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Solutions to the Output Regulation Problem of Linear Singular Systems
2a-17 I
Copyright © 19961FAC 13th Triennial World Congress. San Francisco, USA
SOLUTIONS TO THE OUTPUT REGULATION PROBLEM OF LINEAR SINGULAR SYSTEMS Wei Lin *,1 and Liyi Dai *'"
'" Dept. Systems, Control
Industrial Engi., Case Western Reserve University, Cleveland, OH 44106 •• Dept. Systems Scimce & Math., Washington University, St. Louis, MO 63130 fj
Abstract. Output regulation is an important problem in control theory. In this paper, we revisit this problem for linear singular systems and identify an important case of the general regulation problem where the measurement output is identical to the vector to be regulated. We derive a necessary and sufficient condition for the regulation problem to be solvable via either full information feedback or error feedback. Then we show how full informat ion feedback and error feedback fontrollers can be constructed explicitly. Keywords. Singular systems, regulation, state feedback, measured feedbac·k.
1. INTRODUCTION Singular systems exist, for instance, in large-scale systems, networks (Lewis 1986, Lucnbcrg:cr 1977), circuits (Newcomb and Dziurla 1989), modeling of robot systems (McClamroch 1986), boundary control systems (Pandolfi 1990), modeling of general systems (Willems 1991), and power systems. Singular systems have attracted attention of many researchers since late 19708. Several books and survey paper:s have appeared specially dealing with these systems (they are, to the best of Ollr knowledge, Aplevich 1991, Brcnan et. al 198£>, Campbell1981, 1982, Dai 1US9, Lewis 1986. 1992a,hJ. A large number of important problems ha.ve been treated and solved in a long list of individual research papers, many of which can be found in these L100ks and review articles. The regulation problem can be stated as the following: Consider a system with the presen"e of input (and/or measurement) disturbances or external signals. It is desirable to find a controller such that the closed-loop 1
Research supported
ill
part by A FOSR and NSF
system is internally stable and has desired properties such as (asymptotic) disturbance attenuation and signal tracking. The regulation problem for the conventional state-space systems attracted much attention in the 1970s. A rat.her complcte regulation theory of linear state-space system was established during that period (see, for example, Frauds 1977, Knobloch et al. 1993). The regulation problem for linear singular systems has been investigated, for example, in Dai 1987, 1989, in a quite general setting where the measurement output and the vector to he regulated are not necessarily same. Although sllch general formulation provides beneficial insights for understanding the problem of output regulation, it also has some limitations: The solution to the general problem was not satis:"actory, Existence conditions were expressed in terms of solutions to a set of nonlinear matrix equations that depend on not only system parameters hut also somf' other parameters. Such conditions are difficult to verify. This naturally raises a fundamental question of when such conditions can be further refined. Partial answers to this question have been obtained for several spec'ial cases (see Dai 1989,
1542
Chapter 5 and references there for additional details). In this paper, we revisit the regulation problem for linear singular systems. \Ve examine a special case of the general problem considenxi in Dai 1987 where the output measurement is the same a..~ the vector to be regulated. This has been known to be a. very important case in the literature of th e C'1)nventional st.a.te-space systems (Francis 1977, Kn obloch ct aJ. 1993). Actually, this is the case that has received the most attention and whose solution is the most elegant and complet.e. I~vlaIlY important problems such as signal tracking can be recasted as special cases of the regulation problem considered in this paper. For this case, we are able to provide a satisfactory solution to t.he regulation problem, thus filling a gap in the ~tuciy of the regulation problem for linear singular systems. Under severa l very mild assumptions (which are standard in the regulation theory of state-space systems), we derive a necessary and sufficient condition for the output regulation problem to be solvable.
2. PROBLEM STATEMENT Con~ider a singular linear system described by equations of the form Ex == Ax + B11, + Pw tb = Sw (1) e = Cx+Qw,
where x e nn 1 U E IR:n, and e E IRr are the state, the control input, and the measurable error output, respectively, E,A E 1Rn x fI , BE R llxm , P ERn x s, C E R f x n , S E R Sx ., and Q E IR r x s arp. constant matrices. The signal w E R S is the exogenous input which includes refer..nce signals to be t1UckeJ and/or ,lis/nTbanees tQ be rejected. Throughout this paper, we aSSume that w(t) is generated by a linear autonomous system tU Sw, known as the exosystem. \Vithout loss of generality, we assume t hat E is singular and syst.P.n1 (1) is regular, i.c. , det(AE - A) ;t O. Regularit.y guarantf'es t.hat (1) has a unique solution for any 11 and w.
=
The control action to the system (1) can be provided by either full information feedback or e''1'Or feedback. By full informal.ion feedba.ck we m<>an the feedback control law of the form
,,=[(x+Lw mx n
mxs
(2)
where K E R and L E R are gain matrices to be deSigned. Note that measurement of w makes perfect senSf! in the case where tu is a reference signal. Putting (1) and (2) together, I.he closerl-Ioop system is E x = (A - B1<)x + ( P + BL)w to Sw (3) e = Cx+ Qw .
