Phase Plane analysis of control system non linear control system

1. Non-linear Systems
Phase Plane Analysis

2. Phase Plane Analysis
• Phase plane analysis is one of the most important
techniques for studying the behavior of nonlinear
systems, since there is usually no analytical solution
for a nonlinear system. It has many useful properties.
• As a graphical method, it allows us to visualize what
goes on in a non-linear system starting from various
initial conditions, without having to solve the non-
linear equations analytically.
• It is not restricted to small or piece-wise
nonlinearities.
• However, it is restricted to second-order systems
only. Many practical systems can be adequately
approximated as second-order systems and can be
studied through phase – plane analysis.

3. Background
• The response characteristics (relative speed of
response) for unforced systems were dependent on
the initial conditions.
• Eigen-value/eigen-vector analysis allowed us to
predict the fast and slow (or stable and unstable)
initial conditions.
• Another way of obtaining a feel for the effect of initial
conditions is to use a phase-plane plot.
• A phase-plane plot for a two-state variable system
consists of curves of one state variable versus the
other state variable (x1(t) versus x2(t)), where each
curve is based on a different initial condition.

4. Phase - Plane
• The phase plane method is concerned with the
graphical study of second-order autonomous
systems described by
where,
x1, x2 : states of the system
f1, f2 : nonlinear functions of the states
Geometrically, the state space of this system is
a plane having x1, x2 as coordinates. This plane
is called phase plane.
)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
2
2
1
2
2
2
1
1
1
2
1
1
1
x
x
f
dt
dx
x
x
f
x
x
x
f
dt
dx
x
x
f
x









5. Phase - Plane Trajectory/Portrait
• Given an initial condition, the solution of the above
equation, as time varies from zero to infinity, can be
represented as a curve in the phase plane. Such a
curve is called a phase plane trajectory.
• If different initial conditions are considered, a family
of trajectories starting from respective initial
conditions is called a phase - plane portrait of a
system.
• These phase portraits can be develop by hand or by
means of software packages such as Matlab.

6. Example – Phase Portrait

7. )
,
(
)
,
(
)
,
(
)
,
(
2
1
2
2
2
1
2
2
2
1
1
1
2
1
1
1
x
x
f
dt
dx
x
x
f
x
x
x
f
dt
dx
x
x
f
x










 x
x
x
x 2
1 and
plane.
generally
is
plane
phase
the
So


 x
x
Let us reconsider,
In simpler form, above equations can be written as,
In most of the cases,
)
,
( 2
1
2
2
1
x
x
f
dt
dx
x
dt
dx



8. Singular Points
• A singular point is an equilibrium point in the phase
plane. Since an equilibrium point is defined as a
point where the system states can stay forever, this
implies that dx/dt = 0, i.e.
• For linear systems, there is usually only one singular
point although in some cases there can be a set of
singular points. But a non-linear system may have
more than one singular point.
• The slope of the phase trajectory passing through a
point (x1, x2 ) is determined by
0
)
,
(
0
)
,
( 2
1
2
2
1
1 
 x
x
f
x
x
f and
)
,
(
)
,
(
2
1
1
2
1
2
1
2
x
x
f
x
x
f
dx
dx


9. • where f1, f2 are assumed to be single valued functions.
This implies that the phase trajectories will not
intersect. At singular point, however, the value of the
slope is 0/0, i.e., the slope is indeterminate. Many
trajectories may intersect at such point. This
indeterminacy of the slope accounts for the adjective
“singular”.
• Singular points are very important features in the
phase plane. Examining the singular points can reveal
a great deal of information about the properties of a
system. In fact, the stability of linear systems is
uniquely characterized by the nature of their singular
points.

10. Obtaining first-order differential equation
for second order systems
)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
2
2
1
2
2
2
1
1
1
2
1
1
1
x
x
f
dt
dx
x
x
f
x
x
x
f
dt
dx
x
x
f
x








)
,
(
)
,
(
2
1
1
2
1
2
1
2
x
x
f
x
x
f
dx
dx

Let us reconsider,
Eliminating the independent variable ‘t’
It is a first order differential equation relating x1 to x2 and
actually gives the slope of the tangent to the trajectory
passing through point (x1, x2). The solution for it may be
written in the form )
( 1
2 x
x 

It represents a curve in phase-plane i.e. solution of x2
as a function of x1 and indicates motion of a
representative point on the curve.

