NON-LINEAR SYSTEMS

SUBJECT CODE: EIPE204 M.Tech, Second Semester

Prepared By

Dr. Kanhu Charan Bhuyan Asst. Professor Instrumentation and Electronics Engineering

COLLEGE OF ENGINEERING AND TECHNOLOGY BHUBANESWAR

MODULE 1

Classification of Non-Linearities

Non-linearities can be classified into two types

1. Incidental Non-Linearities: These are the non-linearities which are inherently present in the system. Ex: Saturation Dead Zone, Coulomb Friction, Stiction and Backlash. 2. Intentional Non-Linearities: These are the non-linearities which are deliberately inserted in the system to modify the system characteristics. Ex: Relay

Saturation

O/P Appointment Ch

Actual Ch

I/P

 Output is proportional to the input in the limited range of input signals.  When input exceeds the range, the output tends to become nearly constants. Friction

푉푖푠푐표푢푠 퐹푟푖푐푡푖표푛 퐹표푟푐푒 = 푓xi Where 푓 is constant and 푥 is relative velocity. Viscous Friction is then linear in nature O/P

Force

Slope = f

Velocity

I/P

to the viscous force, there exists two non-linear friction a. Coulomb Friction b. Stiction  Coulomb Friction which is a constant retarding force (always opposing the relative motion).  Stiction Friction which is the force required to initiate motion. 푆푡푖푐푡푖표푛 퐹표푟푐푒 > 퐶표푢푙표푚푏 퐹표푟푐푒  In actual practice, stiction force decreases with velocity and changes over to Coulomb force at reasonably low force.

# In actual practice, Frictional Force will proportional to square and cube of the speed.

Relay (Intentional Non-Linearity)

 A relay is non-linear power amplifier which can provide large power amplification inexpensively and is deliberately introduced in control system.  A relay controlled system can be switched abruptly between several discrete states.  Relay-controlled system has wide applications in the control field.

Jump Phenomenon:

Non- exhibit phenomenon that cannot exist in linear system. The amplitude of variation can increase or decrease abruptly as the excitation frequency 휔 is increased or decreased. This is known as jump phenomenon.

Singular Point:

Time variant system is described by the state equation

푥. = 푓(푥, 푢) . 푥 = 푓(푥)

 A system represented by an equation of the above form called as autonomous system.  The points in the phase-space at which the of all the state variables are zero. Such points are called singular points.

Nodal Points:

Eigen Values (휆1, 휆2) are both real and negative.

푗휔

휆 /휆 = 푘 2 1 1

휆2 휆1 휎

 They all converge to origin which is then called as nodal point. This is stable nodal point.

 When 휆1 & 휆2 are both real and positive the result is an unstable nodal point.

Saddle Point

Both eigen values are real, equal and negative of each other. The origin in this case a saddle point which is always unstable, one eigenvalue being negative.

Focus Point:

휆1, 휆2= 휎 ± 푗휔 = 푐표푚푝푙푒푥 푐표푛푗푢푔푎푡푒 푝푎푖푟

 The origin in the focus point and is stable/unstable for negative/positive real parts of the eigen values.

 Transformation has been carried out for (푥1푥2) to (푦1 푦2) to present the trajectory inform of spiral.

Centre/Vertex Point:

X2 휆1 휆2= 푗휔 = 표푛 푡ℎ푒 푖푚푎푔푖푛푎푟푦 푎푥푖푠 Y2

Y1 X1

Stability of Non-Linear System: i. For Free System: A system is stable with zero input and arbitrary initial conditions if the resulting trajectory tends to the equilibrium state. ii. Force System: A system is stable if with bounded input, the system output is bounded.

There are types of stability condition in literature. Basically we concentrated on the following:

i. Stability, Asymptotic and Asymptotic in the large. 푥 = 푓(푥) Asymptotic in the large

Because 푥(푡0) = 0 푋(푡) remain near the origin for all ′푡′

ii A system is said to be locally stable (stable in the small) if the resign 훿(푡) is small.

 It is stable

 Every initial state 푋(푡0) results in 푋(푡) → 0 as 푡 → ∞  Asymptotically Stability in the large guarantees that every motion will approach the origin.

Limit Cycles:

System Stability has been defined in terms of distributed steady- state coming back to its equilibrium position or at least staying within tolerable limits from it.

 Distributed NL system while staying within tolerable limits may exhibit a special behaviour of closed trajectory/limit cycle.

The limit cycles describe the oscillations of non-linear system. The existence of a limit cycle corresponds to an oscillation of fixed amplitude and period.

Ex: Let us consider the well known Vander Pol’s

2 푑 푥 2 푑푥 2 − 휇(1 − 푥 ) + 푥 = 0 (1) 푑 푦 푑푡

It describes the many non-linearities by comparing with the following linear differential equation:

2 푑 푥 푑푥 2 + 2휁 + 푥 = 0 푑푡 푑푡

휇 2휉 = −휇(1 − 푥2) ⟹ 휉 = − ( ) (1 − 푥2) {Damping Factor} 2

a. |푥| ≫ 1, then damping factor has large positive means system behaves overdamped

system with decrease of the amplitude of 푥1. In this process the damping factor also decreases and system state finally reaches the limit cycle. b. |푥| ≪ 1, then damping factor is negative, hence the amplitude of x increases till the system state again enters the limit cycle.  On the other hand, if the paths in the neighbourhood of a limit cycle diverse away from it, it indicates that the limit cycle is unstable. Ex: Vander Pol Equation 2 푑 푥 2 푑푥 2 + 휇(1 − 푥 ) + 푥 = 0 푑 푦 푑푡  It is important to note that a limit cycle in general is an undesirable characteristics of a control system. It may be tolerated only if its amplitude is within specified limits.  Limit cycles of fixed amplitude and period can be sustained over a finite range of system parameter.

