Question 14.2: Examine the effect of the open-loop gain K on stability and ...

The open-loop sinusoidal transfer function is

G_{(j\omega )}  H_{(j\omega) } = T_{OL } (j\omega ) = \frac{K} {j\omega  (j\omega +1) (j\omega +5)} .

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

Re\left[ T_{OL } (j\omega )\right] =\frac{-6K}{(\omega ^{4}+26\omega ^{2}+25  )}  ,

Im\left[ T_{OL } (j\omega )\right] =\frac{K(\omega ^{2}-5 )}{(\omega ^{5}+26\omega ^{3}+25\omega   )}  .

The stability gain margin k_{g} was defined in Chap. 13 as

K_{gdB} =20\log \frac{1}{\left|T_{OL } (j\omega _{p} )\right| }  ,

where \omega _{P}   is such that

\angle  T_{OL } (j\omega _{P}  )=-180° ,    or     Im\left[ T_{OL } (j\omega )\right] =0 .

For this system \omega _{P} =\sqrt{5} rad/s, and the gain margin in decibels is

K_{gdB} =20\log \frac{30}{K }  .

Now, to examine the steady-state performance of the system subjected to a unit ramp input u(t) = t, the static-velocity error coefficient must be determined. Using Equation (14.22) for a type 1 system yields

K_{V}=\underset{s\rightarrow 0}{\lim } \left[\frac{sK}{s(s+1)(s+5)} \right]  =\frac{5}{K} ,

and hence the steady-state error is

e_{ss } =\frac{5}{K} .

Figure 14.9 shows the system gain margin k_{gdB} , static, velocity error coefficient   K_{V} , and steady-state error e_{ss} as functions of the open-loop gain K. Note that the system is marginally stable for K= 30, at which the minimum steady-state error approaches 0.1667. Selecting values of K less than 30 will improve system stability at the cost of increasing steady-state error.