Parallel Structures Between Continuous and Discrete Time Systems

CT Plant Systems (Generalized)

The dynamics of a typical plant in continuous time can be represented as

$$\sum _{i=0}^n{a}_i\frac{d^{i}y}{d{t}^i}(t)-\sum _{j=0}^m{b}_j\frac{d^{j}u}{d{t}^j}(t)=0$$

$y$ and $u$ here are arbitrary output/input of the plant. $a$ and $b$ are arbitrary coefficients.

We usually take the Laplace transform of this plant to work in the frequency domain.

$$\mathcal{L}={\int }_0^{\infty }x(t)e^{-st}dt$$

The Inverse Laplace Transform is given by

$$x(t)={\mathcal{L}}^{-1}\{X(s)\}=\frac{1}{2\pi j}{\int }_{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds$$

Following that, we can take the Laplace Transform of the dynamical system.

$$\mathcal{L}⟹\sum _{i=0}^n{a}_i{s}^{i}Y(s)-\sum _{j=0}^m{b}_j{s}^{j}U(s)=0$$

And from this obtain the transfer function.

$$Y(s)=\frac{{\sum }_{j=0}^m{b}_j{s}^j}{{\sum }_{i=0}^n{a}_i{s}^i}U(s)$$ $$P(s):=\frac{Y(s)}{U(s)}=\frac{{\sum }_{j=0}^m{b}_j{s}^j}{{\sum }_{i=0}^n{a}_i{s}^i}$$

$:=$ is not the assignment operator like you see in Optimizers and gradient descent, it just mean "is defined as" in formal math terms.

So from this we see that we have derived a nicely defined characteristic equation of the dynamical system in the frequency domain.

Example

Mass spring damper (as seen in Control Systems) would be represented as.

$$m\ddot{y}+c\dot{y}+ky=F$$

Which in terms of the general formula would look something like.

$$a_2\ddot{y}+a_1\dot{y}+a_{0}y-b_{0}u=0$$

where $a_2=m$, $a_1=c$, $a_0=k$, and $F=u\text{(THIS IS OUR INPUT)}),b_0=1$. We can take a Laplace transform of above to then determine the transfer function of the system.

DT Plant Systems (Generalized)

The dynamics of a typical plant in discrete time can be represented as

$$\sum _{i=0}^n{a}_{i}y[k+i]-\sum _{j=0}^m{b}_{j}u[k+j]=0$$

Where again $y$ and $u$ are the output/input of the system respectively. $a$ and $b$ are arbitrary coefficients characterizing the plant.

Just like how in CT we took the Laplace Transform to bring ourselves into the frequency domain to do better processing, we can do a similar thing here. In Discrete-Time, we are doing a Z transform.

$$X[z]=\mathcal{Z}(x[k])=\sum _{k=0}^{\infty }x[k]\frac{1}{z^k}$$

The Inverse Z Transform is given by

$$x[k]={\mathcal{Z}}^{-1}(X[z])=\frac{1}{2\pi j}{\oint }_{\text{unit circle}}X[z]z^{k-1}dz=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X[e^{j\theta }]e^{jk\theta }d\theta$$

Following that, we can take the Z transform of the platnt.

$$\mathcal{Z}⟹\sum _{i=0}^n{a}_i{z}^{i}Y[z]-\sum _{j=0}^m{b}_j{z}^{j}U[z]=0$$

And just like in the frequency domain, we can rearrange to get a discrete transfer function.

$$Y[z]=\frac{{\sum }_{j=0}^m{b}_j{z}^j}{{\sum }_{i=0}^n{a}_i{z}^i}U[z]$$ $$G[z]:=\frac{Y[z]}{U[z]}=\frac{{\sum }_{j=0}^m{b}_j{z}^j}{{\sum }_{i=0}^n{a}_i{z}^i}$$

Parallels

Continuous-Time	Discrete-Time
Differential equation (ODE)	Difference equation
Laplace transform	z-transform
Transfer function $P(s)$	Transfer function $G[z]$
$Y(s)=P(s)U(s)$	$Y[z]=G[z]U[z]$