Question

# Consider a system with P-only control (Gc = Kc) which is two first-order systems in series with a gain of 1,\left(G_{1}=\frac{1}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}\right) \text { and a disturbance, } U,

\text { between the controller and the system. The system also } \text { has measurement with first-order delay }\left(G_{M}=\frac{1}{\tau_{M} s+1}\right) \text {. The set-point of the system is } R \text { and the } controlled variable is C. Sketch a block-flow diagram of the system. 2. Combine the blocks to obtain C/R. 3. Simplify this to show that \frac{C}{R}=\frac{K_{C}\left(\tau_{M} s+1\right)}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)\left(\tau_{M} s+1\right)+K_{C}} \text { Determine the response to the system for a unit-step change in set point if } \tau_{1}=1, \tau_{2}=\frac{1}{2} \text {, } \tau_{\boldsymbol{H}}=\frac{1}{3} \text { for } K_{c}=3,6,9, \text { and } 12 Determine the stability of the system by analyzing roots of the characteristic equation for the conditions described in Problem 4. Use the Routh Test to determine the maximum value of K. that results in a stable system. Add in integral control with ri=1/4- and determine the open-loop transfer function. Put the open-4loop transfer function in a form for use in rlocus. Generate the root locus plot for the system.What value of K. makes the system unstable?

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8

Fig: 9

Fig: 10

Fig: 11

Fig: 12

Fig: 13