All optimal controls for the singular linear-quadratic problem with stability : related algebraic and geometric results

with stability : related algebraic and geometric results

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Geerts, A. H. W. (1988). All optimal controls for the singular linear-quadratic problem with stability : related algebraic and geometric results. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 88-WSK-04). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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FACULTEIT WISKUNDE EN INFORMATICA

DEPARTMENT OF MATHEMATICS AND COMPUTING SCIENCE

All optimal controls for the singular linear-quadratic problem with stability; . related algebraic and geometric results

by A.H.W. Geerts

AMS Subject Classification: 49E20, 93B17, 93B27, 93015

EUT Report 88-WSK-04 ISSN 0167·9708 Coden: TEUEDE

RELATED ALGEBRAIC AND GEOMETRIC RESULTS

ABSTRACT

The present Report is the follow-up of [11, in which the linear-quadratic control problem without stability

(also called the free end-point problem) has been considered. Here, we compute all optimal controls for the problem where the state trajectory is required to converge to zero as time goes to infinity. Our approach is based upon the generalized dual structure algor ithm. an extended version of Silverman's dual structure algorithm. The paper will show that this method yields several interesting by-results as well as more insight in the unique and the non-unique parts of the optimal state trajectories.

KEYWORDS

Linear-quadratic control problem, strongly reachable subspace, dual structure algorithm, rank minimizing solutions of the dissipation inequality.

Autumn 1988

Research supported by the Netherlands organization for scientific research.

Between the periods that we kept watch at my father's dying-bed.

His strong heart stopped beating

only on april 30, a national festive day. The main part of this Report I wrote during the month of April.

And it shows.

1. Introduction

The job of computing all optimal controls for the singular linear-quadratic problem, which was started in [1] for the case without stability ([2, Sec. 2]), is finished here for the case that the state trajectory is required to vanish at infinity. As in [lJ, the underlying foundations of our construction of calculations are formed by the generalized dual structure algorithm ([1, Sec. 4]), a generalization of the concept of dual structure algorithm which is first mentioned in [4]. The basic idea in [1] is, that the algorithm transforms the given control problem into a related problem with a nice structure. Then a "coarse" state space decomposition for the related problem is introduced in [1] and the problem without stability turns out to be fully solvable. Here, we will define a more "subtle" state space decomposition in order to demonstrate where the non-uniqueness of optimal controls comes in and to what extent there is freedom of choice in these inputs without violating the state's behaviour at infinity.

Although the problem stated here has, though in a hidden way, been treated before ([2]), we still believe that our approach has significance because of the following observations.

(i) 80th in [2] and in this paper the structure algorithm is used to compute an optimal control. lIhereas in [2] (a primal version of) the structure algorithm comes up at the end, we start with the (dual) algorithm. This is done for several reasons, most of them can be read between the lines in [1]. The intuitive philosophy is, that the dual algorithm yields an explicit computation of the stronaly reachable subspace. ([4, Def. 3.13], [1, Def. 3.3]). This subspace differs from the zero-subspace if and only if the linear-quadratic problem under consideration is singular ([4, Sec. 4], (1, Sec. 4]). The primal

algorithm, however, concentrates on the determination of the weakly unobservable subspace ~ (e.g. [4, Def. 3.8]), which may be nontrivial in regular as well as in singular problems. In other words, the dual algorithm version seems the most appropriate one to deal with singular control problems. Indeed the dual algorithm has enabled us to give a complete characterization of all so-called rank minimizing solutions K of the dissipation inequality F(K) ~ 0 ([1, Sec. 6]), which turn out to be the possible candidates for defining optimal costs for linear-quadratic problems ([5] I [7] , [15J, [1] , [6], this paper's Th. 4.2). With the dual algorithm we have even shown that every solution K of the dissipation inequality necessarily satisfies the condition 9 c ker(K) (which implies that if

w

=

Rn

then the only real symmetric K that satisfies F(K)

,0

is K

=

O! ) •

(ii) In [2] it is stated that the optimal state trajectory for the problem with stability for t > 0 lies on a linear subspace of dimension n - (rank (T(s» - rank (D». Here T(s)

=

D + Cesl

- A)-18 denotes the transfer function corresponding to the linear system

z,

described by the matrix quadruple (A,

a, c,

D). In the present paper we will show that this state trajectory actually lies on a linear subspace of dimension n - n , where n _{00 00}

denotes the number of infinite zeros of T(s) (see e.g. [14]). This result corresponds to [8], where it is claimed that noo is

associated with the fixed infinite dYnamics of the output nulling trajectories.

(iii) In [4, Sec. 6] a modification of the dual algorithm, damaging the elegance of the method, is proposed to ensure that the state trajectory converges to zero as t tends to infinity. We will show however that no modification whatsoever is needed to solve the problem with stability. What is more, the modification in (4] appears to be unfit for use if the system under consideration is not left invertible (for a definition of left invertibility, see e.g. [1. Def. 3.11]).

(iv) As a by-product, we will establish a somewhat uncustomary

*

characterization of the set of invariant z~ros 0 (X) (see e.g.

[16], [24]). Since it is known that the linear-quadratic problem with stability has for all initial conditions an optimal input

*

if and only if (A, B) is stabilizable and 0 (X) n CO

=

0 ([2])

*

where CO denotes the imaginary axis, it is clear that 0 (X) is a

*

set of interest. Here we will prove that 0 (X)

=

o(A I~alw)'

~ with ~

=

d ~ + wand the n x n matrix A _~ is in [1, Sec. 4]

called the "preliminary closed-loop" matrix.

The most important consideration however for us to choose for the method to be displayed in this (bulky) paper is, that we have constructed a state space transformation based upon the results in [1] and this paper. Not only will it turn out that the transformation matrix can be written down immediately and concisely, but the resulting system appears to have such a structure that an optimal control for the linear-quadratic problem with stability can be calculated with no difficulty. It will be even possible to determine an optimal control that yields full internal stability, i. e. which is such that the resulting closed-loop system matrix is asymptotically stable. These assertions will be made stern in [28] •

Finally, we have adorned the report with a number of additional contributions on linear-quadratic optimal control theory and with some algebraic manipulations concerning matrices of interest.

2. Preliminaries

In order to keep things concise (sic) we will assume the reader to be familiar with [lJ.

Ve consider the finite-dimensional linear time-invariant system

E:

x(U

=

Ax(t) + Bu(t) , x{O)

=

Xo ' yet)

=

Cx(t) + DuCt) ,

and the non-negative cost criterion

(2.1a) (2.lb)

J(X₀₁ u)

=

jlly(t) uZdt • (2.2)

o

As usual, x(t) E f<n, u(t) E IRm, y(t) E

fl.

The symbol II'" stands

for the Euclidean norm and

rn].

[e, D] are assumed to be left and right invertible, respectively.

Vhereas in (1] we dealt with the linear-quadratic control problem (LQCP) without stability, we will study here the LQCP with stability:

(LQCP) +: find J+(x_o) = inf{J(x_o• u) lu E c~ such that

lmp

lim x ( t )

=

0 I ( 2 • J ) t...,oo

and compute, if it exists, an optimal control.

Here c~ _lmp is the m-vector version of c _imp' the class of impulsive-smooth distributions ([4, Def. 3.1], [1, Def. 3.1]). Ve briefly recall the distributional interpretation of (2.la):

px = Ax + Bu + Xo (2.4a)

where p stands for the derivative of the 6-distribution and Xo

=

xo·l

=

Xo·6 (6 itself is denoted by the constant land

convolution by juxtaposition, see [4, Sec. 3J, [1, Sec. 3]). The m

solution of (2.4a) for a given u E C. : lmp x = (pI - A)-l(Bu + x_{o )}

n

is unique and in C_imp' As in [4], [1] I [6J, (pI

(2.4b) A) -1

=

etA

(t ~ 0), or, more precisely, etA'l

R+(t) where IR+(t) denotes the indicator function of [0, ~}.

