652 AUTOMATIC,TRANSACTIONS,ON IEEE CONTROL DECEMBER 1967. where 0 is a zero matrix consistent with the order of. Li F G If Eij denotes the n2X m matrix with 1 as, ijth entry and zeros elsewhere then E is a n m X n. I E I matrix xith the ith row identical to the ith row of and. all other rows zero The matrix EiiQ will be denoted by. Fig 1 Idtivariable feedback system Qi T h e following definition canno be made. T h e matrices F and G with G nonsingular decouple. or the system 1 if,a tz 1 if CiAjB 0 for a l l j, x here Cidenotes the ithrow of C Then a simple calcu. lation shows that,t r L i F GIG, for i I m Application of the state variable feed Note that this is a precise definition t h a t does not. back 2 and repeated differentiation togetherwith 4 involvevaguestatementsaboutinputscontrolling. yield the relations outputs independently,X C A BF OX 111 AIAIS THEOREM. i X C i A BF x, C A IVith the definitions of Section 11 it is no v possible. to state and prove a theorem which gives a necessary. and sufficient condition for decoupling, Then there is a pair of matrices F and G which decouple. the system 1 if and only if,det B 0 13,here the F are scalars depending on F Thus x can. be eliminated from thefinal relation of 5 to give Le if and only if B is nonsingular. Proof Suppose first that B is nonsingular Then it, 3 n p k F y i c k tr L i F GIB 7 is claimed that the pair. k O F B lA, where tr denotes the traceof a matrix P is the mX z G B l. matrix given by here,and Li F G is the n X m matrix given by. C i A BF di BG, Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. FALB AND WOLOVICH DECOUPLING IN MULTIVARIABLE CONTROL SYSTEMS 653. decouples 1 I n view of 4 Since 5 implies t h a t,C i A BF di l C Adi l C Ad. lBF 16 yi di l C i A B F d s l CiAd BGw 25, But CiAdiBis simply the ithrow of B and so i t follows u hich may also be writtenin the form. y A B F x B Go 26,CAdiBF Bi B lA Ai CiAdi l 17, where y is the m vector with components y i l it is. where Bi and Ai are the ithrows of B and A respec clear that the choice F F and G G leads to. tively Thus,Ci A BF di k 0 18,or equivalently to, for any positive integer k I n a similar way i t follows jri di l uj 28. C i A BF diBB l B B l 19 Caution Equation 28 doesnotrepresentthede. coupled system since in general it involves the cancel. and hence that lation of zeros The equations of the decoupled system. are given by 10 or in state form as,1 x A BF x B G o. Bi B l where F G are a decoupling pair, I t has now been established that the nonsingularity. l o of B is a necessaryandsufficientconditionforthe. existence of a pair of matrices F G which decouple 1. However Bi B l e i a row vector with 1 in t h e i t h In the next section the set of all pairs F G which de. place and zeros elsewhere and so couple 1 will be characterized under the assumption. t r L F G Q i l W W i l t h a t B is nonsingular This characterization leads to. Wi n di l 21 answers to the following two questions. tr L i F G a i P O 22 1 the synthesis question namely how many closed. loop poles can be specified for the decoupled sys, In other words F and G decouple 1 tem howarbitrarilycantheybe specified and. Now suppose that there is a pair of matrices F G how easily can an algorithm for specifying these. which decouple 1 Then it follows from 4 t h a t poles be developed. Ci A BF diBG Bi G i 1 m 23 2 the output feedback question namely when can. feedback of the form u H y G w decouple l,Since CiAiB 0 for all j would imply that. t r L i F G SL 0 whichwould contradict the fact that IV CLASSOF DECOUPLIKG RIATRICES. F and G decouple l i t isclear t h a t Bi O for, i l m As G is nonsingular Bi G O forall i Let F be an m x n matrix and let G be a nonsingular. Since 10 is satisfied i t follows t h a t Bi G is an m row m Xm matrix Under the assumption that 1 can be de. vector of theform airi with ai O otherwisethere coupled necessary and sufficient conditions for F G to. would be uj j i terms in t r Li F G SL Thus be adecoupling pairaredetermined in thissection. These conditions turn out to be independent of G so. t h a t i twill make senseto speakof the class C J of matrices. F which decouple 1,Definition,Hence B is nonsingular since G is. T h e theorem just proved shows that B is of para,mountimportance in thedecoupling of 1 bystate. variable feedback T h e basis for the choice of F andG. Let Q F be the nXm matrix given by,Ci A BF n 2B, where 0 is a zero matrix consistent with the order. 