Energy Balance:

\rho V C \frac{d T}{d t}=w C\left(T_{i}-T\right)-U\left(1+q_{c}\right) A\left(T-T_{c}\right)-U_{L} A_{L}\left(T-T_{a}\right)               (1)

Expanding the RHS,

\begin{aligned}\rho V C \frac{d T}{d t}=& w C\left(T_{i}-T\right)-U A\left(T-T_{c}\right) \\&-U A q_{c} T+U A q_{c} T_{c}-U_{L} A_{L}\left(T-T_{a}\right)\end{aligned}            (2)

Linearizing the nonlinear term,

q_{c} T \approx \bar{q}_{c} \bar{T}+\bar{q}_{c} T^{\prime}+\bar{T} q_{c}^{\prime}             (3)

Substituting (3) into (2), subtracting the steady-state equation, and introducing deviation variables,

\begin{aligned}\rho V C \frac{d T^{\prime}}{d t}=& w C\left(T_{i}^{\prime}-T^{\prime}\right)-U A T^{\prime}-U A \bar{T} q_{c}^{\prime}-U A \bar{q}_{c} T^{\prime} \\&+U A T_{c} q_{c}^{\prime}-U_{L} A_{L} T^{\prime}\end{aligned}               (4)

Taking the Laplace transform and assuming steady-state at t =0 gives,

\begin{aligned}\rho V C s T^{\prime}(s) &=w C T_{i}^{\prime}(s)+U A\left(T_{c}-T^{\prime}\right) q_{c}^{\prime}(s) \\&-\left(w C+U A+U A \bar{q}_{c}+U_{L} A_{L}\right) T^{\prime}(s)\end{aligned}                (5)

Rearranging,

T^{\prime}(s)=G_{L}(s) T_{i}^{\prime}(s)+G_{p}(s) q_{c}^{\prime}(s)                (6)

where:

\begin{aligned}&G_{d}(s)=\frac{K_{d}}{\tau s+1} \\&G_{p}(s)=\frac{K_{p}}{\tau s+1}\end{aligned}

\begin{aligned}K_{d} &=\frac{w C}{K}              (7) \\K_{p} &=\frac{U A\left(T_{c}-\bar{T}\right)}{K} \\\tau &=\frac{\rho V C}{K} \\K &=w C+U A+U A \bar{q}_{c}+U_{L} A_{L}\end{aligned}

The ideal FF controller design equation is given by,

G_{F}=\frac{-G_{d}}{G_{t} G_{v} G_{p}}                (21-15)

But, G_{t}=K_{t} e^{-\theta s} and G_{v}=K_{v}            (8)

Substituting (7) and (8) gives,

G_{F}=\frac{-w C e^{+\theta s}}{K_{t} K_{v} U A\left(T_{c}-\bar{T}\right)}             (9)

In order to have a physically realizable controller, ignore the e ^{+\theta s} term,

G_{F}=\frac{-w C}{K_{t} K_{v} U A\left(T_{c}-\bar{T}\right)}                  (10)