Feedback in general discrete time systems

Feedback in general discrete time systems

Citation for published version (APA):

Hautus, M. L. J. (1978). Feedback in general discrete time systems. (Memorandum COSOR; Vol. 7817). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

providing details and we will investigate your claim.

81

COS

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 78-17

Feedback in general discrete time systems

by

M.L.J. Hautus

Eindhoven, september 1978 The Netherlands

TIME SYSTEMS

by M.L.].

HautuA*

ABSTRACT.

The paper presents a description of the feedback problem for general discrete time systems. By an extension of a method developed in [5J for linear systems, a characterization of the (static,state) feedback class of an arbitrary plant is obtained.

*

Dept. of Mathematics Eindhoven University of Technology, Eindhoven, The Netherlands.

1 •

Z ntJwduc;tum

An obvious question 1n control theory is as to how we can change the system dynamics 6f a plant using feedback, in particular, using feedback without dynamics. This problem can be stated more precisely as follows: Determine the feedback class of a system, tha~ __

!::;_!

the class of .~y~_tems

obtained by static feedback. This question has been investigated in detail in literature, in particular for linear systems. (see, however, also [IJ, where feedback invariants for nonlinear systems are given). It has become apparent that reasonable results can only be expected in the case of state feedback. The research in linear system theory resulted in the discovery of a complete sets of invariants by P. Brunovski ([2J, see also [6,10J). It turned out that these feedback invariants have significance not only for problems not directly related to feedback but also for realization theory, in particular for the description of canonical forms of realizations (see [3,4,8,9J). The connection between realization theory and the feedback problem has been revealed in a more direct way in [5J, where the feedback class of a plant has been characterized but not with the aid of canonical forms, but utilizing the fundamental treatment of realization theory by R.E. Kalman, given in [7J. The method used in [5J is of a fundamental nature, and it is not surprising that it can be generalized. It is the purpose of this paper to describe the essential ideas of the method in the setting of "general system theory", that is, on the set theoretic level. Of course, one cannot expect deep results in this setup. However, it is hoped that this paper will give an indication of the method of treatment and the type of result to be expected, in the case of systems with an algebraic or topological structure.

In order to simplify the description we restrict ourselves to discrete time systems, although it is likely that the results can be extended to the continuous time case.

In section 2, we give a definition of general discrete time i/o maps. In section 3, the feedback problem is formulated. Since, as usual, we will restrict ourselves to state feedback, state response maps and (semi)

realizations are defined in section 4. Using an algebraic terminology, intro-duced in section 5, we give necessary and sufficient conditions for a

state response map to be a semirealization of a given response map. This result is used to characterize the feedback class of a plant in section 6.

+

If S is a set,we denote by S the free semigroup generated by S, i.e., the set of sequences 0 = sl",sn with sk E S(k=l",.,n), If we adjoin the

empty sequence e to S+, we obtain S*, the free monoid generated by S. The set S is imbedded ~n S+ and S* in an obvious way. The length

101

of 0 is defined by n, if 0

=

sl ••• sn' sk E S (k=I, ••• ,n). Also, lel:= O.

;-

*

In Sand S multiplication is defined to be concatenation: If 0 = sl.··sa'

• =

tl,··t_m, then 0.:= sl, •• sntt ••• tm'

I n a dd . . ~t~on,eo:= oe := 0, Th en S + . . ~s ~ndeed a se~group . an d S'i1- a monoL • d (= semigroup with unit element). Finally, we define the map A: S+ + S by

s •

m

Let U be a set, called the input alphabet, Y a set called the output alphabet. We do not require these sets to be finite. Consider a map

+

f: U + Y

which will be called the response map,

- .... : _ - : _ _ _ ...al

yaf~w)

The interpretation of this map is as follows: If the input string w E U+ is applied to a system with response map f, then the output after the application of this string will be y

=

f(w), It will be assumed that the system yields the output y at the same instant at which AW is applied, so that there is not necessarily a delay in the system. The set of response

+ .

maps f: U + Y wLll be denoted r(U,Y).