1 i 1 nz 29, in the proof of the theorem is thefollowing observation Q i F Let Pi F be the n X n matrix given by. Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. 634 DECEMBER,TRANSACTIONS,ON IEEE 1967, where the F are the coefficients of the characteristic. polynomial of A BF i e tr L F B Q tr P F Q F B P,n 1 tr Pi F Qi F B Pi 0 39. pk F A BFIk 3 1,so the F B l decouples 1, and I is an identity matrix consistent with the order. of Pi F Corollary 1, Since P F is nonsingular it follon s that the rank If the pairF G decouples l then there is adiagonal. of Pi F Qi F is the same as the rank of Qi F Kote matrix A such that G AB l. also t h a t Proof If F G decoupless l then Q i F is given by. L i F G Pi F Qi F G 3 2,where Li F G is defined by 9 Thus. rank L F GI rank Qi F,since G is nonsingular,In of definition. Z 33 Qi F G 0 1, coupling the follou ing theorem can be established I I I I. Theorem 2 I t follo vs,that X 1 X theand cod, If the pair F G decouples l then the rank of Q i F 1aV is established. is one for all i conversely if the rank of Qi F is one for. all i and if B is nonsingular thenthepair F E Corollary 2. decouples the system 1 If the pair F G decouples l then there is a diagonal. Proof Supposefirstthat F G decouple 1 Thenmatrix such that. tr L FG Q tr L FG Q,z o 34 FB B l A A B, forall i where a is the m X l z matrixgivenbywhere A and A are given by. Since Parbitrary,of L i F G is a,nonzero vector while every other column of Li F G. is the zero vector I t follows that Li F G has rank one Proof T h e corollary is an immediate consequence of. by 33 that,rank Q i F 1 the,Kow suppose that rank Q i F. B is nonsingular Since,1 for all i and that Ci A BF CiAdiB CiAdi BFB 43. C i A BF riCi A BF iB 44,Ci A BF d B CiA iB Bi 0 3 6. In summary thus fari t has been shown that the non. definition,di There is,theith of B of sufficient condition. is a necessary and,i t follows that, for the existenceof a decoupling pairF G Furthermore. the set of all pairs F G which decouple 1 consists of. matrices F such that rank Qi F 1 for all i and G,35 suchthat G AB l where A is diagonalandnon. singular In order to clarify these points an example. will now be presented, Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. FALB iWD WOLOVICH,MULTIVARIABLE,DECOUPLING,IN CONTROL SYSTEMS 655. Example 1 whichindicates t h a t F and G decouple 1 and that. Let m cy l,d i of the closed loop poles can be varied by. I d b varyingthe Mk Inthislight considerthe,o 0 11 0 1 0 0 0 0 0 1. Then 0 0 1 0 0 0 1 0,0 0 0 0 0 1 0 0,Thus B is nonsingularandthesystemcanbede. 0 0 0 0 0 0 1 0, coupled The set of all F which decouple the system. 45 can now beobtainedbydetermining all 2x3,matrices F such that rank Q i F 1 In this example. this implies that the elements of SP must be of the form. V A STXTHESIS, Theorem 2 does provide a procedure for determining N O m. a the class of all feedback matrices F which decouple. 1 However the direct application of the condition,rank Q i F 1 forall i results only in constraints. 