Data-driven input design to maximize information in MIMO system identification

B3 Icke-referentgranskade konferenspublikationer

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Publikationens författare: Kurt E. Häggblom
Publiceringsår: 2019
Förläggare: American Institute of Chemical Engineers
Moderpublikationens namn: 2019 AIChE Annual Meeting Proceedings
ISBN: 978-0-8169-1112-7


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 per­turbed 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 [1].

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 [1]. A drawback of the proposed
method, and various extensions of it summarized in [2], is that dynamics
are not taken into account. Therefore, methods that address dynamics
have been proposed [3–6]. An overview of recent developments is given in
[7]. The objectives of the proposed designs vary, but they tend to
produce nearly uncorrelated outputs, as noted in [8].

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 [8–10]. 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

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 ξ(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 ξi(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) = (k)

that will produce uncorrelated outputs y(k) with desired variances.

Assume that it is known how the perturbation ξj(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 ns×n. Assume Ym, m = 1,...,n2, is known for all combinations of ξj(k) and ui(k). Let x = vecT. The output produced by (2) is then

(3) Y = Y0XT,

where Y0 = [Y1...Yn2], X = xTIn, ⊗ 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.

Obtaining Data

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) = ξj(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) = ξ(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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experiments for robust control. A geometric approach for bivariate
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[2] K. E. Häggblom, “On experiment design for identification of ill-conditioned systems,” IFAC Proceedings Volumes, vol. 47, no. 3, pp. 1428–1433, Aug. 2014.

[3] D. E. Rivera, H. Lee, H. D. Mittelmann, and M. W. Braun, M.W,
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[7] S. Misra, M. Darby, S. Panjwani, and M. Nikolaou, “Design of
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[8] K. E. Häggblom, “Easy ways to design inputs to obtain uncorrelated outputs in MIMO system identification,” IFAC-PapersOnLine, vol. 51, no. 15, pp. 227–232, July 2018.

[9] K. E. Häggblom, “Input designs to obtain uncorrelated outputs in MIMO system identification,” in Proc 13th Int. Symp. on Process Systems Engineering – PSE 2018, San Diego, CA, USA, pp. 637–642, July 2018.

[10] K.E. Häggblom, “A new optimization-based approach to experiment design for dynamic MIMO identification,” IFAC-PapersOnLine, vol. 50, no. 1, pp. 7321–7326, July 2017.

[11] L. Ljung, System Identification: Theory for the User. Upper Saddle River, NJ: Prentice Hall, 1999.

[12] R. Isermann and M. Münchhof, Identification of Dynamic Systems. Berlin and Heidelberg: Springer, 2011.


data-based design, experiment design, ill-conditioned systems, multivariable systems, system identification


Senast uppdaterad 2020-17-02 vid 04:00