On the output-stabilizable subspace

On the output-stabilizable subspace

Citation for published version (APA):

Geerts, A. H. W., & Hautus, M. L. J. (1988). On the output-stabilizable subspace. (Memorandum COSOR; Vol. 8827). Technische Universiteit Eindhoven.

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

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 88-27

On the output-stabilizable subspace

A.H.W. Geerts and M.L.J. Hautus

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven, October 1988 The Netherlands

ON THE OUTPUT·STABILIZABLE SUBSPACE

ABSTRACT

Properties of a certain subspace are linked to well-known problems in system theory.

Keywords:

Output stabilizability, linear-quadratic problem, singular controls, structure algorithm, dissipation inequality.

ON THE OUTPUT-STABILIZABLE SUBSPACE A.H.W. Geerts & M.L.J. Hautus

O.

Introduction

The objective of this paper is to investigate properties of a space to be denoted !f-, which plays a role in a number of problems in system theory.

In [WO 79, Section 4.4], the Output Stabilization Problem is considered (for the case D = 0):

PROBLEM (OSP): Given the system

L:

x = Ax + Bu, Y = Cx + Du

find a feedback map F such that with the input u

=

Fx, we have y(t) ~ 0 (t ~ 00) for any initial value x

o'

If this problem has a solution,

L

is called output stabilizable. A necessary and sufficient condition for output stabilizability was provided by [WO 79, Theorem 4.4]. A slightly different formulation of this condition was given in [HA 80, Theorem 4.10], where it was shown that OSP has a solution iff !f-

=

X. Here X denotes the state space Rn, and

!f was defined in terms of (~,w)-representations. More generally, the space !f also plays a role in the output-stabilization problem under disturbances, i. e. , the problem of achieving BIBO stability in the presence of a disturbance input term Eq. Then, it turns out, the condition

is im E ~ !f-. The space is also relevant for LQ problems.

Consider the general positive semidefinite infinite horizon LQ problem:

PROBLEM (LQ): Determine a function u: [0,00) ~ RID such that for the solution x of the differential equation

·

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

o

the following quantity is minimized:

!XI

J(x ,u) :=

S

w(x(t),u(t»dt,

o 0

where

w(x,u) := x'Qx + 2u'Sx + u'Ru

is a positive semidefinite quadratic form in the variable

(x, u) E IRn+m•

As in [H&S 83], we introduce matrices C and D using the

factorization

[QS R

S'] [C']

=

D' [C D],

which is possible because the matrix in the left-hand side is positive semidefinite. Next, we define the output

y :=

ex

+ Du,

and the integral to be minimized can be written as

S:ly

12dt .

The LQ problem is called regular if D is injective (left

invertible). For regular problems, the problem is well

established and considered classical. Curiously, the problem

of finding necessary and sufficient conditions for the

existence of a solution of this (infinite-horizon) problem

has found little attention even in the regular case. Usually, one is satisfied with the statement that a solution exists if

the system

L

is stabilizable. Of course, this condition is

not necessary. In [GE 88], a necessary and sufficient

condition was given for the regular case (R>O) in terms of

the stabilizability of a suitably defined quotient system.

For singular problems, a reformulation in the style of

[H&S 83] is needed incorporating distributions as possible

inputs, since otherwise, one can not expect that optimal

controls exist even in the finite horizon case. With this

singular problems. It is expressed in terms of the structure algorithm (See [GE 88, Remark 4]).

In the present paper, this issue is also addressed and it

will be shown in particular that output stabilizability (and

hence !f- = X) is necessary and sufficient for the existence

of a solution of the LQ problem. This intuitively rather

obvious condition turns out to be relatively difficult to

prove. In the case that D is left invertible (the regular

case), the condition will be shown to be equivalent to the

existence of a positive semidefinite solution of the

Algebraic Riccati Equation associated with the LQ problem

mentioned above. Finally, the condition !f-

=

X will be

brought into connection with the dissipation inequality, to

be discussed in section 3.

1. The output-stabilizable subspace

Consider the system

r:

x

=

Ax + Bu, Y

=

Cx + Du,

where A e IRnxn, B e IRnxm, C e IRPxn and D e IRPxm•

In connection with the LQ problem, we have to admit

distributions as inputs. These distributions are chosen from

the class ~ of impulsive-smooth distributions on IR with

1mp

support on [O,~) introduced in [H&S 83]. Such a distribution

can be written as a sum of a function which is smooth on

[O,~) and an impulsive distribution with support in {O}.

