How to Obtain Desirable Transfer Functions in MIMO Systems Under Internal Stability Using Open and Closed Loop Control

Transcription

1 How to Obtain Desiable ansfe Functions in MIMO Sstems Unde Intenal Stabilit Using Open and losed Loop ontol echnical Repot of the ISIS Goup at the Univesit of Note Dame ISIS June, 03 Panos J. Antsaklis and Elo Gacia Depatment of Electical Engineeing Univesit of Note Dame Note Dame, IN Intedisciplina Studies in Intelligent Sstems-echnical Repot

2 How to Obtain Desiable ansfe Functions in MIMO Sstems Unde Intenal Stabilit Using Open and losed Loop ontol Panos J. Antsaklis and Elo Gacia Abstact his pape summaizes impotant esults useful fo contolle design of multi-input multi-output sstems. he main goal is to chaacteize the stabilizing contolles that povide a desied esponse. I. INRODUION he aim of this bief pape is to use esults fom [] to show how in Multi-Input Multi-Output (MIMO) sstems, using -degees of feedom compensation and existing techniques, the attainable esponse unde intenal stabilit can be completel chaacteized. hese include the model matching poblem, the invese poblem, the diagonal and block diagonal decoupling poblems, and the static decoupling poblem among othes. he equiement fo stabilit esticts the design when the given sstem is non-minimum phase. Specificall, the compensated sstem tansfe function matix must have among its zeos all the unstable zeos of the given plant, togethe with thei associated diections. In addition, the equiement fo causalit does not allow the compensated sstem tansfe function to have elative degee lowe than the given plant s. his pape also shows constuctivel how to design feedback and feed-fowad -degees of feedom contolles that attain the desiable esponses unde intenal stabilit. In fact all such contolles will be paameticall chaacteized. It is then up to the designe to decide how much of the needed contol should be implemented via feedback compensation and how much via feedfowad compensation. his decision depends on the specifications on feedback popeties such as sensitivit to plant paamete vaiations, distubance eduction, noise attenuation etc. In man cases, the contol achitectue ma not be a full -degees of feedom achitectue but ma be esticted. Fo example it could be the ve common in pactice state feedback achitectue, state feedback with obseve, unit output feedback achitectue etc. he eect of such estictions on the attainable esponses is examined and it is shown in each case how to deive the appopiate contolles.

3 Special attention is being paid to the special case of the tansfe function of the plant P(s) being squae and non-singula and it is shown how expessions ae simplified; note that this case includes the SISO case. Examples ae used thoughout. In the following section, Section II, two basic esults ae fist pesented. he fist chaacteizes all esponses attainable using a -degees of feedom contolle unde intenal stabilit. he second esult paameticall chaacteizes all stabilizing -degees of feedom contolles, which attain the desied esponse, showing explicitl the intepla of feedfowad and feedback contol actions. Achitectues that ealize and implement the contol policies ae also shown. In Section III seveal design poblems ae discussed including the decoupling and invese poblems. he eect of seveal moe esticted common contol achitectues on the sstem esponse is discussed in Section IV. In Section V we stud the special case of the plant tansfe function matix being squae and nonsingula which includes the SISO case and deive esults fo the case when the denominato of the compensated tansfe function is given. II. DESIGN OF ONROL SYSEMS FOR DESIRED RESPONSE onside the MIMO plant with p x m tansfe function matix P = ND ( = Pu), whee N, D ae ight copime polnomial matices. (he esults in this pape ae based on [], pp ). Suppose we want to contol P so that: = u M whee, M ae pope and stable desied tansfe function matices and is an extenal input. Basic Result I. It has been shown in [] (h. 4.3 p. 67; also h. 4.4 fo pope stable factoizations) that the above and M can be ealized via a geneal -degees of feedom contolle with intenal stabilit if onl if thee exist stable ational matix X so that: N = X. M D In [], Fig. 7. p. 67, shows a wa to ealize such and M. See Fig. below whee is an feedback stabilizing contolle fo P.

