3. Closed Loop Robust ID

3.1. Residual Errors
Given a P ?? structure and estimated output noise, ~n, the errors

e :=

ae ey
eu

oe

=

ae y

u

oe

?

ae ~y

~u

oe

(5)

between the predicted (denoted by~) and measured values of the plant outputs and inputs can be related directly to the structured uncertainty. Figure 1 show these errors for

S-KyuTrue plantnSy~ShxP12P11P21P22DKSSSn~reyeurfu~-

Figure 1: General block diagram for robust ID

a general closed loop system ID experiment. It is assumed that the controller, K, and external command, r 2 Rnr , are known and the plant inputs, u 2 Rnu, and outputs, y 2 Rny , are measured. The fictitious signals have dimensions ? 2 Cn? and ? 2 Cn? where

n? =

oX

j=1

nj; n? =

oX

j=1

mj (6)

The symbols, Rnu and Cn? denote real and complex vector spaces of dimension nu, and n? respectively. For the case when external disturbance at the plant input is present, it can be modeled approximately by adding a filtered disturbance at the output.
The predicted outputs and inputs are given by

ae ~y

~u

oe

= T(?)

ae r

~n

oe

(7)

where

T(?) =
h (I ? Fu(P; ?)K)?1Fu(P;?) (I ? Fu(P;?)K)?1

(I ?KFu(P; ?))?1 (I ? KFu(P;?))?1K

i

(8)
Note that T represents the four components of transfer matrices that define internal stability of a general two block feedback system consisting of Fu(P; ?) and K. Since the closed loop ID experiment is internally stable, the system consisting of Fu(P; ?) and K is assumed to be stable also. This translates to robust stability of (P; K) with respect to ?.

In order to solve for the uncertainty, it will be more convenient to rewrite matrix T(?) in equation 8 so that

? appears as an argument in an LFT. This useful form can be summarized as follows:

Lemma 1:

ae ~y

~u

oe

= Fu(R; ?)

ae r

~n

oe

(9)

where

R =

" Fl(P; K) P12(I ?KP22)?1[I K] h I
K

i
(I ? P22K)?1P21 T (0)

#

2

(10)
Proof of Lemma 1: Consider the basic relations:

? = P11? + P12~u (11)

~y = P21? + P22~u + ~n (12)

~u = K~y + r (13)

? = ?? (14)

Equations 11 to 13 can be rearranged to

2

4
I ?P12

(I ? P22K) (I ? KP22)

3

5

8<

:

?

~y

~u

9=

;

=

2

4
P11 P21 P22 I

KP21 I K

3

5

8<

:

?

r

~n

9=

; (15)

Using the partitioned matrix inverse identity the coefficient matrix in the left hand side of equation 15 can be inverted so that
8<

:

?

~y

~u

9=

; = R

8<

:

?

r

~n

9=

; (16)

Equations 14 and 16 gives 9. 2

T (0) denotes the nominal value of T(?) in equation 8

T (0) =

<= (I ? P22K)?1P22 (I ? P22K)?1

(I ? KP22)?1 (I ?KP22)?1K

>=

(17)

Note that T (0) = Fu(R; 0) corresponds to the four component transfer function matrices of the two block nominal feedback system. Hence, internal stability of the nominal closed loop system is equivalent to stability of T (0). Using Lemma 1, the error in equation 5 is written as

e = eo ? R21?(I ?R11?)?1M12 (18)

where

eo =

ae y

u

oe

? T (0)

ae r

~n

oe

(19)

M12 = R12

ae r

~n

oe

(20)

Note that eo is the residual from nominal fit, i.e., when ? = 0.