The set-point of the control system under proportional control $\left(K_{c}=5\right)$ undergoes a step change of magnitude 2 . For

$G_{v} G_{p}=\frac{2}{(s+1)(2 s+1)}$ and $G_{m}=1$

(a) Determine when the maximum value of $y$ occurs.

(b) Determine the offset.

(c) Determine the period of oscillation.

(d) Sketch $y(t)$ as a function of time, showing key characteristics as determined in (a), (b), and (c).

Question

The set-point of the control system under proportional control [latex]\left(K_{c}=5\right)[/latex] undergoes a step change of magnitude 2 . For [latex]G_{v} G_{p}=\frac{2}{(s+1)(2 s+1)}[/latex] and [latex]G_{m}=1[/latex]

(a) Determine when the maximum value of [latex]y[/latex] occurs.

(b) Determine the offset.

(c) Determine the period of oscillation.

(d) Sketch [latex]y(t)[/latex] as a function of time, showing key characteristics as determined in (a), (b), and (c).

Accepted Answer

First we use Eq. 11-26 to get the closed loop transfer function

[latex]\frac{Y}{Y_{s p}}=\frac{K_{m} G_{c} G_{v} G_{p}}{1+G_{c} G_{v} G_{p} G_{m}}[/latex] (11-26)

[latex]\frac{Y}{Y_{s p}}=\frac{\frac{10}{(s+1)(2 s+1)}}{1+\frac{10}{(s+1)(2 s+1)}}=\frac{10}{(s+1)(2 s+1)+10}=\frac{10}{2 s^{2}+3 s+11}[/latex]

Or in standard form

[latex]\frac{Y}{Y_{s p}}=\frac{10 / 11}{\frac{2}{11} s^{2}+\frac{3}{11} s+1} \quad \zeta=\frac{3 \sqrt{22}}{44} \quad au=\frac{\sqrt{22}}{11}[/latex]

The time at which the maximum occurs is given by Eq. [latex]5-52[/latex]

[latex] ext { Decay ratio: } \quad DR =( OS )^{2}=\exp \left(-2 \pi \zeta / \sqrt{1-\zeta^{2}} ight)[/latex] (5-52)

[latex]t_{p}=\pi au / \sqrt{1-\zeta^{2}} \quad t_{p}=1.41[/latex]

(b)

The response is given by

[latex]Y(s)=\frac{20}{s\left(2 s^{2}+3 s+11 ight)}[/latex]

The Final Value Theorem gives the steady state value as

[latex]y(\infty)=\frac{20}{11}[/latex]

Subtracting the steady state value from the set point change gives offset as

[latex] ext { offset }=\frac{2}{11}[/latex]

(c)

The period of oscillation is given by Eq. [latex]5-55[/latex]

P=3060-1000=2060 s (5-55)

[latex]P=\frac{2 \pi au}{\sqrt{1-\zeta^{2}}} \quad P=2.83[/latex]

(d)

Hint: You do not need to obtain the analytical response [latex]y(t)[/latex] to answer the above questions. Use the standard second order model expressed in terms of [latex]\zeta[/latex] and [latex] au[/latex] (see Chapter 5 ).

Question 11.25: The set-point of the control system under proportional contr...

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