A note on "Families of linear-quadratic problems : continuity properties"

A note on "Families of linear-quadratic problems : continuity

properties"

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Geerts, A. H. W. (1988). A note on "Families of linear-quadratic problems : continuity properties". (Memorandum COSOR; Vol. 8805). Technische Universiteit Eindhoven.

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

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Memorandum COSOR 88-05 A note on "Families of linear-quadratic problems: continuity

properties by A.H.W. Geerts

Eindhoven, Netherlands February 1988

A NOTE ON: "Families of Linear-Ouadratic Problems: Continuity properties"

ABSTRACT

In the paper "Families of Linear-Ouadratic Problems: Continuity properties" one-parameter (~) families of these problems with stability are investigated. In this technical note we will show that the set of invariant zeros of the £-problem is contained in the set of invariant zeros of the "boundary" problem (the problem with £

=

0) if the "boundary" system is left invertible. This result is obtained only by applying the assumptions on the continuity and the monotonicy

of the weighting matrices that are made in the

above-mentioned paper. Consequently, we can replace the third assumption in that paper by a much weaker one.

February 1988

Research supported by the Netherlands organization for

1 Introduction

In a recent paper ([1]) one-parameter (~) families of linear time-invariant finite dimensional systems with corresponding linear-quadratic control problems with stability ([1, Sec. II]) are introduced. Then convergence of the optimal cost for the ~-problem (~ ) 0) to the optimal cost for the "boundary" problem (Le. the problem with ~ = 0) is established only by assuming continuity and (in a certain sense) monotonicy of the weighting matrices w.r.t. ~. In addition, convergence (in distributional sense, see [1, Sec. II]) of the optimal inputs for positive ~ to the optimal controls for ~

=

0 is proven, under the assumption that for each initial condition the latter (and thus the former) are unique. In this case also convergence (in distributional sense) of the corresponding state trajectories is obtained and, moreover, the optimal outputs are proven to converge strongly

([1, Sec. IV, p. 325]).

Now, it is well known that for all initial conditions the optimal inputs for the "boundary" problem exist and are unique

(e.g. [1, Prop. 4.1 (ii)]) i f and only i f the "boundary" system (i.e. the system with ~

=

0) is left invertible and there are no

invariant zeros (see e.g. [1, Sec. III, p. 325]) on the imaginary axis. Indeed these assumptions are expressed in assumptions A.3 and A.4 on page 325 of [1]. Also, it is remarked there that if the intersection of the set of invariant zeros for

*

the "boundary" system, a (X_o), and the imaginary axis CO is empty, then in general the intersection of the set of invariant

*

zeros of the &-system, a (X), (& ) 0) and CO need not be empty ~

*

as well. The latter condition, a (x ) n CO

=

0 for small ~ ) 0, ~

is necessary to ensure that also in the ~-problem optimal inputs exist for all initial conditions.

3

-However, in this technical note we will show that if the

"boundary" system Zo is left invertible then the continuity assumption and the monotonicy assumption on the weighting

*

*

matrices imply that a (Z ) c a (Eo) for all £ in a SUfficiently

£

small closed interval [0, 8] with e > O. Thus we establish that

*

the assumption " 3

q > 0 : a (Z£) n CO = 0 for all £ E [0, q] "

of [1] can in fact be replaced by the much weaker assumption

*

2 Basic problea foraulation and our additional results

Consider the finite-dimensional linear time-invariant systems E_E. determined by the system equation

x(t)

=

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

=

X_{o ,}

together with the E.-depending output equations YE.(t)

=

C(E.)x(t) + D(E.)u(t) ,

(2.1a) (2.1b) and with associated cost-functionals

J (x_E. o ' u) = jlly (t) nZdt • (2.1c)

0 E.

Here, u(t), x(t) and yet) are all real vectors of m, nand p components and C(E.), D(E.) are assumed to be linear mappings from Rn to ~ and ~ to RP, respectively, for E. lying in an interval [0, e] where e ) 0 will be specified below. The pair (A, B) is stabilizable.

For fixed E. E [0, e] the linear-quadratic problem with stability

associated with (2.1) is the problem of finding the infimal value of J (x_E. o , u) with respect to an appropriate class of

. ( , [1 S II] stab ( ) ) d t t ' f l't

1nputs 1n , ec. CU

dist Xo an 0 compu e, 1

exists, an optimal control. The optimal cost in [1] is denoted

*

*

*

by J (x_{o). It is proven in [1, Th. 3.2] that lim J} (x_o)

=

_{Jo(x o)}

E. E.~ E.

for all X_{o if the following two assumptions hold:} A.l: E. ~ C(E.) and E. ~ D(E.) are continuous at O. A.2: For all 0 ~ E.. ~ E._z ~

e

we have O(E.I ) ~ O(E.z).

