Almost disturbance decoupling by a proportional derivative state feedback law

0005 1098/86 $3.00+ 0.00 Pergamon Journals Ltd. ~) 1986 International Federation of Automatic Control

Automatica, Vol. 22, No. 4, pp. 449 456, 1986

Printed in Great Britain.

Almost Disturbance Decoupling by a Proportional Derivative State Feedback Law* V. A. A R M E N T A N O t A P D compensator may be synthesized so that unknown disturbances have arbitrarily small influence on the output variables. Key Words--Multivariable control; disturbance decoupling; almost invariant subspaces; PD compensators; structural invariants.

almost disturbance decoupling it is necessary that F~e~o o% where the positive real scalar e is a

Abstract--In this paper the almost disturbance decoupling problem (ADDP) for a multivariable linear system is studied by means of a proportional derivative (PD) state feedback law, u = F x + R~. A condition for solvability of the A D D P by state feedback has been given by Willems ( I E E E Trans. Aut. Control, AC-26, 235, 1981) who also indicated that the solution of the A D D P requires high gain state feedback (F --* oo). It is shown here that under the condition given by Willems, the A D D P also admits a solution by a PD law involving finite maps. It is also shown that the type of PD law considered here preserves some structural properties of linear systems.

measure of the influence of d on z in the closed loop system. In this paper the A D D P is studied by a proportional derivative (PD) state feedback law, u = F x + RYe. It is shown that under the solvability condition given by Willems (1981) it is also possible to solve the A D D P by means of a P D law. The main difference is that the almost disturbance decoupling can be performed with a finite m a p F and a m a p R~ which converges to a finite m a p R as e ~ 0. This is of interest in situations where the derivative can be measured and one wishes to avoid high gain state feedback. For example, Trentelman (1985) has shown that the use of high gain state feedback to solve the A D D P may cause certain state variables in the closed loop system to become unacceptably large.

1. I N T R O D U C T I O N

CONSIDER the time-invariant linear system

(1)

:~ = A x + Bu + Gd z=Ox x~X:=

~n;ue~//:= ~";dE~:=

~;ze~e:=

,~,

where u, d and z are vectors which denote, respectively, the control variables, the unknown disturbances and the to-be-controlled outputs. The almost disturbance decoupling problem (ADDP) is concerned with the existence of a control law, u = F~x, such that in the closed loop system the influence of the disturbances d on the output variables z is arbitrarily small in some mathematical sense. The A D D P has been formulated and solved by Willems (1981) who expressed the solvability condition in terms of a certain almost controlled invariant subspace. He showed that if the disturbance decoupling problem introduced by Wonham (1979) is not solvable then in order to achieve

A feature of the P D law used to solve the A D D P is that (I - BR~) is non-singular for e > 0. Such a law will be termed a non-singular PD law to distinguish it from the regular P D law used by Armentano (1983b) to study the exact disturbance decoupling (influence of d on z is zero). In the final section of the paper it will be shown that a non-singular P D law preserves the structural properties of a linear system represented by the triple (C, A, B). Notation

Throughout the paper lower case letters will be used for vectors, capitals for matrices and maps, and scripts for linear subspaces and vector spaces. Im and ker denote image (range) and kernel (null space) respectively. I f X is a vector space and 50 c X then X (mod 50) or X / 5 0 denotes the quotient space {x + 5 °, x E f } , dim X / 5 0 = dim X - dim 50. If A: X ~ X is a map and 50 is A-invariant then A (mod 50) is the unique

* Received 9 May 1985; revised 21 November, 1985. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor B. A. Francis under the direction of Editor H. Kwakernaak. S This work was carried out while the author was with the Department of Electrical Engineering, Imperial College, University of London. The author is now with the Department of Electrical Engineering, FEC, UNICAMP, C.P. 6122, Campinas, 13100, S.P., Brasil. 449

450

V.A. ARMEN-IAN(~

m a p such that A (mod ~ ) P = PA where P : / / ' ~ / t ' / S is the canonical projection. a(A) denotes the set of eigenvalues of A counting multiplicities. C denotes the open left-half complex plant. R denotes the real line and R the open lefthalf real line. If n is a positive integer then _n stands for the set {1, 2 ..... n~,.