=
If either the state x or the exogenous signal w is not available for the control action, (3) cannot be implemented directly. In tltis case, a mOre realistic and rather common approach is to design a dynamic compensator driven by the measurable error output e. We consider a eontrol1er of the following forOl
{=FH-Ge u
(4)
= H~+Je ,
where EE IRn~ is the state of the dynami c compensator, F E Rncxn~, G E Rnc xf , H E m,rn x nc, and J E IRrnxr are constant matrices to be df'termined. Usually, (4) is called the error feedback controller. Under the error feedback controller (4), the closed-loop system for (1) is Ex = (A + BJC)x + BH~ + (P + BJQ)w
t; = GC", + F{ + GQw tiJ
(5)
= Sw
e = Cx+Qw. The design goal is to achieve internal (asymptotic) stability and out.put regulation. Internal stability means that the closed-loop system (3) Or (5) is asymptotically stable when the exosystem is disconn ected (Le., set w = U) . Output. regulation requires that the error signal e(t) in the closed-loop system (3) or (5) asymptotically decay to zero as time tends to infinity for any initial state x(O) E JRn, w(a) E JR', ~(O) E lR.n, . In summary, wc arc going t.o solve the following two problems. Problem 1 (Output regulation via full information feedback): Find, if possible, two constant matrices 1< and L such that (la) a(E , A+B1<) c(J;- , where, for any square matrices a and 11 of same dimension, q( a , ,9) d~f {>. I det(Ao:11) = O} is the set of gencraH7.ed eigenvalues of the pencil (a , 11) . (lb) for any (x(O), w(O)) E Rn < R ' , the error signal e(/) of the closed-loop system (3) satisfies Hrn e(t) = Hm (Cx(t ) + Qw(t)) = O.
t ...... iXJ
t -+oo
Problem 2 (Output regulation via error feedback): Find, if possible, four matrices F, G,H, J such t.hat (2a)
([Ea 1 [A + BJC BH l)
a:-
a a I n, ' GC F c , (2b) for every (x(O),~(O),w(O)) E Rn X Rn, X R', the error signal e(t) of the closed-loop (5 ) satisfies
Hm e(t) = lim (Cx (t ) + Qw(t)) = O.
l-4 oo
i _oo
The regulation Problem 1 (or Problem 2) is said to be solvable if there exist such matrices K, L (or F ,G , H , J, respectively) . To solve these t\\O problems, we make the following three assumptions:
1543
(HI) a(S) ca;+ ~r p E(£IRe(A) 2 0); (H2) the triple (E, "I, B) is stabilizable in the sense of Cobb 1984, which is equivalent to rank[AE A BJ;n, VAEa; +;
regulation via full information feedback is solvable if and only if there exist two matrica, II and r such that EllS; All + Br + l' (8)
(H3) the triple ( 0 I, ' 0 S' ,[ e Q j) is detectable
o
0; 011 t Q.
[EU] [AP]
in the sense of Dai 1989, which is equivalent to
AE.-A
rank
[
(I
C
-1' ] :\1, - S' ; n + 8, VA ea;+.
Q
The first assumption (HI) causes no loss of generality since asymptotically stable modes in the exosystem decay to zero eventually and, therefore, do not affect the output regulation. The second onc (H2) is rather standard and indeed necessary for the existence of a state feedback control law that achieves asy mptotic internal stability of the closed-loop system. The last assumption (H3) is slightly strong,,, than necessary for asymptotic stabilization of the closed-loop ,ystem by error feedback control, but we prefer this one for technical simplicity.
(9)
Proof. (Sufficiency): By assumption (H2), the triple (E. A, B) is stabilizable. Therefore, there exists a matrix K such that atE, A + BK ) C(£ - , Using this particular K, together with the mat,rices 11 and r satisfying (8)-(9), one is able to construct the following full information feedback control law
u;K(x-IIw)+rw. Algebraic manipulation shows that it solves Problem 1. (Necessity): Suppose that Problem 1 can be solved by the full information feedback control la", (2). Then there exists a matrix 11 satisfying (6)-(7). Equations (8)-(9) follows from setting r ; Kll -t L. 0
4. ERROR FEEDBACK 3. FULL INFORMATION FEEDBACK This section deals with t.he problem of output regulation via f1111 information feedback. \Ve first recall a simple but extremely useful result of Dai 1989.
Lemma 3.1 Consider a linear singular system of the form Ex = ..1.3' + l'w
,v=Sw
,' ; Ox + Qw . Suppose that assumption (HI) holds and <7(E, A) ca;-. Then lim ..... oo e(t) ; 0, V (x(O). w(O» E Rn X R' if and only if there exists a matrix V such that ,11' - E\' S = P CF=Q. o Suppose assumption (HI). Assume that there exists a full information feedback control law (2) such that condition (la) of Problem 1 is satisfied. Then Lemma 3.1 immediately implies that the condition (lb) is also satisfied if and only if there exists a matrix II such that
+ BK)ll + P + BL 0= Cll + Q.
EllS; (A
In this section, a necessary and sufficient condition for the solvability of Problem 2 is first derived under the assumption of normalizability of system (1). Then this assumption is removed via several system decompositions. Normalizability is only a step :1tone in solving Problem 2. The following Lemma 4.1 is a consequence of Lemma 3.1 applied t.o the closed-loop "ystem (5). Lemma 4,1 Assume that (HI) holds. Suppose there exists a dynamic compensator of the form (4) for which condition (2a) of Problem 2 is satisfied. Then condition (2b) of Problem 2 holds if awl only if there exist two matrices II and E such that
EllS' ; All + BHE
ES; FE
(11)
0; ell,. Q.
(12)
4.1 The case of normalizable singular systems A linear singular system
Ex; A" + B" y ;Cx
(7)
Theorem 3.2 Suppose that a linear singular system (1) satisfies (HI) and (H2). Then the problem of output
(10)
o
(6)
Based all the previous lemma, we are now able to derive a necessary and sufficient condition for the solvability oC the regulation Problem 1 with f,,1I information feedback,
+P
is called normali:t.able if ranklE BJ ; n. In this case, we say (E, A , B) is normalizable. If (E, A, B) is nonnalizable, there exists a matrix L such that det(E + BL) oF 0 (Dai 1989, Mukundao and Ayawansa 1983). Choosing the derivative state feedback control 'U -Lx + VI we have (since (E + BL) is invertible)
=
1544
= (E + 8L)-' Ax + (E + BL)- ' Bv y= Gx ,
;i;
that iS 7 the closed-loop system hec:omf~ a normal statespace system! Therefore, the concept of nonnalizability helps us convert a control problem of singular systems into that of normal systems for which many standard results are available. Of course, the main trouble is the realization of the unplea.-'iant and often undesirable derivative state feedback. The follm\'ing theorem presents a necessary and sufficient condition for the solvability of Prohlem 2 under the assnmption of normalizability.
