Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Assignment 4
Problem 1
Suppose that V : [0; 1) ! R is non-increasing and that V (t)
limt!1 V (t) exists and is nite.
Problem 2
Problems 4.1, 4.2, 4.3 in Ioannou & Sun.
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 9
Problem 4.10 from I&S
c) Recall the last assignment. If we select
1
11
0
0
=
22
(1)
;
we have the update laws
m =
_
_
=
_
k =
To enforce m
11 2
22 2
1 1
s2
y
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 13
Problem 1
a) From the transfer function of the plant we get,
s2 y + 2 ! n sy + ! 2 y = ! 2 u:
n
n
Therefore, the parametric model of the system is z =
s2
!2
Department of Engineering Cybernetics
TK17 System Identication
Solution to Subspace Identication Assignment
% [AalongBintoC]=obliqueproj(A,B,C)
% Returns A/_cfw_BC
function [AalongBintoC]=obliqueproj(A,B,C)
[r,j]=size(C);
M=pinv([C*C C*B;B*C B*B]);
Aalong
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 11
Problem 6.1 from I&S
Consider the control law
u=
(1)
k y + l r:
If b is known, k = 3=b and l = 2=b. However, since b is unkown we use the estimate of k
and
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 12
Problem 1
a) We write
Zp
up ;
Rp
Zm
= km
r;
Rm
y p = kp
(1)
ym
(2)
where kp = b1 , Zp = s + b0 =b1 , Rp = s2 + a1 s + a0 , km = 4, Zm = 1, and Rm = s + 5. C
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 10
Problem 4.13 from I&S
Let (s) = (s + 1)2 and lter the equation
y=
(u
m
y
(1)
y) ;
_
by 1= (s). Then, we have
1
y=
(s)
s2
m
y
(s)
1
u
(s)
s
y :
(s)
(2)
Denin
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 6
Problem 4.4 from I&S
We have from (4.3.29) in the book the following relation between the parameter estimation
error and the estimation error
~T
= WL
n2 :
s
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 4
Problem 1
Since V (t) 0, V (t) is bounded from below and thus has an inmum given by
(1)
Vm = inf V (t) :
t2[0;1)
Therefore, for any " > 0, there exists a t 2
Department of Engineering Cybernetics
TTK4215 SYSTEM IDENTIFICATION AND ADAPTIVE CONTROL
SOLUTION OF ASSIGNMENT 5
Problem 2.7:
Part (a)
The plant can be expressed by a differential equation
a2 a1 y a0 y b2u b1u b0u
y
y
(1)
T Y
y
(2)
T b0 , b1 , b2 , a2
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 3
Instrumental Variables Method
Problem 1 (Instrumental Variables Method)
a) From
h
i
1 X
() () + 0 () = 0
=1
we get
=
!1
1 X
1 X
() ()
() 0 ()
=1
=1
Le
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 2
Prediction-Error Methods The Least-Squares Method
Problem 1 (Optimal Predictions)
a) The prediction error is
" (t) =
ay (t
1) + bu (t
1) + e (t) + ce (t
1)
(
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 8
Problem 4.9 from I&S
c) The recursive LS algorithm for generating
is given by
T
_ =P
;
(0) =
0;
_
P = P
P
m2
T
P; P (0) = P0 = P0 > 0;
where
0 and P0 are des
Department of Engineering Cybernetics
TTK4215 System Identication and Adaptive Control
Solution 1
Non-parametric Methods
Problem 1 (Impulse-Response Analysis)
a) Since () = 0 for 6= 0the only non-zero term in the sum is for = Thus, we have
() =
X
=1
0 ()