A new operator is considered, allowing compact formulae and proofs in the context of the derivation of a transfer matrix with respect to another matrix. The problem of the parametric sensitivity matrix calculation is chosen for illustration. It consists in deriving a Multiple Input Multiple Output transfer function with respect to a parametric matrix and is central in robust control theory. Efficient algorithms may be straightforwardly got from the compact analytic formulae using the operator introduced

Chevrel, Philippe

Hilaire, Thibault

HAL-Rennes 1

On the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault Hilaire, Philippe ChevrelTo cite this version:Thibault Hilaire, Philippe Chevrel. On the compact formulation of the derivation of a transfermatrix with respect to another matrix. [Research Report] PI 1916, 2008, pp.13. <inria-00348344>HAL Id: inria-00348344https://hal.inria.fr/inria-00348344Submitted on 18 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.I   R   I   S   AINSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRESP U  B  L  I  C  A  T  I  O  NI  N  T  E  R  N  ENoI R I S ACAMPUS UNIVERSITAIRE DE BEAULIEU - 35042 RENNES CEDEX - FRANCEISSN 1166-86871916ON THE COMPACT FORMULATION OF THE DERIVATIONOF A TRANSFER MATRIX WITH RESPECT TO ANOTHERMATRIXTHIBAULT HILAIRE, PHILIPPE CHEVRELINSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRESCampus de Beaulieu – 35042 Rennes Cedex – FranceTél. : (33) 02 99 84 71 00 – Fax : (33) 02 99 84 71 71http://www.irisa.frOn the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault HILAIRE* , Philippe CHEVREL**Syste`mes nume´riquesProjet CAIRNPublication interne n˚1916 — August 2008 — 11 pagesAbstract: A new operator is considered, allowing compact formulae and proofs in thecontext of the derivation of a transfer matrix with respect to another matrix. The problem ofthe parametric sensitivity matrix calculation is chosen for illustration. It consists in derivinga Multiple Input Multiple Output transfer function with respect to a parametric matrix andis central in robust control theory. Eﬃcient algorithms may be straightforwardly got fromthe compact analytic formulae using the operator introduced.Key-words: sensitivity, derivative, robust control(Re´sume´ : tsvp)* CAIRN Project, INRIA/IRISA** IRCCyN, Institut de Recherche en Communication et Cyberne´tique de NantesCentre National de la Recherche Scientifique Institut National de Recherche en Informatique(UMR 6074) Université de Rennes 1 – Insa de Rennes et en Automatique – unité de recherche de RennesUne expression compacte pour la de´rivation d’unefonction de transfert par une matriceRe´sume´ : Dans ce papier, un nouvel ope´rateur mathe´matique est conside´re´, permet-tant des expressions compactes dans le contexte de la de´rivation d’une matrice de fonctionde transfert par rapport a` une matrice. Le proble`me de la sensibilite´ parame´trique sertd’illustration. Il consiste en la de´rivation d’une fonction de transfert a` plusieurs entre´eset plusieurs sorties par rapport a` une matrice de parame`tres. Ce proble`me est central encommande robuste, notamment pour la recherche de re´alisations eﬃcaces vis-a`-vis de leurimplantation nume´rique.Mots cle´s : sensibilite´, de´rivation, commande robusteOn the compact formulation of the derivation of a transfer matrix 31 MotivationOne important questioning in control theory is robustness. A property associated to agiven system will be said to be robust if it is still satisﬁed when the system is slightlymodiﬁed. Diﬀerent properties may be considered, such as stability or say a certain levelof performance measured, e.g. thanks to system norms. The problem is crucial in thetheory of feedback, because the systems considered are (physical or mathematical) modelswhich are representing the process with some approximations and uncertainties [10, 2].Moreover, the feedback controller itself may be considered as an uncertain system, due tothe inevitable approximation coming from the implementation. In particular, the use ofcomputers introduces Finite Word Length quantiﬁcation of the controller parameters [3, 7].Whatever the case, the computation of the parametric sensitivity of MIMO transferfunction is of particular interest. The problem involves the calculus of a matrix with respectto (w.r.t.) another matrix. But, as far the authors know, it exists no special techniques orspecial properties to simplify the expressions induced.For example, let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m be four matricesdeﬁning the MIMO transfer function H1 [8] :H1 :C → Cp×qz → C(zIn −A)−1B + D (1)The sensitivity measure (in the context of FWL implementation) used (e.g. Gevers andLi [3]) isM ∥∥∥∥∂H1∂A∥∥∥∥22+∥∥∥∥∂H1∂B∥∥∥∥22+∥∥∥∥∂H1∂C∥∥∥∥22+∥∥∥∥∂H1∂D∥∥∥∥22(2)where ‖.‖2 is the transfer function L2-norm.The analytic expression of ∂H1∂A ,∂H1∂B ,∂H1∂C and∂H1∂D are easy to formulate in the SISO1 case(when p = q = 1, so H1(z) ∈ C), but these expressions are less obvious in the MIMO case.After having recalled general deﬁnitions and classical properties on matrix and transferfunction derivatives in section 2, section 3 introduces a new operator  to simplify derivativeexpressions in the MIMO case. Finally, a more complicated example is solved in section 4,before conclusion in section 5.2 DefinitionsLet’s introduce some interesting deﬁnitions about matrix and transfer function derivatives.Definition 1 (Scalar derivative w.r.t. a matrix) Let X ∈ Rm×n be a matrix and f :Rm×n → C a scalar function of X, diﬀerentiable w.r.t. each element of X.1S ingle Input S ingle OutputPI n˚19164 T. Hilaire & P. ChevrelThe derivative of f w.r.t. X is deﬁned as the matrix ∂f∂X ∈ Cm×n such(∂f∂X)i,j ∂f∂Xi,j(3)where Xi,j is the (i, j) element of X.This derivative deﬁnes the sensitivity of f w.r.t. X.This deﬁnition can be extended to functions with values in Cp×l as follow :Definition 2 (Derivative of a matrix w.r.t. a matrix) Let X ∈ Rm×n be a matrixand f : Rm×n → Cp×l a function of X, where each component of f is diﬀerentiable w.r.t.each element of X.The derivative of f with respect to X is a matrix of Cmp×nl partitioned in m × n matrixblocks of Cp×l. Each (i, j)th block is deﬁned by∂f∂Xi,j∈ Cp×l (4)Then∂f∂X∂f∂X1,1∂f∂X1,2. . . ∂f∂X1,n∂f∂X2,1∂f∂X2,2. . . ∂f∂X2,n....... . ....∂f∂Xm,1∂f∂Xm,2. . . ∂f∂Xm,n (5)which can also be written as :∂f∂X=m∑r=1n∑s=1Em,nr,s ⊗∂f∂Xr,s(6)where the matrices En,mr,s of Rn×m are the elementary matrices deﬁned by(En,mr,s)i,j δr,iδs,j (7)and ⊗ is the Kronecker product.Remark 1 When X is a row vector and f(X) a column vector, ∂f∂X is the jacobian matrixof f .Other deﬁnitions of the derivative of a matrix w.r.t. a matrix are sometimes used : ∂A∂X ∂Vec(A)∂Vec(X) or∂A∂X ∂Vec(A)∂Vec(X) , in order to get a jacobian matrix. It exempts to consider blocksof matrices ∂f∂Xi,j . However some useful propositions (like proposition 1 and theorem 1) aremore easy according to deﬁnition 2.IrisaOn the compact formulation of the derivation of a transfer matrix 5Remark 2 ∂X∂X , the derivative of X ∈ Rp×q w.r.t. itself, is a constant matrix of Rp2×q2such that :∂X∂X=p∑r=1l∑s=1(Ep,lr,s ⊗ Ep,lr,s)(8)=p∑r=1l∑s=1Ep2,l2(r−1)p+r,(s−1)l+s (9)The transfer function sensitivity (SISO and MIMO case) is deﬁned belowDefinition 3 (Transfer function sensitivity) Let X ∈ Rm×n be a matrix and H : C→Cp×q the transfer function which associate H(z) to all z ∈ C. H(z) is supposed to beparametrized by X and to be diﬀerentiable w.r.t. each element of X whatever z ∈ C.Finally, the sensitivity function of H w.r.t. X is the transfer function denoted by ∂H∂X , suchthat∂H∂X:C → Cpm×qnz → ∂(H(z))∂X(10)Remark 3 The subsequent properties on matrix derivatives also hold for transfer function,without any modiﬁcations.The General Leibniz product rule for derivative of a product of matrices w.r.t. a matrixis the main property used when dealing with matrix derivative.Proposition 1 Let’s consider X ∈ Rk×l, Y ∈ Rl×m and Z ∈ Rp×l. The derivative of theproduct XY with respect to Z is∂(XY )∂Z=∂X∂Z(Il ⊗ Y ) + (Ip ⊗X)∂Y∂Z(11)Proof:The proof can be found in [1].Remark 4 Proposition 1 applied to∂(Y Y −1)∂Z leads to∂(Y −1)∂Z= − (Ip ⊗ Y −1) ∂Y∂Z(Il ⊗ Y −1)(12)When considering the initial example H1 (eq. (1)), it comes then (∀z ∈ C)∂H1∂A(z) =(In ⊗ C(zIn −A)−1) ∂A∂A(In ⊗ (zIn −A)−1B)(13)∂H1∂B(z) =(In ⊗ C(zIn −A)−1) ∂B∂B(14)∂H1∂C(z) =∂C∂C(In ⊗ (zIn −A)−1B)(15)∂H1∂D(z) =∂D∂D(16)PI n˚19166 T. Hilaire & P. ChevrelThe formulae given in equations (13) to (16) suﬀer from some drawbacks. First, theyare expressed with constant matrices ∂A∂A ,∂B∂B ,∂C∂C and∂D∂D that do not depends on A, B,C or D (they only depend on their dimensions). Secondly, their manipulations may berather tedious when dealing with such complicate expressions as in section 4. Lastly, a morecompact form is possible, as proposed in section 3.3 A new operator for compact derivatives formulaeIn order to simplify the expressions, the new operator  is proposed. Three propositionsshow how to use this operator in classical linear derivative problem.Definition 4 Let G and H be two transfer functions in Cm×p and Cl×n.The operator  is deﬁned byGH  Vec(G).