Theorem 4.2 Suppose (HI), (H2), (1I3), and, in addition , that (E, A , E) is IlorIllalizable. Then the problem of output regulation via error feedback is solvable if and only if there exist two matrices IT ann r that ~olve the linear matrix equations (8)·(9). 0 Proof. (Necessity): A3 in the case of full information feedback , the necessity is an immediate consequence of Lemma 4.1. To see this point, note that if t here exists an error feedback control Law of t.he form (4) t hat solves Problem 2, t hen the c1osp.d-loop system (5) satisfies conditions (2a) and (2 b ). By Lemma 4.1, there exist two matrices IT and l: such that 110)-112) hold. Choose r = HE . Clearly, equatioIls (8) .. (9) follows immediately from (la) ami (12). (Sufficiency ): The proof of sufficiency is constructi ve. Assume that there exist two matrices IT and r satisfying (8)-(9). Since (E ,A, B) is normalizahle, there exists a matrix L such that E + EL is invertible. According to the assumption (H2) , (E, A, B ) is st.ahilizabLe. Then, for any matrix L chosen in such a way, (E + BL, A, B) is also stabilizable (Dai 1989). Therefore, a matrix K can he chosen such that. u(E +EL, A + EK) e (L (13) The assumption (H3) further guarantees the existence of matrices GOI Cl such that
u([~ f] , [~
i]- [~~] [GQ])ect ·
(14)
For the matrices L,K·, GO,G 1 thus obtained, we consider the dynamic compensator (4) with the matrices F,G,H, J chosen in the following manner.
next show that the dynamic compensator (4) with coefficient matrices chosen in (15) indeed solves the problem of output regulation. First of all, by r.onstructi~n (15) , u=H~+Je=(K r-J(IT]~-[L -LIT]~. (17) The substitution of the previous expression into the system (1) results in E:i:+[BL -BLIl]~=Ax+[BJ( B(r-KIT)]E+Pw . (18) On the other hand, for the matrices chosen in (15)-(16), (4) can be written as
[ E+BL-BLIT]~=([A o I, 0
+ [ BoK B(r~KIT)])~ + [g~] (GH Qw).
[g~ 1
(15)
H:= [K r - KIT] - [L -LII]F J:=-(L -LITIG where
t>
= [~ ~.l
+
[~] [ L - LIT]
E[:j:A[;] + Bw
Since E + BL is invertihle, so is the mat.rix tl.. Therefore. the four matrices F , G, H , .J iu (15) are well-defined. We
(20)
w=Sw e=Gx+Qw. wh ere
E=
[
[P] E BL -BLn] 0 E+BL -BLn ,B= GoQ , o 0 I. G,Q
BK B(r - KIT) ] A GoG A-GoG+BK P - GoQ+B(r-KIT) . [ G,G -GIG S-G,Q The set of eigenvalues of the closed-loop system (20) is u(E , A). Direct verification yields
,l=
q{E,A) = u(E+BL,A +BK) P-GoQ]) C
(21)
We thus conclude from (21) th,.t condition (2a) of Problem 2 holds. Next , we prove by using Lemma 4.1 that condition (2b) of Problem 2 is also satisfied. To this end, it suffices to check that the matrices Il and l:
= [Z]
are solutions
to (10)-(12). As a matter of fac t , (12) is identical to (9). As for (11), a direct computation proves that
t>l:S
= [E +0 EL -~~IT 1 [Z] S= [E~S] (22)
t>FE - [AIT+Br+p-Go(GIT+Q)] S-G,(GIT+Q) . By using equation (9), we haY,! t>FE =
(16)
(19)
Putting together (18)-(19) and t.he system (1), we see that the closed-loop system (5) can be expressed equivaLentlyas
.- A_I[A -GoC -rBK P-GoQ+B(r- KIT)] F .-'" -G,G S-{;,Q G := t>-'
P]_[GoCGoQ] S G,G G,Q
[AII + ~r + P] .
(23)
It. is immediate to see from (8) that equation (22) coin-
cides with (23). Note that t> is invertible by construction. Hence FE = ES, which proves that equation (11) holds. Finally, it follows from (15) that
1545
AIl+BH~+P
= All+P+Bf- [BL
-BLll]F~.
Remark 4.3 It is dear that the clooed-Ioop system (5) is singular for any controller (4), Since we are only COIlcerned with asymptotic behavior of the error, the undesirable impulsive part of the behavior associated with a
singular system has b,,,,n ignored (see. e.g., Cabb 1984 on impulsive modes). In order to eliminate the possiimplll~ive
modes, we only Heed t.o require that, the
closed-loop system (5) have no infinite poles. According to (21), it suffices to require that (E + BL, A + BK) CoG GoQ] and ( 0 I,' -GIG S _ G1Q : have no infinite
[EO] [A -
+ B1u + Ptw
+ QLW,
where
The last equation comes from (8). Therefore, equation (10) is satisfied aB welL 0
hIe
EllXl = Anx} el = CII I
Using the relation (In, we get All + BH~ + P = AII + Br + P = EIIS.
P-
and M solution to
PI = PI + E I2 MS, Q, = Q - C 2 M L:~o Ni P2 S'. Note that M is the unique (26)
o Lemma 4.5 The linear matrix equations (8)-(9) are solvable with respect to II and r if and only if EllIIIS = AllIII + Blr + PI 0= CIII I +QI is solvable with respect to ill and r.
o
priate feedback gains L,K,Go.G I . Since the choice of
Now we are ready to remove tILe assumption of normalizability of Theorem 4.3, as illustrated in the following theorem.
these gains ean be obtained using a rather standard approach, wc do not pursue thiR direction in this paper. Interested readers are referred to Dai 1989 (Chapter 3) for additional dp.tails.
