Instance Models
This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.
| Refname | IM:pdEquationIM |
|---|---|
| Label | Computation of the Process Variable as a function of time |
| Input | \({r_{\text{t}}}\), \({K_{\text{p}}}\), \({K_{\text{d}}}\) |
| Output | \({y_{\text{t}}}\) |
| Input Constraints | \[{r_{\text{t}}}\gt{}0\]\[{K_{\text{p}}}\gt{}0\]\[{K_{\text{d}}}\gt{}0\] |
| Output Constraints | \[{y_{\text{t}}}\gt{}0\] |
| Equation | \[\frac{\,d^{2}{y_{\text{t}}}}{\,dt^{2}}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{{y_{\text{t}}}}={r_{\text{t}}}\,{K_{\text{p}}}\] |
| Description |
|
| Source | abbasi2015 and johnson2008 |
| RefBy | FR:Output-Values and FR:Calculate-Values |
Detailed derivation of Process Variable:
The Process Variable \({Y_{\text{s}}}\) in a PD Control Loop is the product of the Process Error (from DD:ddProcessError), Control Variable (from DD:ddCtrlVar), and the Power Plant (from GD:gdPowerPlant).
\[{Y_{\text{s}}}=\left({R_{\text{s}}}-{Y_{\text{s}}}\right)\,\left({K_{\text{p}}}+{K_{\text{d}}}\,s\right)\,\frac{1}{s^{2}+s+20}\]
Substituting the values and rearranging the equation.
\[s^{2}\,{Y_{\text{s}}}+\left(1+{K_{\text{d}}}\right)\,{Y_{\text{s}}}\,s+\left(20+{K_{\text{p}}}\right)\,{Y_{\text{s}}}-{R_{\text{s}}}\,s\,{K_{\text{d}}}-{R_{\text{s}}}\,{K_{\text{p}}}=0\]
Computing the Inverse Laplace Transform of a function (from TM:invLaplaceTransform) of the equation.
\[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{K_{\text{d}}}\,\frac{\,d{r_{\text{t}}}}{\,dt}-{r_{\text{t}}}\,{K_{\text{p}}}=0\]
The Set-Point \({r_{\text{t}}}\) is a step function and a constant (from A:Set-Point). Therefore the differential of the set point is zero. Hence the equation reduces to
\[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{r_{\text{t}}}\,{K_{\text{p}}}=0\]