c M+ u 1

푠2 e - M

The differential equation describes the dynamics of the system which is given by 푐 = 푢 푟 − 푐 = 푒 −푐̇ = 푒̇ ⟹ −푐̈ = 푒̈ (as 푟 constant) Therefore 푒 = −푢

푇 ̇ 푇 Choosing the state vector [푥1 푥2] = [푒 푒] , then we have

푥̇1 = 푥2

푥̇2 = 푒̈ = −푢 The ouput of on/off controller is

푢 = 휇푠푖푔푛(푒) = 휇푠푖푔푛(푥1) The state equation is described as

푥̇1 = 푥2

푥̇2 = −휇푠푔푛(푥1) The of the equation is given by 푑푥 휇푠푖푔푛(푥 ) 2 = 1 푑푥1 푥2 Separating the variables and integrating, we get

푥2 푥1 ∫ 푥2푑푥 2 = −휇 ∫ 푠푖푔푛(푥1)푑푥 푥2(0) 푥1(0)

Carrying out integration and rearranging we get

2 2 푥2 = 2휇푥1(0) + 푥2 (0); 푥1 < 0

Region 퐼(푥1 = 푒 푝표푠푖푡푖푣푒, 푟푒푙푎푦 표푢푝푢푡 + 휇) 2 2 푥2 = 2휇푥1 − 2휇푥1(0) + 푥2 (0); 푥 < 0

Region II (푥1 = 푒 푛푒푎푔푡푖푣푒, 푟푒푙푎푦 표푢푡푝푢푡 − 휇) Construction by Graphical Methods:

1. Isocline Method 2. 휹 − Method

1. Isocline Method:  Trajectory slope at any point is given by

푑푥2 푓2(푥1,푥2) =푠 = (1) 푑푥1 푓1(푥1,푥2)  Slope at a specific point is expressed as

푓2(푥1, 푥2) = 푆1푓1(푥1, 푥2) (2)

 Given 푆1, this is the equation of an isocline where any trajectory crosses at slope 푆1.  If isoclines for different values of S are drawn throughout the phase plane trajectory starting at any point can be constructed by drawing short lines from one isocline to another at average slope of the two adjoining isoclines. 2. 휹 − 푴풆풕풉풐풅:  This method is used to construct a single trajectory for systems with describing differential equation in the form 푥̈ + 푓(푥, 푥̇, 푡) = 0 Converting this equation to the form 푥̈ + 푘2[푥 + 훿(푥, 푥̇, 푡)] = 0

Let 푘 = 푊푛, undamped natural frequency of the system when 훿 = 0 Then the equation is written as 2 푥̈ + 휔푛[푥 + 훿(푥, 푥̇, 푡)] = 0 2 푥̈ + 휔푛(푥 + 훿) = 0 Choose the state variables as 푥̇ 푥1̇ = 푥, 푥2 = 휔푛

Giving state equation

푥1̇ = 휔푛푥2

푥2 = −휔푛(푥1 + 훿) The trajectory slope is expressed as 푑푥 푥 + 훿 2 = − 1 푑푥1 푥2

푋 2

푃(푥1, 푥2)

푋2 Q

B C 훿 푋1 푋1

Any point 푃(푥1, 푥2), 훿 is computed and marked on negative side of 푥1 − 푎푥푖푠

푥  The of slope − 2 is at 900 to CP. The trajectory at P is then identified as a short (푥1+훿) arc at P with centre at ‘C’.  The process is repeated for another ‘P’. shortly away from P(original) on the arc. Then completed trajectory can be drawn.

Phase Plane Method

Unforced linear spring damp system whose dynamics is described by 푑2푥 푑푥 휇 = 2휉휔푛 + 휔2푥 = 0 푑푡2 푑푡 푛

휉 is the damping factor and 휔푛 =undamped natural frequency.

푑푥 Let the state of the system is described by two variables, displacement (x) and the velocity( ) 푑푡 푑푥 in the state variable notation 푥 = 푥, 푥 = , 1 2 푑푡

⟹ 푠푡푎푡푒 푣푎푟푖푎푏푙푒푠 = 푝ℎ푎푠푒 푣푎푟푖푎푏푙푒푠

푑푥 1 = 푥 푑푡 2

푑푥 1 = −휔2푥 − 2휉휔 푥 푑푡 푛 1 푛 2

푑푥 1 = 푥 푑푡 2

푑푥 푓 푘 푥 푘 푥3 2 = − 푥 − 1 1 − 2 1 (if the spring force is NL then 푘 푥 + 푘 푥3) 푑푡 휇 2 휇 휇 1 2

 The coordinate plane with axes that corresponds to the dependent variables 푥1 = 푥 and

its first 푥2 = 푥̈ is called phase-plane.

 The described by the state point (푥1, 푥2) in the phase plane with running parameter is called phase-trajectory.  A plane trajectory can be easily constructed by graphical or analytical technique.  Family of trajectories is called phase-portrait.

