Distributed Control of Linear Multi-Channel Systems

Distrib uted Contr ol of Linear Multi-Channel Systems L. W ang, D. Fu llmer , F . Liu, a nd A. S. Morse Abstract — A solution is given to the basic distributed feed- back control pr oblem f or a multi-channel linear system as- suming only that the system is jointly contr ollable, jointly observ able and h as an associated neighbor graph which is strongly connected. The soluti on is an observer -based control system which is implemented in a distributed manner . Usin g these ideas, a solution is also given to the d istributed set- point control problem fo r a mu l ti-channel linear system in which each and e very agent with access to the system is able to independently adjust its controlled out p ut to any desired set-point value. An example is given to b riefly illustrate h ow network tra nsmission delays mig ht be dealt with. I . I N T RO D U C T I O N A well-k n own application of an observer is to ser ve as a comp onent of a feed back control system fo r regulating a dyn amical process. In particular, for a contro llable and observable linear sy stem y = C x , ˙ x = Ax + B u , su c h a feedback contro l is of the fo rm u = F ˆ x where ˆ x is the state of the observer ˙ ˆ x = ( A + K C ) ˆ x − K y + F ˆ x , an d F an d K are matrices wh ich are u sually chosen so that A + B F and A + K C ar e at least stability matrices. As is also well known, the resulting clo sed-loop system can be described by the equation s ˙ x = ( A + B F ) x + B F e (1) ˙ e = ( A + K C ) e (2) where e is the state estimation error e = ˆ x − x . As is plainly clear , the utility of this ap proach is in no small par t due to th e fact that the error d ynamics d escribed by (2) is an un for ced linear differential eq u ation. Arriving at th is necessitates including in the dif ferential equation d efining the observer, the term F ˆ x . While this is a perfe c tly valid step fo r the centralized ob server un d er d iscussion, an analogo us step for the observer-based distrib uted contro l o f a multi-cha nnel linear system usually cannot b e ca rried without violating distributional requ irements. The primary aim o f this pa p er is to explain how to overcome this difficulty an d in so doing, to p rovide what is a lmost cer tainly the first sy stematic proced u re for con structing a d istributed feedback contr ol for stabilizing or otherwise regulating a multi-chann el linear system. A full-le ngth version of this paper includi ng proofs is curre ntly posted on the archi ve (http:/ /arxi v . org/a bs/1909.11823 ) and will be submitted for journal publica tion in the near future. L. W ang, D. Fullmer , F . Liu, and A. S. Morse are with the Departmen t of Electri cal E ngineer ing, Y ale Uni versity { lili.wang, daniel.ful lmer, fengjiao.liu, as.morse@yale .edu } . This research was s upporte d by A F O S R, AR O, NSF I I . T H E P RO B L E M Perhaps the most basic pro b lem in distributed feedb a ck control is to d evelop a proce d ure which can enable a net- worked family of m > 1 agents to stab ilize or otherwise control in a distributed mann er , a phy sical pr ocess P mod- elled by a multi-c h annel, time-inv ar iant, linear system. By an n -dim e nsional, m - cha n nel linear system is mean t a linear system of the form ˙ x = Ax + m X i =1 B i u i y i = C i x, i ∈ m (3) where, n and m are positive integers, m = { 1 , 2 , . . . , m } , x ∈ I R n , and for i ∈ m , u i ∈ I R m i and y i ∈ I R p i are the contro l in put to chan nel i and the measured outpu t f rom channel i respectively . Here A , the B i , and the C i are real- valued, constan t ma tr ices o f appro priate sizes. W ithout any real loss of g e nerality it is assumed that the system defined by (3 ) is both jointly controllab le and jointly observable; that is, the matrix pairs ( A,  B 1 B 2 . . . B m  ) and           C 1 C 2 . . . C m      , A      are contro llable and observable respectively . For simp licity it is further assumed that B i 6 = 0 , C i 6 = 0 , i ∈ m . It is presumed th at th e system d escribed by (3) is to be contro lled b y m age nts with the understand ing tha t for i ∈ m , agent i can m easure the y i and has access to the control inp ut u i . I n addition, eac h agent i can receiv e informa tio n from its “neighbo rs” wh ere exactly who each agent’ s neighb ors are is specified in th e problem