lect_2 two-dimentwo-dimensional systemssional systems

8/10/2019 Lect_2 Two-DimenTwo-Dimensional Systemssional Systems

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x1 = f 1(x1, x2) = f 1(x)

x2 = f 2(x1, x2) = f 2(x)Let x(t) = ( x1(t), x2(t)) be a solution that starts at initialstate x0 = ( x10 , x20). The locus in the x1x2 plane of thesolution x(t) for all t 0 is a curve that passes through thepoint x0. This curve is called a trajectory or orbit The x1x2 plane is called the state plane or phase plane The family of all trajectories is called the phase portrait The vector eld f (x) = ( f 1(x), f 2(x)) is tangent to the

trajectory at point x becausedx2dx1

= f 2(x)f 1(x)

Nonlinear Control Lecture # 2 Two-Dimensional Systems


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Vector Field diagram

Represent f (x) as a vector based at x; that is, assign to x thedirected line segment from x to x + f (x)

x1

x2

f (x)

x = (1 , 1)

x + f (x) = (3 , 2)

Repeat at every point in a grid covering the plane



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Qualitative Behavior of Linear Systems

x = Ax, A is a 2 2 real matrix

x(t) = M exp(J r t)M 1x0

When A has distinct eigenvalues,

J r = 1 0

0 2 or

x(t) = Mz (t)

z = J r z (t)



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Case 1. Both eigenvalues are real:

M = [v1, v2]

v1 & v2 are the real eigenvectors associated with 1 & 2

z 1 = 1z 1, z 2 = 2z 2

z 1(t) = z 10e 1 t , z 2(t) = z 20e 2 t

z 2 = cz 2 / 11 , c = z 20/ (z 10)

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The shape of the phase portrait depends on the signs of 1and 2



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2 < 1 < 0

e 1 t and e 2 t tend to zero as t

e 2 t tends to zero faster than e 1 t

Call 2 the fast eigenvalue (v2 the fast eigenvector) and 1 theslow eigenvalue (v1 the slow eigenvector)

The trajectory tends to the origin along the curve z 2 = cz 2 / 11

with 2/ 1 > 1

dz 2dz 1= c21

z [( 2 / 1 ) 1]1



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z 1

z 2

Stable Node2 > 1 > 0Reverse arrowheads Unstable Node



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x 2

x 1

v 1

v2

(b)

x1

x2

v1

v 2

(a)

Stable Node Unstable Node



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2 < 0 < 1

e 1 t , while e 2 t 0 as t

Call 2 the stable eigenvalue (v2 the stable eigenvector) and1 the unstable eigenvalue (v1 the unstable eigenvector)

z 2 = cz 2 / 11 , 2/ 1 < 0

Saddle



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z 1

z2

(a)

x 1

x 2 v 1v2

(b)

Phase Portrait of a Saddle Point



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Case 2. Complex eigenvalues: 1,2 = j

z 1 = z 1 z 2, z 2 = z 1 + z 2

r = z 21 + z 22 , = tan 1 z 2z 1r (t) = r0et and (t) = 0 + t

< 0 r(t) 0 as t

> 0 r(t) as t

= 0 r(t) r 0 t



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z1

z2 (c)

z 1

z 2 (b)

z1

z 2 (a)

< 0 > 0 = 0

Stable Focus Unstable Focus Center

x 1

x2(c)

x 1

x 2(b)

x 1

x 2(a)



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Effect of Perturbations

A A + A (A arbitrarily small)The eigenvalues of a matrix depend continuously on itsparameters

A node (with distinct eigenvalues), a saddle or a focus isstructurally stable because the qualitative behavior remainsthe same under arbitrarily small perturbations in A

A center is not structurally stable

1 1 , Eigenvalues =

j

< 0 Stable Focus, > 0 Unstable Focus



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Qualitative Behavior Near Equilibrium Points

The qualitative behavior of a nonlinear system near anequilibrium point can take one of the patterns we have seen

with linear systems. Correspondingly the equilibrium points areclassied as stable node, unstable node, saddle, stable focus,unstable focus, or center

Can we determine the type of the equilibrium point of a

nonlinear system by linearization?



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Let p = ( p1, p2) be an equilibrium point of the system

x1 = f 1(x1, x2), x2 = f 2(x1, x2)

where f 1 and f 2 are continuously differentiableExpand f 1 and f 2 in Taylor series about ( p1, p2)

x1 = f 1( p1, p2) + a11 (x1

p1) + a12(x2

p2) + H .O.T .x2 = f 2( p1, p2) + a21(x1 p1) + a22(x2 p2) + H .O.T .

a11 = f 1(x1, x2)

x1 x= p, a12 =

f 1(x1, x2)

x2 x= p

a21 = f 2(x1, x2)

x1 x= p, a22 =

f 2(x1, x2)x2 x= p



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f 1( p1, p2) = f 2( p1, p2) = 0

y1 = x1 p1 y2 = x2 p2

y1 = x1 = a11 y1 + a12y2 + H .O.T .y2 = x2 = a21y1 + a22y2 + H .O.T .

y Ay

A =a11 a12

a21 a22 =

f 1

x1

f 1

x2

f 2x 1

f 2x 2 x= p

= f x x= p



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Eigenvalues of A Type of equilibrium pointof the nonlinear system

2 < 1 < 0 Stable Node2 > 1 > 0 Unstable Node

2 < 0 < 1 Saddle j, < 0 Stable Focus j, > 0 Unstable Focus

j Linearization Fails



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Example 2.1

x1 = x2 x1(x21 + x22)x2 = x1 x2(x21 + x

22)

x = 0 is an equilibrium point

f x

= (3x21 + x22) (1 + 2x1x2)(1 2x1x2) (x21 + 3x22)

A = f x

x=0

= 0 1

1 0

x1 = r cos and x2 = r sin r = r 3 and = 1

Stable focus when > 0 and Unstable focus when < 0



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For a saddle point, we can use linearization to generate thestable and unstable trajectories

Let the eigenvalues of the linearization be 1 > 0 > 2 andthe corresponding eigenvectors be v1 and v2

The stable and unstable trajectories will be tangent to thestable and unstable eigenvectors, respectively, as theyapproach the equilibrium point p

For the unstable trajectories use x0 = p v1

For the stable trajectories use x0 = p v2

is a small positive number


lect_2 two-dimentwo-dimensional systemssional systems

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