The phase plane method uses state model of the system which is obtained from the differential equation governing the system dynamics.

 This method is powerful generally restricted to systems described by 2nd order differential equation.

G1(S) N G2(S)

 푥 = 푋푠푖푛휔푡  input 푥 to the non-linearity is sinusoidal  Input, output of the non-linear will in general be non-sinusoidal periodic which maybe expressed in terms of Fourier series.

푦 = 퐴0 + 퐴1푠푖푛휔푡 + 퐵1푐표푠휔푡 + 퐴2푠푖푛2휔푡 + 퐵2푐표푠2휔푡  If non-Linearity is assumed to be symmetrical. The average value of Y is zero, so that the outputs 푦 is given by

푦 = 퐴1푠푖푛휔푡 + 퐵1푐표푠휔푡 + 퐴2푠푖푛2휔푡 + 퐵2푐표푠2휔푡 + ⋯  The harmonic of 푦 can be thrown away for the purpose of analysis and the fundamental components of 푦 i.e.

푦1 = 퐴1푠푖푛휔푡 + 퐵1푐표푠휔푡

= 푌1 sin(휔푡 + 휙)  This type of linearization is valid for large signals as well so long as the harmonic filtering condition is satisfied.

 The NL can be replaced by a describing function 퐾푁(푥, 휔) which is defined to be the complex function embodying amplification and phase shift of the fundamental frequency component of 푦 relative to 푥 i.e. 푌 퐾 (푥, 휔) = ( 1) < 휙 푁 푋 When the input to the non-linearity is 푥 = 푥푠푖푛휔푡

y e x G1(푗휔) 퐾푛(푥, 휔) G2(푗휔)

Derivation of Describing Function The describing function of a non-linear element is given by 푌 퐾 (푥, 휔) = ( 1) < 휙 푁 푋

X- amplitude of the sinusoid

푌1= amplitude of the fundamental harmonic component of the output.

휙 = phase shift of the fundamental harmonic component of the output wrt the input.

 To compute the describing function of a non-linear element, simply find the fundamental harmonic component of its output for an input 푥 = 푥푠푖푛휔푡 The fundamental component of the o/p can be written as

푦1 = 퐴1푠푖푛휔푡 + 퐵1푐표푠휔푡

Where 퐴1 & 퐵1 are the coefficients of fourier series. 1 2휋 퐴1 = ∫ 푦푠푖푛휔푡 ∗ 푑(휔푡) 휋 0 1 2휋 퐵 = ∫ 푦푐표푠휔푡 ∗ 푑(휔푡) 1 휋 0 The amplitude and phase angle of the fundamental component of the ouput are given

2 2 −1 퐵1 by 푌1 = √퐴1 + 퐵1 , 휙 = tan ( ) 퐴1

Pell’s method of phase trajectory

The 2nd order non- is considered to be phase trajectory by using Pell’s Method. General form

푑2푥 푑푥 + 휙 ( ) + 푓(푥) = 0 (1) 푑푡2 푑푡

푑푥 푑푥 Now defining 푥 = = 1 2 푑푡 푑푡

The above equation can be re-arranged after dividing by 푥2 to get

푑2푥 ( ) 푑푡2 푥 −휙(푥 )−푓(푥 ) = 2 = 2 1 (2) 푥2 푥1 푥1

 −휙(푥2) with 푥2 and −푓(푥1) with 푥1

For given initial condition 푃[푥1(0), 푥2(0)], the construction of a segment of the trajectory are shown.

휑(푥2) P[푥1(0), 푥2(0)] B

휑

푋 2 휇

O D A

−푓푥1

푋1

1) Given the initial condition 푃[푥1(0), 푥2(0)] as shown in above figure. Draw a

perpendicular from 푃 on the 푥1 푎푥푖푠 to get 푂휇 = 푥1(0) and extend the line 푃푀 to meet

the curve −푓푥1 at N. such that 휇푁 = 푓{푥1(0)}. Now locate the point A on the x1 axis such that 휇퐴 = 휇푁 2) Similarly, for the given 푃[푥1(0), 푥2(0)], draw PB perpendicular on the 푥2 푎푥푖푠 so that

OB=휇P 푥2(0). Now extend PB to meet the – 휑푥2curve at C such that BC = 휑{푥2(0)}.

Extend the line segment 휇퐴 on the x1 axis point D such that AD=BC. Therefore the

segment of the 푥1 axis is between D and M equals to the absolute values of the function

of x1 and x2 . that is DM = DA+A 휇= {푥2(0)} + 푓{푥1(0)}

푃푀 푥2 3) Join the points D and P. the slope of the line DP denoted by 푚1 = = 퐷푀 휑(푥2)+푓(푥1)

4) If the slope of the trajectory in equation (2) is denoted by 푚2, then 푚1푚2 = −1. The solution of equation (2) will move on a segment of a perpendicular to the line DP to get

the next point 휑(푥1(푓1), 푥2(푓2)) as shown in the above figure. The process is repeated with 휑 as the new initial condition to get the complete trajectory.

Advantages of this method

a) The construction is very simple. b) It eliminates the trial and error approach of the delta method.

