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# Phase plane analysis (nonlinear stability analysis)

Phase plane analysis is a nonlinear stability analysis method.

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### Phase plane analysis (nonlinear stability analysis)

1. 1. PHASE PLANE ANALYSIS BINDUTESH V SANER May 5, 2015 BINDUTESH V SANER PHASE PLANE ANALYSIS
2. 2. 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 BINDUTESH V SANER PHASE PLANE ANALYSIS
3. 3. INTRODUCTION ◮ The most important feature of Nonlinear systems is that Nonlinear systems do not obey the Principle of Superposition. ◮ Due to this reason, in contrast to the linear case, the response of nonlinear systems to a particular test signal is no guide to their behavior to other inputs. ◮ Phase plane analysis is a graphical method for studying second-order systems. ◮ The nonlinear system response may be highly sensitive to input amplitude. For example, a nonlinear system giving best response for a certain step input may exhibit highly unsatisfactory behavior when the input amplitude is changed. BINDUTESH V SANER PHASE PLANE ANALYSIS
4. 4. BEHAVIOR OF NON LINEAR SYSTEM ◮ The nonlinear systems may exhibit limit cycles which are self-sustained oscillations of ﬁxed frequency and amplitude. ◮ Once the system trajectories converge to a limit cycle, it will continue to remain in the closed trajectory in the state space identiﬁed as limit cycles. ◮ In many systems the limit cycles are undesirable particularly when the amplitude is not small and result in some unwanted phenomena. BINDUTESH V SANER PHASE PLANE ANALYSIS
5. 5. METHOD OF ANALYSIS ◮ Nonlinear systems are diﬃcult to analyze and arriving at general conclusions are tedious. ◮ However, starting with the classical techniques for the solution of standard nonlinear diﬀerential equations, several techniques have been evolved which suit diﬀerent types of analysis. ◮ It should be emphasized that very often the conclusions arrived at will be useful for the system under speciﬁed conditions and do not always lead to generalizations. BINDUTESH V SANER PHASE PLANE ANALYSIS
6. 6. TYPES OF METHODS FOR ANALYSIS 1. Linearization Techniques 2. Describing Function Analysis 3. Liapunovs Method for Stability 4. Phase Plane Analysis BINDUTESH V SANER PHASE PLANE ANALYSIS
7. 7. CLASSIFICATION OF NON-LINEARITIES 1. Inherent Non-linearities:The nonlinearities which are present in the components used in system due to the inherent imperfections or properties of the system are known as inherent nonlinearities. for eg. Satuation in magnetic,Deadzone, Backlash in gear etc 2. Intentional Non-linearities:In some cases introduction of nonlinearity may improve the performance of the system, make the system more economical consuming less space and more reliable than the linear system designed to achieve the same objective. Such nonlinearities introduced intentionally to improve the system performance are known as intentional nonlinearities. BINDUTESH V SANER PHASE PLANE ANALYSIS
8. 8. CONCEPT OF PHASE PLANE ANALYSIS ◮ phase portraits:The phase plane method is concerned with the graphical study of second-order autonomous systems described by ˙x1 = f1(x1, x2) (1) ˙x2 = f2(x1, x2) (2) ◮ where x1, x2 : states of the system f1, f2 nonlinear functions of the states ◮ Geometrically, the state space of this system is a plane having x1, x2 as coordinates.This plane is called Phase plane. BINDUTESH V SANER PHASE PLANE ANALYSIS
9. 9. ◮ The solution of (1) with time varies from zero to inﬁnity can be represented as a curve in the phase plane. Such a curve is called 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 BINDUTESH V SANER PHASE PLANE ANALYSIS
10. 10. Contd.... ◮ The nature of the system response corresponding to various initial conditions is directly displayed on the phase plane. ◮ In the above example, we can easily see that the system trajectories neither converge to the origin nor diverge to inﬁnity. Indicating the marginal nature of the systems stability. ◮ A major class of second-order systems can be described by the diﬀerential equations of the form ¨x = f (x, ˙x) ◮ In state space form, this dynamics can be represented with x1 = x and x2 = ˙x asfollows : ˙x1 = x2 (3) ˙x2 = f (x1, x2) (4) BINDUTESH V SANER PHASE PLANE ANALYSIS