0 1 0 0 0 0,0 0 0 0 0 0, being placed on certain of the nzn parameters of F W h a t Since B is nonsingular the system can be decoupled. is still required is a procedure for specifying closed loop Setting for example. system poles while simultaneously decoupling 1 using. an appropriate FE Inthislight a synthesispro, cedure will now be presented for directly obtaining a. feedback matrix F e whose parameters are so deter, mined as to yield a desired closed loop pole structure one obtains using 29 the decoupled system. In particular suppose that M k R 0 1 6 are,given m X m matrices Then the choice. F B M Ak A,Note thatin this case,det SI A BF s2 s l s3 s s 1. where the poles representing s s3 ss s 1 have been. specified by the choice of the M k Other choices of the. M kwould lead to other closed loop pole configurations. Therefore if B is nonsingular m d of thesys, tem s closed loop poles canbearbitrarily specified. l a t a time while simultaneously decoupling the,systemusingthesynthesisprocedure Thesynthesis. question is therefore partially resolved although some. points still require clarification In particular it will be. shown thatm zz,d i can never exceed n the number, of system poles a n d t h a t i t is sometimespossible t o. specify more thanm xy l d i poles whilesimultaneously. decoupling the system, Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. 656 IEEE TRANSACTIONS ON AUTOMATIC CONTROL DECEMBER 1967. Lemma Example 3,Let K be the m x X n matrix given by. O o0 10 O 0 I,L o o o IJ 0 1,Then rank K m x,d i and hence m di 4n Then. Proof Let ki denote the ith row of K and let ri be. arbitrary scalars such that B, riki 0 57 and m d i 4 n Thus all the closed looppoles. can be specified by using the synthesis procedure, In order to establish the lemma one need only show hioreover. that application of Theorem 2 shows t h a t 62, 57 implies that each ri O However this follows represents the most general form for a decoupling F so. directly from 57 by successive postmultiplication by t h a t f 4 n T h e general form ofthe decoupled transfer. B AB A6B and the fact that B is nonsingular matrix is. Now let p denote the number of closed looppoles Example 4. which can bespecified while decoupling and letf denote Let. the number of free parameters entries in a decoupling. matrix F for example f 3 in 47 Then the lemma 0 0,and 51 combine to give A B l 0. m 21 31 011 0 1,n z f d i I p I n 1 1 0,Moreover if m f cy. di n then all n of the closed, loop poles can be arbitrarily positioned while simultane. ouslydecouplingthe,system Also if f m di, then 51 gives direct physical significance to the free Thus B is nonsingular and the system 64 can be de. parametersin F If f m di or n then i t may coupled I t can be shown that the elements. of must be,possible to,morethan m di,the such that. closed loop poles In this situation it is often advanta. geous to calculate C s1 A BF lBB l with f entries,in F remainingarbitrary The following examples. illustratetheseideasandsome,volved in their application. of the difficultiesin so t h a t f n 3 2 m,loop transfer matrix is given by. di Moreover the closed, Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. FALB AND WOLOVICH,DECOUPLING,MULTIVARIABLE,IN CONTROL. SYSTEMS 657, C d A BF BB thesystem described by 60 withthemostgeneral. H given by,S S 1 s j2r,S2 fi2f3 S fi f2,f12 3 S fll 2. so t h a t all of the closed loop poles can be specified by Since det sI A BHC s 1 hz2 s3 h11 output. application of Theorem 2 However notethatappli feedback will not be adequate to stabilize the system. cation of thesynthesisprocedure in this casewould although state variable feedback provides a higher de. allow one to specify only two of the three closed loop gree of flexibility 63. I I DECOCPLING FEEDBACK,Consider the system described by 64 I t has been. Since output feedback is only a special case of state shown in 6 7 that state variable feedback can be used. variable feedback i e to decouple the system while simultaneously specifying. u Hy Go HCx Go 68 all three closed looppoles Application of Theorem 2. and 39 imply that any2 x 2 matrix H of the form, with H C replacing F i t follows immediately that 1. can be decoupled using output feedback if and only if 74. 