formula tio n. In this paper it is assum ed th at each age n t’ s neig hbors do not change with time, that N i ∆ = { j i 1 , j i 2 , . . . , j i k i } is the set of labels of age nt i ’ s neigh b ors inc lu ding itself, and that each agent can rec ei ve the curren t state of each o f its neighb or’ s controller s. Ne ighbor relations ca n b e conveniently describe d by a dir ected gra p h N defined on m vertices with a direct arc from vertex j to vertex i ju st in case ag ent j is a neighbor of agent i . It is assum ed throug hout this paper that N is stro ngly connected . It is stra ig htforward to extend what follows to the general case when N is only w e a kly connected. The ba sic distributed control pr oblem for th e m channel system (3) is to develop a systematic procedu re fo r construct- ing m linear time- invariant feedb ack contr o ls, one for each channel, so that th e state of the resulting clo sed-loop system conv erges to zero exponentially fast at a pre-assigned rate. Before addressing this pr o blem, it will be useful to b riefly revie w th e main results fr o m classical d ecentralized contr ol [1], [2]. A. Decen tralized Contr ol The classical d ecentralized co ntrol pro blem for an m - channel linear system is exactly th e same as the distributed control pro blem just formulated, except for o ne impo r tant difference. In the case of decen tralized contr o l, there is no commun ication between agen ts so the only signal available to each agent i is y i . The fundam ental decentralized contr ol question is th is. Und er wha t co nditions do there exist local linear, time-inv ar iant contro llers, o n e for e ach channel, which stabilize P ? In answering th is qu estion it was shown in [1] that no matter wh at the local co ntrollers ar e, the spectr um of the resulting closed-loo p system contains a un iquely determined subset of eigen values which re main unchang ed no matter wh ich loc a l c o ntrollers ar e applied . Th is is the fixed spectrum 1 of P [1 ], [3 ] . Decentr alized stabilizatio n of P by time in variant lin ear con trols thus dem ands that its fixed spectrum contain on ly o pen left h alf plane eig en values. This con dition on th e fixed spectr um of P is ne c e ssary and sufficient for stabiliza tio n with dece ntralized con trol [1]. In addition, it is known that that the necessary and sufficient condition for the closed-loo p spec trum to be freely assignab le with decentra lize d contro l is that P has no fixed eigenv alu e s [2]. The pr eceding pr ompts the following q uestion. Does the distributed control pr o blem formula ted at the beginnin g of this section have a fixed-spec trum constraint analogous to the fixed spectrum c o nstraint encoun tered in the decen tralized control problem? T he findings of this paper establish that it does not. This will be ac complished by explaining how to construct a distributed observer-based control system which solves the distributed sp e c trum assignmen t p r oblem fo r the multi-chan nel system described by ( 3). W e begin with a brief revie w of distributed observers. I I I . D I S T R I B U T E D O BS E R V E R In a series of paper s [4 ]–[13] , a variety of distributed observers have been prop osed for estimating the state o f (3) assuming all o f the u i = 0 . The distributed observer studied in [ 6] will be u sed in this p aper . It is described by the equation s ˙ x i = ( A + K i C i ) x i − K i y i + X j ∈N i H ij ( x i − x j ) + δ iq ¯ C z , i ∈ m (4) ˙ z = ¯ Az + ¯ K C q x q − ¯ K y q + X j ∈N q ¯ H j ( x q − x j ) (5) 1 Referre d to as “fixed modes” in [1] where all x i ∈ I R n , z ∈ I R m − 1 , q ∈ m , δ iq is the Kronecker de lta, and the K i , H ij , ¯ A, ¯ K , ¯ H j , ¯ C are matrices of a pprop r iate sizes. Th e subsy stem consisting of (5) and the signal δ iq ¯ C z is ca lled a channel contr oller of (4). I ts function will be explained in the sequel. The erro r dynamics for this o bserver are described b y the equations ˙ e i = ( A + K i C i ) e i + X j ∈N i H ij ( e i − e j ) + δ iq ¯ C z (6) ˙ z = ¯ Az + ¯ K C q e q + X j ∈N q ¯ H j ( e q − e j ) (7) where for i ∈ m , e i is the i th state estimatio n error e i = x i − x . No te th a t (6), (7) is an ( mn + m − 1) - dimensiona l, unfo rced linear system. It is known that its spectrum can b e f reely assigned by ap propr iately p ick ing the matrices K i , H ij , ¯ A, ¯ K , ¯ H j , ¯ C [6]. T hus by so choosing these matrices, all of the e i and z can be made to converge to zero exponentially fast at a pre-assigned rate. There are se veral step s inv olved in pic k ing these m a trices. First q is chosen; a ny value of q ∈ m su ffices. The n ext step is to temporarily ig nore the channel co ntroller (7) a nd to ch oose matrices ˜ K i and the ˜ H ij so that th e open- loop err or system ˙ e i = ( A + ˜ K i C i ) e i + X j ∈N i ˜ H ij ( e i − e j ) + δ iq ˜ u q , (8) i ∈ m , is contro llable by ˜ u q and observable throug h ˜ y q =      C q e q e q − e j q 1 . . . e q − e j q k q      (9) where { j q 1 , j q 2 , . . . , j q k q } = N q . In fact, the set of ˜ K i , ˜ H ij , j ∈ N i for which the se pr operties h o ld is the compleme n t of a proper algebr aic set in the linea r space of all such matrice s [6 ]. Thus almost any choice for these matrices will accomplish the desired objectiv e. The next step is to pick matrices ¯ A , ¯ B , ¯ C and ¯ D so that so that the c lo sed loo p spectrum of the system consisting of (8), (9) and the channel controller ˜ u q = ¯ C z + ¯ D ˜ y q , ˙ z = ¯ Az + ¯ B ˜ y q has th e prescribed spec trum. On e tech nique for cho osing these matrices can be found in [1 4]. Of course since the system de fin ed by ( 8) and (9) is c o ntrollable and observable, there are many ways to define a ch annel controller and thus the m atrices ¯ A, ¯ B , ¯ C , and ¯ D . In any event, once th ese matrices are chosen, the K i and H ij are defined so that for all i 6 = q , K i ∆ = ˜ K i and H ij ∆ = ˜ H ij , j ∈ N q , while for i = q , K q ∆ = ˜ K q + ˆ K q and H qj ∆ = ˜ H qj + ˆ H qj , j ∈ N q , where h ˆ K q ˆ H qj q 1 · · · ˆ H qj q k q i = ¯ D . Finally ¯ K q and the ¯ H j , j ∈ N q are defined so th at h ¯ K q ¯ H j q 1 · · · ¯ H j q k q i = ¯ B . I V . D I S T R I B U T E D - O B S E RV E R B A S E D C O N T R O L The first step in the development of a distributed obser ver based feed back system f or ( 3) is to devise state feed b ack laws u i = F i x, i ∈ m , w h ich en dow the closed loop system ˙ x = A + m X i =1 B i F i ! x (10) with pr escribed pr operties such as stability and/or op timality with respect to som e per formanc e ind ex. In accord ance with certainty equiv alence, the next step is to imp lement in stead of state feedback laws u i = F i x, i ∈ m , the distributed feedback laws u i = F i x i , i ∈ m , wher e x i is agent i ’ s estimate of x g enerated b y a distributed o bserver . Doing this results in the system ˙ x = Ax + m X i =1 B i F i x i (11) instead of(10). A system which provide s the required estimates x i of x is ˙ x i = ( A + K i C i ) x i − K i y i + X j ∈N i H ij ( x i − x j ) + δ iq ¯ C z + m X j =1 B j F j x j i ∈ m (12) ˙ z = ¯ Az + ¯ K C q x q − ¯ K y q + X j ∈N q ¯ H j ( x q − x j ) (13) since, in this case the associated e r ror system is exactly the same a s before wh en there was no feedba ck to the process to accou nt for . Unfor tu nately this system cannot be used without v iolating the p roblem assumptions sinc e the implementatio n of ( 12) r equires each ag ent to use the state estimates of tho se agen ts which are not its n eighbor s. An alternative system wh ich is imple m entable without violating problem assumptions is the m odified distributed state esti- mator ˙ x i = ( A + K i C i ) x i − K i y i + X j ∈N i H ij ( x i − x j ) + δ iq ¯ C z +   m X j =1 B j F j   x i , i ∈ m (14) ˙ z = ¯ Az + ¯ K C q x q − ¯ K y q + X j ∈N q ¯ H j ( x q − x j ) (15) In the sequel it will b e shown that even with th is modifica- tion, this system can still p rovide th e requir ed estimates o f x . The error dynam ics for (14), (15) are described by the linear system ˙ e i = ( A + K i C i ) e i + X j ∈N i H ij ( e i − e j ) + δ iq ¯ C z + m X j =1 B j F j ( e i − e j ) , i ∈ m (16) ˙ z = ¯ Az + ¯ K C q e q + X j ∈N q ¯ H j ( e q − e j ) (17) while the pro cess dynamics modelled by (11) can be r ewrit- ten as ˙ x = A + m X i =1 B i F i ! x + m X i =1 B i F i e i (18) Since (16), (1 7) is a n unforced linear system, its d y namic behavior is determin ed primarily by its spectru m. In th e sequel it will be explain ed how to cho ose the K i , H ij , ¯ K i and ¯ H i so that the spectru m of (16) and (17) coin cides with a prescr ibed symmetric set of comp lex nu mbers. T o achiev e this, attention will first b e focu