Theorem 4.6, Suppose (HI), (H2), and (H3). Then the problem of output regulation via error feedback. is solvable if and only if there eXIst two matrices n and r that solve the linear matrix equations (8)-(9). 0
4.2 General case
Proof. According to Lemmas ,1.5-4.6, the result of Theorem 4.3, and that normalizability decomposition does not affect the stabilizability, w" need only to verify that the assumption (H3) is equivalent to
eigenvalues, which can be
achi~~ved
by choosing appro-
Normalizability may not be satisfied in practice. In this subsection, the assumption ofnormalizability is removed with the help of appropriate system decompositions. For any singular system (1), there always exist non-singular matrices TI and T2 sHch that. system (1) is re:~tricted system equivalent to (Dai 1989, p.139) EUXl + E 12 i 2 = A l1 X I +B}u+P1w NIX2 = X1 + P'jUJ (24) e = CIXI + C2X2 +Qw where NI is a nilpotent. mat.rix with a nil potent index denoted as I, that is. Ni '10. Nf+l = 0,
x=T1
[XI]'Xl E1R x,
y: ET 2
1-
[ElI0 E12]
T 2 B= [ ~l
NI
].
ill
,X2 EnW.l ,Ill
y: AT
,2
1
(H3')
. I ([ Ell0 I,0]
detectable, i.e.,
AEll-Au rank
T 2 P= [;:] , CTI = [Cl C 2
Cl
[I:;' I~, _~], = [I~' I~, ~';?] [~ f] ,
T3= (25) T, ] ,
and (En, AI, Br) is normalizahle. If (E. A, B) is stahi· lizable and (E, A, C) is detectable. then (Ell, All, Br) is stabilizable and (Ell ..4ll. Cr) is detectable (Dai 1989, p.139).
[~I f]
.)
0
I,
where M satisfies (26) and T" T2 are defined as in (25). Then a direct matrix calculation yields that _
° 0]
I2 EO] [EllE T, [ 0 I T:l = NI 0 o 0 I,
The proofs of the following two lemma"! are given in Lin and Dai (1995).
Lemma 4.4 Assume IHI)-(H31. A feedback controller (4) solves the regulation Problem 2 for system (1) if and only if it solves the regulation problem for t.he following normalizable system
= "I +8, \lA E(C+,
0
[
To this end, define two nonsingular matrices
+n2=n,
= [All 0 ] 0 In2 '
[All -]) is 0 PI] S ' [ Cl QI
the trIP e
[AP] '.
, T, i 0 S T3 =
[GQ]T3 = [CI C2 QI]' Therefore. [ AE - A _ P .] rank 0 AI, - 5
G if and only if
1546
Q
[All 0 PI] 0 In, 0 DOS
AEll -All rank
[
0
-PI ] AI,-8
=
n, + s, \lA Ea:+'
Cl QI The last condition is exactly (H3'). 0 Remark 4.7 Mat.rix equations of the form of (8)-(9) are of the form of generalized Sylvester equations. It is not difficult to prove that (8)-(9) have solutions IT and r for any P and Q if ,",d only if (Dai 1989)
rank [ AEC·4 B] 0 = n+8, A E O"{S).
(27)
Remark 4.8. We con dude from Theorems 3.3 and 4.6 that, under the assumptions of (Hl)-(H3), the output regulation problem is solvable via error feedback if and
only if it is solvable via full information feedback. The conditions for the existence of solutions call be expressed in terms of the solvability of linear matrix equations
(8)-(9). The rank condition (27) provides a sufficient. condition for the solvability of (8)-(9) Ihat is convenient for verification. It thus follows from Theorem 4.6 and Remark 4.7 that Problem 2 is solvable if (27) and (Hl)(H3) are sat.isfied. Remark 4.9 An argument similar to the proof of Theorem 4.2 shows that Theorem 4.6 still holds if (4) is replaced by the following singular controller E,~ = F( +Ce u=H(+.le, in other words, the existence condition of (4) is the same as that of a singular controller.
5. CONCLUSIONS In this paper, we have investigated tlw problem of output regulation for linear :;ingular ~y.stems when the measurement output i~ the same as the vector to be regulated. A necessary and sufficient. condition has been de-
veloped for the solvability of the regulation problem by either full information feed Lack or error feedback. A detailed procedure for the design of controllers is provided. In summary, we have solved thf' output. regulation problem, thus filling a gap in the study of regulation problem for linear singular sy~n'~ms.
6. REFERENCES
Campbell, S. L. (1980). Singular Systems of Differential Equations. Pitman, New York. Camp bell, S. L. (1982). Singul
Proportional and
derivative feedback of the, tate. Int. J. Systems Science, 14, 615-632. Lin, W. and L. Dai (1995). Solutions to the output regulation problem of linear singular systems. Manuscript.
Newcomb, R.W. and B. Dzimla (1989). Some circuits and systems applications of semistate theory, Circuits, Systems and Signal Processing, 8, No. 4, Pandolfi, L. (1990). Generalized control systems, boundary control systems, and delayed control systems. Math. of Control, Signals, and Systems, 3, 165-181.
Willems, J.C. (1991). Paradigms and puzzles in the theory of dynamical systems. IEEE TI-ans. Automatic Control, 28, 423-446.
Aplevich, J.D. (1991). Implicit Linear Systems. Springer-Verlag, New York. Brenan, K.E., S.L. Campbell and L.R. Petzold (1989). Numerical Solution of lnitial- Va.lue Problems in Differential-Algebraic Equations. North-Holland, Amst.erdam.