However, in the LQ problem, only distributions such that the

output is regular are admitted. The class of such inputs is

denoted 'IL._--~ Using the notation y (~) for lim_{t ..~} y (t), we

introduce the following spaces:

r:J

'!J2 := {X 0 E XI 3u e

~: IOJ

Iy (t) 1

2

dt < OJ }

o

and for every F E R~n, we define the space

F

'!J

I

:=

{xo E XI If u

=

Fx then y(OJ)

=

O}

'!J~

:= {xo E XI If u = Fx then IOJjy (t) 12dt < OJ}.

o

Finally, the space ~- is defined as

~-:= <AlimB>

+

X- (A)

+

V L

where <AI imB> is the reachable subspace of

L ,

X- (A) the

stable subspace of A, and V

L denotes the weakly unobservable

space of L, i. e., the space of initial states X

o such that

there exists a regular control (hence without impulses) such

that the output is identically zero.

Our main result is:

THEOREM For every

L,

we

F. In addition, there

i = 1,2.

have '!J

i

exists

= ~- and '!JF s;; ~- for every

i

F such that '!JF = ~- for

i

Because of the relation '!JF

=

~- for some F, we will refer to

I

~- as the output-stabilizable subspace. Some of the

statements of the Theorem are rather obvious. In particular,

if we use the feedback control u

=

Fx, the resulting output

y(t) will have the property that it tends to zero

exponentially fast when either X

o e '!J~ or Xo E '!J~.

Consequently, '!JF

=

'!JF. Furthermore, it is trivial that

1 2

'!JF s;; '!J for all F. Also the inclusion ~-s;; '!J is easily shown.

i i 1

In fact, any element of x E!f- can be decomposed as

x

=

Xl + x

2' where XlE <AI im B> + X-(A) , the stabilizable

subspace and x

2 E VL' It is known (See [WO 79, Ch.3]) that

there exists a feedback F such that with the input u = Fx and

initial state x, the state x (t) tends to zero. For the

initial value x, there exists a smooth input such that y is

2

identically zero. A similar reasoning can be used for proving

the inclusion ~-~ ~ . Finally, the fact that there exists an

2

F such that ~F

=

~- is known (compare [HA 80]). The difficult

1

part of the Theorem is the proof of the inclusions ~. ~

~-1

(i

=

1,2). This proof is given in [G&H 88].

An interesting special case is ~-

=

X.

This condition can be

reformulated as: For each X

o e

X

there exists an input u e ~

such that y(oo) = O. According to the Theorem, an equivalent

formulation is: There exists a feedback u

=

Fx such that the

has the

stable, i.e.,

solution to

resulting closed loop system is output

y(oo)

=

a

for every x_o e X. This is the

problem OSP mentioned in section 1. The equality ~2 =

X

says

that in the optimal control problem, there exists for any X

o

an input u e ~ such that the performance J (x_o' u) < 00. This

will be seen to be a necessary and sufficient condition for

the existence of an optimal control in ~.

In addition to V

L, the strongly reachable subspace WL plays

an important role in the (singular) optimal control problem.

It is defined as the space of points x e X for which there

1

exists an impulsive input u e U

L such that x(O+)

=

Xl' It can

easily be seen that W

L =

a

iff D is injective, which

corresponds to the regular case for the LQ problem.

Furthermore, it can be seen (see [H&S 83, section 3]) that

strongly reachable states are also strongly (null-)

controllable, i.e., there exists for any such state an

impulsive control in ~ for which the state jumps to zero

instantaneously, so that x(O+) = 0, while y remains zero.

Obviously, for such initial states, the minimal cost of the

LQ problem equals zero.

2. Optimal control

It is clear that the existence of a solution of LQ for a

certain X

(see [GE 88, property is semidefinite minimizing J(xo'u). It follows

for the regular problem, this the existence of a positive Algebraic Riccati Equation

of such a solution for all x implies~·

=

X,

hence that

L

is

o 2

output stabilizable. Conversely, it is shown in [GE 88] for the regular case that the existence of u e CUI: such that J (x ,u) < 00 , is also sufficient for the existence of an

o input u e ~ Corr.2] that equivalent to solution of the ARE:~(P) := C/C + A'P + PA - (PB + C/D) (D/D)-l(B/P + D/C) = O.

The smallest positive semidefinite solution of to be denoted by P-, yields the minimal integral:

this equation, value of the

min {J (xo' u)

I

u e ~}

For singular problems, the ARE is not defined, because D'D is

not invertible. However, as is shown in [H&S 83] for the left-invertible case and in [GE 88+] for the general case, a sequence of transformations can be constructed that transform

L

into a system of the form

.

x=Ax+Bu +Bu

o 0 1 1

Y

=

Cx

+

5

u ,

o 0

where the following properties are satisfied:

i) D is left invertible and contains the independent o

columns of D.

ii) rank

5 =

rank T (s), where T (s) o

is the transfer function of

L.

