I have 4 tasks from Nyquist diagram and they are similar. I don't know how to solve then so here is one example which I'm trying to solve.
For a system which transfer function is $\displaystyle W(s)=K\frac{(s+5)}{(s−2)(s+1)}$ , Nyquist diagram for $K=1$ is given on picture. Based on the diagram, find the range of gain for which the system is stable.
Can anybody help me how to solve this?
You have to complete the complex trace of $W(s)$ to see the number of encirclements:
Note that since $K=1$ you can view the plot as the trace of $W(s)/K$, so you can consider encirclements of the point $-1/K+0\cdot i$. The number of clockwise encirclements $N$ of that point equals the number of zeros $Z$ of $1/K+W(s)$ in the right-half plane minus the numbers of poles $P$ of $1/K+W(s)$ in the right half-plane:
$$N=Z-P\tag{1}$$
The term $s-2$ in the denominator contributes one pole in the right half-plane, so in order to have no zeros ($Z=0$), we need to have $N=-1$ clockwise encirclements of the point $-1/K+0\cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0\cdot i$ is inside the right-hand side closed part of the curve, i.e. if $1/K$ is in the range
$$-1<-1/K<0$$
which is equivalent to $K>1$. Consequently, for $K>1$ the corresponding closed-loop transfer function is stable.