Here

Q(E.) = (C(E.), D(E.»'(C(E.), D(E.» (2.2)

and thus the cost-functionals in (2.1c) can be rewritten as JE.(xo • u) =oj[x'(t), u'(t)]O(E.)

[:~~ndt

• (2.3)

5

-(A.2) ': For all 0 ~ £1 ~ £a S e we have Q(£I) SQ(£2) and there exists a subspace N such that ker(Q(£» = N

for all & E (0, e]. This is shown in Leama 2.1. Lemma 2.1.

Assume that A.2 holds. Then there is a el S e and there exists a subspace N such that ker(Q(£»

=

N for all £ E (0, e'l.

Proof. From A.2 we observe that, i f 0 S £1 S £2 S e, then ker(Q(O» ~ ker(Q(£I» ~ ker(Q(£a». Renee ker(Q(£» is constant for £ sufficiently small and positive. Note that not necessarily

ker(Q(O» = N.

Remark.

Of course, the assumption ker(Q(£»

=

N (£ E (0, e]) is

equivalent to: For all £ E (0, e], ker([C(£), D(£)]) = N.

Next, let ...(x) and .(1:) be the weakly unobservable and the

£ £

strongly reachable subspace corresponding to 1: (£ E [0, e]),

£

respectively, see e.g. [2, Def. 3.8 and Def. 3.13]. Then from

the algorithms [2, (3.20)] for computing or(x ) and [2, (3.22)]

£

for computing

.(x )

with the Remark we establish that

£

Proposition 2.2.

Under the assumption (A.2) I we have: There are subspaces ... and.

such that for all & E (0, e], ...(x )

=...

and w(1: )

=

W.

It turns out that these subspaces are contained in their respective versions for the "boundary" system:

Proposition 2.3.

Proof. Let X_o E ~ and u{t) be a smooth input (e.g. [2, Prop. 3.9]) with resulting state trajectory x{t) such that y (t)_E.

=

o.

Then for all t also (see (2.3) and recall that Q{E.) ~ Q(O»

_

_

[X(t)]

[xI (t), uI (t)JQ(0) _ EO, u(t)

i.e. yo(t)

=

0 and X o E ~. Next, if

"0

= 101 and W_{i +}

1 = [A, B]f{;i • R

m) n

NI, then"

=

"n ([2, (3.22)]). If, in addition,

"0

=

(01 and "i+1 = [A, B]("i • Rm) n Nol with No = ker(QCO», then"

=

"n and it is easy to show that for all i

=

0, ••• , n, "i

obvious.

c

"0.

₁ Thus " c W. The last assertion now is

Observe from the previous Proposition that if s = (01 then s = (01 (in [1] s is called the controllable output-nullina subspace

*

*

and denoted by

s

and ~ is denoted by ~ ). Hence if Eo is left invertible ([2, Th. 3.26]) then E_E. is left invertible. This is also remarked in [1] on page 325. Note that not necessarily "

=

• and ~

=

~!

We are now ready to state our main result. Assume, as in [1,

Sec. IV], that Eo is left invertible. Then E_E. is left

invertible. If we denote the set of invariant zeros of E_E. by

*

(2.4a) (2.4b) 7 -Theorem 2.4.

*

*

o (x ) co (X_{o) for all} ~ E [0, ~]. ~

Proof. To start, define for a fixed ~ E (0, ~] the set of

mappings 9=(~)

:=

IF: Rn .. Rml(A + BF); c;, (C(~) + D(~)F)

1;.=

1011. It is well known (e.g: [1]) that the spectrum o(A + BFI.} is independent of F E 9=(~) (~= (01). Now it is easy to see that 9=(~2} c g(~l} (~ ~ ~2 ~ ~l > O) since Q(~2} ~ Q(E._{l ). Therefore}

*

we observe that 0 (x ) does not depend on E. for ~ E (0, ~]. Let

~

_.

F E 9=(E.). It follows from Q(E.) ~ Q(0) that (C (0) + D(0) F) I. = 101. Next, let the regular transformation So be such that D(O)So

• [D._ 0]

w~

th D. of full oolulIln raDk_ aDd write F • So

lr:J

aD~

BS o

=

rB o, Bo]. Then we find that

i\ ,.