11. Examples
x
x 


Consider the first order system described by:
Only one singular point i.e.
origin.
Trajectory may
start from a
given initial condition.

12. Consider the first order systems described by:
3
4 x
x
x 



Three singular points i.e.
0, 2, -2.
If x(0) > 2, x(∞) → ∞ If 2 > x(0) > -2, x(∞) → 0
If x(0) < -2, x(∞) → ∞

13. 0






x
x
x












x
x
x
x
d
x
d
dt
x
d
dt
x
d
x
x
/
/
0
0


x
d
x
d
Consider the system described by:
Rewriting,
Also,
Thus, origin is a singular
point i.e. equilibrium point.
For initial conditions (0, 10),
the trajectory is ABCDO,
while for (2, -7), it is EFO.
In upper half, a representative
point moves in right direction
while in lower half in left direction.

14. Symmetry in phase plane portrait
• Let us consider the second-order differential
equation:
Where, is the slope of trajectories in phase plane.
0
)
,
( 




x
x
f
x








x
x
x
f
x
d
x
d
dt
x
d
dt
x
d )
,
(
/
/
x
d
x
d


15. Since symmetry of the phase portraits also implies
symmetry of the slopes (equal in absolute value but
opposite in sign), we can identify the following
situations:
1. Symmetry about the x - axis.
)
,
(
)
,
(



 x
x
f
x
x
f
2. Symmetry about the - axis.

x
)
,
(
)
,
(




 x
x
f
x
x
f
)
,
(
)
,
(





 x
x
f
x
x
f
3. Symmetry about both the x – axis and the - axis.

x

16. Constructing Phase Portraits
• There are a number of methods for constructing
phase plane trajectories for linear or nonlinear
systems, such as:
1. Analytical methods: are useful for systems having
differential equations which are simple and piece-
wise linear – Integration and Solution in terms of ‘t’ &
then elimination of ‘t’.
2. Graphical Methods: are useful when it is tedious,
difficult or impossible to solve the given differential
equation analytically. Applicable to both linear and
non-linear systems - The isoclines method and The
delta method.
3. Experimental Methods: the trajectories can be
developed graphically.

17.   0
;
)
0
(
; 0 







x
x
x
M
M
x :
conditions
initial
with
Constant
M
x
d
x
d
x
x
d
x
d
x
dt
x
d
dt
x
d
x
dt
x
d
x 














But,
M
x
x
x )
(
2 0
2



Consider the second order differential equation
First method: Integrating the above equation
Second method: Solution can be written as
0
2
2
1
)
(
)
( x
Mt
t
x
and
Mt
t
x 





Analytical methods
Eliminating ‘t’, trajectories can be obtained.

19. Consider the second order differential equation 0




x
x
x
d
x
d
x
dt
x
d
dt
x
d
x
dt
x
d
x










But, 0





x
x
d
x
d
x
First method: Integrating the above equation
2
0
2
2
x
x
x 


Second method:
Solution can be written as
t
x
t
x
t
x
t
x
sin
)
(
cos
)
(
0
0




Eliminating ‘t’,
2
0
2
2
x
x
x 



20. 0
2




x
x 
x
d
x
d
x
dt
x
d
dt
x
d
x
dt
x
d
x










But, 0
2





x
x
d
x
d
x 
Consider the second order differential equation
First method: Integrating the above equation 2
2
2
2
A
x
x




Where A is a constant to be determined by initial conditions.
Second method: Solution x(t) can be written as
)
cos(
)
( 
 
 t
A
t
x
Where A & α are constants determined by initial conditions.
Differentiating w. r. t. ‘t’
)
sin(
)
( 

 



t
A
t
x
Eliminating ‘t’ from the two equations, same solution as the
first method can be obtained i.e.
2
2
2
2
A
x
x





22. Graphical Methods
The Isocline Method: The locus of points where
trajectories have a given slope is called an isocline.
This method is suitable for trajectories having
constant slope. It is applicable to the following type
of first-order differential equation:
where, f1(x1, x2) and f2(x1, x2) are analytic functions.
x1→ Independent variable; x2→ Dependent variable
The following equation gives the locus of constant
slope
)
,
(
)
,
(
2
1
1
2
1
2
1
2
x
x
f
x
x
f
dx
dx

)
,
(
)
,
( 2
1
1
2
1
2
1
2
x
x
f
x
x
f
dx
dx

 


 constant

23. • When this method is used, the entire phase plane is
filled with short line segments fixing directions of the
field. As the slope is indeterminate at singular point,
no line segment is drawn at this point. The trajectory
passing through any given ordinary point can be
constructed by drawing a continuous curve following
the directions of the field.

