stability analysis of nonlinear system by
TRANSCRIPT
Stability Analysis of
Nonlinear System by
Popovโs Stability Criterion
By: Nafees Ahmed
Introduction
It is the first technique given for nonlinear systems analysis. (1962)
It is in frequency domain technique.
It can be seen in several view points.
We will see it with loop transformation
By: Nafees Ahmed
Consider the following closed loop
system
Many nonlinear physical systems can be represented as a
feedback connection of a linear dynamical system and a nonlinear element.
G(S)r=0
NLฮพ ฯ
ฯ-ฮพ
_G(S)
NLฮพ
ฯ-ฮพ
_
ฯ
T.F. of linear system
G(s)=ฯ/-ฮต
NL=> Non linear System
โ
By: Nafees Ahmed
Consider following nonlinearity
Let us define the nonlinearity
NL ั m[K, โ) sector
Above nonlinearity can be transformed to new nonlinearity NL1 ั m[0, โ] sector
Slope=K
ฮต
ฯ
By: Nafees Ahmed
Loop transformation
Above sys can be transformed to a new system as shown on next slide keeping same i/p but different o/p for non linear part.
T.F of new linear plat ๐ฎ๐ ๐ =๐
โ๐๐=
๐
โ๐+๐ฒ๐=
๐ฎ(๐)
๐+๐ฒ๐ฎ(๐)
In this way nonlinearity NL is transformed to NL1 i.e. From m[K, โ) sector to m[0, โ) sector
Note: here i/p & o/p relationship of G(s) and NL are same as the original one.
By: Nafees Ahmed
Loop transformation โฆ
G(S)
NLฮพ
ฯ
-ฮพ_
ฯ
K
ฮพ1
New Nonlinearity
NL1
Where
ฮพ1= ฮพ- ฯK
Nonlinearity of NL1 ั m[0, โ) sector
K
-ฮพ1
_
_
New linear Plant
G1(s)
By: Nafees Ahmed
Loop transformation โฆ
If a transfer function is Passive
=> it is positive real
=> it is asymptotic stable
Interconnection of two passive
systems is also passive system.
By: Nafees Ahmed
Loop transformation โฆ
Now we can say that if original system is asymptotic
stabile then its transformed sys shown on previous slide
will also be asymptotic stable.
New real plant G1(s) must be positive real i.e ๐ฎ๐ ๐ =๐ฎ(๐)
๐+๐ฒ๐ฎ(๐)must be a positive real.
New nonlinearity defined by NL1 ั m[0, โ) sector
By: Nafees Ahmed
Loop transformationโฆ
Same way we can do other loop transformation
Consider following nonlinearity
NL ั m[0, K] sector
Above nonlinearity can be transformed to new nonlinearity NL1 ั m[0, โ) sector
Slope=Kฮต
ฯ
By: Nafees Ahmed
Loop transformationโฆ
Above sys can be transformed to a new system as shown on next slide
keeping same o/p but different i/p for non linear part.
T.F of new linear plat ๐ฎ๐ ๐ =G(s)+1/K
In this way nonlinearity NL is transformed to NL1 i.e. From m[0,K] sector to
m[0, โ) sector
Note: here i/p & o/p relationship of G(s) and NL are same as the original
one.
By: Nafees Ahmed
Loop transformationโฆ
G(S)
NL
ฯ1
-ฮพ_
ฮพฯ1
1/K
1/K
+
New linear Plant
G1(s)
Where
ฯ1= ฯ-(1/K)ฮพ
New Nonlinearity
NL1
+
Nonlinearity of NL1 ั m[0, โ) sector
ฯ
ฯ
By: Nafees Ahmed
Loop transformation โฆ
Similar to previous
Now we can say that if original system is asymptotic
stabile then its transformed sys shown on previous slide
will also be asymptotic stable.
New real plant G1(s) must be positive real i.e ๐ฎ๐ ๐ =๐ฎ ๐ + ๐/๐ฒ must be a positive real.
New nonlinearity defined by NL1 ั m[0, โ) sector
By: Nafees Ahmed
Loop transformation โฆ
Note: In above two loop transformation one portion is
feed forward and other portion is feed backward with
same sign.