Examples of Pell’s Method

푑2푥 푑푥 + 푐 + 휔2(1 + 푏2푥2)푥 = 0 푑푡2 1 푑푡 1

Draw the phase plane trajectory using Pell’s method

2 2 Let 푐1 = 125 휔1 = 25000 푏 = 0.60

We introduce time scaling 휏 = 휔1푡 so that

푑휏 푑휏 = 휔 푑푡 ⟹ 휔 = 1 1 푑푡

푑휏 ⇒ 푑푡 = 휔1

푑2푥 푑푥 + 푐 + 휔2 (1 + 푏2푥2) 푥 = 0 푑휏2 1 푑휏 1 2 휔 휔1 1

푑푥2 푑푥 휔2 + 푐 휔 + 휔2(1 + 푏2푥2)푥 = 0 1 푑휏2 1 1 푑휏 1

2 푑 푥 푐1 푑푥 2 2 2 + + (1 + 푏 푥 )푥 = 0 푑휏 휔1 푑휏 MODULE-2

Stability Study of Non-Linear System

r u G(s) y

휓(. )

 The process for representing a system is in the form of having a feedback connection of a linear dynamics system and non-linear element.  The control system non-linearity is in form of relay or actuator or sensor non-linearity 푢 = −휓(. ) + 푟  Here we assume that the external input 푟 is equal to 0 which indicates the behaviour of an unforced system.  The system is said to be absolutely stable if it has a globally uniformly asymptotically stable equilibrium points are the origin.  For all non-linearities in a given circle, without input, initial condition and asymptotically stable.  For this the Circle and Popov criteria give frequency domain absolutely stability in the form of silent positive realness of certain transfer function.  In the SISO case both criterion can be applied graphically. We assume the external input 푟 = 0 and the behaviour of unforced system can be represented by 푋̇ = 퐴푥 + 퐵푢 푌 = 퐶푋 + 퐷푢 푢 = 휓(. ) Where 푋 ∈ 푅∧ 푢, 푦 ∈

(퐴, 퐵) − 퐶표푛푡푟표푙푙푎푏푙푒

(퐴, 퐶) − 푂푏푠푒푟푣푎푏푙푒

휓(0, ∞) × 푅푃 is a memory less possibly time varying non-linearity and probability continuous in time 푡.

Definition-1

A memory function ℎ(0, ∞)

R is said to belong to the sector

ℎ(0, ∞) → if 푢. ℎ[푡, 푢] ≥ 0

ℎ(∝, ∞) → if 푢[ℎ(푡, 푢) − 훽푢] ≤ 0

ℎ(0, 훽) → 푤푖푡ℎ 훽 > 0 푖푓 [ℎ(푡, 푢) − 훽푢] ≤ 0

Y

푦 = 훽푢

푦 = 훼푢

Y

푦 = 훽푢

푦 = 훼푢 Definition 2

In closed loop system is called absolute lead stable in sector between (0, 훽) if the origin is globally uniformly asymptotically stable. For any non-linearity in the given sector it is absolutely stable in a finite domain if the origin is unifromly asymptoticall stable.

Popov Criterion (Time Invariant Feedback)

Popov criteria is applicable if the following condition are satisfied.

 Time invariant non-linearity 휓: 푅 → 푤ℎ푒푟푒 푅 푠푎푡푖푠푓푖푒푠 푡ℎ푒 푠푒푐푡표푟 푐표푛푑푖푡푖표푛.  ∧ 푤ℎ푒푟푒 푅 푠푎푡푖푠푓푖푒푠 휓(0) = ∞ 푃(푠)  퐺(푆) = where degree {P(s)}< 푑푒푔푟푒푒푠 {푞(푠)} 푠∧푞(푛)  The poles of G(s) are in LHP or on imaginary axis.  The system is marginally stable in singular case.

Theorem-1

The closed loop system is absolutely if 휓 ∈ [0, 푘]

0 < 푘 < ∞

And there exists a constant ‘q’ such that the following condition is satisfied

1 푅푒[퐺(푗휔)]. 푗푞푛 퐼푚[퐺(푗휔)] > − 푘

Where 휔 ∈ [−∞, ∞]

Graphical Representation:

Popoc plot 푃(푗휔) [푍. 푅푒[퐺(푗휔)] + 푗휔퐼푚[퐺(푗휔)] 푤ℎ푒푟푒 휔 > 0

The closed loop system is absolutely stable if "푃" lies to the right of the line that interrupts

1 1 푆 (− + 푗0) will slope 푘 푞 푠푙표푝푒 1/푞 휔 × 퐼푚[퐺(푗휔)]

−1/푘

푅푒[퐺(푗휔)] As per popov criterion the closed loop system in absolute stable if 휓 ∈ [0, 푘], 0 < 푘 < ∞ and their exists a constant 푞 such that the following equation is satisfied.