24. Consider the differential equation
0
2 2






x
x
x 

0
2 2







x
x
x
d
x
d
x 

0
2
;
2







x
x
x
x
d
x
d




Putting,








2
2
x
x
















tan
2
;
tan
,
2
x
x
As
For better accuracy more number
of isoclines must be used i.e. for
every 50 or so. Further, the method
is suitable if isoclines are straight
lines.

25. Slopes:
A to B: (-1-1.2)/2 = -1.1
B to C: (-1.2-1.4)/2 = -1.3
C to D: (-1.4-1.8)/2 = -1.6
D to E: (-1.8-2)/2 = -1.9

26. Delta Method
• The trajectory is obtained as a sequence of circular
arcs whose centers slide along the x - axis. It can be
applied to the equations of the form
where may be either linear or non-linear and
may be time varying but must be continuous and
single valued. Adding a term ω2x to both the sides
term ω2x must be selected such that the values of
function δ are neither too small or too large for the
range of values of , x, and t being considered.
)
,
,
( t
x
x
f
x





)
,
,
( t
x
x
f

x
t
x
x
f
x
x 2
2
)
,
,
( 
 






2
2
)
,
,
(
)
,
,
(



x
t
x
x
f
t
x
x






x
)
,
,
(
2
2
t
x
x
x
x





 



27. )
,
,
( t
x
x


1
1
1
1
1
1
1
1
1
1
1
,
,
,




as
constant
assumed
is
But
and
i.e.
amount
small
a
by
change
they
od,
neighborho
the
In
and
be
states
the
Let
t
t
t
x
x
x
x
x
x
t
t
x
x
x
x


















,

x
The function depends on the variables
x, and t. For small changes in these variables, δ is
considered constant.
0
)
( 1
2







 x
x
The above equation shows a simple harmonic motion.
The trajectories for this system are circles in the x -
normalized phase-plane with

/

x
2
1
1
2
1
1
)
(
)
/
(
;
0
/
,










x
x
x
x
Radius
and
at
Centre

28. The trajectory can be constructed as shown:
1
2
1
)
,
,
(
x
t
x
x
f






Point P represents the
state of the system on
the trajectory at time
t = t1. The value of δ1 is
determined from
Once point Q, the centre of the circular arc is located
on the x - axis, the radius is fixed as . The true
trajectory in the neighborhood of point P is then
approximated by a short circular arc. The arc must be
short enough to ensure that the changes in the
variables are small.
PQ

29. Consider the differential equation
0
2 2
2
2



 x
dt
dx
dt
x
d


0
2 2






x
x
x 

dt
dx
x
dt
x
d

 2
2
2
2








 d
dx
x
d
x
d
t 2
1
, 2
2





Let
y
d
dx
dt
dx
x
y
x










1
, as
axis
Letting
y
x
d
x
d


2
2
2



 


 



 x
d
x
d
y 2
2
2
Let
For y = y1 = constant, the above equation indicates that
the trajectory in the x-dx/dτ plane is a circle with centre
at x = -2ζy1, y = 0. Since the location of centre depends
on y1, it moves along the x-axis as y changes. For the
construction of the trajectory, it is convenient to draw a
line x = δ = -2ζy in the x-y plane.

30. The centre of arc AB is located at point P (x = -2ζyAB, y = 0).
Here yAB is the mean value of y between point A and Point
B. The location of the centre of the circle can be obtained
directly and simply by use of line x = -2ζy drawn in the x-y
plane. The segments of arcs from point B to point C and
from point C to point D have their centers at point Q and
point R, respectively.

31. Classification of Singular Points
0
)
0
,
0
(
,
0
)
0
,
0
( 2
1 
 f
f and
)
,
(
)
,
( 2
1
2
2
2
1
1
1
x
x
f
dt
dx
x
x
f
dt
dx

 and
Consider the system
where f1(x1, x2) and f2(x1, x2) are analytic functions of the
variables x1 and x2 in the neighborhood of the origin. Also
suppose that the origin is a singular or equilibrium point,
so that
Expanding f1(x1, x2) and f2(x1, x2) in Taylor series in the
neighborhood of the origin.
)
,
(
)
,
(
2
1
2
2
2
22
2
1
12
2
1
11
2
2
1
2
2
2
1
1
2
2
22
2
1
12
2
1
11
2
1
1
1
1
x
x
g
x
b
x
x
b
x
b
x
b
x
a
dt
dx
x
x
g
x
a
x
x
a
x
a
x
b
x
a
dt
dx












g1(x1, x2) and g2(x1, x2) involve third and higher order terms.