sed on the proper ties of the open-lo op erro r system descr ibed b y ˙ e i = ( A + K i C i ) e i + X j ∈N i H ij ( e i − e j ) + m X j =1 B j F j ( e i − e j ) + δ iq ˜ u q , i ∈ m (19) and (9). This system is what wh a t r e sults when the ch annel controller appearing in (17) is removed. The main techn ical result of this paper is as follows. Pr opo sition 1: There are matr ices K i , H ij , j ∈ N i , i ∈ m such that for all q ∈ m , the open- loop erro r system described by (19) and (9) is ob servable throug h ˜ y q and controllab le by ˜ u q with controllab ility index m . The implication of this propo sition is clea r . Theor em 1: For any set of f e e dback matrices F i , i ∈ m , any integer q ∈ m , and any sym metric set of mn + m − 1 complex number s Λ , there are m atrices K i , ¯ K i , H ij , ¯ H i for which the spectru m of the closed-loop error system de fin ed by (16) and (17) is Λ . W e will now pro ceed to justify Prop osition 1. First no te that (19) can be written in the compact fo rm ˙ ǫ = ˜ A + m X i =1 ˜ B i ( K i ˆ C i + H i ˜ C i ) ! ǫ + ˜ B q ˜ u q (20) where ǫ = colu mn { e 1 , e 2 , . . . , e m } . Here ˜ A = I m × m ⊗ ( A + P m j =1 B j F j ) − Q w h ere ⊗ is the Kron ecker produ ct and Q is th e nm × nm partitioned m atrix o f m 2 square blo cks, whose ij th b lock is B j F j ; ˜ B i is the matrix ˜ B i = b i ⊗ I n × n , i ∈ m where b i is th e i th unit vector in I R m and ˆ C i = C i ˜ B ′ i ; H i is the matrix H i = h H ij i 1 H ij i 2 · · · H ij i k i i where { j i 1 , j i 2 , . . . , j i k i } = N i ; ˜ C i is the m atrix ˜ C i = column { C ij i 1 , C ij i 2 , . . . , C ij i k i } where C ij = c ij ⊗ I n × n , j ∈ N i , i ∈ m and c ij is the row in the tran sp ose of the incidence matrix of N correspon ding to the arc from j to i . Next obser ve that (20) is what re sults wh en the d istributed feedback law ˜ v i =  K i H i  ˜ y i + δ iq ˜ u q , i ∈ m , is app lied to the m channel linear sy stem ˙ ǫ = ˜ Aǫ + m X j =1 ˜ B j ˜ v j , ˜ y i =  ˆ C i ˜ C i  ǫ, i ∈ m (21) The proof of Pro position 1 depends o n the follo wing lemmas. Lemma 1: The m -ch annel linear system described by (21) is jointly controllable and jointly observable. Proof of Lemma 1: In v iew o f the definition s of the ˜ B i it is clear that  ˜ B 1 ˜ B 2 · · · ˜ B m  is the nm × nm identity . Therefo re that (21) is jointly contro llable. T o estab lish joint observability , suppose that ˜ v is an eigenv ector of ˜ A for which  ˆ C i ˜ C i  ˜ v = 0 , i ∈ m From the relations ˜ C i ˜ v = 0 , i ∈ m , the definitions of the ˜ C i and the assumption that N is stro n gly conn ected it follows that ˜ v = colu mn { v , v, . . . , v } f or some vecto r v ∈ I R n . Meanwh ile f rom th e r elations ˆ C i ˜ v = 0 , i ∈ m a n d the definitions of the ˆ C i it follows that C i v = 0 , i ∈ m . Moreover from th e definition of ˜ A a nd the stru cture of ˜ v it is clear that ˜ A ˜ v = ( I m × m ⊗ A ) ˜ v = colu mn { Av, Av, . . . , Av } . This and the hypothe sis that ˜ v is an eigenvector of ˜ A imp ly that v must be an eigenv ector o f A . But this is impossible because of jo int ob servability of (3) and the fact that C i v = 0 , i ∈ m . Thus (21) has no u nobservable m odes throug h the combined outputs ˜ y i , i ∈ m which means th at the system is jointly observable. Lemma 2: For any g i ven set of app r opriately sized matr i- ces K i , i ∈ m , th e r e exist matrices H i for wh ich the ma- trix pair  ˜ A + P m i =1 ˜ B i ( K i ˆ C i + H i ˜ C i ) , ˜ B q  is controllable with controllab ility index m fo r e very q ∈ m . The proof depends on Lemma 1 an d the following facts. Lemma 3: Ther e are ma trices ˆ H i , i ∈ m , for which  P m i =1 ˜ B i ˆ H i ˜ C i , ˜ B q  is a controllable pair with co ntrolla- bility index m f or every choice of q ∈ m . A proof of this lemma will b e given below . Lemma 4: Fix q ∈ m and let b q denote th e q th unit vector R m . T here exists a m atrix G =  g ij  ∈ R m × m with row sums all equal zero and g ij = 0 whenever a g ent j is not a neighbo r of agent i such that ( G, b q ) is a controllable pair . Proof of Lemma 4: Hautus’ s lemma [15] a ssur es that th e pair ( G, b q ) is contro llable if and only if for each eigenv alue s of G ,  G − sI b q  (22) has fu ll rank. This is eq u iv alent to showing th at if G ′ x = sx , and b ′ q x = 0 then x = 0 . Since N is stro ngly conn ected, there