Triangular Decoupling of Linear. Multivariable Systems. Absfract-A geometric formulation of the state feedback tri- angular decoupling problem is given.
447
SHORT P.4PERS
0.6788 0.6655 0.6655 0.6653
0 1 2 3
-0.1667 -0.3619 -0.3937 0.1423 -0.4420
-0,4400
-0.1464 0.8140 -0.1484
0.6788 0.6656 0.66550.1589 0.0521 -0.0667 0.4544 -0.0445 -0.2023 0.6653
1.36 0.7965 -0,4119 0.62 1.280.6551 -0.1617 -0,5292 0.58 0.4i93
0.44311 0.1022 -0.1161 0.6359 -0.1884
1.49
1.580.0873 0.0346 -0.0341
of t,he optimal open-loop cost,. However, the suboptimal closed-loop cost of [ a ] for t.he same problem with two subintervalsor one switching time is only wit.hin 2 percent of t.he optimal open-loop cost. Apparently, the optimization of the sait.chingtime and the use of the mean-square error crit.erion suggested in this paperas compared to the “averaged” control cost suggested in [ 2 ] partly accounts for the improved performance of the given system.
are fixed:
COXCLUSIONS The synthesis of suboptimal linear control laws having piecewiseconstant, feedback gains and nonperiodic switching times has been achieved. Results show that. t.he met.hod proposed in thispaper compares favorably with otherlinear and nonlinear suboptimal control laws. The met.hod proposed is applicable to the control of nonlinear systems with nonquadratic type performance indices. Further work is needed to evaluate critically some problems associated with t,he proposed met,hod such as stabilit,y and sensitivity to changes in init.ia1 conditions. The examples at,tempt,ed to date indica1.ethat. t.he method is feasible and should be studiedfurt.her.
APPENDIX The suboptimal performance index I and the total variation of Z are given by (11) and (14). The total variation can also be expressed in a slight.117 different, form. The arguments of some functions are omitted in the following equat,ions for notat,ional simplicity:
The necessary conditions for an extremum of this suboptimal problem can be obt.ained by equating t,he first variation of the suboptimal performance index equal to zero and not.ing that t,he feedback gains and switching time variat.ions areindependent. Equat.ions(15) and(16) represent the necessary condit.ions for this suboptimal control problem. ACKNOWLEDGMEST
The authorwould like to thank AI. Y.Tarng for valuable assist.ance in programming the problems contained in t.his paper.
REFERENCES N. E. Nahi and T. S. Bettwy “Synthesis,of near optimal feedback
control systems,” presented a t t h e Hamau Internatl. Conf. System Sciences January 29-31 1968. D L. KleinmanT. Fohmann and &I. At.hans. “Onthe design Of linear systems ’with piecemi&-constant feedback gains,” IEEE Trans. Automatic onf fro^, vol. AC-13. pp. 354-361, August t968. 8. V. Rekasius Suboptimal design of 1nt.entlonallv nonllnear controllers,” I E E E Trans. Automatic Control, vol. ACl9, pp. 380386 October 1964 L. R. Nardizziand G. A . Bekey, “Synt,hesizine opt.ima1 controls for modula.ted discrete-t,ime systems,” Intcrnatl.OJ. Control, vol. 8, pp. 571-590, December 1968.
Triangular Decoupling of Linear Multivariable Systems
The first summation of (28) d be expanded in a Taylor series and only t , e r m up to second order will be ret.ained
Absfract-A geometric formulation of thestate feedback triangular decoupling problem is given. Necessary and sufficient conditions for the existence of decouplhg matrices are presented. A procedure is outlinedforsimultaneously realizing a triangular structure and assigning the poles of the closed-loop system transfer function matrix.
