Prove that a Type-1 system has no steady state error for step input while the steady state

Question

error for ramp input decreases for increase of Velocity Error Constant

Answer 1

A Type-1 system is a system with one integrator in its forward path. The transfer function of such a system can be represented as:

G(s) = K_p * (K_i / s),

where K_p and K_i are the proportional and integral gain constants, respectively.

Let's consider a step input R(s) = A / s, where A is the magnitude of the step.

The steady-state error, E(s), is given by:

E(s) = R(s) - G(s) * R(s) / (1 + G(s)).

For a Type-1 system with a step input:

E(s) = (A / s) - (K_p * K_i / s^2) * (A / s) / (1 + (K_p * K_i / s^2) * (A / s)).

After multiplying the denominator with s^2 and rearranging the terms, we obtain:

E(s) = A * s^2 / (s^2 + K_p * K_i * A).

Taking the limit as s approaches 0 (steady-state error):

lim (s->0) E(s) = lim (s->0) A * s^2 / (s^2 + K_p * K_i * A).

Since the numerator is a quadratic function of s and the denominator has a linear term and a quadratic term, the limit is 0:

E(0) = 0.

Hence, a Type-1 system has no steady-state error for a step input.

Now let's consider a ramp input R(s) = B / s^2, where B is the slope of the ramp. The error function for a Type-1 system with a ramp input is given by:

E(s) = (B / s^2) - (K_p * K_i / s^2) * (B / s^2) / (1 + (K_p * K_i / s^2) * (B / s^2)).

After multiplying the denominator with s^4 and rearranging the terms, we obtain:

E(s) = B * s^2 / (s^2 + K_p * K_i * B).

Taking the limit as s approaches 0 (steady-state error):

lim (s->0) E(s) = lim (s->0) B * s^2 / (s^2 + K_p * K_i * B).

Again, since the numerator is a quadratic function of s and the denominator has a linear term and a quadratic term, the limit can be calculated using L'Hopital's rule:

lim (s->0) (d(B * s^2)/ds) / (d(s^2 + K_p * K_i * B)/ds)
= lim (s->0) (2 * B * s) / (2 * s)
= B.

So the steady-state error for a ramp input is given by the slope of the ramp, B.

The Velocity Error Constant (K_v) is found by taking the limit as s approaches 0 of sG(s), which in this case is:

K_v = lim (s->0) s * (K_p * (K_i / s))
= K_p * K_i.

According to the steady-state error formula for ramp input, E(s) = B. If we replace B with the slope of the sum of multiple ramp inputs, the steady-state error will still be B. Since E(s) = B does not depend on the value of K_v (K_p * K_i), the steady-state error for a ramp input does not decrease with an increase of the Velocity Error Constant K_v.

However, it is essential to note that an increase in K_v results in a reduction of the steady-state error concerning time. In other words, the Type-1 system will settle to the steady-state error value more quickly when the Velocity Error Constant is increased.