The plant in the feedback control system shown in the figure is \( P(s) = \frac{a}{s^2 - b^2} \), where \( a > 0 \) and \( b > 0 \). The type(s) of controller \( C(s) \) that CANNOT stabilize the plant is/are
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A plant with poles in the right-half plane (like $s = \pm b$) is inherently unstable. Controllers are used to shift these poles to the left-half plane for stabilization. Different controller types have varying abilities to achieve this.
Step 1: Analyze the closed-loop characteristic equation for each controller type.
The open-loop transfer function is \( G(s) = C(s)P(s) = C(s) \frac{a}{s^2 - b^2} \). The characteristic equation is \( 1 + G(s) = 0 \).
(A) Proportional (P) Controller: \( C(s) = K_p \)
Characteristic equation: \( s^2 - b^2 + a K_p = 0 \implies s = \pm \sqrt{b^2 - a K_p} \). For stability, we need roots in the left-half plane, which is not possible for all \( K_p > 0 \).
(B) Integral (I) Controller: \( C(s) = \frac{K_i}{s} \)
Characteristic equation: \( s^3 - b^2 s + a K_i = 0 \). The Routh array shows a zero in the first column, indicating instability.
(C) Proportional-Integral (PI) Controller: \( C(s) = K_p + \frac{K_i}{s} = \frac{K_p s + K_i}{s} \)
Characteristic equation: \( s^3 + (a K_p - b^2) s + a K_i = 0 \). The Routh array also shows a zero in the first column, indicating instability.
(D) Proportional-Derivative (PD) Controller: \( C(s) = K_p + K_d s \)
Characteristic equation: \( s^2 + a K_d s + (a K_p - b^2) = 0 \). We can choose \( K_p \) and \( K_d \) positive and large enough such that \( a K_p - b^2 > 0 \), leading to a stable system. Step 2: Identify the controllers that cannot stabilize the plant.
Based on the analysis, Proportional (P), Integral (I), and Proportional-Integral (PI) controllers cannot guarantee stability for the given plant.
