Quiz 4: Linearization

7 thoughts on “Quiz 4: Linearization

  1. 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 that’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.

    • Yes, always the approximation.

      The first derivative of

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

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

    • 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.

  2. 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.

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