Phase plane analysis (nonlinear stability analysis)

PHASE PLANE ANALYSIS

BINDUTESH V SANER

May 5, 2015

BINDUTESH V SANER PHASE PLANE ANALYSIS

CONTENTS

1. INTRODUCTION

2. BEHAVIOUR OF NON-LINEAR SYSTEM

3. METHOD OF ANALYSIS

4. CLASSIFICATION OF NON-LINEARITIES

5. CONCEPT OF PHASE PLANE ANALYSIS

◮ Phase portraits◮ Singular point◮ Phase portraits construction◮ Phase Plane Analysis of Linear Systems◮ Phase Plane Analysis of Non-Linear Systems

6. ADVANTAGE and DIS-ADVANTAGE

7. CONCLUSION

8. REFERENCE


INTRODUCTION

◮ The most important feature of Nonlinear systems is thatNonlinear systems do not obey the Principle ofSuperposition.

◮ Due to this reason, in contrast to the linear case, the responseof nonlinear systems to a particular test signal is no guide totheir behavior to other inputs.

◮ Phase plane analysis is a graphical method for studyingsecond-order systems.

◮ The nonlinear system response may be highly sensitive toinput amplitude.For example, a nonlinear system giving best response for acertain step input may exhibit highly unsatisfactory behaviorwhen the input amplitude is changed.


BEHAVIOR OF NON LINEAR SYSTEM

◮ The nonlinear systems may exhibit limit cycles which areself-sustained oscillations of fixed frequency and amplitude.

◮ Once the system trajectories converge to a limit cycle, it willcontinue to remain in the closed trajectory in the state spaceidentified as limit cycles.

◮ In many systems the limit cycles are undesirable particularlywhen the amplitude is not small and result in some unwantedphenomena.


METHOD OF ANALYSIS

◮ Nonlinear systems are difficult to analyze and arriving atgeneral conclusions are tedious.

◮ However, starting with the classical techniques for the solutionof standard nonlinear differential equations, several techniqueshave been evolved which suit different types of analysis.

◮ It should be emphasized that very often the conclusionsarrived at will be useful for the system under specifiedconditions and do not always lead to generalizations.


TYPES OF METHODS FOR ANALYSIS

1. Linearization Techniques

2. Describing Function Analysis

3. Liapunovs Method for Stability

4. Phase Plane Analysis


CLASSIFICATION OF NON-LINEARITIES

1. Inherent Non-linearities:The nonlinearities which are presentin the components used in system due to the inherentimperfections or properties of the system are known asinherent nonlinearities. for eg. Satuation in

magnetic,Deadzone, Backlash in gear etc

2. Intentional Non-linearities:In some cases introduction ofnonlinearity may improve the performance of the system,make the system more economical consuming less space andmore reliable than the linear system designed to achieve thesame objective. Such nonlinearities introduced intentionally toimprove the system performance are known as intentionalnonlinearities.


CONCEPT OF PHASE PLANE ANALYSIS

◮ phase portraits:The phase plane method is concerned withthe graphical study of second-order autonomous systemsdescribed by

x1 = f1(x1, x2) (1)

x2 = f2(x1, x2) (2)

◮ wherex1, x2 : states of the system

f1, f2 nonlinear functions of the states

◮ Geometrically, the state space of this system is a plane havingx1, x2 as coordinates.This plane is called Phase plane.


◮ The solution of (1) with time varies from zero to infinity canbe represented as a curve in the phase plane. Such a curve iscalled a Phase plane trajectory.

◮ A family of phase plane trajectories is called a phase portrait

of a system.◮ consider a example of Phase portrait of a mass-spring

system.

Figure: Mass-spring system and its portrait


Contd....

◮ The nature of the system response corresponding to variousinitial conditions is directly displayed on the phase plane.

◮ In the above example, we can easily see that the systemtrajectories neither converge to the origin nor diverge toinfinity. Indicating the marginal nature of the systems stability.

◮ A major class of second-order systems can be described bythe differential equations of the form

x = f (x , x)

◮ In state space form, this dynamics can be represented withx1 = x and x2 = x asfollows :

x1 = x2 (3)

x2 = f (x1, x2) (4)


SINGULAR POINTS

◮ A singular point is an equilibrium point in the phase plane.Since an equilibrium point is defined as a point where thesystem states can stay forever, this implies that x = 0.

