Non linear dynamic model equation to linear equation

Question

Could you please show the steps from non linear dynamic model equation to linear equation?

Answer 1

We'll show one step as this linearization is automatically carried out in MATLAB.

Consider the part F(h) = CA_s(2gh)^1/2

To linearize we'll represent this part or F(h) as Taylor series expansion around steady state stock height hs.

F(h) = F(h_s) + (dF/dh)_hs(h – h_s) + ... higher order terms in (h – h_s)

dF/dh = CA_s(1/2)(2g)^1/2 h^-1/2

(dF/dh)_s = CA_s(1/2)(2g)^1/2 hs^-1/2 

F(h) = F(h_s) + (dF/dh)_hs(h – h_s)
       
F(h) = F(hs) + (CA_s(1/2)(2g)^1/2 hs^-1/2)(h – h_s)

F(h) = F(h_s) + K'_h h'
where K'h is a constant and h' is the deviation variable in h.

Of course we have to linearize the first term also in the same way, Taylor series expansion, then we can collect the h' and P'_vd.