Non Linear Control System

1. Department of
Mechatronics Engineering
Modern Control System (MC1655)
1

2. 2
Why Study Non Linear System
All physical System are non linear in nature.
Linearization only valid for a small operational range.
Some models cannot be linearized (discontinuous
nonlinearities).
 Robust designs can sometimes be obtained by introducing
nonlinearity.

3. 3
Differences (Linear/Non Linear)
 Does Not obey Superposition Principle.
y1= f(u1)
y2=f(u2)
y3=f(u1+u2)
i.e. y3=y1+y2
 Does not have closed form of solutions.
 Stability depends on initial conditions, input signals and system
parameters.
 Output frequency may not necessarily be the same as the input
frequency.
 Nonlinearity is the behaviour of a circuit , in which the output signal
strength does not vary in direct proportion to the input signal strength.
E.g.: diode.

4. 4
Mathematical Model of Nonlinear system
𝑥1=𝑓1(𝑡, 𝑥1, 𝑥2, 𝑥3………………… 𝑥𝑛, 𝑢1, 𝑢2, 𝑢3………………, 𝑢𝑛, )
𝑥2=𝑓2(𝑡, 𝑥1, 𝑥2, 𝑥3………………… 𝑥𝑛, 𝑢1, 𝑢2, 𝑢3………………, 𝑢𝑛, )
.
.
.
𝑥𝑛=𝑓3(𝑡, 𝑥1, 𝑥2, 𝑥3………………… 𝑥𝑛, 𝑢1, 𝑢2, 𝑢3………………, 𝑢𝑛, )
x1, x2,x3 = system parameters/system variables
u1,u2,u3………un= system inputs
In vector notation: x=
𝑥1
𝑥2
.
.
.
𝑥𝑛
u=
𝑢1
𝑢2
.
.
.
𝑢𝑛
, f(t,x,u)= =
𝑓1 (𝑡,𝑥,𝑢)
𝑓2(𝑡, 𝑥, 𝑢)
.
.
.
𝑓𝑛 (𝑡, 𝑥, 𝑢)
𝑥 = 𝑓 𝑡, 𝑥, 𝑢 = 𝑠𝑡𝑎𝑡𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑥 = 𝑠𝑡𝑎𝑡𝑒
𝑢 = 𝑖𝑛𝑝𝑢𝑡
𝑦 = ℎ(𝑡, 𝑥, 𝑢)
State Space Model

5. 5
Mathematical Model of Nonlinear system
𝑥= f(t, x, u)
𝑥 =f(t, x)= unforced oscillations
𝑥=f(x)= Autonomous System/time Invariant System
Equilibrium Points
A point x= 𝑥∗ in the state space is said to be an equilibrium point, if
it has the property that whenever the state of the system starts at 𝑥∗,
it will remain at 𝑥∗for all future time.
For autonomous system, the equilibrium points are real roots of
equation f(x)=0

6. 6
Phase Plane Introduction

7. 7
Difficulties in Analyzing Non Linear System
 Superposition principle does not apply.
 Linearization technique can only predict the local behaviour of the non linear
system.
 Some phenomenon are essentially non linear
Some Non Linear Phenomenon:
1. Finite Escape Time: A non linear system can go to infinity in a finite time

8. 8
Difficulties in Analyzing Non Linear System
2. Multiple Isolated Equibilria: The system may converge to one of the steady state
operating points depending on the initial state of the system.
3. Limit Cycle: Non linear system can exhibit stable oscillations of fixed amplitude
and frequency irrespective on initial states. Limit cycle is a closed trajectory; other
trajectories about the limit cycle are spirals from various points of phase plane. •
Limit cycle divides the phase plane in to 2 zones.

9. 9
Difficulties in Analyzing Non Linear System
Stable limit cycle: system will approach an oscillatory state regardless of the initial
condition.
Semi stable limit cycle : in this case maintenance of oscillation depends on initial
condition.

10. 10
Difficulties in Analyzing Non Linear System
Unstable limit cycle:
.
Limit cycles are usually less sensitive to system parameter variations.
Although L.C. can be sustained over a finite range of system parameters.
• E.g.: squealing door hinges, electric wires whistling in the wind ,whirling shafts.