1547
Copyright © 19961FAC 13th Triennial World Congress. San Francisco, USA
SOLUTIONS TO THE OUTPUT REGULATION PROBLEM OF LINEAR SINGULAR SYSTEMS Wei Lin *,1 and Liyi Dai *'"
'" Dept. Systems, Control
Industrial Engi., Case Western Reserve University, Cleveland, OH 44106 •• Dept. Systems Scimce & Math., Washington University, St. Louis, MO 63130 fj
Abstract. Output regulation is an important problem in control theory. In this paper, we revisit this problem for linear singular systems and identify an important case of the general regulation problem where the measurement output is identical to the vector to be regulated. We derive a necessary and sufficient condition for the regulation problem to be solvable via either full information feedback or error feedback. Then we show how full informat ion feedback and error feedback fontrollers can be constructed explicitly. Keywords. Singular systems, regulation, state feedback, measured feedbac·k.
1. INTRODUCTION Singular systems exist, for instance, in large-scale systems, networks (Lewis 1986, Lucnbcrg:cr 1977), circuits (Newcomb and Dziurla 1989), modeling of robot systems (McClamroch 1986), boundary control systems (Pandolfi 1990), modeling of general systems (Willems 1991), and power systems. Singular systems have attracted attention of many researchers since late 19708. Several books and survey paper:s have appeared specially dealing with these systems (they are, to the best of Ollr knowledge, Aplevich 1991, Brcnan et. al 198£>, Campbell1981, 1982, Dai 1US9, Lewis 1986. 1992a,hJ. A large number of important problems ha.ve been treated and solved in a long list of individual research papers, many of which can be found in these L100ks and review articles. The regulation problem can be stated as the following: Consider a system with the presen"e of input (and/or measurement) disturbances or external signals. It is desirable to find a controller such that the closed-loop 1
Research supported
ill
part by A FOSR and NSF
system is internally stable and has desired properties such as (asymptotic) disturbance attenuation and signal tracking. The regulation problem for the conventional state-space systems attracted much attention in the 1970s. A rat.her complcte regulation theory of linear state-space system was established during that period (see, for example, Frauds 1977, Knobloch et al. 1993). The regulation problem for linear singular systems has been investigated, for example, in Dai 1987, 1989, in a quite general setting where the measurement output and the vector to he regulated are not necessarily same. Although sllch general formulation provides beneficial insights for understanding the problem of output regulation, it also has some limitations: The solution to the general problem was not satis:"actory, Existence conditions were expressed in terms of solutions to a set of nonlinear matrix equations that depend on not only system parameters hut also somf' other parameters. Such conditions are difficult to verify. This naturally raises a fundamental question of when such conditions can be further refined. Partial answers to this question have been obtained for several spec'ial cases (see Dai 1989,
1542
Chapter 5 and references there for additional details). In this paper, we revisit the regulation problem for linear singular systems. \Ve examine a special case of the general problem considenxi in Dai 1987 where the output measurement is the same a..~ the vector to be regulated. This has been known to be a. very important case in the literature of th e C'1)nventional st.a.te-space systems (Francis 1977, Kn obloch ct aJ. 1993). Actually, this is the case that has received the most attention and whose solution is the most elegant and complet.e. I~vlaIlY important problems such as signal tracking can be recasted as special cases of the regulation problem considered in this paper. For this case, we are able to provide a satisfactory solution to t.he regulation problem, thus filling a gap in the ~tuciy of the regulation problem for linear singular systems. Under severa l very mild assumptions (which are standard in the regulation theory of state-space systems), we derive a necessary and sufficient condition for the output regulation problem to be solvable.
2. PROBLEM STATEMENT Con~ider a singular linear system described by equations of the form Ex == Ax + B11, + Pw tb = Sw (1) e = Cx+Qw,
where x e nn 1 U E IR:n, and e E IRr are the state, the control input, and the measurable error output, respectively, E,A E 1Rn x fI , BE R llxm , P ERn x s, C E R f x n , S E R Sx ., and Q E IR r x s arp. constant matrices. The signal w E R S is the exogenous input which includes refer..nce signals to be t1UckeJ and/or ,lis/nTbanees tQ be rejected. Throughout this paper, we aSSume that w(t) is generated by a linear autonomous system tU Sw, known as the exosystem. \Vithout loss of generality, we assume t hat E is singular and syst.P.n1 (1) is regular, i.c. , det(AE - A) ;t O. Regularit.y guarantf'es t.hat (1) has a unique solution for any 11 and w.
=
The control action to the system (1) can be provided by either full information feedback or e''1'Or feedback. By full informal.ion feedba.ck we m<>an the feedback control law of the form
,,=[(x+Lw mx n
mxs
(2)
where K E R and L E R are gain matrices to be deSigned. Note that measurement of w makes perfect senSf! in the case where tu is a reference signal. Putting (1) and (2) together, I.he closerl-Ioop system is E x = (A - B1<)x + ( P + BL)w to Sw (3) e = Cx+ Qw .