2. P R E L I M I N A R I E S

In this section the formulation and the solvability criterion for the A D D P by state feedback for the system (1)is reviewed. Definition 1. F o r fixed 1 ~< p ~< 0% the A D D P is said to be solvable if Ve, > 0, there exists F~ such that in the closed loop system with x(0) = 0 there holds [Izllg, ~< clldll~, where II IIz~, denotes the Lpnorm. Let ~;((:= ker D. It has been shown by Willems (1981) that the A D D P has a solution iff Im G c "F*b . , ~

(2)

where ~"*b,.x~ is the supremal Lp-almost invariant subspace "contained" in ~ , 1 ~< p < oc. Suppose that (2) holds• Then a possible construction of a sequence of maps F~, e > 0 which solves the A D D P can be given as follows. First, one defines a sequence of controlled invariant subspaces ",'t' with the property that ~ ~ converges to t ~.x. Next, for every e one defines a m a p F~ such that (A + BF~)'~ c ~ ~. In Trentelman (1983) a construction based on the above was indeed shown to lead to a sequence F, that achieves almost disturbance decoupling. An i m p o r t a n t feature of the above construction is that in general a part of the spectrum, a[(A + BF,)] ~] tends to - ~ along the negative real axis as e tends to zero. Let ~ : = Im B. It has been shown by C o m m a u l t and D i o n (1981) that the subspace ~ * admits the following decomposition b,;f

"~ *

= ~)

® ":t'*

In (5) ;7~ is any subspace such that ;2d = ,~ ca ~' * ® .~

(6)

and ~ , K is the limit of the following sequence ~0 = 0. ,~,,r: =~,"" .~,, o~," = #{ n ( A ~ " - ~ + ~), u ~ n; ,~, (7) It can be easily shown by induction on u that :¢~r n ,~..~ = 0, which implies ,;t< = 0. Let A F A + BF. Since ~ )r n ~,..,r = 0 it now follows that ,@.,,~c n A~ ~ .~ = 0, Let p ' = d i m ~ n ~ ( #

and

VF: .~' ~ o#. •

u

r,, : = d l m ( ~ , / ~ ,

(8) u--I

),

/ A G nI.

Also let rn~:= n u m b e r of integers in the set jr1, r2 . . . . . G} which are ~>i. Then m 1 >/ m 2 >/ "'" ~ m p /> 1, with E m i = d i m ~ j . iEp

It has been shown by A r m e n t a n o (1983a) (see also Armentano, 1983c) that there exists a m a p E / ~ . . ~ r --* ~ , where

BF..~,,,.~c c :Y~

(9)

such that ~O~4'~a,YF:

(~1 ( ~ ~ 2 ( ~ ) ' ' " ( ~ ~ p

(10)

with

S i=t~iGAGdiO'"OA~-ldl,

i~p

and Ar~:= A + BF,. By using (8) and (9) (11) is obtained. Let q" = dim ~ and define n~: = rnz + 1, i~p, and np+l = n p + 2 = ' " = n q = 1. Then from (10) and (I 1)

(3) ;YA'b,,~= -/L1 (~ ,-¢/2 O " " • ,J/q,

(12)

where where (i) ~Q* is the supremal controlled invariant subspace contained in Jr" which is given by '/ '* : =

't "n-

'"=.~nA-~(~"u :t,-o = ,y'.

(ii) ~N

A.~a.,, O 9 .

1 +~),u~n;

(4)

(5)

~f'i = d i ( ~ A Fr~ i ( ~ " " " ( ~ A "dr- 1 d i '

i E q.

The indices ni, i t q , have an interpretation in terms of the transfer matrix G(s)= D(sl-A)-IB. Suppose that B is monic and D is epic. C o m m a u l t and Dion (1982) have shown that G(s) has q infinite zeros of order n,, i e q.

Almost disturbance decoupling 3. A P P R O X I M A T I O N

O F "f'~,x

The following definition simply establishes a terminology to be followed in the text. Definition 2. A P D law, u = F x + R&, is said to be non-singular if (I - BR) is a non-singular map. Note that the system (1) under a n9n-singular PD law can be written as

451

a controlled invariant subspace ~ and a map Fo: "U~. ~ cg such that borg

(14)

¢~,

(A + BFo)~ c ~, a[(A + BF0) I cE] = A~

and (A + BFoYt/'~ c V~c,

5: = (I - B R ) - '(A + BF)x + (I -- B R ) - ~Gd,

a[(A + BFo) I ~U.,~] = A~ w A..

