NECESSARY AND SUFFICIENT CONDITIONS FOR STATE-SPACE NETWORK REALIZATION. Philip E. Paré Masters Thesis Defense

Transcription

1 NECESSARY AND SUFFICIENT CONDITIONS FOR STATE-SPACE NETWORK REALIZATION Philip E. Paré Masters Thesis Defense

2

3 INTRODUCTION

4 MATHEMATICAL MODELING

5 DYNAMIC MODELS A dynamic model has memory

6 DYNAMIC MODELS A dynamic model has memory There are different ways to model dynamic systems Black Box Representation u u x y y u 2 u 2 x 2 y 2 y 2 u 3 u 3 x 3 y 3 y 3

7 DYNAMIC MODELS A dynamic model has memory There are different ways to model dynamic systems State Machine Representation u x y x 4 u 2 x 2 y 2 x 5 x 6 x 7 u 3 x 3 y 3 x 8

8 STATE SPACE MODELS OF LINEAR TIME INVARIANT (LTI) SYSTEMS u x y x 4 u 2 x 2 y 2 x 5 x 6 x 7 u 3 x 3 y 3 x 8 x Ax Bu y Cx Du

9 TRANSFER FUNCTION REPRESENTATION OF LTI SYSTEMS u u x y y u 2 u 2 x 2 y 2 y 2 u 3 u 3 x 3 y 3 y 3 G(s) C(sI A) B D

10 MANY TO ONE RELATIONSHIP D D CX C XB B XAX A ˆ ˆ ˆ ˆ D B A si C D B A si C s G ˆ ˆ ˆ) ˆ( ) ( ) (

11 NETWORK REALIZATION Assume input-output data is produced by the system (A, B, C, D) then what must be known about this system to recover the whole thing from input-output data

12 NETWORK REALIZATION Assume input-output data is produced by the system (A, B, C, D) then what must be known about this system to recover the whole thing from input-output data

13 PROBLEM FORMULATION

14 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices.

15 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices.

16 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices. For a β = P α R P R N N mp N mp N nm n nm n n n n D C B A 2 ) (, ) (, ) (, ) (

17 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices

18 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x

19 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x

20 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x I( )

21 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x I( )

22 RESTRICTION OF THE IDENTITY Assuming elements of (A, B, C, D) are known can restrict the identity parameterization

23 RESTRICTION OF THE IDENTITY Assuming elements of (A, B, C, D) are known can restrict the identity parameterization Let θ R R N encode the known information and the indicator function Θ R be a binary vector indicating the known elements and let the affine restriction of the identity parameterization be represented by I θr

24 CONSISTENCY Consider a system (A, B, C, D) and let θ be the vectorization of the system matrices (A, B, C, D). An affine restriction of the identity parametrization I θr is consistent with θ if θ R(I θr )

25 CONSISTENCY Consider a system (A, B, C, D) and let θ be the vectorization of the system matrices (A, B, C, D). An affine restriction of the identity parametrization I θr is consistent with θ if θ R(I θr )

26 GLOBAL IDENTIFIABILITY Let P α : Ω R q R N be a parameterization of system matrices (A, B, C, D). The parameterization P(α) is globally identifiable from the transfer function G(s) if for all α, α 2 Ω if G s = C α (si A α ) B α = C α 2 (si A α 2 ) B α 2 then α = α 2

27 NETWORK REALIZATION PROBLEM Consider a system (A, B, C, D), with A, B controllable and A, C observable and suppose G s = C(sI A) B + D is given. Find an affine restriction of the identity parameterization, consistent with A, B, C, D, that is globally identifiable from G(s)

28 SOLUTION TO NETWORK REALIZATION PROBLEM

29 SUFFICIENT CONDITION AX ˆ XB CX ˆ XA Bˆ C

30 SUFFICIENT CONDITION C CX B XB XA AX ˆ ˆ ˆ ) ( ˆ) ( ˆ ˆ C vec B vec x C I I B A A n n T T x b A

31 ANOTHER SUFFICIENT CONDITION X Aˆ AX BX ˆ B CX Cˆ

32 ANOTHER SUFFICIENT CONDITION C CX B BX AX A X ˆ ˆ ˆ ˆ) ( ) ( ˆ ˆ C vec B vec x C I I B A A n n T T x b A

33 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. = k, and

34 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. A = k, and N(A) N(A T ) x R(A T ) b R(A)

35 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. A = k, and N(A) N(A T ) R(A T ) R(A) x b

36 COMBINING SUFFICIENT CONDITIONS

37 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. = k, and Also let N A = span{x n,, x nk } and N A = span{x n,, x n l }, where x ni = vec(x ni ).

38 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).

39 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).

40 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).

41 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).

42 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).

43 PHARMACOKINETICS EXAMPLE

44 PHARMACOKINETICS EXAMPLE

45 PHARMACOKINETICS EXAMPLE x y k k k 2 x k k 2 x b u

46 PHARMACOKINETICS EXAMPLE x y u b x k k k k k x 2 2 x y u x x S

47 PHARMACOKINETICS EXAMPLE x y u x x x y u x x S S

48 PHARMACOKINETICS EXAMPLE Assume C = [ I ] is known So for I C=[ I ] we see that k = l = 2

49 PHARMACOKINETICS EXAMPLE Assume C = [ I ] is known So for I C=[ I ] we see that k = l = w w w w

50 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2

51 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2 X

52 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2 X x y x.227 x.25 u

53 THANK YOU

54 THANK YOU Questions?