For the dynamic system
𝐺(𝑠) =1/3𝑠 + 1
with closed loop control performance objectives of:
- Zero steady state error to step inputs
- Maximum percentage overshoot to step inputs of at most 10%
Design an appropriate controller. Also, choose an appropriate sample time and
write a discrete time implementation finite difference model.
Hi Ahmad Keshavarzi, the 1st-order linear plant Gp = 1/(3*s + 1) can achieve zero steady-state error without overshoot. If you want to achieve the performance objectives in a closed-loop control system, then you may consider the PI (proportional–integral) controller, given by
Gc = Kp + Ki/s
where Kp and Ki are the proportional gain and integral gain, respectively. From the closed-loop transfer function, Gcl = Gp*Gc/(1 + Gp*Gc), you can obtain the characteristic equation as
s2 + [(Kp + 1)/3]*s + Ki/3 = 0.
From the control textbooks, you can find that the desired characteristic equation for a 2nd-order linear system which satisfies the performance objectives is given by s2 + 2*ζ*s + 1 = 0, where ζ = 0.6 is the damping ratio. Comparing and solving both equations simultaneously yield the PI gains.
(Kp + 1)/3 = 2*ζ
Ki/3 = 1
However, the closed-loop control system (for ζ = 0.6) produces the overshoot more than 10%. Hence, the damping ratio ζ has to be tuned again until the objectives are achieved. It is discouraged to tune Kp and Ki manually, when you can easily tune a single parameter ζ. In your case, the optimal ζ is found to be ≈ 0.8, which leads to Kp = 3.8 and Ki = 3.
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