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Chapter 20

Q. 20.1

For the transfer functions

G_{P}(s)=\frac{3 e^{-2 s}}{(15 s+1)(10 s+1)} \quad G_{v}=G_{m}=1

(a) Derive an analytical expression for the step response to a unit step change. Evaluate the step-response coefficients, \left\{S_{i}\right\}, for a sampling period of \Delta t=1.

(b) What value of model horizon N should be specified in order to ensure that the step-response model covers a period of at least 99 \% of the open-loop settling time? (That is, we require that N \Delta t \geq t_{99} where t_{99} is the 99 \% settling time.)

Step-by-Step

Verified Solution

a) The unit step response is

Y(s)=G_{p}(s) U(s)=\left(\frac{3 e^{-2 s}}{(15 s+1)(10 s+1)}\right)\left(\frac{1}{s}\right)=3 e^{-2 s}\left[\frac{1}{s}-\frac{45}{15 s+1}+\frac{20}{10 s+1}\right]

Therefore,

y(t)=3 S(t-2)\left[1+2 e^{-(t-2) / 10}-3 e^{-(t-2) / 15}\right]

For \Delta t=1,

S_{i}=y(i \Delta t)=y(i)=\{0,0,0.0095,0.036,0.076,0.13 \ldots\}

b) Evaluate the expression for y(t) in part (a)

y(t)=0.99(3) \approx 2.97 \text { at } t=87

Thus, N=87, for 99 \% complete response.