Fast and robust online dynamic system identification

Fast and robust online dynamic system identication Andrzej Latocha

AGH University of Science and Technology Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Krakow, Poland.

Abstract A new method is proposed for black-box linear model identication of a dynamic system embedded at a nearly Gaussian noise. The Gaussian process can highlight areas of the output spaces where the prediction quality is poor due to the lack of data or its complexity, by indicating the higher variance of the predicted mean; the input spaces in which we can reconstruct data represent the expected values. This paper proposed a new approach for the on-line system identication for non-zero initial conditions in the moving window.

Keywords: 1

identication, ltration, estimation, modeling, control.

Introduction

To solve many of the problems in the design, implementation, and operation of automatic control systems, relatively precise mathematical models for the static and dynamic behavior of industrial processes are required. If the underlying physical laws are not known or are only partially known, or if signicant parameters are not known precisely enough, one has to perform an experimental modeling, which is called process or system identication. There are dierent ways to identify systems when the input and output of the system is known. In real systems, the signals are always more or less subject to interference with noise. The expected values can only be estimated.

Most preferably, identication algorithms are fast algorithms that

allow for the identication of dynamic systems around the operating point in real time. There are dierent methods that model dierent situations with respect to the noise. This study was based on the identication methods by the least squares estimation (LSE) [1], [2] for the black-box model (Fig.

2.1).

The work is of use

in eld of Control Systems Engineering. An innovation in the paper is a fast and independent of the initial conditions high precision identication of linear dynamic system.

1

2

Problem formulation

Describing the process using equations with an acceptable error margin is very difcult due to the complexity of the systems structure and the noise distortion. This data [1] has been obtained from an experiment. For identication, we assume that the structure of the black box system under test (Fig. 2.1) will be approximated by the parametric model.

Fig. 2.1: Output error model For this reason, we dene a cost function that minimizes the error between the tested system and the approximated model. The test signal satises the assumptions [1] of zero initial conditions, and the system is embedded in nearly Gaussian noise [1], [2], [3]. The test signal u(k) is permanently changeable, sampling sequences are equal distance in time [1]. For the above assumptions we can solving problem fast and robust identication of linear dynamic system.

3

Least Squares Estimator (LSE)

Assuming the parametric model (3.1),

y(t) = ϕT (t)θ where

y(t) measure the value of the ϕ(t) n-dimensional vector of the θ n-dimensional vector of the unknown coecient. For the

and the

(3.1) data samples measurement

data, we can write equation (3.2).

Y = ΦT θ

(3.2)

Due to account noise, and the inaccuracy of the model, it is better to do an overly large number of samples, additional data improves the accuracy of estimation. For

N  n,

the system is overdetermined, and there is no exact solution. For oversized

samples, the data matrix will not be a square matrix. In this case, the samples matrix can be replaced by a pseudo square matrix. Taking into account the inaccuracy of samples (3.3) [1], [2], [3]

ε(k) = y(k) − ϕT (k)θ, k ∈ N, k > 0 The least squares error (LSE) estimator

θˆ is

cost function (3.4). 2

(3.3)

dened as a vector that minimizes the

N

V (θ) = ||  ||

1 1 1X 2 ε (k) = εT ε = ||ε||2 2 t=1 2 2

(3.4)

is the Euclidean vector norm. For the positive denite matrix

ΦT Φ,

the cost

function (3.4) has a minimum as follows:

ˆ = minV (θ) = V (θ)

1 T [Y Y − Y T Φ(ΦT Φ)−1 ΦT Y ], 2

(3.5)

E = Y − Φθ, 0=

(3.6)

dV = −Y T Φ + θT (ΦT Φ), dθ

(3.7)

θˆ = (ΦT Φ)−1 ΦT Y.

(3.8)

The equation (3.8) in eld of control systems engineering can consider that good or bad numerical task conditioning (3.9) for computing.

(ΦT Φ)(ΦT Φ)−1 = I˜ ≈ I

(3.9)

Perturbations outside the main diagonal show poor conditions for the numeric task. For signicant perturbations outside of the main diagonal obtained numerically, a pseudo-square matrix can be close to the losing row, although it is reversible. If the matrix

ΦT Φ

is known as the Gramian matrix of

Φ,

which possesses several

correct properties such as being a positive semi-denite matrix, and the matrix is known as the moment matrix of regressand by regressors. Finally,

θˆ is

ΦT Y

the coe-

cient vector of the least-squares hyperplane, expressed as (3.8). For this reason, we consider an equation in the eld of discrete time on the moving window. Systems can be described by the autoregressive moving average model with exogenous inputs (ARMAX) (3.10).

y(k) = z −n

C(z −1 ) B(z −1 ) u(k) + ε(k) −1 A(z ) A(z −1 )

(3.10)

We assume the following:

ε=

C(z −1 ) ε(k). A(z −1 )

(3.11)

y(k), u(k) andε(k) are a series of discrete data equally distant in time.