[Vec(H)](17)where Vec is the usual operator that vectorize a matrix. It corresponds to the product of eachelement of G with each element of H, in a particular order.This operator is used to state the main proposition of this paper, which encompass andsimplify the Leibnitz rule of proposition 1 :Theorem 1 Let X be a matrix in Rp×l and G, H be two transfer functions with valuesrespectively in Cm×p and Cl×n. G and H are supposed to be independent w.r.t. X. Then∂(GXH)∂X= GH (18)∂(GX−1H)∂X= (GX−1) (X−1H) (19)Proof:From proposition 1 and equation (9),∂(GXH)∂X=∑r,s(Ip ⊗G)Ep2,l2(r−1)p+r,(s−1)l+s (Il ⊗H) (20)Considering relation(AEp2,l2i,j B)= A•,iBj,• where A•,i denotes the ith column of A andBj,• the jth row of B, then∂(GXH)∂X=∑r,s(Ip ⊗G)•,(r−1)p+r (Il ⊗H)(s−1)l+s,• (21)The term (Ip ⊗G)•,(r−1)p+r corresponds to(Ip ⊗G)•,(r−1)p+r =(0 . . . G•,r 0 . . .↑ rth block)(22)IrisaOn the compact formulation of the derivation of a transfer matrix 7so it is possible to write(Ip ⊗G)•,(r−1)p+r (Il ⊗H)(s−1)l+s,• = Ep,lr,s ⊗ (G•,r.Hs,•) (23)=0G•,r.Hs,•0(r, s)th block (size m× n)(24)Finally, (G•,r.Hs,•)i,j = Gi,r.Hs,j ; so(∂(GXH)∂X)(r−1)m+i,(s−1)n+j= Gi,r.Hs,j (25)and then,∂(GXH)∂X= Vec(G).[Vec(H)](26)Using the previous result leads equations (13) to (16) to the simpliﬁed expressions :∂H1∂A=(C(zIn −A)−1)((zIn −A)−1B)(27)∂H1∂B=(C(zIn −A)−1) In (28)∂H1∂C= In ((zIn −A)−1B)(29)∂H1∂D= In  In (30)Practically, the additional following properties are useful to simplify the derivative taskand get compacter expressions. The proofs are trivial.Proposition 2(Ip ⊗G) (X  Y ) (Il ⊗H) = (GX) (Y H) (31)Proposition 3 (A C ADB  C B D)=(A B)(CD)(32)PI n˚19168 T. Hilaire & P. Chevrel4 Application to the case of closed-loop transfer func-tionIn this section the problem consisting in deriving the redheﬀer product [9], and its special-ization the lower linear fractional transformation of two transfer functions, is studied. Theproblem has an important practical interest in the context of robust control theory [10],when considering the model uncertainties of the process or even of the controller in thesense of FWL implementation [3].Let’s consider a plant P controlled by a controller C in a standard form [10] (see ﬁg. 1).W (k) ∈ Rp1 and Z(k) ∈ Rm1 are the exogenous inputs and outputs (to control), whereasU(k) ∈ Rp2 and Y (k) ∈ Rm2 are the control and measure signals.Figure 1: Closed-loop system consideredThe plant P is deﬁned by the recurrent relationXP(k + 1) = AXP(k) + B1W (k) + B2U(k)Z(k) = C1XP(k) + D11W (k) + D12U(k)Y (k) = C2XP(k) + D21W (k)(33)where A ∈ RnP×nP , B1 ∈ RnP×p1 , B2 ∈ RnP×p2 , C1 ∈ Rm1×nP , C2 ∈ Rm2×nP , D11 ∈Rm1×p1 , D12 ∈ Rm1×p2 , D21 ∈ Rm2×p1 . Note that the D22 term is null.The controller is deﬁned by{X(k + 1) = AZX(k) + BZY (k)U(k) = CZX(k) + DZY (k)(34)with AZ ∈ Rn×n, BZ ∈ Rn×m2 , CZ ∈ Rp2×n and DZ ∈ Rp2×m2 .The transfer function of the closed-loop system S is then (lower linear fractional transfor-mation)H¯ : z → C¯ (zInP+n − A¯)−1 B¯ + D¯ (35)IrisaOn the compact formulation of the derivation of a transfer matrix 9with A¯ ∈ RnP+n×nP+n, B¯ ∈ RnP+n×p1 , C¯ ∈ Rm1×nP+n and D¯ ∈ Rm1×p1 andA¯ =(A + B2DZC2 B2CZBZC2 AZ)B¯ =(B1 + B2DZD21BZD21)C¯ =(C1 + D12DZC2 D12CZ)D¯ = D11 + D12DZD21(36)Last point, the matrices AZ , BZ , CZ and DZ depends on matrices J , K, L, M , N , P , Q,R and S (J ∈ Rl×l, K ∈ Rn×l, L ∈ Rp2×l, M ∈ Rl×n, N ∈ Rl×m2 , P ∈ Rn×n, Q ∈ Rn×m2 ,R ∈ Rp2×n, S ∈ Rp2×m2) that contain the exact coeﬃcients for the realization of C [5], withAZ = KJ−1M + P, BZ = KJ−1N + Q, (37)CZ = LJ−1M + R, DZ = LJ−1N + S. (38)Those parameters are grouped in a single matrix Z ∈ Rl+n+p2×l+n+m2 asZ −J M NK P QL R S (39)In order to evaluate the sensitivity of H¯ w.r.t Z (this sensitivity is linked to the goodperformance of the global scheme in the FWL context [3, 4, 6]), the problem is then tocompute ∂H¯∂Z (or equivalently∂H¯∂J ,∂H¯∂K ,∂H¯∂L , ...,∂H¯∂S ).Proposition 4 The sensitivity of H¯ with respect to Z is given by∂H¯∂Z=[C¯(zI − A¯)−1 M¯1 + M¯2][N¯1(zI − A¯)−1 B¯ + N¯2](40)withM¯1 =(B2LJ−1 0 B2KJ−1 In 0)N¯1 =J−1NC2 J−1M0 InC2 0 (41)M¯2 =(D12LJ−1 0 D12)N¯2 =J−1ND210D21 (42)Proof:Proposition 1 on equation (35) gives∂H¯∂Z=(Il+n+p2 ⊗ C¯(zI − A¯)−1) ∂A¯∂Z(Il+n+m2 ⊗ (zI − A¯)−1B¯)+(Il+n+p2 ⊗ C¯(zI − A¯)−1) ∂B¯∂Z+∂C¯∂Z(Il+n+m2 ⊗ (zI − A¯)−1B¯)+∂D¯∂Z(43)PI n˚191610 T. Hilaire & P. ChevrelThen, let’s denote Θ =(DZ CZBZ AZ), A¯ can be reformulate byA¯ =(A 00 0)+(B2 00 In)Θ(C2 00 In)(44)So, with theorem 1∂A¯∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(45)By similar process, it is obvious that∂B¯∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(D210))(46)∂C¯∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(47)∂D¯∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(D210))(48)Then, the derivatives of Θ w.r.t. J , K, L, M , N , P , Q, R and S are∂Θ∂J = −(LJ−1KJ−1)(J−1N J−1M)∂Θ∂K =(0I)(J−1N J−1M)∂Θ∂P =(0I)(0 I)∂Θ∂M =(LJ−1KJ−1)(0 I)∂Θ∂N =(LJ−1KJ−1)(I 0)∂Θ∂L =(I0)(J−1N J−1M)∂Θ∂Q =(0I)(I 0)∂Θ∂R =(I0)(0 I)∂Θ∂S =(I0)(I 0)(49)So, with proposition 3 :∂Θ∂Z=(LJ−1 0 Ip2KJ−1 In 0)J−1N J−1M0 InIm2 0 (50)With property 2, ∂A¯∂Z ,∂B¯∂Z ,∂C¯∂Z and∂D¯∂Z are obtained and lead to equation (40).5 ConclusionIn order to simplify the calculus of derivative of matrices with respect to another matrix, theoperator has been introduced, and some important properties associated have been stated.IrisaOn the compact formulation of the derivation of a transfer matrix 11Its interest has been illustrated by application in robust control. Not only the expressionsare made more compact, but also the related numerical computations of transfer functionsensitivity are made more tractable. This has led in practical to a successful application inthe context of FWL implementation [6].References[1] J. Brewer. Kronecker products and matrix calculus in system theory. IEEE Trans. onCircuits and Systems, 25(9):772–779, September 1978.[2] M. Eslami. Theory of Sensitivity in Dynamic Systems. Springer-Verlag, 1994.[3] M. Gevers and G. Li. Parametrizations in Control, Estimation and Filtering Probems.Springer-Verlag, 1993.[4] T. Hilaire, P. Chevrel, and J-P. Clauzel. Low parametric sensitivity realization de-sign for FWL implementation of MIMO controllers : Theory and application to theactive control of vehicle longitudinal oscillations. In Proc. of Control Applications ofOptimisation CAO’O6, April 2006.[5] T. Hilaire, P. Chevrel, and Y. Trinquet. Implicit state-space representation : a unifyingframework for FWL implementation of LTI systems. In P. Piztek, editor, Proc. of the16th IFAC World Congress. Elsevier, July 2005.[6] T. Hilaire, P. Chevrel, and J.F. Whidborne. A unifying framework for ﬁnite wordlengthrealizations. IEEE Trans. on Circuits and Systems, 8(54), August 2007.[7] R. Istepanian and J.F. Whidborne, editors. Digital Controller implementation andfragility. Springer, 2001.[8] T. Kailath. Linear Systems. Prentice-Hall, 1980.[9] R. Redheﬀer. On a certain linear fractional transformation. J. Math. Phys, 39:269–286,1960.[10] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall, 1996.PI n˚1916

