Derivation of Feedforward Gains for current mode control in CCM converter
There are different ways to obtain a small-signal equation from a large-signal one. The expression \$R_i&lt;i_L&gt;=v_c-DT_sS_e-\frac{S_fD'T_s}{2}\$ is a large-signal one and describes the current leaving terminal c in the PWM switch model scaled by the sense resistance \$R_i\$. You can either perturb each of the variables and then sort out dc and ac equations, the latter being the one you want. The other option - which I always use - is partial differentiation.
The equation in question is a function of the duty ratio \$D\$. If you replace \$D\$ by the expressions you gave, you obtain: \$I_L=\frac{v_c}{R_i}-\frac{v_{off}}{v_{on}+v_{off}}T_s\frac{S_e}{R_i}-\frac{\frac{v_{off}R_i}{L}(1-\frac{v_{off}}{v_{off}+v_{off}})}{2R_i}T_s\$. So this function depends on \$v_c\$, \$v_{on}\$ and \$v_{off}\$. If you consider \$v_c\$ and \$v_{off}\$ ac-silent (their derivative is 0), then you can differentiate \$I_L\$ with respect to \$v_{on}\$ and obtain:

Factor the on- and off-voltages to reveal the duty ratio \$D\$ again. You see that a term remains and this is \$(v_{on}+v_{off})\$. If you look back at the PWM switch model, you realize that the dc on-time voltage is labeled \$V_{ac}\$ while the off-time voltage is labeled \$V_{cp}\$. Therefore, substituting these values in the on and off sum leads to: \$V_{on}+V_{off}=V_{ac}+V_{cp}=V_{ap}\$. It leads to:

If you carry on the exercise, you obtain the invariant gain \$k'_f\$ as follows:

You can now repeat the exercise for \$k'_r\$. If you want to simulate the entire model, there you go:

As you can see from the above, Ridley's model cannot find its dc operating point and is therefore constrained to ac simulations only. If you want to explore transient simulations with a current-mode model, I would recommend Vatché Vorpérian's version which predicts sub-harmonic oscillations with only two current sources. It is unbeatable in terms of simplicity. It has been extensively described in the book I published (with an original auto-toggling CCM-DCM model) but also through many APEC seminars (2013 and 2014) I taught.