PT1 Filter without Derivative

Question

I am currently integrating a C-Code (written by a third person) in a Simulink model. In the C-Code various sensor signals are filtered, using a PT1 filter with the following equation:

$$y(t) = \dfrac{u(t) + Cy(t-1)}{C + 1}$$

Where $y(t)$, $u(t)$ and $C$ are the output (the filtered signal), input and a factor applied for each signal individually, respectively.

Looking at the step function response, I can see a typical PT1-behaviour with a gain factor of 1.

I cannot figure out the derivation of this function from the general formulation (the differential equation) of a PT1, which is:

$$Ty'(t) + y(t) = Ku(t)$$

Can any of you guys help me out?

Answer 1

The equation being used in code is a discrete-time implementation of the filter. The second equation you gave is a continuous-time expression of the filter dynamics. In order to create a discrete-time filter from the continuous-time dynamics the person who wrote the code used a first order approximation (backwards difference, specifically) of the derivative of $y$, like so:

$$\frac{dy}{dt} \approx \frac{y(t)-y(t-\Delta t)}{\Delta t}$$

Now, if we use the notation $k$ to denote a sample taken at time $t$ and $k-1$ to denote a sample taken at time $t-\Delta t$, then we can derive the equation used in code as follows:

$$T\frac{dy}{dt} + y(t) = Ku(t)$$ $$T \frac{y(k)-y(k-1)}{\Delta t} + y(k) = Ku(k)$$ $$\left(\frac{T}{\Delta t} + 1\right) y(k) = K u(k) + \frac{T}{\Delta t} y(k-1)$$

Now let $K=1$ and $C = \frac{T}{\Delta t}$, and you get:

$$\left(C + 1\right) y(k) = u(k) + C y(k-1)$$ $$ y(k) = \frac{u(k) + C y(k-1)}{C + 1}$$