On the compact formulation of the derivation of a transfer matrix with respect to another matrix

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HAL Id: inria-00345508https://hal.inria.fr/inria-00345508Submitted on 9 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.On the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault Hilaire, Philippe ChevrelTo cite this version:Thibault Hilaire, Philippe Chevrel. On the compact formulation of the derivation of a transfer matrixwith respect to another matrix. [Research Report] RR-6760, INRIA. 2008. ￿inria-00345508￿appor t  de  r ech er ch eISSN0249-6399ISRNINRIA/RR--6760--FR+ENGThème NUMINSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUEOn the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault HILAIRE — Philippe CHEVRELN° 6760August 2008Centre de recherche INRIA Rennes – Bretagne AtlantiqueIRISA, Campus universitaire de Beaulieu, 35042 Rennes CedexTéléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71On the compact formulation of the derivation ofa transfer matrix with respect to another matrixThibault HILAIRE∗, Philippe CHEVREL†Thème NUM — Systèmes numériquesÉquipe-Projet CAIRNRapport de recherche n° 6760 — August 2008 — 11 pagesAbstract: A new operator is considered, allowing compact formulae and proofsin the context of the derivation of a transfer matrix with respect to anothermatrix. The problem of the parametric sensitivity matrix calculation is chosenfor illustration. It consists in deriving a Multiple Input Multiple Output transferfunction with respect to a parametric matrix and is central in robust controltheory. Efficient algorithms may be straightforwardly got from the compactanalytic formulae using the operator introduced.Key-words: sensitivity, derivative, robust control∗ CAIRN Project, INRIA/IRISA† IRCCyN, Institut de Recherche en Communication et Cyberntique de NantesUne expression compacte pour la drivationd’une fonction de transfert par une matriceRésumé : Dans ce papier, un nouvel oprateur mathmatique est considr, perme-ttant des expressions compactes dans le contexte de la drivation d’une matricede fonction de transfert par rapport une matrice. Le problme de la sensi-bilit paramtrique sert d’illustration. Il consiste en la drivation d’une fonctionde transfert plusieurs entres et plusieurs sorties par rapport une matrice deparamtres. Ce problme est central en commande robuste, notamment pour larecherche de ralisations efficaces vis--vis de leur implantation numrique.Mots-clés : sensibilit, drivation, commande robusteOn the compact formulation of the derivation of a transfer matrix 31 MotivationOne important questioning in control theory is robustness. A property asso-ciated to a given system will be said to be robust if it is still satisfied whenthe system is slightly modified. Different properties may be considered, such asstability or say a certain level of performance measured, e.g. thanks to systemnorms. The problem is crucial in the theory of feedback, because the systemsconsidered are (physical or mathematical) models which are representing theprocess with some approximations and uncertainties [10, 2]. Moreover, thefeedback controller itself may be considered as an uncertain system, due to theinevitable approximation coming from the implementation. In particular, theuse of computers introduces Finite Word Length quantification of the controllerparameters [3, 7].Whatever the case, the computation of the parametric sensitivity of MIMOtransfer function is of particular interest. The problem involves the calculus ofa matrix with respect to (w.r.t.) another matrix. But, as far the authors know,it exists no special techniques or special properties to simplify the expressionsinduced.For example, let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m be fourmatrices defining the MIMO transfer function H1 [8] :H1 :C → Cp×qz 7→ C(zIn − A)−1B + D(1)The sensitivity measure (in the context of FWL implementation) used (e.g.Gevers and Li [3]) isM ,∥∥∥∥∂H1∂A∥∥∥∥22+∥∥∥∥∂H1∂B∥∥∥∥22+∥∥∥∥∂H1∂C∥∥∥∥22+∥∥∥∥∂H1∂D∥∥∥∥22(2)where ‖.‖2 is the transfer function L2-norm.The analytic expression of ∂H1∂A, ∂H1∂B, ∂H1∂Cand ∂H1∂Dare easy to formulate in theSISO1 case (when p = q = 1, so H1(z) ∈ C), but these expressions are lessobvious in the MIMO case.After having recalled general definitions and classical properties on matrixand transfer function derivatives in section 2, section 3 introduces a new oper-ator ⊛ to simplify derivative expressions in the MIMO case. Finally, a morecomplicated example is solved in section 4, before conclusion in section 5.2 DefinitionsLet’s introduce some interesting definitions about matrix and transfer functionderivatives.Definition 1 (Scalar derivative w.r.t. a matrix) Let X ∈ Rm×n be a ma-trix and f : Rm×n → C a scalar function of X, differentiable w.r.t. each elementof X.1S ingle Input S ingle OutputRR n° 67604 T. Hilaire & P. ChevrelThe derivative of f w.r.t. X is defined as the matrix ∂f∂X∈ Cm×n such(∂f∂X)i,j,∂f∂Xi,j(3)where Xi,j is the (i, j) element of X.This derivative defines the sensitivity of f w.r.t. X.This definition can be extended to functions with values in Cp×l as follow :Definition 2 (Derivative of a matrix w.r.t. a matrix) Let X ∈ Rm×n bea matrix and f : Rm×n → Cp×l a function of X, where each component of f isdifferentiable w.r.t. each element of X.The derivative of f with respect to X is a matrix of Cmp×nl partitioned in m×nmatrix blocks of Cp×l. Each (i, j)th block is defined by∂f∂Xi,j∈ Cp×l (4)Then∂f∂X,∂f∂X1,1∂f∂X1,2. . . ∂f∂X1,n∂f∂X2,1∂f∂X2,2. . . ∂f∂X2,n....... . ....∂f∂Xm,1∂f∂Xm,2. . . ∂f∂Xm,n(5)which can also be written as :∂f∂X=m∑r=1n∑s=1Em,nr,s ⊗∂f∂Xr,s(6)where the matrices En,mr,s of Rn×m are the elementary matrices defined by(En,mr,s)i,j, δr,iδs,j (7)and ⊗ is the Kronecker product.Remark 1 When X is a row vector and f(X) a column vector, ∂f∂Xis thejacobian matrix of f .Other definitions of the derivative of a matrix w.r.t. a matrix are sometimesused : ∂A∂X,∂Vec(A)∂Vec(X) or∂A∂X,∂Vec(A)∂Vec(X)⊤, in order to get a jacobian matrix. Itexempts to consider blocks of matrices ∂f∂Xi,j. However some useful propositions(like proposition 1 and theorem 1) are more easy according to definition 2.Remark 2 ∂X∂X, the derivative of X ∈ Rp×q w.r.t. itself, is a constant matrixof Rp2×q2 such that :∂X∂X=p∑r=1l∑s=1(Ep,lr,s ⊗ Ep,lr,s)(8)=p∑r=1l∑s=1Ep2,l2(r−1)p+r,(s−1)l+s (9)INRIAOn the compact formulation of the derivation of a transfer matrix 5The transfer function sensitivity (SISO and MIMO case) is defined belowDefinition 3 (Transfer function sensitivity) Let X ∈ Rm×n be a matrixand H : C → Cp×q the transfer function which associate H(z) to all z ∈ C.H(z) is supposed to be parametrized by X and to be differentiable w.r.t. eachelement of X whatever z ∈ C.Finally, the sensitivity function of H w.r.t. X is the transfer function denotedby ∂H∂X, such that∂H∂X:C → Cpm×qnz 7→ ∂(H(z))∂X(10)Remark 3 The subsequent properties on matrix derivatives also hold for trans-fer function, without any modifications.The General Leibniz product rule for derivative of a product of matricesw.r.t. a matrix is the main property used when dealing with matrix derivative.Proposition 1 Let’s consider X ∈ Rk×l, Y ∈ Rl×m and Z ∈ Rp×l. Thederivative of the product XY with respect to Z is∂(XY )∂Z=∂X∂Z(Il ⊗ Y ) + (Ip ⊗ X)∂Y∂Z(11)Proof:The proof can be found in [1].Remark 4 Proposition 1 applied to∂(Y Y −1)∂Zleads to∂(Y −1)∂Z= −(Ip ⊗ Y−1) ∂Y∂Z(Il ⊗ Y−1)(12)When considering the initial example H1 (eq. (1)), it comes then (∀z ∈ C)∂H1∂A(z) =(In ⊗ C(zIn − A)−1) ∂A∂A(In ⊗ (zIn − A)−1B)(13)∂H1∂B(z) =(In ⊗ C(zIn − A)−1) ∂B∂B(14)∂H1∂C(z) =∂C∂C(In ⊗ (zIn − A)−1B)(15)∂H1∂D(z) =∂D∂D(16)The formulae given in equations (13) to (16) suffer from some drawbacks.First, they are expressed with constant matrices ∂A∂A, ∂B∂B, ∂C∂Cand ∂D∂Dthat donot depends on A, B, C or D (they only depend on their dimensions). Secondly,their manipulations may be rather tedious when dealing with such complicateexpressions as in section 4. Lastly, a more compact form is possible, as proposedin section 3.RR n° 67606 T. Hilaire & P. Chevrel3 A new operator for compact derivatives for-mulaeIn order to simplify the expressions, the new operator ⊛ is proposed. Threepropositions show how to use this operator in classical linear derivative problem.Definition 4 Let G and H be two transfer functions in Cm×p and Cl×n.The operator ⊛ is defined byG ⊛ H , Vec(G).