## Dataacquisition and identification procedure

The modified step-response test consists of three successive measurements:

1. Q-measurement. Rotor positioned such that the quadrature axis is excited;

Stator

2. D-measurement. Rotor positioned such that the direct axis is excited, while the field winding is short-circuited;

3. Rf-measurement. Stepwise excitation of uf and measuring if;

The "D" and "Q"- measurements have been preceded by remagnetising the system by either a large negative or positive current depending on the direction of the step in order to fix the initial magnetic state on the low boundary of the hysteresis loop (location R in Fig. 4.23).

It should be noted that none of the measurements incorporated anti-aliasing filters. The "Q"-measurement data were collected with a sample rate of 5 kHz and have a 3.1 second measurement period. Both the "D" and "Rf"-measurement data were collected with a sample rate of 1 kHz and a 10.1 second measurement period. Each measurement is repeated at least three times. Example input output data of the "Q"-measurement is shown in Fig. 4.25. Notice that the measured voltage appears to be a modified step instead of an exact step due to the battery's internal voltage drop. This drop, in turn, is caused by the significant current taken from the battery.

Parameter estimation procedure System identification or parameter estimation deals with constructing mathematical models of dynamical systems from experimental data. The parameter estimation procedure picks out the "best" model within the chosen model structure according to the measured input and output sequences

10 8

20 0

Figure 4.25: Time-domain MSR input-output signals for estimation of transfer function Yq(s) (excitation uq and response iq).

and some identification criterion. A common and general method of estimating the parameters in system identification is the prediction-error method [166]. In this method, the parameters of the model are chosen so that the difference between the model's (predicted) output and the measured output is minimized.

Black-box model structures In the system identification approach [166], it is assumed that the "true" system description is given in the following form y(k) = G0(q)u(k)+v(k) (4.13)

where y(k) is the (measured) output signal, G0 (q) is a proper, rational, stable transfer function, q the shift operator, u(k) the (measured) input signal, and v(k) the disturbance signal. The disturbance v(k) is modeled as a filtered sequence of zero-mean, identically distributed, independent random variables (i.e. white noise e(k))

The transfer function H0(q) is restricted to be monic (H0(0) = 1) and minimum phase (i.e. H0-1(q) has a stable inverse). The whole system specification is thus given by specifying the two transfer functions (or filters) G0(q) and H0(q).

Analogous to the true system description given by Eq. (4.13), the model is determined by the relation y(k, 0) = G(q, 0)u(k) + H(q, O)e(k) (4.15)

A particular model corresponds thus to the specification of G(q, e) and H(q, e). One way to parametrize the transfer functions G(q, e) and H(q, 0) is to represent them

Figure 4.25: Time-domain MSR input-output signals for estimation of transfer function Yq(s) (excitation uq and response iq).

as rational functions and let the parameters be the numerator and denominator coefficients. These coefficients are collected in the parameter vector 9, which is to be estimated.

For single-input, single-output systems, the general linear, time invariant, blackbox model structure is given by

where A, B, C, D, and F are polynomials in the delay operator q-i

A(q) = |
1 + aiq - |
1 + •• |
• + ana q |
-na |

B(q) = |
bo + biq |
-i + • |
• • + bnb q |
-nb |

C (q) = |
1 + ciq - |
^ + •• |
• + Cnc q - |
-nc |

D(q) = |
1 + diq - |
^ + •• |
• + dnd q |
nd |

F (q) = |
1 + fiq - |
^ + •• |
• + fnf q~ |
-nf |

The numbers na, nb, nc, nd, and nf are the orders of the respective polynomials. The number nk is the pure time delay (the dead-time) from input to output. Notice that for a sampled data system, nk is equal to 1 if there is no dead-time.

Within the structure of Eq. (4.16) all the usual linear black-box model structures are obtained as special cases. For example, the ARX (Autoregressive with external input) model structure is obtained for nc = nd = nf = 0.