=
If either the state x or the exogenous signal w is not available for the control action, (3) cannot be implemented directly. In tltis case, a mOre realistic and rather common approach is to design a dynamic compensator driven by the measurable error output e. We consider a eontrol1er of the following forOl
{=FH-Ge u
(4)
= H~+Je ,
where EE IRn~ is the state of the dynami c compensator, F E Rncxn~, G E Rnc xf , H E m,rn x nc, and J E IRrnxr are constant matrices to be df'termined. Usually, (4) is called the error feedback controller. Under the error feedback controller (4), the closed-loop system for (1) is Ex = (A + BJC)x + BH~ + (P + BJQ)w
t; = GC", + F{ + GQw tiJ
(5)
= Sw
e = Cx+Qw. The design goal is to achieve internal (asymptotic) stability and out.put regulation. Internal stability means that the closed-loop system (3) Or (5) is asymptotically stable when the exosystem is disconn ected (Le., set w = U) . Output. regulation requires that the error signal e(t) in the closed-loop system (3) or (5) asymptotically decay to zero as time tends to infinity for any initial state x(O) E JRn, w(a) E JR', ~(O) E lR.n, . In summary, wc arc going t.o solve the following two problems. Problem 1 (Output regulation via full information feedback): Find, if possible, two constant matrices 1< and L such that (la) a(E , A+B1<) c(J;- , where, for any square matrices a and 11 of same dimension, q( a , ,9) d~f {>. I det(Ao:11) = O} is the set of gencraH7.ed eigenvalues of the pencil (a , 11) . (lb) for any (x(O), w(O)) E Rn < R ' , the error signal e(/) of the closed-loop system (3) satisfies Hrn e(t) = Hm (Cx(t ) + Qw(t)) = O.
t ...... iXJ
t -+oo
Problem 2 (Output regulation via error feedback): Find, if possible, four matrices F, G,H, J such t.hat (2a)
([Ea 1 [A + BJC BH l)
a:-
a a I n, ' GC F c , (2b) for every (x(O),~(O),w(O)) E Rn X Rn, X R', the error signal e(t) of the closed-loop (5 ) satisfies
Hm e(t) = lim (Cx (t ) + Qw(t)) = O.
l-4 oo
i _oo
The regulation Problem 1 (or Problem 2) is said to be solvable if there exist such matrices K, L (or F ,G , H , J, respectively) . To solve these t\\O problems, we make the following three assumptions:
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(HI) a(S) ca;+ ~r p E(£IRe(A) 2 0); (H2) the triple (E, "I, B) is stabilizable in the sense of Cobb 1984, which is equivalent to rank[AE A BJ;n, VAEa; +;
regulation via full information feedback is solvable if and only if there exist two matrica, II and r such that EllS; All + Br + l' (8)
(H3) the triple ( 0 I, ' 0 S' ,[ e Q j) is detectable
o
0; 011 t Q.
[EU] [AP]
in the sense of Dai 1989, which is equivalent to
AE.-A
rank
[
(I
C
-1' ] :\1, - S' ; n + 8, VA ea;+.
Q
The first assumption (HI) causes no loss of generality since asymptotically stable modes in the exosystem decay to zero eventually and, therefore, do not affect the output regulation. The second onc (H2) is rather standard and indeed necessary for the existence of a state feedback control law that achieves asy mptotic internal stability of the closed-loop system. The last assumption (H3) is slightly strong,,, than necessary for asymptotic stabilization of the closed-loop ,ystem by error feedback control, but we prefer this one for technical simplicity.
(9)
Proof. (Sufficiency): By assumption (H2), the triple (E. A, B) is stabilizable. Therefore, there exists a matrix K such that atE, A + BK ) C(£ - , Using this particular K, together with the mat,rices 11 and r satisfying (8)-(9), one is able to construct the following full information feedback control law
u;K(x-IIw)+rw. Algebraic manipulation shows that it solves Problem 1. (Necessity): Suppose that Problem 1 can be solved by the full information feedback control la", (2). Then there exists a matrix 11 satisfying (6)-(7). Equations (8)-(9) follows from setting r ; Kll -t L. 0
4. ERROR FEEDBACK 3. FULL INFORMATION FEEDBACK This section deals with t.he problem of output regulation via f1111 information feedback. \Ve first recall a simple but extremely useful result of Dai 1989.
Lemma 3.1 Consider a linear singular system of the form Ex = ..1.3' + l'w
,v=Sw
,' ; Ox + Qw . Suppose that assumption (HI) holds and <7(E, A) ca;-. Then lim ..... oo e(t) ; 0, V (x(O). w(O» E Rn X R' if and only if there exists a matrix V such that ,11' - E\' S = P CF=Q. o Suppose assumption (HI). Assume that there exists a full information feedback control law (2) such that condition (la) of Problem 1 is satisfied. Then Lemma 3.1 immediately implies that the condition (lb) is also satisfied if and only if there exists a matrix II such that
+ BK)ll + P + BL 0= Cll + Q.
EllS; (A
In this section, a necessary and sufficient condition for the solvability of Problem 2 is first derived under the assumption of normalizability of system (1). Then this assumption is removed via several system decompositions. Normalizability is only a step :1tone in solving Problem 2. The following Lemma 4.1 is a consequence of Lemma 3.1 applied t.o the closed-loop "ystem (5). Lemma 4,1 Assume that (HI) holds. Suppose there exists a dynamic compensator of the form (4) for which condition (2a) of Problem 2 is satisfied. Then condition (2b) of Problem 2 holds if awl only if there exist two matrices II and E such that
EllS' ; All + BHE
ES; FE
(11)
0; ell,. Q.
(12)
4.1 The case of normalizable singular systems A linear singular system
Ex; A" + B" y ;Cx
(7)
Theorem 3.2 Suppose that a linear singular system (1) satisfies (HI) and (H2). Then the problem of output
(10)
o
(6)
Based all the previous lemma, we are now able to derive a necessary and sufficient condition for the solvability oC the regulation Problem 1 with f,,1I information feedback,
+P
is called normali:t.able if ranklE BJ ; n. In this case, we say (E, A , B) is normalizable. If (E, A, B) is nonnalizable, there exists a matrix L such that det(E + BL) oF 0 (Dai 1989, Mukundao and Ayawansa 1983). Choosing the derivative state feedback control 'U -Lx + VI we have (since (E + BL) is invertible)
=
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= (E + 8L)-' Ax + (E + BL)- ' Bv y= Gx ,
;i;
that iS 7 the closed-loop system hec:omf~ a normal statespace system! Therefore, the concept of nonnalizability helps us convert a control problem of singular systems into that of normal systems for which many standard results are available. Of course, the main trouble is the realization of the unplea.-'iant and often undesirable derivative state feedback. The follm\'ing theorem presents a necessary and sufficient condition for the solvability of Prohlem 2 under the assnmption of normalizability.
