθ) changes from uu(tt kk 1 ) to uu(tt kk ) at the time tt = tt kk + θθ. Because of this, it is useful to split the integral into two parts so that the input is constant in each part. KEH Dynamics and 10 19

20 State-space model with a time delay Sampling Splitting the integral in the indicated way yields xx tt kk+1 = e AAh xx tt kk + e AAtt kk+11 ttkk tt kk +θθ e AAττ dττ BBBB(tt kk 1 ) + e AAtt tt kk+11 kk+1 ttkk e AAττ dττ BBBB(tt +θθ kk ) (10.40) The solution can be written where xx tt kk+1 = FFxx tt kk + GG 0 uu(tt kk ) + GG 1 uu(tt kk 1 ) (10.41) FF = e AAh, GG 0 = 0 h θθ e AAττ dττ BB, GG 1 = e AA(h θθ) 0 θθ e AAττ dττ BB (10.42) The integrals can be calculated numerically according to SS tt 0 tt e AAττ dττ = II + 1 2! AAtt + 1 3! AA2 tt ! AA3 tt 3 + tt (10.43) from which FF = II + AASS h, GG 0 = SS h θθ BB, GG 1 = SS θθ BB (10.44) If the time delay LL is such that LL = NNN + θθ, NN is a non-negative integer (10.45) uu(tt kk ) is replaced by uu(tt kk NN ) and uu(tt kk 1 ) is replaced by uu(tt kk 1 NN ). KEH Dynamics and 10 20

21 10.3 Sampling of time-delay models State-space model Pulse transfer operator Generally, a state-space model with a time delay LL = NNN + θθ, 0 θθ < h, for all inputs has the form xx tt kk+1 = FFFF tt kk + GG 0 uu(tt kk NN ) + GG 1 uu(tt kk 1 NN ) yy tt kk = CCxx tt kk + DD 0 uu(tt kk NN ) + DD 1 uu(tt kk 1 NN ) The shift operator q operating on a signal ff(tt kk ) is defined (10.46) qff tt kk = ff(tt kk+1 ) q 1 ff tt kk+1 = ff(tt kk ) (10.47) Application of the shift operator to (10.46) yields from which qxx tt kk = FFFF tt kk + GG 0 q NN uu(tt kk ) + GG 1 q 1 NN uu(tt kk ) yy tt kk = CCxx tt kk + DD 0 q NN uu(tt kk ) + DD 1 q 1 NN uu(tt kk ) xx tt kk = qii FF 1 GG 0 + GG 1 q 1 q NN uu(tt kk ) yy tt kk = CC qii FF 1 GG 0 + GG 1 q 1 + DD 0 + DD 1 q 1 q NN uu(tt kk ) Since the pulse transfer operator is defined by yy tt kk = HH(q 1 )uu(tt kk ), the sampled state-space model has the pulse transfer operator (10.48) (10.49) HH q 1 = CC qii FF 1 GG 0 + GG 1 q 1 + DD 0 + DD 1 q 1 q NN (10.50) KEH Dynamics and 10 21

22 10. Digital 10.3 Sampling of time-delay models Second-order model with real poles and zero Diagonal state-space form A second-order time-delay system with two distinct real poles and a possible zero has the transfer function where GG ss = KK(TT 3ss+1)e LLLL (TT 1 ss+1)(tt 2 ss+1) = kk 1 ss λλ 1 kk 2 ss λλ 2 e LLLL (10.51) λλ 1 = 1/TT 1, λλ 2 = 1/TT 2 kk 1 = KK(TT 1 TT 3 ) TT 1 (TT 1 TT 2 ), kk 2 = KK(TT 3 TT 2 ) TT 2 (TT 1 TT 2 ) The system can be written on diagonal state-space form as xx tt = ΛΛxx tt + bbuu(tt LL) yy tt = cc T xx(tt) ΛΛ = λλ λλ 2, bb = kk 1 kk 2, cc = 1 1 (10.52a) (10.52b) (10.53) (10.54) KEH Dynamics and 10 22

