Question

Consider the first-order transfer function G(s) = 1 s/s− 2 controlled by a P controller Gc(s) = Kp in a basic servo loop.

1. Determine the values of Kp that render the closed-loop system asymptotically stable. The closed-loop system is said to be asymptotically stable if the transmissibility transfer function T(s) is assymptotically stable, where
T(s) ≙Gᵧᵣ(s) = GG/1+GG

Answer 1

The values of Kp that render the closed-loop system asymptotically stable are those greater than 2, as this results in the poles of the closed-loop transfer function having negative real parts.

Step-by-step explanation:

The student is asking about the stability of a closed-loop control system given a certain transfer function and a proportional (P) controller. Stability is determined by the location of the poles of the closed-loop transfer function, which are the solutions to the characteristic equation obtained from det(sI - A) of the system, where A is the system matrix. To ensure a system is asymptotically stable, all the poles of the closed-loop transfer function must have negative real parts.

The closed-loop transfer function, T(s), for the given system with transfer function G(s) = 1/(s - 2) and a P controller Gc(s) = Kp, is T(s) = Gc(s)G(s) / (1 + Gc(s)G(s)). We need to determine the values of Kp that make the system asymptotically stable. The characteristic equation derived from 1 + Gc(s)G(s) = 0 is s - 2 + Kp = 0, which implies that s = 2 - Kp. In order for the real part of s to be negative, and therefore for the system to be asymptotically stable, Kp must be greater than 2.