exists a directed spanning tree of N who se root is vertex q with all arcs oriented away fro m q , which we denote using T q . Since T q is a directed tre e, each vertex i ∈ m has a set o f out-neigh b ors C i ⊂ m , and each vertex i 6 = q has a uniq ue in-neig hbor ρ i ∈ m . Choose v ∈ R m so that v q = 0 and for each i 6 = q , v i is a d istinct non zero real value. By “distinct”, we requ ire that v i 6 = v j for any i 6 = j . Choose G so that, for each i, j ∈ m , g ij =      v i if i = j − v i if j = ρ i 0 otherwise (23) It is clear that g ij = 0 if j is not a n eighbo r of i as T q is a subgrap h of N , and g ii = − m P j =1 ,j 6 = i g ij . Now , sup pose G ′ x = sx and b ′ p x = 0 . From G ′ x = sx and (23), v i x i − X j ∈C i v j x j = sx i i ∈ m (24) written alternatively , ( v i − s ) x i = X j ∈C i v j x j i ∈ m (25) Additionally , since b ′ q x = 0 , x q = 0 . (26) Since eac h v i , i ∈ m is distinct, there is at most o ne i ∈ m with v i = s . If there is o ne, cho ose r ∈ m so that v r = s , otherwise, cho ose r ∈ m arbitr a rily . Since T q is a tree there m ust b e a u nique path f rom q to r , let P den ote the set of labels of vertices along that path, excludin g r , and for e a ch i ∈ P , let c i ∈ C i ∩ P denote the unique vertex along this path. For each nonnegative real nu mber d , let V d ⊂ m co nsist of the set o f vertices o f depth d in T q . By indu c tion, we show that for each d ≥ 0 and i ∈ V d , x i =      x r if i = r v c i v i − s x c i if i ∈ P 0 otherwise (27) Suppose D is the maximum dep th of T q . No vertex i ∈ V D may have any children C i , since oth erwise it would not be a vertex of max imum depth. I f i = r th en (27) is clear ly true. Otherwise i 6 = r , and fro m (2 5), ( v i − s ) x i = 0 , i ∈ V D . But then ( v i − s ) 6 = 0 since ea c h v i is distinct and v i 6 = v r = s , so v i = 0 . Since V D ∩P = ∅ an d x i = 0 for ea c h i ∈ V D , i 6 = r , so (27) must hold for vertices i ∈ V D . Next, suppose for some 0 < d < D , (27) is true for vertices in V d . Supp ose also that i ∈ V d − 1 . A g ain, if i = r then (27) is clear ly true. Otherwise i 6 = r . No ting that ea c h j ∈ C i is in V d , the ind uctive hyp othesis ( 27) assures that for any j ∈ m with j 6 = r and j / ∈ P , v j = 0 . So, from (2 5), ( v i − s ) x i = v c i x c i (28) if i ∈ P and ( v i − s ) x i = 0 (29) otherwise. Note that v i − s 6 = 0 since only v r = s and i 6 = r . Thu s, (27) ho lds for vertices in V d − 1 as well. So b y induction , it hold s fo r each d ≥ 0 an d i ∈ V d . Repeated substitution of (27) alo ng the vertice s in the unique path from q to r reveals that x q = x r Y i ∈P v c i v i − s (30) Howe ver, it is already known that x q = 0 from ( 26). Since for each i ∈ P , v i − s 6 = 0 (since r / ∈ P ) , and v c i 6 = 0 (since only v q = 0 and fo r no i ∈ m does q = c i ), if follows that x r = 0 as well. From this, (26), and (27), it follows that for all i ∈ m , x i = 0 . Proof of Lemma 3: Let ˜ c i = co lumn { c ij i 1 , c ij i 2 , · · · , c ij i k i } . Thus ˜ C i = ˜ c i ⊗ I n × n . By Lemma 4 ther e are matrices h i such that ( P m i =1 b i h i ˜ c i , b q ) is controllab le with contro llability index m for every choice of q ∈ m . By definition ˜ B i = b i ⊗ I n × n , ˜ C i = ˜ c i ⊗ I n × n for i ∈ m . Choose ˆ H i = h i ⊗ I n × n . From this, m X i =1 ˜ B i ˆ H i ˜ C i = m X i =1 b i h i ˜ c i ! ⊗ I n × n Thus Q , h ˜ B i  P m i =1 ˜ B i ˆ H i ˜ C i  ˜ B i . . .  P m i =1 ˜ B i ˆ H i ˜ C i  m − 1 ˜ B i i = h b i  P m i =1 b i h i ˜ c i  b i . . .  P m i =1 b i h i ˜ c i  m − 1 b i i ⊗ I n × n Since ( P m i =1 b i h i ˜ c i , b q ) is controllab le with controllab ility index m for e very choice of q ∈ m , rank Q = mn . Theref ore for each q ∈ m ,  P m i =1 ˜ B i ˆ H i ˜ C i , ˜ B q  is a contr ollable pair with contro llab ility index at most m . On the oth er hand, n o te that the matrix Q has exactly nm co lumns, m is the smallest possible contr ollability index. Th us for each q ∈ m ,  P m i =1 ˜ B i ˆ H i ˜ C i , ˜ B q  is a controllable pair with controllab ility index m . Lemma 5: Let ( A n × n , B n × r ) be a real-valued c o ntrol- lable ma tr ix pair with contro llability index m . For each real matrix M n × n , th e m atrix pair ( M + g A, B ) is contr ollable with controllability in dex no greater than m f or all but at most a finite number of v alues of the