23 10.3 Sampling of time-delay models Second-order model Sampling Because ΛΛ is diagonal, sampling and calculation of e AAh is easy. With LL = NNN + θθ, the result is xx tt kk+1 = FFFF tt kk + GG 0 uu(tt kk ) + GG 1 uu(tt kk 1 ) yy tt kk = cc T xx(tt kk ) where (10.55) FF = e ΛΛh = eλλ 1h 0 0 e λλ 2h (10.56) GG 0 = 0 h θθ e ΛΛττ dττ bb = λλ 1 1 (e λλ 1 h θθ 1) 0 0 λλ 2 1 (e λλ 2 h θθ 1) GG 1 = e ΛΛ(h θθ) = kk 1λλ 1 1 (e λλ 1 h θθ 1) kk 2 λλ 2 1 (e λλ 2 h θθ 1) 0 θθ e ΛΛττ dττ bb = eλλ 1(h θθ) 0 0 e λλ 2(h θθ) λλ 1 1 (e λλ 1θθ 1) 0 0 λλ 2 1 (e λλ 2θθ 1) = kk 1λλ 1 1 ee λλ 1h (1 ee λλ 1θθ ) kk 2 λλ 2 1 ee λλ 2h (1 ee λλ 2θθ ) kk 1 kk 2 kk 1 kk 2 (10.57) (10.58) KEH Dynamics and 10 23

24 10.3 Sampling of time-delay models Second-order model Pulse transfer operator According to (10.50), the pulse transfer operator is HH q 1 = cc T qii FF 1 GG 0 + GG 1 q 1 q NN (10.59) Substitution of ( ) into (10.59) gives a pulse transfer operator of the form Using the parameters HH q 1 = bb 1+bb 2 q 1 +bb 3 q 2 1+aa 1 q 1 +aa 2 q 2 qq 1 NN (10.60) bb 11 = kk 1 λλ 1 1 e λλ 1 h θθ 1, bb 12 = kk 2 λλ 2 1 e λλ 2 h θθ 1 bb 21 = kk 1 λλ 1 1 ee λλ 1h 1 ee λλ 1θθ, bb 22 = kk 2 λλ 2 1 ee λλ 2h 1 ee λλ 2θθ the parameters of (10.60) are given by (10.61) aa 1 = ee λλ1h + ee λλ2h, aa 2 = ee (λλ 1+λλ 2 )h, bb 1 = bb 11 + bb 12 bb 2 = ee λλ2h bb 11 + ee λλ1h bb 12 + bb 21 + bb 22 (10.62) bb 3 = ee λλ2h bb 21 + ee λλ1h bb 22 KEH Dynamics and 10 24

25 Second-order model Pulse transfer operator The case θθ = 0 If the time delay is an integer multiple the sampling interval, i.e., LL = NNN, θθ = 0, the expressions for the second-order pulse transfer operator are simplified. In this case HH q 1 = bb 1+bb 2 q 1 1+aa 1 q 1 +aa 2 q 2 q 1 NN (10.63) In (10.61), bb 21 = bb 22 = 0 and bb 11 = kk 1 λλ 1 1 e λλ 1h 1, bb 12 = kk 2 λλ 2 1 e λλ 2h 1 (10.64) The parameters of (10.63) are then given by aa 1 = ee λλ 1h + ee λλ 2h, aa 2 = ee (λλ 1+λλ 2 )h bb 1 = bb 11 + bb 12, bb 2 = ee λλ 2h bb 11 + ee λλ 1h bb 12 (10.65) Exercise Determine the pulse transfer operator HH(q 1 ) for the system GG ss = KKe LLLL (TT 1 ss+1)(tt 2 ss+1) when KK = 1, LL = 1 min, TT 1 = 1 min, TT 2 = 0.5 min, h = 0.1 min. KEH Dynamics and 10 25

