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Abstract
In the present work, the equating coefficient method (ECM) of designing proportional integral derivative (PID) controllers is applied to critically damped stable second-order plus time delay (SOPTD) systems. Four simulation examples representing the higher-order models and based on the identified critically damped SOPTD model are considered to show the effectiveness of the proposed method. A good improvement in the closed-loop system performance is achieved for the ECM when compared to that the internal model control (IMC) method. A significant improvement is obtained in the performance of the control systems based on the critically damped SOPTD model over that of the first-order plus time delay (FOPTD) model. The ECM of designing PID controllers is also given for stable general SOPTD systems (under damped or over damped).
Nomenclature
| s | = | Laplace variable |
| θ | = | process time delay |
| τ | = | process time constant |
| Kp | = | process gain |
| ζ | = | damping coefficient |
| K1 | = | = KcKp |
| K2 | = | |
| K3 | = | |
| q | = | = τs |
| α1, α2, α3 | = | tuning parameters |
| ε | = | |
| Gc | = | controller transfer function |
| Gp | = | process transfer function |
| kc | = | controller gain |
| τI | = | integral time |
| τD | = | derivative time |
| τ1, τ2 | = | process time constants of overdamped system |
| t1 | = | time taken to reach fractional response (of 0.14 for SOPTD model), (of 0.353 for FOPTD model) |
| t2 | = | time taken to reach fractional response (of 0.55 for SOPTD model), (of 0.853 for FOPTD model) |
| t3 | = | time taken to reach fractional response of 0.91 for the SOPTD model |
| t* | = | time taken to reach fractional response of 0.73 after subtracting the time delay for the SOPTD for Harriott's method |
| y1 | = | fractional response of 0.14 for the SOPTD model fractional response of 0.353 for the FOPTD model |
| y2 | = | fractional response of 0.55 for the SOPTD model fractional response of 0.853 for the FOPTD model |
| y3 | = | fractional response of 0.91 for the SOPTD model |
| ▵y∞ | = | change in steady-state output |
| ▵u∞ | = | change in steady-state input |
| ISE | = | Integral of the square of the error |
| ECM | = | Equating coefficient method |
| IMC | = | Internal Model Controller method |
| m | = |
Greek Letters
| α | = | |
| β | = | |
| X1 | = | dimensionless biomass cell concentration |
| X2 | = | dimensionless substrate concentration |
| X2f | = | dimensionless substrate feed concentration |
| D1 | = | dilution rate |
| µ | = | specific generation rate |
| µm | = | maximum specific generation rate |
| Km | = | constant in the expressions for µ |
| Γ | = | yield factor |