Question

H(S)= S + S + 25 s'+ 100S Perform a stability analysis of the system. Draw bode diagrams by creating the amplitude and Phase table of this system.. Design a controller to stabilize the system, add it to the system via the block diagram and calculate the gain value to stabilize the system.

Answer 1

The transfer function of the system is s?+s + 25 H(S) = s +100s To check the stability of the system, we need to find for the location of poles of H(s). The location of poles can be detemined from the denominator dynamics H(s) Solving for s3 + 100s = 0, we get the location of poles are Poles = [0,-10j, +10j] As the polesare at onthe imaginary axis, hence, we can state that H(s) is a marginally stable system.

28 location of poles Zeros. -jw² + login location of Loros : [-0.5-4.911, -0.5 +4.93] To - 10j, + 10; As the zeros are complex Conjugate, we need to get the absolute valus; [5] Now, ! Now, inserting = jw, we get; Haw) = -W?+ jw + 25 (+25-2) + jw jw( 100 w2) | How)) = ((25-62)² + w2. √(25-62)²+w2 VW (100-w2f wx (100-w2) & Hiw) = tan'la 25-w2 :)- Mangnitude tude table & phase table 5 - 12.5 -24.5 -37.5 -30 130 - 8 <lców) -876 - 79.5 -5.5° 58.9 82.4° -96.4-90° this, the the magnitude & phase plots are Shown an 909 3 16 THw) de =>90 below,

Howl og 190 + Woo <Heins) log(w) -98 10° 5 102 UN we are log(w) Now, we need to design a Controller to Stabilizing the system, In this aspect using a proportional Controller whose It is Gels) up the corrupaning is shown below; RCS) T(s) the characteristics equation can be written as, Ge(s) It G(s)H(s) - 0. It Kpx ('s+5+25) black diagra Tam > H(5) -0. s +1005 we S3/L (100 + kp) Now, to get the value of up too → 5°+ kps + (100+ Kp) 5 + 25 lep = 0. the stability of the system, need to create a Routh table as shown below, the stability of the system would be, (25 up) (kp + 75) to up - 75 25 kp >0 po s! Hea(2+75) hence; for up to the above system System would be Stable. so 32 up > 25up O