32. In the neighborhood of the origin, where x1 and x2 are
very small, above equations may be approximated as
2
2
1
2
2
2
1
1
1
1
x
b
x
a
dt
dx
x
b
x
a
dt
dx



 and
0
0
)
(
)
(
0
1
2
2
1
2
1
2
1
2
1
2
1
1
2
1
1
2
2
1
1
1
1
1
2
2
1
1
1
1
2
2
1
1
1





























 









 

























bx
x
a
x
x
b
a
b
a
x
b
a
x
x
b
a
x
b
x
b
a
x
a
x
x
b
b
a
x
b
b
x
a
x
b
a
b
x
b
x
a
x
b
x
a
b
x
a
x
dt
d
b
x
a
x
x
x
b
x
a
x
x
x
Let

33. The above equation may be written as
0
2


 b
a

Here, a and b are constants with b ≠ 0. The locations
of two roots λ1 and λ2 of above characteristic equation
in the complex plane determine the characteristics of
the singular point.
1. λ1 & λ2 are complex conjugate and lie in LH s-plane.

34. 2. λ1 & λ2 are complex conjugate and lie in RH s-plane.
3. λ1 & λ2 are real and lie in LH s-plane.

35. 4. λ1 & λ2 are real and lie in RH s-plane.
5. λ1 & λ2 are complex conjugate and lie on jω-axis.

36. 6. λ1 & λ2 are real; λ1 lies in LH s-plane, while, λ2 lies in
RH s-plane.

37. Limit Cycles
• Limit cycles occur frequently in non-linear systems.
• They have a distinct geometric configuration in the
phase plane portrait i.e. that of an isolated closed path.
• A given system may exhibit more than one limit cycle.
• It represents a steady-state oscillation, to which or from
which all trajectories nearby will converge or diverge.
• It divides the plane into an inside and an outside. No
trajectories inside (outside) a particular limit cycle ever
cross the limit cycle to enter outside (inside) region.
• In a non-linear system, it describes the amplitude and
period of a self-sustained oscillation.
• Since limit cycles are isolated from each other, there
can be no other limit cycles in the neighborhood of any
limit cycle.

38. A limit cycle is called stable if the trajectories near the
limit cycle, originating from outside or inside, converge
to that limit cycle. In this case, the system exhibits a
sustained oscillation with constant amplitude. The
inside of the limit cycle is an unstable region in the
sense that the trajectories diverge to the limit cycle,
and the outside is a stable region in the sense that the
trajectories converge to limit cycle.

39. A limit cycle is called unstable if the trajectories near
the limit cycle, originating from outside or inside,
diverge from that limit cycle. The inside of the limit
cycle is a stable region in the sense that the
trajectories diverge from the limit cycle and converges
to the singular point within the limit cycle. The outside
is an unstable region in the sense that the trajectories
diverge from the limit cycle and increases to infinity
with time.

40. A limit cycle is called semi-stable if the trajectories
originating at points outside the limit cycle diverge from
it, while the trajectories originating at points inside the
limit cycle converge to it. Both inside and outside are
unstable regions. OR
A limit cycle is called semi-stable if the trajectories
originating at points outside the limit cycle converge to
it, while the trajectories originating at points inside the
limit cycle diverge from it. Both inside and outside are
stable regions.

