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

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Abstrakt

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

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 *ξ*(*k*), *k* = 1,...,*n*_{s}, where *n*_{s}

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 *ξ _{i}*(

*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*) = *Tξ*(*k*)

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

Assume that it is known how the perturbation *ξ _{j}*(

*k*) applied to the input

*u*(

_{i}*k*), with other inputs constant, affects the outputs. Let

*Y*

_{m},

*m*=

*i*+

*n*(

*j*-1), be the obtained output data matrix of size

*n*

_{s}

*×*

*n*. Assume

*Y*

_{m},

*m*= 1,...,

*n*

^{2}, is known for all combinations of

*ξ*(

_{j}*k*) and

*u*(

_{i}*k*). Let

*x*= vec

*T*. The output produced by (2) is then

(3) *Y* = *Y*_{0}*X*^{T},

where *Y*_{0} = [*Y*_{1}...*Y*_{n2}], *X* = *x*^{T}⊗*I** _{n},* ⊗ is the Kronecker product, and

*I*

*is the*

_{n}*n*-dimensional identity matrix. The output covariance matrix is

(4) *P* = *XP*_{0}*X*^{T},

where *P*_{0} = cov*Y*_{0}.

Assume output variances var* y _{i}* = 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 *Y*_{0} can be obtained by doing *n*^{2} experiments. This many experiments is undesirable, of course. An alternative is to do *n* experiments with u_{i}(*k*) = *ξ*_{j}(*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 *Y*_{0}. 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 *Y*_{0},

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.

*References*

[1] C.-W. Koung and J. F. MacGregor, “Design of identification

experiments for robust control. A geometric approach for bivariate

processes.” *Ind. Eng. Chem. Res.*, vol. 32, no. 8, pp. 1658–1666, Aug. 1993.

[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,

“High-purity distillation: using plant-friendly multisine signals to

identify a strongly interactive process,” *IEEE Control Syst. Mag.*, vol. 27, no. 5, pp. 72–89, Oct. 2007.

[4] T. Li and C. Georgakis, “Dynamic input signal design for the identification of constrained systems,” *J. Process Control*, vol. 18, no. 3–4, pp. 332–346, March 2008.

[5] M. L. Darby and M. Nikolaou, “Identification test design for multivariable model-based control: An industrial perspective,” *Control Eng. Pract.*, vol. 22, pp. 165–180, Jan. 2014.

[6] A. Kumar and S. Narasimhan, “Optimal input signal design for identification of interactive and ill-conditioned systems,” *Ind. Eng. Chem. Res.*, vol. 55, no. 14, pp. 4000–4010, March 2016.

[7] S. Misra, M. Darby, S. Panjwani, and M. Nikolaou, “Design of

experiments for control-relevant multivariable model identification: An

overview of some basic recent developments,” *Processes*, vol. 5, no. 3, 30 pp., Sept. 2017.

[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 13 ^{th} 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.