Draw a block diagram with transfer functions and get the closed-loop transfer function with step-by-step reduction. Check stability with the Routh–Hurwitz criterion and the stable range of K, the steady-state error, the step response, the root locus with its rules, the Bode and Nyquist plots with gain and phase margins, and design controllers.
All closed-loop poles must have a negative real part. Routh's criterion checks it without computing the roots.
The paths of the closed-loop poles as a gain K goes from 0 to infinity.
A classic way is Ziegler–Nichols: find the ultimate gain Ku and period Pu, then Kp = 0.6Ku, Ti = Pu/2 and Td = Pu/8.
A feedback control system compares the output with the reference and corrects the error. Analyzing it means answering three questions: is it stable? How does it respond over time (overshoot, settling time)? And what error remains at steady state? This guide explains the tools to answer them and works a complete example.
With a plant G(s) and unity negative feedback,
T(s) = C(s)/R(s) = G(s) / (1 + G(s))
The poles of T(s), the roots of the characteristic equation 1 + G(s) = 0, determine the whole behavior. More complex block diagrams are reduced step by step: blocks in series multiply, blocks in parallel add, and a feedback loop is closed with the formula above.
The system is stable if every closed-loop pole has a negative real part. The Routh–Hurwitz criterion decides this without computing the roots: build the Routh array from the coefficients of the characteristic equation, and the number of sign changes in the first column is the number of poles in the right half-plane.
Many systems behave like the standard second-order system ωn²/(s² + 2ζωns + ωn²), with natural frequency ωn and damping ratio ζ. For 0 < ζ < 1, under a step input:
It depends on the system type (the number of integrators, poles at s = 0, in G) and on the input. With the constants Kp = lim G, Kv = lim s·G and Ka = lim s²·G as s → 0: for a step ess = 1/(1 + Kp), for a ramp ess = 1/Kv and for a parabola ess = 1/Ka.
G(s) = 10 / (s(s + 2)) with unity feedback.
The tool gives Mp = 35.09 %, tp = 1.045 s and the same errors. For the settling time it gives 3.54 s, measured on the simulated response: the 4/(ζωn) formula is only an approximate bound. It also computes the phase margin, about 34.9° at 2.86 rad/s.
The root locus shows how the closed-loop poles move as a gain K varies, and lets you choose K for a desired ζ. Bode and Nyquist plots analyze the frequency response and give the gain and phase margins, a measure of how far the system is from becoming unstable. With them, lead or lag compensators are designed and PID controllers are tuned.
K. Ogata, Modern Control Engineering (Pearson); N. S. Nise, Control Systems Engineering (Wiley). Simuveliq is not affiliated with these authors or publishers.
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