Applied Nonlinear Control - Chapter 2: Phase Plane Analysis - Nguyễn Tân Tiến

oordinates. We now to describe two techniques 
for computing time history from phase portrait. Both of 
techniques involve a step-by step procedure for recovering 
time. 
Obtaining time from xxt &/∆≈∆ 
In a short time t∆ , the change of x is approximately 
txx ∆≈∆ & (2.8) 
where x& is the velocity corresponding to the increment x∆ . 
From (2.8), the length of time corresponding to the 
increment x∆ is xxt &/∆≈∆ . This implies that, in order to 
obtain the time corresponding to the motion from one point to 
another point along the trajectory, we should divide the 
corresponding part of the trajectory into a number of small 
segments (not necessarily equally spaced), find the time 
associated with each segment, and then add up the results. To 
obtain the history of states corresponding to a certain initial 
condition, we simply compute the time t for each point on the 
phase trajectory, and then plots x with respects to t and x& 
with respects to t . 
Obtaining time from dxxt ∫≈ )/1( & 
Since dtdxx /=& , we can write xdxdt &/= . Therefore, 
∫≈− xx dxxtt 0 )/1(0 & 
where x corresponding to time t and 0x corresponding to 
time 0t . This implies that, if we plot a phase plane portrait 
with new coordinates x and )/1( x& , then the area under the 
resulting curve is the corresponding time interval. 
2.4 Phase Plane Analysis of Linear Systems 
The general form of a linear second-order system is 
211 xbxax +=& (2.9a) 
212 xdxcx +=& (2.9b) 
Transform these equations into a scalar second-order 
differential equation in the form )( 1112 xaxdxcbxb −+= && . 
Consequently, differentiation of (2.9a) and then substitution of 
(2.9b) leads to 111 )()( xdabcxdax −++= &&& . Therefore, we 
will simply consider the second-order linear system described 
by 
0=++ xbxax &&& (2.10) 
To obtain the phase portrait of this linear system, we solve for 
the time history 
tt ekektx 21 21)(
λλ += for 21 λλ ≠ (2.11a) 
tt etkektx 21 21)(
λλ += for 21 λλ = (2.11b) 
whre the constant 21,λλ are the solutions of the characteristic 
equation 
0))(( 21
2 =−−=++ λλ ssbass 
The roots 21,λλ can be explicitly represented as 
2
42
1
baa −+−=λ and 
2
42
2
baa −−−=λ 
For linear systems described by (2.10), there is only one 
singular point )0( ≠b , namely the origin. However, the 
trajectories in the vicinity of this singularity point can display 
quite different characteristics, depending on the values of 
a and b . The following cases can occur 
• 21,λλ are both real and have the same sign (+ or -) 
• 21,λλ are both real and have opposite sign 
• 21,λλ are complex conjugates with non-zero real parts 
• 21,λλ are complex conjugates with real parts equal to 0 
We now briefly discuss each of the above four cases 
Stable or unstable node (Fig. 2.9.a -b) 
The first case corresponds to a node. A node can be stable or 
unstable: 
0, 21 <λλ : singularity point is called stable node. 
0, 21 >λλ : singularity point is called unstable node. 
There is no oscillation in the trajectories. 
Saddle point (Fig. 2.9.c) 
The second case ( 21 0 λλ << ) corresponds to a saddle point. 
Because of the unstable pole 2λ , almost all of the system 
trajectories diverge to infinity. 
Applied Nonlinear Control Nguyen Tan Tien - 2002.3 
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Chapter 2 Phase Plane Analysis 
5 
ωj
σ
ωj
σ
ωj
σ
ωj
σ
ωj
σ
ωj
σ
center point
stable node
unstable node
saddle point
stable focus
unstable focus
x
x&
x
x&
x
x&
x
x&
x
x&
x
x&
)(a
)(b
)(c
)(d
)(e
)( f 
Fig. 2.9 Phase-portraits of linear systems 
Stable or unstable locus (Fig. 2.9.d-e) 
The third case corresponds to a focus. 
0),Re( 21 <λλ : stable focus 
0),Re( 21 >λλ : unstable focus 
Center point (Fig. 2.9.f) 
The last case corresponds to a certain point. All trajectories 
are ellipses and the singularity point is the centre of these 
ellipses. 
⊗ Note that the stability characteristics of linear systems are 
uniquely determined by the nature of their singularity points. 
This, however, is not true for nonlinear systems. 
2.5 Phase Plane Analysis of Nonlinear Systems 
In discussing the phase plane analysis of nonlinear system, 
two points should be kept in mind: 
• Phase plane analysis of nonlinear systems is related to 
that of liner systems, because the local behavior of 
nonlinear systems can be approximated by the behavior 
of a linear system. 
• Nonlinear systems can display much more complicated 
patterns in the phase plane, such as multiple equilibrium 
points and limit cycles. 
Local behavior of nonlinear systems 
If the singular point of interest is not at the origin, by defining 
the difference between the original state and the singular point 
as a new set of state variables, we can shift the singular point 
to the origin. Therefore, without loss of generality, we may 
simply consider Eq.(2.1) with a singular point at 0. Using 
Taylor expansion, Eqs. (2.1) can be rewritten in the form 
),( 211211 xxgxbxax ++=& 
),( 212212 xxgxdxcx ++=& 
where 21, gg contain higher order terms. 
In the vicinity of the origin, the higher order terms can be 
neglected, and therefore, the nonlinear system trajectories 
essentially satisfy the linearized equation 
211 xbxax +=& 
212 xdxcx +=& 
As a result, the local behavior of the nonlinear system can be 
approximated by the patterns shown in Fig. 2.9. 
Limit cycle 
In the phase plane, a limit cycle is defied as an isolated closed 
curve. The trajectory has to be both closed, indicating the 
periodic nature of the motion, and isolated, indicating the 
limiting nature of the cycle (with near by trajectories 
converging or diverging from it). 
Depending on the motion patterns of the trajectories in the 
vicinity of the limit cycle, we can distinguish three kinds of 
limit cycles. 
• Stable Limit Cycles: all trajectories in the vicinity of the 
limit cycle converge to it as ∞→t (Fig. 2.10.a). 
• Unstable Limit Cycles: all trajectories in the vicinity of 
the limit cycle diverge to it as ∞→t (Fig. 2.10.b) 
• Semi-Stable Limit Cycles: some of the trajectories in 
the vicinity of the limit cycle converge to it as 
∞→t (Fig. 2.10.c) 
2x
1x
converging
trajectories 2
x
1x
diverging
trajectories 2
x
1x
converging
diverging
limit cycle limit cycle limit cycle
)(a )(b )(c 
Fig. 2.10 Stable, unstable, and semi-stable limit cycles 
Example 2.7 Stable, unstable, and semi-stable limit cycle___ 
Consider the following nonlinear systems 
(a) 