11. 11
Difficulties in Analyzing Non Linear System
4. Sub-harmonics, harmonic or periodic oscillations: A non linear system under
periodic excitation can oscillate with the frequencies that are submultiples or
multiples of input frequency.
Nonlinear system may contain frequencies other than forcing frequency in input
• These frequencies are multiple of forcing frequency.
If f(t)= A sin ωt
then oscillatory phenomenon of frequency ᾤ may appear at certain points
here ω not equal to ᾤ
when ᾤ > ω it is superharmonic
when ᾤ < ω it is subharmonic
5. Chaos: Nonlinear system exhibit motion that are random, although they may
seem deterministic. complex , irregular motion that are extremely sensitive to initial
conditions.

12. 12
Harmonics/Subharmonics

13. 13
Difficulties in Analyzing Non Linear System
6. Multiple Modes of Behaviour:
• May have more than one limit cycle
• May exhibit Sub-harmonics, harmonic or more steady state behaviour depending
on the amplitude and frequency of the input.
• May exhibit discontinuous jumps known as Jump Phenomenon
E.g. : hard spring •
As input frequency is increased gradually from 0 ,response follow curve ABC .
• At ‘C’- increment in frequency resulted in discontinuous jump down to ‘D’.
• Further increase – response curve follows through DE.
• If frequency is now decreased – curve EDF is followed with a jump up to B from the point F
and then to A .

14. 14
Difficulties in Analyzing Non Linear System
7. Beat Frequency: The system is sensitive to some frequency only e.g. Guitar, Sonomter.

15. 15
Difficulties in Analyzing Non Linear System
8. Frequency Entrainment or Synchronisation : When excited by a sinusoidal input of
constant frequency and amplitude is increased from lower values, the put frequency at same
point exactly matches with input frequency and continue to remain as such thereafter.
 It is because of the physical absorption of the particular frequencies.

16. 16
Classification of Non Linearities
 Inherent/Incidental or Intentional non Linearities: Those which are
inherently present in the system like saturation, dead zone, friction etc., and
Those which are deliberately inserted into the system to modify the
system characteristics i.e, to improvement the system performance or/and to
simplify the construction of the system.
 Static or Dynamic System
 Functional or piecewise Linear

17. 17
Common Physical Non Linearities
 The output is proportional to input for a limited range of input signals, when input exceeds
this range, the output tends to become nearly constant. This phenomenon is called
saturation or Limiter.
All devices when driven by large signals, exhibit the phenomenon of saturation due to
limitations of their physical capabilities. Example are electronics amplifier, output of
sensors which measuring the position, velocity, temperature etc.

18. 18
Common Physical Non Linearities
 A relay is a nonlinear power amplifier which can provide large power amplification
inexpensively and is therefore deliberately introduced in control systems. A relay controlled
system can be switched abruptly between several discrete states which are usually
off, full forward and full reverse. Relay controlled systems find wide applications in the
control field.
 Friction: It comes in to existence when mechanical surface comes in sliding contact.
Viscous Friction
Coulomb Friction
Stiction Friction

19. 19
Common Physical Non Linearities
 Many physical devices do not respond to small signals, i.e., if the input amplitude is less
than some small value, there will be no output. The region in which the output is zero is
called dead zone.
Output becomes zero when input crosses certain limiting value.
Effect of dead zone:
 system performance degradation
 reduced positioning accuracy •
 may destabilize system
• E.g.: • Dead zone in actuators, such as hydraulic servo valves, give rise to limit cycle and
instability. • Electronic devices like diode.

