Question

Problem 5: Suppose that you are to design a unity gain feedback controller for a first order plant. The plant and controller respectively take the form ,s+ p where K> 0, p. z are parameters to be specified. (a) Using root-locus methods, specify some p and z for which it is possible to make the closed-loop system strictly stable. Include a sketch of the closed-loop root locus, as well as the corresponding range of gains K for which the system is strictly stable (b) Suppose p and z are fixed to the values chosen in (a). Design K to meet the following specifications: The closed-loop system must be strictly stable. .The damping ratio must be between 0.4 and 0.6. Given these constraints, minimize te natural frequency Problem 5: Suppose that you are to design a unity gain feedback controller for a first order plant. The plant and controller respectively take the form s+2 where K> 0, p, z are parameters to be specified. (a) Using root-locus methods, specify some p and z for which it is possible to make the closed-loop system strictly stable. Include a sketch of the closed-loop root locus, as well as the corresponding range of gains K for which the system is strictly stable (b) Suppose p and z are fixed to the values chosen in (a). Design K to meet the following specifications: .The closed-loop system must be strictly stable. .The damping ratio must be between 0.4 and 0.6. Given these constraints, minimize te natural frequency

Answer 1

K-1: Figure 1 File Edit View Insert Tools Desktop Window Help Gp tf ( [1-2], [1 0]); rlocus (Gp) cloop feedback (sys,1) Root Locus 0.25 %figure 0.2 %subplot (2,1,1) rlocus (cloop) 0.15 sstep (cloop) CL3ys-feedback (K*sys, 1) %subplot (2,1,2) 0.05 ty-step (clsy3) plot (y) %xlabel ( 'sec' ) -0.05 ylabel ('Step Response') %figure (2) EK [1] E -0.1 #31 feedback (tf ( [K], [1 2 01) , 1) -0.15 #33 feedback (tf( [3*K), [1 2 01),1) #35-feedback (tf( [5*K), [1 2 01),1) -0.2 #37 -feedback (tf( [7*K), [1 2 01), 1) %bode (31, s3,s5,s7); -0.25 0.5 0.5 2.5 %grid Real Axis (seconds) The bandwidth 3dB is approximately obtained %[Grn, Prn, wog , wcp]=margin([7], [1 2 01)

Here by trail and error method the values are choosen as below:

p=2,z=-2,K is varied from 1 to 70, the pole reaches to zero and from there it is stable at that point whatever the value of K

Editor - C:UserslSindhu\lead_compensator.m File Edit Text Go Cell Tools Debug Desktop dowFigure 1 G3 I 参看▼ A←→ fu File Edit View Insert Tools Desktop Window Help K70: Root Locus z =-2 ; 0.25 Gp tf ( [1-2], [1 0]); 0.2 rlocus (Gp) 0.15 System: cloop Gain: 0.841 cloop feedback (sys,1) figure Pole: -1.98 10 %subplot (2,1,1) Damping:1 Overshoot (%): 0 Frequency (rad/s): 1.98 0.05 rlocus (cloop) 12 sstep (cloop) 13 14 clsys-feedback (K*sys, 1) 0.05 15 %subplot (2,1,2) 16 ty-step (clsy3) 17 plot (y) 0.15 18 %xlabel ( 'sec' ) 19 ylabel ('Step Response') -0.2 20 %figure (2) 21 AK [1] 0.25 2.5-2 -1.5-1 -0.5 31 feedback (tf ( [K], [1 2 0] ) , 1) 0.5 2.5 23 %33 feedback (tf ( [3*K) , [1 2 0] ) , 1) Real Axis (seconds) 24 %35 feedback (tf( [5*K), [1 2 0]), 1)

0:120ノ a3 astO 见- ced 03a

K at this value of pole is calculated.

丐 16+163 3-2