chapter 2: second order systems - lehigh...
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Lecture 1Chapter 2: Second Order Systems
Eugenio Schuster
Mechanical Engineering and Mechanics
Lehigh University
Lecture 1 – p. 1/26
Behavior of Second Order Systems
Consider the following linear system
x = Ax(1)
The solution of (1) for an initial condition x0 is given by
x(t) = M exp(Jrt)M−1x0
where Jr is the real Jordan form of A and M is a realnonsingular matrix such that
Jr = M−1AM
Lecture 1 – p. 2/26
Behavior of Second Order Systems
There are three possible Jordan forms for A:
Different real eigenvalues
Equal real eigenvalues
Complex conjugate eigenvalues
[
λ1 0
0 λ2
] [
λ k
0 λ
] [
α −β
β α
]
In addition, we need to consider the case where at leastone of the eigenvalues is zero.
Lecture 1 – p. 3/26
Behavior of Second Order Systems
Different real eigenvalues: λ1 6= λ2 (non-zero)In this case,
M = [ v1 v2 ]
where v1 and v2 are the real eigenvectors of A associatedwith λ1 and λ2, respectively.The change of coordinate z = M−1x transforms the systeminto two decoupled first-order differential equations, i.e.,
z1 = λ1z1, z2 = λ2z2
with solution
z1(t) = z10eλ1t, z2(t) = z20e
λ2t ⇒ z2 =z20
(z10)λ2/λ1
zλ2/λ1
1
Lecture 1 – p. 4/26
Behavior of Second Order Systems
Stable Node: λ1, λ2 < 0
Figure 1: Modal (left) and original (right) coordinates.
Lecture 1 – p. 5/26
Behavior of Second Order Systems
Unstable Node: λ1, λ2 > 0
Figure 2: Original coordinates.
Lecture 1 – p. 6/26
Behavior of Second Order Systems
Saddle Point: λ1 > 0, λ2 < 0
Figure 3: Modal (left) and original (right) coordinates.
Lecture 1 – p. 7/26
Behavior of Second Order Systems
Complex conjugate eigenvalues: λ1,2 = α± jβ
In this case,M = [ v1 v2 ]
where v1 and v2 are the real eigenvectors of A associatedwith λ1 and λ2, respectively.The change of coordinate z = M−1x transforms the systeminto the form
z1 = αz1 − βz2, z2 = βz1 + αz2
Defining the change of coordinates
r =√
z21 + z22 , θ = tan−1
(
z2
z1
)
Lecture 1 – p. 8/26
Behavior of Second Order Systems
we can write the dynamic equations in polar coordinates as
r = αr, θ = β
with solution
r(t) = r0eαt, θ(t) = θ0 + βt
Lecture 1 – p. 9/26
Behavior of Second Order Systems
Stable Focus: α < 0
Figure 4: Modal (left) and original (right) coordinates.
Lecture 1 – p. 10/26
Behavior of Second Order Systems
Unstable Focus: α > 0
Figure 5: Modal (left) and original (right) coordinates.
Lecture 1 – p. 11/26
Behavior of Second Order Systems
Center: α = 0
Figure 6: Modal (left) and original (right) coordinates.
Lecture 1 – p. 12/26
Behavior of Second Order Systems
Equal real eigenvalues: λ1 = λ2 = λ (non-zero)In this case,
M = [ v1 v2 ]
where v1 and v2 are the real eigenvectors of A associatedwith λ1 and λ2, respectively.The change of coordinate z = M−1x transforms the systeminto the form
z1 = λz1 + kz2, z2 = λz2
with solution
z1(t) = (z10+kz20t)eλt, z2(t) = z20e
λt ⇒ z1 = z2
[
z10
z20+
k
λln
(
z2
z20
)]
Lecture 1 – p. 13/26
Behavior of Second Order Systems
Case 1: k = 0
Figure 7: (a) λ < 0, (b) λ > 0.
