and the relation of the Blawith the noise-free output 𝑦given by (3.5), one writes: 𝑧(𝑡) = (𝑔Bla∗ 𝑢)(𝑡) + 𝜈(𝑡) + 𝑛(𝑡) (4.1) where 𝜈 accounts for the nonlinear distortions. In the literature, the noise 𝑛 is often modelled as (filtered) additive white Gaussian noise (Awgn) and being sta- tionary and ergodic, such that despite being a stochastic process, it has well-defined properties such as a fixed variance.
The property of the nonlinear distortion term𝜈depends on the input𝑢used. Its behaviour, when the input is either random-phased multisines or Gaussian noise, has been studied by Pintelon and Schoukens (2012, p. 87), who showed that it has several properties—the most important of which are listed as follows:
1. 𝜈 has zero mean and is uncorrelated but not independent of𝑢.
2. Given the system is periodicity-invariant (see Section 2.1), 𝜈 is also periodic with the same period as 𝑢.
3. 𝜈 is asymptotically normally distributed with a circular complex normally distributed frequency spectrum.
Because environment noise tends to be aperiodic, while the nonlinear dis- tortions are periodic, it is possible to estimate their variances independently. This has advantages, for example, in identification algorithms that rely on the knowledge of noise variances to yield better estimates (e.g. ELiS from the MatlabFdident
Toolbox, see their respective glossaries entries), or in enabling better modelling choices by allowing the user to ascertain the levels of nonlinear distortions. A ro- bust method proposed in Pintelon and Schoukens (2012, sec. 4.3.1) is reproduced in the next section.
4.2
Robust method for estimating the B
The most widely used method innon-parametric modelidentification of systems in the frequency domain is to use the following relationship:
𝐺(j𝜔) = 𝑆ZU(j𝜔)
𝑆UU(𝜔) (4.2)
where𝐺(j𝜔)is the frequency response function,𝑆ZU(j𝜔)is the cross power spectrum between the measured output𝑧(𝑡)and the input𝑢(𝑡), and𝑆UU(𝜔)is the auto power spectrum of the input (Bendat & Piersol,1980; Rake, 1980).
For a linear system,𝐺(j𝜔)is estimated through averaging the power spectra over several periods or experiments to minimise the influence of output noise; for a nonlinear system however, even in the absence of environment noise, averaging
4.2. Robust method for estimating theBla 𝑧 𝑧 ... 𝑧 𝑧 𝑧 ... 𝑧 ⋮ 𝑧 𝑧 ... 𝑧 𝑀 realisations 𝑃periods transients
Figure 4.1: The robust procedure for estimating the BLA
must be performed to obtain 𝐺(j𝜔)due to the dependence of𝜈 on ⟨𝑢⟩in (4.1). To estimate 𝑆ZU(j𝜔)the spectra obtained from a series of𝑀experiments, each with a different input segment, are averaged:
̂
𝑆ZU,M(j𝜔) = Avg r𝑍 (j𝜔) ⋅ 𝑈 (j𝜔)z (4.3) where𝑌 (j𝜔)is thediscrete Fourier transform (Dft)of a period of output record of the𝑚th experiment, and𝑈 (j𝜔)is the complex conjugate of the Dftof a period of input record of the 𝑚th experiment. It is known that under weak conditions (Pintelon & Schoukens,2012, p. 50),
lim 𝑆ZU,M̂ (j𝜔) = 𝑆ZU(j𝜔). (4.4) Typically, the averaging would be performed by conducting𝑀 experiments, each using a different independent realisation of the input, over 𝑃 periods of the input-output data in steady-state (i.e. after any initial transient has died away and this can be checked by looking at the correlation between inter-period data records in the output). Fig.4.1illustrates the averaging strategy. In the robust procedure, for each experiment𝑚, a different periodic input realisation𝑢 excites the system and 𝑃 periods of data are collected. If the system is entirely linear and in steady-state under a periodic input, it does not matter whether one sets for example, 𝑀 = 10 and 𝑃 = 1, or 𝑃 = 10 and 𝑀 = 1. In nonlinear system identification however, the periodic nonlinear distortions can only be reduced by averaging over𝑀realisations of different input signals, rather than over multiple periods 𝑃 of the same signal.
4.2. Robust method for estimating the Bla
For a nonlinear system, the globalBla estimate is given by:
̂ 𝐺Bla(j𝜔) ≜ ̂ 𝑆ZU (j𝜔) 𝑃 𝑆̂UU(𝜔) (4.5)
where the cross- and auto-power spectrum are 𝑆̂ZU = 𝑍 𝑈 and 𝑆̂UU = 𝑈 𝑈 respectively. Note that since within each experiment the input is periodic such that 𝑆UU is not a function of the period 𝑝, there is no averaging performed across 𝑝. If the input has an invariant (fixed) auto-power spectrum (Section 4.3) across different input realisations, such as the case with, for instance, multisines with user-specified magnitude spectrum (Section 2.3.3) or pseudorandom binary sequences (Prbs’s) such as Maximum Length Binary Sequences (Mlbs’s) (Sec- tion 2.3.4), 𝑆UU is known exactly. The denominator then becomes 𝑀𝑃 𝑆UU and averaging is hence not required across 𝑚. In practice it is not advisable to assume 𝑆UU would be a constant across different realisations due to nonlinear effects from signal generators or actuators. This estimator is robust against noise disturbances and is unbiased if input noise levels are small.
For known inputs, the environment and measurement noise variance 𝜎n as a function of frequency 𝜔 is estimated by (Pintelon & Schoukens, 2012, eqs. 4.18 & 4.19):
̂
𝜎n(𝜔) = 1
𝑀 𝑃 (𝑃 − 1) ⏐⏐⏐𝐺̂ −𝐺“ ⏐⏐⏐ (4.6)
where the block estimate𝐺̂ and realisation estimate𝐺“ of theBlaare defined as: ̂ 𝐺 ≜ 𝑆ZÛ ̂ 𝑆UU (4.7) “ 𝐺 ≜ 𝐺̂ . (4.8)
The total variance𝜎̂n+ , encompassing the variance introduced by nonlinear distor- tions𝜈 and noise 𝑛is estimated by (Pintelon & Schoukens, 2012, eq. 4.19):
̂
𝜎n+ (𝜔) = 1
𝑀(𝑀 − 1) ⏐⏐⏐𝐺“ − ̂𝐺Bla⏐⏐⏐ (4.9) It is important to note that (4.5) is not equal to the averaged (4.8) if the
4.3. Stochastic and deterministic power spectra