Theorem 4.2 Suppose (HI), (H2), (1I3), and, in addition , that (E, A , E) is IlorIllalizable. Then the problem of output regulation via error feedback is solvable if and only if there exist two matrices IT ann r that ~olve the linear matrix equations (8)·(9). 0 Proof. (Necessity): A3 in the case of full information feedback , the necessity is an immediate consequence of Lemma 4.1. To see this point, note that if t here exists an error feedback control Law of t.he form (4) t hat solves Problem 2, t hen the c1osp.d-loop system (5) satisfies conditions (2a) and (2 b ). By Lemma 4.1, there exist two matrices IT and l: such that 110)-112) hold. Choose r = HE . Clearly, equatioIls (8) .. (9) follows immediately from (la) ami (12). (Sufficiency ): The proof of sufficiency is constructi ve. Assume that there exist two matrices IT and r satisfying (8)-(9). Since (E ,A, B) is normalizahle, there exists a matrix L such that E + EL is invertible. According to the assumption (H2) , (E, A, B ) is st.ahilizabLe. Then, for any matrix L chosen in such a way, (E + BL, A, B) is also stabilizable (Dai 1989). Therefore, a matrix K can he chosen such that. u(E +EL, A + EK) e (L (13) The assumption (H3) further guarantees the existence of matrices GOI Cl such that
u([~ f] , [~
i]- [~~] [GQ])ect ·
(14)
For the matrices L,K·, GO,G 1 thus obtained, we consider the dynamic compensator (4) with the matrices F,G,H, J chosen in the following manner.
next show that the dynamic compensator (4) with coefficient matrices chosen in (15) indeed solves the problem of output regulation. First of all, by r.onstructi~n (15) , u=H~+Je=(K r-J(IT]~-[L -LIT]~. (17) The substitution of the previous expression into the system (1) results in E:i:+[BL -BLIl]~=Ax+[BJ( B(r-KIT)]E+Pw . (18) On the other hand, for the matrices chosen in (15)-(16), (4) can be written as
[ E+BL-BLIT]~=([A o I, 0
+ [ BoK B(r~KIT)])~ + [g~] (GH Qw).
[g~ 1
(15)
H:= [K r - KIT] - [L -LII]F J:=-(L -LITIG where
t>
= [~ ~.l
+
[~] [ L - LIT]
E[:j:A[;] + Bw
Since E + BL is invertihle, so is the mat.rix tl.. Therefore. the four matrices F , G, H , .J iu (15) are well-defined. We
(20)
w=Sw e=Gx+Qw. wh ere
E=
[
[P] E BL -BLn] 0 E+BL -BLn ,B= GoQ , o 0 I. G,Q
BK B(r - KIT) ] A GoG A-GoG+BK P - GoQ+B(r-KIT) . [ G,G -GIG S-G,Q The set of eigenvalues of the closed-loop system (20) is u(E , A). Direct verification yields
,l=
q{E,A) = u(E+BL,A +BK) P-GoQ]) C
(21)
We thus conclude from (21) th,.t condition (2a) of Problem 2 holds. Next , we prove by using Lemma 4.1 that condition (2b) of Problem 2 is also satisfied. To this end, it suffices to check that the matrices Il and l:
= [Z]
are solutions
to (10)-(12). As a matter of fac t , (12) is identical to (9). As for (11), a direct computation proves that
t>l:S
= [E +0 EL -~~IT 1 [Z] S= [E~S] (22)
t>FE - [AIT+Br+p-Go(GIT+Q)] S-G,(GIT+Q) . By using equation (9), we haY,! t>FE =
(16)
(19)
Putting together (18)-(19) and t.he system (1), we see that the closed-loop system (5) can be expressed equivaLentlyas
.- A_I[A -GoC -rBK P-GoQ+B(r- KIT)] F .-'" -G,G S-{;,Q G := t>-'
P]_[GoCGoQ] S G,G G,Q
[AII + ~r + P] .
(23)
It. is immediate to see from (8) that equation (22) coin-
cides with (23). Note that t> is invertible by construction. Hence FE = ES, which proves that equation (11) holds. Finally, it follows from (15) that
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AIl+BH~+P
= All+P+Bf- [BL
-BLll]F~.
Remark 4.3 It is dear that the clooed-Ioop system (5) is singular for any controller (4), Since we are only COIlcerned with asymptotic behavior of the error, the undesirable impulsive part of the behavior associated with a
singular system has b,,,,n ignored (see. e.g., Cabb 1984 on impulsive modes). In order to eliminate the possiimplll~ive
modes, we only Heed t.o require that, the
closed-loop system (5) have no infinite poles. According to (21), it suffices to require that (E + BL, A + BK) CoG GoQ] and ( 0 I,' -GIG S _ G1Q : have no infinite
[EO] [A -
+ B1u + Ptw
+ QLW,
where
The last equation comes from (8). Therefore, equation (10) is satisfied aB welL 0
hIe
EllXl = Anx} el = CII I
Using the relation (In, we get All + BH~ + P = AII + Br + P = EIIS.
P-
and M solution to
PI = PI + E I2 MS, Q, = Q - C 2 M L:~o Ni P2 S'. Note that M is the unique (26)
o Lemma 4.5 The linear matrix equations (8)-(9) are solvable with respect to II and r if and only if EllIIIS = AllIII + Blr + PI 0= CIII I +QI is solvable with respect to ill and r.
o
priate feedback gains L,K,Go.G I . Since the choice of
Now we are ready to remove tILe assumption of normalizability of Theorem 4.3, as illustrated in the following theorem.
these gains ean be obtained using a rather standard approach, wc do not pursue thiR direction in this paper. Interested readers are referred to Dai 1989 (Chapter 3) for additional dp.tails.