[Vec(H⊤)]⊤(17)where Vec is the usual operator that vectorize a matrix. It corresponds to theproduct of each element of G with each element of H, in a particular order.This operator is used to state the main proposition of this paper, whichencompass and simplify the Leibnitz rule of proposition 1 :Theorem 1 Let X be a matrix in Rp×l and G, H be two transfer functions withvalues respectively in Cm×p and Cl×n. G and H are supposed to be independentw.r.t. X. Then∂(GXH)∂X= G ⊛ H (18)∂(GX−1H)∂X= (GX−1) ⊛ (X−1H) (19)Proof:From proposition 1 and equation (9),∂(GXH)∂X=∑r,s(Ip ⊗ G) Ep2,l2(r−1)p+r,(s−1)l+s (Il ⊗ H) (20)Considering relation(AEp2,l2i,j B)= A•,iBj,• where A•,i denotes the ith columnof A and Bj,• the jth row of B, then∂(GXH)∂X=∑r,s(Ip ⊗ G)•,(r−1)p+r (Il ⊗ H)(s−1)l+s,• (21)The term (Ip ⊗ G)•,(r−1)p+r corresponds to(Ip ⊗ G)⊤•,(r−1)p+r =(0 . . . G•,r 0 . . .↑ rth block)(22)INRIAOn the compact formulation of the derivation of a transfer matrix 7so it is possible to write(Ip ⊗ G)•,(r−1)p+r (Il ⊗ H)(s−1)l+s,• = Ep,lr,s ⊗ (G•,r.Hs,•) (23)=0G•,r.Hs,•0(r, s)th block (size m × n)(24)Finally, (G•,r.Hs,•)i,j = Gi,r.Hs,j ; so(∂(GXH)∂X)(r−1)m+i,(s−1)n+j= Gi,r.Hs,j (25)and then,∂(GXH)∂X= Vec(G).[Vec(H⊤)]⊤(26)Using the previous result leads equations (13) to (16) to the simplified ex-pressions :∂H1∂A=(C(zIn − A)−1)⊛((zIn − A)−1B)(27)∂H1∂B=(C(zIn − A)−1)⊛ In (28)∂H1∂C= In ⊛((zIn − A)−1B)(29)∂H1∂D= In ⊛ In (30)Practically, the additional following properties are useful to simplify thederivative task and get compacter expressions. The proofs are trivial.Proposition 2(Ip ⊗ G) (X ⊛ Y ) (Il ⊗ H) = (GX) ⊛ (Y H) (31)Proposition 3(A ⊛ C A ⊛ DB ⊛ C B ⊛ D)=(A B)⊛(CD)(32)RR n° 67608 T. Hilaire & P. Chevrel4 Application to the case of closed-loop transferfunctionIn this section the problem consisting in deriving the redheffer product [9],and its specialization the lower linear fractional transformation of two transferfunctions, is studied. The problem has an important practical interest in thecontext of robust control theory [10], when considering the model uncertaintiesof the process or even of the controller in the sense of FWL implementation [3].Let’s consider a plant P controlled by a controller C in a standard form [10](see fig. 1). W (k) ∈ Rp1 and Z(k) ∈ Rm1 are the exogenous inputs and outputs(to control), whereas U(k) ∈ Rp2 and Y (k) ∈ Rm2 are the control and measuresignals.Pm1m2Cp1p2W (k) Z(k)U(k) Y (k)SFigure 1: Closed-loop system consideredThe plant P is defined by the recurrent relationXP(k + 1) = AXP(k) + B1W (k) + B2U(k)Z(k) = C1XP(k) + D11W (k) + D12U(k)Y (k) = C2XP(k) + D21W (k)(33)where A ∈ RnP×nP , B1 ∈ RnP×p1 , B2 ∈ RnP×p2 , C1 ∈ Rm1×nP , C2 ∈ Rm2×nP ,D11 ∈ Rm1×p1 , D12 ∈ Rm1×p2 , D21 ∈ Rm2×p1 . Note that the D22 term is null.The controller is defined by{X(k + 1) = AZX(k) + BZY (k)U(k) = CZX(k) + DZY (k)(34)with AZ ∈ Rn×n, BZ ∈ Rn×m2 , CZ ∈ Rp2×n and DZ ∈ Rp2×m2 .The transfer function of the closed-loop system S is then (lower linear fractionaltransformation)H̄ : z → C̄(zInP+n − Ā)−1B̄ + D̄ (35)with Ā ∈ RnP+n×nP+n, B̄ ∈ RnP+n×p1 , C̄ ∈ Rm1×nP+n and D̄ ∈ Rm1×p1 andĀ =(A + B2DZC2 B2CZBZC2 AZ)B̄ =(B1 + B2DZD21BZD21)C̄ =(C1 + D12DZC2 D12CZ)D̄ = D11 + D12DZD21(36)Last point, the matrices AZ , BZ , CZ and DZ depends on matrices J , K, L,M , N , P , Q, R and S (J ∈ Rl×l, K ∈ Rn×l, L ∈ Rp2×l, M ∈ Rl×n, N ∈ Rl×m2 ,INRIAOn the compact formulation of the derivation of a transfer matrix 9P ∈ Rn×n, Q ∈ Rn×m2 , R ∈ Rp2×n, S ∈ Rp2×m2) that contain the exactcoefficients for the realization of C [5], withAZ = KJ−1M + P, BZ = KJ−1N + Q, (37)CZ = LJ−1M + R, DZ = LJ−1N + S. (38)Those parameters are grouped in a single matrix Z ∈ Rl+n+p2×l+n+m2 asZ ,−J M NK P QL R S (39)In order to evaluate the sensitivity of H̄ w.r.t Z (this sensitivity is linked tothe good performance of the global scheme in the FWL context [3, 4, 6]), theproblem is then to compute ∂H̄∂Z(or equivalently ∂H̄∂J, ∂H̄∂K, ∂H̄∂L, ..., ∂H̄∂S).Proposition 4 The sensitivity of H̄ with respect to Z is given by∂H̄∂Z=[C̄(zI − Ā)−1M̄1 + M̄2]⊛[N̄1(zI − Ā)−1B̄ + N̄2](40)withM̄1 =(B2LJ−1 0 B2KJ−1 In 0)N̄1 =J−1NC2 J−1M0 InC2 0 (41)M̄2 =(D12LJ−1 0 D12)N̄2 =J−1ND210D21 (42)Proof:Proposition 1 on equation (35) gives∂H̄∂Z=(Il+n+p2 ⊗ C̄(zI − Ā)−1) ∂Ā∂Z(Il+n+m2 ⊗ (zI − Ā)−1B̄)+(Il+n+p2 ⊗ C̄(zI − Ā)−1) ∂B̄∂Z+∂C̄∂Z(Il+n+m2 ⊗ (zI − Ā)−1B̄)+∂D̄∂Z(43)Then, let’s denote Θ =(DZ CZBZ AZ), Ā can be reformulate byĀ =(A 00 0)+(B2 00 In)Θ(C2 00 In)(44)So, with theorem 1∂Ā∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(45)RR n° 676010 T. Hilaire & P. ChevrelBy similar process, it is obvious that∂B̄∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(D210))(46)∂C̄∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(47)∂D̄∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(D210))(48)Then, the derivatives of Θ w.r.t. J , K, L, M , N , P , Q, R and S are∂Θ∂J= −(LJ−1KJ−1)⊛(J−1N J−1M)∂Θ∂K=(0I)⊛(J−1N J−1M)∂Θ∂P=(0I)⊛(0 I)∂Θ∂M=(LJ−1KJ−1)⊛(0 I)∂Θ∂N=(LJ−1KJ−1)⊛(I 0)∂Θ∂L=(I0)⊛(J−1N J−1M)∂Θ∂Q=(0I)⊛(I 0)∂Θ∂R=(I0)⊛(0 I)∂Θ∂S=(I0)⊛(I 0)(49)So, with proposition 3 :∂Θ∂Z=(LJ−1 0 Ip2KJ−1 In 0)⊛J−1N J−1M0 InIm2 0 (50)With property 2, ∂Ā∂Z, ∂B̄∂Z, ∂C̄∂Zand ∂D̄∂Zare obtained and lead to equation (40).5 ConclusionIn order to simplify the calculus of derivative of matrices with respect to anothermatrix, the operator ⊛ has been introduced, and some important propertiesassociated have been stated. Its interest has been illustrated by application inrobust control. Not only the expressions are made more compact, but also therelated numerical computations of transfer function sensitivity are made moretractable. This has led in practical to a successful application in the context ofFWL implementation [6].References[1] J. Brewer. Kronecker products and matrix calculus in system theory. IEEETrans. on Circuits and Systems, 25(9):772–779, September 1978.[2] M. Eslami. Theory of Sensitivity in Dynamic Systems. Springer-Verlag,1994.[3] M. Gevers and G. Li. Parametrizations in Control, Estimation and Filter-ing Probems. Springer-Verlag, 1993.INRIAOn the compact formulation of the derivation of a transfer matrix 11[4] T. Hilaire, P. Chevrel, and J-P. Clauzel. Low parametric sensitivity re-alization design for FWL implementation of MIMO controllers : Theoryand application to the active control of vehicle longitudinal oscillations. InProc. of Control Applications of Optimisation CAO’O6, April 2006.[5] T. Hilaire, P. Chevrel, and Y. Trinquet. Implicit state-space representation :a unifying framework for FWL implementation of LTI systems. In P. Piztek,editor, Proc. of the 16th IFAC World Congress. Elsevier, July 2005.[6] T. Hilaire, P. Chevrel, and J.F. Whidborne. A unifying framework forfinite wordlength realizations. IEEE Trans. on Circuits and Systems, 8(54),August 2007.[7] R. Istepanian and J.F. Whidborne, editors. Digital Controller implemen-tation and fragility. Springer, 2001.[8] T. Kailath. Linear Systems. Prentice-Hall, 1980.[9] R. Redheffer. On a certain linear fractional transformation. J. Math. Phys,39:269–286, 1960.[10] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall, 1996.RR n° 6760Centre de recherche INRIA Rennes – Bretagne AtlantiqueIRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence CedexCentre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-IsmierCentre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’AscqCentre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy CedexCentre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay CedexCentre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay CedexCentre de recherche INRIA Sophia Antipolis – Méditerranée : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis CedexÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)http://www.inria.frISSN 0249-6399