20. 20
Common Physical Non Linearities
 Backlash:The difference between the tooth space and tooth width in mechanical system ,
which is essential for rapid working gear transmission is known as backlash. It is present in
most of the mechanical and hydraulic systems
Effect of backlash :
 gear backlash may cause sustained oscillation or chattering phenomenon
 system may turn unstable for large backlash

21. 21
Methods of Analysis
 Non Linear System are difficult to Analyse
 Each technique has its own advantage and disadvantage.
 Does not guarantee generalisations.
 Linearization Techniques
Involves linearization of a non linear phenomenon near a operating point.
Validity depends on how well the linearized system represent the actual non linear
system
 Phase Plane Analysis
Applicable to second order linear or non linear systems.
Aids in the study of nature of phase trajectories near equilibrium points.
 Describing Function Analysis
principle involved is that under sinusoidal exciation,the harmonics present in the
output of the nonlinear element is ignored and the functional component of the
output is compared with the input sinusoid.
Accuracy is better for higher order system.

22. 22
Methods of Analysis
Lypunov’s Method for Stability
 Developed by AM Lyapunov
 Allows us to conclude about the stability without solving the system
equations.
Popov’s Criterion
Circle Criterion

23. 23
Phase Plane Method
 First developed by French Mathematician Poincare.
 As a new method to analyse the Three Body Problem
 Phase plane analysis is a graphical method for studying the first order and
second-order linear or nonlinear systems.
 Phase plane method is used for obtaining the graphical solutions of the two
simultaneous equations of an autonomous system given by:
𝑥1=𝑓1 (𝑥1, 𝑥2, )
𝑥2=𝑓2 (𝑥1, 𝑥2, )
Where 𝑓1 and 𝑓2 are functions of state variables of 𝑥1 𝑥2, respectively

24. 24
Concept of Phase Plane Method
Phase plane, phase trajectory and phase portrait :- the second-order system by the
following ordinary differential equation:
Where is the linear or non-linear function of x and 𝑥
In respect to an input signal or with the zero initial condition
The state space of this system is a plane having x and as coordinates which is called as the
phase plane.
As time t varies from zero to infinity, change in state of the system in x-ẍ plane is represented
by the motion of the point. The trajectory along which the phase point moves is called as
phase trajectory.

25. 25
Properties of Phase Trajectory
 The slope of phase trajectory
The slope of the phase trajectory passing through ( ) a point in the phase plane is
determined by
𝑥0, 𝑥 0
The slope of the phase trajectory is a definite value unless ẋ=0 and
Thus, there is no more than one phase trajectory passing through this point, i.e. the phase
trajectories will not intersect at this point. These points are normal points in the phase
plane.
 Singular point of the phase trajectory
In the phase plane, if ẋ=0 and are simultaneously satisfied at a point,
thus there are more than one phase trajectories passing through this point, and the slope of the
phase trajectory is indeterminate. Many phase trajectories may intersect at such points, which
are called as singular points.

26. 26
Direction of the Phase Trajectory
In the upper half of the phase plane, ẋ >0 , the phase trajectory moves from left to right along
the x axis, thus the arrows on the phase trajectory point to the right; similarly, in the lower half
of the phase plane, , ẋ < 0 the arrows on the phase trajectory point to the left. In a word, the
phase trajectory moves clockwise.
Direction of the phase trajectory passing through the x axis.
The phase trajectory always passes through x axis vertically. All points on the x axis satisfy
ẋ=0. thus except the singular point ƒ(x, ẋ)=0 , the slope of other points is
indicating that the phase trajectory and x axis are orthogonal.

27. 27
Sketching Phase Trajectories
 The sketching of the phase trajectory is the basis of phase plane analysis.
 Analytical method and graphical method are two main methods for plotting the phase
trajectory.
 The analytical method leads to a functional relationship between x and ẋ by solving the
differential equation, then the phase trajectory can be constructed in the phase plane.
 The graphical method is used to construct the phase trajectory indirectly.
Analytical Method
 The analytical method can be applied when the differential equation is simple.
 consider a second-order linear system with non zero initial condition.
When ζ=0 , the above equation becomes
Integration of this equation by variable separation method yields

28. 28
Sketching Phase Trajectories
Equ (1) represents the ellipse with the center at
the origin in the phase plane. When the initial
conditions are different, the phase trajectories
are a family of ellipses starting from the point
(𝑥0, 𝑥 0) The phase portrait is shown in Fig.
indicting that the response of the system is
constant in amplitude. The arrows in the figure
indicate the direction of increasing t . Phase portrait of Undamped second order system

29. Sketching Phase Trajectories
Graphical Method
There are a number of graphical methods, where method of isoclines is easy to realize and widely
used.
 The method of isoclines is a graphical method to sketch the phase trajectory.
From Equation.
the slope of the phase trajectory passing through a point is determined by
An isocline with slope of α can be defined as
Equation.(2) is called as an isocline equation. the points on the isoclines all have the same
slope α By taking α to be different values, we can draw the corresponding isoclines in the
phase plane.
The phase portrait can be obtained by sketching the continuous lines along the field of
directions.