real scalar gain g . Moreover if mr = n , th e n m is the contr o llability index o f ( M + g A, B ) for all but a finite nu mber of values of g . Proof of Lemma 5: The assumed properties of the pair ( A, B ) imply that mr ≥ n and that there must be a minor of o rder n of th e matrix  B AB · · · A m − 1 B  which is nonzer o . Let 1 , 2 , . . . q be a labeling of the n th order minors of  B AB · · · A m − 1 B  and supp ose that the k th such minor is n onzero . Let µ : I R n × n ⊕ I R n × r → I R deno te that function which assigns to any matr ix pair ( ¯ A n × n , ¯ B n × r ) , the value of the k th m inor of  ¯ B ¯ A ¯ B · · · ¯ A m − 1 ¯ B  . Thus µ ( A, B ) 6 = 0 and if ( ¯ A, ¯ B ) is a matrix p air f or which µ ( ¯ A, ¯ B ) 6 = 0 , then ( ¯ A, ¯ B ) is a contro llable pair w ith controllab ilty index no greater th an m . Since µ ( A, B ) 6 = 0 , it must be true that µ ( g A, g B ) 6 = 0 provided g 6 = 0 . Note that µ ( λM + A, B ) is a polyno mial in the scalar variable λ . Since µ ( λM + A, B ) | λ =0 6 = 0 , µ ( λM + A, B ) is not the zero polynom ial. It follows tha t th ere are at most a finite numb er of values of λ for which µ ( λM + A, B ) vanishes and λ = 0 is not one of them. Let g be any n umber for which µ ( 1 g M + A, B ) 6 = 0 . Th en µ ( M + g A, g B ) 6 = 0 and since g 6 = 0 , µ ( M + g A, B ) 6 = 0 . Therefor e ( M + g A, B ) is a con trollable pair with controllab ility index no greater th an m . Let m g denote the controllab ility index of ( M + g A, B ) ; then m g r ≥ n . Suppo se that mr = n . It follows tha t m g r ≥ mr and thu s that m g ≥ m . But for all but at mo st a finite set of values of g , m g ≤ m .Therefor e m g = m for all but at most a finite set o f values of g . Proof of Lem ma 2: As an immediate conseque n ce of Lemma 5 it is clear tha t for any K i , i ∈ m an d for all but a finite n u mber of values o f g , th e matrix pair  ˜ A + P m i =1 ˜ B i ( K i ˆ C i + g ˆ H i ˜ C i ) , ˜ B q  is con trollable with controllab ility index m fo r e very q ∈ m . Setting H i = g ˆ H i thus gi ves the d esired result. Proof of P ropositio n 1: The existence of th e H i which makes the m atrix p air  ˜ A + P m i =1 ˜ B i ( K i ˆ C i + H i ˜ C i ) , ˜ B q  controllab le for every q ∈ m implies that all of the co m- plementary subsy stems of (21) are co m plete { cf. Th eorem 1 of [2] } . From this, the jo in t controllab ility an d joint observability of (21), it n ow follows from Corollary 1 o f [2] that there exist m a trices K i and H ij for which (21) is contro llable a nd observable fo r any value of q ∈ m . The matrix pair  ˜ A + P m i =1 ˜ B i ( K i ˆ C i + H i ˜ C i ) , ˜ B q  also has controllability index m . Moreover th e set of K i and H ij for wh ich this is true is the complem ent of a p roper algeb r aic set in the linear space of all such ma tr ices so almost any choice fo r such matrices will h av e the required prop erties. V . D I S T R I B U T E D S E T - P O I N T C O N T RO L This aim o f this section is to explain how the ideas discussed in the pr eceding sections can be used to solve th e “distributed set-po int co ntrol pr oblem. ” This pr oblem will be formu late d assum in g that each agen t i senses a scalar output y i = c i x with the goal of ad justing y i to a pre scribed number r i which is agent i ’ s d esired set-point value. The distributed set-point con tr ol pr ob lem is then to develop a d istributed feedback contr o l system for a pro cess modelled by the multi- channel system (3) which , when applied will enable each and ev ery agent to in d ependen tly ad just its o utput to any desired set-point value. T o construct such a contro l system, each agen t i w ill make use of integrator d ynamics of th e form ˙ w i = y i − r i , i ∈ m (31) where r i is th e desired { co nstant } value to which y i is to be set. Th e comb ination of these integrator equatio n s p lu s the multi-chan nel system described by ( 3), is thus a system of the form ˙ ˜ x = ˜ A ˜ x + m X i =1 ˜ B i u i − ˜ r w i = ˜ c i ˜ x, i ∈ m (32 ) where ˜ x = colu mn { x, w 1 , w 2 , . . . , w m } , ˜ A =  A 0 C 0  , ˜ B i =  B i 0  , i ∈ m , ˜ r =  0 r  C = colum n { c 1 , c 2 , . . . , c m } m × n , r = column { r 1 , r 2 , . . . , r m } , and ˜ c i =  0 v ′ i  , v i being the ith unit vector in I R m . Thu s (32) is an n + m dimensiona l, m channe l system with measur able outpu ts