Higher-order functions can be simplified by applying the truncated Taylor series that we learned about for time delay functions (e-s = 1/(s+1)). When applying this approach in the opposite direction, the more dominate modes (high1er values) must be maintained and others can be substituted. For example, the function

can be approximated by Consider a process including a sensor, control valve, and heated tank that can be modeled by the fourth-order transfer function below. Determine (a) PI controller parameters based on approximating this expression with a first- order plus dead time model and (b) PID controller parameters based on approximating this expression with a second-order plus dead time model. In both cases, use c = 3. Then, build a Simulink simulation to compare the two controller settings. How do they compare?

G(s)=(2s+1)(0.1s+1)1 G(s)=2s+1e0.1s G(s)=(s+1)(0.2s+1)(0.12s+1)(0.009s+1)0.15