Realizability and Internal Model Control on Networks
RRealizability and Internal Model Control on Networks
Anders Rantzer
Abstract — It is proved that network realizability of con-trollers can be enforced without conservatism using convexconstraints on the closed loop transfer function. Once a networkrealizable closed loop transfer matrix has been found, acorresponding controller can be implemented using a networkstructured version of Internal Model Control.
I. I
NTRODUCTION
The importance of closed loop convexity in the theoryfor control system design has long been recognized [2]. Alarge number of specifications, both in time and frequencydomain, can be stated as convex constraints on the closedloop system. The convexity opens up for efficient synthesisalgorithms, as well as for computation of rigorous boundson achievable performance. However, there are also manyimportant specifications that cannot be expressed in a closedloop convex manner. A notable example is controller com-plexity, measured by the number of states needed in therealization.During the past decade, growing attention has been paid tolarge scale networks and distributed control. In this context,it is common to consider controller transfer matrices with apre-specified sparsity pattern. Such sparsity constraints aregenerally not closed loop convex, but a number of importantclosed loop convex structural constraints have been derivedand summarized under the framework known as quadraticinvariance [6]. However, with exception for positive systems[5], solutions based on sparsity restricted transfer matriceshave a tendency to become computationally expensive andpoorly scalable. It was therefore an important discovery inthe theory for large-scale control when [8] recently provedthat optimization with finite impulse response constraints canbe used for scalable synthesis of distributed controllers.The objective of this short note is to isolate an ideaused in “system level synthesis” [8], [1], to show thatnetwork realizable controllers in the sense of [7] can besynthesized using convex optimization. Moreover, we willdemonstrate that the classical idea of internal model control[3] is useful to convert the optimization outcome into anetwork compatible controller realization.II. N
OTATION A transfer matrix denotes a matrix of rational functionsthat can be written on the form G ( z ) = C ( zI − A ) − B + D ,where A ∈ R n × n , B ∈ R p × n , C ∈ R n × m and D ∈ R p × m .It is said to be strictly proper if D = 0 . The author is affiliated with Automatic Control LTH, Lund University,Box 118, SE-221 00 Lund, Sweden.
III. N
ETWORK R EALIZABILITY
Following [7], we make the following definition.
Definition 1.
Given a graph G = ( V , E ) with N nodes, atransfer matrix G is said to be network realizable on G if ithas a stabilizable and detectable realization (cid:20) A BC D (cid:21) = A . . . A N B ... ... . . . A N . . . A NN B N C . . . C n D ... ... . . . C N . . . C NN D N (1)where A ij = 0 and C ij = 0 for ( i, j ) (cid:54)∈ E . Such a realizationis said to be compatible with G . We need A ij ∈ R n i × n j , B i ∈ R n i × m i , C ij ∈ R m i × n j and D i ∈ R p i × m i , where n i , p i and m i are the number of states, outputs and inputs innode i respectively.Given a transfer matrix and a graph, no simple test fornetwork realizability is known. However, given a realization,it is of course straightforward to verify the conditions ofDefinition 1. Theorem 1:
Let the transfer matrices G and G benetwork realizable on G . Then the following statements hold:( i ) G + G is network realizable on G .( ii ) If G and G are stable, then G G is networkrealizable on G .( iii ) If G ( ∞ ) is invertible, then G − is network real-izable on G . Remark 1.
