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Control Loop Stability Analysis

Ensuring the closed-loop system remains stable. This requires analyzing the system's transfer function G(s)H(s)G(s)H(s) using tools like Bode plots and determining the phase and gain margins.
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The statement of the theorem

Consider the closed-loop system described by the transfer function G(s)H(s)G(s)H(s). Stability is guaranteed if and only if all poles of the closed-loop characteristic equation 1+G(s)H(s)=01 + G(s)H(s) = 0 lie in the left half of the complex ss-plane (Re(s)<0\text{Re}(s) < 0). This is formally verified using the Nyquist stability criterion, which requires the encirclement of the critical point (−1,0)(-1, 0) by the locus plot G(jω)H(jω)G(j\omega)H(j\omega) to be zero, ensuring sufficient phase and gain margins.