Corresponding to a response map f we define the i/o-map (input/output map)

-

f: U

*

+ Y

*

by: f(e):= e

-

_{f(wu):= f(w)f(wu)}

-

_{(w E U ,}

*

_{U E U)}

(

_{Here we have denoted the empty sequences of U and Y both}

*

*

_w~th

.

_{t e same} h symbol e). The sequence few) gives the output string (rather than only the last element) due to the input sequence w. Obviously, the response map is uniquely determined by the i/o-map:

few)

=

Xf(w) (w E U ), +

In order that a map F: U* + Y* be an i/o-map corresponding to some

response map it is necessary and sufficient that F satisfy the following properties:

i) F is length preservin~, that is, IF(w)!

=

Iwl for all w €

u*,

ii) F is causal, that is, if F(w

tw2)

=

YtY2, ly11

=

IwII, then Y1

=

F(wI). Notice, that i) implies F(e)

=

e. It is easily seen that the composition of two causal maps and the inverse of an invertible causal map is causal. DEFINITION. A pesponse map f: U+ + Y

is

caZZed stpictZy causal iff(~), whepe

00

=

u ••• u , depends only on u1""'u 1 and not on u

=

AW

n n- n

The map f

is

called static if few) depends only on AW

*

If f is a strictly causal map, we define f : U + Y by

(w):= f(wu)

for any u € U. In particular,

where u E U is arbitrary.f_ is called the strict response map corresponding

to the strictly causal response map f~ For every map h: U* + Y, there exists

exactly one strictly causal map f: U+ + Y such that f_

=

h.

A static map is uniquely determined by the map fa: U + Y: u ~ feu).

Every map F: U + Y determines uniquely a static response map U+ + Y,

which we also denote by F (it there is no danger of confusion) and which is defined by

3. Feedbac.k.

Our objective is to study feedback in the general setting of the previous section. Suppose we are given input alphabets U and V and an output

alphabet Y. Let f: V+ + Y ,

+

L: (Y x U) + V

be response maps. We say that L defines a feedback connection for the

+

plant (= given system) f, if there exists a unique response map f

L: U + Y,

+

such that for every W E U we have:

(3. 1) where (3.2) <p

=

L(y,w) r -I I j

'-I

L

1

--

-I

<1'.

:

f

-

-

---

....,

I

r

y I II

I

I

..

I

- -

_ ...J

The diagram gives a symbolic representation of equations (3.1) and (3.2). If we use the notation w

=

u1, .. ·,u

n' <P

=

v1'· .. ·,vn ' y

=

yl'····'yn' then these equations read as follows:

Yk = f(vIo •• v k)

vk

=

L(Yl"'Yk' ul···uk) for k

=

1, ••• , n.

(3.3) LEMMA If f:

v+

+ Y is strictly causal then every L E r(Y x U,V)

defines a feedback connection.

0

If £ ~ r(Y x U,V), then we want to investigate the following problem:

Given f:

u+

+ Y, determine the

£-

feedback class of f, i.e. {fLI L E

£}.

In view of lemma (3.3) we restrict ourselves to the case where f is strictly causal. It will be seen in a moment that the problem is rather trivial

in the case £ = r(Y x U,V). A more interesting class £ is the set of all

It follows from (3.1) and (3.2) that

(3.4) qJ

=

L (f(qJ),w)

By the strict causality of f, we may solve qJ from this equation, which

yields,

qJ

=

~(w)

+

for some response map ~: U + Y. Hence the feedback connection is equivalent

to the cascade connection

w

..

-y

=

f 0 i(w),

f

...

y

that is, fL = f 0 i. It follows that every response map obtainable by a feedback connection is also obtainable by a cascade connection. The problem of the determination of the £-feedback class of f can be

rephrased as: "For what response maps R, does these exis t L E £ such that fL

=

foR,. If £

=

r(Y x U,V) then every cascade connection is obtainable

by feedback: Just take L(y,w)

= R,(w)!