1 푅푒[퐺(푗휔) − 푗푞휔퐼푚[퐺(푗휔)] > − 푘

푤ℎ푒푟푒 휔 ∈ [−∞, ∞]

Circle Criterion: Applicable to non-linear time variant system which gives information absolutely stability. Definition we define 퐷(훼, 훽) to be closed disk in the complex plane whose diameter is the line segment 1 1 connected the points (− + 푗0) and (− + 푗0) 훼 훽

퐷(훼, 훽)

1 − + 푗0 훽

1 − + 푗0 훼

Definition 2 Consider a scalar system

푋̇ = 퐴푥 + 퐵푢 푌 = 퐶푥 + 퐷푢 푢 = −휓(푦, 푡) → non-linear time invariant Where 퐺(푠) = 푓(퐴, 퐵, 퐶, 퐷) 푎푛푑 휓 ∈ [훼, 훽] The system is absolutely stable if one of the three following conditions :

1) If 표 < 훼 < 훽 , the nyquist plot 퐺(푗휔) doesn’t enter 퐷[훼, 훽] and encircles it in counter clockwise directions where 휔 is no of poles of 퐺(푠) with the real points. 2) If (훼 = 0) < 훽, 퐺(푠) is therefore Hurwitz and Nyquist Plot 퐺(푗휔) lies to right of the 1 vertical line defined by 푅푒(푠) = − 푝 3) If 훼 < 0 < 훽 is Hurwitz and Nyquist plot 퐺(푗휔) the interior part of 퐷[훼, 훽]

Describing Function 푥 = 푥푠푖푛휔푡

푦 = 푎0 + 푎1푠푖푛휔푡 + 푏1푐표푠휔푡 + 푎2푠푖푛2휔푡 + 푏2푐표푠2휔푡 + ⋯ + ⋯ + 푎푛 sin(푛휔푡) + 푏푛 cos(푛휔푡) ∞

= 푎0 + ∑ 푌푛 sin(푛휔푡 + 휙푛) 푛=1

= 푎0 + 푌1 sin(휔푡 + 휙1) + 푌2 sin(2휔푡 + 휙2) + ⋯ 1 2휋 1 2휋 Where 푎 = ∫ 푓(휃)푑휃 = ∫ 푦 푑휃 0 2휋 0 2휋 0 1 2휋 1 2휋 푎푛 = ∫ 푓(휃) cos(푛휃) 푑휃 = ∫ 푦푐표푠(푛휃) 푑휃 휋 0 휋 0 1 2휋 1 2휋 푏푛 = ∫ 푓(휃)sin (푛휃)푑휃 = ∫ 푦푠푖푛(푛휃) 푑휃 휋 0 휋 0

2 2 푌푛 = √푎푛 + 푏푛

−1 푎푛 휙푛 = tan ( ) 푏푛

y

-m y

x

x

Linear Characteristics -mNon -Linear Characteristics 2 2 푌1 = √푎1 + 푏1

−1 푎1 휙1 = tan ( ) 푏1 푌 Describing Funtion (DF) = 1 < 휙 = 퐾 (푋, 휔) 푋 1 푁

C(S)

G1(S) N G2(S R(S) )

N- Non-Linear Element

N must be linearized by the help of DF represented as 퐾푁(푋, 휔) 푌 Where 퐾 (푋, 휔) = 1 < 휙 푁 푋 1

C(S)

퐾 (푋, 휔) G1(S) 푁 G2(S R(S) )

Non-linear Element

Incidental Intentional

Ex: Saturation, Dead Ex: Ideal Relay, Ideal

Zone, Saturation & relay with dead Dead Zone zone, Ideal relay with hysteresis. Intentional Non-Linearity a) Ideal Relay +M

Y

X

-M

b) Ideal Relay with Deadzone

Y

-X/2

X/2 X

c) Ideal relay with hysteresis

+M

Y

-x/2 x/2 X

-M

Ideal Relay

Y

X

Y +m

0

휋 0 2휋

휃

휋 푌1 퐾 (푋, 휔) = < 휙 푁 푋 1

2휋

Y Ideal Relay (LVDT)

+m

0 휋 2휋 휃

+m

0

X -m

0

X

X

휋

2휋

Here 푥(푡) = 푥푠푖푛휔푡 푦 = 푀, 0 ≤ 휃 ≤ 휋 = −푀, 휋 ≤ 휃 ≤ 2휋

푌 DF= 퐾 (푥, 휔) = 1 < 휙 (1) 푁 푥

2 2 −1 푎1 푌1 = √푎1 + 푏1 & 휙 = tan ( ) 푏1 1 2휋 1 휋 2휋 푎0 = ∫ 푌 푑휃 = [∫ 푀 푑휃 + ∫ (−푀) 푑휃] 2휋 0 2휋 0 0 1 = 푀[(휃)휋 − (휃)휋] 2휋 0 0 1 = . 푀[휋 − 0 − 2휋 + 휋] = 0 2휋 1 2휋 1 휋 2휋 푎1 = ∫ 푌푐표푠휃 푑휃 = [∫ 푀푐표푠휃 푑휃 + ∫ (−푀)푐표푠휃 푑휃] 휋 0 휋 0 0 1 = 푀[[푠푖푛휃휋] − [푠푖푛휃2휋]] 휋 0 0 푀 = [푠푖푛휋 − 푠푖푛0 − 푠푖푛2휋 + 푠푖푛휋] = 0 휋 1 2휋 1 휋 2휋 푏1 = ∫ 푌푠푖푛휃 푑휃 = [∫ 푀푠푖푛휃 푑휃 + ∫ (−푀)푠푖푛휃 푑휃] 휋 0 휋 0 0 푀 = [[−푐표푠휃휋] − [−푐표푠휃2휋]] = 0 휋 0 0 푀 4푀 = [−푐표푠휋 + 푐표푠휋 + 푐표푠2휋 − 푐표푠휋] = 휋 휋 푎 0 휙 = tan−1 ( 1) = tan−1 ( ) = 0 푏1 4푚/휋

