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

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. Key-words: sensitivity, derivative, robust control ∗ CAIRN Project, INRIA/IRISA † IRCCyN, Institut de Recherche en Communication et Cyberntique de Nantes in ria -0 03 45 50 8, v er si on 1 9 D ec 2 00 8 Une expression compacte pour la drivation d’une fonction de transfert par une matrice Résumé : Dans ce papier, un nouvel oprateur mathmatique est considr, permettant des expressions compactes dans le contexte de la drivation d’une matrice de fonction de transfert par rapport une matrice. Le problme de la sensibilit paramtrique sert d’illustration. Il consiste en la drivation d’une fonction de transfert plusieurs entres et plusieurs sorties par rapport une matrice de paramtres. Ce problme est central en commande robuste, notamment pour la recherche de ralisations efficaces vis--vis de leur implantation numrique. Mots-clés : sensibilit, drivation, commande robuste in ria -0 03 45 50 8, v er si on 1 9 D ec 2 00 8 On the compact formulation of the derivation of a transfer matrix 3 1 Motivation One important questioning in control theory is robustness. A property associated to a given system will be said to be robust if it is still satisfied when the system is slightly modified. Different properties may be considered, such as stability or say a certain level of performance measured, e.g. thanks to system norms. The problem is crucial in the theory of feedback, because the systems considered are (physical or mathematical) models which are representing the process with some approximations and uncertainties [10, 2]. Moreover, the feedback controller itself may be considered as an uncertain system, due to the inevitable approximation coming from the implementation. In particular, the use of computers introduces Finite Word Length quantification of the controller parameters [3, 7]. Whatever the case, the computation of the parametric sensitivity of MIMO transfer function is of particular interest. The problem involves the calculus of a 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 expressions induced. For example, let A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rp×m be four matrices defining the MIMO transfer function H1 [8] : H1 : C → Cp×q z 7→ C(zIn −A)−1B +D (1) The sensitivity measure (in the context of FWL implementation) used (e.g. Gevers and Li [3]) is M , ∥∥∥∥∂H1 ∂A ∥∥∥∥2 2 + ∥∥∥∥∂H1 ∂B ∥∥∥∥2 2 + ∥∥∥∥∂H1 ∂C ∥∥∥∥2 2 + ∥∥∥∥∂H1 ∂D ∥∥∥∥2 2 (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 SISO 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 transfer function derivatives in section 2, section 3 introduces a new operator ~ to simplify derivative expressions in the MIMO case. Finally, a more complicated example is solved in section 4, before conclusion in section 5. 2 Definitions Let’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 Output RR n° 6760 in ria -0 03 45 50 8, v er si on 1 9 D ec 2 00 8 4 T. Hilaire & P. Chevrel The derivative of f w.r.t. X is defined as the matrix ∂f ∂X ∈ C m×n such ( ∂f ∂X )