(13)

z = Dx.

In this section a sequence of non-singular P D laws, u = F x + R ~ are shown to exist, such that i for the sequence of subspaces "U~ ~ o ~ , ~ c con-

Note from (5) and (7) that ~ . ~ r n Yg = ~ , . x and let .~..,r be any subspace such that .~b.~¢ = Thus from (14) and (3)

structed by Trentelman (1983), ( I - BR~)(A + B F ) ~ c ~ and R ~ o R , where R is a finite

Y" = V A@ ~ . , x ® ~b,,~,

map. In the following some sets of complex numbers are defined. Let:

where V ~: = U x • (g. It has been shown by Armentano (1983b) that there exists an extension of F0, denoted by F, F: ~ ® Rb,x such that

(a) A~ be a symmetric set of d i m ~ complex numbers, where ~ x is the supremal controllability subspace contained in ~ff. (b) A~: = a[(A + B F , ) [ ~ x ( m o d ~ x ) ] , where F~• F ( U ) ) denotes the set of maps F for which (A + BF)~V"x c ~Fx. It is well known (Wonham, 1979) that A~ is fixed for all F ~ F

(~)). (c) A~ be a symmetric set of n - dim ~/~,x complex numbers. (d) Ai,~:= {2i.i(e), j • n i , i • q } , where 2j,i(e)•R is such that [2j,i(e)l --* o~. The following result can now be stated. Theorem 1. Let the pair (A, B) be controllable and consider the subspaces U~,,x and ~,#-. Let A r, A~, A¢, A~.~ be the sets defined above. Then there exist: (a) a sequence of controlled invariant subspaces I ~/~ such that ~ ~ 0 ~ , z ¢ (which implies U s +

~

o~o ~'"~)'

(b) maps F and R~ ~ R such that: ~ 0

(i)

(I - BR~)

is

non-singular

and

(I -- B R ~ ) - 1 A F Y/~ c ~u, q

(ii) a[(I - BR~)- I AF[~//'~] = U Ai,t and i=1

(iii) (I -- BR)~o, x c Arab. x and (I - BR) is singular. Furthermore, q

(c) a [ ( I - B R ~ ) - I A F ]

= U Ai,~wA~wA~wAc i=1

Proof It involves two steps: (1) Definition of the map F. Since the pair (A, B) is controllable, it follows from a result due to Trentelman (1985) that there exist

(15)

f = ~F"A G Av~tb,x.

The reason for the decomposition (15) will become clear later. Finally, let F l ~ , , x = F,, where Fr is as in (9 10). (2) Construction of ~ and definition of ,~. Let Ae: = A + BF, where F is as in step 1, let 6j,i(e) : = l/,~j,i(g ) and for i • _q,j • {2. . . . . hi} , consider the vectors v~,i(e) defined by vj,i(e): = (I - 6j,i(OAr)- 1Arvj_ 1,i(e),

(16)

v m ( 0 : = (I -- 6Li(e)AF)- lbi. Note that ( 1 - 6~,~(e)Av) is invertible for [6j.~(e)[ 0 and that vj,i(e) ~oA)F-lbi,

ieq,

jen_,.

(17)

Moreover, for e sufficiently small, the vectors vj,i(e), defined in (16), are linearly independent (see Jaffe and Karcanias, 1981 on the convergence of subspaces). Let ~Ui,~:= span{vj,i(~)}, j e n i and ~F~:= V1, ~ @ . . . @ "F~,~. Since 2~,i(e) is real then the subspace ~U~is spanned by real vectors and observe from (12) and (17) that ~ ~o~b.X = Jgl O ' " + J//q" It is also easy to see that ArCJ'~ c ¢'~ + ~ , i.e. ¢/'~ is a controlled invariant subspace. Since f = ~//~A®.~b..g, it follows that for E sufficiently small we have f = "F~a G ~f~. Define R~: f ~ q/ by BR~vl,i(g) = bl,

R~vj,~(e) = 0, R~ [ ~FA = 0.

ieq_;

i e p_,

j e {2..... nl};

(18)

