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  • EE3302 Lab Report:

    Ziegler-Nichols & Relay Tuning Experiments

    Phang Swee King

    October 29, 2009

    1 Objectives

    1. Estimate a first-order plus dead-time process model from an open-loop step response test for theZiegler-Nichols open-loop method of tuning.

    2. Obtain the ultimate gain and ultimate period from a relay experiment for the Ziegler-Nichols ultimatecycling method of tuning.

    3. Estimate a transfer function model of the process from the ultimate gain and ultimate period.

    4. Implement on-off temperature control.

    2 Equipment

    1. DIGIAC 1750 Transducer and Instrumentation Trainer

    2. Connecting Wires

    3. Digital Multi-meter

    4. Personal Computer

    5. Matlab

    3 Open-Loop Step Test

    Table 1: Open-Loop Step Test (Ambient Temperature = 25.2C)Time (minute) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Voltage v, (V) 2.982 2.988 3.000 3.018 3.034 3.049 3.062 3.075 3.085 3.095

    Temperature above Ambient (C) 0 0.6 1.8 3.6 5.2 6.7 8.0 9.3 10.3 11.3Unit Step Response y (C) 0 0.008 0.025 0.050 0.072 0.093 0.111 0.129 0.143 0.157

    Time (minute) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5Voltage v, (V) 3.103 3.110 3.115 3.120 3.124 3.127 3.129 3.133 3.135 3.138

    Temperature above Ambient (C) 12.1 12.8 13.3 13.8 14.2 14.5 14.7 15.1 15.3 15.6Unit Step Response y (C) 0.168 0.178 0.185 0.191 0.197 0.201 0.204 0.209 0.212 0.216

    Time (minute) 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5Voltage v, (V) 3.142 3.144 3.145 3.148 3.150 3.150 3.149 3.149 3.150 3.151

    Temperature above Ambient (C) 16.0 16.2 16.3 16.6 16.8 16.8 16.7 16.7 16.8 16.9Unit Step Response y (C) 0.222 0.225 0.226 0.230 0.233 0.233 0.232 0.232 0.233 0.234

    1

  • Time (minute) 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5Voltage v, (V) 3.151 3.151 3.153 3.154 3.153 3.154 3.155 3.152 3.153 3.154

    Temperature above Ambient (C) 16.9 16.9 17.1 17.2 17.1 17.2 17.3 17.0 17.1 17.2Unit Step Response y (C) 0.234 0.234 0.237 0.239 0.237 0.239 0.240 0.236 0.237 0.239

    Time (minute) 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5Voltage v, (V) 3.156 3.155 3.154 3.154 3.154 3.155 3.155 3.156 3.156 3.155

    Temperature above Ambient (C) 17.4 17.3 17.2 17.2 17.2 17.3 17.3 17.4 17.4 17.3Unit Step Response y (C) 0.241 0.240 0.239 0.239 0.239 0.240 0.240 0.241 0.241 0.240

    4 Closed-Loop Relay Experiment

    From the result in Part 3, steady-state voltage, vss = 3.155V

    Table 2: ON-OFF Interval of the Electronic Switch

    No. tON (sec) tOFF (sec)1 86 862 88 913 84 85

    Average 86 87.3

    Table 3: Maximum and Minimum Temperatures of the Oscillation

    No. Maximum Temperature (C) Minimum Temperature (C)1 43.9 41.42 44.1 41.53 43.9 41.4

    Average 44.0 41.4

    5 ON-OFF Control

    The temperature of 10C above ambient, v10 = 3.082V

    2

  • Table 4: ON-OFF Period of the Electronic Switch

    No. tON (sec) tOFF (sec)1 50 1472 49 1363 51 144

    Average 50 142.3

    Table 5: Maximum and Minimum Temperatures of the Oscillation

    No. Maximum Temperature (C) Minimum Temperature (C)1 37.5 34.92 37.2 34.93 37.0 34.9

    Average 37.2 34.9

    6 Discussions

    6.1 Experimental Unit Step Response

    From Table 1, the experiment unit step response is plotted as shown in Figure 1.

    Figure 1: Unit Step Response from the Experiment

    3

  • 6.2 Ziegler-Nichols Step Response Method - First-Order Plus Dead-TimeModel

    The system is approximated by a first-order plus dead-time model as shown in Figure 2.

