Week
nr. 
Date 
Topics 
2

9.1. 
Introduction to system
identification. Notations, state space models,
realization theory, hankel matrix, impulse response
matrices. Identification of impulse responses.
Hankel matrices and system order.
Lecture 1: Lecture notes
Ch.1, Section 2.2.5,
Exercise 1. Exercise 1.
Solution
1. Matlab script for
solution of the numerical part of exercise
Video Lectures: Introduction
Realization_theory

3

16.1 
Realization theory.
Singular value decomposition (SVD). Different state
space model realizations; output normal, input
normal and balanced realizations. System
identification of autonomeous systems.
Lecture 2: Lecture notes: Ch. 1, Sec. 2.2.5, Sec.
4.10
Exercise 2: Taken as Task 1 in exercises.
Video Lectures:
Lecture2a
Lecture2b
Realization_etc
Sid_autonomeous_syst
SVD_div
2019 Lecture2:
Problem description
Note: A link between impulse responses and
discrete zplane (or qplane) transfer function
models mentioned, as in in Ch. 2.3.7 in MPC_lect_notes

4

23.1

PCA, PCR og SVD. System
identification of steady state systems. The Least
squares (LS) method. Partial Least Squares (PLS)
regressjon.
Lecture 3: OLS, PCA and SCD, PCR
Exercise 3: Exercises 8 and 7 in Note
Video Lectures: Lecture3a
Lecture3b
Lecture3_ols
Lecture3_ols_b
Lecture3_PCA

5

30.1

Identification of
deterministic systems. Identification of system
order, the extended observability matrix of the
system and the dynamic properties, i.e.
identification of, n, O_L, A and D. Orthogonal
projection matrices.
Lecture 4: SID of deterministic systems. Autonomeous
systems intro
Exercise 4: Exercises 11 and 12 in Note
Video Lectures: Lecture
4a Lecture4b
Lect4_intro
Lect4_id_det_syst
Lect4_orth_pro

6

6.2 
Identification of
combined deterministic and stochastic systems.
Week 6. Basically computation of B and E. Week 7
basically how to handling noisey data. Subspace
system identification. Litle about the Kalman
filter.
Lecture5: SID of the general problem, i.e.
Combined deterministic and Stochastic Systems.
Lecture notes Ch. 3, Ch. 4, Ch. 8.3.2, 8.3.4
Video Lectures: Lecture
5a Lecture5b
Lecture_combined
Lecture_Kalman_intro

78

13.2

Subspace identification
of combined deterministic and stochastic systems,
and identification of the innovations process and
closed loop systems.
Also theory as described previous week.
1) Closed
and Open Loop SID of Kalman Filter. Ch. 6
2) Paper, "On sys id of Kalman filer..", Lemma
3.9, page 15 and Ch 5. Se item 5 in syllabus list
and micjournal paper: http://www.micjournal.no/ABS/MIC200923.asp
Video Lecture: Lecture6a
Lecture
6 Lecture6b
(Closed Loop Subspace ID)
Remark: Unfortunately without sound records.
Other videos: Se
link

8

20.2 
State estimation and the
kalman filter for linear systems. Innovations
formulations and aprioriaposteriori formulation of
the kalman filter for discrete time linear systems.
Lecture notes: State estimation
and Kalman filter, Ch. 2.2, 2.6.2, 2.6.3
Video Lectures: Lecture7a
Lecture7b
Lecture_Kalman_intro
Lecture7_2019
(Continuos Kalman filter, Ch.2.2)

9

27.2
7.3

Prof of equation for kalman gain matrix.
Video Lecture: Lecture8
Exercise: Calculating the Kalman
gain matrix.
Exam 2014: Tasks 4 kalman filter. Task 5 and 6.
Susbspace SID, shift invariance principle and
feedback in data. Details in Ch. 6 in micjournal
paper
http://www.micjournal.no/PDF/2009/MIC200923.pdf
MATLAB: Kalman filter example: main_ex_march6.m

10 
7.3 
Prof of some Kalman
filter equations. Lecture notes: State estimation
and Kalman filter, Ch. 2.6.2, 2.6.3.
State estimation and the Kalman filter for non
linear systems, the Extended Kalman Filter (EKF). Lecture notes:
Ch. 3
Video Lectures: Lecture9
(The EKF)
Lecture78_EKF

11 
13.3 
Kalman filter and
introduction to prediction error methods for system
identification.
Video Lecture: Lecture10
(Introduction to PEM
Exercise : Kalman filter exercise. Work through
parts of Kalman filter exercise 2.

12 
20.3 
Prediction error methods
for system identification, parameter estimation and
Kalman filter. Polynomial, ARMAX and state space
models.
The prediction error method and linear regression
models. Ch. 2,3, 2.3.2
Video Lecture: Lecture11
Exercise: Curve fit example using MATLAB polyfit.m
and polyval.m functions to fit a polynomial to data
X and Y, and MATLAB plotting facilities. main_ex_polyfit.m

13 
27.3 
TOPICS
1. Prediction error methods. More about polynomial
models, ARX, OE, BOXJenkins etc. as in Sec 3 in lecture
notes. This is also the topic in Week 15.
2. SSPEM Toolbox
for MATLAB, Section 6.1 in lecture
notes.
3. A MIMO (m=2 and r=2) system with n=3 states
syntetic example. MATLAB mfiles linex2n2.m utype.m
4. Estimate State Space (SS) model using DSR Toolbox for MATLAB
and the MIMO example mentioned above.
5. EXAMPLE (video
record): SISO 1st order general linear model.
State Space model, Linear Regression model, ARMAX
and ARX models). Unfortunately noisy sound on the
record.

14 
3.4 
The prediction
error method and the Ordinary Least Squares (OLS)
method. ARX models and the OLS method. Statistical
analysis of the OLS estimate. The Best Linear
Unbiased Estimator (BLUE).
The recursive OLS method, Section
7, p. 26
Video Lecture: Lecture12
Exercise: Illustration of ROLS method Ex. 7.2
with mfile: main_rols_ex.m ,
prbs1.m

15 
10.4 
Input and Output Model
structures (polynomial models), lecture
notes Ch.3 p. 14. ARX, ARMAX models etc.
Video Lecture: Lecture13




16


Easter




17 
24.4 
Topics for Lecture14:
1. How to handle trends.Ch.9
2. Model validation.Ch.10
3. Input experiment design. Ch. 11. MATLAB
function prbs1.m to
generate Pseudo Random Binary (PRBS) input signal
experiments.
The above chapters is in the "subspace" lecture
notes.
Lecture14

17 
24.4 
EXERCISE
1) Work with earlier exercises
2) System identification by first identification of
a higer order ARX model and following by model
reduction. MATLAB function: harxmr.m This
function is using mfile: hank_m.m
Quote from Ljung (1999): "A high order ARX model
is capable of approximating any linear system
arbitrarily well."
3) Identification of nonlinear systems.
reformulating as a linear regression problem. Lecture notes.

17 
24.4 
Summing up the main
topics:
1) Realisation theory.
2) Subspace based methods for system
identification.
3) Optimal state estimation and the Kalman
Filter.
4) Prediction error methods for system
identification.
Summary on Web



