The document discusses different types of singular points in control systems: 1. A nodal point occurs when both eigenvalues are real and negative, causing all trajectories to converge to the origin in a stable manner. 2. A saddle point occurs when the eigenvalues are real and equal but opposite in sign, making the origin unstable with some trajectories converging and others diverging. 3. A focus point occurs when the eigenvalues are complex conjugates with negative real parts, causing the trajectories to spiral inward in a stable manner towards the origin. 4. A center or vortex point occurs when the eigenvalues are purely imaginary, causing the trajectories to travel in closed paths around the origin in a limitedly stable manner.

1. Presented By:
Pathak Anish (130410117044)
Guided by:
Prof. Krupa Narwekar
Instrumentation & Control
ADVANCE CONTROL THEORY (2181705)

2.  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 ẋ = 0 .
 Consider a general time invariant system described by
the state equation
ẋ = f(x,u) …[1 ]
SINGULAR POINT

3.  If the input vector u is constant, it is possible to write the
above equation in the form
ẋ = F(X) …[2]
 A system represented by an equation of this form is called
autonomous system.
 For such system, consider the points in the phase-space at
which the derivate of the all state variable are zero. Such
points are called singular points.

4.  Singular points are very important features on the phase
plane.
 Consider a time invariant second order system described by
eq of the form
ẋ=X2 ; ẋ2 = f(x1,x2)
 Now using Taylor series expansion , eq can be
ẋ=X2
ẋ2 = ax1+bx2+g2(x1,x2)
Where g2(.) contains higher order terms

5. The higher ordered terms can be neglected and therefore,
the nonlinear system trajectories essentially satisfy the
linearized eq
ẋ=X2
ẋ2 = ax1+bx2
 Transforming these eq into a scalar second order eq
𝒙1 =ax1+bẋ1
 Therefore, we will consider the second order linear
system described by

7. • In the following ,we discuss the behaviors of the trajectories of this
point with undefined slope.

8.  Eigenvalues (λ1, λ2) are both real and negative are
shown in fig (a). The nature of the trajectories is
shown in fig (b).
 They all converge to origin, which then called a nodel
point.
Nodal Point

9.  Both eigenvalues are real, equal and negative of each
other shown in fig(a). The phase portraits are shown
(b) and (c).
 The origin in this case a saddle point which is always
unstable, one eigenvalues being negative.
Saddle Point

10.  The origin is the focus point and stable/unstable for
negative/positive real parts of the eigenvalues.
 Let the eigenvalues be
λ1, λ2 = σ ± j ω
= complex conjugate pair
Focus Point

11.  The phase portrait has closed paths trajectories
shown in fig (a) & (b) and the origin is called a Centre.
 The system is limitedly stable.
λ1, λ2 = ± j ω
= on the imaginary axis
Centre or Vortex Point

12. THANK YOU