452

V.A. ARMENTANO It remains to show that II - BR~) is non-singular. F r o m (18)

From (16) and (18)

(]

BR~:)t,l,i(l; ) = OI.i(g)AFVl,i(C )

(19)

Im(l - BR~) = ~ ~ + ( ~ ) @ ~,.A),

(21

and where

(I - BR~)vj.i(e) = 6j.i(e ) Avvs,i(e ) + AFVs_ 1.i(~). ~ "1" = span{vl.i(c) Assume for the m o m e n t that (I - BR~) is nonsingular. Then from (19) it follows that Mat[(/- BRy

~f2 "= span{vj./(e)},

itp,

i t [ 2 . . . . . nil.

F r o m (16) it follows that

~AF I3Q,] = M,,

UI.i(C, ) -- b i =/51,i(~:)AvvLi(e),

where

mi: =

itq:

-- bi},

Ili,,u2,1,, uni1, 0

0

and

vs,~(~:) = Avvi ~,i(e) + 5~.~(~)Arvs ~(e), itp, Since

with

Us,k(e) = (--1)J-k2.i,i(e)... £k,i(e), for k < j: (20b) //j,j(~,) = 2j i(~])j t ~i" Since the subspaces ~i,~, i t q, are independent it follows that

(I - BR~)-1Av3r'e, ~ "~, with q

= U Ai.~. i=1

F r o m the definition of F and R~ (I -

BR~)

1Av~..A

~ ~:-a,

with

a[(I - BR~,) ' A v I ~t:~] = A¢ w A, w A= is also obtained which establishes the claim on the configuration of eigenvalues in b (ii). Since vl.i(~:)~ob i, it follows that R ~ o R such that (I - -

BR)b i = O,

itq

and

RAJF-lbi=O, so that ( I -

itp,

j t 12 ..... n~).

(20a)

u.,,.,(OJ

a[(l - B R ~ ) - I A v I ~ ]

itq

jtni,

BR) is singular and ( I - - B R ) ~ b j c

span

{AFVs,i(e ), ieq,

.jtni} ~ ~ span

{A~b~} = A r a b , x, it follows from (15) that for e sufficiently small the vectors {Avvs,i(e)} are linearly independent and also independent of t-a. This implies that the sum in (21) is a direct one, i.e. I m ( I - - B R ~ ) = f ~ @ ~ 2 ® ' ~ ' a = ' ~ " and thus (I - BR~) is non-singular. [] Comments (1) T h e o r e m 1 also holds if the sets Ai. ~, iEp, are taken as sets of n i symmetric complex numbers. In this case the vectors vs,/(0 in (16) are in general complex. In order to avoid the definition of complex feedback maps T r e n t e l m a n (1983) has suggested a nice procedure to c o m p u t e real vectors bs,i(e) from the vectors vsj( O. The only modification in the p r o o f of theorem is that the m a p R~ is then defined on the real vectors ~s.M'). (2) An intuitive interpretation for the result of T h e o r e m 1 is as follows: the velocities of the trajectories on ;~ are extremely high as ~ ' ~ - - * ' ~ b : - Thus the relevant information to be used by the control u is the derivative 2 and not the state on ~i,. 4. A L M O S T

DISTURBANCE

DECOUPLING

The main objective of this section is to show that if lm G m ¢J'~,,. then almost disturbance decoupling can also be achieved by means of a non-singular P D law involving finite maps. This version is denoted by ( A D D P ) ° and its formal formulation is as follows. Definition 3: F o r fixed 1 ~< p ~< oo, the almost disturbance decoupling problem, ( A D D P ) °, is said to be solvable if Y'~:> 0, there exists a non-singular P D law, represented by the pair of m a p s (F,R~), such that in the closed loop system (13) with x(0) = 0 there holds [IZ[IL, ~< g][dl[L v.

Almost disturbance decoupling Let A, : = (I -- BR~)-~ (A + BF). A preliminary result is required, which has been pointed out by Willems (1981). L e m m a 1. Fix 1 ~< p ~< Go. Suppose there exists a sequence of non-singular P D laws (F, R~) such that IlDe A"' ( I - B R ~ ) -1 G I I L ~ 0 . Then ( A D D P ) ° is solvable. Proof. Since the L F i n d u c e d n o r m of a convolution operator is b o u n d e d by the L~-norm of its kernel there holds IIZlIL~ ~ [IDe&'(I -- BR~) -1 GILL, IldllL,.

