Chapter 3. Tohru Katayama

Transcription

1 Subspace Methods for System Identification Chapter 3 Tohru Katayama Subspace Methods Reading Group UofA, Edmonton Barnabás Póczos May 14, 2009

2 Preliminaries before Linear Dynamical Systems Hidden Markov Models y(t-1) y(t) y(t+1) x(t-1) x(t) x(t+1) (have the same graphical model ) Our goal is the identification of the unknown parameters. 2

3 Identification Approaches LDS identification: Batch (offline) HMM identification: Recursive Prediction Error methods Extended Kalman Filter Dual Kalman Filter Batch (offline) EM Variational EM (Beal) Prediction Error Methods Subspace methods Recursive (online) EM (Baum - Welch) Variational EM (MacKay) Prediction Error Methods OOM (Jaeger) Recursive (online) Recursive Maximum Likelihood (Collings; LeGland and Mével) Recursive Prediction Error (Collings, Krishnamurthy) Recursive Kullback-Leibler (Moore, Krishnamurthy) 3

4 Contents Chapter 3 reviews Discrete-time LTI systems Lyapunov stability State-space equations Reachability Observability Canonical structure Balanced realization 4

5 z-transform Theorem: 5

6 z-transform Theorem: Note: the one-sided z-transform is a special case of the 2-sided z-transform 6

7 Convergence domains Lemma 3.1 Similarly, 7

8 Examples Step function: ) One-sided exponential function: ) Two-sided exponential function: ) 8

9 Inverse transform of F(z) Definition: (residue) Note Theorem 9

10 Properties of z-transform Linearity Definition (shift-operator) Time shift (for one-sided ) Time shift (for two-sided ) 10

11 Properties of z-transform Convolution Partial sum Difference of f Matrix case 11

12 Discrete-time LTI systems Assumptions: 1. The system is causal: (the future control doesn t influence the present) 2. The system is at rest for t = -1,-2, Convolution with a convolution kernel g of size t 12

13 Transfer function Lemma: We will investigate systems of this form. Example: 13

14 Transfer function Definition: (proper transfer function) Note: Definition: Bounded Input Bounded Output (BIBO) stable system Theorem: Proof: page

15 Transfer function Theorem: Proof: page 46. Note: Proof: linearity + Example 3.1(b) 15

16 Norms of signals and systems Definition: (2-norm in the time domain) Norm of two-sided infinite sequence: Definition: Fourier transform: Definition: (2-norm in the frequency domain) 16

17 Norms of signals and systems Definition: Definition: Definition: max singular value 17

18 Norms of signals and systems It is a bit confusing that we don t know when we are in the timeand when in the frequency-domain Theorem: Proof: page

19 State Space Systems Note: zero-input response zero-state response 19

20 State Space Systems Definition: zero-input response Definition: zero-state response Definition: 20

21 State Space Systems 21

22 State Space Systems Theorem: Example: Proof: page

23 Lyapunov stability Definition: homogeneous system Definition: equilibrium points Note: Definition: Theorem: Note: Lyapunov equation (page 51) 23

24 Reachability, Controllability, Observability, Detectability Definition: reachability Definition: Controllability Definition: reachability matrix 24

25 Reachability, Controllability Theorem: Proof: page 52. Note: Note: 25

26 Reachability, Controllability, Theorem: Definition: (stabilizable) Lemma: 26

27 Observability Definition: (observable) Note: Definition: (observability matrix) 27

28 Observability Theorem: Theorem: (3.28) 28

29 Detectability Definition: (detectability) Detectability is weaker than observability. Theorem: Theorem: Proof: page 54. unobservable modes are stable 29

30 Canonical Decomposition of Linear Systems Note: Note: 30

31 Canonical Decomposition of Linear Systems Theorem

32 Canonical Decomposition of Linear Systems Note: only the 2nd subsystem counts. (It is reachable and observable) Definition: (minimal realization) A realization with the least dimension Theorem: Theorem: 32

33 Realization Theory Definition: (infinite block Hankel matrix) Definition: Definition: Definition: (Markov parameters) 33

34 Realization Theory Definition: k-observability matrix Definition: l-reachability matrix Definition: observability matrix = n-observability matrix reachability matrix = n-reachability matrix k > n ) extended observability matrix l > n ) extended reachability matrix 34

35 Realization Theory Lemma 3.10 Note Lemma 3.11 Proof: page 66. Definition: (rank of infinite block Hankel matrix H) 35

36 Realization Theory Definition: Theorem 3.13 Proof: page 67. Note: 36

37 Realization Theory Theorem 3.14 Thanks! 37