Substitution of (2.4b) in y

=

Cx + Du

yields

y

=

T(p)u + C(pI- A)-'xo

(2.4C)

(2.4d) and y is an element of c~ . Here T(p)

=

D + C(pI - A) -18 is the

lmp

result of writing s

=

p in the transfer function

T(s)

=

D + C(sI - A)-18 , (2.S)

a rational matrix in the indeterminate complex s.

The interpretation of x(~) := lim x(t) for an impulsive-smooth

t...,co

distribution is, of course, x(~)

=

lim x2(t)

t...,co

(2.6)

where Xz stands for the smooth part of x. For further details,

see [4J, [2J, [lJ.

A standina assumption throughout the paper is that (LQCP)+ is Singular, i.e. that ker{D) _ 101 (see e.g. [4, Sec. 2]), for if it is regular (ker{D)

=

(01) then the problem is well established (e.g. (2], [4]' [5], [13], [17]). Also we restrict ourselves to those inputs u in e~ that yield regular outputs

lIlIP

y, i.e. outputs for which the impulsive part y. equals

lmp zero

([4, Sec.3]). In [2] J(xo ' u) is defined to be +~ if Yimp - O.

The space of these admissible inputs in e~ is, due to system lmp

dependency, denoted by ~l ([4], [1]).

Also we remind the reader of the following important subspace ([4, Def. 3.13]).

Definit ion 2.1.

A state Xl is called strongly reachable (from the origin) if there exists an impulsive input u E ~X such that for the

corresponding trajectory we have x(O, u) (0+)

=

Xl (Here x(O+) .-lim x,(t), see (2.6». The space of strongly reachable states is t.l.O

denoted 9

=

9(X).

Now consider X «2.4a), (2.4c}). As in [1], [6] we apply to E

the generalized dual structure algorithm ([1, Sec. 4]). In short, this algorithm yields a final system E of the following form:

px

=

Ax + B u + B u + Xo , Y

=

Cx + D u ,

and it holds that ([1, Sec. 4J)

(2.7a) (2.7b)

- D is left invertible, D contains a selection of linearly independent columns of D that spans im(D),

im(B) c "

c.

c imCD)

u =

~(p)

[t

jOher.

~(p)

i. the •• trix-valued

distribution obtained by substituting s

=

p into a known invertible polynomial matrix xCs),

(2.7c) (2.7d) (2.7e)

- Vt>o:

(x(x,.

u~ (~)

_ -

(~(X,.

tj)(tJ) •

w. (2.71) - Le tAo :

=

A - B (D • D) -I D • C, Co' - (I - D (D 'D) -1 D ' ) C ,

-

-then A_{o (') c "} C_{o (.)} = 0 , (2.7g) - "rd = cy + 'W

= (

ker(C_{o) lAo)} (2.7h)

One may observe from [1, Sec. 4] that, it ker(D) = 10J, then D =

D, B

=

B and that B is not appearing. Moreover, 'W

=

101 if and

only if ker(D)

=

101. We have assumed that ker(D) _ 101, and hence • _ 101.

For completeness, we mention that ~ = ~(I) is the subspace of initial conditions Xo for which there exists a smooth input u such that the resulting output y

=

y(x_{o '} u) is identically zero

(e.g. [4, DeL 3.8) and that Xo e #f

d = #fd(I) c:::. 3u e

cD!

lmp y{x01 u) :: 0 «(1, Prop. 3.7J).

Next, let us define the linear-quadratic control problem with stability modulo

!i!l:

(LQCP)!: find J!(xo ) = inf IJ(xo ' u) lu e

cD!

such that

"W' ' " lmp

(x(xo ' u) '.,) (00) = 01 (2.8)

and compute, if it exists, an optimal control. Here, (x(xo • u)/",) (00) := lim (x(xo ' u)/_fI) (t) t with (xl.,) (t)

t....,oo

=

P(x(t», P denoting the canonical projection of Rn on Rn, . (see

e.g. [24, Ch. 0]).

This LQCP (of interest of its own since it represents a LQCP with linear end-point constraints) will turn out to be strongly related to (LQCP)+ «2.3». Getting ahead of things to come, we state:

Proposition 2.2.

Assume that v .. n J!(x_{o ) (} 00. Then v ..,n J + (x_{o ) (} 00

Xo e '" '" Xo e '"

and J+(xo> = J;(xo)'

If the set ot optimal controls for (LQCP); is non-empty, it contains an optimal control for (LQCP) +.

Remark.

The claims in Proposition

«(15, Lemma 3]) _{that J+(x o}}

2.2 are not too surprising. Assume

=

xo'K ·x o and that J~(xo) :: Xo 'K;Xot then both K+ and

K;

satisfy the dissipation inequality F(K) ~ 0

([5], (7], (15], [1, Sec. 6]). Now it is shown in [1, Sec. 6J that for any real symmetric K such that F(K) ~ 0 necessarily Kw

=

101. Hence K+W :: 101 and K~W

=

101. This suggests that the

optimal cost for (LQCP) + is equal to J~(xo) (although "x (00) = 0" is more restrictive than" (xl) (00)

=

0").

W

Hence, from the observations made above, it is of importance to know whether (LQCP)~ is easily solvable or not. Now it appears that this is indeed the case, due to the results in (2.7) which are established in [1]. In order to demonstrate the power of the dual algorithm, we will solve (LQCP)~ in outline below and then, in the next Section, put things on a more rigid basis.

We decompose the state space Rn as follows: Let $1

=

w(zl

and

let $2 be a subspace such that $1 • $2

=

Rn, Then there are left invertible matrices Wand W such that

w

=

im(W) and $2

=

C

im(W). Then W := [W, W ] is invertible with inverse, say,

c c L

=

[~el'

Next, from (2.7f),

LeX

=

LeX

and thus

we

establisb that necessarily (L xl (00) :: _{0 if u is optimal for (LQep) ;. Hence we}

c

may just as well solve the problem: Find

-=

infl fOOy.y dtl[~'1

J~(xo) u 'J I such that (xl wl (00)

=

O},

IJ

This is done in the usual way (e.g. [1], [2], (6]): Applying the feedback transformation u = - (D'D)D'Cx + v in (2.7a) - (2.7b) yields px =

_-

Aox + B v + B u + Xo '

-

-." Y = Cox + D v

Writing x

=

Wx, + W Xl gives, with (2. 7d), (2. 7g),

c _ _ N _ N (2.9) (2.10a) (2.10b)

p[~1]

= [A011

~012][~1]

+

[~l]~

+

[8

1

]~

+ [X01] Xl 0 A022 _{x 2} Ba 0 X02 y = ( 0

c"

l[~:]

+.D V." • = infl rrX2'C02'C02X2 + v' D'D v]dt o

II such that x.a(oo)

=

01,

(2.11a)

(2.11b)

(2.12a) and we observe that we can confine ourselves to the subsystem

pXa

=

A022xa + Ba v + X02 (2.12b)

of (2.11a) for determining J~(Xo). But (2.12a) - (2.12b) now is a reaular linear-quadratic control problem and it is well known

(e.g. [2]) that, if _{(A o22 ,} B_{2) is stabilizable (which will be}

the case if (A, 8) is stabilizable ([1, Lemma 5.6]», then (2.13a)

where _K2l+ is the largest positive semi-definite solution of the

Algebraic Riccati Equation N _ _

o

= CO2' C 02 + A 0 12 'K 22 + K 22A 0 12 - K 228 2 (D • D) -18 :.I • K 12 •

(2.13b) Koreover, if for every XO:.l an optimal input v for the subproblem

(2.12a) - (2.12b) exists, then v = - (D'D)-18:.1'K_a2+X₂

and the resulting closed-loop matrix for xa' Aou(Ku+) = Aou - B,(D'D) -1B2'Ku+ , is asymptotically stable.