The n X n matrix [&/&Y] is evaluated along the nominal closedloop trajectory. The other terms of (28) are expanded in a similar manner and the first variation of the performance index Z has the following form if i t is observed t,hatthe initial and final times
Manuscript received September 29, 1969. Paper recpmmended by D. G . Luenberger Chairman of the IEEE G-AC Lmear Systems Commit.tee. This rksearch mas supported in part by KASA under a n NItC Postdoctoral Resident, Research Associateship held bv IT. hf. Wonham. A. S. Morse y a s with t,he Office of Control Theory and +pplicat.ion, NASA Electromcs Research Center, Cambridge., Mass. H e 1s now l p t h the Depart.ment of Engneenng and Applled Sclence, kale UIllX7ersltF, h-ew Haven Corn. V.M . Teonham was with the OmceofCont.ro1 Theory and Applicat.ion h-ASA ElectronicsResearchCenter,Cambridge, Mass. He is nom &th t.he Department of Applied Science and Engineering, University of Toronto, Toronto, Ont., Canada.
448
WEE TflANEi4CTIONS ONAUTOMATICCONTROL,
I. IXTRODCXTION
yi(r) =
HiZ(t),
+ Bu(t)
i E J = (1,2...,kj .
ai (1) (2)
Here z is an n-vector, u is an m-vector of controls, yi is a p,-vector output., i E J , and A , B, Hi, i E J are real constant matrices of appropriat,e size. For certain applicat.ions it is useful to modify the st.ructure of (1) using feedback and feedforward control. For example, one might look for matrices C and Ki, i E J , so that with xu(t) = Cz(t) KiVi(t)
+
i=l
11. FORKULATIOX The problem will be formulat,ed using the geomet.ric ideas of [I]; pertinent results are summarized in the Appendix. Write & for the n-dimensional state space of (1). If C & is the controllability subspace of ( A $ ) associated with vi, i E J, t,henl =
(A
+ BC I { B K i J ) , i E J ,
Hi& = { H i ) , i E J .
(5)
If vi does not influence yj, for i E J‘ = (1,,..,k - I}, j E J i = (i I,.-.,k), t.hen
+-
Ji.
(6)
Let X ; den0t.e the null space of H i . The problem may now be stat.ed in t.he following manner. Given A , B , X i , i E J , find a nmtrix C and controllability subspaces ai, i E J, such that
a i = ( ~ + ~ c l a n ~a ~ E )J , @i+Xi=&, i E J
(7) (8)
Here (8) and (9) are equivalent to (5) and (6), respectively. The equivalence of (7) and (4) follows from theproperties of control1abilit.ysubspaces (see t.he Appendix).
111.
&I
a},
Theorem 1 T h e ezist C and ai, i E J , satisfying (7)-(9) if and only if
i E J.
(10)
1 For an arbitrary matrix M, 3n or (A41 denotes the range of M . The symbol ( A I 13J denotes the subspace [ A1 I 3 1 = Ca AB +A”-’@.
+ ...
c ax-.
c& c
(12)
Write go= 0 and let Gi, i f J , be any subspaces such that
e ai+ i E J. (13) It follows from (12) 2nd (13) thzt the are mutually independent subspaces. Since A a i c 63 + ai,i E J , (see the Appendix), there
& = ~i
&i
c a + E; +
A E ~
i E J.
&-I,
+
Since the E i are independent, theree_xistsC such that ( A BC) Ei Ei i gi-1, i E J . Thus ( A BC) (F11 and by induction
+
(A
+ BC)&
=
(A
c
+ BC) ( E +~
c gi +
=
c
&, i E J ~ .
This C has the desired properties (see the Appendix). 1x7. POLEASSIGNXEXT
A procedure will be given for assigning the eigenvalues of A
+
BC while maintaininga +angular st.ructure for the system. For simplicity arit.e cili = ai,i E J , and assume ( A I (53) = E ; i.e., the system is completely controllable. As the first step, choose any Co such t,hat (A f BC0)ai C i E J . Write A O = A BCOand let. C = (C: (A0 BC)CRi ai, i E J ) . It. is clear t,hat.C1 E C if and only if
+
(Ail
+-
+ BCI)&i c E i +
ac1,
c
i E J.
ai,
(14
Let Pi, i E J , be the projection on &i along Then Pi(Ao BC1) Gi C E;, i E J , for C1 E C . characteristic polynomial For h e d C1 E C, let ai@) bethe of the restriction of Pi(& BCl) to G i . Similarly, let, Bi(h) be the characteristic polynomial of -40 BC1 on ai. Clearly B1(X) = al(X), and from (14),Bi(X) = a i ( X ) p i - ~ ( X )i, E J1. That is,
+
+-
+
1
B ~ ( x )=
Haj(h),
i E J.