One-step-ahead prediction error A model obtained by identification can be used in many ways, depending on the intended use of the model. For both simulation as well as control design purposes, it is valuable to know at time (k — 1) what the output of the system is likely to be at time k in order to determine the input at time (k — 1). Therefore, the parameter estimate 0 is usually determined so that the one-step-ahead prediction error e(k, 0) = y(k) — y(k\k — 1,0) k = 1, ■■■,N (4.17)

is small for every time instant. In Eq. (4.17) y(k\k — 1,0) denotes the one-step-ahead prediction of y(k) given the data up to and including time (k — 1) based on the parameter vector 0. Observe that the prediction error can only be calculated a posteriori, when measurement y(k) has become available. In Ljung [166] it is shown that the one-step-ahead prediction of y(k) is given by y(k\k — 1,0) = H-1(q,0)G(q,0)u(k) (4.18)

Recall that H -1(0,0) = 1, which means that the predictor depends only on previous output values. Substituting Eq. (4.18) in Eq. (4.17), the prediction error becomes e(k,0) = H -1(q, 0) [y(k) — G(q,0)u(k)] (4.19)

The prediction error is thus exactly that component of y(k) that could not have been predicted at time instant (k — 1). Obviously, in case of a consistent model estimate (i.e. if the estimated model G(q,9), H(q,9) is equal to the true system G0(q), H0(q)), then the prediction error becomes a white noise signal (e(k) = e(k)).

Identification criterion The most simple and most frequently applied identification criterion is a quadratic function on e(k,9), denoted as

fc=i where Zn := {y(1), u(1), y(2), u(2), , y(N), z(N)}.

The estimated parameter vector 0N is now defined as the minimizing element of the criterion Eq. (4.20), i.e.

This criterion is known as the "least squares criterion".

For the FIR (finite impulse response; na = nc = nd = nf =0) and the ARX model structure, the one-step-ahead prediction y(k|k — 1) is a linear function of the polynomial coefficients that constitute the parameter vector 9 (the so-called linear-in-the-parameters property). A consequence of this linearity is that a least squares identification criterion defined on the prediction errors e(k) is a quadratic function in 9. As a result, there will be an analytical expression for the optimal parameter 9 that minimizes the quadratic criterion. For all other model structures, on the other hand, the parameter estimation involves an iterative, numerical search for the best fit.

The developed identification procedure consists of three successive steps:

• Step 1: Pretreatment of data. In general, when the data have been collected from the identification experiment, they have to be pretreated to avoid problems during the parameter identification. The necessary pretreatment of the time-domain data, however, is limited since the MSR-test results in relatively clean signals. This is mainly because the machine has been taken out of operation. The only pretreatment of the data that is required is compensating for the (slight) static non-linearity of the sensors and removal of the offset;

• Step 2: Model structure and order selection. It is trivial that a bad model structure cannot offer a good, low order model, regardless the amount and quality of the available data. The measured input-output data is imported into SITB (graphical user interface to the System Identification Toolbox) [287]. First, an initial model order estimate is made by estimating 1100 ARX-models.

Recall that a linear regression estimate ARX is generally the most simple model to start with, particularly because of its computationally simplicity [166]. If the resulting model, however, produces an unsatisfactory simulation error and/or if the input is correlated with the residual, the model is rejected and another model structure (or order) is selected. This continues until the model produces a satisfactory simulation error and results in zero cross-covariance between residual and past inputs. In that case it can be concluded that a consistent model estimate has been obtained;

• Step 3: Model validation. Model validation is highly important when applying system identification. The parameter estimation procedure picks out the "best" model within the chosen model structure. The crucial question is whether this "best" model is "good enough" for the intended application: timedomain simulation, analysis of dynamic loads, or control design purposes. To this end, the identified models should be confronted with as much information about the process as practical. Here, the outputs of the identified model are compared to the measured ones on a data set that was not used for the fit (the so-called "validation data set").

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