Transfer Function Sensitivity & Complete Derivations

I have a transfer function block which looks like the following in SIMULINK:

Ignore the temporary error messages as 'a', 'k1', & 'k2' are not defined. When I find the CLTF of this system or T(s) I get this expression:

I am unsure how to derive these steps (if they are even correct!) in MATLAB.

My second problem is that I cannot determine the following sensitivity paramaters, definitely not in MATLAB and unsure about my own derivations:

=

&

(chain rule!)

I have = , however I did not expect the denominator powers and I remain unsure. For I simply could not figure this problem out myself.

I would appreciate any help on this subject matter, thank you!

Answer 1

[Sam Chak]

Hi @Richard

I followed the formulas you provided and arrived at these solutions.

syms s a k1 k2

Gp   = 1/(s^2 + a*s);
Gc   = k1 + 1;
H    = k2;
G    = Gc*Gp;

% Closed-loop transfer function
Gcl  = G/(1 + G*H);
T    = simplify(Gcl)

T = 

% sensitivity of T to changes in Gp
STG  = 1/(1 + (Gc*Gp)*H);
STG  = simplify(STG, 'steps', 100)

STG = 

% sensitivity of Gp to changes in a
dGda = diff(Gp, a);
dGda = simplify(dGda, 'steps', 100);
SGa  = dGda*a/G;
SGa  = simplify(SGa, 'steps', 100)

SGa = 

% sensitivity of T to changes in a
STa  = STG*SGa;
STa  = simplify(STa, 'steps', 100)

STa =