4푀 2 4푀 푌 = √0 + ( ) = 1 휋 휋

푌 4푀 DF = 퐾 (푥, 휔) = 1 < 휙 = < 0 푁 푥 휋푥 Relay with Hysteresis (Schmitt-trigger)

+m

휃 휋 + 휃1 2휋 + 휃 1 1

-m

Y

X

-M

0

휃1

X1

휋 + 휃1

2휋 + 휃 1

Here 푥 = 푥푠푖푛휃 ℎ 휃 = sin−1 ( ) 1 2휋

푌 = 푀, 휃1 ≤ 휃 ≤ 휋 + 휃1

= −푀, 휋 + 휃1 ≤ 휃 ≤ 2휋 + 휃1 푌 DF = 1 < 휙 푋 1 2휋 1 휋+휃1 2휋+휃1 푎 = ∫ 푌 푑휃 = [∫ 푀 푑휃 − ∫ 푀 푑휃] 0 2휋 2휋 0 휃1 휋+휃1

푀 2휋+휃1 = [휃 | 휋+휃1 − (휃)| ] 휃1 2휋 휋+휃1 푀 = [휋 + 휃 − 휃 − 2휋 − 휃 + 휋 + 휃 ] = 0 2휋 1 1 1 1 1 2휋 1 휋+휃1 2휋+휃1 푎 = ∫ 푌푐표푠휃 푑휃 = [∫ 푀푐표푠휃 푑휃 + ∫ (−푀푐표푠휃 푑휃) ] 1 휋 휋 0 휃1 휋+휃1

푀 휋+휃1 2휋+휃1 = [∫ 푐표푠휃 푑휃 − ∫ (푐표푠휃 푑휃) ] 휋 휃1 휋+휃1

푀 2휋+휃1 = [푠푖푛휃 | 휋+휃1 − (푠푖푛휃)| ] 휃1 휋 휋+휃1 푀 = [sin (휋 + 휃 ) − 푠푖푛휃 − sin (2휋 − 휃 ) + sin (휋 + 휃 )] = 0 휋 1 1 1 1 푀 4푀sin휃 4푀 ℎ = [−sin휃 − 푠푖푛휃 − sin 휃 − sin휃 )] = 1 = − ( ) 휋 1 1 1 1 휋 휋 2푥 1 2휋 푏1 = ∫ 푌푐표푠휃 푑휃 휋 0

1 휋+휃1 2휋+휃1 = [∫ 푀푠푖푛휃 푑휃 + ∫ (−푀푠푖푛휃 푑휃) ] 휋 휃1 휋+휃1

푀 2휋+휃1 = [(−푐표푠휃) | 휋+휃1 + 푐표푠휃| ] 휃1 휋 휋+휃1 푀 = [− cos(휋 + 휃 ) + 푐표푠휃 + 푐표 푠(2휋 + 휃 ) − cos (휋 + 휃 )] = 0 휋 1 1 1 1 1/2 푀 4푀cos휃 4푀 ℎ 2 = [cos휃 + 푐표푠휃 + cos 휃 + cos휃 )] = 1 = [1 − ( ) ] 휋 1 1 1 1 휋 휋 2푥

1/2 4푀 ℎ 2 4푀 ℎ 2 푌 = √푎2 + 푏2 = √( ) − ( (1 − ) ) 1 1 1 휋 2푥 휋 2푥

4푀 ℎ 2 ℎ 2 4푀 = √( ) + 1 − ( ) = 휋 2푥 2푥 휋

4푀 푠푖푛휃 −1 푎1 −1 휋 1 −1 휙 = tan ( ) = tan [4푀 ] = tan tan 휃1 푏1 cos휃 휋 1 ℎ 4푀 ℎ 휙 = sin−1 ( ) 푎푛푑 퐷퐹 = 퐾 (푥, 휔) = < sin−1 ( ) 2푥 푁 휋 2푥 MODULE-III

Liapunov Function for Non-Linear System Krasovskii’s Method: Consider the system 푥̇ = 푓(푥); 푓(0) = 0 (1) [푠푖푛푔푢푙푎푟 푝표푖푛푡 푎푡 표푟푖푔푖푛]

Define a Liapunov Function as 푣 = 푓푇푃푓 (2) Where P= Symmetric +ve definite matrix

Now 푣̇ = 푓푡̇ 푝푓 + 푓푡푝푓 (3) 휕푓 휕푥 (푓̇ = ) = 퐽푓 휕푥 휕푡 휕푓 휕푓 휕푓 1 2 … … … … … … … 푛 휕푥1 휕푥2 휕푥푛 휕푓 휕푓 휕푓 2 2 … … … … … … … 2 휕푥 휕푥 휕푥푛 퐽 = 1 2

휕푓 휕푓 휕푓 푛 푛 … … … … … … 푛 [ 휕푥1 휕푥2 휕푥푛 ] Replace 푓̇ in equation (3) we have 푉 = (퐽푓)푇푃푓 + 푓푡푃퐽푓 = 푓푇퐽푇푃퐹 + 푓푡푃퐽푓 = 푓푇(퐽푇푃 + 푃퐽)푓 푄 = 퐽푇푃 + 푃퐽 Since V is +ve definite for the system to be asymptotically stable, Q should be positive definite. If V(x)→ ∞ 푎푠 ‖푥‖ → ∞ then the system is asymptotically stable in large. Variable Method For the autonomous system

