A process consists of an integrating element operating in parallel with a first-order element (Fig. E6.6). (a) What is the order of the overall transfer function, G(s) = Y(s)/U(s)? (b) What is the gain of G(s)? (c) What are the poles of G(s)? Where are they located in the complex s-plane? (d) What

Question

A process consists of an integrating element operating in parallel with a first-order element (Fig. E6.6).

(a) What is the order of the overall transfer function, [latex]G(s)=[/latex] [latex]Y(s) / U(s) ?[/latex]

(b) What is the gain of [latex]G(s)[/latex] ?

(c) What are the poles of [latex]G(s)[/latex] ? Where are they located in the complex [latex]s[/latex]-plane?

(d) What are the zeros of [latex]G(s) ?[/latex] Where are they located? Under what condition(s) will one or more of the zeros be located in the right-half [latex]s[/latex]-plane?

(e) For what conditions, will this process exhibit both a negative gain and a right-half plane zero?

(f) For any input change, what functions of time (response modes) will be included in the response, [latex]y(t)[/latex] ?

(g) Is the output bounded for any bounded input change, for example, [latex]u(t)=M ?[/latex]

Accepted Answer

[latex]\begin{aligned}&Y(s)=\frac{K_{1}}{s} U(s)+\frac{K_{2}}{ au s+1} U(s)=\left[\frac{K_{1}}{s}+\frac{K_{2}}{ au s+1} ight] U(s) \&\frac{Y(s)}{U(s)}=\frac{K_{1} au s+K_{1}+K_{2} s}{s( au s+1)}=\frac{\left(K_{1} au+K_{2} ight) s+K_{1}}{s( au s+1)}\end{aligned}[/latex]

Put in standard [latex]K / au[/latex] form for analysis:

[latex]G(s)=\frac{Y(s)}{U(s)}=\frac{K_{1}\left[\left( au+\frac{K_{2}}{K_{1}} ight) s+1 ight]}{s( au s+1)}[/latex]

a) Order of [latex]G( s )[/latex] is 2 (maximum exponent on [latex]s[/latex] in denominator is 2 )

b) Gain of [latex]G( s )[/latex] is [latex]K_{1}[/latex]. Gain is negative if [latex]K_{1}<0[/latex].

c) Poles of [latex]G( s )[/latex] are: [latex]s_{1}=0[/latex] and [latex]s_{2}=-1 / au[/latex]

[latex]s_{1}[/latex] is on imaginary axis; [latex]s_{2}[/latex] is in the left hand plane.

d) The zero of [latex]G( s )[/latex] is:

[latex]s_{a}=\frac{-1}{\left( au+\frac{K_{2}}{K_{1}} ight)}=\frac{-K_{1}}{K_{1} au+K_{2}}[/latex]

If [latex]\frac{K_{1}}{K_{1} au+K_{2}}<0[/latex], the zero is in right half plane.

Two possibilities:

[latex]K_{1}<0[/latex] and [latex]K_{1} au+K_{2}>0[/latex]

e) Gain is negative if [latex]K_{1}<0[/latex]

Then the zero is RHP if [latex]K_{1} au+K_{2}>0[/latex]. This is the only possibility.

f) Constant term and [latex]e ^{-t / au}[/latex] term.

g) If input is M/s, the output will contain a t term that is not bounded.

Question 6.6: A process consists of an integrating element operating in pa...

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