11. 11. SINGULAR POINTS ◮ 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 ˙x = 0. ◮ For a linear system, there is usually only one singular point although in some cases there can be a set of singular points. BINDUTESH V SANER PHASE PLANE ANALYSIS
12. 12. ◮ 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 characteristic equation 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) BINDUTESH V SANER PHASE PLANE ANALYSIS
13. 13. ◮ For linear systems there is only one singular point 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 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. BINDUTESH V SANER PHASE PLANE ANALYSIS
14. 14. STABLE AND UNSTABLE NODE ◮ The ﬁrst case corresponds to a node. A node can be stable or unstable: 1. λ1, λ2 < 0: singularity point is called stable node. 2. λ1, λ2 > 0: singularity point is called unstable node. BINDUTESH V SANER PHASE PLANE ANALYSIS
15. 15. SADDLE POINT ◮ The second case (λ1 < 0 < λ2 ) corresponds to a saddle point. Because of the unstable pole λ2, almost all of the system trajectories diverge to inﬁnity. Figure: BINDUTESH V SANER PHASE PLANE ANALYSIS
16. 16. STABLE OR UNSTABLE LOCUS ◮ The third case corresponds to a focus. 1. Re (λ1, λ2) < 0: stable focus 2. Re (λ1, λ2) > 0: unstable focus BINDUTESH V SANER PHASE PLANE ANALYSIS
17. 17. CENTRE POINT ◮ The last case corresponds to a certain point. All trajectories are ellipses and the singularity point is the centre of these ellipses. 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. BINDUTESH V SANER PHASE PLANE ANALYSIS
18. 18. PHASE PLANE ANALYSIS OF NON-LINEAR SYSTEM ◮ phase plane analysis of nonlinear system,has two important points as follow: 1. 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. 2. Nonlinear systems can display much more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles. ◮ LOCAL BEHAVIOUR OF NON-LINEAR SYSTEM: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. ◮ As a result, the local behavior of the nonlinear system can be approximated by the patterns shown for linear system. BINDUTESH V SANER PHASE PLANE ANALYSIS
19. 19. 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. 1. Stable Limit Cycles: All trajectories in the vicinity of the limit cycle converge to it as t → ∞ (Fig a). 2. Unstable Limit Cycles: All trajectories in the vicinity of the limit cycle diverge to it as t → ∞ (Figb) 3. Semi-Stable Limit Cycles: some of the trajectories in the vicinity of the limit cycle converge to it as t → ∞(Figc) BINDUTESH V SANER PHASE PLANE ANALYSIS
20. 20. STABLE,UNSTABLE AND SEMI-STABLE LIMIT CYCLE Figure: Stable, unstable, and semi-stable limit cycles BINDUTESH V SANER PHASE PLANE ANALYSIS
21. 21. MERITS AND DEMERITS ◮ MERITS: 1. Phase Plane Analysis is on second-order, the solution trajectories can be represented by carves in plane provides easy visualization of the system qualitative behavior. 2. Without solving the nonlinear equations analytically, one can study the behavior of the nonlinear system from various initial conditions. 3. It is not restricted to small or smooth nonlinearities and applies equally well to strong and hard nonlinearities. 4. There are lots of practical systems which can be approximated by second-order systems, and apply phase plane analysis. ◮ DEMERIT: 1. It is restricted to at most second-order and graphical study of higher-order is computationally and geometrically complex. BINDUTESH V SANER PHASE PLANE ANALYSIS
22. 22. CONCLUSION ◮ Phase plane analysis is a graphical method used to study second-order dynamic systems. ◮ A number of useful classical theorems for the prediction of limit cycles in second-order systems are also presented. BINDUTESH V SANER PHASE PLANE ANALYSIS
23. 23. 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. BINDUTESH V SANER PHASE PLANE ANALYSIS
24. 24. THANK YOU BINDUTESH V SANER PHASE PLANE ANALYSIS