Nyquist diagram stability
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 $displaystyle W(s)=Kfrac{(s+5)}{(s−2)(s+1)}$ , 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?
complex-analysis optimization control-theory
edited May 31 '18 at 11:09
caverac
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
|
show 1 more comment
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 $displaystyle W(s)=Kfrac{(s+5)}{(s−2)(s+1)}$ , 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?
complex-analysis optimization control-theory
edited May 31 '18 at 11:09
caverac
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30
|
show 1 more comment
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 $displaystyle W(s)=Kfrac{(s+5)}{(s−2)(s+1)}$ , 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?
complex-analysis optimization control-theory
edited May 31 '18 at 11:09
caverac
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
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 $displaystyle W(s)=Kfrac{(s+5)}{(s−2)(s+1)}$ , 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?
complex-analysis optimization control-theory
complex-analysis optimization control-theory
edited May 31 '18 at 11:09
caverac
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
edited May 31 '18 at 11:09
caverac
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
edited May 31 '18 at 11:09
caverac
13.8k21030
edited May 31 '18 at 11:09
caverac
13.8k21030
edited May 31 '18 at 11:09
caverac
13.8k21030
13.8k21030
asked May 31 '18 at 10:52
Mišel Ademi
1
asked May 31 '18 at 10:52
Mišel Ademi
1
asked May 31 '18 at 10:52
Mišel Ademi
1
1
I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30
|
show 1 more comment
I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30
I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30
|
show 1 more comment
1 Answer
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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 $-1/K+0cdot i$. 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:
$$N=Z-Ptag{1}$$
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 $-1/K+0cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0cdot i$ 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.
answered Nov 28 '18 at 10:25
Matt L.
8,871822
add a comment |
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1 Answer
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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 $-1/K+0cdot i$. 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:
$$N=Z-Ptag{1}$$
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 $-1/K+0cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0cdot i$ 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.
answered Nov 28 '18 at 10:25
Matt L.
8,871822
add a comment |
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 $-1/K+0cdot i$. 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:
$$N=Z-Ptag{1}$$
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 $-1/K+0cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0cdot i$ 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.
answered Nov 28 '18 at 10:25
Matt L.
8,871822
add a comment |
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 $-1/K+0cdot i$. 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:
$$N=Z-Ptag{1}$$
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 $-1/K+0cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0cdot i$ 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.
answered Nov 28 '18 at 10:25
Matt L.
8,871822
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 $-1/K+0cdot i$. 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:
$$N=Z-Ptag{1}$$
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 $-1/K+0cdot i$, which is equivalent to $1$ counter-clockwise encirclement.
From the Nyquist plot, a counter-clockwise encirclement is achieved if the point $-1/K+0cdot i$ 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.
answered Nov 28 '18 at 10:25
Matt L.
8,871822
answered Nov 28 '18 at 10:25
Matt L.
8,871822
answered Nov 28 '18 at 10:25
Matt L.
8,871822
answered Nov 28 '18 at 10:25
Matt L.
8,871822
8,871822
add a comment |
add a comment |
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I tried to draw the inverse part of this diagram and then to mark point on left side of real axis and then using this formula N=p-z to determine stability but I'm making mistake somewhere.
– Mišel Ademi
May 31 '18 at 16:22
So what should N, P and Z be? And when is that the case?
– Kwin van der Veen
May 31 '18 at 18:12
@KwinvanderVeen N is number of encirclements, P is number of poles on right side and if Z is 0 than in that part system is stable, otherwise is unstable.
– Mišel Ademi
May 31 '18 at 18:37
For N in what direction and around which point do you consider the encirclements? Also note that in order to count the encirclements you also have to plot the negative frequencies as well.
– Kwin van der Veen
Jun 1 '18 at 18:03
I use 1+jw point
– Mišel Ademi
Jun 1 '18 at 18:30