26 10. Digital 10.3 Sampling of time-delay models First-order model For a strictly proper first-order system with a time delay, the transfer function is GG ss = KKe LLLL TTss+1 The sampling result and the pulse transfer operator can be directly obtained from the solution for the second-order system using kk 1 = KK/TT, λλ 1 = 1/TT, λλ 2 = (10.66) For the general time-delay case (i.e., arbitrary θθ), (10.61) and (10.62) yield the pulse transfer operator HH q 1 = bb 1+bb 2 q 1 1+aa 1 q 1 q 1 NN (10.67) with aa 1 = e h/tt, bb 1 = KK(1 + aa 1 e θθ/tt ), bb 2 = KKKK 1 (1 e θθ/tt ) (10.68) KEH Dynamics and 10 26

27 10. Digital 10.4 ler synthesis using discrete-time model PID controller Loop-transfer specification If the controller and the system have the pulse transfer operators HH c (q 1 ) and HH(q 1 ), respectively, the loop transfer operator is resulting in the closed-loop pulse transfer operator HH l q 1 = HH(q 1 )HH c (q 1 ) (10.69) HH r q 1 = HH l q 1 1+HH l q 1 (10.70) For a system with the time delay LL = NNN + θθ, 0 θθ < h, the loop transfer HH l q 1 = q 1 NN (2NN+1)(1 q 1 ) (10.71) gives a closed-loop system having approximately a 4 % overshoot and a rise time equal to 3 4 time delays. Such a control performance can be considered good. KEH Dynamics and 10 27

28 10.4 ler synthesis using discrete model PID controller First-order model with LL = NNN According to ( ), a first-order system with a time delay being an integer multiple of the sampling interval has the pulse transfer operator HH q 1 = bb 1 1+aa 1 q 1 q 1 NN (10.72) The loop transfer operator (10.71) is achieved with the controller HH c q 1 = HH q 1 1 HH l q 1 (10.73) = 1+aa 1q 1 bb 1 q 1 NN q 1 NN 2NN+1 1 q 1 = 1 (2NN+1)bb 1 1+aa 1q 1 1 q 1 (10.74) Since HH c q 1 = UU(q 1 )/EE(q 1 ), this is a controller implementing the incremental control law uu tt kk = uu tt kk (2NN+1)bb 1 ee tt kk + aa 1 ee(tt kk 1 ) (10.75) where ee tt kk = rr tt kk yy(tt kk ) is the standard control error. Comparison with ( ) shows that this is a PI controller with bb i1 + KK c = 1 (2NN+1)bb 1, bb i2 KK c = aa 1 (2NN+1)bb 1 KEH Dynamics and 10 28

29 PID controller First-order model with LL = NNN If the trapezoidal approximation is used, this yields a controller with KK c = 1 aa 1 2(2NN+1)bb 1, bb i1 = bb i2 = 1+aa 1 2(2NN+1)bb 1 TT i = h 2 1 aa 1 1+aa 1 (10.76) Implementation The controller can be implemented in various ways using SIMULINK. One way is to implement the positional controller as explained in Section As this is a PI controller, DD = 0 should then be used. Another way is to implement the incremental form using the Discrete Zero- Pole block. The variable zz used in the block definition is obtained by replacing the operator q by zz. Since HH c q 1 = 1 1+aa 1q 1 HH (2NN+1)bb 1 1 q 1 c zz = 1 zz+aa 1 (2NN+1)bb 1 zz 1 the pole is +1, the zero is aa 1 and the gain is 1 (2NN + 1)bb 1 for this block. The Discrete Transfer Fcn block can also be used. In this block, the numerator and denominator polynomials are defined. In this case, the numerator parameters are 1 aa 1 and the denominator parameters are 1 1. KEH Dynamics and 10 29

