Example: Transport systems

Transport systems are typical candidates for systems. They can be described by the one-dimensional partial differential equation (or simply transport equation)

$$\frac{\partial x(t,z)}{\partial t}+\nu \frac{\partial x(t,z)}{\partial x}=u(t,x)\text{for}z\in (0,L),t>0$$

with velocity $\nu >0$, left boundary condition $x(t,0)=u_0(t)$ for $t>0$ and the initial data

$$x(0,z)=x_z(0)\text{for}z\in [0,L].$$

Function $u(t,x)$ is the distributed control input along the complete length and function $u_0(t)$ is the boundary input on the left side ($x=0$).

For simplicity, it is assumed that control input $u(t,x)$ is constant along the whole length.

Solution in Laplace-domain

Further the output is assumed at the right side of the system ($x=L$) as $y(t)=x(t,L)$. A Laplace transform of the PDE and further calculations lead to

$$\hat{y}(s)=\exp (-s{T}_t){\hat{u}}_0(s)+\frac{1}{s}[1-\exp (-s{T}_t)]\hat{u}(s)$$

with $T_t=\frac{L}{\nu }$.

Test cases

Two scenarios are considered next:

• Fluid flow in a pipe with single boundary input $u_0(t)>0$, thus $u(t,x)=0$ and

• Conveyor system with distributed input $u(t)=u(t,x)>0$, thus $u_0(t)=0$.

Fluid flow in a pipe

The transfer function is noted as

$${\hat{g}}_0=\frac{\hat{y}(s)}{{\hat{u}}_0(s)}=\exp (-s{T}_t)\text{with}{T}_t=\frac{L}{\nu }.$$

The ideal delay is approximated by

$${\hat{g}}_{approx}=\frac{a}{s+a}\exp (-s{T}_t)$$

with $a>0$.

Step response

Bode plot