In above loop transformation we introduced a constant
K (=slop of nonlinearity), to change the nonlinearity.
By: Nafees Ahmed
Loop transformation โฆ
With same nonlinearity NL ั m[0, K] sector
The above loop transformation was done
in feedback and feed forward loop, it can
also be done in forward path only as
shown on next slide
By: Nafees Ahmed
Loop transformation โฆ
G(S)ฯ1
_-ฮพ
NL ฯ1
New linear Plant
G1(s)
+
ฮพ1/K
ฯ
+
Where
ฯ1= Kฯ-ฮพ
New Nonlinearity
NL1
Nonlinearity of NL1 ั m[0, โ) sector
ฯ
K
By: Nafees Ahmed
Loop transformation โฆ
Similar to previous
Now we can say that if original system is asymptotic
stabile then its transformed sys shown on previous slide
will also be asymptotic stable.
New real plant G1(s) must be positive real i.e๐ฎ๐ ๐ = ๐ +๐๐ ๐ฌ must be a positive real.
New nonlinearity defined by NL1 ั m[0, โ) sector
By: Nafees Ahmed
Loop transformation โฆ
Now use mix of above two methods
And instead of K we can also use some transfer
function
With same nonlinearity NL ั m[0, K] sector
By: Nafees Ahmed
Loop transformation โฆ
๐พ๐๐๐๐ ๐๐ = ๐ ๐ + ๐ โ๐
๐ฒ๐ป
Nonlinearity of NL1 ั m[0, โ) sector
G(S)ฯ1
_-ฮพ
NL ฯ1
New linear Plant
G1(s)
+
ฯ
1/(1+as)ฮพ +
(1+as)
ฯ
New Nonlinearity
NL1
1/K
1/K
By: Nafees Ahmed
Loop transformation โฆ
Similar to previous
Now we can say that if original system is asymptotic
stabile then its transformed sys shown on previous slide
will also be asymptotic stable.
New real plant G1(s) must be positive real i.e ๐ฎ๐ ๐ =๐ + ๐๐ฌ ๐ ๐ฌ + ๐/๐ must be a positive real.
By: Nafees Ahmed
Interconnection of 2 passive systemes
If G1(s) is passive and NL1 is passive then there interconnection will
also be passive. (Passive=>positive real)
So passivity for new linear system G1(s) ๐ฝ๐ ๐ฟ โค โ๐๐ฮพ
We have to prove that new nonlinear system NL1 will also be
passive. So that overall interconnection will also be passive.
For this make the differential equation of NL1 assuming NL=f as
shown on next slide
Note: We know Lyapunov function of a passive system will have ๐ฝ ๐ฟ โค ๐๐where u = i/p & y=o/p (passivity in term of Lyapunov function)
By: Nafees Ahmed
Interconnection of 2 passive
systemesโฆ
Let ฮพ = ๐(๐) & f ั m[0, K] sector
๐๐ = ๐ ๐ + ๐ โ๐
๐ฒฮพ
โ ๐๐ = ๐ ๐ + ๐ โ๐
๐ฒ๐(๐)
Now take ๐ as state and ๐๐ as input
So we can write the state equ
๐ ๐ = โ๐ +๐
๐ฒ๐ ๐ + ๐๐ state equ
ฮพ = ๐ ๐ output equ
f ฯ11/(1+as)ฮพ +ฯ
New Nonlinearity
NL1
1/K
By: Nafees Ahmed
Interconnection of 2 passive
systemesโฆ
Take ๐=x and ฮพ=y & ๐๐=u sate equ