30 10.4 ler synthesis using discrete model PID controller Second-order model with LL = NNN A second-order system with a time delay equal to an integer multiple of the sampling interval has the pulse transfer operator (10.63). Substitution into (10.73) yields HH c q 1 = 1+aa 1q 1 +aa 2 q 2 q 1 NN (bb 1 +bb 2 q 1 )q 1 NN 2NN+1 1 q 1 where = 1 1+aa 1q 1 +aa 2 q 2 (2NN+1)bb 1 (1 q 1 )(1+ddq 1 ) (10.77) dd = bb 2 bb 1 (10.78) This is a controller implementing the incremental control law uu tt kk = uu tt kk (2NN+1)bb 1 ee tt kk + aa 1 ee tt kk 1 + aa 2 ee tt kk 2 dd uu tt kk 1 uu tt kk 2 (10.79) Comparison with ( ) shows that this is a PID controller with a derivative filter. KEH Dynamics and 10 30

31 PID controller Second-order model with LL = NNN Implementation Using SIMULINK there are again various implementation options Easiest is to use the Discrete Transfer Fcn block. In this case the numerator parameters are 1 aa 1 aa 2, the denominator parameters are 1 (dd 1) dd, and the gain is 1 (2NN + 1)bb 1. It is also possible to determine the controller parameters KK c, TT i, TT d and TT f and to use the Discrete PID ler block as explained in Section If the trapezoidal approximation is used aa d = dd TT f = h/ ln dd TT i = h 1 aa 2 1 dd 1+aa 1 +aa 2 2(1+dd) KK c = TT i h 1+aa 1 +aa 2 (1+dd)(2NN+1)bb 1 TT d = h h TT i 1 1+aa 1 +aa 2 1 2(1+dd) (10.80a) (10.80b) (10.80c) (10.80d) KEH Dynamics and 10 31

32 10. Digital 10.4 ler synthesis using discrete model Time-delay compensation Closed-loop specification A more direct way of specifying the desired control performance is to specify the desired closed-loop performance instead of the loop transfer. Eq. (10.69) and (10.70) yield the closed-loop pulse transfer operator HH r q 1 = HH(q 1 )HH c (q 1 ) 1+HH(q 1 )HH c (q 1 ) (10.81) where HH(q 1 ) and HH c (q 1 ) are the pulse transfer operators of the process and the controller. From (10.81) the controller needed to achieve HH r q 1 can be derived as done in direct synthesis, i.e., HH c q 1 = 1 HH(q 1 ) HH r (q 1 ) 1 HH r (q 1 ) (10.82) The closed-loop system cannot have a smaller time delay than the open loop system. Assume that the open-loop system has the time delay LL = NNN + θθ, 0 θθ < h. According to (10.67), a first-order closed-loop system with this time delay and the gain KK = 1, which is desired for setpoint tracking, has a pulse transfer operator of the form KEH Dynamics and 10 32

33 Time-delay compensation Closed-loop specification HH r q 1 = 1 ββ+(ββ αα)q 1 1 ααq 1 q 1 NN (10.83) where αα = e h/tt r, ββ = ααe θθ/tt r, and TT r is the desired closed-loop time constant. Note that ββ = αα if θθ = 0, i.e., if the time delay is an integer multiple of the sampling interval, LL = NNN. This choice of HH r q 1 yields HH r q 1 1 HH r q 1 = 1 ββ+(ββ αα)q 1 1 ααq 1 1 ββ+(ββ αα)q 1 q 1 NN q 1 NN (10.84) First-order time-delay system A first-order time-delay system has the pulse transfer operator given in (10.67). Substitution of (10.67) and (10.84) into (10.82) gives HH c q 1 = 1+aa 1q 1 bb 1 +bb 2 q 1 1 ββ+(ββ αα)q 1 1 ααq 1 1 ββ+(ββ αα)q 1 q 1 NN (10.85) where = cc 0 (1+aa 1 q 1 )(1+dd 2 q 1 ) 1+dd 1 q 1 [1 ααq 1 1 ββ 1+dd 2 q 1 q 1 NN ] cc 0 = 1 ββ, dd bb 1 = bb 2, dd 1 bb 2 = ββ αα 1 1 ββ (10.86) (10.87) KEH Dynamics and 10 33

