Feedback Control of Ohio, LLC, hires you to work on a closed loop system that it

Question

Feedback Control of Ohio, LLC, hires you to work on a closed loop system that it is investigating for future use. They ask you, the company's best engineer, to determine the range of the parameter, K, for which the system is stable. You are given the plant function, GP(s), and the feedback function, H(s) and find the overall system transfer function, T^s), namely, The plant function is and the feedback transfer function is What result for T0(s) do you find and what is the range of the gain factor, K, for the system to be stable?

Explanation / Answer

T0(s) = Gp(s)/1+Gp(s)H(s)

[ k/(s+1)2(s+2) ] / [1 + k2/(s+1)2(s+2)s ]

= ks / [(s+1)2(s+2)s + k2 ]

= ks / [ s4 + 4s3 + 5s2 + 2s + k2]

For T0(s) to be stable we need the denominator i.e. s4 + 4s3 + 5s2 + 2s + k2 to have all its zeroes(or poles of T0(s)) in the Left Half Plane

We do it by routh's stability criteria

s4        1        5        k2

s3        4        2       

s2

s1

s0

First row of s2 = ( 5*4 - 2 )/4 = 18/4 = 9/2

second row of s2 = 2k2/2 = k2

Hence now our table becomes-

s4        1        5        k2

s3        4        2       

s2       9/2      k2

s1

s0

First row of s = ( (9/2)*2 - 4k2 )/9/2 = (2/9)(9-4k2)

also First row of s0 = (2/9)(9-4k2)( k2) / (2/9)(9-4k2) = k2

Hence now our table becomes-

s4        1        5        k2

s3        4        2       

s2       9/2      k2

s1     (2/9)(9-4k2)

s0        k2

Now ,

The number of changes in the sign in first row determines the number of roots in the right half plane

Since 1,4,9/2 >0 and k2 >0 for every k , hence for stability (i.e. no zeroes in RHP) (2/9)(9-4k2) should be greater than 0

(2/9)(9-4k2) >0

9-4k2 >0

k2 < 9/4 or -3/2<k<3/2

-1.5<k<1.5

Let know if anything is unclear

Cheers :)

Harshit

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