By describing

the system using a dierence equation, the following equation is obtained:

y(k) + a1 y(k − 1) + ... + an y(k − n) + ε = b1 u(k − 1) + ... + bm u(k − m); k  n, n ≥ m; m, n ∈ N; u ∈ R; y ∈ R; ε ∈ R where the linearization error

ε = 0, b0 , b1 , ..., bm ; a0 , a1 , ..., an

(3.12)

search coecients. By

applying discrete Z-transform, the transfer functions of zero initial condition and

ε=0

obtained as follows:

ˆb1 z m−1 + ˆb2 z m−2 + ... + ˆbm−1 z + ˆbm Y (z) ˆ G(z) = = , U (z) 1+a ˆ1 z n−1 + a ˆ2 z n−2 + ... + a ˆn−1 z + a ˆn 3

(3.13)

The discrete transfer function from the denition discrete  z  operator requires the assumption that the signal does not grow faster than the exponential function (3.14).

Z[f ∗ (t)] = Z[f (kT )] = F (z), F (z) =

∞ X

f (kT )z −k

k=−∞ 2

k ∈ N, T ∈ R, f (k) < k!, f (k) < eak ; a > 0, a ∈ R.

4

(3.14)

Non-zero initial condition

A linear system without noise fulll the principle of causality and can be identied by the LSE in any state. For the non-zero initial condition, we have non-continuous function. The problem appears when the system is exposed to noise, such system does not fulll the principle of causality. The goal satisfy zero initial condition on input of the system is the arbitrary imposition zero initial condition on an input signal, an output error is added to the noise. For this reason the discontinuity on the input is modeled as a nonlinearity

f (.).

(4.1).

Fig. 4.1: The system structure of identication model. A nonlinear function

f (.)

(Fig. 4.1) is estimated by the proposed function (4.1),

which optimally carries out the

u ˜(k) (4.2) signal from zero initial condition to its ac-

tual state on interval of data collecting and has insignicant impact on the dynamics of the system (4.4).

h(z) =

1 ; ηN, η > 1 zη

(4.1)

A proposed function (4.1) is dened as the zero input initial reconstructor (ZIIR).

u ˜(k) = Z −1 [h(z)U (z)]; kN; k > 0; u ˆR

(4.2)

k = j, j + 1, ..., N ; N N; jN

(4.3)

N − j  n,

(4.4)

y(k) = z −n

C(z −1 ) B(z −1 ) u ˜(k) + ε(k), −1 A(z ) A(z −1 ) 4

(4.5)

ε=

C(z −1 ) ε(k), A(z −1 )

(4.6)

where (4.6) includes an output error.

y(k)+a1 y(k−1)+...+an y(ki −n)+ε = b1 u ˜(k−1)+...+bm u ˜(k−m); yR, ε ∈ R θˆi = (Φ˜Ti Φ˜i )−1 Φ˜Ti Yi .

(4.7)

(4.8)

By applying discrete Z-transform obtained as follows:

ˆ i (z) = G

ˆb1 z m−1 + ˆb2 z m−2 + ... + ˆbm−1 z + ˆbm 1+a ˆ1 z n−1 + a ˆ2 z n−2 + ... + a ˆn−1 z + a ˆn ˆ u(z)] yˆ(k) = Z −1 [G(z)˜

(4.9)

(4.10)

The mean squar error (MSE) on the window (4.11, 4.12)

ei =

N −j 1 X (yj+k − Eyj+k )2 N −j

(4.11)

N −j 1 X (yj+k − E yˆj+k )2 N −j

(4.12)

k=0

eˆi =

k=0

The optimality function (4.1) can be calculated as follows:

η = f (inf (ei (1(t))), inf (ei (δ(t)))),

(4.13)

optimal identication we obtain for the minimum of error (4.14)

ˆ G(z) = arg inf (ei ).

(4.14)

ˆ i (z) G

5

Numerical experiments

5.1 Example system identication A discrete example system is described by equation (5.1), where sampling discretization step

4t = 0.1[s]. G(z) =

−0.3832z 2 − 0.2338z + 0.06683 z 3 − 1.127z 2 + 0.494z − 0.1129

(5.1)

Using the relationship from section 3, section 4 can identify the model of the system for dierent cases.

5

5.1.1 System without noise A plant that is not subject to noise is identied by the LSE in any state by a minimum number of samples (Fig. 5.1). Oversizing data in relation to the system dimensions is a result of numerical errors.

Fig. 5.1: Response identied model without noise.

5.1.2 System with unit distorted input If introducing into the system unit noise on the input e.g.

u(N −5) = 0, for k > N −5

samples the principle of causality for dynamic system will not be satised.

