1. Linear dynamic systems

2. Introduction
Example of a complex dynamic system

3. Example of a complex dynamical system

4. Example of a complex dynamical system (cnt’d)

5. Example of a complex dynamical system (cnt’d)? ?

6. Oregon interaction data
69 normal and 72 depressed adolescents in laboratory
Several 9 minute interaction tasks: two positive tasks (e.g., discuss nice time together), two reminiscence tasks (e.g., discuss salient aspects of adolescenthood and parenthood) and two conflict tasks (e.g., discuss familial conflicts)

7. Second-to-second physiological data collected (heart rate, blood pressure, skin conductance, etc.) and observational data (i.e., behavior coding: neutral, happy, angry, dysphoric)

9. Oregon interaction data set

10. Problem: cardiovascular system is nonlinear  If we look at heart rate, it contains 1/f noise and is self-similar (--> fractals)

11. Question: Why study a linear dynamical system if the world is nonlinear?

12. Question: Why study a linear dynamical system if the world is nonlinear?
you need to understand a linear system first
linear methods are often basis of nonlinear ones
they often work well as approximations
often easier to apply in a noisy context
measurement error
unidentified influences
computational efficient methods are developed
 Feynman: linear systems are important because we can solve them! (Lect on Physics)

13. Other examples

18. Overview of dynamical systems

20. What will we study today?

21. Simple dynamic system
Falling ball: distance  time2 t = time x(t) = position at time t = velocity at time t = acceleration at time t

22. x=0
x(t)

23. using Newton’s second law (no friction, small heigths)

24. thus, it follows that
 this is a linear second-order ordinary differential equation (ODE)

25. solution can be found analytically by integration
first, consider it to be a first-order ODE in v(t) :
if we know v(0), then C = v(0)

26. solution now, we have a first-order ODE in x(t)
if we know x(0), then K = x(0)
solution

27. this system is very simple analytically, but somewhat more complex to study in general because it is non-homogeneous
Again, we can go from a second-order ODE to a system of first-order ODEs
How? By defining the state variables x(t) = position v(t) = x’(t) = velocity

28. if we know the state variables, then we know the system
in general, we can write
this is a system of first order ODEs (replacing the single second-order ODE)

29. or also (using vector notation)

30. Another (more interesting) dynamic system
mass-spring system
x=0
x(t)

31. Newton’s second law
combine this with Hooke’s law where k is the spring constant (large for stiff springs)

32. the ODE becomes or
again a second-order ODE
assume

33. How to study this dynamic system?
Again, we can go from a second-order ODE to a system of first-order ODEs
How? By defining the state variables x(t) = position v(t) = x’(t) = velocity
if we know the state variables, then we know the system

34. dynamic system now becomes
we can plot again a vector field in the state space or phase plane (i.e., the plane with x and v as coordinate axes)
each point (x,v) is associated with (x’,v’) = (v, -2 x)

35. Exercise
Plot the vector field.

37. imagine a flow of an imaginary fluid and place an imaginary particle in it
watch how it is carried away
if (x’(t),v’(t)) = (0,0), then this is the fixed point (for the mass-spring system this happens at (x,v)=(0,0)
starting anywhere else leads to circulation in a closed orbit  phase portrait of this system

38. Questions
Based on this information, graph schematically the solution of the second-order ODE for x(t).
How does the solution for v(t) look like? (also schematically)

39. for the general homogeneous system
or several different kinds of phase portraits

40. to show that our the mass-spring system is an example hence

41. center
star (unstable)
saddle point
stable spiral
stable node
degenerate node

42. we need to check the eigenvalues of A
solve the characteristic equation: det(A-I)=0

43. p
=(1-2)2 q

44. for 2 = 1
eigenvalues are pure imaginary complex numbers: 1=+i and 2=-i

45. Another more interesting dynamic system
mass-spring system with damping

46. Now the dynamics are a little bit more complex where
ODE:

47. Exercise Rewrite the ODE as a two-dimensional linear system
What is the fixed point?
Explore the possible phase portraits? Take in all cases m=1 and let c and k vary such that
c2 > 4k
c2 < 4k
optional: c2 = 4k
Relate this to “overdamped”, “critically damped”, and “underdamped”

48. Discrete time
Almost all our measurements are in discrete time t0 < t1 < t2 < … < tk-1 < tk < tk+1 < …
Possibly equally spaced t = tk – tk-1, for all k
In such a case, the differential equation becomes a difference equation

49. As said earlier, this transition is not trivial
instead of we (hope to) get

50. t t +t Let us do the exercise for the falling ball
What happens when going from t to t+t ?
initial position = x(t) initial velocity = v(t) t
t +t
new position = x(t+t) new velocity = v(t+t)

51. the transition equations for the state variables become:
or in matrix notation

52. So in general

53. For the mass-spring system, the calculations are a little bit more complex, but not impossible
In that case with where v1 and v2 are the eigenvectors of A and 1 and 2 are the corresponding eigenvalues

54. to conclude: we may use a discrete dynamical model for the linear system if we observe it at discrete times
the systems we have talked about now can be represented schematically as follows
u1(t)
x1(t)
xi(t)
uj(t)
xI(t)
uJ(t)

55. the systems we have talked about now can be represented schematically as follows
x1(t+t)
xi(t +t)
uj(t)
xI(t +t)
uJ(t)

56. Tom will make it a little bit more complicated:
adding an observation equation
adding error to both equations

57. Kalman filter for the local level model

58. Remember the local level model with

59. define
Can we find (recursive) expressions for t and Vt? The solution will be the celebrated Kalman filter

60. The problem is
we have observations and a current estimate of t-1, now yt becomes available
how should we update our estimate of t

61. Let us first remark that since all distributions we start from are normal, all derived distributions will be normal as well
Then using Bayes’ theorem

63. Let us focus on (*)

64. Inserting this result then gives
You may recognize the traditional Bayesian structure here: a normal observation yt with mean t and a normal prior on t

65. This gives for the posterior

66. Using some elementary algebra, this can be formulated as follows:
these are the Kalman recursive formulas with Kt being called the Kalman gain

67. Exercise:
Simulate data from the local level model and try to apply the Kalman filter.
Plot the data and draw the filtered states through it.
Plot the filtered state variance Pt.
Plot the errors.