−+−−=
−+−=
)1(
)1(
2
2
2
1212
2
2
2
1121
xxxxx
xxxxx
&
&
 (2.12) 
(b) 



−++−=
−++=
)1(
)1(
2
2
2
1212
2
2
2
1121
xxxxx
xxxxx
&
&
 (2.13) 
(c) 



−+−−=
−+−=
22
2
2
1212
22
2
2
1121
)1(
)1(
xxxxx
xxxxx
&
&
 (2.14) 
Applied Nonlinear Control Nguyen Tan Tien - 2002.3 
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Chapter 2 Phase Plane Analysis 
6 
By introducing a polar coordinates 
2
2
2
1 xxr += 


= −
1
21tan)(
x
xtθ 
the dynamics of (2.12) are transformed as 
)1( 2 −−= rr
dt
dr 1−=
dt
dθ 
When the state starts on the unicycle, the above equation 
shows that 0)( =tr& . Therefore, the state will circle around the 
origin with a period π2/1 . When 1r& . This 
implies that the state tends to the circle from inside. 
When 1>r , then 0<r& . This implies that the states tend to the 
unit circle from outside. Therefore, the unit circle is a stable 
limit cycle. This can also be concluded by examining the 
analytical solution of (2.12) 
tec
tr
2
01
1)( −+
= and tt −= 0)( θθ , where 112
0
0 −=
r
c 
Similarly, we can find that the system (b) has an unstable limit 
cycle and system (c) has a semi-stable limit cycle. 
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2.6 Existence of Limit Cycles 
Theorem 2.1 (Pointcare) If a limit cycle exists in the second-
order autonomous system (2.1), the N=S+1. 
Where, N represents the number of nodes, centers, and foci 
enclosed by a limit cycle, S represents the number of enclosed 
saddle points. 
This theorem is sometime called index theorem. 
Theorem 2.2 (Pointcare-Bendixson) If a trajectory of the 
second-order autonomous system remains in a finite region 
Ω , then one of the following is true: 
(a) the trajectory goes to an equilibrium point 
(b) the trajectory tends to an asymptotically stable limit 
cycle 
(c) the trajectory is itself a limit cycle 
Theorem 2.3 (Bendixson) For a nonlinear system (2.1), no 
limit cycle can exist in the region Ω of the phase plane in 
which 2211 // xfxf ∂∂+∂∂ does not vanish and does not change 
sign. 
Example 2.8________________________________________ 
Consider the nonlinear system 
2
2121 4)( xxxgx +=& 
2
2
112 4)( xxxhx +=& 
Since )(4 22
2
1
2
2
1
1 xx
x
f
x
f +=∂
∂+∂
∂ , which is always strictly 
positive (except at the origin), the system does not have any 
limit cycles any where in the phase plane. 
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