[]

The following definition is also needed. Definition 4. Let e > 0 and fix i~q. Let A." : = {2x,i(e) . . . . . 2., ,(e)} be a set of n i real numbers such that 2,.i(e) = 2~.i(e):= 2/(e) for r ~ s, r~_n i and s~nl. Let A.~ be a set of infinite root loci with c o m m o n growth cq and asymptotic direction 21, ;(i~R-, if there exists a real a i > 0 such that ~'2,~(e) = ~ , e --, 0. Fix i~q_ and consider the vectors vj.i(e) in (16) with 6j,i(e):= 1/21(e), j¢n_ i, and 21(e)~A~',~. Let ~/,~ : = span{v~,i(e),

jeni}.

Hence from the proof of T h e o r e m 1 it follows that ~Vi,, ~o J / l , and there exist maps F and R, such that ( I - BR~.) -~ Av~/~i., a ~Us., and Mat [ ( I - B R ~ ) - I A v [ ~Yi,,] = T~, where Ti is a matrix identical to M~ in (20a) with elements uj,~(~) given by u~,k(e) = ( - 1)i-k2j-g+~(e), u~,j(e) = 21(e).

k
(22a)

Proof. Let li and Ni be the matrices defined above. Then liNi = Nili,

Hence e a~' vj,i(~) = eritvj,i(e) tnl- 1

= et~'(I + tNi + "'" + -(n i--- 1)! ST'- X)vi'i(e)" The result follows from L e m m a 2.

(23)

which implies (Chen, 1970) that e u' + u,), = e,,,eU,,.

[]

Comment N o t e that the commutativity in (23) plays an important role. It is for this reason that the set A~" t,£ has been chosen. In the paper by Trentelman (1983) the set Ai~, has been replaced by the more general set ~i,~:= {21,i(g).... ,2,,,i(e)}, where ~i,~ is a symmetric set of n~ complex numbers and the elements of the matrix T~ are given by (20b). It is easy to see that if l i : = diag (2LJe) .... ,2~,,i(e)) with 2r,i(l?, ) # 2s,i(~ ) and N i : = T i - I i then N i l i and l i N i do not commute. This discussion shows that there is not so much freedom in the choice of asymptotes to obtain the result of L e m m a 3. The next theorem states a combination of the results obtained in Theorem 1, L e m m a 3 and is a key one for the solution of (ADDP) °. Theorem 2. Let the pair (A, B) be controllable and consider the subspaces ~//~.. and ~b.X. Let A~, A~ and A.. be sets as in Theorem 1 with A~ c C - . For i~q, let Ai~~ be a set of n~ infinite root loci with c o m m o n growth a~ and asymptotic direction

2iER-. Then the statements a, b and c of Theorem 1 are true (just replace Ai,~ by A~.). Moreover IIDeA~'~'*h,~IIL, ~ 0 0,

(22b)

Let Ii: = diag(uLl(e ). . . . . u.,.~(e)) and let N i denote the nilpotent matrix defined by N~ = T,. - I~. N o t e that NT' = 0. Then the following results. L e m m a 2. ][OelitttNlvLi(g)llLpe~o 0 for j E n i and I~{0,1 . . . . . n i - 1}. Proof. See Trentelman (1983). The next lemma establishes that the L F n o r m of the closed loop impulse response in the direction of the vectors via(e), j ~ n_i, can be made arbitrarily small. L e m m a 3. Let A~:= ( I - B R O- ~A v, where F and R~ are the maps defined in T h e o r e m 1. Then, for jE ni and 1 ~< p < ~ there holds liD e a'' V~,i(e)llL~~ o 0.

453

where Proof. A~Vx that it

1 ~< p < ~ ,

(24)

A~ is as in T h e o r e m 1. It is clear from the proof of T h e o r e m 1 that c y .,. Then from (3) and (24) it follows must be shown that IIDeA~',~b,,,IIL,~o0 ,

1 ~< p < m.

