Determine the range of k for stability of a unity feedback control system
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The 11 comes from the closed-loop transfer function.
See these WikiBook Routh-Hurwitz-Criterion notes for details.
From G(s)=Ks(s+1)(s+5)G(s)=Ks(s+1)(s+5), we form:
1+Ks(s+1)(s+5)=0⟹1+Ks3+6s2+5s=01+Ks(s+1)(s+5)=0⟹1+Ks3+6s2+5s=0
If we find a common denominator and multiply through, we arrive at:
s3+6s2+5s+K=0(1)(1)s3+6s2+5s+K=0
The Routh table (see linked site above) is:
s3s2s1s01630−K6K5K0−s315s26Ks130−K60s0K−
Routh's criterion states that the number of sign changes in the first column is equal to the number of roots in the right-half plane. For stability, the first column musthave all entries positive.
Clearly 11 and 66 are positive, but the next two values are dependent on the gain parameter KK.
See these WikiBook Routh-Hurwitz-Criterion notes for details.
From G(s)=Ks(s+1)(s+5)G(s)=Ks(s+1)(s+5), we form:
1+Ks(s+1)(s+5)=0⟹1+Ks3+6s2+5s=01+Ks(s+1)(s+5)=0⟹1+Ks3+6s2+5s=0
If we find a common denominator and multiply through, we arrive at:
s3+6s2+5s+K=0(1)(1)s3+6s2+5s+K=0
The Routh table (see linked site above) is:
s3s2s1s01630−K6K5K0−s315s26Ks130−K60s0K−
Routh's criterion states that the number of sign changes in the first column is equal to the number of roots in the right-half plane. For stability, the first column musthave all entries positive.
Clearly 11 and 66 are positive, but the next two values are dependent on the gain parameter KK.
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