: =

D + C(sI - A) -lB is the derivative of is an invertible by the structure iii) im B1 ~ WI:. iv) C(WI:) ~ im

50.

v) u

=

1/(p)ii, where ii :

=

[~:],

p

o

and hence acting as distributions. Finally, ~

polynomial matrix, computed

algorithm.

The transformed system

f

can be reduced further by means of a feedback transformation, i.e., a transformation of the form u = Fx + u, where F :=

-0 (0'0)

-lO/C. This yields the system

o 0

(A ,:S,c

,0),

where

o 0

-:=C+DF._o

The result is the following system:

x=Ax+:Su_o +:su,

0 0 1 1

y = C x + o u .

0 0 0

The following properties can be proved:

A_o(WI:) S;; W

t ,

VI: + WI: = <ker Co lAo> = V~,

O/C

= O.

o 0

It follows from these properties together with the properties iii and iv mentioned before that the output y is independent of u. Hence u represents the nonuniqueness of the optimal

1 1

control. Selecting in particular u

1 = 0 leaves us with a regular control problem, since D is left invertible.

o

Consequently, the optimal control can be expressed in terms of the Riccati equation corresponding to system

r

:= (A,:So'C,Oo)· Let us denote this equation by i(p) =

O.

It is shown in [GE 88+] that the smallest positive semidefinite solution of this equation determines the minimal cost of the problem. It follows from these results that the existence for each x of an input u such that J (x , u) < 00, i. e., the

o 0

condition ~2 =

X,

implies the existence of an optimal control. Consequently, output stabilizability of

L

is equivalent to the existence of a positive semidefinite solution of the equation ~ (P) = O. Finally we note that ker P = VI: + WI:' in particular, WI: S;; ker P where P

~(P) = O. This property is an easy consequence of the equality V~ + W~ = V_f .

3. The dissipation inequality

Another way of characterizing the optimal controls can be expressed in terms of the dissipation inequality ([WI 71]).

To this extent we define the dissipation matrix

_ [C' C+A' P+PA

F (P) :

=

B'P+D'C

PB+C'

OJ .

0'0

The dissipation inequality or the linear matrix inequality

then reads:

F(P) ?; 0

For every solution P of the dissipation inequality the following holds:

rank F(P) ?; rank T(s),

where the right-hand side denotes the global rank of the rational matrix T (s) (see [SCH 83]). Solutions of the dissipation inequality for which equality holds in the above inequality are called rank-minimizing solutions of the dissipation inequality.

We consider the set:

r := {PI P=P', F(P) ?;

a}.

Then r ¢

t

because 0 e r. We are more particularly interested

in the rank-minimizing solutions of the dissipation inequality. Therefore we also introduce

rmin:= {P e r

I

rank F (P) = rank T (s) } .

The following results hold: (See [GE 88+]) i) F(P) ?; 0 implies W~ ~ ker P.

ii) P e r . iff P e r , ~ (P) = 0 and W~ s;; ker P.

m~n w

iii) P e r . (Recall that W~ ~ ker P ).

m~n w

The following theorem relates the dissipation inequality with the rest of this paper:

THEOREM

L

is output stabilizable i f and only i f there exists

P e r such that P ?;

o.

In addition, i f such a P exists,

min

the smallest such P equals the matrix P-, representing the minimal cost of the optimal-control problem.

The dissipation inequality enables us to formulate the

conditions for the existence and the actual computation of

the cost of the optimal-control problem directly in terms of

the original data of the problem. The characterization of P

in terms of the dissipation inequality is more general than

the characterization using the ARE, because it includes the

singular case. Note that the ARE is not defined in the

singular case, because the matrix D'D is not invertible.

Hence one has to use the ARE of the transformed system in

order to express P-.

REMARK It is shown in

y (t) ~ 0 for t ~ 00

equivalentlyI (x/ker

does not hold. It

[G&H 88] that if u is such that either

or S;lyl2dt < 00 then P-x(t) ~ 0, or

P-) (t) ~

O.

Remarkably, the converse

is possible that P-x(t) ~ 0 (t ~ 00),

whereas y(t) does not converge to zero. _

Finally, we note that a condition for output stability can

gi ven in the spirit of the results of [HA 70]. In fact, a

more general formulation is:

PROPOSITION Let ~ be an (A,B)-invariant subspace of X. Then

<Alim B> + X- (A) + ~ = X i f and only i f

VA

e

C, Re A ?;

OV~I

e

cn[~[A

-

AI,B]

=

0 A

~~

=

0

~ ~

=

0].