(25)

As in Theorem 1 let Y/~'= Y/~I., 0 ' " • ~sq.~, where "t'~i,~:= span{vj, i(e), i e q , jen_i} with vi,i(e) Aje- 1 bi"

~ "-40

Let Ai.~ = A~[~.~, i~q. Hence, from L e m m a 3 HD e a~'`t Uj,i(e) llL p ~ O, i ~ q, j e n i is obtained. F r o m the proof of Theorem 1 A~lCg = AvlCg, with a[AFICK] = A,, c C - . T o prove (24) it follows from (25) that it suffices to show that IIDeA~' A~-1 b s l l L ~ 0 0, For

e. sufficiently small

s~q,

l~n~.

f = ~-

• cg • ~U~,

V. A. ARMENTANO

454

which implies that the vector A~z lb~ can be written u = Fx + ~ Ffl "),

as

i

A~-~/~ = ~ ~ ~j ,(~)~,/,,(~)+ w(~) + c(~:), i~q j~n i

where ei.i(e)~ R, w(e)e ~ i ) and c(e)e ~. Since v~~(0 --* A/- ibm, it must be that ~j.~(~) ~ 0 for (j, i) # (1, s), ~t.~(~,)~o 1, w(e) --* 0 and e(e) ~ O. Also note that since c(e) ~ 0 and a[AF I ~"] ~ ~ - , then liD eA~"C(e)IIL.~FO O.

0

which also achieves exact disturbance decoupling. The term d ") represents the ith order derivative of d and N + 1 is the largest order of an infinite-zero of D(sl - A) IB when D is epic and B is monic. It is clear that the above law is applicable to the sohrtion of the disturbance decoupling problem when d ~ C "~, the class of functions with derivatives up to order N.

Hence lID e A~'a~v ~ b~llr, ~< ~ IO~j,i(~)[ IIDeAi'~tvj,i(e)[ILp + IID e Art c(e)I[t,p ~,~o 0, i~q jEn_i

which proves (24). [] Finally, the following is obtained. Corollary 1. Suppose that l m G c ~ • x . Then (ADDP) ° is solvable. Proof. Note that Im G c ~"*~,.~rimplies Im G + ~ c I m ( l -- BR~,)- a G ~ l m G + ~ ~ ~ b.X.

The result then follows from Theorem 2 and L e m m a 1. [] Comments (1) It might be thought that a non-singular P D law would introduce a new angle to the disturbance decoupling problem (DDP). In order to see that this is not true let 4 : = ( I - - B R ) -~ (A + B F ) , :=(I-BR)-IG and ~ / / ' a : = I m ( ~ + A I m ( ~ + ... + ~ , - 1 Im (~. It is easy to see that if the system (13) is disturbance decoupled, namely the transfer matrix from d to z is zero, then ~//~ is a controlled invariant subspace contained in X and therefore

5. PRESERVATION OF STRUCTURAL INVARIANTS In this section a multivariable linear system Z:2 = A x + Bu; y = Cx, where A : f ~ :T, B:~// :~', C: °2" ~ 7~J, d i m , T = n, d i m ~ = m, d i m l y = r i s considered. W o n h a m (1979) has shown that the controllability indices of a controllable pair (A,B) are invariant under the group transformation ~p induced by a state feedback control law. The action of ~p on the pair (A, B) is described by f~p: (A, B) --, ((A + BF), BG), where F : f ~ 0/l is arbitrary and G is an automorphism of ~ . In the sequel the group transformation f#pa induced on the pair (A, B) by a non-singular PD law is considered• The action of ffpa on the pair (A, B) is given by ~pd: (A, B) --* ((I - B R ) - I(A 3- BF), (I - B R ) - ' B G ) ,

(2) It is also interesting to note that the condition (2) corresponds to the existence of other types of control laws which achieve exact disturbance decoupling. Armentano (1983b) has shown that if (2) holds then there exists a PD law

where F is arbitrary, R is any map such that (I - BR) is non-singular and G is an automorphism

of '~#. Proposition 1. Let q5u = ~ + A B + . - - + A " - I ~ , u ~ n. Then 4)" is invariant under ffp~. Proof. First note that Im BG = ~ and

u = F x + R2 + Td, (I - B R ) - ~ ( ~

such that the transfer matrix from d to z of the closed loop system (I - BR)2 = (A + BF)x + Td, z =Dx

is zero, where R and F are the maps defined in Theorem 1 and the pencil s(l - BR) - (A + BF) is regular. Willems (1982b) has shown that condition (2) is equivalent to the existence of a control law given by

+ 3--) = ~ + ,Y--

for any subspace ,Y- = .~'. L e t / ] : = (I - B R ) - t (A + BF). Hence (I -- B R ) - ' ~

+ 4 ( 1 -- B R ) - I ~

+ A"-1(1 -- BR) = (I

+ ...