41. Phase Plane Analysis Of Linear Systems












 r
r
T
Ke
e
e
T
c
r
e
Since,
Ke
c
c
T
Ke
cs
Tcs
Ts
s
K
e
c











2
)
1
(
0
0
.
0
0
)
(















t
Ke
e
e
T
t
r
r
R
t
r
for
for
input
step
the
For
Step Response:
Assume that the system
is initially at rest. For this
system, we have
Since the system is assumed to be initially at rest, the
initial conditions for the error signal are
.
0
)
0
(
)
0
( 


e
R
e and

42. system.
the
of
point
singular
the
origin,
the
to
converges
and
point
at
starts
plane
the
in
trajectory
The )
0
,
(R
e
e


0
,
0
,
0
0
0
/
/




















e
e
Ke
e
T
e
e
e
dt
e
d
dt
e
d
e
d
e
d
For
For
point
Singular
ess= 0

43. Ramp Response:
0
)
(
0
0
0
)
(
)
(



























K
V
e
K
e
e
T
t
V
Ke
e
e
T
t
V
r
r
R
Vt
t
r
Vt
t
r
for
for
and
input
step
ramp
OR
input
ramp
For
V
e
R
e 


)
0
(
)
0
( and
are
conditions
Initial
K
V
e
e
K
V
e
e
T
e
e
e
dt
e
d
dt
e
d
e
d
e
d




















,
0
,
0
0
0
/
/
For
For
point
Singular
0









Kx
x
x
T
x
K
V
e
Let,

44. ess= V/K

45. Impulse Response:
The unit impulse response of a system is given by it’s
transfer function,
T
K
K
s
Ts
K
s
c
K
s
Ts
sK
c
s
s














 2
2
2
lim
)
0
(
,
0
lim
)
0
( and
0
0
0
0














t
Ke
e
e
T
t
r
r
for
for
input,
impulse
unit
For
Using initial value theorem,
T
K
e
e 

 

 )
0
(
0
)
0
( and
are
conditions
initial
The
0
,
0
,
0
;
0
0
















e
e
Ke
e
T
e
e
e
For
For
point
Singular

46. ess= 0

47. • Signal-dependent second order non-linear systems
can be approximated by several piecewise-linear
systems.
• The entire phase-plane is divided into several sub-
regions, each corresponding to an individual linear
operation.
• If a singular point lies within its sub-region, then it is
called an actual singular point. But, if the singular
point lies outside its sub-region and the trajectory
can never reach it, it is called a virtual singular point.
• The phase-plane portrait of each sub-region is that
of a linear system. The composite trajectory is, then,
obtained by joining trajectories at the boundaries of
each operating region.
Phase Plane Analysis Of Non-linear Systems

48. 0
0
e
e
ke
m
e
e
e
m




for
for
Km
c
c
T
Ts
s
K
m
c








)
1
(













r
r
T
Km
e
e
T
c
r
e
Since,
Consider the system with a non-linear gain GN.
The system is assumed to be initially at rest.
Step Response:
0
0
.
0
0
)
(















t
Km
e
e
T
t
r
r
R
t
r
for
for
0
0
0
0
e
e
kKe
e
e
T
e
e
Ke
e
e
T














for
for


49. It is assumed that for ‫׀‬e‫׀‬ >e0, the system is underdamped
and thus the nature of the trajectory is that of a stable
focus. While, for ‫׀‬e‫׀‬ < e0, the system is overdamped (the
value of k is such) and thus the nature of the trajectory
is that of a stable node. The trajectories for two linear
operations will be:

50. The trajectory of the error signal
of the system for a unit step input
is shown.
The system constants
are taken as:
T =1,
K = 4,
k = 0.0625,
e0 = 0.2
.
e
,
e 0
)
0
(
1
)
0
( 


:
Conditions
Initial
ess= 0

51. The singular point for first equation, representing
a stable focus, is at point X (V/K, 0) and that for second
equation, representing
a stable node, is at
Point Y (V/kK, 0).
0
0 0
0
0
0
0
)
(
)
(
e
e
kKe
e
e
T
e
e
Ke
e
e
T
t
V
Km
e
e
T
t
V
r
r
R
Vt
t
r
Vt
t
r


































for
and
for
for
for
and
input
step
ramp
OR
input
ramp
For
Ramp Response:

52. The trajectory of the error signal
of the system for a ramp input is
shown.
The system constants
are taken as:
T =1, K = 4,
k = 0.0625, e0 = 0.2
R = 0.3, V = 0.04
)
0
,
16
.
0
(
16
.
0 

 Y
kK
V
)
0
,
01
.
0
(
01
.
0 

 X
K
V
.
V
e
,
R
e 04
.
0
)
0
(
3
.
0
)
0
( 




:
Conditions
Initial
ess= 0.16

58. K
s
Ts
sK
s
c
s
r
s
r
K
s
Ts
sK
s
c








 2
2
)
(
1
)
(
);
(
)
( But