34 Closed-loop specification First-order time-delay system Eq. (10.86) gives a control law in difference form as uu tt kk = cc 0 ee tt kk + (aa 1 +dd 2 )ee tt kk 1 + aa 1 dd 2 ee tt kk 2 + αα dd 1 uu tt kk 1 + ααdd 1 uu tt kk 2 (10.88) + (1 ββ) uu tt kk 1 NN + (dd 1 +dd 2 )uu tt kk 2 NN + dd 1 dd 2 uu tt kk 3 NN This is a PID-like controller (because of the ee terms) with delayed control signals to compensate for the time delay. If the time delay is an integer multiple of the sampling interval, i.e., if θθ = 0, then bb 2 = 0, dd 1 = dd 2 = 0, ββ = αα, and the control law reduces to the PI-like control law uu tt kk = cc 0 ee tt kk + aa 1 ee tt kk 1 + ααuu tt kk 1 + (1 αα)uu tt kk 1 NN (10.89) The discrete-time control laws (10.88), (10.89) and (10.91), with explicit compensation for the time delay, are known as Dahlin controllers (or Dahlin- Higham s algorithms). Because quite old information is used in the form of delayed control signals, they can be expected to be sensitive to model errors, especially concerning the time delay. KEH Dynamics and 10 34

35 Time-delay compensation Closed-loop specification Second-order time-delay system with LL = NNN A second-order time-delay system, where the time delay is an integer multiple of the time delay, has the pulse transfer operator (10.63). Substitution of (10.63) and (10.84) into (10.82) yields HH c q 1 = 1+aa 1q 1 +aa 2 q 2 bb 1 +bb 2 q 1 1 αα 1 ααq 1 (1 αα)q 1 NN (10.90) This gives the PID-like control law uu tt kk = cc 0 ee tt kk + aa 1 ee tt kk 1 + aa 2 ee tt kk 2 + αα dd 1 uu tt kk 1 + ααdd 1 uu tt kk αα [uu tt kk 1 NN + dd 1 uu tt kk 2 NN ] (10.91) with cc 0 and dd 1 as given in (10.87). Note that the control law (10.91) for a second-order system with LL = NNN is less complicated than the control law (10.88) for a first-order system with LL > NNN. Implementation In SIMULINK, the control laws (10.88), (10.89) and (10.91) can be implemented by a Discrete Transfer Fcn block. Note that the denominator polynomial will have many zeroes for the missing inputs up to uu tt kk 1 NN. KEH Dynamics and 10 35

36 10. Digital 10.4 ler synthesis using discrete model Dead-beat control In the controller synthesis according to Dahlin-Higham s method, the time constant TT r of a desired first-order closed-loop system is specified. How short can TT r be? The theoretical minimum is TT r = 0, of course. This results in the closed-loop transfer operator as a pure time delay Substitution into (10.82) yields the controller as HH r q 1 = q 1 NN (10.92) HH c q 1 = 1 HH(q 1 ) q 1 NN 1 q 1 NN (10.93) where q 1 NN can be cancelled against the same operator in the pulse transfer operator HH(q 1 ) of the model. This kind of controller is called a dead-beat controller. There is no corresponding continuous-time controller since TT r = 0 would require an infinite controller gain in the continuous-time case. KEH Dynamics and 10 36