(2.13c) (2.13d)

These observations, however, immediately yield those u and u in (2.11) that actually attain J;(x_o): If for every Xo an optimal u and u exist, then the optimal u is unique and follows from (2.9) and C2.13c) and for u we may take any impulsive-smooth input. But with (2.7e) we then also have all optimal u for the problem in (2.8)! Without further ado (see for the missing details Prop. 3.4), we will assume that for every Xo an optimal input for

(LQCP); exists and state the next

Proposition 2.3.

If ~L+ denotes the set of optimal inputs u for the problem opt, ., in (2.8), then ~ + l opt,

=

lu

=

K(p) - [ - g(K+}(B • u + Xo)

1

u impulsive-smooth where g(K+) is the matrix-valued distribution defined by

...

-

..

..

g(K+) := (D'D) -l(D'C + _{B'K+) (pI - Ao(K+»-1}

with

Ao(K+) := Ao - B(D'D)-1B'K+

u

and K+ := L 'K₂₂+L, where K_2a+ is the largest positive

c c

semi-definite solution of (2.13b). Remark.

3. A "subtle" state space decomposition and some algebraic machinery

In the present Section we will start with translating (2.7) in terms of [lJ and then define the "subtle" state space decomposition that we have announced in the Introduction. Having done this, we return for a moment to (LQCP)~ and give a (perhaps not very customary) characterization of the set of invariant

*

zeros 0 (X) (see e.g. [16], [24]) of the system X. Next, a

number of algebraic manipulations are displayed. These computations will be of use in Section 4 where we solve (LQCP) +

in its full generality.

The state space decomposition.

We first write all assertions in (2.7) in the terminology of

[lJ. The dual algorithm yields a terminal system E

=

E (where a

a

is an integer, a ~ 1), that can be described as follows: px

=

Ax +

8

w

+ BC we + xo (3.1a)

<l <l ~ a a a

y

=

Cx + D w

<l ~ a

D.lb)

with the integer

on

e (0,

...

, al ([1, (4.53)]). The matrix D

-on

is left invertible, contains a selection of linearly independent columns of D that spans im(D) and has rank ~ ~ qo

=

rank (D). Here ~

=

qo + ql +

E and E are related by

a u

=

;c

(p) a w a wc a * w a

+ q with all q. ) O. The controls for

on

1

where

i

(s) is a known polynomial matrix in s ([1, (4.57)]),

a

invertible as a polynomial matrix, and w 1 we and w* are

a a a

impulsive-smooth distributions of respective dimensions n

:Lan'

PI + PI + ••• + P and 0 = 0, + 02 + ••• + 0 (all P

l" 01' ~ 0).

a a a

*

Note that w does not appear in (3.1a) - (3,lb)! The states x

a

and x are linked by

a

x _a = x - W v 1 ' _a-a- () .ld) see [1, (4.55)J, where W is a basis matrix for wand it can be

a

given in the partition ([1, (4.48»))

[B o' B,P,Vl I , •• , Ba-lPa-1Va-1J • (3.1e)

Here Bo has rank ro} 0, BiPiV_i has rank r_i > 0 (i

=

1, ••• , (a-1» and thus rank (W_a)

=

ro + r l + ••• + r_a-₁

=

ta-I' The

matrix BC has the partition a

[8

,P

IV 11 • • • 1 B IP IV l ' B

P ]

a- a- a- a a (3.1f)

where Bi Pi Vi (V _a the identity matrix) is left invertible with rank p. and BC

=

W HI with

1 a a HI

=

( [1, (4.26d) 1 App. 1, Lemma 2])

'"

*

*

*

*

N, N21 Nu N _a-1 1 N

*

*

a, 0 _{Nu N} J 2

*

0 0 _N" (3.1g)

*

*

0 0 0 N a-I N a-1 ~-1

'"

0 0 0 0 N a 0:

an upper triangular matrix. For further details, we refer to (1, Sec. 4, App. 1J. See also [14].

Next, we split up the output y in the same way as in (1, (4.59) (4.64)]: Write D

=

U G with U orthogonal and G

'1>

~~ ~ ~

invertible and let the orthogonal matrix U be such that U := c

[U

c;,'

U c

1

is orthogonal and invertible. Then, after having applied the preliminary feedback

; = G -1 ( - U· ex + w ]

Cl

c;,

c;,

Cl Cl

the system X transforms into

Cl

px

=

A x +

B

G-1 W + BC wC + xo ,

Cl ~ Cl

'1>

c;,

Cl Cl Cl

Y2

=

U'y _c

=

U'cx _c

Cl'

and we have lIy"a = lIy 1 ,,2 + lIy 2112 •

Here A

=

A -

B

(D 'D ) -'D' C

c;,

~ ~

-c;,

-~ (3.1h) (3.H) (3.1j) (3.1k) (3.11) (3.1m) is in [1] called the "preliminary closed-loop" matrix and Ao in (2.7) stands for this A . Indeed both wand 0/

=

0/ + • (with 0/

c;,

d

the weakly unobservable subspace, see Sec. 2) are A -invariant

c;,

([1, Lemmas 4.2, 4.4]) and ( [1, App. 1, Lemma 2])

-A W

=

V All

c;,Cl Cl

(3.1n)

with -All

=

All + H,Z - c (3.10)

i f Zc

=

diag (Z~) , (3.1p)

Z7

₌

_Vi

_0i

* -

_Ri

(i

=

1,

...

, Cl) , (3.1q)

0 0 0 0 0 A, 0 0 0 0 0 ..12 0 0 0 and All

₌

_{0 0 A} (J.lr) J 0 0 0 0 0 0 0 0 0 0 0 A _0.-1 0 * -Here, as in [1]. A.

=

V! Q. R! (i = 1,

.

..

, (a - 1» and 1 1 1 1 *

[~i

l=p",

[Pi' p .) -* 1 (i =1,

...

, a) (3.1s) p, 1 1 1

As is claimed in (2.7d), CW

=

D C for some N x r 1 matrix C

a -~ :a.~

-0.-([1, App. 1, Lemma 1, (iii)]). Hence C:: (D 'D )-lD' C'Il =

-~-~ ~ a

*

K K

*

a *1~-1 al~ (L1t)

where for i

₌

1,

...

, _~, the matrix ~.

1 is such that ( [14J )

*

Note that

().lv)

with CO counting qo rows and C1 is right invertible.

Moreover, it R • 1 0 0 0 0 R 2' 0 0 0 Cit

.-

.-

0 f (3.1w) 0 0 0 0 R I

c;,-I r 0

c;,.

[ CC<

1

then _ • is invertible with inverse [ ~lf ~a ] where

R. 0 0 _Ra ~1

=

_{0 0} and ~a =

* - *

_{1 -R lK2} 11 0 0 0

o

*

R,

• R

0 0 (3.1x)

c;,

*

-

*

-

*

R -R K •• -R K

c;,

c;, c;,

+ 11

c;,

0.1

____________________

~---c;,

c;,

Ir

o

c;,.

• Ir 0.-1 () .1y)

The matrices F, and Fa divide the strongly reachable subspace , in two.

Lemma 3.1.

Let ' a · '

n

_{ker(C o)} = ,

n

(C-limCD», where Co

=

P_D C with P_D

o 0

:= (I - Do (Do 'Do) -'Do t) (Do is left invertible and contains a selection of columns of D that spans imCD), see [1, (4.2) J), and

WI be any subspace such that, •

'1

+

'2'

Then COC"'l) = cot,)

and 'a

=

im(W Fa" A suitable choice for

w,

is WI = im(W ) with

a ~

W~

=

W~F1' Moreover, for this choice we have

w

=

.1 ~ .2'

(A.6a)]) =

«J.lv». Hence CoW Fa = 0 _{(and ker(C o)} = C-1_im(D». ~

Next, it is clear that for any

.1

such that

.1

+ .a = , it holds rank CW F1 )

=

~

-

qo

=

dim(C o('»

=

dime,) - dime. n ker(C o»

~ ~

=

dim(.) - dim(.2)

=

~1 - dim(w_a)· Remarks.

