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I have the following system enter image description here

where $$G(s)=\frac{0.5}{s+1}+\frac{5}{s+10}$$

How can I design the C(z) controller so that the steady state error for a step input r(t)=1(t) is zero?

I know that this has to do with the system type and in this case we have to deal with a type 0 system which for a step input will give a finite steady state error. Adding an integrator we make the type 1 getting the desired result. Now how would we deal with this in a discrete system? Do I get my constant time controller and convert it to a discrete controller?

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    $\begingroup$ It'd be good to see what you have researched and tried so far. Showing some effort on your part will motivate others to help. $\endgroup$ – Juancho Jun 24 '17 at 23:51
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You can obtain the steady state error using Finite value theorem (http://www.engr.iupui.edu/~skoskie/ECE595_f05/handouts/fvt_proof.pdf) and then you can compute your controller to make it zero. or if you have sampling fast enough, you can just discretize your continous version of controller.

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Note that G(s) is simply two first order low pass filters in parallel. A simple accumulator (which is a digital integrator): y[n] = K(x[n]+ y[n-1]), would bring the steady state error to zero. K is a gain factor that affects the Loop gain.

Note that we must also ensure T is short enough such that 1/T is well above the cutoff of the higher bandwidth filter (the second with a pole at s= -10) in order to model it as a linear control loop with G(s). In this case we can safely solve for G(z) using method of impulse invariance and from that determine best K from the root-locus for desired settling time and phase and amplitude stability constraints.

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