30. Sketching Phase Trajectories

31. Sketching Phase Trajectories

32. Sketching Phase Trajectories

33. Example 1
Consider the system whose differential equation is ẍ+ẋ+x=0 ,sketch the phase portrait of the
system by using method of isoclines.
Where, Equ.(1) is straight line equation with slope of
We can obtain the corresponding slope of the isoclines with different values of α.
The slope of isoclines and the angle between the isoclines and x axis can be found in table
(1).

34. Example 1

35. Example 1
1) Fig. shows the isoclines with different values of α and the short line segments which
indicate the directions of tangents to trajectories.
2) A phase trajectory starting from point A can move to point B and point C gradually along
the direction of short line segments.

36. Stability of Non Linear System
 A system is stable if with bounded input, the system output is bounded with unstable if
the output is unbounded.-Revisit
 The system is stable if with zero input and arbitrary initial conditions, if the resulting
state trajectories tend towards the origin and unstable, if the resulting trajectory tends to
infinity.
 Superposition principle does not apply.
 Has several equilibrium points.
 Free and forced behaviour of system can vary.
Equilibrium Points
Equilibrium point is a state of the system where it will stay forever. Also called stationary
points, fixed points , critical points.
ẋ= f(x)=0

37. Stability of Non Linear System
Stability of an Equilibrium Point
 Stability in a region in the immediate neighbourhood of the equilibrium point is called
the stability in the small.
 Stability in a larger region around the equilibrium point is called stability in the large.

38. Lyapunov Stability
 Aleksandr Mikhailvoich Lyapunov in his doctoral thesis on the general problem of the
stability of motion, 1892
Concept of Lyapunov Stability
 Stability of system is determined without solving differential equation.
 Based on the concept of energy.
Energy
Stable Systems

39. Lyapunov Stability

40. Lyapunov Stability

41. Lyapunov Stability

42. Lyapunov Stability

43. Lyapunov Function

44. Lyapunov Function

45. Lyapunov Function

46. Lyapunov Function

47. Lyapunov Function

48. Lyapunov Function

49. Lyapunov Function

50. Lyapunov Function

51. Lyapunov Function

52. Lyapunov Function

53. Steps to Solve Non Linear Problem
Consider nonlinear state space equation
 Select Lyapunov or energy function V(x)
 Check V(x) is positive definite or not
 Determine derivative of V(x)
 Check 𝑉(𝑥) is negative definite function or not
 Check stability area

54. Steps to Solve Non Linear Problem
 Syms - Create symbolic variables and functions - syms x y
 Assume- Set permanent assumption assume(condition)
 Jacobian- Jacobian matrix jacobian(f , v)
 lyap- Continuous Lyapunov equation solution lyap(A,Q)
 eig- Eigenvalues and eigenvectors eig(A)
 transpose- Transpose vector or matrix transpose(A)
 det- Matrix determinant det(A)
 disp - Display value of variable disp(‘X’)

55. Steps to Solve Non Linear Problem

56. Steps to Solve Non Linear Problem

57. Steps to Solve Non Linear Problem

58. Steps to Solve Non Linear Problem

59. Steps to Solve Non Linear Problem

60. MATLAB Solution

61. MATLAB Solution

62. MATLAB Solution

63. MATLAB Solution

64. MATLAB Solution

65. Exercise
1. Check the stability of the equilibrium state of the system
described by:
𝑋1 =𝑋2
𝑋2= -𝑋1 − 𝑋12𝑋2

69. Digital control System