Consider the graph with V = { , , , } and E = { (1 , , (2 , , (3 , , (4 , , (1 , , (1 , , (2 , , (2 , } .Notice that both the two transfer matrices G ( z ) = z − z − G ( z ) = z − are network realizable on G , but, as was pointed out in [4],this is not the case with their product G ( z ) G ( z ) = z − z − z − z − . This shows that the stability assumption is essential forstatement ( ii ) in Theorem 1. a r X i v : . [ m a t h . O C ] M a r roof of Theorem 1 Let G i ( z ) = C i ( zI − A i ) − B i + D i where (cid:20) A i B i C i D i (cid:21) = A i . . . A i N B i ... ... . . . A iN . . . A iNN B iN C i . . . C i n D i ... ... . . . C iN . . . C iNN D iN , Then G ( z ) = G ( z ) + G ( z ) provided that A kl = (cid:20) A kl A kl (cid:21) B kl = (cid:20) B k B k (cid:21) C kl = (cid:2) C kl C kl (cid:3) for all k and l . The sparsity conditions, as well as stabiliz-ability and detectability, follow trivially and ( i ) holds.Similarly, G ( z ) = G ( z ) G ( z ) provided that (cid:20) A kl B k C kl D k (cid:21) = A kl B k B k C kl A kl B k D k D k C kl C kl D k D k , (2)so G satisfies the sparsity conditions. Both factors areassumed to be stable, so stabilizability and detectability holdtrivially. Hence ( ii ) follows.If G ( z ) = C ( zI − A ) − B + D and D is invertible, then G − has the realization (cid:20) A − BD − C BD − − D − C D − (cid:21) . where the needed sparsity structure, as well as stabilizabilityand detectability, follow from network realizability of G .This proves ( iii ) . (cid:50) IV. N
ETWORK R EALIZABLE C ONTROLLERS
The following theorem shows that in a number of cases,network realizability conditions on the controller can bemapped into similar conditions on closed loop transfer func-tions. From Theorem 1, we know that such constraints areconvex, so they can be conveniently included in synthesisprocedures based on convex optimization.
Theorem 2:
Consider P and C such that P is strictlyproper and define the closed loop matrix H = (cid:20) I − PC I (cid:21) − = (cid:20) ( I + PC ) − P ( I + CP ) − − C ( I + PC ) − ( I + CP ) − (cid:21) . Then the following two statements are equivalent:( i ) Both P and C are network realizable on G .( ii ) H is network realizable on G .The following two statements are also equivalent:( iii ) Both PC and C are network realizable on G .( iv ) Both ( I + PC ) − and C ( I + PC ) − are networkrealizable on G .Suppose in addition that P is stable and network realizableon G , while H is stable. Then the following are equivalent:( v ) C is network realizable on G .( vi ) C ( I + PC ) − is network realizable on G . Remark 2.
The presentation in [8] is focusing on finiteimpulse response representations of the closed loop maps.However, nothing excludes the use of other denominatorswhen finite-dimensional parametrizations of closed loop dy-namics are needed for computations. In fact, there is a richliterature on heuristics for selection of closed loop poles.
Remark 3.
The statement and proof of Theorem 2 is com-pletely independent of how the set of stabilizing controllers isparametrized. The Youla-Kucera parametrization is the mostwell known option, but the parametrization suggested in [8]appears to give simpler formulas for unstable plants.
Remark 4.
Stability of the closed loop transfer matrix H means that all poles should be strictly inside the left halfplane. In most applications the poles can actually restrictedto a smaller subset Ω of the complex plane. Such a strongerassumption can be used to also get a stronger conclu-sion, namely that the controller has a network compatiblerealization with no uncontrollable or unobservable modescorresponding to poles outside Ω . Proof of Theorem 2.