4 Fig.. Fig.. Basic Result II. he geneal -degee of feedom configuation is shown in Fig., whee u = =, (see [] p. 63 Fig. 7.0 and h. 4.), with an feedback stabilizing contolle ( u = intenall stabilizes = Pu) and such that X = D ( I P) stable o = ( I P) DX = ( I PM ). Hee = = NX and u = M = DX. Feedback ontolle. all stabilizing contolles ([] h. 4. p. 64): can be an stabilizing contolle. o see the flexibilit we chaacteize [ ] [ ] =, = ( I + QP) Q, M = ( I + LN) D L, X whee Q = DL, M = DX ae pope with L, X, exists and is pope). D ( I + QP) = ( I + LN) D stable (also ( I + QP) = X, KN ( X + KD), X whee K, X stable; K is the Youla paamete. Othe expessions ma be found in [] h. 4. p.64. 3

5 III. RESPONSE. In ou case X is given (o pat of X is specified). Note that, M cannot be chosen independentl fo a solution to () to exist, but the must be in the ange of N D, o equivalentl, ank N D M = ank N = m D, full column ank. he also must be stable, which is to be expected. So, the onl significant estictions imposed (othe than popeness) ae due to the unstable zeos of P, (in N), if an. Since = NX and, X stable, must contain all unstable zeos of the plant P. ( s )( s+ ) he example 4. in [], on p. 68 whee P = ( s ) illustates all these points. It is also shown in the next section what additional estictions ae imposed if instead of a geneal - degees of feedom configuation the moe esticted unit feedback is used. ( s )( s+ ) Example. We conside P = ( s ) and wish to chaacteize all pope and stable tansfe functions (s) that can be ealized b means of some contol configuation with intenal stabilit. Let ( s) ( s ) P = = N' D' ( s+ ) ( s+ ) all such must satisf be an c MFD in RH. hen in view of [], heoem 4.4 on p. 67, ( s + ) N' = = X ' RH ( s ) heefoe, an pope with a zeo at + can be ealized via a -degees of feedom feedback contolle with intenal stabilit. Diagonal Decoupling. Hee the desied tansfe function (s) is not completel specified but is equied to be diagonal, pope and stable. hat is N ' X ' = diag( n / d ) i i whee N', X ' ae pope and stable. If P exists (see also the last section of this pape) then X N diag n d ' = ' ( i / i). If P(s) has onl stable zeos then no additional estictions ae imposed on (s). When diagonal decoupling is accomplished using moe esticted contol configuations (e.g. unit feedback) additional estictions ae imposed on X ' due to the contol achitectue (see next section). 4

6 When linea state feedback u = Fx+ G is used fo diagonal decoupling = N D G M = D D G and ' ' f, ' ' f, X ' = D G. ' f So, when P exists X D G N diag n d ' ' = f = ' ( i / i) fo a solution to exist (see Execise 4.7, 4.0, hapte 4 of []). Invese poblem. In this case (s)=i and so = I = N' X ' which gives the conditions on the stable and pope paamete hapte 4 of []). X '(see also Execise 4.7, 4.8 in Static decoupling. Hee (s) is squae, pope and stable and (s)=λ, a eal nonsingula diagonal matix, so that a step change in the fist input will onl aect the fist output at stead state. Fom = N' X ' in view of (s)=λ, a necessa and suicient condition is that P(s) does not have a zeo at the oigin (so det N '(0) 0). hen '( ) X s must satisf X '( s) ( N' ( s) ) = Λ. his is also suicient and static decoupling in this case can be achieved with just pe-compensation b a eal gain matix his can be seen fom the expession in Section II fo fom which = P () s Λ. = ( I P) D' X ' = D' X ' ( = 0) s = D s X s = P s Λ () '() '() (). IV. RESRIED ONROLLER ONFIGURAION. If the configuation in Fig. 3 ([], Fig. 7.3 p. 69) is chosen, then ' ' u =, =, R = D ' c N, N a lc factoization is RH (pope and stable). 5

7 Fig. 3 Fig. 4 hen, if =, is given one could choose ' R = N, =, ' D c ' N. = Note that R, ae stable; exists and it is stable. In tems of the pope and stable paametes X ', K ' the can be elated as R = X ', X K N = ( ', ' '), = ( X ' + K' N '). If the configuation in Fig. 4 (unit feedback) is chosen, ([], Fig. 7.5 p. 63), then i.e. = and in view of (4.58) (o 4.6) Hee (fom 4.58) u =, =, L = X ( L' = X ') o ( Q = M). 6