(15)
j=1
+
Kotethat p k ( X ) is the characteristic polynomial of A0 BC. Although the polynomials a i ( h ) , i E J , are &xed in degree for all CI E C, their coefficients may be freely adjustedwith suit,able CI E c. A simple calculat.ion shows t.hat {PiAo I P i @ } = P i & = E i , i E J ; hence Ei is a cont.rollability subspace of ( P i A , , P i B ) . By [l, Theorem 4.21, there exists D i such that,the coefficients of the characteristic polynomial of P i ( A o BDi) on E i have any preassigned values. Having determined suitable Di,define CI by
+
k
CI
DiPi.
= i=l
Let, &, i E J’, denote the maximal contr0llabilit.y subspace of ( A $ ) satisfying (9). A const.ructive procedure for finding ma@nal controllabilit; subspaces is outlinedin the Appendix. Let. ( R h = ( A1 the controllable space of ( 1 ) .
+
(11)
Proof: The necesity of (IO) follows from the maximality of the aC for which (7) holds wit.h (Ri = ai,i E J . From (9) it. is clear that,
MMN RESULT
Gi + xi = E ,
i E J.
&i,
6s.For sufficiency, it will be elough t.o show the existence of
(4)
For vi to control yi, i E J ,
H j a i = 0 , i E J’, j E
=
follows
each new input vi (a p,-vector) cancontrol y i ait,hout affecting yj, for j # i. An extensive theory for t,he synthesis of such “diagonally” decoupled systems is available in [1]-[4]. Inthe present correspondence an easier problem of system synthesis is considered. I n particular we seek a control of t,he form (3) such that each vi controls y i without irduencing yj, for j > i. Namely, the problem is to find matrices C and Ki, i E J , such that. the transfer-function mat.rix relating the y and p! has an uppertriangular form. Such a configuration is appropriateto cert,ain applicat.ions in proem control and power regulation.
ai
1970
I(’urthernzore,if (IO) holds, one may choose
Consider the system described by 5(t) = A z ( t )
AUGUST
+
Clearly C1 E C and Pi(A0 BC1) E; = Pi(Ao -I-BDi) Gi, i E J . Thus the matrix C = Co C1 will provide the desired pole distribution while maintaining a triangular struct.ure. The results of this procedure are summarizedin t,he following theorem.
+-
449
SHORT P A P E X
1-:
is nonempty. One can easily show t.hat,
V. EXAMPLE
Consider the system wit.h
C(U) # + e A U C c B + V .
1 -1
*-I
=
0
c
0 if
1 ,
--I
(16)
A subspace 6i E is called a wnt.rollabilitysubspace C(B)# + and if, for some C E C((R),
a = 1-4 + B c I a n a ) .
B = [
0 0
0
,
HZ = [l
of (A$)
(17)
If is acontrollabilitysubspace, (17) holds for all C E C((K). The coefficients of the characteristic polynomial of A BC rest.ricted t.o a maybe freely assigned with suitable C E C ( a ) . The control1abilit.y subspace (K in (17) coincides with the set of reachable st,ates for the system
+
1 0 0 01.