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On the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault Hilaire, Philippe ChevrelTo cite this version:Thibault Hilaire, Philippe Chevrel. On the compact formulation of the derivation of a transfermatrix with respect to another matrix. [Research Report] RR-6760, INRIA. 2008. <inria-00345508>HAL Id: inria-00345508https://hal.inria.fr/inria-00345508Submitted on 9 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.appor t  de  r ech er ch eISSN0249-6399ISRNINRIA/RR--6760--FR+ENGThème NUMINSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUEOn the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault HILAIRE — Philippe CHEVRELN° 6760August 2008Centre de recherche INRIA Rennes – Bretagne AtlantiqueIRISA, Campus universitaire de Beaulieu, 35042 Rennes CedexTéléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71On the compact formulation of the derivation ofa transfer matrix with respect to another matrixThibault HILAIRE∗, Philippe CHEVREL†The`me NUM — Syste`mes nume´riquesE´quipe-Projet CAIRNRapport de recherche n° 6760 — August 2008 — 11 pagesAbstract: A new operator is considered, allowing compact formulae and proofsin the context of the derivation of a transfer matrix with respect to anothermatrix. The problem of the parametric sensitivity matrix calculation is chosenfor illustration. It consists in deriving a Multiple Input Multiple Output transferfunction with respect to a parametric matrix and is central in robust controltheory. Efficient algorithms may be straightforwardly got from the compactanalytic formulae using the operator introduced.Key-words: sensitivity, derivative, robust control∗ CAIRN Project, INRIA/IRISA† IRCCyN, Institut de Recherche en Communication et Cyberntique de NantesUne expression compacte pour la drivationd’une fonction de transfert par une matriceRe´sume´ : Dans ce papier, un nouvel oprateur mathmatique est considr, perme-ttant des expressions compactes dans le contexte de la drivation d’une matricede fonction de transfert par rapport une matrice. Le problme de la sensi-bilit paramtrique sert d’illustration. Il consiste en la drivation d’une fonctionde transfert plusieurs entres et plusieurs sorties par rapport une matrice deparamtres. Ce problme est central en commande robuste, notamment pour larecherche de ralisations efficaces vis--vis de leur implantation numrique.Mots-cle´s : sensibilit, drivation, commande robusteOn the compact formulation of the derivation of a transfer matrix 31 MotivationOne important questioning in control theory is robustness. A property asso-ciated to a given system will be said to be robust if it is still satisfied whenthe system is slightly modified. Different properties may be considered, such asstability or say a certain level of performance measured, e.g. thanks to systemnorms. The problem is crucial in the theory of feedback, because the systemsconsidered are (physical or mathematical) models which are representing theprocess with some approximations and uncertainties [10, 2]. Moreover, thefeedback controller itself may be considered as an uncertain system, due to theinevitable approximation coming from the implementation. In particular, theuse of computers introduces Finite Word Length quantification of the controllerparameters [3, 7].Whatever the case, the computation of the parametric sensitivity of MIMOtransfer function is of particular interest. The problem involves the calculus ofa matrix with respect to (w.r.t.) another matrix. But, as far the authors know,it exists no special techniques or special properties to simplify the expressionsinduced.For example, let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m be fourmatrices defining the MIMO transfer function H1 [8] :H1 :C → Cp×qz 7→ C(zIn −A)−1B +D(1)The sensitivity measure (in the context of FWL implementation) used (e.g.Gevers and Li [3]) isM ,∥∥∥∥∂H1∂A∥∥∥∥22+∥∥∥∥∂H1∂B∥∥∥∥22+∥∥∥∥∂H1∂C∥∥∥∥22+∥∥∥∥∂H1∂D∥∥∥∥22(2)where ‖.‖2 is the transfer function L2-norm.The analytic expression of ∂H1∂A, ∂H1∂B, ∂H1∂Cand ∂H1∂Dare easy to formulate in theSISO1 case (when p = q = 1, so H1(z) ∈ C), but these expressions are lessobvious in the MIMO case.After having recalled general definitions and classical properties on matrixand transfer function derivatives in section 2, section 3 introduces a new oper-ator ⊛ to simplify derivative expressions in the MIMO case. Finally, a morecomplicated example is solved in section 4, before conclusion in section 5.2 DefinitionsLet’s introduce some interesting definitions about matrix and transfer functionderivatives.Definition 1 (Scalar derivative w.r.t. a matrix) Let X ∈ Rm×n be a ma-trix and f : Rm×n → C a scalar function of X, differentiable w.r.t. each elementof X.1S ingle Input S ingle OutputRR n° 67604 T. Hilaire & P. ChevrelThe derivative of f w.r.t. X is defined as the matrix ∂f∂X∈ Cm×n such(∂f∂X)i,j,∂f∂Xi,j(3)where Xi,j is the (i, j) element of X.This derivative defines the sensitivity of f w.r.t. X.This definition can be extended to functions with values in Cp×l as follow :Definition 2 (Derivative of a matrix w.r.t. a matrix) Let X ∈ Rm×n bea matrix and f : Rm×n → Cp×l a function of X, where each component of f isdifferentiable w.r.t. each element of X.The derivative of f with respect to X is a matrix of Cmp×nl partitioned in m×nmatrix blocks of Cp×l. Each (i, j)th block is defined by∂f∂Xi,j∈ Cp×l (4)Then∂f∂X,∂f∂X1,1∂f∂X1,2. . . ∂f∂X1,n∂f∂X2,1∂f∂X2,2. . . ∂f∂X2,n....... . ....∂f∂Xm,1∂f∂Xm,2. . . ∂f∂Xm,n (5)which can also be written as :∂f∂X=m∑r=1n∑s=1Em,nr,s ⊗∂f∂Xr,s(6)where the matrices En,mr,s of Rn×m are the elementary matrices defined by(En,mr,s)i,j, δr,iδs,j (7)and ⊗ is the Kronecker product.Remark 1 When X is a row vector and f(X) a column vector, ∂f∂Xis thejacobian matrix of f .Other definitions of the derivative of a matrix w.r.t. a matrix are sometimesused : ∂A∂X,∂Vec(A)∂Vec(X) or∂A∂X,∂Vec(A)∂Vec(X)⊤, in order to get a jacobian matrix. Itexempts to consider blocks of matrices ∂f∂Xi,j. However some useful propositions(like proposition 1 and theorem 1) are more easy according to definition 2.Remark 2 ∂X∂X, the derivative of X ∈ Rp×q w.r.t. itself, is a constant matrixof Rp2×q2 such that :∂X∂X=p∑r=1l∑s=1(Ep,lr,s ⊗ Ep,lr,s)(8)=p∑r=1l∑s=1Ep2,l2(r−1)p+r,(s−1)l+s (9)INRIAOn the compact formulation of the derivation of a transfer matrix 5The transfer function sensitivity (SISO and MIMO case) is defined belowDefinition 3 (Transfer function sensitivity) Let X ∈ Rm×n be a matrixand H : C → Cp×q the transfer function which associate H(z) to all z ∈ C.H(z) is supposed to be parametrized by X and to be differentiable w.r.t. eachelement of X whatever z ∈ C.Finally, the sensitivity function of H w.r.t. X is the transfer function denotedby ∂H∂X, such that∂H∂X:C → Cpm×qnz 7→ ∂(H(z))∂X(10)Remark 3 The subsequent properties on matrix derivatives also hold for trans-fer function, without any modifications.The General Leibniz product rule for derivative of a product of matricesw.r.t. a matrix is the main property used when dealing with matrix derivative.Proposition 1 Let’s consider X ∈ Rk×l, Y ∈ Rl×m and Z ∈ Rp×l. Thederivative of the product XY with respect to Z is∂(XY )∂Z=∂X∂Z(Il ⊗ Y ) + (Ip ⊗X)∂Y∂Z(11)Proof:The proof can be found in [1].Remark 4 Proposition 1 applied to∂(Y Y −1)∂Zleads to∂(Y −1)∂Z= −(Ip ⊗ Y−1) ∂Y∂Z(Il ⊗ Y−1)(12)When considering the initial example H1 (eq. (1)), it comes then (∀z ∈ C)∂H1∂A(z) =(In ⊗ C(zIn −A)−1) ∂A∂A(In ⊗ (zIn −A)−1B)(13)∂H1∂B(z) =(In ⊗ C(zIn −A)−1) ∂B∂B(14)∂H1∂C(z) =∂C∂C(In ⊗ (zIn −A)−1B)(15)∂H1∂D(z) =∂D∂D(16)The formulae given in equations (13) to (16) suffer from some drawbacks.First, they are expressed with constant matrices ∂A∂A, ∂B∂B, ∂C∂Cand ∂D∂Dthat donot depends on A, B, C or D (they only depend on their dimensions). Secondly,their manipulations may be rather tedious when dealing with such complicateexpressions as in section 4. Lastly, a more compact form is possible, as proposedin section 3.RR n° 67606 T. Hilaire & P. Chevrel3 A new operator for compact derivatives for-mulaeIn order to simplify the expressions, the new operator ⊛ is proposed. Threepropositions show how to use this operator in classical linear derivative problem.Definition 4 Let G and H be two transfer functions in Cm×p and Cl×n.The operator ⊛ is defined byG⊛H , Vec(G).