CompensatorLecture - ME 279 Automatic Control of Dynamic...

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Compensators ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers
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P Compensator 2 Compensator Design – objectives are to improve transient and steady state performance and maintain stability. Compensators Dr. Robert G. Landers Start with G C (s) = 1. This is known as a proportional compensator since the control signal is proportional to the error. In this case, u(t) = Ke(t). Add poles and zeros to improve closed–loop performance. G(s) KG C (s) U(s) R(s) + - Y(s) E(s)
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PI Compensator 3 Compensators Dr. Robert G. Landers The control signal is proportional to the error and the integral of the error. The integral portion helps to eliminate or reduce steady state error. The zero and gain are adjusted to improve transient performance. ( 29 ( 29 ( 29 0 t u t Ke t Ka e d τ τ = + Proportional plus integral (PI) compensator transfer function ( 29 C s a G s s + = Control signal
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Lag Compensator 4 Compensators Dr. Robert G. Landers For a lag compensator, z c > p c . The lag compensator is similar to a PI compensator and becomes a PI compensator if p c = 0. Lag compensators are used to reduce steady state error. ( 29 ( 29 ( 29 ( 29 c c u t p u t Ke t Kz e t + = + & & Lag compensator transfer function Differential equation describing control signal ( 29 c C c s z G s s p + = +
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Lag Compensator 5 Compensators Dr. Robert G. Landers The steady state error for a unit step input is ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 s s z s z G s K s p s p + + = + + L L Let the system be given by ( 29 ( 29 ( 29 0 1 2 1 2 1 1 1 lim 1 1 s c C s c e s z z z G s G s s KK p p p ∞ = = + + L L Select the ratio z c /p c to be large to reduce e(∞). Adjust the pole (or zero) to improve transient performance. Typically the pole and zero are to the right of the system poles and zeros and are close to the origin.
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PD Compensator 6 Compensators Dr. Robert G. Landers The control signal is proportional to the error and the derivative of the error. Typically the zero is to the left of the system poles and zeros. The zero and gain are adjusted to improve transient performance. The drawback of PD compensators is that differentiation is a noisy process. ( 29 ( 29 ( 29 u t Ke t Kae t = + & Proportional plus derivative (PD) compensator transfer function Control signal ( 29 C G s s a = +
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PD Compensator 7 Compensators Dr. Robert G. Landers ( 29 ( 29 ( 29 x t t x t v t t + ∆ - = 0 2 4 6 8 10 -2 -1 0 1 2 time (s) output derivative v(t) v m (t) 0 5 10 -1 -0.5 0 0.5 1 time (s) output x(t) x m (t) The derivative is computed with a first order forward finite difference
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Lead Compensator 8 Compensators Dr. Robert G. Landers For a lead compensator, z l < p l . The lead compensator is similar to a PD compensator. The noise due to differentiation is reduced. The pole and zero, which are typically to the left of the system poles and zeros, and the gain are adjusted to improve transient performance. ( 29 ( 29 ( 29 ( 29 l l u t p u t Ke t Kz e t + = + & & Lead compensator transfer function Differential equation describing control signal ( 29 l C l s z G s s p + = +
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PID Compensator 9 Compensators Dr. Robert G. Landers The control signal is proportional to the error, the integral of the error, and the derivative of the error. The integral portion helps to eliminate
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