푥̇ = 푓(푥); 푓(0) = 0 (1) Let V(x) be considered for a LF The time derivative of V can be expressed as

휕푣 휕푣 휕푣 푉̇ (푥) = 푥̇1 + 푥̇2 + ⋯ … … … … … … . + 푥̇푛 (2) 휕푥1 휕푥2 휕푥푛 Which an be expressed in terms of the gradient of V As 푉 = (∇푉)푇푋̇ (3) 휕푣 = ∇푉1 휕푥1 휕푣 = ∇푉2 휕푥2 . ∇푉 = : : : : 휕푣 = ∇푉푛 [휕푥푛 ] The LF can be generated by integrating w.r.t. time both sides equation (3) 푥 휕푣 푥 푉 = ∫ 푑푡 = ∫ (∇푣)푇푑푥 (4) 0 휕푡 0  The above integral is a line integral whose result is independent of the path.

 The integral can be evaluated sequentially along the component directions (x1, x2,

……..xn) of the state vector.

푥 That is 푉 = (∇푣)푇푑푥 ∫0

푥1 푥2 푉 = ∫ ∇푉1(푥1, 0,0,0 … … .0)푑푥1 + ∫ ∇푉2(푥1, 푥2, 0,0,0 … … .0)푑푥2 0 0

푥3 + ∫ ∇푉3(푥1, 푥2, 푥3, 0,0,0 … … .0)푑푥3 + ⋯ … … … 0

푥푛 + ∫ ∇푉푛(푥1, 푥2, … … . 푥푛)푑푥푛 (5) 0

Let us define

1 0 0 0 1 0

푒1 = ⋮ 푒2 = ⋮ … … … … … … … . 푒푛 = ⋮ 0 0 0 [⋮] [⋮] [1]

The integral given in equation (5) states that the path starts from the origin and moves along the vector 푒1 to 푥1. From this point the path moves in the direction of the vector 푒2 to 푥2. In this way the path finally reaches the point (푥1, 푥2, … … . 푥푛)  From the function V to be unique thecurl of its gradient must be zero ∇ × (∇푉) = 0 (6) 푛  This results in (푛 − 1) equation to be satisfies by the components of the 2 gradient 휕∇푉 휕∇푉푗 ( 푖 = ) 푓표푟 푎푙푙 푖, 푗 (7) 휕푥푗 휕푥푖 To begin with a completely general form given below is assumed for the gradient vector ∇푉

푎11푥1 + 푎12푥2 + ⋯ … … … … . . 푎1푛푥푛

푎21푥1 + 푎22푥2 + ⋯ … … … … . . 푎2푛푥푛 : : ∇푉1 ∇푉 : 2 ∇푉 = ⋮ = : ⋮ : [∇푉푛] : : 푎푛1푥1 + 푎푛2푥2 + ⋯ … … … … . . 푎푛푛푥푛

[ ] ′ 푎푖푗푠 푎푟푒 푐표푚푝푙푒푡푒푙푦 푢푛푑푒푡푒푟푚푖푛푒푑 quantity and could be constant of fucntions of both state variable and t. it is convenient to choose as a constant Advantages: Easy to apply Disadvantages:  Only asymptotic stability can be investigated.  F(x); continuously differentiable  Domain of attraction is unknown

The determination of Liapunov stability via Liapunov direct method centres around the choice of positive definite function v(x) called the Liapunov function. There is no universal method for selecting Liapunov function whch is unique for a special problem. Theorem-1

푥̇ = 푓(푥); 푓(0) = 0 Real scalar function v(x) satisfies the following properties for all 푥 on the region ‖푥‖ < 푡 a) 푉(푥) > 0; 푥 ≠ 표 b) 푉(0) > 0 c) 푉(푥) has continuous partial derivatives with respect to all components x. 푑푣 푑푣 d) < 0 (푖. 푒 푖푠 − 푣푒 푠푒푚푖 푑푒푓푖푛푖푡푒 푠푐푎푙푎푟 푓푢푛푐푡푖표푛) 푑푡 푑푡

Then the system is stable.

Theorem-2 a) 푉(푥) > 0; b) 푉(0) > 0 c) 푉(푥) has continuous partial derivatives with respect to all components x.

푑푣 푑푣 d) < 0 (푖. 푒 푖푠 − 푣푒 푠푒푚푖 푑푒푓푖푛푖푡푒 푠푐푎푙푎푟 푓푢푛푐푡푖표푛) 푑푡 푑푡

푋̇ = 푓(푥); 푓(0) = 0

There exists a real scalar function 푤(푥)

1) 푤(푥) > 0 ; 푥 ≠ 0 2) w(0)= 0 3) 푤(푥) has continuous partial derivatives with respect to all components x. 푑푤 4) ≥ 0 푑푡

Then the system is unstable at the origin

Analysis of Stability by Liapunov Direct Method

 This method is useful for determining the stability. It is also applicable for linear systems.  In this method it can be determined without actually solving the differential equation. So it is called as direct method.

Concept of Definiteness

Let V(x) is a real scalar function. The scalar function v(s) is said to be positive definite if the function V(x) has always +ve sign in the given region about the origin except only at origin where it is zero

 The scalar function V(x) is the +ve in the given region if V(x)>0 for all non-zero states x in the region and v(0) = 0  The scalar function V(x) is said to be negative definite if the function V(x) has always –ve sign in the given region about the origin, except only at the origin where it is zero or a scalar function.

