Feedback Control Closed-Loop Transfer Function

Figure source:

B. Bequette, Process control: modeling, design. and simulation, Prentice Hall Press, Upper Saddle River, NJ 2002

Block diagram (Laplace domain) variables:

c(s) = controller output

e(s) = error

r(s) = setpoint (reference signal)

u(s) = manipulated variable

l(s) = load disturbance

y(s) = process output

ym(s) = measured output

Transfer functions:

g_c(s) = controller

g_v(s) = valve

g_p(s) = process

g_m(s) = measurement

g_CL(s) = closed-loop

g_d(s) = disturbance

Calculations:

y(s) = l(s)g_d(s) + u(s)g_p(s)

u(s) = c(s)g_v(s)

c(s) = e(s)g_c(s)

e(s) = r(s) – y_m(s)

y_m(s) = y(s)g_m(s)

Lump all expressions together, we can get the following:

y(s) = l(s)g_d(s) + c(s)g_v(s)g_p(s)

y(s) = l(s)g_d(s) + e(s)g_c(s)g_v(s)g_p(s)

y(s) = l(s)g_d(s) + (r(s) – y_m(s))g_c(s)g_v(s)g_p(s)

y(s) = l(s)g_d(s) + (r(s) – y(s)g_m(s))g_c(s)g_v(s)g_p(s)

then:

Equation (1)

If there is no disturbance:

Equation (2)

Equation (3)