1~

BR)-I [~ + 4(...(M + A~))...]

= ~ + (A + BF)M + ... + (A + BF) u - l ~

=~M+A~+...+A"

l~=gb".

[]

(26)

Almost disturbance decoupling ~'u

Corollary 2.

(a) The controllable subspaces ~ + A ~ + • .. + A " - 1 ~ is invariant under ~ n . (b) Let the pair (A, B) be controllable and B be a monic map. Then the controllability indices are invariant under %a.

~.~ is the supremal controllability subspace contained in 3((" and is given by the sequence : ~ , : = :~0, 72" = 3V', c~ ( A ~ " - ~ + ~), u e n ; ~ ° = 0; - - ~ , ~ is the supremal almost controllability subspace contained in ~{ and is given by the sequence

u e n;,¢~° = 0; - - ~O,, x*

is the supremal L~, 1 ~< p < oc, almost controllability subspace "contained" in oU and is computed by the sequence

~h,X: =

,~t~, ~ ,

A ( ~ , - 1 (-~ ~l/') "Jr ~ ,

u e n ; , ~ ° = 0. See Wonham (1979) and Willems (1981) for a detailed exposition on the significance of the above subspaces. In the following the action of ~pa on the triple (C, A, B) is considered, which is defined by ~,a: (C, A, B) ~ (HC, (I - B R ) - ~(A + BF), (I -- B R ) - IBG),

where H is an automorphism of ~ and G is an automorphism of ~. The next proposition extends the property of invariance under aJ~d for the sequences above described and the sequence (4). Proposition 2. The subspaces ~ " , N'~, ~," and N*~are invariant under ~ a Proof. Invariance of 3g'". Note that ker H C = 3ft. Recall the notation ,4=(1--BR)-~ ( A + B ~ F ) and consider the sequence

+ (I - - B R ) - - 1 ~ ) ,

= o,~ ~ .,4- l(~'u-1

uen;

~,-o=f.

(27)

From (26) it follows that

Proof. (a) It is obvious from Proposition 1. (b) The statement follows on noting that the controllability indices are determined from dim &, u e n (see (Wonham, 1979)). [] In the sequel some subspaces of the state space, which play a key role in a linear system theory, are considered. Let ~ : = ker C. Then

455

;~" = X c~/]- 1(, .,-~ + ~).

(28)

From (4) and (28) ~ 1 = ~.a = 9ff is obtained. Suppose that ~U"-1 = ,/~-,-1. Hence

~ " = { x e o f [(l - B R ) - I ( A + B F ) x = v + b,

ve~"-l,be

o2},

which implies A x e ~ " - 1 + ~ and thus ~-u c ~'u. Now let x e~V". Then A x = v + b , for some ve~V u-1 and some beM. Hence, (A + B F ) x = v + b', where b' = B F x + b, VF: .T ---, ql. Moreover, (A + B F ) x = (1 -- BR) (v + ~), for any R such that (I - BR) is non-singular and b = (I - B R ) - 1 (b' + BRv). Thus ( I - B R ) - I

(A + B F ) x e $ ' u

1 +~,

i.e.

$-u c ~ u and the result follows. The invariance of ~u, ~ and ~ is proved in a similar way. [] Corollary 3. Let B be monic and C be epic. Then the infinite-zero structure of G(s) = D(sI - A ) - 1 B is invariant under ~pd' Proof. Commault and Dion (1982) have shown that the infinite-zero structure of G(s) is determined from dim[(Rg + f .~)/~ x]. The claim then follows from Proposition 2. [] Now consider the map A,:= [(A + B F ) ] ~ I ~ (mod ~.x~)] for F ~ F('t/~-). Theorem 5.7 by Wonham (1979) has shown that this map is fixed for all F ~ F ( ~ , ) . It is also known that A~ is invariant under ffp (see Morse, 1973). We then have the following result. Proposition 3. At is invariant under ~pa. From (27) we obtain (I - B R ) - 1 (A + B F ) ~ ' ~ c ~ ~ + (I - BR)-~:~. Thus there exists F,: ~f~ ~ q/ such that (I - BR) (A + B F + B F , ) U x c "t ~ and write AFo := (I - B R ) - I (A + BF + BF.). Since A f ' ~ ~ f'~- + ~, then there exists Fb: U.x¢ --* ~' such that (A + BFb)~t ,, ~ U , and let Avh:= A + BFh.

