Control Engineering by C. C. Bissell (auth.)

By C. C. Bissell (auth.)

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We can approximate this in practice by taking measurements only when sufficient time has elapsed after switching on the input. A good rula of thumb for first-order systems is to allow a time equivalent to 5 time constants before taking measurements of phase shift and amplitude ratio. Q = kv both Q and Q will also be sinusoidal in nature. Then the left-hand side can be combined into a single sinusoid, which must correspond to the applied sinusoidal voltage multiplied by the gain k. This is a fundamental result, which applies to any linear differential equation with constant coefficients.

Hence compare the closed-loop step response of the motor speed control system for a controller gain of 1, 5 and 10. 9 Fig. 0 tls The effect on closed-loop step response of increasing controller gain. 17 s. The unit step responses for K = 1, 5 and 10 are as shown in Fig. 2. For the general case consider a first-order process with transfer function G/(1 + jwr), where G is now the steady-state gain. 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 .

An equation of this type can model many lumped parameter systems which possess inertia, stiffness, and viscous friction. In mechanical terms such models are often known as 'massspring-damper' models. 2 = kwnei 37 ~ System output 80 Lumped second moment of mass of antenna 8§ _ks ~ Fig. 20 Note that two independent parameters are required for a fi rst-order lag, and th ree for a second-order lag. The significance of the parameters ( and wn will be considered below. 9 Lumped torsional stiffness of structure Sy"em ;cp

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