[Vec(H⊤)]⊤(17)where Vec is the usual operator that vectorize a matrix. It corresponds to theproduct of each element of G with each element of H, in a particular order.This operator is used to state the main proposition of this paper, whichencompass and simplify the Leibnitz rule of proposition 1 :Theorem 1 Let X be a matrix in Rp×l and G, H be two transfer functions withvalues respectively in Cm×p and Cl×n. G and H are supposed to be independentw.r.t. X. Then∂(GXH)∂X= G⊛H (18)∂(GX−1H)∂X= (GX−1)⊛ (X−1H) (19)Proof:From proposition 1 and equation (9),∂(GXH)∂X=∑r,s(Ip ⊗G)Ep2,l2(r−1)p+r,(s−1)l+s (Il ⊗H) (20)Considering relation(AEp2,l2i,j B)= A•,iBj,• where A•,i denotes the ith columnof A and Bj,• the jth row of B, then∂(GXH)∂X=∑r,s(Ip ⊗G)•,(r−1)p+r (Il ⊗H)(s−1)l+s,• (21)The term (Ip ⊗G)•,(r−1)p+r corresponds to(Ip ⊗G)⊤•,(r−1)p+r =(0 . . . G•,r 0 . . .↑ rth block)(22)INRIAOn the compact formulation of the derivation of a transfer matrix 7so it is possible to write(Ip ⊗G)•,(r−1)p+r (Il ⊗H)(s−1)l+s,• = Ep,lr,s ⊗ (G•,r.Hs,•) (23)=0G•,r.Hs,•0(r, s)th block (size m× n)(24)Finally, (G•,r.Hs,•)i,j = Gi,r.Hs,j ; so(∂(GXH)∂X)(r−1)m+i,(s−1)n+j= Gi,r.Hs,j (25)and then,∂(GXH)∂X= Vec(G).[Vec(H⊤)]⊤(26)Using the previous result leads equations (13) to (16) to the simplified ex-pressions :∂H1∂A=(C(zIn −A)−1)⊛((zIn −A)−1B)(27)∂H1∂B=(C(zIn −A)−1)⊛ In (28)∂H1∂C= In ⊛((zIn −A)−1B)(29)∂H1∂D= In ⊛ In (30)Practically, the additional following properties are useful to simplify thederivative task and get compacter expressions. The proofs are trivial.Proposition 2(Ip ⊗G) (X ⊛ Y ) (Il ⊗H) = (GX)⊛ (Y H) (31)Proposition 3(A⊛ C A⊛DB ⊛ C B ⊛D)=(A B)⊛(CD)(32)RR n° 67608 T. Hilaire & P. Chevrel4 Application to the case of closed-loop transferfunctionIn this section the problem consisting in deriving the redheffer product [9],and its specialization the lower linear fractional transformation of two transferfunctions, is studied. The problem has an important practical interest in thecontext of robust control theory [10], when considering the model uncertaintiesof the process or even of the controller in the sense of FWL implementation [3].Let’s consider a plant P controlled by a controller C in a standard form [10](see fig. 1). W (k) ∈ Rp1 and Z(k) ∈ Rm1 are the exogenous inputs and outputs(to control), whereas U(k) ∈ Rp2 and Y (k) ∈ Rm2 are the control and measuresignals.Pm1m2Cp1p2W (k) Z(k)U(k) Y (k)SFigure 1: Closed-loop system consideredThe plant P is defined by the recurrent relationXP(k + 1) = AXP(k) + B1W (k) + B2U(k)Z(k) = C1XP(k) + D11W (k) + D12U(k)Y (k) = C2XP(k) + D21W (k)(33)where A ∈ RnP×nP , B1 ∈ RnP×p1 , B2 ∈ RnP×p2 , C1 ∈ Rm1×nP , C2 ∈ Rm2×nP ,D11 ∈ Rm1×p1 , D12 ∈ Rm1×p2 , D21 ∈ Rm2×p1 . Note that the D22 term is null.The controller is defined by{X(k + 1) = AZX(k) +BZY (k)U(k) = CZX(k) +DZY (k)(34)with AZ ∈ Rn×n, BZ ∈ Rn×m2 , CZ ∈ Rp2×n and DZ ∈ Rp2×m2 .The transfer function of the closed-loop system S is then (lower linear fractionaltransformation)H¯ : z → C¯(zInP+n − A¯)−1B¯ + D¯ (35)with A¯ ∈ RnP+n×nP+n, B¯ ∈ RnP+n×p1 , C¯ ∈ Rm1×nP+n and D¯ ∈ Rm1×p1 andA¯ =(A+B2DZC2 B2CZBZC2 AZ)B¯ =(B1 +B2DZD21BZD21)C¯ =(C1 +D12DZC2 D12CZ)D¯ = D11 +D12DZD21(36)Last point, the matrices AZ , BZ , CZ and DZ depends on matrices J , K, L,M , N , P , Q, R and S (J ∈ Rl×l, K ∈ Rn×l, L ∈ Rp2×l, M ∈ Rl×n, N ∈ Rl×m2 ,INRIAOn the compact formulation of the derivation of a transfer matrix 9P ∈ Rn×n, Q ∈ Rn×m2 , R ∈ Rp2×n, S ∈ Rp2×m2) that contain the exactcoefficients for the realization of C [5], withAZ = KJ−1M + P, BZ = KJ−1N +Q, (37)CZ = LJ−1M +R, DZ = LJ−1N + S. (38)Those parameters are grouped in a single matrix Z ∈ Rl+n+p2×l+n+m2 asZ ,−J M NK P QL R S (39)In order to evaluate the sensitivity of H¯ w.r.t Z (this sensitivity is linked tothe good performance of the global scheme in the FWL context [3, 4, 6]), theproblem is then to compute ∂H¯∂Z(or equivalently ∂H¯∂J, ∂H¯∂K, ∂H¯∂L, ..., ∂H¯∂S).Proposition 4 The sensitivity of H¯ with respect to Z is given by∂H¯∂Z=[C¯(zI − A¯)−1M¯1 + M¯2]⊛[N¯1(zI − A¯)−1B¯ + N¯2](40)withM¯1 =(B2LJ−1 0 B2KJ−1 In 0)N¯1 =J−1NC2 J−1M0 InC2 0 (41)M¯2 =(D12LJ−1 0 D12)N¯2 =J−1ND210D21 (42)Proof:Proposition 1 on equation (35) gives∂H¯∂Z=(Il+n+p2 ⊗ C¯(zI − A¯)−1) ∂A¯∂Z(Il+n+m2 ⊗ (zI − A¯)−1B¯)+(Il+n+p2 ⊗ C¯(zI − A¯)−1) ∂B¯∂Z+∂C¯∂Z(Il+n+m2 ⊗ (zI − A¯)−1B¯)+∂D¯∂Z(43)Then, let’s denote Θ =(DZ CZBZ AZ), A¯ can be reformulate byA¯ =(A 00 0)+(B2 00 In)Θ(C2 00 In)(44)So, with theorem 1∂A¯∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(45)RR n° 676010 T. Hilaire & P. ChevrelBy similar process, it is obvious that∂B¯∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(D210))(46)∂C¯∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(47)∂D¯∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(D210))(48)Then, the derivatives of Θ w.r.t. J , K, L, M , N , P , Q, R and S are∂Θ∂J= −(LJ−1KJ−1)⊛(J−1N J−1M)∂Θ∂K=(0I)⊛(J−1N J−1M)∂Θ∂P=(0I)⊛(0 I)∂Θ∂M=(LJ−1KJ−1)⊛(0 I)∂Θ∂N=(LJ−1KJ−1)⊛(I 0)∂Θ∂L=(I0)⊛(J−1N J−1M)∂Θ∂Q=(0I)⊛(I 0)∂Θ∂R=(I0)⊛(0 I)∂Θ∂S=(I0)⊛(I 0)(49)So, with proposition 3 :∂Θ∂Z=(LJ−1 0 Ip2KJ−1 In 0)⊛J−1N J−1M0 InIm2 0 (50)With property 2, ∂A¯∂Z, ∂B¯∂Z, ∂C¯∂Zand ∂D¯∂Zare obtained and lead to equation (40).5 ConclusionIn order to simplify the calculus of derivative of matrices with respect to anothermatrix, the operator ⊛ has been introduced, and some important propertiesassociated have been stated. Its interest has been illustrated by application inrobust control. Not only the expressions are made more compact, but also therelated numerical computations of transfer function sensitivity are made moretractable. This has led in practical to a successful application in the context ofFWL implementation [6].References[1] J. Brewer. Kronecker products and matrix calculus in system theory. IEEETrans. on Circuits and Systems, 25(9):772–779, September 1978.[2] M. Eslami. Theory of Sensitivity in Dynamic Systems. Springer-Verlag,1994.[3] M. Gevers and G. Li. Parametrizations in Control, Estimation and Filter-ing Probems. Springer-Verlag, 1993.INRIAOn the compact formulation of the derivation of a transfer matrix 11[4] T. Hilaire, P. Chevrel, and J-P. Clauzel. Low parametric sensitivity re-alization design for FWL implementation of MIMO controllers : Theoryand application to the active control of vehicle longitudinal oscillations. InProc. of Control Applications of Optimisation CAO’O6, April 2006.[5] T. Hilaire, P. Chevrel, and Y. Trinquet. Implicit state-space representation :a unifying framework for FWL implementation of LTI systems. In P. Piztek,editor, Proc. of the 16th IFAC World Congress. Elsevier, July 2005.[6] T. Hilaire, P. Chevrel, and J.F. Whidborne. A unifying framework forfinite wordlength realizations. IEEE Trans. on Circuits and Systems, 8(54),August 2007.[7] R. Istepanian and J.F. Whidborne, editors. Digital Controller implemen-tation and fragility. Springer, 2001.[8] T. Kailath. Linear Systems. Prentice-Hall, 1980.[9] R. Redheffer. On a certain linear fractional transformation. J. Math. Phys,39:269–286, 1960.[10] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall, 1996.RR n° 6760Centre de recherche INRIA Rennes – Bretagne AtlantiqueIRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Centre de recherche INRIA Bordeaux – Sud Ouest : Domaine Universitaire - 351, cours de la Libération - 33405 Talence CedexCentre de recherche INRIA Grenoble – Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-IsmierCentre de recherche INRIA Lille – Nord Europe : Parc Scientifique de la Haute Borne - 40, avenue Halley - 59650 Villeneuve d’AscqCentre de recherche INRIA Nancy – Grand Est : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy CedexCentre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay CedexCentre de recherche INRIA Saclay – Île-de-France : Parc Orsay Université - ZAC des Vignes : 4, rue Jacques Monod - 91893 Orsay CedexCentre de recherche INRIA Sophia Antipolis – Méditerranée : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis CedexÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)http://www.inria.frISSN 0249-6399