Henceforth we will only consider

+

the case where £ is the class of static response maps (Y x U) + V,

and we denote the corresponding map LO (see section 1) simply by L. Hence, L: Y x U + V. In addition, we require that for every y E Y the

map u

*

L(y,u) be invertible. The equations of the system with feedback read

Yk

=

_{f(vl,···,vk),} vk

=

L(Yk'~)'

The class of feedback maps L of this form will henceforth be denoted by £. If L E £ we say that L is a regular static feedback~

REMARK. For lineair equations, the feedback equation reads

The invertibility condition imposed on L(.,.) corresponds to the invertibility of the linear map V.

The inverse map of ut+ L(y,u) will be denoted by VH- M(y,v). It follows

from (3.4) that, if L E £, the i/o-map R, is invertible and that the

inverse is given by (3.5) where (3.6) --1 -9..

=

m

The invertibility of

i

is a necessary but not a sufficient condition for 9.. to be obtainable by regular static feedback. In order to formulate a sufficient condition we introduce a new concept:

+ +

DEFINITION. Let g: U ~ Y, f: U ~ Z and H: Y x U ~ Z. We say that (g,H)

is a model of f if

few)

=

H(g(W),AW) (w E: U ) +

+ +

If g: U ~ Y, f: U ~ Z, we say that f is a semimodel of f is there

exists H such that (g,H) is a model of f.

u

r

I I I I I I

-

Y g

-

- ':-1

f

I

I

I

H

I

I .... _____ - - - I

The following is an immediate consequence of (3.5) and (3.6): z

.

PROPOSITION: Given a strictly causal response map f: v+ ~ Y,

and

a invertible

i/o-map"1: U* ~ v*, there exists L E: .t such that fL

=

f 0

i

iff f is a

- --1

semimode l of m" where m: = 9..

+

Thus, our problem is reduced to the following problem: Given f: V ~ Y,

We will answer the question posed at the end of the previous section for the case where f is a state response map.

(4.1) DEFINITION

A strictZy causal map

f: u+ + X

is calZed a state response

o

map if

f_(wu)

only depends on

f(w)

and

u,

that

is~

if there exist

x € X

+

and

F: X x U + X

such that for every

w

=

u

t" •• un € U ,

we have

o

xl ::: X (k = I, ••• , n-1 )

-where

xl" ,x

n .,. ~

=

f(w).

An

i/o'-map

f

corresponding to a state response

map

f

will be called an i/s-map.

(4.2) DEFINITION.

Let

U+ + Y

be a response map. A pair

(g,H),

where

g: u+ + X

and

H: X x U + Y

is caUed a realization of

fis g is

a state

+

response map and

(g,H)

is a modeZ of

f.

A state response

map g: U + X is

called a semirealizationof

f

if it

is

a semimodel of

f.

(4.3) DEFINITION.

A state response map

f: u+ + X

is calledreachabZe

if it is surjective.

For reachable state response maps g a condition on f for g to be a semirealization of f can be given in terms of Nerode equivalence. In

order to facilitate the formulation, we introduce an algebraic terminology.

5. U

*

-modui.u

In this section, U will be an fixed set, called the input alphabet. (5.1) DEFINITION.

A pair

(S,~)~

where

S

is a set and

~: S x U* + S,

*

is called a (right)

U

-module if

~(s,e) = s

~(s,wlw2) .,. ~(~(s,wl),w2)

*

Usually we speak of the U -module S rather than (S,~). Also we denote the action ~ by juxtaposition: ~(s,w) ::: sw. Consequently, the following properties hold

(5.2) EXAMPLES

1) U* 1.'tself 1.'S a U* -mo u e, were, 1. d 1 h of S"" W we d fO e 1ne SW

.-I.- ww 1. Here the juxtaposition in the right-hand side denotes concatenation.

2) If X is a set and F: X x U + X a map, then the formulas

xe _{:= x,} xu := F(x,u) x(wu):= (xw)u (u E U), + (w E U , U E U),

*

define a U -module structure on X.

3) If Y is a set, then r(U,Y) is a U*-module with fw defined by

o

The usual algebraic concepts can be defined for U*-modules: homomorphisms submodules, congruences, quotient modules. We omit the details.

