Conveyor Belt Case Study

Which has the following State Space Representation: Where $Q$ is torque

Example With Numbers

Given

$$T=1,j=10,r=1,d=1,m=1$$

Where $T$ is the time constant, $j$ is the intertia, and $m$ is usually unknown

$$\left[\begin{array}{c}\dot{Q}\\ \ddot{x}\\ \dot{x}\end{array}\right]=\left[\begin{array}{ccc}-1& 0& 0\\ \frac{1}{11}& -\frac{1}{11}& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}Q\\ \dot{x}\\ x\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]u$$

Gives us $\lambda =0,-1,-\frac{1}{11}$, marginally stable

We use an observer model where $m=0$, even though this is wrong, the observer will make up for it.

$$\left[\begin{array}{c}\dot{Q}\\ \ddot{x}\\ \dot{x}\end{array}\right]=\left[\begin{array}{ccc}-1& 0& 0\\ \frac{1}{10}& -\frac{1}{10}& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}Q\\ \dot{x}\\ x\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]u$$

We have three different cases, depending on how many states we are measuring: Every case here is observeable

Case 3

Using Measurement Equation Inversion:

$$y_m=C_{3}x+n,C_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

With $C_3={\mathbb{I}}_3$:

$$\hat{x}=C_3^{-1}{y}_m=x+n$$

And the error will be:

$$e=x-\hat{x}=-n$$

Which yields the following results

Using Open-loop Estimator, yields following results

Using Closed-loop Estimator - Luenberger Observer, yields following results