HAL Id: inria-00348344https://hal.inria.fr/inria-00348344Submitted on 18 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.On the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault Hilaire, Philippe ChevrelTo cite this version:Thibault Hilaire, Philippe Chevrel. On the compact formulation of the derivation of a transfer matrixwith respect to another matrix. [Research Report] PI 1916, 2008, pp.13. ￿inria-00348344￿I   R   I   S   AINSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRESP U  B  L  I  C  A  T  I  O  NI  N  T  E  R  N  ENoI R I S ACAMPUS UNIVERSITAIRE DE BEAULIEU - 35042 RENNES CEDEX - FRANCEISSN 1166-86871916ON THE COMPACT FORMULATION OF THE DERIVATIONOF A TRANSFER MATRIX WITH RESPECT TO ANOTHERMATRIXTHIBAULT HILAIRE, PHILIPPE CHEVRELINSTITUT DE RECHERCHE EN INFORMATIQUE ET SYSTÈMES ALÉATOIRESCampus de Beaulieu – 35042 Rennes Cedex – FranceTél. : (33) 02 99 84 71 00 – Fax : (33) 02 99 84 71 71http://www.irisa.frOn the compact formulation of the derivation of atransfer matrix with respect to another matrixThibault HILAIRE* , Philippe CHEVREL**Systèmes numériquesProjet CAIRNPublication interne n˚1916 — August 2008 — 11 pagesAbstract: A new operator is considered, allowing compact formulae and proofs in thecontext of the derivation of a transfer matrix with respect to another matrix. The problem ofthe parametric sensitivity matrix calculation is chosen for illustration. It consists in derivinga Multiple Input Multiple Output transfer function with respect to a parametric matrix andis central in robust control theory. Efficient algorithms may be straightforwardly got fromthe compact analytic formulae using the operator introduced.Key-words: sensitivity, derivative, robust control(Résumé : tsvp)* CAIRN Project, INRIA/IRISA** IRCCyN, Institut de Recherche en Communication et Cybernétique de NantesCentre National de la Recherche Scientifique Institut National de Recherche en Informatique(UMR 6074) Université de Rennes 1 – Insa de Rennes et en Automatique – unité de recherche de RennesUne expression compacte pour la dérivation d’unefonction de transfert par une matriceRésumé : Dans ce papier, un nouvel opérateur mathématique est considéré, permet-tant des expressions compactes dans le contexte de la dérivation d’une matrice de fonctionde transfert par rapport à une matrice. Le problème de la sensibilité paramétrique sertd’illustration. Il consiste en la dérivation d’une fonction de transfert à plusieurs entréeset plusieurs sorties par rapport à une matrice de paramètres. Ce problème est central encommande robuste, notamment pour la recherche de réalisations efficaces vis-à-vis de leurimplantation numérique.Mots clés : sensibilité, dérivation, commande robusteOn the compact formulation of the derivation of a transfer matrix 31 MotivationOne important questioning in control theory is robustness. A property associated to agiven system will be said to be robust if it is still satisfied when the system is slightlymodified. Different properties may be considered, such as stability or say a certain levelof performance measured, e.g. thanks to system norms. The problem is crucial in thetheory of feedback, because the systems considered are (physical or mathematical) modelswhich are representing the process with some approximations and uncertainties [10, 2].Moreover, the feedback controller itself may be considered as an uncertain system, due tothe inevitable approximation coming from the implementation. In particular, the use ofcomputers introduces Finite Word Length quantification of the controller parameters [3, 7].Whatever the case, the computation of the parametric sensitivity of MIMO transferfunction is of particular interest. The problem involves the calculus of a matrix with respectto (w.r.t.) another matrix. But, as far the authors know, it exists no special techniques orspecial properties to simplify the expressions induced.For example, let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m be four matricesdefining the MIMO transfer function H1 [8] :H1 :C → Cp×qz → C(zIn − A)−1B + D (1)The sensitivity measure (in the context of FWL implementation) used (e.g. Gevers andLi [3]) isM ∥∥∥∥∂H1∂A∥∥∥∥22+∥∥∥∥∂H1∂B∥∥∥∥22+∥∥∥∥∂H1∂C∥∥∥∥22+∥∥∥∥∂H1∂D∥∥∥∥22(2)where ‖.‖2 is the transfer function L2-norm.The analytic expression of ∂H1∂A ,∂H1∂B ,∂H1∂C and∂H1∂D are easy to formulate in the SISO1 case(when p = q = 1, so H1(z) ∈ C), but these expressions are less obvious in the MIMO case.After having recalled general definitions and classical properties on matrix and transferfunction derivatives in section 2, section 3 introduces a new operator  to simplify derivativeexpressions in the MIMO case. Finally, a more complicated example is solved in section 4,before conclusion in section 5.2 DefinitionsLet’s introduce some interesting definitions about matrix and transfer function derivatives.Definition 1 (Scalar derivative w.r.t. a matrix) Let X ∈ Rm×n be a matrix and f :Rm×n → C a scalar function of X, differentiable w.r.t. each element of X.1S ingle Input S ingle OutputPI n˚19164 T. Hilaire & P. ChevrelThe derivative of f w.r.t. X is defined as the matrix ∂f∂X ∈ Cm×n such(∂f∂X)i,j ∂f∂Xi,j(3)where Xi,j is the (i, j) element of X.This derivative defines the sensitivity of f w.r.t. X.This definition can be extended to functions with values in Cp×l as follow :Definition 2 (Derivative of a matrix w.r.t. a matrix) Let X ∈ Rm×n be a matrixand f : Rm×n → Cp×l a function of X, where each component of f is differentiable w.r.t.each element of X.The derivative of f with respect to X is a matrix of Cmp×nl partitioned in m × n matrixblocks of Cp×l. Each (i, j)th block is defined by∂f∂Xi,j∈ Cp×l (4)Then∂f∂X∂f∂X1,1∂f∂X1,2. . . ∂f∂X1,n∂f∂X2,1∂f∂X2,2. . . ∂f∂X2,n....... . ....∂f∂Xm,1∂f∂Xm,2. . . ∂f∂Xm,n (5)which can also be written as :∂f∂X=m∑r=1n∑s=1Em,nr,s ⊗∂f∂Xr,s(6)where the matrices En,mr,s of Rn×m are the elementary matrices defined by(En,mr,s)i,j δr,iδs,j (7)and ⊗ is the Kronecker product.Remark 1 When X is a row vector and f(X) a column vector, ∂f∂X is the jacobian matrixof f .Other definitions of the derivative of a matrix w.r.t. a matrix are sometimes used : ∂A∂X ∂Vec(A)∂Vec(X) or∂A∂X ∂Vec(A)∂Vec(X) , in order to get a jacobian matrix. It exempts to consider blocksof matrices ∂f∂Xi,j . However some useful propositions (like proposition 1 and theorem 1) aremore easy according to definition 2.IrisaOn the compact formulation of the derivation of a transfer matrix 5Remark 2 ∂X∂X , the derivative of X ∈ Rp×q w.r.t. itself, is a constant matrix of Rp2×q2such that :∂X∂X=p∑r=1l∑s=1(Ep,lr,s ⊗ Ep,lr,s)(8)=p∑r=1l∑s=1Ep2,l2(r−1)p+r,(s−1)l+s (9)The transfer function sensitivity (SISO and MIMO case) is defined belowDefinition 3 (Transfer function sensitivity) Let X ∈ Rm×n be a matrix and H : C →Cp×q the transfer function which associate H(z) to all z ∈ C. H(z) is supposed to beparametrized by X and to be differentiable w.r.t. each element of X whatever z ∈ C.Finally, the sensitivity function of H w.r.t. X is the transfer function denoted by ∂H∂X , suchthat∂H∂X:C → Cpm×qnz → ∂(H(z))∂X(10)Remark 3 The subsequent properties on matrix derivatives also hold for transfer function,without any modifications.The General Leibniz product rule for derivative of a product of matrices w.r.t. a matrixis the main property used when dealing with matrix derivative.Proposition 1 Let’s consider X ∈ Rk×l, Y ∈ Rl×m and Z ∈ Rp×l. The derivative of theproduct XY with respect to Z is∂(XY )∂Z=∂X∂Z(Il ⊗ Y ) + (Ip ⊗ X)∂Y∂Z(11)Proof:The proof can be found in [1].Remark 4 Proposition 1 applied to∂(Y Y −1)∂Z leads to∂(Y −1)∂Z= − (Ip ⊗ Y −1) ∂Y∂Z(Il ⊗ Y −1)(12)When considering the initial example H1 (eq. (1)), it comes then (∀z ∈ C)∂H1∂A(z) =(In ⊗ C(zIn − A)−1) ∂A∂A(In ⊗ (zIn − A)−1B)(13)∂H1∂B(z) =(In ⊗ C(zIn − A)−1) ∂B∂B(14)∂H1∂C(z) =∂C∂C(In ⊗ (zIn − A)−1B)(15)∂H1∂D(z) =∂D∂D(16)PI n˚19166 T. Hilaire & P. ChevrelThe formulae given in equations (13) to (16) suffer from some drawbacks. First, theyare expressed with constant matrices ∂A∂A ,∂B∂B ,∂C∂C and∂D∂D that do not depends on A, B,C or D (they only depend on their dimensions). Secondly, their manipulations may berather tedious when dealing with such complicate expressions as in section 4. Lastly, a morecompact form is possible, as proposed in section 3.3 A new operator for compact derivatives formulaeIn order to simplify the expressions, the new operator  is proposed. Three propositionsshow how to use this operator in classical linear derivative problem.Definition 4 Let G and H be two transfer functions in Cm×p and Cl×n.The operator  is defined byG  H  Vec(G).