37 10.4 ler synthesis using discrete model Dead-beat control First-order time-delay model A first-order time-delay system has the pulse transfer operator given in (10.67). Substitution of (10.67) into (10.93) gives HH c q 1 = 1+aa 1q 1 bb 1 +bb 2 q 1 1 = bb 1 1 (1+aa 1 q 1 ) 1 q 1 NN 1+dd 1 q 1 (1 q 1 NN ) (10.94) where dd 1 is as defined in (10.87). The control law in difference form is uu tt kk = bb 1 1 ee tt kk + aa 1 ee tt kk 1 + uu tt kk 1 NN dd 1 [uu tt kk 1 uu tt kk 2 NN ] (10.95) This is PI-like control law where the dominating old control signal is uu tt kk 1 NN instead of uu tt kk 1. Note that dd 1 = 0 if LL = NNN (i.e., θθ = 0). Second-order time-delay system with LL = NNN For a second-order time-delay system with LL = NNN, the control law becomes which is a PID-like controller. uu tt kk = bb 1 1 ee tt kk + aa 1 ee tt kk 1 + aa 2 ee tt kk 2 + uu tt kk 1 NN dd 1 [uu tt kk 1 uu tt kk 2 NN ] (10.96) KEH Dynamics and 10 37

38 10. Digital 10.4 ler synthesis using discrete model Improving intersample behaviour Intersample ripple A possible problem with discrete-time control of a continuous-time system is the behaviour between the sample points, as illustrated by the figure. If only the sample points of the output yy are considered, the step response looks like the response of a first-order system. The true continuous-time value of yy oscillates; this is called intersample ripple (or ringing). The input signal oscillates at sample points (and in continuous time). = sample point, full line = continuous-time value [SEMD] KEH Dynamics and 10 38

39 Improving intersample behaviour Intersample ripple In particular, intersample ripple can occur when a discrete-time controller has been designed based on the inverse of a discrete-time (sampled) model. When a continuous-time model is sampled using standard techniques, the transfer function of the sampled model often receives a zero close to 1 (but slightly > 1) even if the continuous-time model has no zero; e.g. GG ss = KKe LLLL HH (TT 1 ss+1)(tt 2 ss+1) q 1 = bb 0+bb 1 q 1 1+aa 1 q 1 +aa 2 q 2 q 1 NN, bb 1 < 1. bb 0 If the controller design is based on the inverse of HH q 1, bb 0 + bb 1 q 1 becomes a pole of the controller, i.e., a controller pole close to 1, which causes intersample ripple. KEH Dynamics and 10 39

40 10.4 ler synthesis using discrete model Intersample behaviour Elimination of intersample ripple There are basically two ways to eliminate intersample ripple. Dahlin s modification According to Dahlin s modification, the denominator bb 1 + bb 2 q 1 of the model used in the controller synthesis is replaced by bb 1 + bb 2. Instead of the model (10.63), for example, HH q 1 = bb 1 +bb 2 1+aa 1 q 1 +aa 2 q 2 q 1 NN (10.97) is used. The figure illustrates the control performance with such a modification. There is no intersample ripple no (extensive) control signal oscillations KEH Dynamics and 10 40

41 Improving intersample behaviour Elimination of intersample ripple Vogel-Edgar modification Vogel and Edgar have suggested that bb 1 + bb 2 q 1 be included in the numerator of the desired closed-loop model. Instead of, for example, the desired first-order model HH r q 1 = 1 αα 1 ααq 1 q 1 NN (10.98) the controller synthesis is done to obtain the closed-loop transfer operator HH r q 1 = (1 αα)(bb 1+bb 2 q 1 ) 1 ααq 1 q 1 NN (10.99) The figure shows the control performance with this modification. There is no intersample ripple no control signal oscillations smother but slightly slower response than with Dahlin s modification KEH Dynamics and 10 41

42 10. Digital 10.5 Example: comparison of various controllers Example system In Exercise the pulse transfer operator was determined for the system GG ss = KKe LLLL (TT 1 ss+1)(tt 2 ss+1) (10.100) with KK = 1, LL = 1 min, TT 1 = 1 min, TT 2 = 0.5 min, h = 0.1 min. The result was with HH q 1 = bb 1+bb qq q 1 1+aa 1 q 1 +aa 2 q 2 q 1 NN (10.101) aa 1 = , aa 2 = , bb 1 = , bb 2 = , NN = 10. Four different discrete-time controllers are to be compared: a discretized PID controller tuned by Ziegler-Nichols s method a discrete PID controller tuned for small overshoot (4 %) a Dahlin controller with TT r = 0.5 min and wrong model a Dahlin controller with TT r = 0.5 min and correct model KEH Dynamics and 8 42

