Resume for Ladispe

8/10/2019 Resume for Ladispe

1/18

Politecnico

diTorino

14.1Cosimo Greco

THE OVERALL DEVELOPMENT SYSTEM

Overall technological structure (13/14)


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Politecnico

diTorino

24.1Cosimo Greco

n1n11i

1i

bbaa

e)ni(u)1i(u)ni(y)1i(y

herew)i(y

Model #1 can be rewritten as

n2mis defined information set (m 1) with

is the parameters vector(m 1).The model tf is:

n

1n

1

n

n

1n

1

n

n

1

1

n

n

1

1

azaz

bzb

zaza1

zbzb)z(G

ARX model (2/3)

#2


3/18

Politecnico

diTorino

34.1Cosimo Greco

ii

ii

||

1i

2i

Ni

||

yA

wichfrom

Ay

)i(y

)1i(y

)1Ni(y

From #1 (or #2), by ordering N samples backward from i,

we can obtain:

#3

N

m 1

mN:conditionNecessary

ARX model (3/3)


4/18

Politecnico

diTorino

44.1Cosimo Greco

Problem: inversion of ; IMPORTANT: is it invertible?

A possible solution: use of the QR factorization:

it is possible to determine a matrix QNN such that:

with Ri upper triangular and Q ortonormal

LS estimation (2/4)

iiAA

0

RQA

i

i

1QQ,IQQ

#5Q

N m

m

N-m0

Ai

Ri

m


5/18

Politecnico

diTorino

54.1Cosimo Greco

IMPORTANT from #6: is it true that ???0y~i

Premultiplying equation #3 by Q we obtain:

N.B: solution #6 can be also obtained from #4 by

substituing Ai from #5 and remembering that Q isortonormal.

i

1

ii

i

ii

yRwichfromy~

y

0

R

#6

1

m

LS estimation (3/4)

IMPORTANT: is Ri invertible?


6/18

Politecnico

diTorino

64.1Cosimo Greco

Primary aspects:

1. estimation of n (or of m)

2. inversion of Ri3. insertion of additional samples

4. real-time implementation

5. models of different structure

6. effect of the uncertainties

In the following, tools are provided only for items 14

LS estimation (4/4)


7/18

Politecnico

diTorino

74.1Cosimo Greco

RLS Solution KE The solution KE is derived from the Kalman estimator.

It can be demonstrated(+) that an efficient solution for the

problem #8 is the following:

________________* see next slides

iii

2

iiiii21i

ii

i1ii1i

P

PPP

1P

andi1iy

with)i1i(y)1i(yP

#9

m

1 m

proof


8/18

Politecnico

diTorino

84.1Cosimo Greco

Note: from the present generic instant k backward,

the forgetting factor acts by weighing the information

as follow:

at k by 0

at k1 by 1

at k2 by 2

at k3 by 3

at kN by N

is therefore said exponential forgetting factor

Recursive Least Squares RLS (3/4)

k-40 k-35 k-30 k-25 k-20 k-15 k-10 k-5 k

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

1


9/18

Politecnico

diTorino

94.1Cosimo Greco

L1

P

PL1

LPP

PPlim

cc1

c

cc

1

c

cc

1

c

1

c

1

c

1

ii

From the last result it is evident that the matrix Pc isnot computable as

1ccc L1P

because of the matrix is not invertible due to thefact that its dimensions are mm but its rank is unitary;this fact implies the covariance windup, that is:

cc

Solution KE in steady state conditions (2/2)

cP accuracy and convergence problems for


10/18

Politecnico

diTorino

104.1Cosimo Greco

Solution KE : initialization IMPORTANT: the recursive solution #9 requires the

initial values P0 and 0 of P() and () respectively

Three types of initialization are proposed in the following:

1.

2.

3. P0 and0 from previous estimation experiments carriedout on the same system

00001

000 yAP,AAP

...10000or1000or100with mrand;IP 00


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12/18


13/18

Politecnico

diTorino

134.1Cosimo Greco

m = n 1 (in general m < n)

Specifications:

closed loop poles: arbitrary placement r close loop zeros

system zeros to cancel +

system zeros to maintain

zeros to add a

closed loop DC gain KDC

POLEZERO PLACEMENT (2/7)


14/18

Politecnico

diTorino

144.1Cosimo Greco

NB: + and are roots of B(z), then we define:

Cancellation of B+

:

Then, referring to the closed loop zeros: B

+

iscancelled; B in maintained; additional zeros are the

roots of T


SBRATBW

SBBBRA

BTB

SBBAR

BTB

BSAR

TBW

BRR

Bcancelwantweif

mandBBB

withzbB;zB

BBB

1jjm

1ii

BB


15/18

Politecnico

diTorino

154.1Cosimo Greco

Note: the closed loop tf has the following form


r

r

r

r

r

rr

r

r

r

r

r

ASBRA

BTB

SBRA

TB

A

B

W

then,BfactorcommonthehaveBandA;SBBBRABSARA

BTBTBB

withA

BW

BSAR

TBW


16/18

Politecnico

diTorin

o

164.1Cosimo Greco

Without loss of generality, we can assume that the

polynomialsA, B+ and R are monic; T, B- and S are not

monic The unknowns of the design problem are T, R and S

Given the na zeros a to add, the not monic polynomial T,

of degree T, is given by

with KT to compute


T

1i

i,aT

aT

z)z(Twith),z(TK)z(T

n


17/18


18/18

Politecnico

diTorin

o

184.1Cosimo Greco

1. Given from the system model

2. given from the previous inequalities

3. we can fix the sets and then

4. compute

5. from is derived a system of equations

in unknowns, so R and S polynomials are computed

6. given the closed loop DC gain KDC, the coefficient KT of

T(z) is computed from


rASBRA

BBBA ,,,

TARS and,, r

ar and

Ar

1i

i,rr z)z(A

T

1i

i,az)z(Tand

rA

rA

1z

r

DCT BT

AKK

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