The strict properness of P implies that H (0) is invertible, so the equivalence between ( i ) and ( ii )follows immediately from Theorem 1.The equivalence between ( iii ) and ( iv ) follows fromstatement ( iii ) in Theorem 1 and the identity (cid:20) I + PC C I (cid:21) − = (cid:20) ( I + PC ) − − C ( I + PC ) − I (cid:21) , since the strict properness of P gives both matrices aninvertible direct term.Assume that ( v ) holds. Then ( iii ) follows and thereforealso ( iv ). This proves ( vi ).Conversely, suppose that ( vi ) holds. Then C ( I + PC ) − and P are both stable and network realizable on G , so thesame holds for their product PC ( I + PC ) − . The identity ( I + PC ) − = I − PC ( I + PC ) − gives stability andnetwork realizable on G for ( I + PC ) − . Hence ( iv ) holdsand the equivalence with ( iii ) proves that C is networkrealizable on G , so the proof is complete. (cid:50) A common situation in applications is that a networkrealizable Q = C ( I + PC ) − has been designed and acorresponding controller needed. The equivalence between( v ) and ( vi ) proves existence, but the proof of Theorem 2 isnot convenient for construction of a corresponding controller.Instead, as will be seen in the next section, the classicalInternal Model Control [3] approach is useful for this task.V. I NTERNAL M ODEL C ONTROL ON N ETWORKS
Given P and Q , consider a map from process output y andreference value r to control input u , defined by the equation u = Q [ r + P u − y ] . (3)See Figure 1. Here P is the “internal model” that is usedby the controller to predict the measured process output.The difference P u − y denotes a comparison between thepredicted output P u and the measurement y . In the ideal Q ( z ) Process P ( z ) r u y + − Controller
Fig. 1. If P and Q are networks realizable, then a network realization ofthe controller can be obtained using Internal Model Control. case that the difference is zero, the control law reduces to u = Q r , so Q defines the desired map from reference toinput and PQ is the resulting map from reference r to output y . The transfer matrix from r − y to u , given by (3), is C = Q ( I − PQ ) − .Our interest in Internal Model Control stems from the factit generates network realizable controllers in a very naturalmanner. Suppose that P ( z ) = C ( zI − A ) − B Q ( z ) = G ( zI − E ) − F + H Then the controller C = Q ( I − PQ ) − has the realization (cid:20) ˆ x + ξ + (cid:21) = (cid:20) A + BHC BGF C E (cid:21) (cid:20) ˆ xξ (cid:21) + (cid:20) BHF (cid:21) ( r − y ) u = (cid:2) HC G (cid:3) + H ( r − y ) . It is easy to see that if B , F and H are diagonal, while A , C , E and G have a sparsity structure compatible with thegraph G , this realization is compatible with G after properordering of the states and block partitioning of the matrices.More specifically, if node i of the graph is hosting the processstate x i , it should also host the controller state (ˆ x i , ξ i ) . Example 1.
Consider a simple model for control of waterlevels in dams along a river: x ( t + 1) = 0 . x ( t ) − u ( t ) x ( t + 1) = 0 . x ( t ) + 0 . x ( t ) + u ( t ) − u ( t ) x ( t + 1) = 0 . x ( t ) + 0 . x ( t ) + u ( t ) − u ( t ) Each state represents the water level in a dam and the controlvariables are used to control the release of water from onedam to the next. In this case, the transfer function from ( u , u , u ) to ( x , x , x ) is P ( z ) = ( zI − A ) − B where A = . . . . . B = − − − A graph G , corresponding to downwards flow of information,is defined by the node set V = { , , } and the linkset E = { (1 , , (2 , , (2 , , (3 , , (3 , } . The realizationabove does not have diagonal B -matrix, but the transferfunction from u to x is still network realizable on G , as shown by the (non-minimal) realization ¯ A = . . . . . . . . ¯ B = − − − ¯ C = (cid:34) (cid:35) ¯ D = 0 . Theorem 2 tells us that in order to find river dam controllersthat only exchange information along the graph G , it suf-ficient to consider Q = ( I + CP ) − C that are networkrealizable on G . In particular, let Q have the state realization (cid:20) E FG H (cid:21) = E F E E F E E F G G G Then C = ( I − QP ) − Q mapping ( e , e , e ) to ( u , u , u ) can be implemented as the Internal Model Controller ˆ x +1 ξ +1 ˆ x +2 ξ +2 ˆ x +3 ξ +3 = . − G F E . G . − G E F E . G . − G E F E ˆ x ξ ˆ x ξ ˆ x ξ − e e e This realization has a structure compatible with the graph G (in spite of the fact that it is based on the original statematrices A, B rather than ¯ A, ¯ B ).VI. A CKNOWLEDGEMENT
Financial support from the Swedish Research Council andthe Swedish Foundation for Strategic Research is gratefullyacknowledged, The author is a member of the Linnaeuscenter LCCC and the excellence center ELLIIT.R
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