Apply the controller synthesis formula (19.2.6) for the following process transfer func- tions: (a) G(s) =...

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Apply the controller synthesis formula (19.2.6) for the following process transfer func- tions: (a) G(s) = (b) G(s) = (c) G(s) = k(1+tos) (t₁s +1)(₂+1)(T38+1) ' k(1-tos) (t₁s+1)(₂+1)(t3s k (T₁s+1)(T₂s+1) -0s e +1) ' To >0, T₁0, T₂ > 0, T3 > 0 To > 0, T₁ > 0, T₂ > 0, T₂ > 0 T₁ > 0, T₂ > 0, 0>0 3 For the last transfer function, under the assumption of small dead time, apply a first- order Padé approximation to derive an approximate rational controller transfer function. In all cases, determine if the controller transfer function is PID. Apply the controller synthesis formula (19.2.6) for the following process transfer func- tions: (a) G(s) = (b) G(s) = (c) G(s) = k(1+tos) (t₁s +1)(₂+1)(T38+1) ' k(1-tos) (t₁s+1)(₂+1)(t3s k (T₁s+1)(T₂s+1) -0s e +1) ' To >0, T₁0, T₂ > 0, T3 > 0 To > 0, T₁ > 0, T₂ > 0, T₂ > 0 T₁ > 0, T₂ > 0, 0>0 3 For the last transfer function, under the assumption of small dead time, apply a first- order Padé approximation to derive an approximate rational controller transfer function. In all cases, determine if the controller transfer function is PID.