w i , i ∈ m , con trol inputs u i , i ∈ m , and constant exogenou s input ˜ r . Note that any lin e ar constant feed back control, d istributed or no t, which stabilizes this system , will enable each agent to attain its d esired set-point value. The reason fo r this is simple. Fir st note th at any such control will boun d the state of the resulting closed loo p system and cause the state to ten d to a co nstant limit as t → ∞ . Therefo re, since eac h w i is a state variable, eac h must tend to a finite limit. Similarly each y i must also tend to a finite limit. In view of (3 1), th e o nly way this can happen is if each y i tends to agent i ’ s desired set-point value r i . T o solve the distributed set-point co ntrol pro blem it is enoug h to devise a distributed controller wh ich stabilizes (32). This can be accomplished using the id eas discussed earlier in this p aper pr ovided (3 2) is b oth jointly controllab le by the u i and jointly observable throug h the w i . Accor ding to Hautus’ s lemma [ 15], th e condition fo r joint o bservability is that rank  sI − ˜ A ˜ C  = n + m for all complex number s where ˜ C = column { ˜ c 1 , ˜ c 2 , . . . , ˜ c m } . In other word s what is required is that rank   sI − A 0 − C sI 0 I   = n + m (33) But ( C, A ) is an observable pa ir because (3) is a jointly observable system. From th is, the Hautu s cond ition, a n d the structure of the m a trix penc il appearin g in (3 3) it is clear that the requir e d rank co ndition is satisfied and thus that (32) is a jointly observable system. T o estab lish joint controllability of (3 2), it is en o ugh to show that ra nk  sI − ˜ A ˜ B  = n + m for all comp lex number s where ˜ B =  ˜ B 1 ˜ B 2 · · · ˜ B m  . In othe r words what is required is that rank  sI − A 0 B 1 B 2 · · · B m − C sI 0 0 · · · 0  = n + m (34) But since (3) is a jointly con trollable system, rank  sI − A B  = n for all s , where B =  B 1 B 2 · · · B m  . Thu s (34) ho lds for all s 6 = 0 . For s = 0 , ( 34) will also h old provided rank  A B C 0  = n + m (35) In other words, ( 35) is the co ndition fo r (32) to be jointly controllab le and thus stabilizable with distributed contr ol. It is po ssible to g ive a simple inter p retation of co ndition (35) for th e ca se when each B i is a single c o lumn. In this c a se the tran sf e r matrix C ( sI − A ) − 1 B is square and co ndition (3 5) is equivalent to the requiremen t tha t its determinan t has no zeros at s = 0 { cf, [16] } . No te that if the transfer matrix wer e no n singular but had a zer o at s = 0 , this would lead to a pole zero can cellation at zero because of the integrators. Suppose conditio n ( 35) is satisfied. The proc ess o f con- structing an observer-based distributed c o ntrol to stabilize (32) is a s follows. Th e first step would be to construct an observer-based distributed co ntrol to stabilize th e r e ference signal free system ˙ ˜ x = ˜ A ˜ x + m X i =1 ˜ B i u i w i = ˜ c i ˜ x, i ∈ m (36) using the technique discussed earlier in the paper . This w ould result in a feedback control system of the fo rm u i = F i x i , i ∈ m ˙ x i = ( ˜ A + k i ˜ c i ) x i − k i w i + X j ∈N i H ij ( x i − x j ) + δ iq ¯ C z + m X j =1 ˜ B j F j x i ˙ z = ¯ Az + ¯ k ˜ c q x q − ¯ k w q + X j ∈N q ¯ H j ( x q − x j ) Application of this contr ol system to (32) would stabilize (32) an d thus provide a solu tio n to the d istributed set-point control problem despite the fact that the signals x i would not be asymptotically correct estimates of ˜ x . V I . T R A N S M I S S I O N D E L A Y S An important issue n ot discussed so far is th e effect of network transmission d elays on the co n cepts d iscussed in this paper . While th is topic is inv olved enoug h to warrant treatment in a separa te paper, to illu strate o ne way to deal with d elays a brief d iscussion w ill be g i ven here using a specific exam ple. Suppo se the system to be co ntrolled is a three ch annel, jointly controllab le, jointly observable discrete-time system of the form x ( t + 1) = Ax ( t ) + 3 X i =1 B i u i ( t ) , y i ( t ) = C i x ( t ) , (37) for i ∈ { 1 , 2 , 3 } , t ∈ [0 , 1 , 2 , . . . ) . Suppose in additio n that N is the directed graph determined by n e ig hbor sets N 1 = { 1 , 2 } , N 2 = { 1 , 2 , 3 } , and N 3 = { 2 , 3 } . A ssum e that u i = F i x i , i ∈ { 1 , 2 , 3 } where x i is agent i ’ s estimate