[1]

[2}

Observe that Dadd

=

CW ([6, (A.5) - (A.6)]), that COF!

'1>

o

and thus that CO = _{COF2Ca (Ir} = F1CI + FaCt)' -~-1

.

*

In [2] '2 1S denoted as R •

a

Our state space decomposition, of course, will be based upon the Aan-invariances of • and .d and the nice structure of All

({l.ln». We might ask ourselves the question, whether the division of. into

'1

and ., yields more structure or not. From [1, (Al.4)] it is easily seen that A W?1

=

O. However, A W?,

ana ana

Therefore we will keep the two pieces of • together in the state space decomposition. But before coming to this decomposition, we derive some interesting by-results.

Corollary 3.2.

Let (D'D) + denote the Koore-Penrose inverse of D'D. Then

rank (T(s)}

=

dim«I - D(D'D) +D'}e(.» + rank (D)

Proof. In [1, Lemma 5.3] it is stated that the rank of the transfer function «2.5}) over the field of rational functions is ~. From Lemma 3.1 we establish that dim(C o

('»

=

~

-

qo and qo

=

rank (D). Finally it is proven in Appendix 1 that Do(Do'Do)-lDo' = D(D'D)+D'.

Corollary 3.3.

Let n denote the number of infinite _ao ~ (e.g. [16], [8]) for T(s) and (D'D)· denote the Koore-Penrose inverse of D'D. Then

rank (T(s)}

=

rank (D) ~ n

=

0 ~

ao

(I - D (D 'D) +D t) C (.) = ( 0 I .

Proof. In [14, Prop. 3.1] it is stated that n

=

0 if and only 00

if C_{o (')}

=

101 _{and Ao(')} c . and.

= (

_{ker(C o) lAo} >. Here Ao = A

- B

0 (D 0' Do) -ID 0' C and Co

=

P D C ( [1 , (4.2) - (4. J) ], [14,

(3.18)J, Lemma 3.1). Next, it follows from [4, (3.14)] that Ao(w_{2 )} <.:. and from [4, 'Th. 3.10, (3.11)J that ( ker(C o) lAo) <.:

~ <.: ker(C_o) and Ao(~) <.: ~ +

w.

Hence, if C_{o (.)}

=

101 then

w

=

.2

and thus Ao(') <.:, and ~ <.: ( ker(C_o) lAo >. Therefore, noo

=

0 t=»

Co (') =

lot.

Now apply Corollary 3.2 and App. 1.

Remarks.

1. Corollary 3.2 clearly expresses the strong relation between the rank of the transfer function and the subspace •• Actually the result is rather surprising because it connects the rank of a rational matrix function in the complex s with the dimension of a subspace!

2. Note that the matrix Ao in the proof of Corr. 3.l in general is unequal to A - B(D'D)+D'C (See Appendix 1).

n

Now we decompose R as follows. Let S,

=

w,

let S2 be a subspace such that S l .

s.

=

~d and let $, be a subspace such that Sl •

Sa • S,

=

~n. If W a r e left invertible matrices such that C"2

$." = im(W ), then W := [W , W , W J is invertible; let L

cl'a 0. c 1 c 2

.- [L' L' , L' ], denote its inverse. After defining

0.' c 1 C 2

xa := Lx,

x~'a

:= L x (3.2)

0.0. cp 2 0.

we get the equations «3.1i) - (l.lk)} a All Au Au a

I

H, x x b 0 _{Au Au} b + H2 + P Xl = Xl W 0. b 0 0 _A" b H, xa XI HI xao 0 wc + bo (l.3a) 0. Xl 0 xbo 2

Yl

=

w _ex while «2.2), (3.11» (3.3b) C3.3c) J(x_{o ,} u) = r[lI~ 112 + IIC,X~1I2]dt • (3.3d) o a

Moreover, (C" A,,) is observable. Observe that (3.3) is equal to [1, (5.7)], also compare (2.11). The "subtlety" of our state space decomposition is hidden in the division of W «J.1e».

a

Whereas in [1] this division has not been used in the determination of all optimal controls for the LQCP without stability, we will make use of the specific forms of All and HI

«J.1g), (l.lo) - (l.lr» in this paper. For further use, observe that «3.1d), (J.2»

b L x

=

xl' 2 (J.4a) c I ' 2 a -L _ex x

=

x + v 1 ' (J.4b)

-0:-and thus that the trajectory of x is known if -0:-and only if the

b a

-trajectories for X"2 and (x + ~1) are known.

All optimal controls for (LQCP>;.

Here, we will give the complete solution of the problem

(2.8) , (LQCP);. As in Section 2, we establish that

J;(x_o) = inf I

r[

II~

112 + IIC,X~1I2]dt

I

[

x~ (~)

] = 01

o a _X

2 (00) (3.5a)

subject to the subsystem

P [:; ]. [ A;. :::

w +

ex

[

:~:

].

(3.5b) in

This problem, derived from (LQep);, is regular (compare the one

in (2.12» and therefore we arrive at the next result which is a combination ot [2, Prop. 11] and the last part of Sec. 2.

Proposition 3.4.

For all Xo it holds that J;(x o) ( 00 if and only if the pair

[[ 1;,

A,,].

[ :: ]1

(3.6a)

Au

is stabilizable. Assume this to be the case. Then the Algebraic Riccati Equation

-

-[ 0 0 ] +

o

C I'C I

o

][

~::.

::: 1

+

[ A;,

Au

1

[~

..

K12

][:J

[K .. K

..

]

H2 ' H ' J ]

-

-Au K12 I Ku K_{12 '} K22 = 0 (3.6b)

has a largest positive semi -de finite solution K+ =

-[~

..

,

_

K .. ' ]

_{• If K+}

₌

_{[ L' L' J -}

[K .. ' K .. '

_-

We

_I

1

_{, then} K +, Kaa+ c ' c a Ku+' K₂₂+ LC a 1 12

J;(x o) = [ _x bo , I , Xa bo, ]K+

-

[X~O 1

x~o

=

xo'K+xo· For every Xo there

exists an input u such that both J(X01 u) = xo'K+xo and (x (x o' u) I.,,) (00) = 0 if and only if the intersection of the set of eigenvalues of A22 , a(A22 ) , and the imaginary axis CO := Is E

eIRe(s)

=

01 is empty. Assume this to be the case, i.e. oCA,,) n eO

=

0. Then, with

A (K+) := A -

8

CD 'D )-18 'K·

the set of optimal controls for (LQCP);, ~E+ , equals opt, W lu E e~ lu =

i

(p) lmp a - g(K+)(Bc wC + xo) a a wC a ". W a m-g (3. 6e)

"n

E e. , arbitrary ,. lmp Remarks.

1. The assertions in Proposition 3.4 are analogous to the statements in [1, Th. 5.2] and [6, Th. 4.5]. Therefore we would like to refer to [1] or [6] for the outline of a proof. The statement on the set ~ .. is nothing more

E_{opt ,} W than a reformulation of Prop. 2.3.

2. Define C := (1 - D (D 'D ) -lD ')e

c;, . " -c;,." -c;, (J.7a) (and observe «2.7h» that "'d = < ker(cc;,) lAc;, )} and C (I(+):= C - D {D 'D }

-'8

'K+ (3.7b)

c;, c ; , . " . " . " . "

(analogously to (J.6e», then one establishes easily that the optimal output in Prop. 3.4 equals

y = C (K+) (pI - A (K+» -IX (3.8)

c;, c;, 0

since im(8c) c

w,

C (w)

=

[01, A (W) c wand K+W = (01.

a c;, c;, Therefore A '(I(+)t A (K+)t Kt =

r(e

c;, )C • (I(+)C (K+) (e c;, )dt o c;, c;, (3.9)

and clearly this formula is a generalization of similar results in regular linear-quadratic optimal control.