◮ For a linear system, there is usually only one singular pointalthough in some cases there can be a set of singular points.


◮ The general form of a linear second-order system is

x1 = ax1 + bx2 (5)

x2 = cx1 + dx2 (6)

◮ After solving the above equation we can have a characteristicequation which can be further solved to have the roots λ1, λ2

can be explicitly represented as

λ1 =−a+

√a24b

2(7)

λ2 =−a −

√a2 − 4b

2(8)


◮ For linear systems there is only one singular point namely theorigin.However, the trajectories in the vicinity of this singularitypoint can display quite different characteristics, depending onthe values of a and b.The following cases may occur:

1. λ1, λ2 are both real and have the same sign(+ or -).2. λ1, λ2 are both real and have opposite sign.3. λ2 are complex conjugates with non-zero real parts.4. λ1, λ2 are complex conjugates with real parts equal to 0.

◮ Lets briefly discuss above cases.


STABLE AND UNSTABLE NODE

◮ The first case corresponds to a node. A node can be stableor unstable:1. λ1, λ2 < 0: singularity point is called stable node.2. λ1, λ2 > 0: singularity point is called unstable node.


SADDLE POINT

◮ The second case (λ1 < 0 < λ2 ) corresponds to a saddlepoint. Because of the unstable pole λ2, almost all of thesystem trajectories diverge to infinity.

Figure:


STABLE OR UNSTABLE LOCUS

◮ The third case corresponds to a focus.1. Re (λ1, λ2) < 0: stable focus2. Re (λ1, λ2) > 0: unstable focus


CENTRE POINT

◮ The last case corresponds to a certain point. All trajectoriesare ellipses and the singularity point is the centre of theseellipses.

Figure:

◮ 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.


PHASE PLANE ANALYSIS OF NON-LINEAR SYSTEM

◮ phase plane analysis of nonlinear system,has two importantpoints as follow:

1. Phase plane analysis of nonlinear systems is related to that ofliner systems, because the local behavior of nonlinear systemscan be approximated by the behavior of a linear system.

2. Nonlinear systems can display much more complicated patternsin the phase plane, such as multiple equilibrium points and

limit cycles.

◮ LOCAL BEHAVIOUR OF NON-LINEAR SYSTEM:If thesingular point of interest is not at the origin, by defining thedifference between the original state and the singular point asa new set of state variables, we can shift the singular point

to the origin.

◮ As a result, the local behavior of the nonlinear system can beapproximated by the patterns shown for linear system.


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 bytrajectories converging or diverging from it).

◮ Depending on the motion patterns of the trajectories in thevicinity of the limit cycle, we can distinguish three kinds oflimit cycles.

1. Stable Limit Cycles: All trajectories in the vicinity of thelimit cycle converge to it as t → ∞ (Fig a).

2. Unstable Limit Cycles: All trajectories in the vicinity of thelimit cycle diverge to it as t → ∞ (Figb)

3. Semi-Stable Limit Cycles: some of the trajectories in thevicinity of the limit cycle converge to it as t → ∞(Figc)


STABLE,UNSTABLE AND SEMI-STABLE LIMIT CYCLE

Figure: Stable, unstable, and semi-stable limit cycles


MERITS AND DEMERITS

◮ MERITS:

1. Phase Plane Analysis is on second-order, the solutiontrajectories can be represented by carves in plane provides easyvisualization of the system qualitative behavior.

2. Without solving the nonlinear equations analytically, one canstudy the behavior of the nonlinear system from various initialconditions.

3. It is not restricted to small or smooth nonlinearities and appliesequally well to strong and hard nonlinearities.

4. There are lots of practical systems which can be approximatedby second-order systems, and apply phase plane analysis.

◮ DEMERIT:

1. It is restricted to at most second-order and graphical study ofhigher-order is computationally and geometrically complex.


CONCLUSION

◮ Phase plane analysis is a graphical method used to studysecond-order dynamic systems.

◮ A number of useful classical theorems for the prediction oflimit cycles in second-order systems are also presented.


REFERENCES

1. Nguyen Tan Tien (2002-03 ) Applied Nonlinear

Control :chapter 2 Phase Plane Analysis

2. K.T. Alligood, T.D. Sauer, J.A. Yorke (1996).Chaos: An

Introduction to Dynamical Systems. Springer.


THANK YOU


Engineering

Phase plane analysis (nonlinear stability analysis)