Control-System Mathematics

Transfer functions, stability and state space. The mathematics of feedback control: transfer functions and block diagrams, characteristic equations and stability (Routh–Hurwitz), root locus, Bode and Nyquist frequency-response methods, and state-space models with eigenvalue stability.

• Transfer Functions — Transfer functions describe a linear system's input–output behaviour in the Laplace domain. Learn how to derive them, poles and zeros, first- and second-order systems, time-domain specs and the final-value theorem.
• Block Diagrams — Block diagrams show how signals flow through a system. Learn series, parallel and feedback reduction rules, moving summing points and take-off points, and Mason's gain formula for complex signal-flow graphs.
• Stability & Routh–Hurwitz — A stable system returns to equilibrium after a disturbance. Learn BIBO stability, pole locations, the Routh–Hurwitz criterion, marginal stability and critical gain, and relative stability measures.
• Root Locus — The root locus plots how closed-loop poles move as a gain varies. Learn the sketching rules, asymptotes, breakaway points and jω crossings, and how to pick a gain for a desired damping ratio.
• Bode Plot — Bode plots show gain (dB) and phase versus log frequency. Learn asymptotic sketching, corner frequencies, bandwidth, gain and phase margins from the open-loop plot, and loop-shaping design.
• Nyquist — The Nyquist criterion determines closed-loop stability from the open-loop frequency response — even for open-loop unstable plants and systems with time delay. Learn the polar plot, encirclements of −1, and margins.
• State Space — State-space models describe systems with first-order matrix equations. Learn states, the A, B, C, D matrices, conversion to transfer functions, eigenvalues and stability, controllability, observability, pole placement, observers and LQR.