The

results of such a disturbance are presented in (Table. 1); the quality of these results depend on the number of data gathered before disturbance. Where: the (4.11), of the response identied model on the step signal, the the response identied model on the impulse signal, the signal by function

h(z),

and the

e(j..N )

eδ(t)

eu(1..N )

e1(t)

MSE

MSE (4.11) of

MSE distorted

u

MSE (4.11) response identied model on the

identication window. k N − 10 N − 50 N − 100 N − 1000 N − 3000

e(j..N )

e1(t)

eδ(t)

eu(1..N )

396.8864 423.0187 493.6943 1451.8 3.30e+10

219.9079 219.9079 219.9079 219.9079 220.3036

93.5970 93.5970 93.5970 93.5970 93.5978

0.0151 0.0151 0.0151 0.0151 0.0151

Table 1: Distorted input

u(N − 5) = 0

for

k >N −5

samples.

By comparing the results of (Table.1) and (Table.2), it can be seen that the disturbance of zero at the rst position on the input signal for a large number of samples fullls the principle of causality and zero initial conditions and gives acceptable results.

6

k N − 10 N − 50 N − 100 N − 1000 N − 3000

e(j..N )

e1(t)

eδ(t)

eu(1..N )

114,1275 29.7687 19.6988 0.5552 7.336

0,0142 0.0218 0.0218 4.89e-4 4.90e-4

3,5951 5.6054 5.5959 0.1163 0.1164

0.0156 0.0203 0.0270 0.0076 0.0305

Table 2: Distorted input

u(N − j) = 0

on the rst position input samples.

5.1.3 System with noise on output The next experiment identies the system (5.1) embedded in Gaussian noise in the output.

It was assumed that the output system signal was exposed to a noise of

covariance (5.3).

ErrCov =

N 1 X (Exi − xi )2 N i=1

yErrCov = 0.0494

(5.2)

(5.3)

Table (Table. 3) shows the comparison error (4.11) of the response identied model by the Matlab System Identication Toolbox (MIT) and the proposed algorithm. k eu(1..N ) e(j..N ) e(j..N ) (M IT ) e1(t) e1(t) (M IT ) eδ(t) eδ(t) (M IT )

N − 100

N − 1000

opt(N −2698)

N − 3000

0.0807 1089.6 6.7094 346.5430 3.2970 105.5208 194.5097

0.0235 2.3990 2.5969 22.1021 35.0409 54.2538 63.7391

0.0238 0.6006 0.4274 9.4401 1.5530 41.0381 170.8675

0.0928 5.4661 0.5190 53.5393 2.3846 70.1437 120.9573

Table 3: Response error of identied model with noise on the output. Table (Table. 4) displays the dependence of estimated model coecients on the horizon of data and that of the noise on the output.

coe\k a1 a2 a3 b1 b2 b3

Model

N − 1000

opt(N −2698)

N − 3000

-1.1269 0.4940 -0.1129 -0.3832 -0.2328 0.0668

-0.3604 -0.3185 -0.2609 -0.2139 0.0207 0.0623

-0.3267 -0.2953 -0.223 -0.2843 -0.0998 0.0034

-0.3572 -0.3442 -0.2614 -0.0875 -0.0485 0.0559

Table 4: Dependence of estimated coecients on the discrete samples.

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Fig. 5.2: Response identied model with noise on the output.

k eu(1..N ) e(j..N ) e(j..N ) (M IT ) e1(t) e1(t) (M IT ) eδ(t) eδ(t) (M IT )

N − 100

N − 1000

0.0832 1136.6 76.2574 358.3644 42.2193 108.5663 68.2047

0.0415 2.8476 3.5377 28.4760 54.8410 58.6032 90.8851

opt(N −2713)

N − 3000

0.0651 0.7436 2.6045 12.9394 55.5994 46.8358 92.9625

0.1526 11.6981 4.2952 101.9493 54.7985 46.8358 92.0620

Table 5: Response error of identied model with noise on the input and output.

5.1.4 System with noise on input and output The next experiment identies the system (5.1) embedded in Gaussian noise on the input and output. It was assumed that the input and output system signals were exposed to a noise of covariance (5.4), (5.5).

uErrCov = 0.0494

(5.4)

yErrCov = 0.0494

(5.5)

The gures (Fig. 5.3) show the time constants of the identied system by proposed algorithm and MIT.

coe\k a1 a2 a3 b1 b2 b3

Model

N − 1000

opt(N −2713)

N − 3000

-1.1269 0.4940 -0.1129 -0.3832 -0.2328 0.0668

-0.5878 -0.0964 -0.2650 -0.1129 0.0020 0.0045

-0.5854 -0.0679 -0.2074 -0.1455 -0.0797 0.0765

-0.5950 -0.1212 -0.2642 -0.0306 -0.0123 0.0011

Table 6: Dependence of estimated coecients on the discrete samples. 8

Fig. 5.3: The time constants of the response identied model by proposed algorithm and MIT.

