Subspace Methods for System Identification
Chapter 3
Tohru Katayama
Subspace Methods Reading Group
UofA, Edmonton
Barnabás Póczos
May 14, 2009
Preliminaries before…
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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…)
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Our goal is the identification of the unknown
parameters.
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Identification Approaches
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LDS identification:
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Batch (offline)
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HMM identification:
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Recursive Prediction Error
methods
Extended Kalman Filter
Dual Kalman Filter
Batch (offline)
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EM
Variational EM (Beal)
Prediction Error Methods
Subspace methods
Recursive (online)
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EM (Baum - Welch)
Variational EM (MacKay)
Prediction Error Methods
OOM (Jaeger)
Recursive (online)
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Recursive Maximum
Likelihood (Collings; LeGland
and Mével)
Recursive Prediction Error
(Collings, Krishnamurthy)
Recursive Kullback-Leibler
(Moore, Krishnamurthy) 3
Contents
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Chapter 3 reviews
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Discrete-time LTI systems
Lyapunov stability
State-space equations
Reachability
Observability
Canonical structure
Balanced realization
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z-Transform
Theorem:
5
z-Transform
Theorem:
Note: the one-sided z-transform is a special case of the 2-sided z-transform
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Convergence domains
Lemma 3.1
Similarly,
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Examples
Step function:
)
One-sided exponential function:
)
Two-sided exponential function:
)
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Inverse transform of F(z)
Definition: (residue)
Note
Theorem
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Properties of z-Transform
Linearity
Definition (shift-operator)
Time shift (for one-sided )
Time shift (for two-sided )
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Properties of z-Transform
Convolution
Partial sum
Difference of f
Matrix case
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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
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Transfer function
Lemma:
We will investigate systems of this form.
Example:
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Transfer function
Definition: (proper transfer function)
Note:
Definition:
Bounded Input Bounded Output (BIBO) stable system
Theorem:
Proof: page 45.
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Transfer function
Theorem:
Proof: page 46.
Note:
Proof: linearity + Example 3.1(b)
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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)
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Norms of signals and systems
Definition:
Definition:
Definition:
max singular value
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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 48.
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State Space Systems
Note:
zero-input response
zero-state response
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State Space Systems
Definition: zero-input response
Definition: zero-state response
Definition:
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State Space Systems
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State Space Systems
Theorem:
Example:
Proof: page 50.
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Lyapunov stability
Definition: homogeneous system
Definition: equilibrium points
Note:
Definition:
Theorem:
Note:
Lyapunov equation
(page 51)
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Reachability, Controllability, Observability, Detectability
Definition: reachability
Definition: Controllability
Definition: reachability matrix
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Reachability, Controllability
Theorem:
Proof: page 52.
Note:
Note:
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Reachability, Controllability,…
Theorem:
Definition: (stabilizable)
Lemma:
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Observability
Definition: (observable)
Note:
Definition: (observability matrix)
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Observability
Theorem:
Theorem:
(3.28)
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Detectability
Definition: (detectability)
Detectability is weaker than observability.
Theorem:
Theorem:
Proof: page 54.
unobservable modes are stable
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Canonical Decomposition of Linear Systems
Note:
Note:
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Canonical Decomposition of Linear Systems
Theorem 3.11.
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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:
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Realization Theory
Definition: (infinite block Hankel matrix)
Definition:
Definition:
Definition:
(Markov parameters)
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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
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Realization Theory
Lemma 3.10
Note
Lemma 3.11
Proof: page 66.
Definition: (rank of infinite block Hankel matrix H)
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Realization Theory
Definition:
Theorem 3.13
Proof: page 67.
Note:
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Realization Theory
Theorem 3.14
Thanks! … 
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