The following problems consider approaches to stabilize an unstable system.

(a) An unstable system can be stabilized by using negative feedback with a gain Kin the feedback loop. For instance consider an unstable system with transfer function

which has a pole in the right-hand s-plane making the impulse response of the system, h(t) grow as t increases. Use negative feed-back with a gain K > 0 in the feedback loop, and put H(s) in the forward loop. Draw a block diagram of the system. Obtain the transfer function G(s) of the feedback system and determine the value of K that makes the overall system BIBO stable, i.e., its poles in the open left-hand s-plane.

(b) Another stabilization method consists in cascading an all-pass system with the unstable system to cancel the poles in the right-hand s-plane. Consider a system with a transfer function

which has a pole in the right-hand s-plane, s = 1, so it is unstable.

i. The poles and zeros of an all-pass filter are such that if p₁₂ = ˆ’σ ± jΩ₀ are complex conjugate poles of the filter then z₁₂ = σ ± jΩ₀ are the corresponding zeros, and for real poles p₀ = ˆ’σ there is a corresponding zero z₀ = σ. The orders of the numerator and the denominator of the all-pass filter are equal. Write the general transfer function of an all-pass filter H_ap(s) = KN(s)/D(s).

ii. Find an all-pass filter H_ap(s) so that when cascaded with the given H(s) the overall transfer function G(s) = H(s) H_ap(s) has all its poles in the left-hand s-plane.

iii. Find K of the all-pass filter so that when s = 0 the all-pass filter has a gain of unity. What is the relation between the magnitude of the overall system |G(s)|and that of the unstable filter|H(s)|.

Transcribed Image Text:

H(s) = H(s) = (s – 1)(s + 2s + 1) s+1