Principles of System Identification

26

Linear Multivariable Identification

In this chapter we present a preview of multivariable identification, specifically two aspects:

(i) a time delay estimation method as an extension of the frequency-domain techniques pre-

sented in §22.5, and (ii) a method for linear multivariable identification using principal com-

ponent analysis (PCA). A tutorial introduction to multivariable data analysis using PCA is

provided first. The presentation of this technique is in the context of model identification. Il-

lustrative examples demonstrate the application of PCA to identification of linear steady-state

and dynamic processes.

26.1 MOTIVATION

A large class of real-life processes present themselves as multiple-input, multiple-output (MIMO)

systems. These processes not only include chemical plants but also other ones such as manufacturing

processes, atmospheric process, etc. For the rest of the discussion in this chapter, we shall consider

a MIMO system with n

u

inputs and n

y

outputs under open-loop conditions and term it as a n

u

× n

y

system. What is the main challenge in identification of MIMO systems? It is the correlation among

inputs that precludes us from a direct application of the delay estimation techniques described in

Chapter 22 or the SISO input-output identification methods in Chapters 20 and 21.

Example 26.1: A 2 ×1 system

Consider the following 2 × 1 process:

y

1

[k] = G

11

(q

−1

)u

1

[k] + G

12

(q

−1

)u

2

[k] =

n=∞

X

n=−∞

g

11

[n]u

1

[k − n] +

n=∞

X

n=−∞

g

12

[n]u

2

[k − n]

Treating the MISO system as two separate SISO systems for estimating the IR coefficients

is marked with confounding since

σ

y

1

u

1

[l] =

n=∞

X

n=−∞

g

11

[n]σ

u

1

u

1

[l − n] +

n=∞

X

n=−∞

g

12

[n]σ

u

2

u

1

[l − n]

Therefore, the estimation of IR coefficients for the individual sub-systems is decoupled only

when the inputs are uncorrelated, i.e., σ

u

2

u

1

[l] = 0, ∀l. This has a direct impact on the delay

estimation in the sub-systems.

In the frequency domain, one has

Y

1

(ω) = G

11

(ω)U

1

(ω) + G

12

(ω)U

2

(ω)

Once again estimating the individual FRFs is a coupled problem since the cross-spectral

density of the output with an individual input is affected by the correlation between the two

inputs:

γ

y

1

u

1

(ω) = G

11

(ω)γ

u

1

u

1

(ω) + G

12

(ω)γ

u

2

u

1

(ω)

The same situation applies to the identification of parametric models as well.

Thus, in general, the response functions or the parameters of the individual sub-systems

have to be estimated simultaneously. More importantly, the determination of user-specified

parameters such as orders and delays can be fairly cumbersome.

790

Linear Multivariable Identification 791

Most of the challenges highlighted by the above example are easily overcome by the subspace

identification methods of Chapter 23. However, as we learnt therein, these methods are not neces-

sarily optimal, especially for delay estimation, and are formulated to handle the case of noise-free

inputs. Nevertheless, certain ideas in subspace identification, particularly, that of order determina-

tion can be still used.

In this chapter, we shall outline ideas for handling multivariable system identification using the

ideas of decoupling or de-correlation, however, in a transform space. At first, a method for delay

estimation is presented as an extension of the frequency-domain technique described in §22.5.3

using partial coherence. The concept of partial coherence was described in §11.5 as a method

for discounting for the effects of extraneous or confounding variables in assessing the correlations

between two variables in the frequency domain.

Subsequently, a method for identification based on the principal component analysis (PCA) is

presented. PCA is a tool for multivariate statistical analysis that identifies relations between vari-

ables by projections (transformations) of variables onto a basis space spanned by the eigenvectors

of the covariance (correlation) matrix1. These projections are constrained to be orthogonal to each

other and known as principal components, giving the technique its name.

Thus, the objectives of this chapter are two-fold, namely, to describe a

1. Method for time-delay estimation by numerically decoupling the multivariable system in the

frequency-domain using the partial coherence technique.

2. Method for identification of input-output models using PCA, a multivariate data analysis tech-

nique that works in a transformed space.

PCA belongs to the class of methods for the so-called errors-in-variables (EIV) case where both

input and output measurements are corrupted with errors. An introduction to principal component

analysis from an identification as well as a statistical perspective is presented in §26.3. The time-

delay estimation method, although expressly does not make any assumption on the inputs, is still

suited for the EIV case as well.

26.2 ESTIMATION OF TIME DELAYS IN MIMO SYSTEMS

Section 22.5 described the approach to the delay estimation problem for a SISO system. The method

involved the computation of the phase spectrum, discounting the contributions of time-constant

using the Hilbert transform relations followed by a minimization problem:

D

?

:= sol min

D

X

ω

W (ω) cos  (ω) (26.1a)

 (ω) = φ(ω) − arg

¯

G(ω) − Dω (26.1b)

W (ω) =

|κ(ω)|

2

1 − |κ(ω)|

2

(26.1c)

where |κ

2

(ω)| is the squared coherence function. The quantities φ(ω) and arg

¯

G(ω) are the phases of

the overall and delay-free transfer functions, respectively. Estimates of these quantities are obtained

1Traditionally, PCA is introduced in most parts of the literature as a method for dimensionality reduction by exploiting

inter-variable correlations.