    Figure 2: Ziegler-Nichols Step Response Approximation

    The unit step response of the experiment data is approximated by

    Gp(s) =Kp

    sT + 1esL

    with Kp = 0.24, L = 0.5 and T = 4.5.

    Unit step response for the transfer function,

    Gp(s) =0.24

    4.5s+ 1e0.5s

    is superimposed with the unit step response from the experiment data (Figure 3).

    4

  • Figure 3: Unit Step Response of Experiment Data (Solid Line) Approximated by First-Order Plus Dead-TimeModel (Dashed-Line)

    6.3 Method of Areas - First-Order Plus Dead-Time Model

    By calculating areas from the unit step response of the experiment data shown in Figure 1, we can obtain

    A0 = 1.02 = Kp(T + L)

    A1 = 0.31 = KpTe1

    Substituting Kp = 0.24, we got T = 3.51 and L = 0.739. Unit step response for the transfer function,

    Gp(s) =0.24

    3.51s+ 1e0.739s

    is superimposed with the unit step response from the experiment data (Figure 4).

    5

  • Figure 4: Unit Step Response of Experiment Data (Solid Line) Approximated by First-Order Plus Dead-TimeModel (Dashed-Line)

    6.4 Least-Square Estimation - First-Order Model

    Consider a first-order model,

    Y (s)

    U(s)=

    KpTs+ 1

    TsY (s) + Y (s) = KpU(s)

    T.y(t) + y(t) = Kpu(t)

    With sampling interval 0.5s,

    Ty(tk) y(tk1)

    0.5+ y(tk) = Kpu(tk)

    y(tk) =T

    T + 0.5y(tk1) +

    0.5KpT + 0.5

    u(tk)

    = 1y(tk1) + 2u(tk)

    where

    Kp =2

    1 1(1)

    T =0.511 1

    (2)

    Using least-square approximation method on the experiment data in Table 1, we obtain 1 = 0.9074 and2 = 0.0230.

    6

  • Substituting 1 = 0.9074 and 2 = 0.0230 into equation 1 and 2 gives us the approximate first-order transferfunction,

    Gp(s) =0.2484

    4.9016s+ 1

    The unit step response from the estimated first-order model is superimposed with the unit step responsefrom the experiment data (Figure 5).

    Figure 5: Unit Step Response of Experiment Data (Solid Line) Approximated by First-Order Model (Dashed-Line) using Least-Square Approximation

    6.5 Relay Feedback - Second-Order Plus Dead-Time Model

    The following parameters are obtained from Table 2 and Table 3 in Section 4.

    Tu = 173.3

    a = 1.3

    d =122

    2= 72

    Ku =4d

    a= 70.52

    By inspection,

    L = 0.25

    Kp = 0.24

    We knowarg Gp(j!u) =

    7

  • and

    Gp(j!u) =1

    Ku

    where

    Gp(s) =Kp

    (sT1 + 1)(sT2 + 1)esL

    We have

    arctan!uT1 arctan!uT2 !uL = (3)Kp

    !2uT21 + 1

    !2uT

    21 + 1

    =1

    Ku(4)

    Solving equation 3 and 4 simultaneously after substituting Ku = 70.52, Kp = 0.24, L = 0.25 and !u =2Tu 60 = 2.175, we got

    T1 = 2.2

    T2 = 1.5

    The approximated second-order plus dead-time step response of

    Gp(s) =0.24

    (2.2s+ 1)(1.5s+ 1)e0.25s

    is superimposed with the step response of the experiment data (Figure 6).

    Figure 6: Unit Step Response of Experiment Data (Solid Line) Approximated by Second-Order Plus Dead-Time Model (Dashed-Line)

    8

  • 6.6 ON-OFF Control

    Using Table 4 and 5, the approximate average temperature,

    Tave =50 37.2 + 142.3 34.9

    50 + 142.3= 35.5C

    7 Conclusions

    1. A system can be estimated by a first-order plus dead-time process model using Ziegler-Nichols open-loop method of tuning by obtaining the parameters of the open-loop step response.

    2. The system can also be estimated using Ziegler-Nichols ultimate cycling method of tuning by obtainingthe ultimate gain and ultimate period from the relay system.

    3. Alternately, method of area and least square estimation can be used to estimate the system.

    9

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