Question $1$: PID Implementation

$[100-$marks$]$

The pseudocode of an adaptive pid controller algorithm is given as follows $1.$

Algorithm $1$ Adaptive PID Algorithm

procedure ADAPTIVE PID CONTROLLER

Step $1$: Initialization:

Initialize $K_{p0}=0\mathrm{.}75,K_{i0}=0\mathrm{.}1,K_{d0}=0\mathrm{.}25,u_{\text{m}\text{a}\text{x}}=1,u_{\text{m}\text{i}\text{n}}=0$

$n=3($Iteration Index$)$

Step $2$: Compute error

$e(n)=r(n)-y(n)$

Step $3$: PID Control Signal

$u(n)=u(n-1)+Kp(n)[e(n)-e(n-1)]+Ki(n)e(n)+Kd(n)[e(n)-2e(n-1)+e(n-2)]$

Step $4$: Saturate Control Signal

if $u(n)>u_{\text{m}\text{a}\text{x}},u(n)=u_{\text{m}\text{a}\text{x}}$

if $u(n)y(n+1)=f_{CSTR}(x_1(n),x_2(n),x_3(n),u(n))Kp(n+1)=K_{p0}+e(n)Ki(n+1)=K_{i0}+e(n)Kd(n+1)=K_{d0}+e(n)nnlarrn+1u(n)$

Step $5$: Apply $u(n)toCSTR$ system

$y(n+1)=f_{CSTR}(x_1(n),x_2(n),x_3(n),u(n))$

Step $6$: Update PID gain

Update: $Kp(n+1)=K_{p0}+e(n)$

Update: $Ki(n+1)=K_{i0}+e(n)$

Update: $Kd(n+1)=K_{d0}+e(n)$

Step $7$: Increment $n$

nlarrn$+1$ goto Step $2$