The matrix pair in (3.6a), the equation (3.6b) and the condition 0(A22 ) n CO

=

0 in Prop. 3.4 depend obviously on the state space

decomposition that we have applied. Of course we are interested in equivalent formulations that are independent of the choice for W. These are given in the next Proposition. We will need the concept of induced map there. Recall (see e.g. [24, Ch. OJ) that

if A: 'fRn .... fRn is a linear map and :1' is an A-invariant subspace

(A(~) c ~), then the induced map

A:

i

~

i

(with

i

:= fRn/~) is defined by

A x := Ax

where

i

=

x + f. Analogously, if 8: fRm ~ fRn is a linear map,

- m -

-then 8: fR .... $ is defined by 8 u := Bu.

Hence, since A (W) c

w,

the map A : fRn_{/w ....} fRn/

w

is defined and

'iJ

'iJ

so is 8

-c:;,

Proposition 3.5. Consider A

c:;,

I

~

- n

and let $ := R I •. Then:

[[

Au A2J

]. [

Hz

II

(A I

B )

(a) stabilizable 4=t 0 _An H,

c:;, -c:;,

shbilizable 4=t (A, B) stabilizable.

(b) Let K :=

[~11 ~121'

Ie

:=

[0

0

OK

1

(with dimension of

K

1Z '

K:u

the upper left zero block equal to dim(w» and K := L'KL. Then K satisfies (3.6b) 4=t Kia

=

0 and ~(K)

=

0 where ~(K)

*

(c) Let 0 (x) denote the set of invariant zeros corresponding

to t (e.g. [16] I

11<

[24]). Then 0 (x)

=

o(A I~afw" Hence

ely,

Proof. Ad (a): The first equivalence is obvious; the second one is stated in (1, Remark 2 at the end of Section 5]. Ad (b): [1, Corr. 6.3]. Ad (c): The equivalences are clear if the first assertion is valid. In Appendix 2 we prove this first claim for

the case that ~ ) O. Here we will study the (pathological)

situation that ~

=

O. In [14, page 11] it is stated that ~ = 0

*

«:::$n = 0 «:::$"= ~n w. Next, (see e.g. [16], [24J) 0 (x)

=

o(Ao

00

+ _{BoFol~/(~ n w) ) for any Fo for which (Ao} + BoFo)~ c ~ ([1, (4.2)

-

(4.3)] and Ao

₌

A - BoCDo'Do> -IDo'C. Now ( [14, Prop. 3.1]) ~

= (

_{ker(C o) lAo} >if ~

=

0 and hence o 11< (X)

=

_{o(A o} I~I _{w' •}

The claim then follows since A

=

Ao and ~d

=

~. ~

Remarks.

1.

2.

Combining Prop. 2.2, Prop. 3.4 and Prop. For all Xo it holds that J + (x_{o )} (

3.5 (a) yields:

()O «:::$ (A, B' is

stabilizable. This is also stated in [2, Th. 3 (i)]. In [18, Remark 5] a necessary and sufficient condition is presented for the statement: For all Xo it holds that

J-(x_o)

=

_{inflJ(x o1 u) lu} E c~ I ( co.

ap

The set r_min = _IKIK

=

K', KW =

o

and .p(K) = 01 turns out

ex

to be the set of the solutions of the dissipation inequality F(K) ~ 0 for which the rank of the dissiRation

The set

r.

is of importance hecause it contains the only mln

possihle candidates for representing optimal costs of LQ control prohlems under any conditions on the long-term hehaviour of the state trajectory (see Th. 4.2 (a) in Sec. 4 and [15]).

3. If we introduce for any K satisfying F(K) ~ 0 the matrix cpo (K) : = Co' C 0 + A 0 I K + KA 0 - KS 0 (D 0 I Do) - !

B

0 I K, ( 3 .11) then (Schur's Lemma, [7], Corr. 3.2) rank (F(K»

=

rank

(cpo (K» + rank (D) ~ rank (T (s»

=

rank (D) + dim (C 0 (w»

and thus rank (CPo(K)) ~ dim(Co(w». In particular, K Ermin ~ rank (CPoCK»

=

dim(Co(w». In several papers, a real symmetric K satisfying F(K) ! 0 is called a solution of the Alaehraic Riccati Equation corresponding to the system

x

when cpo(K)

=

O. Let r := tKIK

=

K', F(K} ~ 0, CPo(K)

=

01. Then it is easily seen that

r

=

r.

~ Co(w)

=

(01. Thus,

mln

with Corr. 3.3,

r

=

r.

_mln ~ n : O. Hence, in particular,

r

(10

=

r

min if ker(D) : 101 (i.e. w

=

101). This result was already estahlished in 1971 ([5]). Ohserve that, if ker(D)

_.

_ fOI, then generically n ) 0 which implies

r

=

0 (if Ko e

(10

r

_{then rank (F(K o}»

=

rank (D) and thus, necessarily, Co(w)

=

/01).

4. Again consider cpCK) «3.10» and cpo(K} «3.11}). Since rank (cpo(K» = rank (F(K» - rank D and rank (cp(K})

=

rank (F(K» - rank (T(s)} ([1, (6.10)J), it holds that rank (cpCK» = rank (CPo(K» - rank (W' ) (Lemma 3.1, Corr.

3.2>-~

Hence dim ker (cp(K»

=

dim im(W' ) + dim ker (cpo(K». Since

~

also ker (~o(K» c ker (cp(K» ([6, App., proof of Lemma 1]), im(W ) ewe ker (cp(K» (C (w) = 101, K'W

=

tOl) and

~ ~

see proof of Lemma 3.1 and Remark 1 below this Lemma), we conclude that ker (~(K» = ker (~o(K» e im(W

on

)

= ker

(~o(K» + W (w2

=

w n ker (Co) c ker (~o(K». Hence (Remark 2) the real symmetric solution K of the dissipation inequality is in

r.

if and only if ker (~o(K» + W

=

Rn.

mln '1'

Compare with [6, App., Corollary 3] and [7].

5. Note that ~o(K) = C'(I - O(O'O)+O')C + (A - 8(O'O)+O'C) 'K +

K(A - B(O'O) +O'C) - KB(O'O)+B'K, see Appendix 1.

6. The expression o(A '~dlw) makes sense because of the

on

7.

invariances A (w)

'1>

c w, A

'1>

(~d) c ~d. Observe that G :=

-B

(0 '0 )-10 ' «3.1m» is an element of the set G(x) :=

-'1> -'1> -'1>

-'1>

IG: Rn ~ Rml(A + GC)w C 19, im(B + GO) c 19/ and it is known (e.g. [16]) that for every G e G(X), (A + GC)~d c ~d and o(A + GC I~

I

W) is fixed. Since ([16], [19] - [22]. [24])

*

m

o (x) = o(A + BFI~/(~ n 19» for every F e ~(x) := IF: R ~ Rnl(A + BF)~ c ~, (C + OF)"

=

1011, the result o*(x) =

o(A

'''dIW)

looks plausible. In App. 2 a constructive proof

on

is provided.

If

A

(K+) denotes the

on

induced map of «3.6c) )

Awe w!), then all eigenvalues of

'1>

*

are in e- := Is e eIRe(s) < 01 if 0 (x) n eO = 0

(Prop. 3.4 and Prop. 3.5 (c». Thus the integral in (3.9) is finite since also C (K+)W = 101 «3.7b».

Some alaebraic machinery.

This subsection concerns some new definitions and a few algebraic calculations that yield extra insight in matrices like A , C • Recall that by means of a regular transformation the

<;,

<;,

system E looks like ([1, (4.5)], also Appendix 1) px

=

Ax + Bowo + Bowo + Xo '

Y = Cx + Dowo '

where Do is left invertible (qo

=

rank (Do)

=

_.

rank (D».

_.