๐ ๐ = โ๐ +๐
๐ฒ๐ ๐ + ๐ state equ
y= ๐ ๐ output equ
Let us define a Laypunov function for NL1 as
๐ฝ๐ ๐ = ๐ ๐๐๐ ๐ ๐ ๐ which is always >0 โ ๐
โ ๐ฝ๐ ๐ = ๐๐(๐) ๐
Put the value of ๐ ๐ from above equation
๐ฝ๐ ๐ = โ๐ ๐ ๐ +๐
๐ฒ๐ ๐ ๐ + ๐๐(๐)
So ๐ฝ๐ ๐ โค ๐๐ ๐ โค ๐๐ โ NL1 is also passive
Slope=Kฮต=f(x)
ฯ=x
Note: ๐๐๐ ๐ ๐ ๐ is always +ve for above non linearity By: Nafees Ahmed
Interconnection of 2 passive
systemesโฆ
We know Lyapunovโs function of new linear system (Passive) G1(s)
V1(x)=xTPx with ๐ฝ๐ ๐ฟ โค โ๐๐ฮพ P is positive definite matrix
And Lyapunovโs function of new nonlinear system (Passive) NL1
๐ฝ๐ ๐ = ๐ ๐๐๐ ๐ ๐ ๐ ๐๐๐๐ ๐ฝ๐ ๐ฟ โค ๐๐ฮพ
So Lyapunovโs function of overall system
V=V1+V2= xTPx+๐ ๐๐๐ ๐ ๐ ๐ with ๐ฝ = ๐ฝ1+ ๐ฝ2 โค โ๐๐ฮพ+ ๐๐ฮพโค0
Here V>0 and ๐ฝ โค0 and by Lyapunov this is the condition of
Asymptotic stability
By: Nafees Ahmed
Popovโs criterion
Consider a system shown in following figure
G(S)
NLฮพ
ฯ-ฮพ
_
ฯ
Nonlinearity of NL1 ั m[0, โ) sector
G(S)ฯ1
_-ฮพ
NL ฯ1
New linear Plant
G1(s)
+
ฯ
1/(1+as)ฮพ +
(1+as)
ฯ
New Nonlinearity
NL1
1/K
1/K
โ
Nonlinearity of NL1 ั m[0, K] sector
By: Nafees Ahmed
Popovโs criterion โฆ
This system will be asymptotically stable
If ๐ฎ๐ ๐ = ๐ + ๐๐ ๐ฎ ๐ + ๐/๐ฒ is positive real i.e
Re(G1(s)โฅ0 โ Re[ ๐ + ๐๐ ๐ฎ ๐ + ๐/๐ฒ ] โฅ0
For this G1(s) must be strictly proper [i.e. |deg(num)-deg(den)|โค1], necessary condition
By: Nafees Ahmed
Popovโs criterion โฆ
How to check condition Re[ ๐ + ๐๐ ๐ฎ ๐ + ๐/๐ฒ ] โฅ0
=> Re[ ๐ + ๐๐๐ (๐น๐๐ฎ ๐๐ + ๐๐ฐ๐ฆ๐ ๐ฃ๐ )] + ๐/๐ฒ โฅ0
=> ๐น๐๐ฎ ๐๐ โ ๐๐๐ฐ๐ฆ๐ ๐ฃ๐ + ๐/๐ฒ โฅ0
For above draw the following plot called Popovโs plot
By: Nafees Ahmed
Popovโs criterion โฆ
Re(G(jฯ))=x
ฯIm(G(jฯ))=y
Popovโs Plot
By: Nafees Ahmed
Popovโs criterion โฆ
๐น๐๐ฎ ๐๐ โ ๐๐๐ฐ๐๐ฎ ๐๐ + ๐/๐ฒ โฅ0
โx-ay+1/K โฅ0
Re(G(jฯ))=x
ฯIm(G(jฯ))=y
Popovโs Plot
-1/K
Slope=1/a
x-ay+1/K=0
By: Nafees Ahmed
Popovโs criterion โฆ
Re(G(jฯ))=x
ฯIm(G(jฯ))=y
Popovโs Plot
-1/K
Slope=1/a
x-ay+1/K=0
By: Nafees Ahmed
Popovโs criterion โฆ
Hence to check the condition Re[ ๐ + ๐๐ ๐ฎ ๐ + ๐/๐ฒ] โฅ0 just draw Popovโs plot and draw the line as
shown above, if Popovโs plot is to the right (below) of
line that means above condition is satisfied. So the
overall system will be asymptotically stable.
Note: Difference b/w Popovโs plot and Nyquist plot
Popovโs is Re(G(jฯ) Vs ฯIm(G(jฯ)
Nyquist plot is Re(G(jฯ) Vs Im(G(jฯ)
By: Nafees Ahmed
Reference
Prof. Harish K Pillai, IIT Bombay, through NPTEL
By: Nafees Ahmed
Thanks
?By: Nafees Ahmed