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bu˙+ku=mx¨+bx˙+kx
(bs+k)u(s)=(ms2+bs+k)x(s)
G(s)=u(s)x(s)=ms2+bs+kbs+k
taking m = 1 kg, spring constant k = 32 N/m, damping coefficient b = 8 N/(m/sec)
the plant transfer function becomes:
G(s)=s2+8s+328s+32
let's design a closed loop system using a simple proportional controller with unity feedback.
T.F.=1+GHG=1+K(8s+32)/(s2+8s+32)K(8s+32)/(s2+8s+32)=s2+8s+32+K(8s+32)K(8s+32)
the open loop function becomes:
GH(s)=s2+8s+32K(8s+32)
we have two poles at −4±4i and one zero at -4.
the asymptotic location is 1−4−4=−8 and the asymptotic angle is 180∘
the break in point is: 8(s2+8s+32)−(8s+32)(2s+8)=0 so s=−8
the departure angle for the pole at −4+4i is: 90−90+θd=180 so θd=180∘
the characteristic equation is s2+8s+32+K(8s+32)=0
the value of k for s=−8 is −−64+3264−64+32=1
the system is underdamped for k<1, critically damped when k = 1 and overdamped for k>1
G(s)=s2+8s+32K(8s+32)
G(jω)=32(1+jω/4+(jω/√32)2)32K(1+jω/4)
the corner frequencies are 4 and √32=5.66 rad/s.
the magnitude at ω=1 is 20logK dB
the starting phase is 0 degrees.
the end phase is -90 degrees.
there's no phase crossover frequency so the gain margin is infinity.