of x and like befor e, th at state feedback matrices F i have been chosen so th at the matrix A + B 1 F 1 + B 2 F 2 + B 3 F 3 has desired properties such as discrete-time stability . Like bef ore, the g oal is to construct a discrete-time distributed observer with state estimates x i , i ∈ { 1 , 2 , 3 } so that for each i , th e e stimation erro r e i ( t ) = x i ( t ) − x ( t ) , conv erges to zero in d iscrete time, e xpon entially fast at a prescribed rate. Suppo se that at tim e t , agen t 1 receives the delayed state x 2 ( t − 2 ) fr om neig hbor 2 rather th an x 2 ( t ) , that agent 2 receiv es x 3 ( t ) and the delayed states x 1 ( t − 1) from its neigh bors 3 and 1 respectively , and that agent 3 receives the delayed state x 2 ( t − 2) from neighbor 2 . One way to obtain a distributed observer wh ich delivers the d esired behavior , is to u se the 2 -un it delayed states x i ( t − 2) , i ∈ { 1 , 2 , 3 } in the local estimators. In particular, sup pose tha t for i ∈ { 1 , 2 , 3 } , the u pdate e q uation for agent i ’ s estimator is x i ( t + 1) = ( A + K i C i ) x i ( t ) − K i y i ( t ) + 3 X j =1 B j F j ! x i ( t ) + 3 X i =1 H ij ( x i ( t − 2) − x j ( t − 2)) + δ iq ¯ C z ( t ) x i z ( t + 1) = ¯ Az ( t ) + ¯ K C q e q ( t ) + X j ∈N q ¯ H j ( x q ( t − 2) − x j ( t − 2)) for some fixed q ∈ { 1 , 2 , 3 } . Here δ iq is the Kron ecker delta, just like before. After picking q ∈ { 1 , 2 , 3 } , the first step in defin ing the the K i and H ij is to define the “ lif ted” er ror states set e ik = e i ( t − k ) , i ∈ { 1 , 2 , 3 } , k ∈ { 1 , 2 } . In this ca se, th e open-lo op erro r system is e i ( t + 1) = ( A + ˜ K i C i ) e i ( t ) + 3 X i =1 ˜ H ij ( e i 2 ( t ) − e j 2 ( t )) + 3 X j =1 B j F j ( e i ( t ) − e j ( t )) + δ iq ˜ u q ( t ) e i 1 ( t + 1) = e i ( t ) e i 2 ( t + 1) = e i 1 ( t ) , i ∈ { 1 , 2 , 3 } , tog ether with the outpu t ˜ y q ( t ) =        C q e q ( t ) e q 2 ( t ) − e j 1 2 ( t ) e q 2 ( t ) − e j 2 2 ( t ) . . . e q 2 ( t ) − e j k q 2 ( t )        where { j 1 , j 2 , . . . , j k q } = N q . Using exactly the same technique s th at were used earlier in th is paper, it can be checked th at the ˜ K i and ˜ H ij can be chosen to m ake th is system contro llable thro u gh inp ut ˜ u q and ob servable through output ˜ y q . Having so chosen these matrices, the r e m aining steps to be taken to define the K i , H ij , ¯ A, ¯ K and ¯ H j are exactly the same as befor e in th e delay- free case. What results is a d istributed observer with the r e quisite pr operties. In su mmary , th e ke y ch ange need ed to account for th e transmission d elays is to use the terms x i ( t − 2 ) − x j ( t − 2) , j ∈ N i in the u pdate equations for the x i , r ather th an terms of the fo rm x i ( t ) − x j ( t ) , j ∈ N i . The significance of the 2 -un it d e la y used here is that it is also is th e maximum of all transmission d e lays across the n etwork. Th is idea generalizes and will be elaborated on in another paper . V I I . C O N C L U D I N G R E M A R K S The discussion in § VI explains by example one way to deal with network tr ansmission delays. Contrary to intuition , the example illustrate s that transmission delays do no t preclu de achieving a r bitrarily fast estimation er ror co n vergence with a suitably defined o bserver . What delays certainly will effect is estimator “ perform ance” in the face o f sen sor no ise and disturbanc e s. Qu a ntifying this is a majo r ope n problem for future research. While th e algorithms d iscu ssed in this paper can be implemented in a distributed manner, all require “central- ized d e signs. ” Centralized designs are implicitly assumed in essentially all decentralized control a nd distributed contr o l research including , for example, the work in [1], [2]. In our view it is h ig hly un likely , if not impo ssible, to a void centralized d esigns u nless very re stricti ve assumptions are added to the pr o blem fo rmulation s. Of co urse ther e are so me distributed algorithms such as those studied in [1 7 ], [18] which d o no t call fo r centr a lized designs; but these are not feedback control algorithms. Algorithms based on centralized designs tend to be “frag - ile” in that they will typ ically fail if there is a single break in the network or perhaps a single com ponen t failure. 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