Week nr. 
Date 
Topics 
2 
11.113.1 
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
Lecture notes on Finite Impulse Response (FIR) models
Video Lectures: Introduction
Realization_theory
prbs1.m for experiment design (download here or on exercise page)
MATLAB example main_ex1_lecture1.m

3 
18.120.1 
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
Lecture notes on Finite Impulse Response (FIR) models
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
MATLAB example: Realization theory

4 
25.127.1 
PCA, PCR og SVD. System identification of steady state systems. The Least squares (LS) method. Partial Least Squares (PLS) regression. PLS only mentioned and not in details.
Lecture 3: OLS, PCA and SVD, PCR
Exercise 3: Exercises 8 and 7 in Note
Video Lectures: Lecture3a Lecture3b
Lecture3_ols
Lecture3_ols_b
Lecture3_PCA

5 
1.23.2 
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 Solution proposals for Exercise11/Task 11 here: losn_oppg10.m
Video Lectures: Lecture 4a Lecture4b
Lect4_intro
Lect4_id_det_syst
Lect4_orth_pro 
6 
8.210.2 
Identification of combined deterministic and stochastic systems.
Computation of B and E from tilde(B) matrix. How to hande noisy data. Subspace system identification. Little 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

7

15.217.2

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 system identification of the Kalman filter..", Lemma 3.9, page 15 and Ch 5 in 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
MATLAB mfile example, main_clopid_ex1,m

89 
22.224.2 (1.33.3) 
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)
Example: main_oppg13_ext.m

9 
1.33.3
3.3 
Prof of equation for kalman gain matrix.
Video Lecture: Lecture8
Exercise: Calculating the Kalman gain matrix. (MATLAB mfile)
Simple proof of 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

9 
2.33.3 
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 
10 
8.310.3 
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 as described in Exercise plan, Week 1213, with mfile solution proposals.

11 
15.317.3 
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. (Syllabus from Item 4 in syllabus page. )
Work out the proof of Eq. (55) on page 11.
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

12 
22.324.3 
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. 
13 
29.331.3 
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
MATLAB demo

14 
Easter 

15 
12.414.4 
Input and Output Model structures (polynomial models), lecture notes Ch.3 p. 14. ARX, ARMAX models etc.
Quote from Ljung (1999): "A high order ARX model is capable of approximating any linear system arbitrarily well."
A view on closed loop subspace system identification, the DSR_e algorithm and MIC paper.







16 
19.421.4 
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
Exercises: As in ExercisePlan

17 
26.428.4 
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 
26.428.4 
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 