i(t)
By an easy computation
=
( A
+ BC)s(t) + BKv(t)
(18)
provided K is defined so that ( B K J = 63 fl a.Conversely, for each fixed pair ( C , R ) , the corresponding set of reachable states for (18) is a cont.rollability subspace of ( A $ ) . There exist, uniquemaximal (largest)invariantand controllability subspaces contained in a &xed subspace X. The m a ~ l a l invariant subspace 3,such t,hat. % C X, is given by % =Vu, where ' 0 0 = X, Up+l=UpnA-1(Up+(R),
Clearly (IO) is satisfied, so the system may be put. into triangular form. In particular, if
and that
0:
=
p=O,I,**', A-W=
(z:AxEW}
dim X. The maximal control1abilit.y subspace X is given by
&,
such
(il C
&=
{ A +BcIanu)
for C E C(U). ACKNOWLEDGMENT
and .fl
c=[
fz
0 0
-f2
-91
Theauthors would like tothank W . A . Wolovich for several useful discussions contributing to this work.
"I
-92
RXFEREKCES
with fj,gi arbitrary, t.he system will have triangular st.ructure. C of this form, al(X) and a2(X)are given by
For
+ + 91
a1 (X) =
X2
a2(X) =
X* + f Z X
g2X
+jl.
VI. COKCLUDIXG REMARKS
It might. be conjectured that the triangular structure could be realized for a larger class of systems if a control of more general form than (3) were used. A simple argument, similar to that given in [2], can be used to shox thatthis is not the case. Although any system which may be decoupled may also be put into t.riangular form, the converse st.atement is false. For example, itmay easily be checked t,hatthe preceding systemcannot be decoupled [Z]. Even if a system can be decoupled, dynamic extension may be required for either realization or stabilization [ Z ] . Since for such systems a stable triangular structure can a h a y s be achieved without. additional dpamics, one maythink of this procedure as "poor man's decoupling." Finally, it should be mentioned that t.he geometric theory applied here and in [I], [2] can be used to st,udy other problems of system st.ruct.ure. For such problems, realizat.ion and pole assignment conditions may easily be determinedusingargumentssimilar to those in [2]. Realization may of course necessitate the adjunction of dynamicsfor certain systemstruct.ures. HOT to achieve such realizations using minimal orderdynamics remains, hoaever,an unsolved problem. APPENDIX
A subspace U C E is called an invariantsubspace if the class
C(V) = IC: (A
+ BC)V CUI
of (A,B)
[ l ] W. M. Wonham and A. S. Morse, "Decoupling and pole assignment in linear multivariable systems: a geometrlc approach, NAS-4 Tech. ReDt. ERCPM-66. October 1968; also, S I A M J. CoflfroZ, vol. 8,pp- 1-18, February 1970. [2] A. S. Morse and W. 31. Wor$am "Decoupling and pole assignment by dynamic compensation NkSA Tech. Rept. ERC PM-86, July 1969;also, S I A M J.' Control, vol. 8 , August 1970 (to be published). [3] p. L. Falb and W. A. Wolovich, "Decoupling i n the design and synthesis of multivariable control systems," I E E E Trans. Automafic Control. y l . AC-12, pp.. 661-659, December 1967. [4] E. G . Gilbert The decoupllng of mult.lrarlable SySt,emS by State feedback," S I h M J. Confrol, vol. 7 , no. 1, 1969.
Average Gain of Feedback System with One Time-Varying Gain
Abstract-The feedback system considered has a transfer function in the forwardpath and a time-varying gain in the reverse path. Bounds on the average of the system output will be established for a constant input.
INTRODUCTION The system to be st.udied here can be represented by the block diagram in Fig. 1. The relationship between t.he input and output of the g(s) unit can be expressed by a convolution integral or, as will be done here, by a differential equation. The k ( t ) unit is a time-varying g i n . The theorem that will be presented will establish bounds on the average of the output y n-hen t,he input u is constant. Manuscript received May 21 1969, Paper recommended by P. Dorato Chairman of theIEEE'G-AC Opt,imal Systems Committee. This wbrk was supported in part by the U. S. Army Research Office, Durham N C. -The aithbr is ~ t Bell h Telephone Laboratories, Inc., Holmdel, N. J.
