By C. C. Bissell (auth.)
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Extra info for Control Engineering
Estimate the time 24 Hydraulic valve and actuator Controller Demanded position 6; Actual position eo Position transducer Fig. 7 A position control system. ," 8 Fig. """__,.. 0 t/s Response to a step change in demanded position constant and steady-state gain of the closed-loop system, and write down a differential equation modelling the system dynamics. Solution An estimate of the time constant can be made by determining how long the system output takes to reach 63% of its steady-state value. The steady-state output is 10 mm; from Fig.
In unity feedback configuration with a proportional controller of gain K the closed-loop transfer function becomes KG + jwr KG + 1 + jwr no ni KG (1 + KG) + jwr Dividing numerator and denominator by 1 + KG to get the standard form gives KG flo = _ _1_+_K_G_ _ 1 . 1 + (1 + KG) JWr In other words, the unity feedback closed-Ioop system behaves like a first-order lag, but with a steady-state gain of In theory the time constant and steady-state error of this system will be reduced indefinitely as the controller gain is increased.
The Bode plot of Fig. 18 is therefore identica! to that of Fig. 17, but raised by 32dB corresponding to the gain of 41. As in Fig. 17, the phase shift is -90° for ali frequencies. lncidentally, it is worth remembering that the amplitude ratio corresponding to an integrator model G(jw) = ~ JW is OdB at a value of w numerically equal to K. Integrator models are also often used in the form G(jw) = . 1T JW i where Ti is a constant. In this case 1 G 1 becomes OdB when w models will be used in this form in Chapter 4.