controls - Characteristic Equation of A Closed Loop System in Terms of PI Controller -

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just wondering if guide me on how find characteristic equation of trasfer function g(s) (see below g(s)) in terms of coefficients in pi controller?

g(s) = 45/(5s + 2)

no sure here, i'm used multiplying error proportional gain - there's no error value provided.

any advice appreciated. in advance ;)

given: g(s) = 45/(5s + 2) (plant transfer function) c(s) = kp + ki/s (pi controller transfer function)

and assuming system looks like: https://www.dropbox.com/s/wtt4tvujn6tpepv/block_diag.jpg

the equation of closed loop transfer function is: 

gcl(s) =  c(s)g(s)/(1+c(s)g(s)) = cg/(1+cg)

in general, if had transfer function on feedback path, h(s), the closed loop transfer function becomes:

cg / (1 + cgh)

if plug in g(s) , c(s) shown above following closed loop transfer function after algebraic simplification:

45*[kp*s + ki] / [5*s*s + (2 + 45*kp)*s + 45*ki]

and characteristic equation is

5*s*s + (2 + 45*kp)*s + 45*ki = 0

notice how integral term adds pole system has side effect of adding 0 produce unwanted transient behaviour if kp not chosen correctly. presence of kp in s term in denominator shows value of kp determine damping ratio of system , therefore determine transient response.

more information on poles, zeros, , system dynamics: http://web.mit.edu/2.14/www/handouts/polezero.pdf


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