Question 13.5: Determine the stability of the closed-loop system that has t...

The sinusoidal transfer function of the open-loop system, T_{OL}\left(j\omega \right) , is

T_{OL}\left(j\omega \right) =\frac{K}{j\omega \left(j\omega +1\right)\left(2j\omega +1\right) } .           (13.14)

The real and imaginary parts of T_{OL}\left(j\omega \right) are

Re\left[T_{OL}\left(j\omega \right)\right] =-\frac{3K}{9\omega ^{2} +\left(2\omega ^{2}-1 \right) ^{2} } .      (13.15)

Im\left[T_{OL}\left(j\omega \right)\right] =-\frac{K\left(2\omega ^{2}-1 \right) }{9\omega ^{3} +\omega \left(2\omega ^{2}-1 \right) ^{2} } .      (13.16)

According to the Nyquist criterion, the closed-loop system will be stable if

where \omega _{p} is the frequency at the point of intersection of the polar plot with the negative real axis. The imaginary part of T_{OL}\left(j\omega \right) at this point is zero, and so the value of \omega _{p} can be found by the solution of

Im\left[T_{OL}\left(j\omega \right)\right] =0.         (13.17)

Substituting into Eq. (13.17) the expression for Im\left[T_{OL}\left(j\omega \right)\right] , Eq. (13.16), yields

2\omega ^{2}_{p} -1=0 ,

and hence \omega _{p}={1}/{\sqrt{2} } rad/s.To ensure stability of the closed-loop system, the real part of T_{OL}\left(j\omega \right) must be less than 1 at \omega =\omega _{p},i.e.,

\left|Re\left[T\left(j\omega _{p} \right) \right] \right| \lt 1.      (13.18)

Now substitute into Eq. (13.13) the expression for the real part, Eq. (13.15), to obtain

and solve for K:

The closed-loop system will be stable for K less than 1.5. The polar plot of the open-loop system transfer function for K = 1.5 is shown in Fig. 13.7.