Question

A proportional controller with transfer function, K_p is used with a first—order system having its transfer function as \(G(s)=\frac{K}{(1+sτ)}\) in unity feedback structure. For step inputs, an increase in K_p will 
1. Increase the time constant and decrease the steady state error 
2. Decrease the time constant and decrease the steady state error
3. Decrease the time constant and increase the steady state error 
4. Increase the time constant and increase the steady state error

Answer 1

Correct Answer - Option 2 : Decrease the time constant and decrease the steady state error

Concept:

LTI system in time constant form is defined as:

\(C\left( s \right) = \frac{{K\left( {1 + s{τ _1}} \right)\left( {1 + s{τ _2}} \right) \cdots }}{{{s^n}\left( {1 + s{τ _a}} \right)\left( {1 + s{τ _b}} \right) \cdots }}\)

τ1, τ2 ⋯ , and τa, τb ⋯ are time constants.

Steady-state error:

K_p = position error constant

= \(\mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)

Kv = velocity error constant

= \(\mathop {\lim }\limits_{s \to 0} sG\left( s \right)H\left( s \right)\)

Ka = acceleration error constant

= \(\mathop {\lim }\limits_{s \to 0} {s^2}G\left( s \right)H\left( s \right)\)

Steady-state error for different inputs is given by

Input	Type -0	Type - 1	Type -2
Unit step	\(\frac{1}{{1 + {K_p}}}\)	0	0
Unit ramp	∞	\(\frac{1}{{{K_v}}}\)	0
Unit parabolic	∞	∞	\(\frac{1}{{{K_a}}}\)

Calculation:

Given:

\(G(s)H(s)=\frac{K}{(1+sτ)}\)

G_C(s) = K_P

The close loop transfer function is given as:

\(CLTF=\frac{KK_P}{1+sT+KK_P}\)

\(CLTF=\frac{\frac{KK_P}{1+KK_P}}{1+\frac{T}{KK_P}s}\)

Comparing with the standard forms we get time constants as:

\(τ = \frac{T}{1+KK_P}\)

\(K_p=\mathop {\lim }\limits_{s \to 0} G\left( s \right)H\left( s \right)\)

\(K_p=\mathop {\lim }\limits_{s \to 0} \frac{KK_P}{1+sT}\)

\(K_p=KK_P\)

Steady-state error is calculated as:

\(e_{ss}=\frac{1}{{1 + {K_p}}}\)

\(e_{ss}=\frac{1}{{1 + {KK_P}}}\)

Hence, If K_P is increased then both time constant and e_ss decreases.

Note: Don't get confused with K_P and K_p.

K_P: Transfer function of a proportional controller.

K_p: position error constant