Next, if D

-on

=

[Do' D dd] with D dd = CW , W = V ?l ([6,

a a

<;,

<;,

eX

(A.6a)], Lemma 3.1 and Remark 1 below Lemma 3.1) then Ll :=

DaddPDoDadd > _{0 with PDo}

=

(I -D(D'D) +D'). Also, if ~on

=

[Bo, Badd1

(A.6b)]). Define P := W L -IV 'C'P C

<;,

1

<;,

Do (I -then Badd

=

AW

on

( [6, (3.12) then it holds with Ao

=

A - Bo(Do'Do)-lDo'C and Co

=

P

D o C ([14,

(3.18)], see also Appendix 1) that «l.lm), (l.7a»

Lemma l.6.

A = Ao (I - P)

<;,

In addition, if for any K such that FeK} dissipation matrix

Bo(Do'Do} -IBo'K, CoCK)

( [7]) ) we define (3.1la) (3.13b) (with F (K) AotK) := the Ao -A (K) ,

<;,

C (K) analogously to (l.6c), (3.7b), then we have that

A CK)

=

Ao{K)(I - P{K» , <;,

C<;, (K)

=

Co(K){I - P{K» ,

where P(K)

=

W LI·1W '~o(K) {(3.1l». <;, <;, (3.14a) (3.l4b) - L .ID 'D L ·1 I add 0 0 - L - 'D 'D o 0 add L 1 -1

1

L -1 1 (J.IS) where Lo

=

Do'Do and therefore

_.

8 (D 'D ) -18 '

=

8 L -18 ' + AoWa... LI-1Wa.:Ao' ,

'1> '1>'1> '1>

0 0 0 1J 1J

(3.16a)

D (D 'D ) -'D • = DoLo-'DO' + CoW L1-1W 'Co' •

'1>

-~'1> -~ ~ ~ (3.16b)

All claims now follow immediately by noting that AoW~

=

Badd -BoLo·lDo'Dadd and CoW<;,

=

Dadd - DoLo·1Do·Dadd

=

PDoDadd'

The matrix P in (3.12) stands for a projector since p2

=

_{P and}

we have imCP)

=

im(w<;,), kerCP)

=

ker{WariCo'Co}

=

ker(DaddPDoC) (see Remark 1 below Lemma 3.1~ _{We establish that ker(C o)} c ker(P} and thus that ~d

n

ker(C o)

=

~ +

w

2 c ker(P) (Lemma 3.1).

Also observe that imCW ) $ kerCC o)

=

kerCC )

=

ker(Co(I - P»

~ ~

since rank (C )

=

r - rank (D )

=

r - n

=

r - q - (g,

-~ _~ 1~ o~

qo)' But then ker(C )

=

C-1_{im(D) + w since w}

2 c ker(C o)'

<;,

Compare this with Remark 4 below Th. 3.5.

An interesting special case occurs when r

=

g, (i.e. the system

~

Proposition 3.7

x is right invertible c:::. im(C o) = Co(w) c:::. C -lim (D) + W = p'n.

Furthermore, if X _{is right invertible then ker(C o)} = v + w2

where

w

2

=

w n ker(C o) and hence ker(P)

=

v + w2 •

Proof. Combine the fact that dim(im(C_o»

=

r - rank (D) with Corr. 3.2. Next, from [1, Prop. 3.9, Corr. 5.5J and the fact that rank (C )

~ = r

-

~' we establish that X is right

invertible if and only if C

=

0, i.e. C-1im(D) + W

=

Rn.

'1>

Further, observe that V + w₂ = Vd _{n ker(C o)} c _{ker(C o) and we}

have that im(W ) $ (v + w_{2 )}

=

V

d. Thus dim(v + w2 )

=

dim(V_d) -~

(~ - qo) and hence, if X is right invertible, then dim(v +

w

2 )

=

n - (r - qo)

=

dim(ker(C_o» and therefore v + W_z

=

ker(Co)'

ker(P)

=

ker(Co)

=

v +

w

2 and dim{ker (P»

=

(n - (r - qo»'

Remarks.

1. One might be tempted to believe that the condition: "v + w2

= ker(Co)" also is sufficient for right invertibility of x,

because then we find, with the observations above, that ker(c'1» = vd = ( ker(c'1» IA~ ) _{and hence dim(vd)} = n - (r - ~~). However, it is easily established that for X

=

([~

n, [n,

[~~],

[n)

the optimal cost without stability

([1], [18]) is represented by K- =

[~ ~2

_] with K2 - = 1 + 2% _ 0 and hence ([1, Lemma 3.10]) X not right invertible whereas 'f'd

=

V

= (

ker(C) IA )

=

ker(e) (C~

=

e, A~

=

A).

2. In fact Proposition 3.7 is a new kind of relation as the ones given in [1, Prop. 3.9]. Observe that if E is right invertible, then ~ + 'If : Rn and hence indeed im(C o} == Co(~

+ '19) == Co(19). However, if Co(') == cor,. + 'Pi) = im(C o)' then

it is not immediately clear that,. + 'Pi == Rn! Note that if 'Pi

=

10} (i.e. ter(D)

=

101) then E is right invertible if and only if Co :: 0 which is a well-known result (e.g. [18]).

More generally, the transfer function T(s} is both right invertible as a rational matrix and has no infinite zeros (Corr. 3.3) if and only if Co

=

0 (if Co

=

0 then r

=

qo '" rank (D)

=

rank (T(s» and hence T(s) has full row rank, n ₀₀

=

0).

3. Note that" E right invertible ~ C-1im(D) + • = IRn .. looks

but is not weaker than" E right invertible ~ "d(E)

=

IRn " (recall (previous Remark) and that if 'If

=

101 then E is

right invertible if and only if ker(C o)

=

C-1im(D)

=

IRn). Also, observe that if rank (T(s» '" r, then the smallest positive semi-definite solution of ,(K) '" 0 «3.10» indeed is K-

=

0 ([1, Lemma 3.10]). Conversely, if K-

=

0, then C

=

0 and E is right invertible.

~

4. In [16J it is stated that "d

=

(C-1im(D) + 'IA + GC >

where G E G(E) (Remark 6 below Prop. 3.5). Hence indeed E

right invertible ~ C-1im(D) + '19

=

!Rn•

5. Observe Lemma 3.6. It is easily found that C 'C

=

CotCo(I

~~

- P), that W 'C 'C (I - P) :: 0 and that (of course) (I -~ 0 0

P)W

=

O. Furthermore it holds that (1 - P(K»

=

(I P)

-~

W L -lW '(A (K» 'K and thus that A (Kl == Ao(K) (I P)

-~ 1 ~ 0 ~

" A

A (K)W L -lW '(A (K» 'K. Finally, .... (K) '" 0 «l.lO)} may be

rewritten as 0 = (I - P')Co'Co(1 - P) + KAo(K) (I - P) + (I - P')(Ao(K)}'K + KBL -IB 'K=KA (K)W L -IW '(A (K»'K

0 0 0 0 ~1 ~o

(KV

=

0). Now compare the main result in [12]. Then one

~

should note that I, 0, 8 and A in [12J play the role of our V and AoCK) (1). Moreover, the matrix

~

quadratic rewriting addition,

equation mentioned there is the result of ,(K) = 0 tor the special case treated in [12J. In the linearized equation in [12] is the interpretation at A '(K)K + KA el) = O. Together with KV

~ ~ a

= 0, this pair of equations has for K

=

K+ only the trivial

-

*

solution K

=

0 if a (Z) n CO

=

0 (see Remark 7 below Prop.

3.5). Thus, our approach covers the case in [12] and, in

fact, is the most general one using the transformations of Moylan and Hoore ([27]).

