(324c) Data-Driven Input Design to Maximize Information in MIMO System Identification
A problem in the identification of multiple input multiple output (MIMO) systems is that the system outputs in an identification experiment may be strongly correlated if the inputs are perÂturbed by uncorrelated signals, as is standard practice. Such a correlation reduces identifiability because the different outputs essentially contain the same information. In such a case, there is a significant risk that the identification results in a model with different controllability properties than the true plant .
To address the issue of the potential problem with integral controllability, an input design procedure based on an estimate of the static gain matrix was introduced in . A drawback of the proposed method, and various extensions of it summarized in , is that dynamics are not taken into account. Therefore, methods that address dynamics have been proposed [3â6]. An overview of recent developments is given in . The objectives of the proposed designs vary, but they tend to produce nearly uncorrelated outputs, as noted in .
Uncorrelated outputs can be considered to maximize the information content of output data. This is similar to principal component analysis (PCA), where the maximum amount of information is extracted from a data matrix X into a number of components less than the number of columns in X. The component data vectors (score vectors) are norm-bounded linear combinations of the X columns with maximum sample variance and no sample correlation between component vectors (they are orthogonal). In line with PCA, the aim of the input design proposed here is to maximize the information content of the outputs by explicitly producing uncorrelated outputs with sample variances at some maximum desired level.
This author has previously presented a set of model-based input design methods that explicitly minimize the output correlation with specified variances [8â10]. In this contribution, a data-based input design method is proposed. Naturally, this is preferable from a practical point of view. The data are obtained from one or more preliminary experiments with the system to be identified. Various ways of obtaining the required data are considered and the effect of measurement noise is given particular attention. The results are illustrated by application to some ill-conditioned distillation column models.
Outline of the Input Design Procedure
Let the MIMO system to be identified have n inputs and n outputs with numerical values u(k) and y(k), respectively, at sampling instant k. The dynamics of the system can be described by the model
(1) y(k) = G(q)u(k),
where G(q) is a matrix of pulse transfer operators defined through the shift operator q. The input design does not require knowledge of this model, and it is not implied that a model of this form is to be identified; it is mainly introduced to facilitate the description below.
The input design is based on the use of an n-dimensional perturbation signal Î¾(k), k = 1,...,ns, where ns is the sequence length. Usually, this signal is a random binary sequence (RBS), a pseudo-random binary sequence (PRBS), or a multi-sinusoidal signal (MSS). The individual signals Î¾i(k), i = 1,...,n, should preferably be uncorrelated with one another as in a standard MIMO identification experiment. The input design is done by deriving a linear combination
(2) u(k) = TÎ¾(k)
that will produce uncorrelated outputs y(k) with desired variances.
Assume that it is known how the perturbation Î¾j(k) applied to the input ui(k), with other inputs constant, affects the outputs. Let Ym, m = i + n(j-1), be the obtained output data matrix of size nsÃn. Assume Ym, m = 1,...,n2, is known for all combinations of Î¾j(k) and ui(k). Let x = vecT. The output produced by (2) is then
(3) Y = Y0XT,
where Y0 = [Y1...Yn2], X = xTâIn, â is the Kronecker product, and In is the n-dimensional identity matrix. The output covariance matrix is
(4) P = XP0XT,
where P0 = covY0.
Assume output variances var yi = 1, with no correlation between different outputs, is desired. This corresponds to P = I. Various ways of solving (4) to obtain P = I will be considered.
For an n×n system, the output covariance matrix P is defined by n(n+1)/2 parameters. This implies that the same number of independent elements of T is sufficient to determine P. This can be achieved by a triangular or symmetric/skew-symmetric T matrix (including row and column permutations), for example. If a full T matrix is used, some additional property besides output correlation can be optimized. It is possible, for example, to minimize input or output peak values, or the input crest factor.
The matrix Y0 can be obtained by doing n2 experiments. This many experiments is undesirable, of course. An alternative is to do n experiments with ui(k) = Î¾j(k), i = 1,...,n, j = 1. For each experiment, a finite-impulse response (FIR) model can easily be determined, and these models can be used to calculate all submatrices of Y0. The FIR models are an intermediate step; they are not considered to be proper model.
It is also possible to do only one experiment with u(k) = Î¾(k), where all inputs are perturbed simultaneously. If the sequence length is long enough, and the signal components are not too strongly correlated with one another, it is possible to determine n FIR models, required to calculate Y0, from this single experiment. This kind of experiment is the standard identification experiment recommended in textbooks [11, 12], but here the data is used to design a better experiment.
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