43 10. Digital 10.5 Example: comparison of controllers Discretized PID with Ziegler-Nichols tuning The system (10.100) has the phase shift φφ = LLLL arctan TT 1 ωω arctan TT 2 ωω = 1ωω arctan 1ωω arctan 0.5ωω which for the phase shift φφ = ππ yields the critical frequency ωω c = rad/min. The amplitude ratio at this frequency is GG(ωω c ) = KK = 1 = (TT 1 ωω c ) 2 1+(TT 2 ωω c ) from which the ultimate gain KK c,max = 1/ According to Ziegler-Nichols s recommendations for a PID controller, this yields KK c = 0.6KK c,max = 1.36, TT i = ππ ωω c = 2.08 min, TT d = 0.25TT i = 0.52 min. Combination of ( ) and the parameters for the backward-difference approximation gives for an ideal PID controller uu tt kk = KK c 1 + h + TT d ee tt TT i h kk 1 2TT d h which here results in ee tt kk 1 + TT d h ee(tt kk 2) + uu(tt kk 1 ) uu tt kk = 8.51ee tt kk 15.5ee tt kk ee tt kk 2 + uu(tt kk 1 ) KEH Dynamics and 10 43

44 10. Digital 10.5 Example: comparison of controllers Discrete PID tuned for small overshoot The pulse transfer operator (10.101) yields the controller (10.79), i.e., uu tt kk = 1 (2NN+1)bb 1 ee tt kk + aa 1 ee tt kk 1 + aa 2 ee tt kk 2 + uu tt kk 1 +dd uu tt kk 1 where dd = bb 2 bb 1. Numerically, this results in uu tt kk 2 uu tt kk = 5.26ee tt kk 9.06ee tt kk ee tt kk 2 + uu tt kk uu tt kk 1 uu(tt kk 2 ) KEH Dynamics and 10 44

45 10. Digital 10.5 Example: comparison of controllers Dahlin controller and wrong model Assume that GG ss = KKe LLLL TTTT+1 with KK = 1, LL = 1 min, and TT = 1.5 min is thought to be the correct model. The corresponding pulse transfer operator is then HH q 1 = bb 1 1+aa 1 q 1 q 1 NN with aa 1 = e h/tt , bb 1 = KK(1 + aa 1 ) , NN = 10. According to (10.89), the control law is uu tt kk = 1 αα bb 1 ee tt kk + aa 1 ee tt kk 1 + uu tt kk 1 + (1 αα) uu tt kk 1 uu tt kk 1 NN With αα = e h/tt r = e 0.1/ , this gives uu tt kk = 2.81ee tt kk 2.63ee tt kk 1 + uu tt kk uu tt kk 1 uu(tt kk 11 ) KEH Dynamics and 10 45

46 10. Digital 10.5 Example: comparison of controllers Dahlin controller and correct model With the correct model, the Dahlin controller is given by (10.91): uu tt kk = cc 0 ee tt kk + aa 1 ee tt kk 1 + aa 2 ee tt kk 2 + αα dd 1 uu tt kk 1 + ααdd 1 uu tt kk αα [uu tt kk 1 NN + dd 1 uu tt kk 2 NN ] where cc 0 = (1 αα) bb 1 and dd 1 = bb 1. αα = e h/tt r gives bb 2 uu tt kk = 20.0ee tt kk 34.5ee tt kk ee tt kk uu tt kk uu tt kk uu tt kk uu(tt kk 12 ) Simulations Ziegler-Nichols PID ( ) Small overshoot PID ( ) Dahlin controller with wrong model ( ) Dahlin controller with correct model ( ) KEH Dynamics and 10 46