1 B is nonsingular and 2 there is an m X m matrix H. such that rank Q HC 1 for i 1 m These, conditions provide a suitable test for whether or not a will define anoutputfeedback n hichdecouplesthis. system can be decoupled using output feedback system From 7 4 it follows t h a t. det sl A BHC s l h s2 h 3 s hll 2, and hence that the system can be stabilized using out. put feedback e g hZ2 1 kll 5 although the,poles are not completely arbitrary. is nonsingular so that the systemdefined by 69 can be. decoupled However i t is not possible to decouple this. system using output feedback T o see this observe that Then. Theorem 2 and 39 imply that an F which decouples,must be of the form B CB 1 0. m ai 2 3 n,a n d t h a t H C m u s tbe of the form. I t can be shown using Theorem 2 that any decoupling. F is of the form,Equations 71 and 7 2 lead to the contradictory re. quirement that flz 0 and flz 1 This example illus, trates the point that decoupling by state variable feed. back need not imply decoupling by output feedback,I t should be noted that although a system may be. decoupled using output feedback someof the flexibility. of specifying closed loop poles as with state variable. feedback wil1 in general be lost For example consider. Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. 658 AUTOMATIC,TRAiiSACTIONS,ON IEEE CONTROL DECEMBER 1967. so t h a t p 2 i e only two of the closed loop poles can In this example. be specified I t can also be shown for this example that. output feedback leads to the transfer matrix, 1 1 79 and is nonsingular since it is assumedZcsl f nfXcr. so that output and state variable feedback are equiva 3. lent This as previous examples illustrate is not true rz 4 di 6. in general, and hence all six of the closed loop poles can be arbi. VII A PRACTICAL, EXAMPLE trarily specifiedwhile simultaneouslydecouplingthis. An area in which decoupling techniques may be of system I t can be shown using Theorem 2 that a de. interest is the design of flightcontrolandstability coupling F has 6 Le f 6 free parameters Thus the. augmentation systems Consider for example the fol synthesis procedure Section 1 can be directly applied. lowing linearized longitudinal equations of motion for to give physical significance to these free parameters. a lift fan V STOL vehicle in a hovering condition Forexample supposethatindependentpitch trans. lation and altitude control is desired i e,X X e O 0 0 0. 0 0 1 0 0 0,AX rzzoAx l n2 Af 4 u2,Mu 0 Me l t i 0 0. 0 0 Z e Z O 0,1Z3 h n 3 W3 83, 1 0 0 0 0 0 According to the synthesis procedure F can be set equal. 0 0 0 1 0 0,I t can be shown t h a t for this decoupling F. zcs n2 WZ20,0 0 0 1 0 0 0,24 incrementallongitudinal x velocity change. 8 incremental pitch angle 0 0 0 1 0 0,e pitch rate. If G is now set equal to B l the closed loop transfer. w incremental vertical 2 velocity change matrix is. Ax incremental position error,AZ incremental altitude error C d A BF BB. 6 incremental collective fan input,incremental nose fan input. 6 incremental fan stagger input, The relevant outputs in this example are 8 A x and. Az and the subscripted capitals e g X are the rele. vant stability derivatives,S2 22 71s 20 s2 m31s 230. S 17t11S Z10 m31S 30,0 s2 nzllS nzlo s2 p S m20,s2 z11s 10 s2 z21 1n20 s2 m31s 30. If the m i i are suitably chosen then in effect the pilot. The output matrix C is thus defined as, will be faced with the task of controlling three highly. 0 1 0 0 0 0 stable second order systems This example serves only. to indicate a potential practical area of application for. c 0 0 0 0 1 0,the ideas presented in this paper, o o o o o l l 81 Theaboveexamplesillustratethetechniquesde. velopedfor,synthesizing decoupling,controllers,1 Similar to the XV 5 4 multivariable systems. Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply. F 4LB AND WOLOVICH,DECOUPLING IN MULTIVARIABLE,SYSTEMS 659. VIII CONCLCSIONS 4ssociateProfessorin theDe, Theproblem of decoupling a time invariantlinear partment of Aeronautics and. systemusingstatevariablefeedbackhas been con Astronautics at Stanford Uni. sidered Necessaryandsufficientconditionsforde versity Stanford Calif In 1966. couplinghavebeendetermined in terms of the non hewas Associate Professor of. Information and Control a t t h e, singularity of a matrix B T h e class of all feedback. University of Riichigan Ann Ar, matrices n hich decouple a system has been character. ized and a synthesis technique for the realizationof de bor CurrentlyheisXssociate. closed loop, pole configurations has been de Professor of Applied 3lathe. veloped In essence the major theoretical questions re matics a t Brown University. lating to decoupling via state variable feedback have Providence R I and a Consultant for XASAAand Bolt. been resolved for time invariant linear systems Beranek and Newman Inc His research interests are. 4 number of interesting potential areas of future re in control and applied mathematics and he is coauthor. search arise from the results obtained here In particu of the book Optiwzal Control McGraw Hill 1966 Heis. lar the question of extending the theory to the time a Member of the Editorial Board of the S I A X J o u r m l. varying situation is of considerable interest Some pre on Control. liminary results relating to stabilization have already Dr Falb is a member of Phi Beta Kappa Sigma Xi. beenobtained r61 T h e design of aircraft and V STOL the American Mathematical Society and the Society for. stabilityaugmentationsystemsviadecouplingtech Industrial and Applied Mathematics. niques is a potential practical areaof application asmas. mentioned in Section VII Practical implementation of. the techniques presented in this paper has begun but William A Wolovich 31 64 was. muchremainstobedonebeforethetheory is trans bornin. Hartford Conn on, formed into a practical design technique October15 1937 H e received. REFERENCES theB S E E degree fromthe,University of Connecticut. 111 B S Morgan The synthesis of linear multivariable systems. bystatevariablefeedback Preprints JointAutomaticControl Storrs in 1959 and the M S E E. Conf Stanford Calif June 1964 pp 468472 Also I E E E T a n s degreefrom. Worcester Poly, dzctonzatic Corttrol vol AC 9 pp 405 411 October 1964. r21 Z V Rekasius Decoupling of multivariablesystemsby technic. means of state variable feedback Proc 3rd 4nn Allerton Covf on 3 lass in 1961. Circui and System TJzeoFy Urbana Ill 1965 pp 439 447. P L Falb and 11 A 37010vich On the decoupling of multi From 1959 to 1961 he was with. variable systems Preprints Joint Automatic Control Conf Phila the Research Laboratories of the United Aircraft Corp. delphia Pa June 1967 pp 791 796, I L A Zadeh and C A Desoer L h z e a S s l e mTheory New H e served as Ground Electronics Officer in the U S. York McGraw Hill 1963 Air Forcefrom1961to 1964 In 1964 hejoinedthe. P I E Seckel Stability and Con trol of A i r p l a m s and Helicopters. New York AcademicPress 1964 staff of the Electronics Research Center in the Control. 1 A A olovich On the stabilization of controllable systems andInformationSystemsLaboratory Hehastaken. IEEE Tfans Azrtonzdu Confrol to be p u b l i s k d. graduate courses attheMassachusettsInstitute of,Technology Cambridge and Northeastern University. Peter L Falb M 64 was born in New York N ET on Boston Mass and he is currently a graduate student. July 26 1936 H e received t h e A B M A and Ph D in the Department of Engineering Brown University. degrees all in mathematics from Harvard University Providence R I working in the area of dynamic sys. Cambridge lass in1956 1957 and 1961 respectively tems theory. From 1960 t o 1965 he was at the M I T Lincoln Lab Mr Wolovich is a member of Tau Beta Pi and Eta. oratory Lexington Mass In 1965 hewasa Visiting Kappa Nu. Authorized licensed use limited to Rensselaer Polytechnic Institute Downloaded on October 9 2009 at 00 34 from IEEE Xplore Restrictions apply.

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