When Physics and Control Theory Collide

Transcription

1 When Physics and Control Theory Collide Bruce Francis

2 Thanks, Malcolm, for the invitation

3 The difficulty in speaking last is that many accolades have already been said about Keith Professor Young said Keith has a great memory Let s see

4

5 My topic was inspired by the paper Distributed control of spatially invariant systems Bamieh, Paganini, Dahleh 2002 Also relevant is String stability of interconnected systems Swaroop and Hedrick 199 Infinitely many subsystems, indexed by the (nonnegative) integers I searched for a meaning of such things

6 My points 1 There are systems out there, especially in physics, with a very large number of components Eg, N = Avogadro s number, e23 2 A typical question in lattice dynamics is wave velocity Therefore, a dynamic model is needed 3 Then there are boundary conditions Should one approximate by N = infinity, or make the boundary conditions periodic, or what? 4 Perhaps one can look to Aristotle and Willems

7 A puzzle

8

9 b a Infinite lattice of 1 ohm resistors What is the equivalent R from a to b?

10 popular answer: R equiv = 2 eg, Atkinson and Steerwijk Infinite resistive lattices American J Physics, 1999 But

11 What is an infinite resistive lattice? Certainly it isn t real Therefore it doesn t necessarily obey the laws of physics I take it to be a picture of infinitely many coupled KCLs

12 What is equivalent resistance? (1, 1) (0, 0) (1, 0) Insert a 1 A current source The equivalent resistance is the voltage drop (V=IR) Is this a meaningful definition? Ideal current sources aren t real

13 Correct answer is, equivalent resistance can be anything

14 (1, 1) (0, 0) (1, 0) Label the nodes (m, n) Apply a 1A current Label the voltages to ground v m,n Define the relative voltages w m,n = v m,n v m+1,n+1 Compute w 0,0 R equiv = w 0,0

15 KCL (1, 1) (0, 0) (1, 0) v m,n v m,n 1 + v m,n v m,n+1 + =0

16 V = 2 4 v 2, 2 v 2, 1 v 2,0 v 2,1 v 2,2 v 1, 2 v 1, 1 v 1,0 v 1,1 v 1,2 v 0, 2 v 0, 1 v 0,0 v 0,1 v 0,2 v 1, 2 v 1, 1 v 1,0 v 1,1 v 1,2 v 2, 2 v 2, 1 v 2,0 v 2,1 v 2,2 3 5 W 2 3

17 4 A = AW + WA = Y, Y =

18 4 AW + WA = Y A = E matrix of all 1 s EA = AE =0 If W is a solution, so is W + ce Therefore, R equiv = anything

19 So where does 2 come from?

20 One way to get uniqueness is to look for a Hilbert-Schmidt solution: X wm,n 2 < 1 m,n The set of Hilbert-Schmidt matrices forms a Hilbert space under the inner product hx, Y i = trace (X Y )

21 A matrix is a function of two variables Fourier theory is 2D (z,µ)-transform ˆX(z,µ) X X = x m,n z m µ n m n 2D Fourier transform ˆX e j, e j X X = x m,n e j m e j m n n inversion formula

22 Apply these tools to AW + WA = Y 2 z 1 z +2 µ 1 µ Ŵ (z,µ) =2 zµ 1 zµ Ŵ e j, e j = 1 cos( + ) 2 cos( ) cos( ) w 0,0 = 1 (2 ) 2 Z Z 1 cos( + ) 2 cos( ) cos( ) d d = 2 by Mathematica

23 But is there a justification for the Hilbert-Schmidt solution? Remember, the system is not physically realizable and infinity is much bigger than any finite number

24 conversation with Atkinson (Physics, Groningen) A lesson in Aristotle s Physics

25 I completely agree with you that the infinite network of resistors yields a system of equations that possess an infinity of solutions As we noted in the introduction of our paper, (p 48), we tacitly obtain uniqueness by requiring that the currents at infinity vanish,

26 In your note you object that an infinite network is not physical, so appeal to physics is disallowed Indeed, if you stick to the mathematics, the equations do admit multiple solutions, unless you add the requirement at infinity, in which case there is only one solution, namely the one we published If you further press the point as to why we should be interested in this particular solution, then physics can be brought in via the Aristotelian distinction between actual and virtual infinity

27 In an actually infinite network (without a boundary condition at infinity), there is an infinity of solutions of the equations In a virtually infinite network, which means that you consider a finite network of size N, and then let N tend to infinity, there is only one solution, and moreover it is one in which the currents at the periphery do indeed vanish in the limit This is the physical solution, according to the Aristotelian canon

28 Conclusion of the puzzle: A resistive grid is a system with infinitely many components A question arises about the meaning of such a thing At least one physicist thinks the real thing is a mathematical limit

29 A resistive grid has no dynamics We turn to a dynamical system example

30 Brillouin, 1948, Wave Propagation in Periodic Structures a circuit model of a crystal i n n i n+1 + v + n v n+1

31 unit cell is the 2-port

32 I had some correspondence with Jan Willems about this system

33 Dear Jan, I have a question The physicist Brillouin, in his book Wave Propagation in Periodic Structures, studied a lattice that is an infinite chain of inductors and capacitors I attached a sketch I wonder what you would do with this, how you would model it There are no boundaries -- it's a chain infinite in both directions Thus the voltages and currents are indexed by the integers There are no voltage or current sources, hence no inputs I have some ideas but I'd like to hear your thoughts first Thanks, Bruce

34 Dear Bruce, I doodled a bit with your circuit Here are some reflections 0 Infinity in one direction is mystic, in two directions it is metaphysics 1 I tried to calculate the impedance of a half-infinite transmission, say looking from node n all the way to infinity on the right

35 Z Z (Z + Ls) 1 Cs Z + Ls + 1 Cs = Z L C = Z2 + LsZ Jan took the positive-real solution

36 4 I feel very insecure about this answer Cheers, Jan

37 My approach with Abie Feintuch

38 Cauchy system Take L = C =1 i n n i n+1 + v + n v n+1 KCL i n = i n+1 + d dt v n KVL v n + d dt i n+1 + v n+1 =0 I = zi + sv V + szi + zv =0 =) I = s 1 z V =) I = 1 z sz V equate

39 s 1 z V = 1 z sz s 2 V = (1 z)2 z V V = z + 1 z 4 2 V v = Av A = Valid or not?

40 Reviewer 2: The fact that infinite chains may be simpler than finite chains to analyze is not a practical reason to study them An application of a true infinite chain would be interesting, but is not required

41 Thanks for listening