Consider the maps "4vo and ,'IF~ (induced in * /-, y.* * x / ~ and let P: ~ . , ~ ~ ,r/~.,r be the canonical projection. Then /,*

Arox - Arbx = PAr~x -- PAF~x = P(AF~ -- Avb)x,

456

V.A. ARMENTANO

Let ( I - B R ) ~ A x ' = Ax = BRw. Hence

w, which implies

P(Av, - Av~)x = P [ B ( R w -

~....

FhW)

+ (I -- B R ) - I(BFx + BF,x)] • P ( ~ c~ ~ "*v),

by using (26) and the fact that (A~.,.- Av,)xe ~ ~. But P(~ c~ ~ ~) c P:J~ = 0 and the result follows. [] Remark. Corfmat and Morse (1976) have shown that the eigenvalues of A, (also called the transmission zeros) coincide with the finite zeros of the pencil P(s) =

C

. From the above prop-

osition it then follows that the finite-zeros of P(s) are invariant under ~pa. CONCLUSIONS

In this paper the role of a non-singular PD law has been examined in the context of the almost disturbance decoupling problem. The ideas presented should be useful in the study of synthesis of PID compensators for the almost disturbance decoupling problem by measurement feedback (see Willems, 1982a). REFERENCES Armentano, V. A. (1983a). Almost invariant subspaces and generalized linear systems. Ph.D. Thesis, Department of Electrical Engineering, Imperial College, London.

Armentano, V. A. (1983bl. Exact disturbance dccouplmg by a proportional-deriwltivc state feedback lay,. Control Report No. EE/CON'83.19. Department of Electrical Engineering. lmperial College, London. See also 24th IEEE ('1)(, f'orl Lauderdale ( 1985/. Armentano. V. A. i 1983cl. Sliding subspaces and the assignmcnl of asymptotes [\~r invcrtible linear s,,stems. Mat. tpli~. O,nlput., 3. 281. Chen, C. T. (1970). Introduction Io Linear System Theory. Holt, Rinehart and Winston, New York. Commault, C. and J. M. Dion (1981). Structure at infinity of linear mulfivariable systems: a geometric approach. 20th IEEE CDC, San Diego, California. Commaull, C. and J. M. Dion 11982). Structure at inlinity of linear multivariable systems: a geometric approach. IEEE 7?ans. Aut. Control, AC-27, 693. Corfmat, J. P. and A. S. Morse (1976i. Control of linear systems through specified input channels. S I A M .I. Control Opt. 14, 163. Jaffe, S. and N. Karcanias ( 1981 ). Matrix pencil characterization of almost (A, B)-invariant subspaces: a classification of geometric concepts. Int..1. Control, 33, 51. Morse, A. S. (1973). Structural invariants of linear muhivariable systems. S I A M J. Control Opt., I, 446. Trentelman, H. (1983). On the assignability of infinite root loci in almost disturbance decoupling, Int. J. Control, 38, 147. Trentelman, H. (1985). Almost invariant subspaces and high gain feedback. Doctoral dissertation, Mathematics Department. University of Groningen, The Netherlands. Willems, J. C. (1981). Almost invariant subspaces: an approach to high gain feedback design - Part I: almost controlled invariant subspaces. IEEE Trans. Aut. Control, AC-26, 235. Willems, J. C. {1982a). Almost invariant subspaces: an approach to high gain feedback design Part II: almost conditionally invariant subspaces. IEEE 7?arts. Aut. Control, AC-27, 1071. Willems, J. C. (1982b). Feedforward control, PID control laws, and almost inwmant subspaces. Syst. Control Lett., I. 277. Wonham, W. M. [1979). Linear Multivariable Control: a Geometric Approach. Springer. New York.