= (-

(Do'D o) .1Do'C(0» I. and thus with

C(O) 0 := _{(I - Do(Do'D o) -lDo')C(O)} A(O} 0 := _{A - Bo(Do"D o) .1Do'C(0)} it holds that

(A(O) 0 + _{BoF o).} C • and C(O) o. = 101 • (2.5)

Let AE o(A + BF) and v E • such that (A + BF)v

=

Av (i.e. A is an invariant zero of

x ).

Then from the above (A(O)o + BoFo)v

=

~

_ .

.

AV. Now (Prop. 2.3) • c • and it is always possible to extend Fo

_.

_{. .}

such that for the extension FOl it holds that (A(O)o + _{BoF ol ).} C

- -

--., C(O)o.

=

101, and F01 I.

=

Fol. (see e.g. [2, (3.12)]; the

extension for

F

_o, ~f course, is Fol.

= (-

(Do"Do)-lDo'C(~)~ I~). But since for any F01 ' for which it holds that (A(O)o + _{BoF ol ).}

-

-C • and C(O) 0'-

=

101, o(A(O) 0 + B_oF₀₁ I'-). is fixed (~= ~O~) we

thus have shown that there is a v E'- such that (A(O)o + BoFol)V

*

*

*

From Theorem 2.4 we conclude that if 0 (Eo)

n

CO = 0 then 0 (E )

E.

n CO == 0, provided that Eo is left invertible. Hence the assumption A.3 in [1, Sec. IV] can be replaced by:

*

(A.3) ': 0 (Eo) n CO

=

0.

In addition, in Theorem 4.3 of [1] it suffices to assume that

A.l, (A.2) , (or A.2), (A.3)' and 1.4 hold in order to prove the stated convergence results of inputs, state trajectories and outputs. Remark. In [I, and in

*

o (Eo)

Sec. V] the special case of [1, Lemma 5.1] it is stated

*

n CO == III and 0 (E ) E.

"cheap control" is considered that 1.3 holds if and only if

*

n CO == III with 0 (E) =

E.

0(11<

ker(e)

IA

».

Observe that i f Eo is left invertible (Le •

•

A.4 holds) then indeed 0(1

1<

ker(e)

11

»

c 0 (Eo) as follows

9

-Discussion

In [1, Remark 3.4] it is noted (by means of a counterexample) that in general there is no convergence of the optimal cost for

the 6-problem without stability (i.e. with no end-point

conditions) to the optimal cost for the "boundary" problem without stability. In a future paper ([5]) we will show that this is indeed the case. By starting from the assumptions A.l,

(A.2) ., (A.3)' and A.4 we will prove in [5] that the limit of the optimal cost for

_.

_.

the 6-problem without stability equals xo'Ix o where X turns out to be the largest element in the set of real sfllIIlletric solutions K of the dissipation inequality that satisfy (. + W) C ker(K). This dissipation inequality is in [3,

Sec. 6] written in the form

reX) = [A'X + Xl + c·(O)C(O) XB + C'{O)D(O)] ~ 0 B'X + D'(O)C(O) D'(O)D(O)

and it can be shown that X even is the largest real symmetric X for which r{X) ~ 0, (. + w) c ker(K) and rank (r(X» is minimal. Also convergence results concerning inputs, states and outputs will be obtained in [5], similar to the ones in [1, Th. 4.3]. A first contribution to the general problem of determining the limits of optimal cost, inputs, states and outputs for the 6-problem without stability sketched above is given in [4], where the "cheap control" problem without stability is studied.

References

[1] B.L. Trentelman, "Families of linear-quadratic control problems: continuity properties", IEEE Trans. Automat. Contr., vol. AC - 32, pp. 323 - 329, 1987.

[2] K.L.J. Bautus & L.M. Silverman, "System structure and singular control", Lin. ~ i mL., vol. 50, pp. 369 -402, 1983.

[3] A.R.V. Geerts, "All optimal controls for the singular

linear-quadratic problem without stability; a new

interpretation of the optimal cost", Memorandum COSOR 87 -14, Eindhoven University of Technology, 1987, submitted for publication.

[4] A.B.V. Geerts, "Continuity properties of the cheap control problem without stability", Memorandum COSOR 87 - 17, Eindhoven University of Technology, 1987, submitted for publication.

[5] A.R.V. Geerts, "Continuity properties of one-parameter families of linear-quadratic problems without stability", in preparation.