Fig. 5.4: Response identied model with noise on the input and output.

5.2 Laboratory system distillation column 5.2.1 Identication laboratory subsystem of distillation column Identication for the non-Gaussian distribution of noise is done by using registered data from the measurement level from the point point

y = L176,

where:η

= 3, k = 6000

u = L175 to the measurement level 4t = 0.1[s].

samples, discretization step

Fig. 5.5: Direct identication at the operating point.

9

Fig. 5.6: Identication by biased operating points to zero initial condition.

Fig. 5.7: Identication by biased to zero initial condition with optimal ltering.

Fig. 5.8: Identication of around operating point with optimal ltering.

5.2.2 Comparison proposed algorithm and Matlab System Identication Toolbox (MIT) A comparison of identied model transfer function response by proposed algorithm and MIT. (Fig. 5.9).

10

Fig. 5.9: Comparison response identied model by proposed algorithm and MIT, data biased to zero initial condition (deviation model).

GF 1 (z) =

−0.0001766z 2 + 2.856e − 05z + 0.0001745 z 3 − 2.953z 2 + 2.928z − 0.9745

GM IT F 1 (z) =

0.0001183z 2 − 0.0002365z + 0.0001183 z 3 − 3z 2 + 3z − 0.9998

(5.6)

(5.7)

where the error of identication (4.12) obtained

eF 1 = 8.3425, eM IT F 1 = 140.7838

(5.8)

A comparison of an identied model transfer function laboratory subsystem by proposed algorithm and MIT around the operating point (Fig. 5.10). Biased the operating point to the neighborhood of the zero initial condition and using optimal lters the proposed algorithm produces acceptable results. The state matrix almost does not change, but it has a changed coecient of control matrix, impact on the system.

Fig. 5.10: Comparison response identied model by proposed algorithm with optimal ltering and MIT around the operating point.

GF 2 (z) =

2.671e − 05z 2 − 6.632e − 07z + 5.943e − 07 z 3 − 2.954z 2 + 2.93z − 0.976 11

(5.9)

GM IT F 2 (z) =

−0.3832z 2 − 0.2338z + 0.06683 z 3 − 1.127z 2 + 0.494z − 0.1129

(5.10)

where error of identication (4.12) obtained

eF 2 = 14.5630, eM IT F 2 = 153.6657

6

(5.11)

Conclusion

The proposed algorithm gives exact, stable, and repeatable results for systems embedded at a nearly Gaussian noise on input and output.

Is independent of the

initial conditions. The algorithm allows for time constants correction of the identied model by the function. (4.1). In the literature is a lot of theoretical proposals concepts which based on mathematics of dynamics systems [4], [7], [8], [9].

The

problem appear when using these concepts in control systems engineering which require more generalized assumptions: limited precision of data representation, perturbed Gaussian distribution on input and output example in Section 5.2. The study demonstrated that the proposed algorithm is an innowation in the eld of control systems engineering, applied mathematics, returns acceptable quality indexes for real system identication, independent of the system state.

References [1] Soderstrom, T.,Stoica, P.: System Identication. Prentice-Hall, Hemel Hempstead, U.K. 1989. [2] Keesman, K.J.: System Identication. An Introduction. Springer-Verlag London Limited 2011 .DOI 10.1007/978-0-85729-522-4. [3] Isermann R., Munchhof, M.: Identication of Dynamic Systems. An Introduction with Applications. Springer-Verlag Berlin Heidelberg 2011. DOI 10.1007/978-3540-78879-9. [4] Nied¹wiecki, M., Meller, M., Pietrzak, P.: System identication based approach to dynamic weighing revisited. Elsevier. .Mechanical Systems and Signal Processing 80 (2016). [5] Sastry, S.:

Nonlinear Systems Analysis, Stability, and Control. Springer Sci-

ence+Business Media New York 1999 . DOI 10.1007/978-1-4757-3108-8. [6] Greblicki, W., Pawlak, M.:

Nonparametric system identication. Cambridge

University Press 2008. [7] Shijian Donga, Tao Liua, Wei Wanga, Jie Baob, Yi Caoc.:

Identication of

discrete-time output error model for industrial processes with time delay subject to load disturbance. Elsevier. Journal of Process Control. Volume 50, February 2017, Pages 4055. 12

[8] Tao

Liu,

Ke

Yao,

and

Furong

Gao.:

Identication

and

Autotuning

of

Temperature-Control System With Application to Injection Molding. IEEE Transactions On Control Systems Technology, Vol. 17, No. 6, November 2009. [9] Aljanaideh, Khaled F., Bernstein, Dennis S.: Closed-loop identication of unstable systems using noncausal FIR models. International Journal Of Control. Volume: 90 Issue: 2 Pages: 184-201 Published: 2017.

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