Liapunov stability Theorem

푋̇ = 푓(푥);

If there exist a scalar function V(x) which is real, continuous and has continuous first partial derivatives with

a) 푉(푥) > 0; 푥 ≠ 0 b) 푉(0) > 0 c) 푉(푥) < 0 for all 푥 ≠ 0

Then the system is asymptotically stable. V(x) is the Liapunov Function.

Ex: The system is given by

푥̇1 = 푥2

3 푥2 = −푥1 − 푥2

Investigate the system by Liapunov method using

2 2 푉 = 푥1 + 푥2

Solution:

휕푣 휕푣 푉̇ (푥) = 푥̇1 + 푥̇2 휕푥1 휕푥2

3 푉̇ (푥) = 2푥1 푥̇1 + 2푥2 푥̇2 = 2푥1푥2 + 2푥2(−푥1 − 푥2 )

4 = −2푥2 (-ve)

푉̇ (푥) < 0 푓표푟 푎푙푙 푛표푛 − 푧푒푟표푒푠 푖. 푒. −푣푒 푑푒푓푖푛푖푡푒

So asymptotically stable.

 A system is describes by 푥̇ = 푓(푥) If there exists a scalar function V(x) which is real, continuous and has continuous first partial derivatives with

a) 푉(푥) < 0; 푥 ≠ 0 b) 푉(0) = 0 c) 푉(푥) > 0 for all 푥 ≠ 0

Then system is unstable.

 The direct method of Liapunov and Linear System:

Let the system is given by

푥̇ = 퐴푥 (1)

Select the Liapunov function as

푉(푥) = 푥푇푃푥

푉̇ (푥) = 푥̇ 푇푃푥 + 푥푇푃푥̇

Substitute the value of 푥̇ from equation (1)

푉̇ (푥) = (퐴푥)푇푃푥 + 푥푇푃퐴푥

= 푥푇퐴푇푃푥 + 푥푇푃퐴푥

= 푥푇[퐴푇푃 + 푃퐴]푥

⇒ 푉̇ (푥) = 푥푇푄푥 where 푄 = 퐴푇푃 + 푃퐴 (2)

Q is +ve definite matrix

Select a +ve definite Q and find out P from equation (2). If P is +ve definite then the system will be stable

Procedure:

Step-1 Select Q as positive definite.

푛(푛+1) Step-2 Obtain P from the equation (2) from this we will have to solve number of 2 equation where n is the order of matrix A. Step-3 Using Sylvester’s theorem determine the definiteness of P. if P is +ve definite the system is stable otherwise unstable. Generally, Q can be taken as Identity Matrix.

Sylvester’s Theorem:

This theorem states that the necessary and sufficient condition that the quadratic form V(x) be +ve definite are that all the successive principal minor of P be +ve i.e

푃11 > 0

푃 푃 푃 … … … … … . | 11 12 | > 0 [ 11 ] > 0 푃21 푃22 푃1푁………………………..

A) V(x) is –ve definite if –V(x) is +ve definite. B) V(x) = 푥푇푃푥 is +ve semi definite if P is singular and all the principal minors are non- negative.

Ex: Determine the stability of the system describe by the following equation

−1 −2 푥̇ = 퐴푥 퐴 = [ ] −1 −4

퐴푇푃 + 푃퐴 = −푄 (푄 = 퐼 )

−1 1 푃 푃 푃 푃 −1 −2 −1 0 [ ] [ 11 12] + [ 11 12] [ ] = −1 = [ ] −2 −4 푃21 푃22 푃21 푃22 1 4 0 −1

We have taken 푃12 = 푃21

Because solution matrix P is known to be +ve definite real symmetric for stable system.

−푃 + 푃 −푃 + 푃 −푃 + 푃 −2푃 + 4푃 −1 0 [ 11 21 12 22 ] + [ 11 12 11 12] = [ ] −2푃11 − 4푃21 2푃12 − 4푃22 푃21 + 푃22 −2푃21 + 4푃22 0 −1

−2푃11 + 2푃12 = −1

−4푃21 − 8푃22 = −1

−2푃11 − 5푃12 + 푃22 = 0

Solving for 푃, 푤푒 푔푒푡

23/60 −7/60 푃 푃 푃 = [ 11 12] = [ 7 11 ] 푃21 푃22 − 60 60

The necessary and sufficient conditions for a matrix

푞11 푞12 푞13 … … … … … … … . 푞1푛

푞21 푞22 푞23 … … … … … … … … 푞2푛 . .

푄 = . . . . [푞21 푞22 푞23 … … … … … … … … 푞2푛 ]

To be +ve definite are that all the successive principal minors of Q be +ve i.e

푞11 푞12 푞13 푞11 푞12 푞11 > 0, [ ] > 0, [푞21 푞22 푞23] > 0 푞21 푞22 푞31 푞32 푞1

Def[푄] > 0

 The matrix Q is semi definite if any of the above determinants is zero.  The matrix Q is negative definite (semi-definite) if the matrix –Q is positive definite (Semi-definite)  If Q is positive definite so 푄2 푎푛푑 푄−1. It should be noted that the definiteness of quadratic form scalar function is global.