[Vec(H)](17)where Vec is the usual operator that vectorize a matrix. It corresponds to the product of eachelement of G with each element of H, in a particular order.This operator is used to state the main proposition of this paper, which encompass andsimplify the Leibnitz rule of proposition 1 :Theorem 1 Let X be a matrix in Rp×l and G, H be two transfer functions with valuesrespectively in Cm×p and Cl×n. G and H are supposed to be independent w.r.t. X. Then∂(GXH)∂X= G  H (18)∂(GX−1H)∂X= (GX−1)  (X−1H) (19)Proof:From proposition 1 and equation (9),∂(GXH)∂X=∑r,s(Ip ⊗ G)Ep2,l2(r−1)p+r,(s−1)l+s (Il ⊗ H) (20)Considering relation(AEp2,l2i,j B)= A•,iBj,• where A•,i denotes the ith column of A andBj,• the jth row of B, then∂(GXH)∂X=∑r,s(Ip ⊗ G)•,(r−1)p+r (Il ⊗ H)(s−1)l+s,• (21)The term (Ip ⊗ G)•,(r−1)p+r corresponds to(Ip ⊗ G)•,(r−1)p+r =(0 . . . G•,r 0 . . .↑ rth block)(22)IrisaOn the compact formulation of the derivation of a transfer matrix 7so it is possible to write(Ip ⊗ G)•,(r−1)p+r (Il ⊗ H)(s−1)l+s,• = Ep,lr,s ⊗ (G•,r.Hs,•) (23)=0G•,r.Hs,•0(r, s)th block (size m × n)(24)Finally, (G•,r.Hs,•)i,j = Gi,r.Hs,j ; so(∂(GXH)∂X)(r−1)m+i,(s−1)n+j= Gi,r.Hs,j (25)and then,∂(GXH)∂X= Vec(G).[Vec(H)](26)Using the previous result leads equations (13) to (16) to the simplified expressions :∂H1∂A=(C(zIn − A)−1)((zIn − A)−1B)(27)∂H1∂B=(C(zIn − A)−1) In (28)∂H1∂C= In ((zIn − A)−1B)(29)∂H1∂D= In  In (30)Practically, the additional following properties are useful to simplify the derivative taskand get compacter expressions. The proofs are trivial.Proposition 2(Ip ⊗ G) (X  Y ) (Il ⊗ H) = (GX)  (Y H) (31)Proposition 3 (A  C A  DB  C B  D)=(A B)(CD)(32)PI n˚19168 T. Hilaire & P. Chevrel4 Application to the case of closed-loop transfer func-tionIn this section the problem consisting in deriving the redheffer product [9], and its special-ization the lower linear fractional transformation of two transfer functions, is studied. Theproblem has an important practical interest in the context of robust control theory [10],when considering the model uncertainties of the process or even of the controller in thesense of FWL implementation [3].Let’s consider a plant P controlled by a controller C in a standard form [10] (see fig. 1).W (k) ∈ Rp1 and Z(k) ∈ Rm1 are the exogenous inputs and outputs (to control), whereasU(k) ∈ Rp2 and Y (k) ∈ Rm2 are the control and measure signals.Figure 1: Closed-loop system consideredThe plant P is defined by the recurrent relationXP(k + 1) = AXP(k) + B1W (k) + B2U(k)Z(k) = C1XP(k) + D11W (k) + D12U(k)Y (k) = C2XP(k) + D21W (k)(33)where A ∈ RnP×nP , B1 ∈ RnP×p1 , B2 ∈ RnP×p2 , C1 ∈ Rm1×nP , C2 ∈ Rm2×nP , D11 ∈Rm1×p1 , D12 ∈ Rm1×p2 , D21 ∈ Rm2×p1 . Note that the D22 term is null.The controller is defined by{X(k + 1) = AZX(k) + BZY (k)U(k) = CZX(k) + DZY (k)(34)with AZ ∈ Rn×n, BZ ∈ Rn×m2 , CZ ∈ Rp2×n and DZ ∈ Rp2×m2 .The transfer function of the closed-loop system S is then (lower linear fractional transfor-mation)H̄ : z → C̄ (zInP+n − Ā)−1 B̄ + D̄ (35)IrisaOn the compact formulation of the derivation of a transfer matrix 9with Ā ∈ RnP+n×nP+n, B̄ ∈ RnP+n×p1 , C̄ ∈ Rm1×nP+n and D̄ ∈ Rm1×p1 andĀ =(A + B2DZC2 B2CZBZC2 AZ)B̄ =(B1 + B2DZD21BZD21)C̄ =(C1 + D12DZC2 D12CZ)D̄ = D11 + D12DZD21(36)Last point, the matrices AZ , BZ , CZ and DZ depends on matrices J , K, L, M , N , P , Q,R and S (J ∈ Rl×l, K ∈ Rn×l, L ∈ Rp2×l, M ∈ Rl×n, N ∈ Rl×m2 , P ∈ Rn×n, Q ∈ Rn×m2 ,R ∈ Rp2×n, S ∈ Rp2×m2) that contain the exact coefficients for the realization of C [5], withAZ = KJ−1M + P, BZ = KJ−1N + Q, (37)CZ = LJ−1M + R, DZ = LJ−1N + S. (38)Those parameters are grouped in a single matrix Z ∈ Rl+n+p2×l+n+m2 asZ −J M NK P QL R S (39)In order to evaluate the sensitivity of H̄ w.r.t Z (this sensitivity is linked to the goodperformance of the global scheme in the FWL context [3, 4, 6]), the problem is then tocompute ∂H̄∂Z (or equivalently∂H̄∂J ,∂H̄∂K ,∂H̄∂L , ...,∂H̄∂S ).Proposition 4 The sensitivity of H̄ with respect to Z is given by∂H̄∂Z=[C̄(zI − Ā)−1 M̄1 + M̄2][N̄1(zI − Ā)−1 B̄ + N̄2](40)withM̄1 =(B2LJ−1 0 B2KJ−1 In 0)N̄1 =J−1NC2 J−1M0 InC2 0 (41)M̄2 =(D12LJ−1 0 D12)N̄2 =J−1ND210D21 (42)Proof:Proposition 1 on equation (35) gives∂H̄∂Z=(Il+n+p2 ⊗ C̄(zI − Ā)−1) ∂Ā∂Z(Il+n+m2 ⊗ (zI − Ā)−1B̄)+(Il+n+p2 ⊗ C̄(zI − Ā)−1) ∂B̄∂Z+∂C̄∂Z(Il+n+m2 ⊗ (zI − Ā)−1B̄)+∂D̄∂Z(43)PI n˚191610 T. Hilaire & P. ChevrelThen, let’s denote Θ =(DZ CZBZ AZ), Ā can be reformulate byĀ =(A 00 0)+(B2 00 In)Θ(C2 00 In)(44)So, with theorem 1∂Ā∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(45)By similar process, it is obvious that∂B̄∂Z=(Il+n+p2 ⊗(B2 00 In))∂Θ∂Z(Il+n+m2 ⊗(D210))(46)∂C̄∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(C2 00 In))(47)∂D̄∂Z=(Il+n+p2 ⊗(D12 0)) ∂Θ∂Z(Il+n+m2 ⊗(D210))(48)Then, the derivatives of Θ w.r.t. J , K, L, M , N , P , Q, R and S are∂Θ∂J = −(LJ−1KJ−1)(J−1N J−1M)∂Θ∂K =(0I)(J−1N J−1M)∂Θ∂P =(0I)(0 I)∂Θ∂M =(LJ−1KJ−1)(0 I)∂Θ∂N =(LJ−1KJ−1)(I 0)∂Θ∂L =(I0)(J−1N J−1M)∂Θ∂Q =(0I)(I 0)∂Θ∂R =(I0)(0 I)∂Θ∂S =(I0)(I 0)(49)So, with proposition 3 :∂Θ∂Z=(LJ−1 0 Ip2KJ−1 In 0)J−1N J−1M0 InIm2 0 (50)With property 2, ∂Ā∂Z ,∂B̄∂Z ,∂C̄∂Z and∂D̄∂Z are obtained and lead to equation (40).5 ConclusionIn order to simplify the calculus of derivative of matrices with respect to another matrix, theoperator  has been introduced, and some important properties associated have been stated.IrisaOn the compact formulation of the derivation of a transfer matrix 11Its interest has been illustrated by application in robust control. Not only the expressionsare made more compact, but also the related numerical computations of transfer functionsensitivity are made more tractable. This has led in practical to a successful application inthe context of FWL implementation [6].References[1] J. Brewer. Kronecker products and matrix calculus in system theory. IEEE Trans. onCircuits and Systems, 25(9):772–779, September 1978.[2] M. Eslami. Theory of Sensitivity in Dynamic Systems. Springer-Verlag, 1994.[3] M. Gevers and G. Li. Parametrizations in Control, Estimation and Filtering Probems.Springer-Verlag, 1993.[4] T. Hilaire, P. Chevrel, and J-P. Clauzel. Low parametric sensitivity realization de-sign for FWL implementation of MIMO controllers : Theory and application to theactive control of vehicle longitudinal oscillations. In Proc. of Control Applications ofOptimisation CAO’O6, April 2006.[5] T. Hilaire, P. Chevrel, and Y. Trinquet. Implicit state-space representation : a unifyingframework for FWL implementation of LTI systems. In P. Piztek, editor, Proc. of the16th IFAC World Congress. Elsevier, July 2005.[6] T. Hilaire, P. Chevrel, and J.F. Whidborne. A unifying framework for finite wordlengthrealizations. IEEE Trans. on Circuits and Systems, 8(54), August 2007.[7] R. Istepanian and J.F. Whidborne, editors. Digital Controller implementation andfragility. Springer, 2001.[8] T. Kailath. Linear Systems. Prentice-Hall, 1980.[9] R. Redheffer. On a certain linear fractional transformation. J. Math. Phys, 39:269–286,1960.[10] K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall, 1996.PI n˚1916

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