Noise Supression

We want to estimate x using an observer to suppress noise in the measurement

$$\dot{x}=-\frac{1}{T}x+\frac{1}{T}u,y_m=x+n$$

Which is on State Space Representation form, where $A=-\frac{1}{T}$, $B=\frac{1}{T}$, $C=1$

Example Using Closed Loop Estimator

Closed-loop Estimator - Luenberger Observer Estimate update equation will be:

\dot{\hat{x}} &= -\frac{1}{T}\hat{x}+\frac{1}{T}u+L(y_{m}-\hat{x}) \\ \dot{\hat{x}}&=\left( -\frac{1}{T}-L \right)\hat{x}+\frac{1}{T}u+Ly_{m} \end{align}

And the error dynamics is

$$\dot{e}=\left(-\frac{1}{T}-L\right)e-Ln$$

And we can place our pole, ${\lambda }_1=-\frac{1}{T_d}$, like this

-\frac{1}{T}-L &= -\frac{1}{T_{d}} \\ L &= \frac{1}{T_{d}}-\frac{1}{T} \end{align}

Where $L$ is a scalar, and our observer gain (should’ve used $l$ instead to clarify)

We can compare the transfer function, $H(s)=\frac{e}{n}(s)$, with and without our observer respectively

$$\hat{H}(s)=\frac{\left(\frac{1}{T}-\frac{1}{T_d}\right)}{s+\frac{1}{T_d}}$$ $$\hat{H}(s)=-1$$

Meaning that without our observer, where $\hat{H}(s)=-1$, all noise will be let through. And with our observer we lowpass the noise.