As a consequence of example 2, state response maps can be characterized as follows:

PROPOSITION. A response map g: U+ + X is a state response map iff it is strictly causal and X can be endowed with a u*~odule structure such that

*

g_: U + X is a homomorphism.

6. Nvwde map.6

In this section U* and r(U,Y) will be considered U*-modules as described in the example of §5.

(6.1) DEFINITION. Let f E r(U,Y). Then ~

*

f: U + r(U,Y): w ~ fw

o

is called the Nerode map of f. Here fw denotes the result of the right

*

U -moduZe action of w on f. Hence '" f(w)(w 1) "" f(wwl) (w E U

*

,w 1 E U ). +

If Sand T are sets and F: S + T is a map, we denote by ker F the

(6.2) DEFINITION. If f: U+ + Y is a response map then ker

f

is aaZZed

the Nerode equivaZenae reZation of f

(6.3) PROPOSITION. If f E r(U,Y), then

1:

u* + r(U,Y) is a U*-homomorphism

~

*

and ker f is a congruenae on U •

We have the following characterization of reachable state response maps:

+

(6.4) THEOREM. Let g: U + X be a strictZy causaZ response map. Then g

i~ a reaahabZe state response map iff

(I) im g_ == X (2) ker g_ == ker g.

PROOF. Condition (I) is equivalent to the surjectivity of g. It is easily verified that the inclusion

ker

g

Eo ker g_

o

is always valid. Suppose that g is a reachable state response map. Then

X is a U*-module and g_ a homomorphism. If g_(w

l)

=

g_(w2) for some

U

*,

h +

w

1,w2 E t en we have for all W E U :

where we have used the U*-module action on X. Hence g(w1w) == g(w₂w)

+ ~ ~

for all w E U , that is g(w

t)

=

g(w2).

Conversely assume that ker g_ == ker

g

and

* .

there exists an isomorphism : X + U /ker g_

that g_ is surjective. Then

*

'"

== U /ker g such that the diagram U* II

*

~ U /ker g 11"

commutes. Here n denotes the canonical projection. The isomorphism~

*

-1

defines a U -module structure on X and g == ~ on is a homomorphism.

0

The next result gives a necessary and sufficient condition for a

state response map to be a semirealization of an arbitrary response map.

+ +

THEOREM. Let f: U + Y be a response map and g: U + X a reachabZe state

response map. Then the foZZowing conditions are equivaZent.

1) g is a semireaZization of f 2) ker g_

=

ker

g

£ ker

f.

u* _____

f _ _ _ --l ...

_r

(U, Y)

aommutes.

PROOF. 1 - 2. Let (g,H) be a realization of f. Then few)

=

H(g(w),w) for w E U+. Suppose that for some WI w

2 E U* we have S(wt)

=

S(w

z)'

,

Then, g(wtw) = g(wZw) for all w E U+ and hence

2) - 3) This is a well-known situation: For x E X, choose any w E U*

with g(w) = x and set S(x):= few). Condition 2) guarantees that

e

is a well defined homomorphism.

3) - 1). Define H: X x U + Y by

H(x,u):= 8(x)(u)

We show that (g,H) is a (reachable) realization of f, that is, we show

+

that for each w E U , U E U the equality f(wu)

=

H(g_(w),u)

holds. In fact:

H(g_(w),u) = 8(g_(w»(u)

=

f(w)(u)

=

f(wu) D

""

We see that for a state response map g the condition ker g~ ker f is necessary and sufficient for g to be a semirealization of f. The following theorem shows that this property is characteristic for reachable state response maps.

THEOREM. Let g:

u+

+ X be a surjeative striat~y aausa~ response map.

Then g is a (reaahab~e) state response map iff g is a semimode~ of every

+ ~ ""

response map f: U + X satisfying ker g ~ ker f.

PROOF. The "if" part has been shown in the previous theorem. Assume that g is a surjective strictly causal response map which is a semimodel of

+ I"o<J - . J

*

""-J

every response map f: U + X with ker g ~ ker f. Define S:= U /ker g

and let f: U+ + S be the strictly causal response map such that f is

the canonical projection U* + S. Then f is a state response map (see

£ is the canonical projection we have ker f - ker

g.

Consequently, ...

ker f - ker g. By assumption, g is a semimodel of f, i.e., there exists

*

R: X x U + S such that for every u E U, W E U :

f

_-

(w)

=

f(wu)

= R(g (w),u).

-'"

This equality implies ker g_ ~ ker

£

=

ker f. Now the result follows

from Theorem 6.4. 0

7.

Feedback equivalence

Using the results of the preceding section we can give a solution to the problem formulated in section 3, that is, the determination of all response maps t such that f 0

t

=

fL for some t E l .

(7. 1) THEOREM. If g: V + + X is a reachab Ze state response map and

+

t: U + V, 'then there exists L E l such that g 0 1

=

gL iff the foUowing

conditions are satisfied

i) 1: U* + v* is invertibZe.

ii) kerg

g

£ ker ;, where m:=

t-

l •

In the remaining part of this section we pay attention to the l-feedback class of g, that is, to the set of response maps gL where

L E l .

(7.2) THEOREM. Let g be a reachable state response map and Zet L E l.

Then gL is a reachabZe state response map.

o

This result follows easily from the definition. If the state transition

1.S given by

~ + 1 = F (~ ,v_{k ) ,}

then the state feedback v_k

=

L(~ ,~) yields

with

FL(x,u):= F(x,L(x,u».

The reachability of gL is a well-known a straightforward property. In the following we assume that V

=

U. In view of our invertibility assumption on L this is not a real restriction. The~ we can define a product relation on l by:

M 0 L(x,u):== L(X,M(x,u».

It is easily seen that with this multiplication £ is a group, with identity I(x,u) == u. This group acts on the set of reachable state response maps and satisfies:

For this reason we can speak of feedback equivalent state response maps and we might investigate invariance properties. There are two reasons to expect that such a direct approach might be difficult.

(7.4) In the linear case, we have a feedback of the form

v == -Lx + Vu

where V is invertible. In turns out (see [2J) that a simple description of the feedback invariants can only be given, if in addition to £

one also has the group of state space isomorphisms x ~

Ax,

with A: X ~ X nonsingular, acting on g. Accordingly, in the general case one must expect, that simple results are only obtainable if one admits in

addition to £ the group A of bijections A: X ~ X acting on g. That is; for every A E A one has the map:

-1

g~ gA: x ~ A(F(A (x) ,u)

The group obtained by combination of £ and

A,

however, is rather complicated.

0

(7.5) If we consider two state response maps simultaneously, the algebraic

terminology introduced in section 5 cannot be used, unless we allow two different U*-module structures on X. It seems unlikely, that this

is a convenient description (see also [3J).

0

We will call two reachable state response maps gl: U+

~

X , g2! U+ ~ X

2 isomorphic, if there exists a U-isamorphism a: Xl ~ X2 such

that the diagram X

~l

u~la

. 2

*

commutes. It is allowed that the U -modules Xl and X₂ have the same carrier set, like in (7.4).

Rather than dealing with the state space X itself we use the Nerode equivalences of the response map. This approach is based on

(7.6) PROPOSITION.

*

(i) If

C

is a oongruenoe on U ~ there exists a (reaohabLe) state

+

response map g: U ~

x,

suoh that ker g_

=

C (ii) If gl: U+

~

Xl' g2 : U+

~

X

z are reaohable state response maps satisfying ker g1- = ker gz-' then gl and gz are isomorphio.

PROOF. (i) Define XC:= U*jC and let TI: U* ~ X be the canonical projection.

If gC: U+

~

X is the strictly causal response map such that(gC)_= TI,

then gc is a reachable state response map and ker(gC)_

=

C.

(ii) We note that every reachable state response map g: U+

~

X with ker g_

=

C is isomorphic to gC' This follows from the existence of a U*-isomorphism a such that the following diagram commutes

It follows that if we work with Nerode equivalence classes instead of state spaces, then the state space isomorphisms mentioned in (7.4)

o

are built in. In addition, all Nerode equivalence classes to be considered are submodules of one U*-module, viz., U* x

u*,

so that we do not encounter the difficulty mentioned in (7.5)

(7.7) DEFINITION. Two oongruenoes C₁ and C

z on U* are oaLLed feedback

equivaZent if there exist feedbaok equivaLent state response maps

+ +

gl: U ~ X, gZ: U ~ X such that ker gl- = C_1' ker g2- = _C2• We introduce the following notation: If: f: S ~ T is a map, then

2

f : S x S ~ TxT is defined by

then we can state the following result:

(7.8) THEOREM. Two oonvergenoes C 1 and C

2 on U* are feedbaok equivaZent

+

iff there exists a map m: U ~ U BUoh that

-

*

*.

(i) m: U ~ U ~s invertibLe ( ii)

-z

_m (C) C I =~ Z (iii) C 1 c ker m

PROOF. Suppose that there exists a map m satisfying (i), (ii) and (iii). Define X:= U*/C

1 and let gl: U+ + X be a response map such that

*

gl_: U + X is the canonical projection. Then gl is a reachable state response map satisfying ker gl

=

C

1 (see (7.6». De;fine t: U+ + U by

- --1

t:= m

Condition (iii) guarantees that g2:= gl G £ is a reachable state response

map obtainable from gl by regular static feedback. It remains to be shown that_ker g2- = C2~ Now, (OOl'~~) E kerg 2_ ~ff g2:(OO₁₎

=

g2-(oo2) iff

gl_(I(00₁))

=

gl_(I(00₂)) iff t (00

1,002)

=

(1(001),1(002» E ker gl-

=

C

1

iff (00

1,002) E

C

2

(by (ii».

The converse statement is immediate. D

(7.9) REMARK. It is easily seen that the map m mentioned in theorem (7.8)

- --1

satisfies the following property.

C

2 c ker 1, where 1:= m

0

Finally, we give a result about the existence of a static feedback which is not necessarily regular. Here we drop the assumption

u

=

V.

+

(7.19) THEOREM. Let g_:V + X be a reac/habte state response map and Let

+

1: U + V. Let gl:= g o t . These exists a statia feedbaak L: X x U + X

suah that gL

=

gl iff

i) gl is a state response map

ii) ker

8

1 c ker

1.

15

-REFERENCES

[ 1 J Brockett, R. W., "Feedback invariants for nonlinear sys tems", IFAC VII World Congress, Preprints, 1978, pp.lII5-1120.

[2J Brunovsky, P., "A classification of linear controllable systems". Kybernetica 6 (1970), pp. 173-188.

[3J Eckberg, A.E., "A characterization of linear systems via polynomial matrices and module theoryl l . M.I.T. Electronic Systems Laboratory Rep. ESL-R-528, Mass. Inst. of Techn., Cambridge, MA, 1974.

[4J Forney, G.D. "Minimal bases of rational vector spaces". SIAM J. Control 13, 1975, pp. 493-520.

[5J Hautus, M.L.J. and Heymann Michael, "Feedback - An algebraic approach". SIAM. J. Control, 16, 1978, pp. 83-105.

[6J Kalman, R.E., "Kronecker invariants and feedback", Proc. Conf.on Ord. diff. Eqns., NRL Mathematics Research Center, 1971; in L. Weiss (Ed). Ordinary Differential Equations, Academic Press New York, 1972.

[7J Kalman, R.E., Falb P.L., Arbib. M.A., "Topics in Mathematical Systems Theory". McGraw-Hill, New York, 1969.

[8J Popov V.M., "Invariant description of linear time-invariant controllable sys tems", SIAM J. Control 10, (1972), pp. 252-264.

[9J Wolovich, W.A., "Linear Multivariable Systems", Applied Mathematical Sciences Series no. II, Springer-Verlag, Berlin, 1974.

[10J WoDham, W.M., "Linear Multivariable Control: A Geometric Approach", Lecture Notes in Economics and Mathematical Systems, no. 101, Springer-Verlag, Berlin, 1974.