Now let K be any real symmetric solution of the dissipation inequality F(K) ~ 0 ([1, Sec. 6]). Then it is easily found with

(3.15) and [6, (A.6)] that

_ 0 0 [ L -In (K)

1

(D 'D )-l(D 'C + l 'K) =: " "

,,~

Ll-lnadd(K) (3.11a) (L -1 = (D 'D ) -1 L -1

=

(D 'P D ) -1)

o 0 0 , I add Do add where

(3.17b) no(K) = [(Do'C + Bo'K) - Do'CP(K)]

and nadd(K)

=

W~'o(l) = DaddPDoC + (Sadd - DaddDoLo -IBo')K (3.17c) (P(I)

=

VonLI-IW~to(K) appears in Lemma 3.6, with to(K) defined

in (3.11». Observe that P(I)

=

W~Ll-lnadd(K). The matrix

For turther use, let us partition L₁- I as tallows:

I

ll'

I

L.-··

:~:

.

~

0.18a)

with Ii (i

=

1, ~) counting qi columns (recall ~on = qo

+ q ), and also we will use the short-band

~

2, ... ,

notation

.

..

,

(3.l8b)

Analogously, we introduce (el)i (i '" 1, 2, •.• , ~) being the (qi + ••• + q~) x ~1 lower part of the right invertible

-matrix C, {(3.lt), ().1v); (C_I)l =

e,).

If «3.1s»

*

Zi := P!R! (i = 1, .•• , a) (3.19a) 1 1

*

*

and Z

=

diag(Zi) (3.19b)

(compare (3.1p) - (3.lq», then (Zc)i (i

=

1, •.. , a) stands for the (p. +

1

C C C

+ P ) _a x r 1 lower part at Z «3.1p); (Z }1 _-a-

=

Z )

*

*

*

and {Z )i for the (oi + ••• + 0a) x

r

_{a- 1} lower part at Z «Z)l • Z*). Finally, let eli (i

=

1,

...

,

on)

be the i-th block row

-

. *'

ot C_I, and ZCJ , Z J (j

=

1,

*

.

Z , respectively (with ZCJ

..• , a) be the j-th block row at ZC, counting Pj and z*j counting

OJ

rows). Then we define the regular matrix E: I

E '" [(E_{I )} I, (E_{2 ) ' , • • •} (Eon)

I

(Ean+l) , ••• , (E

a)']'

with «3.1n» - - i-1 (Cl)i(~ll). Ei := (zc) . (All) 1-1 for i = 1, • • • I ~

*

1 - i-1

[

_{(Z )i(A} l1 )

1

_(3.20b)

[

(ZC).

(1

11)i-l

1

for i • ("»+1), and E. :=

_{* -}

1 i-1 ... , a . 1 (Z ) i (A 11)

The matrices E112 " , · · , +1 and E count respectively

~~ a

• •• + q ) + (p. + ••• + p ) +

~' a

+ 0 }

=

r., .. , q + (p + ••• + P ) + (0 +

a ~ ~ a ~ +

°a)

=

r~_1' (P~+l + ••• + Pa' + (0~+1 + ••• + 0a) = r~ rows and, finally, E has P + 0 = r 1 rows. Thus, indeed, E is

a a a

a-square. But why is E invertible? Actually the invertibility of E is straightforward if one takes a closer look at its structure (see Appendix 3), because E has everything to do with the

*

_[Vi'

regular transformations S. «3.1u», Pi «3.1s» and V.

=

1 1

Vil. A full proof, however, should be based upon induction and

will take too much of our time. Therefore we propose an illustrative "proof" for the case ~

=

2, a

=

3 in the third Appendix and leave the challenge of the complex proof to the mathematical die-hards among our readers. The paper is already bulky enough.

These last definitions have completed the third Section. In the last one, Section 4, the full solution of (LQCP) + is given, as

promised. Another promise: Sec. 4 will not be as comprehensive as its predecessor. Rang on, will you?

4. The determination of all optimal controls for the linear-quadratic control problem with stability

The generalized dual structure algorithm ([1, Sec. 4]) has transformed our system

x

into the system X «3.1a) (3.1b»

( l

and, after having applied the feedback in (3.lh), the state space has been decomposed «3.3». From (3.4), then, we have

a - b

x (00)

=

0 c:::t «x + Ya-l) (00)

=

0 and x l ' Z ( (0) = 0) (4.1) and it is stated in Prop. 3.4 that all controls in (3.6e) are

b

such that xI,a(oo}

=

O. In particular, the optimal trajectory for

x

b :=

[x~" x~·]·

(4.2a)

satisfies the differential equation pxb

=

L A (K+)V xb + L Xo

C ~ C C

where L = [L' L']' V

=

(W' , W' ].

C ct' c_z ' c c₁ c₂

and Aan (K+) is defined in (3.6c). Note that actually

px

=

A (K+)x + BC wC + Xo

( l ~ a a a

(4.2b)

(4.2c)

(4.3)

after having applied the optimal wa for (LQCP); in (3.1a), and

(4.2) then is immediate from BC

=

V

H11

K+W

=

0 and A W

=

a a a ana

VaAl" For w~, w: we may take any impulsive-smooth input without violating the trajectory for xb•

Next, let us apply preliminary feedbacks for w_aC ' wa as well lit (recall that we applied a preliminary feedback

(3.lh}). Ve propose «3.lp) I (3.lq) I (3.19» wC

=

- ZC x a + w 'c , ( l a i.e. w. =

-

Z. x. 1 c -a + w. (j = 1,

...

, a)

,

J_c J J- J_c

,.

'"

xa + 'lIt and w = - Z w a a

-for w in a (4.4a) (4.4b) (4.4c)

i.e. 1f,

=-J. (4.4d) where ;c = a w

'.

= [w1' I a •

...

, WI] , a. (4.4e) ;~

" ... , ;a

1 'j '. Now, we will

a- show that for

, 'c '.

certaln wand w also E(xa +

~-1) (~)

=

0 ( (3. 20» and

a a

therefore «4.1» x(~)

=

o.

Actually we will give relations for wc

a

,

.

and w that turn out to be necessary and sufficient

a

conditions for x(~) to be equal to zero. We will need the two rational matrices below. Let (1' ). (i = 1, ••• , a) denote the

P_a 1

+ + p}

a x Pa lower part of the identity matrix of

dimension P

a ({I~ _a ) 1

=

I' ) and let _Po. {I~ _a )i analogously stand for

the (0

1, + ••• + a ) x a lower part of

r

{(I~ )1 = l ' ) a '

a

0. 0.

compare the Section 3. Then

•

", (s) :=

a

a a

(4.5)

Also, we introduce an impulsive-smooth distribution (depending on x_{o )} with the regular part coming from (4.2b) (where we establish that xb

=

L (pI - A (K+»-lX since L A (K+}W

=

0)

c ~ 0 c"D a

and the impulsive part is of order j-l (e.g. [4J):

, b (j-l). (. . 1)

9.(X_o} := (A (K+»J W x + I (pl(A (K+» J-1- x_o)

J ex... C ex...

u i=O u (4.6)

Ve state our main result of this Section.

Theorem 4.1.

1}. Consider the problem (LQCP)+ «2.3». Then

V n : J+(x_{o )} < w ~ (A, B) stabilizable. Xo E IR

2). Assume this to be the case. Then J+(x o) = xo'K+x_o where K+ is the largest rank minimizing solution of the dissipation inequality ([1, Sec. 6]). Furthermore, for all Xo an

*

optimal input exists if and only if a (I) n CO

=

0.

3). Assume this to be the case. Then it holds that u is such that x(~)

=

0 and J(x_{o ,} u)

=

J+(x_o) if and only if

c Ac

*

1<.*

." (p) w (00) = 0 and ." (p) w (00)

=

0 (4. 1)

<X <X

Here ."c(p), .,,*(p} are the matrix-valued distributions

c *

obtained by writing s

=

p in ." (s), ." (s) «4.5}).

