achapt 7 - REGULATING THE REFERENCE PATH 7 REGULATING THE...

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REGULATING THE REFERENCE PATH 7 REGULATING THE REFERENCE PATH (CONTINUOUSLY-ACTING ACTUATORS) This Chapter considers the regulation problem restricted to the case in which the control force is produced by a continuously-acting actuator. The control force is a continuous function of time. We’ll consider in this chapter control forces that are linearly proportional to displacements, integrals of displacements, velocities, and linear combinations of them. In each case, we’ll see how the dynamic performance is regulated by the control. 1. Displacement Feedback Consider an undamped single degree of freedom system. In the absence of a control force, the equation governing the motion and the uncontrolled system response are (7 – 1) ) sin( ) cos( , 0 0 0 0 0 0 t v t x x kx x m u u u ω + = = + & & MAE 461: DYNAMICS AND CONTROLS
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REGULATING THE REFERENCE PATH In the presence of displacement feedback, we get (7 – 2) ) sin( ) cos( , , 0 0 0 t v t x x gx f kx x m β + = = = + & & where g is called the displacement feedback control gain . In Eq. (7 – 1) and Eq. (7 – 2), we have m g k m k + = = ω 0 in which is called the closed-loop frequency of the system (See Fig. 7 – 1). ) 3 , 1 , 0 ( ). ( and ) ( 1 7 Figure 0 0 = = = = = g k m v x t x t x u 0 10 20 30 40 -1 -0.5 0 0.5 1 Since displacements are continuous functions of time, so too are displacement feedback control forces. Therefore, displacement feedback can be relatively easy to produce in devices that can generate forces that are continuous functions of time. MAE 461: DYNAMICS AND CONTROLS
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REGULATING THE REFERENCE PATH Now let’s see how displacement feedback changes system performance. First, consider peak- overshoot. From Eq. (7 – 1), the peak-overshoot of the uncontrolled system is (7 – 3) 2 0 0 2 ) / ( ω v x PO o u + = and from Eq. (7 – 2), the peak-overshoot of the controlled system is (7 – 4) u o o o PO v x v x v x PO 2 0 0 2 2 0 2 2 0 2 ) / ( ) / ( ) / ( β + + = + = In Eq. (7 – 4) we find that peak-overshoot can not be reduced significantly if . 0 0 0 v x >> Indeed, if a system is initially at rest when it’s initially displaced, its peak-overshoot will be equal to its initial displacement regardless of the stiffness in the system. On the other hand, if , 0 0 = x Eq. (7 – 4) reduces to (7 – 5) u PO PO 0 = This case also arises when the system is intermittently subjected to impulsive forces. From Eq. (7 – 5), the desired closed-loop frequency is (See Fig. 7 – 2) (7 – 6) 0 = PO PO u MAE 461: DYNAMICS AND CONTROLS
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REGULATING THE REFERENCE PATH Equation (7-6) is applied to lightly damped systems as an approximation. (Recall that Eq.(7 – 1) is undamped, so Eq. (7 – 6) is exact only for undamped systems.) Next, let’s examine the effect of displacement feedback on settling time. From Eq. (7 – 1) and (7 – 2) displacement feedback clearly has no effect of settling time. Finally, turning to steady-state error, let’s subject the system to a unit step function in order to create a steady-state error. The steady-state response in the absence of displacement feedback and the steady-state response in the presence of displacement feedback are (7 – 7) 2 1 1 , 1 = + = = u u PO PO k g k x k x
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achapt 7 - REGULATING THE REFERENCE PATH 7 REGULATING THE...

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