In response to the following question: “I have a question about part 4 of quiz 4, namely what exactly do you mean by controller gain in this case? At first I thought maybe it was the coefficient on the NTC resistor, but I doubt thatâ€™s the case.”:

Agreed, $\kappa$ would not change. Assume you have a P-controller with gain $k_p$. If the sensor gain increases, we need to reduce $k_p$ by the same amount to keep $L(s)$ the same.

This can be seen more general. For example, if we consider a PI-controller $H(s)=k_p + k_I/s$, this can be rewritten to feature a general controller gain $k$ as

where $k$ is now the overall controller gain and $\tau_I$ the integrator time constant. In this alternative form, the controller zero at $z=-1/\tau_I$ becomes independent from the gain. Here, $k$ would be reduced by the same amount that $k_s$ increases.

We are looking at the sensor only, with only a vague idea about the control loop itself (see above).

Part 2 follows the example in the book, Section 8.1, notably Eqns 8.1 and 8.2 — however, you need to take Eq. 2 from the quiz instead of Eq. 8.1.

One thing that helps me a lot is to plot it. Plot the $V_O$ over $T$ curve. Sketch the operating point. Sketch the tangent and draw it out to where it intercepts the $V_O$-axis. Then see if the equation you get matches this line.

For part 4 of the quiz, do we need to include the summation point from the intercept in our loop when finding the ratio or fact for the controller gain?

In response to the following question: “I have a question about part 4 of quiz 4, namely what exactly do you mean by controller gain in this case? At first I thought maybe it was the coefficient on the NTC resistor, but I doubt thatâ€™s the case.”:

Agreed, $\kappa$ would not change. Assume you have a P-controller with gain $k_p$. If the sensor gain increases, we need to reduce $k_p$ by the same amount to keep $L(s)$ the same.

This can be seen more general. For example, if we consider a PI-controller $H(s)=k_p + k_I/s$, this can be rewritten to feature a general controller gain $k$ as

$$

H(s)=k \cdot \left( 1 + \frac{1}{\tau_I s} \right)

$$

where $k$ is now the overall controller gain and $\tau_I$ the integrator time constant. In this alternative form, the controller zero at $z=-1/\tau_I$ becomes independent from the gain. Here, $k$ would be reduced by the same amount that $k_s$ increases.

Hi Dr.Haidekker,

For part 2 of Quiz 4, do we consider the approximated equation in equation 2 for calculating ks?

Yes, always the approximation.

The first derivative of

$$

\frac{1}{a+e^{-kt}}

$$

is doable, but for the purposes of this homework unnecessary.

Can anyone help with part 2? I am confused on where to even start. Is voltage the input and temperature the output?

We are looking at the sensor only, with only a vague idea about the control loop itself (see above).

Part 2 follows the example in the book, Section 8.1, notably Eqns 8.1 and 8.2 — however, you need to take Eq. 2 from the quiz instead of Eq. 8.1.

One thing that helps

mea lot is to plot it. Plot the $V_O$ over $T$ curve. Sketch the operating point. Sketch the tangent and draw it out to where it intercepts the $V_O$-axis. Then see if the equation you get matches this line.Dr. Haidekker,

For part 4 of the quiz, do we need to include the summation point from the intercept in our loop when finding the ratio or fact for the controller gain?

No. The intercept is an additive constant (i.e., a signal). Conversely, $k_s$ is a system coefficient. Only $k_s$ is of interest in this question.