4). Assume this to be the case. The optimal trajectory for xb follows from (4.2b) and the optimal trajectory for (xa +

v I' from (see Sec. 3, (3.17) and further)

-a-(C1)1(All)O

-

-(C₁)2(A_{1I , l}

ex

a + v )

=

~ : _ (~-1) (C 1) (A ,,) -(1-1

<i>

(ZC'l(1_{11 'O} (ZC)

2(1

11) I (Ll -I ) ln add(K+)9o(Xo) (L 1-1) 2nadd (K +) 9 I (x_{o )} (xa + v )

=

-(1-1 , (4.8a) (4.8b)

and

*

-(Z )1(A_{11 )O} * -(Z ) 2(A_{l l}) 1 (xa ₊

_Ya-1

)

=

-*

'.

+ 'P (p}w • a (4. Se)

Let So be tbe regular transformation from [1, (4.2) -(4.3)] (see also App. 1, (14]). Then the optimal inputs u are u '

s.~:l.ith

((l.h»

; = _

CO(xa

+; ) -

L

-In

(K+)W xb

o -a-I 0 0 c

(4.9a) j

and Wo following from (4.9b) - (4.9d), where A(j) := H A.

i=1 1

= (A~AI 1 .. . 11'1,) and .,1(0) != I (see (3.1r)):

J J- fo 11 'aadd (K +) 91 (xo) lZ:1'2add(K+)9a(Xo) ₊ e l l (A 1 1) 1

e

_l2_(A I1)! _(xa _{+ v 1)' (4.9b)} -a-Wo >= -ZC A(a-l) a ZC1

(A

l l) 1 Z~2(All) 2 (xa + ) + Ya-l Zca(A ll)a

o

o

o

o

0. 1 P P a (4.9c)

I P:"(OI

l~·

I .,

I

+

P L 9₁(X_O)

*

*

ot

and P aA(1}

= -

P IIL 92 (x 0>

P:,,(a-ll

• ot p*~ 9 (x o) otot

*

IP

'(A ..

I'

p'l 0 0 * - 0 1 P a(A_l1)2 (xa + 0 pal 0 '*

.

V 1) + O 2 w

.

-a-

.

ot p*a(.~l1)a 0 0 _p aI _a a (4.9d) The optimal regular output is given in (3.8) and tends to zero exponentially fast as time goes to infinity. Finally, we find that the optimal state trajectory for positive times lies on the linear subspace of dimension n - n~ determined by the n~ linearly independent equations

(Ll-l),Radd(K+}(A (Kt)} 0 (L -1) R (K+) (A

~(K+»

1 1 a add 0-J) X

=

0

<1>-1

(L -1) n (K+) (A (K+» 1 ~ add ~ (4.10)

where n denotes the number of infinite zeros (e.g. [14]).

00

Proof.

Ad 1}. Let for every x_o' J+(x_{o} (} 00. Then, obviously, J;(x_o) < 00

and hence (Prop. 3.4, Prop. 3.5 (a» (A, B) is stabilizable. Conversely, if (A, B) is stabilizable, then for all Xo there is

an input u such that x(oo}

=

_{0 and therefore J+(x o} (} ~.

Ad 2). Assume that (A, B) is stabilizable. Then (Prop. 3.4, Prop. 3.5 (b) and Remark 2 thereafter) J;<xo) = xo'K+xo with Kt the largest rank minimizing solution of the dissipation inequality. Now it is shown in [15] that J+(x o}

=

xo'Kx o with K some real symmetric positive semi-definite matrix such that F(K)

One even establishes that I is a rank minimizing solution of the dissipation inequality (Prop. 4.2 (a». On the other hand, we have that for every I for which F(I) ~ 0 it holds that inf(J(x o, u) lu e e~mp' (x/ker(I» (co)

=

01 == xo'Koxo with Ko ~ I

(Prop. 4.2 (b)). But then we have proven that K == K+, for J+(x o>

=

xo'Kxo ~ _{J;(x o)} == xo'l+xo ~ infIJ(xo , u) l(x/ker(K» (00)

=

OJ ~

xo'ixo (recall that for every K that satisfies F{K) ~ 0 it holds that 9 c _{ker(K)!). Thus J+(x o}>

=

J;(x_o>

=

xo'K+x o. Since from

(4.1) it follows that xb(t) has to converge to zero for large t in order to obtain xCco)

=

0, we find (Prop. 3.4, Prop. 3.5 (c»

•

that necessarily a (E)

n

CO == 0.

*

Ad 3), 4). Assume that a (E) nco == 0. We will show that for all Xo an optimal input u for (LQCP) t exists and, moreover, that the

set of optimal controls for (LQCP)+ is a subset of the set of optimal inputs for (LQCP); (see (3.6e». For

Wa

we take (Prop. 3.4) w

= -

(D 'D ) -l(D 'e +

B

'K+)x which yields (4.3) and

a -~-~ -~ -~ a

J(x_{o ,} u) == xo'K+x_{o •} The resulting trajectory for xb is regular

'c '*

and equals Lc(PI - A (1+»-lXO' Now we want to choose w , w

~ a a

such that E(xa

+;

1) (co)

=

0 which implies with (4.4) and

-0:-(3.le) that we have found an optimal u.

First, observe that pxa == pL x

=

L [A (K+)x + Bewc + _{x o]}

=

aa a ~ a aa

- a b - e

A11X + L A (I+)W x + H1w + L Xo (with the optimal

aO

D C a a

xb-trajectory given above). From App. 4. Lemmas 1, 2 (ii) and (4.6) it is then found that

p(xa

+;

1) -0:-since wC == Zc; a -a-1 4, Lemma 2 (i). - a -= Al l(x + !a-1' + L_a81(XO ) + Iowo , (4.11)

and All == All +

HIZ

C «3.10». Next, from App.

Iowo

=

wa - C !a-1

= -

C(xa +

i

_{a- 1 )} +

(K+" = 0) and thus (C3.1v) , (3.17a» we get (4. 9a) : a

Wo = - _{CO (xa + !.a-1) - L o} -In CX+)" x _{0 c} b (4.13a) and C, (xa + !.a-1)

= -

L ₁ -In _add (X+)" _c xb (4.13b)

v 1) (00)

=

O. Differentiating (4.1Jb) yields, with (4.11), (4.6)

-a-- -a-- a

C,A,,(x + !a-1} + C.Iowo

= [-

_{CILa - L,-l1'2add (X+)"cLc]'}

9 • (x 0)

(observe from (4.6) that p90(X o)

=

"cLc9,(Xo) and that 9,(Xo)

=

(D 'D )-'D 'CW (and thus

~-~ -~ a

«3.17a» C,

=

L,-lnadd(K+)"a) we get

Le. and Hence

C,A,,(xa + !.a-1) + C,Iowo = - L,-'n_add(K+)9,(X o)'

*-

-

-

a

-.d,W o = - I, '1'2_{add (Kt)e, (x o) - C,IA" (x} + !a-l) , (C,)aAll(Xa + !.a-1) = - _{(L,-1)a nadd(K+)9}1(XO ) •

(4.14) (4.15a) C4.15b) !a-l) (00)

=

O. Differentiating (4.15b) then

=

A (K+)p9_o(X_o) + pXo and 9₂(X_{O )}

=

~

A~

(X+)9,(X o) + pxo'

(C,)

a(AI1 ) 2(X

a

+

~a-1)

+ (e,) zAl lI owo

=

-(C.)aA"L + (L,-')z1'2 ddCX+)A CXt

) " L ]9,(X_{o )} +

a a ~ c c

- p(L,-') a1'2

add(K+)X o

= -

(LI-')21'2add(K+)6aCXo) and thus

.:1:A,W

o = - la'1'2add(K+)9aCXo) - C,Z(!,,) Z(xa + !.a-l)

(4.16a) and (C,).(All )2(Xa + Ya-1) = -(L,-1)'''add{K+)9 z(X o) • (4.16b)

In this manner the formulas (4.8a) and (4.9b) are found. Next, from (4.4a),

(4.17)