Page 145 - Mechatronics with Experiments
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CLOSED LOOP CONTROL  131
                             the DC motor model,
                                                            T (s) = K ⋅ i(s)                    (2.248)
                                                                   t
                                                             m
                                                             1
                                                       (s) =      ⋅ (T (s) − T (s))             (2.249)
                                                                           l
                                                                     m
                                                             2
                                                          (J s + cs)
                                                            t
                                                                                                  1
                                                                                                   J .
                             Let the parameters of the components be as follows: K = 1.0, K = 1.0, c = 10.0, J = J +  N 2 2
                                                                                               1
                                                                                           t
                                                                             t
                                                                      a
                             J = 0.1, N = 5, J = 1.0.
                              1           2
                                1. Find the closed loop system transfer function between the commanded position signal, load
                                   torque, and actual position signal. Select the gains of the PI controller K and K so that the
                                                                                       p     i
                                   step response setting time is about 100 ms with overshoot of about no more than 5%,andthat
                                   the steady state error in response to a step disturbance load, T (t) = 1.0 Step(t), should be zero
                                                                              l
                                   (assuming no actuator saturation).
                                2. Implement the PI controller digitally. Use the trapezoidal rule for the digital approximation.
                                   Neglect the effect of quantization errors due to finite resolution of ADC and DAC. Let sampling
                                   period to be T sample  = 1.0ms = 0.001 s. Obtain the digital version of the PI controller in the
                                   z-domain as a transfer function, and in the time domain as a difference equation expression.
                                3. Simulate in the time domain the response of the closed loop system to a step input command
                                   and disturbance torque condition,
                                                              (t) = 1.0 for t > 1.0 s           (2.250)
                                                            d
                                                           T (t) = 1.0 for t > 2.0 s            (2.251)
                                                            l
                                   for both analog controller implementation and digital controller implementation.
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