The proof is straightforward. The condition for output

stabilizability is obtained by taking ~

=

V

L• We obtain an

equivalent formulation by choosing a basis matrix V for V

L•

REFERENCES

[GE 88] Ton Geerts, A necessary and sufficient condition for

solvability of the linear-quadratic control problem without stability, Syst. & Contr. Lett., vol. 11, pp. 47 - 51, 1988

[GE 88+] A.H.W.Geerts, All optimal controls for the singular

linear-quadratic problem without stability; a new interpretation of the optimal cost, Memorandum CeSOR

87 - 14, Eindhoven University of Technology, 1987,

to appear in Linear ~g. &Appl.

[G&H 88] A.H.W.Geerts & M.L.J.Hautus, The output-stabilizable

subspace and linear optimal control, in preparation.

[HA 70] M.L.J.Hautus, Stabilization, controllability and

observability of linear autonomous systems, Nederl.

Akad. Wetensch. Proc. Ser. A, 73, pp. 448 - 455,

1970.

[HA 80] M.L.J.Hautus, (A,B)-invariant and stabilizability

subspaces, a frequency domain description,

Automatica, vol. 16, pp. 703 - 707, 1980.

[H&S 83] M.L.J.Hautus & L.M.Silverman, System structure and

singular control, Lin. ~g. & Appl., vol. 50, pp.

369 - 402, 1983.

[SCH 83] J .M. Schumacher, The role of the dissipation matrix

in singular optimal control, vol. 2, pp. 262 - 266, 1983

[WI 71] J.C.Willems, Least squares stationary optimal

control and the algebraic Riccati equation, IEEE

Trans. Automat. Contr., vol. AC - 16, pp. 621 - 634,

1971.

[WO 79] W.M.Wonham, Linear Multivariable Control: A

EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

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List of COSOR-memoranda - 1988

Number Month Author Title

M 88-01 January F.W. Steutel, Haight's distribution and busy periods. B.G. Hansen

M 88-02 January J. ten Vregelaar On estimating the parameters of a dynamics model from noisy input and output measurement

M 88-03 January B.G. Hansen, The generalized logarithmic series distribution. E. Willekens

M 88-04 January J. van Geldrop, A general equilibrium model of international trade with C. Withagen exhaustible natural resource commodities.

M 88-05 February A.H.W. Geerts A note on "Families oflinear-quadratic problems": continuity properties.

M 88-06 February Siquan. Zhu A continuity property of a parametric projection and an iterative process for solving linear variational inequalities.

M 88-07 February J. Beirlant, Rapid variation with remainder and rates of convergence. E.K.E. Willekens

M 88-08 April Jan v. Doremalen, A recursive aggregation-disaggregation method to approxi-1.Wessels mate large-scale closed queuing networks with multiple job

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2-Title

M 88-09 April J. Hoogendoom, The Vax/VMS Analysis and measurement packet (VAMP):

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Marcelis, a case study.

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M 88-10 April E.Omey, Abelian and Tauberian theorems for the Laplace transform E. Willekens of functions in several variables.

M 88-11 April E. Willekens, Quantifying closeness of distributions of sums and maxima SJ. Resnick when tails are fat.

M 88-12 May E.E.M. v. Berkum Exact paired comparison designs for quadratic models.

M 88-13 May J. ten Vregelaar Parameter estimation from noisy observations of inputs and outputs.

M 88-14 May L.Frijters, Lot-sizing and flow production in an MRP-environment. T. de Kok,

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M 88-15 June J.M. Soethoudt, The regular indefinite linear quadratic problem with linear H.L. Trentelman endpoint constraints.

M 88-16 July lC. Engwerda Stabilizability and detectability of discrete-time time-varying systems.

M 88-17 August A.H.W. Geerts Continuity properties of one-parameter families of linear-quadratic problems without stability.

M 88-18 September W.E.J.M. Bens Design and implementation of a push-pull algorithm for manpower planning.

M 88-19 September A.I.M.Driessens Ontwikkeling van een informatie systeem voor het werken met Markov-modellen.

3

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M 88-21 October A. Dekkers Global optimization and simulated annealing. E. Aarts

M 88-22 October J. Hoogendoom Towards a DSS for performance evaluation of VAX/VMS-c1usters.

M 88-23 October R.de Veth PET, a performance evaluation tool for flexible modeling and analysis of computer systems.

M 88-24 October J. Thiemann Stopping a peat-moor fire.

M 88-25 October H.L. Trentelman Convergence properties of indefinite linear quadratic J.M. Soethoudt problems with receding horizon.

M 88-26 October J. van Geldrop Existence of general equilibria in economies with natural Shou Jilin enhaustible resources and an infinite horizon.

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M 88-27 October A. Geerts On the output-stabilizable subspace. M. Hautus