Theorem 4.6, Suppose (HI), (H2), and (H3). Then the problem of output regulation via error feedback. is solvable if and only if there eXIst two matrices n and r that solve the linear matrix equations (8)-(9). 0
4.2 General case
Proof. According to Lemmas ,1.5-4.6, the result of Theorem 4.3, and that normalizability decomposition does not affect the stabilizability, w" need only to verify that the assumption (H3) is equivalent to
eigenvalues, which can be
achi~~ved
by choosing appro-
Normalizability may not be satisfied in practice. In this subsection, the assumption ofnormalizability is removed with the help of appropriate system decompositions. For any singular system (1), there always exist non-singular matrices TI and T2 sHch that. system (1) is re:~tricted system equivalent to (Dai 1989, p.139) EUXl + E 12 i 2 = A l1 X I +B}u+P1w NIX2 = X1 + P'jUJ (24) e = CIXI + C2X2 +Qw where NI is a nilpotent. mat.rix with a nil potent index denoted as I, that is. Ni '10. Nf+l = 0,
x=T1
[XI]'Xl E1R x,
y: ET 2
1-
[ElI0 E12]
T 2 B= [ ~l
NI
].
ill
,X2 EnW.l ,Ill
y: AT
,2
1
(H3')
. I ([ Ell0 I,0]
detectable, i.e.,
AEll-Au rank
T 2 P= [;:] , CTI = [Cl C 2
Cl
[I:;' I~, _~], = [I~' I~, ~';?] [~ f] ,
T3= (25) T, ] ,
and (En, AI, Br) is normalizahle. If (E. A, B) is stahi· lizable and (E, A, C) is detectable. then (Ell, All, Br) is stabilizable and (Ell ..4ll. Cr) is detectable (Dai 1989, p.139).
[~I f]
.)
0
I,
where M satisfies (26) and T" T2 are defined as in (25). Then a direct matrix calculation yields that _
° 0]
I2 EO] [EllE T, [ 0 I T:l = NI 0 o 0 I,
The proofs of the following two lemma"! are given in Lin and Dai (1995).
Lemma 4.4 Assume IHI)-(H31. A feedback controller (4) solves the regulation Problem 2 for system (1) if and only if it solves the regulation problem for t.he following normalizable system
= "I +8, \lA E(C+,
0
[
To this end, define two nonsingular matrices
+n2=n,
= [All 0 ] 0 In2 '
[All -]) is 0 PI] S ' [ Cl QI
the trIP e
[AP] '.
, T, i 0 S T3 =
[GQ]T3 = [CI C2 QI]' Therefore. [ AE - A _ P .] rank 0 AI, - 5
G if and only if
1546
Q
[All 0 PI] 0 In, 0 DOS
AEll -All rank
[
0
-PI ] AI,-8
=
n, + s, \lA Ea:+'
Cl QI The last condition is exactly (H3'). 0 Remark 4.7 Mat.rix equations of the form of (8)-(9) are of the form of generalized Sylvester equations. It is not difficult to prove that (8)-(9) have solutions IT and r for any P and Q if ,",d only if (Dai 1989)
rank [ AEC·4 B] 0 = n+8, A E O"{S).
(27)
Remark 4.8. We con dude from Theorems 3.3 and 4.6 that, under the assumptions of (Hl)-(H3), the output regulation problem is solvable via error feedback if and
only if it is solvable via full information feedback. The conditions for the existence of solutions call be expressed in terms of the solvability of linear matrix equations
(8)-(9). The rank condition (27) provides a sufficient. condition for the solvability of (8)-(9) Ihat is convenient for verification. It thus follows from Theorem 4.6 and Remark 4.7 that Problem 2 is solvable if (27) and (Hl)(H3) are sat.isfied. Remark 4.9 An argument similar to the proof of Theorem 4.2 shows that Theorem 4.6 still holds if (4) is replaced by the following singular controller E,~ = F( +Ce u=H(+.le, in other words, the existence condition of (4) is the same as that of a singular controller.
5. CONCLUSIONS In this paper, we have investigated tlw problem of output regulation for linear :;ingular ~y.stems when the measurement output i~ the same as the vector to be regulated. A necessary and sufficient. condition has been de-
veloped for the solvability of the regulation problem by either full information feed Lack or error feedback. A detailed procedure for the design of controllers is provided. In summary, we have solved thf' output. regulation problem, thus filling a gap in the study of regulation problem for linear singular sy~n'~ms.
6. REFERENCES
Campbell, S. L. (1980). Singular Systems of Differential Equations. Pitman, New York. Camp bell, S. L. (1982). Singul
Proportional and
derivative feedback of the, tate. Int. J. Systems Science, 14, 615-632. Lin, W. and L. Dai (1995). Solutions to the output regulation problem of linear singular systems. Manuscript.
Newcomb, R.W. and B. Dzimla (1989). Some circuits and systems applications of semistate theory, Circuits, Systems and Signal Processing, 8, No. 4, Pandolfi, L. (1990). Generalized control systems, boundary control systems, and delayed control systems. Math. of Control, Signals, and Systems, 3, 165-181.
Willems, J.C. (1991). Paradigms and puzzles in the theory of dynamical systems. IEEE TI-ans. Automatic Control, 28, 423-446.
Aplevich, J.D. (1991). Implicit Linear Systems. Springer-Verlag, New York. Brenan, K.E., S.L. Campbell and L.R. Petzold (1989). Numerical Solution of lnitial- Va.lue Problems in Differential-Algebraic Equations. North-Holland, Amst.erdam.
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