plotting phase portraits with matlab:
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Matlab code to plot phase portraits. [x1, x2] = meshgrid(4:0.2:1, 2:0.2:2); x1dot = x2; x2dot = 0.6*x23*x1x1.^2; quiver(x1,x2,x1dot,x2dot) xlabel('x_1') ylabel('x_2'). range for x 1. range for x 2. PLOTTING PHASE PORTRAITS WITH MATLAB:.  PowerPoint PPT PresentationTRANSCRIPT

PLOTTING PHASE PORTRAITS WITH MATLAB:Phase portraits is a useful graphical tool to understand the stable or unstable behavior of the equilibrium points of a nonlinear systems. Consider the previous example,

Example: Consider the undamped simple pendulum

For l=1 mAt the equilibrium, all derivatives are zeroConsider the small perturbations around the equilibrium point d=0 Nonlinear terms can be linearized using the Maclaurin series.

For d=0Aans =
0 + 3.1321i 0  3.1321iclc;clear;A=[0 1;9.81 0];eig(A)For d=pHigher order termclc;clear;A=[0 1;9.81 0];eig(A)ans =
3.1321 3.1321Marginally stableUnstable

Example:Mathematical model of a nonlinear system is given by the equationWhere f(t) is the input and x(t) is the output of the system. The state variables are chosen as x1=x and x2=dx/dt=dx1/dtFind the equilibrium points for f=80 and linearize the system for small deviations from the equilibrium points. Find the response of the system For the equilibrium condition>>solve(64000*x1^2/(x1+2)=1.2)x1d=0.00613, x2d=0x1=x1d+e1=0.00613+e1x2=x2d+e2=e2Karaglle, System Modeling and Analysis

x1=x1d+e1=0.00613+e1x2=x2d+e2=e2f=fd+u

clc;clear;A=[0 1;390.52 9];eig(A)
ans =
4.5000 +19.2424i 4.5000 19.2424iStable system

clc;clear a=[0 1;390.52 9];b=[0 0.015]';c=[1 0];d=0; sys=ss(a,b,c,d); t=[0:.025:2]; [y,t,x]=step(sys*2,t); plot(t,y,'','Linewidth',2);axis([0 2 0 0.00015]);grid; xlabel('time (sec)');ylabel('Y output');title('Step Response')
Matlab code for step input with magnitude 2c=[0 1]

We can obtain the same result using Simulink.2tu(t)

Example: Phase portrait of a linear system. x1=xx2=dx/dt=dx1/dtInitial conditions:x1(0)=0.2(dx/dt)t=0=x2(0)=1x1=xx2=dx/dt=dx1/dt

At equilibriumDamping ratio

Damping ratio

Damping ratio

Example:x1=xx2=dx/dt=dx1/dtAt equilbrium

m=60;k=500;mu=0.3;g=9.81;[x1, x2] = meshgrid(0.5:0.1:0.5, 2:0.2:2);x1dot = x2; x2dot = (k/m)*x1sign(x2)*mu*g;quiver(x1,x2,x1dot,x2dot)

Example: Van der Pol Equationm=10, c=20, k=1000Massspringdamper system with a position dependent damper.

The Van der Pol equation can be regarded as describing a massspringdamper system with a positiondependent damping coefficient 2c(x21) (or, equivalently, an RLC electrical circuit with a nonlinear resistor). For large values of x, the damping coefficient is positive and the damper removes energy from the system. This implies that the system motion has a convergent tendency. However, for small values of x, the damping coefficient is negative and the damper adds energy into the system. This suggest that the system motion has a divergent tendency. Therefore, because the nonlinear damping varies with x, the system motion can neither grow unboundly nor decay zero. Instead, it displays a sustained oscillation independent of initiao conditions. This socalled limit cycle is sustained periodically releasing energy into and absorbing energy from the environment, through the damping term. This is in contrast with the case of conservative massspring system, which does not exchange energy with its environment during its vibration. Slotine and Weiping, Applied Nonlinear Control.

Of course, sustained oscillations can also be found in linear systems, in the case of marginally stable linear systems (such as a massspring system without damping) or in the response to sinusoidal inputs. However, limit cycles in nonlinear systems are difefrent from linear oscillations in a number of fundamental aspects. First, the amplitude of the selfsustained excitation is independent of initial consitions, while the oscillation of a marginally stable linear system has its amplitude determined by its initial conditions. Second, marginally stable linear systems are very sensitive to changes in system parameters (with a slight change capable of leading either to stable convergence or to instability), while limit cycles are not easily affected by parameter changes.
Limit cycles represent an important phenomenon in nonlinear systems. They can be found in many areas of engineering and nature. Aircraft wing fluttering, a limit cycle caused by the interaction of aerodynamic forces and structural vibrations, is frequently encountered and is sometimes dangerous. Slotine and Weiping, Applied Nonlinear Control.

0.200Limit cycle

00.6Limit cycle

20.6(0.6,2)Limit cycle

Important Behaviors of Nonlinear Systems:Bifurcations:As the parameters of nonlinear dynamic systems are changed, the stability of equilibrium point can change and so can the number of equilibrium points.Values of these parameters at which the qualitative nature of the systems motion changes are known as critical or bifurcation values.The phenomenon of bifurcation, i.e., quantitaive change of parameters leading to qualitative change of system properties, is the topic of bifurcation theory. Let us consider the system described by the cocalled undamped Duffing equation (massspring system with a hardening spring).Pitchfork bifurcationHopf bifurcationSlotine and Weiping, Applied Nonlinear Control.

Chaos:For stable linear systems, small differences in initial conditions can only cause small differences in output. Nonlinear systems, however, can display a phenomenon called chaos, by which we mean that the system output is extremely sensitive to initial conditions. The essential feature of chaos is the unpredictability of the system output. Even if we have an exact model of a nonlinear system and an extremely accurate computer, the systems response in the longrun still cannot be well predicted. As an example of chaotic behavior, let us consider the simple nonlinear systemwhich may represent a lightlydamped, sinusoidally forced mechanical structure undergoing large elastic deflections. Consider two almost identical initial conditions, namely x(0)=2, (dx/dt)0=3 and x(0)=2.01, (dx/dt)0=3.01. Due to the strong nonlinearity in x5, the two responses are radically different after some time. Slotine and Weiping, Applied Nonlinear Control.

Displacement

Velocity