Is the solution provided below correct? Can you correct this one, and tell how to solve it$?$ We didn't work in class with imaginary parts. Exercise $5$

$11\mathrm{.}13$ A process is described by the transfer function

$G(s)=\frac{K}{(5s+1)(s+1)}$

Find the range of controller settings that yield stable closed$-$

loop systems for:

$($a$)$ A proportional$-$only controller.

$($b$)$ A proportional$-$integral controller.

$($c$)$ What can you say about the effect of adding the

integral mode on the allowable "control range" of Kc:

does it narrow or widen the allowable range of $Kc?$ Exercise $5$

$($a$)$ PROPORTIONAL$-$ONLY CONTROL

requirement: $1+G_{OL}=0$ with $G_{O_L}=G_{C}G$

$=>1+\frac{K_{c}k}{(5s+1)(s+1)}=0$

$(P)=>K_c$

control

$=>(5s+1)(s+1)+k_{c}k=0$

$>5s^2+6s+(1+k_{c}k)=0$

$=>,D=b^2-4ac$

$=36-4\mathrm{.}5(1+K_{c}K)$

$=36-20(1+k_{c}k)$

$=4(1-5k_{c}k)$

$=>S_1=\frac{-b-\sqrt[\phantom{2}]{D}}{2a}$ and $S_2=\frac{-b+\sqrt[\phantom{2}]{D}}{2a}$

$(a{x}^2+bx+c=0)$

$2^{nd}$ order equation

C$)$ find noots by $D=b^2-4$acdots

We know that only noots with real parts $0$ are stable!

S$1,\frac{-6-\sqrt[\phantom{2}]{D}}{10}0$

$$ for whatever value of $D,$

this noot will always be $-$

$($however$....)$

$4$ only positive values of $D$

are allowed by $\sqrt[\phantom{2}]{?}$

$,\frac{-6+\sqrt[\phantom{2}]{D}}{10}0$

$>,-6+\sqrt[\phantom{2}]{D}0$

$>,-D36$

$=>,4(1-5K_{c}K)36$

$>,-5K_{c}K9-1$

$>,K_c-\frac{8}{5K}$

This conodition is more smingent than $k_c\frac{1}{2}$ and thes prevails

$k_c\frac{1}{}$ and thes prevails $($b$)$ PROPORTIONAL $-$ INTEGRAL CONTROC

requinement: $1+{}_{O_L}=0$ with $G_{Q_c}=G_{c}G$

PI$)=>k_c(1+\frac{1}{{}_{\text{I}}S})$

$=>1+(1+\frac{1}{{}_{\text{I}}s})*\frac{k_{c}k}{(5s+1)(s+1)}=0$

$=>1+\frac{k_{c}k}{(ss+1)(s+1)}+\frac{k_{c}k}{{}_{\text{I}}s(5s+1)(s+1)}=0$

$=>(5s+1)(s+1)+k_{c}k+\frac{K_{c}k}{{}_{\text{I}}s}=0$

$=>(5s+1)(s+1){}_{\text{I}}s+k_{c}k{}_{\text{I}}s+k_{c}k=0$

$>5{}_{\text{I}}{s}^3+6{}_{\text{I}}{s}^2+{}_{\text{I}}s+k_c{K}_{\text{I}}s+k_{c}K=0$

$=>5{}_{\text{I}}{s}^3+6{}_{\text{I}}{s}^2+{}_{\text{I}}(1+k_{c}k)s+k_{c}K=0$

We know that only roots with real parts $0$ are stable!

$$ For polynomials with order $>2,$ this holds twe only if

all coefficients in the polynomial are positive $($rulefrome

values are alway positive by

definition $)$

The condition of $k_c$ is more stringeut

than $k_c-\frac{1}{k}=>$ only $K_{c}0$ gnarantees stable conditions!$11\mathrm{.}21$ A process control system contains the following transfer functions: $2$ e$-15(60$ s$+105$s $1)$ Gp$($s$)0\mathrm{.}5$ e$-03$s $-02$s $3$ e G$(21($a$)$ Show how Gol$($s$)$ can be approximated by a FOPTD model; Ke$$ Find K$,$ T$,$ and fer function for the open$-$loop process trans$-($b$)$ Use Routh stability methods and your FOPTD model to find the range of Ke values that will yield a stable closed$-$loop system. Repeat for the full$-$order model using MATLAB. $($c$)$ Determine Kem and the corresponding w