The ScaLAPACK software, Blackford and others. (1997), has been designed specifically to achieve high efficiency for a wide range of modern distributed-.
Proceedings of the American Control Conference Chicago, Illinois June 2000
Parallel And Distributed Computational Data Modelling Via Verhaegen & Dewilde's Subspace Method Celso P. Bottura ; Gilmar Barreto; Mauricio JosB Bordon and Annabell D.R. Tamariz gbarreto0dmcsi.fee.unicamp.br LCSI/DMCSI/FEEC/UNICAMP - C.Posta1: 6101 - 13.083-970 - Campinas - SP - Brazil Abstract Input Noke
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Computational data modelling and state-space realization theory for linear discrete time systems are closely related. Verhaegen & Dewilde's identification algorithm is an efficient subspace data modelling method. For increasing its computational performance, in this work we propose a parallelization structure for such an algorithm as well as we present some results for an experiment made on a benchmark we developed.
I Output
1 INTRODUCTION
Figure 1: Schematic Diagram Modeled Computational modelling of multiple input multiple output linear time invariant dynamical systems fiom input-output data sequences is a central problem in signal processing and in control systems analysis and design. Roughly speaking, this problem is equivalent to find a realization for a dynamical system in order to represent input output data sequences. In this way a realization is a choice of a quadruple of matrices that can represent input output data within an acceptable error. Further, the formulation presented in this work leads to a nice approach for engineering applications. In this work, we propose a parallel and distributed computational methodology for faster processing of Verhaegen & Dewilde's algorithm for state space data modelling. It is a high performance algorithm for system identification using linear time invariant finite dimensional state space modelling for multivariable inputoutput data. This algorithm is classified as a subspace based model identification scheme from dynamical system input-output data. This structure is then explored in the obtention of a realization. A common characteristic in the organization of these algorithms is the execution of a factorization, such as QR, followed by a singular value decomposition and the solution of a superdetermined set of linear equations. The scheme to be presented supposes that a sequence of input- output measures of the dynamical system to be identified is available. The figure 1 represents schematically the dynamical 0-7803-5519-9/00 $10.00 0 2000 AACC
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of the System to be
system we are intended to model. State variables representation of a dynamical system is quite useful in engineering, for multivariable industrial processes can be well described using this class of model, and further, nowadays, many control systems design tools are based on this approach. Mathematically, the dynamical system can be described by the following set of difference equations:
where the vectors ut E !RP and yt E W" are, respectively, p inputs and m outputs of the process, sampled at time t. The vector zt E !Rn is the state vector of the process in the discrete time t and contains the n numerical values for the state.
A E Wnxn, B E W n x p , C E S m X na n d D E are the system matrices.
?Rmxp
The problem we are dealing in this work is to identify the state space model, including the order of the system and the matrix quadruple, from an algorithm based on subspace properties, figure 2 .
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the first 2(L 1) columns of a unitary matrix Q with dimension N x N . Tt- - - - - - - Algorithm
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so,
I
I Using the shift invariance property of the matrix I?: the application of singular value decomposition of the matrix R22: yields:
Figure 2: Schematic Diagram of a Procedure for State Space Model Computation 2 HIGH PERFORMANCE IMPLEMENTATION OF A SUBSPACE METHOD
Then the matrices A and C are given by:
Verhaegen & Dewilde's algorithm can be classified as a systems identification procedure based on subspaces: and so: it requires the processing of a matrix associated with subspaces generated by input-output data. The matrix with the system data is constructed in a particular form: Hankel matrix. Using Numerical Linear Algebra operations such as RQ factorization and singular values decomposition on the Hankel matrix we obtain a superdetermined system of matrix linear equations whose solution yields a realization for the supposed dynamical system which gives the input-output data.
CT = U,(1 : 1: :)
(8)
where Uk') is a submatrix composed by the first (i - 1) columns of the matrix U, and Ui2) is a submatrix composed by the last (i - 1) columns of the matrix
U,.
2.1 Matrices A and C Calculation
2.2 Matrices B and D Calculation
Suppose that we apply to the system to be identified a sequence of N independent inputs ui,i = 0,1, 2,. . . ,N - 1, but we can measure a set from t = 0 till t = L , where d 5 L
