Steady state error and PID controller

In Lab 4, We used a PD compensator to control our ball and beam apparatus. The transfer function of our PD compensator was as follows:

However, we did not use the compensator in this form. The transfer function we used in lab was as follows:

Question

Solve for $K_C$ and $z$ in terms of $K_P$ and $K_D$.

Given

Or it can be written as:

To solve for the characteristic equation, we can set the coefficients of $s$ and the constant term to zero:

Thus, we can solve for $K_C$ and $z$ as follows:

Given that the transfer function of our Ball and Beam plant used in the previous lab is as follows:

And given that the controller is applied to the plant in cascade configuration, find:

2.a

Static error constant for position (position constant)

This is a Type-2 system, thus position constant $K_p = \infty$

2.b

Static error constant for velocity (velocity constant)

Velocity constant $K_v = \lim_{s\to 0} sG(s) = \lim_{s\to 0} \frac{0.419s}{s^2} = \infty$

2.c

Static error constant for acceleration (acceleration constant)

Acceleration constant $K_a = \lim_{s\to 0} s^2G(s) = \lim_{s\to 0} \frac{0.419s^2}{s^2} = 0.419$

2.d

Steady-state error for a step input $u(t)$

For a step input $R(s) = \frac{1}{s}$, the steady-state error is given by:

2.e

Steady-state error for a ramp input $tu(t)$

For a ramp input $R(s) = \frac{1}{s^2}$, the steady-state error is given by:

2.f

Steady-state error for a parabolic input $t^2u(t)$

For a parabolic input $R(s) = \frac{1}{s^3}$, the steady-state error is given by:

We will be augmenting our controller to include an integrator. The transfer function of our new PID compensator will be as follows;

Given that the transfer function for our plant has not changed, and given that this controller is also applied to the plant in cascade configuration.

The closed-loop transfer function is

3.a

Static error constant for position (position constant)

3.b

Static error constant for velocity (velocity constant)

3.c

Static error constant for acceleration (acceleration constant)

3.d

Steady-state error for a step input $u(t)$

For a step input $R(s) = \frac{1}{s}$, the steady-state error is given by:

3.e

Steady-state error for a ramp input $tu(t)$

For a ramp input $R(s) = \frac{1}{s^2}$, the steady-state error is given by:

3.f

Steady-state error for a parabolic input $t^2u(t)$

For a parabolic input $R(s) = \frac{1}{s^3}$, the steady-state error is given by:

Ideally you want your controller design to reject a step disturbance input at $D(s)$. This means that in the steady state for $D(s) = \frac{1}{s}$, the output $Y(s)$ is unchanged.

4.a

Ignoring the input $R(s)$, what is the transfer function $\frac{E(s)}{D(s)}$ in terms of $G_1(s)$ and $G_2(s)$?

To find the transfer function $\frac{E(s)}{D(s)}$, then the transfer function $\frac{E(s)}{D(s)}$ is given by:

4.b

For $G_1(s) = K_C(s+z)$ and $G_2(s) = \frac{0.419}{s^2}$ what is the steady state error resulting from step inputs $R(s) = \frac{A}{s}$ and $D(s) = \frac{B}{s}$

The steady-state error to step input $R(s) = \frac{A}{s}$ is given by:

with $L(s) = G_1(s)G_2(s) = \frac{0.419K_C(s+z)}{s^2}$

The steady-state error to step input $D(s) = \frac{B}{s}$ is given by: $e_{ss}(D) = \frac{B}{0.419K_C}$

Thus, the total steady-state error is $e_{ss} = e_{ss}(R) + e_{ss}(D) = \frac{A+B}{0.419K_C}$

4.c

For $G_1(s) = K_Ds + K_P + \frac{K_I}{s}$ and $G_2(s) = \frac{0.419}{s^2}$ what is the steady state error resulting from step inputs $R(s) = \frac{A}{s}$ and $D(s) = \frac{B}{s}$

The steady-state error to step input $R(s) = \frac{A}{s}$ is zero: