Discrete time H ∞ controllers satisfying a minimum entropy criterion

Systems & Control Letters 14 (1990) 275-286 North-Holland

275

Discrete time controllers satisfying a minimum entropy criterion P.A. IGLESIAS, D. M U S T A F A and K. G L O V E R Department of Engineering, Trumpington Street, Cambridge University, Cambridge CB2 1PZ, England Received 7 August 1989 Revised 3 January 1990

Abstract: A discrete time, minimum entropy ~'~ control problem is solved and state space formulae are derived. The solution is obtained from first principles and so avoids the reformulation of the problem to a continuous time problem via a bilinear transformation.

Keywords: Minimum entropy; ,,~oo control; discrete time systems; state space control; suboptimal extensions.

1. Introduction

This paper deals with a particular type of discrete time suboptimal . ~ controller: one which is required to satisfy a minimum entropy condition. While it has been the continuous time 9~'oo control problem that has been given the most attention in the literature, see [5] and the references therein, suboptimal controllers have attracted some interest of late. In particular, suboptimal controllers satisfying a minimum entropy condition have been shown to have some advantages over optimal controllers; see

[8,10,111. We begin by motivating this particular choice of suboptimal controllers in a general input-output framework. Consider the standard plant as in Figure 1, where [Pll

P= LP ,

P12]

P22J

is partitioned conformally with x, y, z, and w, such that

Let G be the closed loop operator from the external input w to the external output z. Then G can be expressed by a lower linear fractional map: G =o~l(P, K ) =." Pll + P12K( I - P22K)-IP21 .

Z I

I P Ii w Fig. 1. Standard plant.

0167-6911/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

P.A. Iglesias et aL / Minimum entropy ~

276

controllers

The optimal 9f'~ problem is then to find, among all stabilizing controllers K, the particular Kop t that minimizes the infinity-norm of ~ ( P , K), i.e. Problem 1.1. Find

Kop t

such that

i n f ( I1o ~ ( P , K ) I 1 ~ : K stabilizes P ) =:3'0 = II ~ t ( P , Kopt)II K

~"

In the suboptimal ~g'~ problem, the bound 7o is relaxed and the requirement becomes that, given 3' > 3'0, the suboptimal controller Ksopt satisfy l[ ~ ( P ,

K~opt) II ~ < 3'.

As will be seen in the sequel, this problem does not have a unique solution, and it is this non-uniqueness that allows some flexibility in the particular choice of Ksopt. For discrete time systems G(z), the infinity norm is defined by IIGII~ := suPl~l=as(a(z)) where 8(G(z)) is the largest singular value of G(z). A function G(z) is in ~g'~ if II a II ~ < ~ and is analytic in ( z: I z I > 1 ). Denote by ~J~ff~ the real-rational element of )g'~. Discrete time controllers satisfying the optimal and suboptimal infinity norm bounds can be obtained directly from the continuous time problem. This is done using the well known bilinear mapping z = ( 1 + ½sT)~(1- ½sT), with inverse s = (2/T)(z - 1 ) / ( z + 1), which maps the unit disk conformally to the left half plane. As shown in [6], the infinity norm is invariant under this transformation, so a discrete time controller can be obtained from the corresponding continuous time problem. For minimum entropy suboptimal controllers, however, care must be taken when applying this transformation. To show this, one must first define what is meant by the entropy of a function.

Definition 1.2 (Continuous

time entropy [11]). Let G(s)E.g¢'~ be any transfer function such that supRe~ >06((~(s)) < 3'. The entropy of G at a point s o ~ (s: Re s > 0) is defined by Re s o

3'2 J/.~ Ic(t~; 3'; So)'= - 2~r _~ lnldet(I-3'-2G*(i°~)G(i~°))l

Iso-V~l

]2

d~0.

(1)

Definition 1.3

(Discrete time entropy [2]). Let G(z)~)f'o~ be any transfer function such that II a II ~ < 3'The entropy of G at a point z o ~ ~ : = ( z: [ z I < 1} is defined by

3'2 f~.

1-

[z012

Ia(G; 3'; Zo):= - -~--~J_1, In det(I-3"-2G*(ei'~)G(ei'°))llzo_ei,Ol 2 d~0.

(2)

It is easy to see that both entropms are well defined, unitarily invariant, and nonnegative. Furthermore, Ic(G; 3'; So) = Ic(G; Y; - go) and Id(G; 7; Zo) = Id(G; Y; ~ol). From (2), it is also clear that Id(G; Y; z 0 ) = 0

¢~ G = 0 .

The entropy can be interpreted as a measure of the closeness of 6(G) to the upper bound y. This is easy to see after rewriting (2) as

32

~

1-Izol

2

Id(G; 3'; Zo) = --~-~f'_~,~ lnll-3"-ao,2(G(ei~'))[ [Zo--ei~12 d ~ . If 82(G(ei'~)) > 3'2 _ e2 for ~o1 < ~0 < ¢o2, then Id(G; 3'; z0) ~ oo as e ~ 0. A similar argument applies to the continuous time case. In the discrete time case, the term (1 - [ z0 [ 2)/( [z 0 - ei~'] 2) acts as a frequency weighting. Different values of z 0 ~ ~ will give different weights. In particular, if z o = 0, then all frequencies will be weighted

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evenly. In that case Id(G; y; 0) =

~/2 f'rr In I d e t ( I -2,~,/_

- y-2G*(ei'° )G (e i'° ))Idto.

A power series expansion of the integral (as in [11]) leads to

Il

Id(G; 7; 0) >_ ~

trace(G*(ei°~)G(ei~)) de0 = IIGIh 2

and so it can be seen that the 'entropy at the origin' is an upper bound on the Z#2 norm of G; see [10]. For these reasons, the special case of z 0 = 0 will be the case of most interest. For continuous time controllers, minimizing entropy at s o --* oo plays the same role. Using the bilinear transformation it can be shown that, if G(z).'= G((2/T)(z- 1 ) / ( z + 1)) and s o .'= (2/T)(z o - 1 ) / ( z 0 - 1), (2) is equal to y2 oo 2 Re s o l n l d e t ( I 7-2G*(i~o)G(i0~)) [ [s0_--~-12 do:, 2---~

(3)

which is equivalent to (1) up to some constants. Thus the discrete time and continuous time problems are equivalent. For the continuous time problem the most explicit solutions are for the So --' oo case [8,11]. Unfortunately, under the bilinear transformation this point is mapped to z o = - 1 and not to the origin, which is where the discrete time problem has its most natural interpretation. Hence a direct discrete time solution is desired. In this note the minimum entropy controller for a discrete time system will be derived, and values for the minimum entropy given. This will be done in Section 2, first for the general case where Zo ~ ~ and then for the special case of z 0 = 0. For this latter case state space formulas will be given. We note that while the general approach of [4] could be used, as was done in [9], the derivation presented here will be an adaptation of the results of Arov and Krein [1], as was done in [11] where the continuous time results are derived. This makes for a relatively straightforward and self contained derivation. Proofs will be supplied for completeness.

2. Derivation of minimum entropy controller As with optimal problems, the Youla et al. parametrization of all stabilizing controllers (see Section A.2), and the unitary equivalence of the infinity-norm, can be used to reduce the original J:oo-suboptimal problem to a general distance problem (see [5]). This amounts to finding Q ~ ~o~ffoo such that

,o II G II oo < V

[R n - Q

R'2] ~ < 7

[

R2 2

R21

where R* ~,,~°oo is fixed. This paper we will only look at the simpler 'one block' problem, where R q = 0, (i, j ) ~ ((1, 2), (2, 1), (2, 2)}. This corresponds to P12 and P21 both square. Since the entropy integral (2) is invariant under a unitary transformation, then, if E .'= R - Q (R := Rn), it can be shown that

I (o; y; Zo)= I (E; v; and the problem can be reformulated as follows. Problem 2.1. Find the particular EMEzo that minimizes Id(E; 7; Zo) at a point z 0 ~ 2 , over the set of all closed loop error system matrices E satisfying II E II oo < "/.

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In order to solve this problem, a basic lemma is needed. For convenience, let

and Yx(q v) := (XHq" + Xlz)(Xza~ + )(22) -1 where X is a square matrix, partitioned conformally with q'. I.emma 2.2. Suppose that

= / x,l(z)

X12(z)-

[X21(z) is J-unitary (i.e. X(z)*JX(z) = J). If (I + X22(z)-1X21(z)¢t ") is a unit of ~.)~aoo for all rt" ~ ~)Voo, then Ia(v~Y-x(q'); V; Zo)= vZld(kO; 1; z0)+V2Id(X12X£za; 1; Zo) + 2V 2 lnldet(I + X~'(zo)X21(Zo)'~(Zo)) I. Proof. Throughout the proof the argument z will be dropped wherever its omission will not arouse confusion. One must first show that the entropy is well defined. To do this, it is enough to prove that I - 3-x(~')* J'x(~" ) > 0.

(4)

By expanding the linear fractional trasformation we have that this is equivalent to ( Xl~ + ~*Xl~ )( X,I~" + )(12) < ( X~ + ~'*X~ )( X~I~ + )(22 ).

(5)

It is straightforward to show, after expanding (5) and using the J-unitary property of X, that this implies, and is implied by g,* ~/, < I, which is an true since '/" ~ ~#?-~oo- Thus, (4) is true, and the entropy is well defined. To prove the lemma statement, again use the J-unitary property of X to show that

1-y-2[y~Y-x(q)]*[y3-x(q)]=X~z*(I+X~'X2,q)

*(I--q*'ik)(l+X~21X2,q)

'X22'.

(6)

Using the fact that for a square transfer function matrix G, In [det(G* (ei~)G(ei'~)) I = In [det(G(ei'~))[ +In [det(G* (ei°')) [ = 2 In [det(G(ei'~))[, and the J-unitary property of X, (6) becomes lnl det ( 1 - 3'- 2[ Y-Y-x( ~ ) ] * [ T'V-x( ,iv)])[ = In [det( I - ~ * ko)] + ln[ det( X~* X~I)[ - 2 lnldet(I + X~21X2]'II")[, = In [ d e t ( I - q'* g') [ + ln] det( I - X~*X1~X12X221)] - 2 lnldet(1 + X~lX21~o) 1. It is then easy to see that Id(7Oq-x(ff'); ~,; Zo)= ~'2Id(q'; 1; Z0)+yZla(X12X2z'; 1; z0)

+vz/"~ lnldet(I+ g~a(e"°)Xz1(e"°),/,(ei'°)) 'h" d _ . n

1 - Izot 2 I Izo-_-ei---~i-2 &o.

(7)

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P.A. Iglesias et at / Minimum entropy ,'~oo controllers

Since I + X~21X21• is a unit in #~,°oo, then so is its determinant, and the Poisson integral formula (see [12], p. 337) can be used to evaluate the integral in (7). Thus I o ( Y J ' x ( ~ ) ; ~; Zo)= yEIo(ff'; 1; Zo)+~,2Id(XlzX~zl; 1; Zo) + 2y 2 ln] d e t ( I + x ~ l ( z o ) X21(zo)q" (zo))] as required.

[]

With this lemma, and using the parametrization of all suboptimal error systems given in the Appendix, we are able to state the main result. For convenience, let ~UE,o denote the choice of • which minimizes the discrete time entropy at a point z 0 ~ ~ . Theorem 2.3. Given the parametrization E -- "/3-o(~ ) of all suboptimal error systems E (see Section A.1), the unique choice of qb which minimizes Id(E; ),; z0) over this class is given by

~MEzo=--(0221(Z0)021(Z0))

(8)

*.

Proof. The proof of this theorem follows directly from that of Arov and Krein [1]. It is presented here for completeness. A slightly different presentation can be found in [11] for continuous time systems, where the parametrization of all error systems is given in terms of a lower linear fractional transformation of an inner matrix (as opposed to a J-unitary matrix function). From Lemma A.1, all error systems E are given by E = V.Uo(~), where, in particular, from (10), O is J-unitary. Thus, Lemma 2.2 can be applied to give

Id(E; y; Zo) = y21d(¢; I; Zo) +~2Ia(O,zO~';1;Zo) + 2"/2 Inl det(I

+O~'(Zo)O21(Zo)¢(Zo))l.

(9)

Let W:= -6~1(Zo)021(Zo) and define the Julia operator ([13], p. 148) --- of W as

(I-- W'W) -1/2 "~:=

-(I-

WW*)-WEW

-(I-

W*W)-I/Zw*]

( I - WW*) -1/2

"

It is easy to see that .~ is J-unitary and that ~ _ . - ( 0 ) = - W * . Also, if ~:=J_.-(ff)), then ~ = 0 = W*. Since the elements of .~ are all constant, Lemma 2.2 can be applied again to give

Id(¢; 1; Zo)= Id(~; 1; z0)+ Id(--'12~'; 1; Z0) + 2 lnldet(I +

¢:,

.~'(Zo).~21(Zo)~(Zo))l

= Id( ~; 1; Zo) + Id(O~'(Zo) 021(z0); 1; z0) + 2 ln] det( I + 0231(Zo)02~(Zo)¢(Zo)) I. (a0) To get (10) we have used the fact that ~-12.~ 1 = - W *, and .~1.~21 = - W = 0~1(z o) 821(z o). Premultiplying (10) by y: and then eliminating the common terms from equations (9) and (10) gives I d ( E ; ~l'~ Z o ) =

~/2Id(~ ; 1; Z0)--'~2Id(O221(Zo)Ozl(Zo)" ~ 1;

Z0)+ yEId(8128~l;

1; z0).

(11)

Notice that the last two terms in (11) are independent of the free parameter ~. The only choice affecting the entropy of E in equation (11) lies in the first term; and it is obvious that the unique entropy minimizing choice is ~ = 0. This in turn corresponds to

*ME~o---- w , = as claimed.

-(o;?(Zo)Oz,(Zo))*

[]

A value for the minimum entropy at a general point z 0 ~ ~ can now be given.

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P.A. lglesias et al. / Minimum entropy ~o~ controllers

Corollary 2.4. With notation as in Theorem 2.3, if EME., ' := 3'o~0(~MEzo), then Id(EMEzo; 7; Z0) = 27 2 lnldet(0~(z0)) I - 7 z lnldet(O~(zo)O22(Zo))l.

(12)

Proof. From equation (11), and setting ~ = 0, it can be seen that Id(EME; 7; Z0)= --y2Id(O~I(zo)O21(Zo); 1; Zo)+ yZld(O,2Ofzl; 1; Zo),

(13)

where, by the J-unitary property of 0, the last term equals _~,2 ~ 1-1Zol 2 2¢r f~ ln[det(O~*(ei~o)o=l(ei~))l ].z. 0. .- ~TJ? d~0. Since 02: is a unit of ~ o ~

(see Remark A.2), then Poisson's integral formula can be used to give ]/2

I-

~

y2Id(OlzO~21; 1; Zo)= 7 f_',~ lnldet(0~(ei'~))I

IZol 2

IZo- -ei-d] "2 d°)= 2]t2

ln[det(0~(Zo)) I.

(14)

The first term of (13) can also be evaluated easily: -V2*d(O£21(zo)O21(Zo); 1; Zo) = V2 l n ] d e t ( 1 - O~(zo)O~*(zo)O£2'(zo)O2,(Zo))[ = 7 2 lnldet( O~za(zo)O6*( Zo)) ]

= 7 2 In Idet( 022" (Zo) 022~(z0))1 = _,/2

lnldet(O22(zo)O~(zo) ) I

(15)

The first equality comes from the fact that the argument of In Idet (.) ] is constant; the second from the J-inner property of O; the third from the identity det(I + N M ) = det(I + M N ) and the fourth is obvious. Finally, combining (13), (14) and (15) gives the required result. [] A further corollary can be proved which tackles the very important case where the point of interest is at the origin. Corollary 2.5. Suppose that z o = O. Then ~MEO = 0,

and a state space realization of the EME0 is given by

EME0(Z ) = R ( z ) - QME0(Z) =

A-V + 7-2A-TQA-I~BBT A -T

y-2 A - T Q A - I ~ B

CP + CA - I~B

CA - 1~B

,

where ~ := ( I - y - 2 p A - T Q A - 1 ) - I , Z := (1 - y - z Q p ) - i and all other terms are as in Lemma A.1. Moreover, the value for the entropy is given by

Id(EMeo; 3'; 0) =V 2 lnldet(ZT~ 1)1. Proof. We begin by noting that, since the original function R ( z ) is antistable, the solutions P and Q to equations (21) and (22) are strictly negative definite. This makes the (1,1) block of /" positive definite, and

P.A. Iglesias et al. / Minimum entropyag'oo controllers

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SO allows for e in equation (23) to be upper block triangular, i.e. e21 = 0. Thus

~]~ME0 m. --(02~ 1(0) 021(0)) * ~---OT(oo)O~2T(o0)=

e21e22T -T = 0.

Evaluating EMEo = Y,~a(0) gives

[ C I el2J --.y-2BTA-T

[ C [ el2J[

=

T-2e~21BTA-T

e22

e221

A

y - 2Ble221BTA - T

Ble221

0

A-r+y-2B2e221BTA -T

B2e221

C

./- 2e12e221BTA - T

e12e221

(16)

where B 1 := y-EpzA-TCTel2 q- ZTBeE2

B E : = ZA-TCTe12 "4- ZQBeE2.

and

From (16) and the definitions of B 1 and B2, the matrix e appears only as e12e221. This can be evaluated explicitly by noting that, with e21 = 0, (23) is equivalent to

[1-'111"12]=[ el-lTelll r=:

r2~

r22]

-Tr

-7-1

-e22 e12ell eal

-- ellTellle12e221 -Tr

-7-1

e12e2-21 = - F~ 1/12 = (I- T-2CA-1pzA-TcT)-1CA-1ZB = CA - ' ( I - y - z Q p _ y - z p A - T c T c A -1 ) -1 B = CA-I(I - 3,-:pA-TQA-1)-IB =: CA - I~B. The following simplifications are then possible: =

y-2pZA-TCTe12e221 + ZTB y - 2 Z T p A - T C T C A - I ~ B + ZTB z T ( I - y - z p A - T Q A - 1 + y-zpA-TCTCA-1)g;B z T ( I - y - EpQ )I~B ~B,

and B:e221 = Z A - TCTe12e221 -t- ZQB = ZA -TCTCA - a~B + ZQB = Z(A-TcTcA-1 + O(I - y-2pA-TQA-a))~B = Z( A-TQA - 1 - ),-2QPA-TQA-1)~B = A - TQA - I~B.

]

e22 e12eH en e12e22 - e2-zTe221 "

Hence,

Ble221

-1

(17)

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P.A. Iglesias et al. / Minimum entropy 9ffo~ controllers

Thus

E~o(Z)=

A

y 2~BBTA - T

o

A - * + A - ~ Q A - ~ B ~ T a -T

C

y 2CA-I~BBTA-T

A - XQA - 1~B . CA -I~B

Applying a similarity transformation

gives the desired state space formulae. It remains to compute a state space formula for the entropy, evaluated at z o = O. We will require the following expression, again from (17): e22Te221 = F21Fll 1F12 -/"22

= BTZA-TcT(I

-- . y - 2 C A - 1 p Z A - T c T ) - I c A - I z T

= BTZA-TcTcA-1(i_

7-2pA-TQA-1)-a

o + ")121q- B T Z Q B

B + y21 + B T Z Q B

= 721 + BTA -TQA-I~B" From (12), /d(EME0; V; 0) = 23, 2 lnldet(02~(0)) I - 7 z ln[det(022(0)0~(0)) = 27 2 ln[ d e t ( 0 ~ ( o e ) ) ] - y2 lnl det (022(0)0~ (oc)) 1. The first part of (18) can be evaluated as 23, 2 ln] d e t ( 0 ~ (oo)) I = - ~,2 ln[ det( ~,- ae22Ve£2') I = _~,2 In Idet(I + B X A - T Q A - I ( ' y 2 I -

PA-TQA-I)-IB)I

= -- y21n [det(I + BBXA - r O A - l ( y 2 1 _ P A - V Q A - 1 ) - 1 ) 1 = _ 7 2 In [det(72I - P A - r Q A -1 + BBTA - TQA- 1) d e t ( y 2 i _ p A - T Q A -1)-11 = -3, 2 In Idet(~/2I - A P Q A -1) det(v2I - P d - T Q A - ~ ) - I = _ v2 In t d e t ( ( V 2 / _

eQ)(y21

I

_ eA - T Q A - ~ ) - a) l

= V2 In Idet(Zr~ -1) I, where (21) has been used to get the fifth equality. We now show that the second part of (18) is 0. 022(0)0~ ( ~ ) = (3'e22 + ~'- aBT( Z A - TCTe12 + ZQBe22 ) )3'e~2 = (3' 21 + B T Z Q B ) ez2e~2 + BTZA - XC TCA - T~Be22e~2 = (3' 21 + BXA - VQA - I~B ) e22e~2 =I.

Thus In Idet(022(0)0~(oe)) I = 0 and the corollary is proved.

[]

(18)

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283

Remark 2.6. Note that the minimum entropy extension is in fact the central ( ~ = 0) in the parametrization of all error systems. Remark 2.7. In the formulae above, the existence of A-1 is guaranteed by the assumption in Lemma A.1 that the transfer function R has a state space representation given by

Since R ( z ) is antistable then A will have no eigenvalues in N and is therefore invertible. This assumption excludes any transfer functions with a polynomial part in z. Remark 2.8. For stable transfer functions R(z), the point of interest will be z 0 ~ oo rather than the origin. In the case, instead of e21 = 0 in (17), we can show that e12 = 0. A similar derivation to that presented above will show that the A invertibility requirement is no longer necessary. In that case, the minimum entropy extension, QME0 will in fact be strictly causal.

3. Conclusions In this note we have presented and solved a discrete time minimum entropy 9~o~ control problem. The . , ~ suboptimal control problem is equivalent to that of a suboptimal error distance problem. As shown in [3], all suboptimal error systems are parametrized as a linear fractional map of a matrix and an arbitrary stable contraction ~. The unique choice of • which minimizes the discrete time entropy at a general point z 0 ~ ~ has been obtained. A formula for the entropy is also given. For the case of entropy at the origin it is shown that • = 0; and state space formulae are presented.

A. Appendix A. 1. Suboptimal extensions on the unit disk In this section we look at some results due to Ball and Ran [3] on the characterization of suboptimal stable extensions of a discrete time, antistable matrix function. Note that since the function R ( z ) is antistable, it is analytic in ~ . Hence, assuming that it has a realization R(z)=[~DB

],

(19)

ensures that A will have no eigenvalues in ~ , and is therefore invertible.

Lemma A.1 (Ball and Ran [3]). Let R ( z ) in (19) be a p x q matrix transfer function such that R* ~ . ~ o ~ . Then the set of all E ( z ) := R ( z ) - Q( z ), Q E ~Jt~'o~, such that II E II oo < "/can be characterized by a linear fractional transformation E = ' t ~ ( ~ ) where ~9l,Yg'oo, II~lo~ < 1 ; and the block ( p + q) x ( p + q) matrix 0 satisfies the following two conditions: O(z)*JO(z) =J and

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284

Moreover, the block matrix 0 has a state space realization given by Ou( z ) = el, + C( z I - A ) - I( 7 - ZPZA -TCTell

-]- ZTBe21 ),

01z(z ) = e12 + C ( z I - A)-*('/-ZPZA-TC"Ce12 + ZXBe22),

O2,(z )

----- y e 2 ,

--

y-'BTA -T(zI -- A -T)-'(ZA-TCTe,, + ZQBet,t),

= ve== - ¥ - ' B T A -

T(zI -

+

ZQBe==),

where P and Q are the unique solutions to the Lyapunov equations P = A P A v + BB v,

(21)

Q = ArQA + CrC;

(22)

Z := ( I - ~/-2Qp)-~, and the constant ( p + q) × ( p + q) matrix

Ie,,e,21 e = [ e2 '

e2 2 j

satisfies (23)

1~-I = eJpqe T,

where

F :=

lip -- y-2CA-IpZA-TCT -BrZA _ T c T

- CA-1ZTB ] __]t2iq __ BTZQ B .

[]

Remark A.2. Using property (20) of 0 it can be shown that 022 is a unit of ~gffo~. In addition, for all ~ ~ . g / ' ~ the matrix I + 0 ~ 1 0 ~ is also a unit of ~ . See [5], Section 8.3 for a proof. A.2. Discrete time minimum entropy controller In this section we set up the general distance problem that arises in discrete time )g'oo control. This allows one to use the formulas of Section 2 to obtain the discrete time minimum entropy controller KME. The presentation here will only outline the major steps; the details are analogous to the continuous time case as presented in [7]. Let

P:=

A

Bl

B2

]

Cl

0

DI~

1

C2 D21

0

be a state space realization of the standard plant, where D12 and D2a are both square and full rank. For simplicity we assume that the problem has been scaled so that D12D~2 = I , D~Dz~ = I and "r = 1. Furthermore, we define

Let X and Y be the unique stabilizing solutions to the following two Riccati equations:

r= r(I+ cTc2Y)-aff. T

P.A. Iglesiaset al. / Minimum entropy~oo controllers

285

and let ¢p =

+ = z + c2rc[,

I + BTXB2,

n = -(B1D:, + A r c : ) ~ - ' , A14=A + HC 2.

A F = A + B2F

Using these definitions, it can be shown that all stabilizing controllers can be expressed as a linear fractional transformation ~ ( K 0 , Q) where

Ko :=

F -Cz

0 I

,

and Q ~ Mgf'~. Moreover, using this fact, the 9f'~ optimization problem can be transformed to a general distance problem by noting that II~(P,K)II~
'~

IIR-QII~
where

AF T

XBaD~C2

- XB1DT~pa/2 ] !

d)/2BTE-T

¢I/2Fy

o

j

The minimum entropy extension QME can then be obtained using the formula of Corollary 2.5. This leads directly to the minimum entropy controller given by KME---~(Ko, QME)" The controller obtained will contain uncontrollable and unobservable modes which can be eliminated in a manner analogous to the continuous time case to give the foUowingminimal state space representation for KME:

Ldl b where

D = - ( I + BTXooB2)-I(BTX~oB,),

B= -Z~a(Lz~-(B2+L,~)b), Z==l-Y=x=,

d=F2o~-b(Cz+ F,o~), .4=A+BF~-B(Cz+F,~),

X~ and Y~ are the solutions (assuming they exist) to the Riccati equations

X~ = ETxo~(I + (BzB T - B, BT)Xoo)-IE,

and

F.

= -(J+B~X.B)

-'

I.F2~ J

Lo~=[Lloo

L2c~I=-[AY~C T

sTx=A ] DT2c1 + BTX~A

'

AY~C T + BAD2,] ( J + CYooCT) - ' .

286

P.A. Iglesias et al. / Minimum entropy aug controllers

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