8 = ( I + MP) M = M( I + PM) M I I XN D X = ( + ) = ( + ) whee D ( I + MP) = ( I + XN) D stable which imposes a estiction on the allowed X (o X ). onside again the example above, if a single degee of feedom contolle must be used, the class of ealizable (s) unde intenal stabilit is esticted. In paticula, if the unit feedback configuation { I;, } again given b I in Fig. 4 is used, then all pope and stable that ae ealizable unde intenal stabilit ae whee X ' = L' RH and in addition ( s ) = N' X ' = X ' ( s + ) I + X N D = + X ( s ) ( s+ ) RH ( ' ') ' ' ( ) ( ) s+ s i.e., X ' = n / d is pope and stable and should also satisf x x s+ d + s n = s p s ( ) x ( ) x ( ) ( ) fo some polnomial p(s). his illustates the estictions imposed b the unit feedback contolle, as opposed to a -degees of feedom contolle. Notice that these additional estictions ae imposed because the given plant has unstable poles. he unit feedback case is studied in [] Poblem 7.3 p V. P EXISS. Example. Next it is shown how the expessions ae simplified when Poblem 7.3 pat (c) p In the unit feedback configuation when onl if: is stable and if is chosen so that P P exists. his is fom [] exists the closed loop sstem is intenall stable if and N ( I ) P, N 7

9 = NX, X stable, then this is the same as ( I ) + XN D stable (in Fig. 7.9 the sign is dieent). In the Single-Input Single-Output (SISO) case, fom (d) p. 643 when P and ae scala, the conditions become ( ) d = Sd and n stable. he contolle Applications of esults. is still given b the fomula ( ). = I + XN D X Suppose then that the desied stable tansfe function matix needs to have onl some desied denominato. In = ND, completel, like in decoupling, ae biefl discussed). Fom = NX, X stable implies that we can select exist whee fo example X D is given (note that in [] p. 633, poblems whee is not specified N = NN. = ND x and X X N = D, assuming that Fom the pevious discussion, fo stabilit (in unit feedback) we need = N ; othe options X, D, ( I ) + XN D stable, o D ( D + N) D stable and since D is assumed stable, the onl condition is ( D ) + N D stable and the contolle is given b onside now Fig. 4 and veif esponse = ( + ) I D N D D = ( D ( D + N) D ) D = DD ( + N) DD = DD + ( N). = P + = I P P ( ) ( ). Hee P= ND, = D( D + N) 8

10 Fig. 5. = P ( I P ) = ND D( D + N) I N D = N ( D + N) ( D + N) = ND [ ] D( D + N) D + N N as expected. Note that hee + P = D N + DN = ( D + D) N = + = ( D D) D DN P. If the configuation in Fig. 5 is chosen (see also [], Fig. 7.8 p. 63), then u =, =, I and then (4.58) p.64 whee L is esticted to satisf = ( I + LN) D L X = I + LN D XD= I + LN ( ) ( ). In the special case when P exists ( exists) X ( XD I) N X L = =. So the stabilit condition is just X must be stable. Hee, 9

11 = ( I P ) P = P( I P) ND I I LN D LND = ( ) [( + ) ] = ( ND ) (( I + LN) D ) [( I + LN ) LN = ( ND ) D ( ) I + LN D = NX X ] D as desied. Assuming P and exist, when X = stable fom pevious page, fo intenal stabilit D = ( D ) D D I N = D D [ D D ] N ( ). = D D N leal, the closed loop tansfe function with as above as expected. = P( I P) = ( ND ) I ( D D ) N N D = ( ND ) D D D = ND + D So given NX ND = = stable ( D desied) select = ( D D ) N and onl wo about popeness. Note that hee P = D N DN = ( D D) N =. VI. ONLUDING REMARKS Seveal esults fom [] on -degee of feedom contolles wee used to solve a set of poblems whee the denominato of the desied tansfe function is given. he contol design poblems whee o pats of it ae specified can be addessed in simila wa. Dieent contol configuations can also be studied; fo example, in the linea stable feedback M = DD and X = D. 0

12 REFERENES [] P. J. Antsaklis and A. N. Michel, Linea Sstems, Spinge/ Bikhause, 006.