Lecture 1 – p. 14/26
Behavior of Second Order Systems
Case 2: k = 1
Figure 8: (a) λ < 0, (b) λ > 0.
Lecture 1 – p. 15/26
Behavior of Second Order Systems
One or both zero eigenvalue: λ1 = 0, λ2 6= 0 or λ1 = λ2 = 0In this case A has a non-trivial null space.When λ1 = 0 and λ2 6= 0,
M = [ v1 v2 ]
where v1 and v2 are the real eigenvectors of A associatedwith λ1 and λ2, respectively. Note that v1 spans the nullspace of A.The change of coordinate z = M−1x transforms the systeminto the form
z1 = 0, z2 = λ2z2
with solutionz1(t) = z10, z2(t) = z20e
λ2t
Lecture 1 – p. 16/26
Behavior of Second Order Systems
When λ1 = λ2 = 0,
M = [ v1 v2 ]
where v1 and v2 are the real eigenvectors of A associatedwith λ1 and λ2, respectively.The change of coordinate z = M−1x transforms the systeminto the form
z1 = z2, z2 = 0
with solutionz1(t) = z10 + z20t, z2(t) = z20
Lecture 1 – p. 17/26
Behavior of Second Order Systems
Case 1: λ1 = 0 and λ2 6= 0
Figure 9: (a) λ1 = 0, λ2 < 0, (b) λ1 = 0, λ2 > 0.
Lecture 1 – p. 18/26
Behavior of Second Order Systems
Case 2: When λ1 = λ2 = 0
Figure 10: λ1 = λ2 = 0.
Lecture 1 – p. 19/26
Qualitative Behavior Near Equilibria
Given the nonlinear system
x1 = f1(x1, x2)
x2 = f2(x1, x2)(2)
let us assume p = (p1, p2) is an equilibrium point of (2), i.e.,
f1(p1, p2) = f2(p1, p2) = 0
Let us know expand the right-hand side of (2) around theequilibrium point p, i.e.,
x = f(p) +∂f(x)
∂x
∣
∣
∣
∣
x=p
(x− p) +H.O.T.
Lecture 1 – p. 20/26
Qualitative Behavior Near Equilibria
where
x =
[
x1
x2
]
, f(x) =
[
f1(x)
f2(x)
]
and∂f(x)
∂x
∣
∣
∣
∣
x=p
=
[
∂f1(x)x1
∂f1(x)x2
∂f2(x)x1
∂f2(x)x2
]
x=p
is the Jacobian evaluated at the equilibrium point p. Sincewe are interested in the behavior near p, we define
A ≡∂f(x)
∂x
∣
∣
∣
∣
x=p
, y = x− p
and we obtain
Lecture 1 – p. 21/26
Qualitative Behavior Near Equilibria
y ≈ Ay
which represents the Jacobi linearization.
Q: Is the system’s linearization a good approximation of its localbehavior?A: Yes, but provided the linearization has no eigenvalue on the imaginaryaxis, i.e., provided the equilibrium is hyperbolic.
Therefore, as long as f1(x) and f2(x) have continuous firstpartial derivatives, we can conclude that
stable/unstable node stable/unstable nodestable/unstable focus remains stable/unstable focussaddle point saddle point
Lecture 1 – p. 22/26
Qualitative Behavior Near Equilibria
Example: hyperbolic case - Pendulum with friction
θ = −b
ml2θ −
g
lsin(θ)
Lecture 1 – p. 23/26
Qualitative Behavior Near Equilibria
−6 −4 −2 0 2 4 6−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x1
x2
Figure 11: Pendulum: Saddle + Stable Focus.
Lecture 1 – p. 24/26
Qualitative Behavior Near Equilibria
Example: non-hyperbolic case
x1 = −x2 − µx1(x21 + x22)
x2 = x1 − µx2(x21 + x22)
Lecture 1 – p. 25/26
Qualitative Behavior Near Equilibria
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x1
x2
Figure 12: Stable/Unstable Focus.
Lecture 1 – p. 26/26