Time varying generalised frequency response functions

Investigating the frequency response of time varying non-linear systems is a dif- ficult task. Methods that are available for time invariant systems mainly depend on taking transforms over a sufficient number of data points from a time domain data record. Capturing rapid changes in system dynamics may therefore not be possible. However, For a non-linear system that can be represented by a paramet- ric model, many methods exist for tracking the parameters as they evolve over time, for example, Kalman filters [57], recursive least squares [75] and wavelets

[131]. If the use of a fixed model structure can be justified over the entire data record then the GFRF based frequency domain analysis methods discussed in the previous chapter can easily be extended to the TV case, greatly simplifying the analysis procedure [49].

The GFRFs obtained from NARX models are a function of the model structure and parameters. Therefore, assuming a fixed model structure leads to time varia- tions in the parameter vector only. The n’th order GFRF, given by equation 4.35, can then be extended to the TV case by explicitly including a dependency on time t such that Hn(ω₁, . . . , ωn, t) = Hnu (ω₁, . . . , ωn, t) +Hnuy(ω₁, . . . , ωn, t) +Hny(ω₁, . . . , ωn, t) 1₋∑ p θp(t)e −j(ω1,...,ωn)kp , (5.2) where again, the time dependency of the functions Hnu(ω1, . . . , ωn, t),

Hnuy(ω1, . . . , ωn, t)and Hny(ω1, . . . , ωn, t)is solely due to the time varying param- eters.

As discussed in Section 4.3, the GFRF based analysis of non-linear systems is largely based on the identification of peaks and ridges in the first and higher order GFRFs respectively. This is because the peaks and ridges in the frequency response indicate frequencies and combinations of frequencies of the input spectrum that produce strong non-linear effects in the output [20]. The relative directions and magnitudes of the peaks and ridges can provide a description of how the non- linear behaviour exhibited by the system.

[49] show that the direction of the ridges in the gain of n’th order GFRF are mainly dependent on the NARX model structure and that there is always a ridge in the ω1+ω2+. . .+ωn =Cidirection with the possible existence of extra ridges in other directions. The position Ci of the ridge is found at the minimum of the denominator in Equation (5.2) such that

Ci = arg min ω1+ω2+...+ωn      1−

∑

Visualisation of TV non-linear systems with TV-GFRFs for the n’th order GFRF requires a(n+2)dimensional space. Therefore, for n>1, so that the dimension-

ality of space required is>3, visualisation of the TV-GFRF is difficult. However,

it is often the case that only a small amount of ridge directions are observed, and the direction depends on the NARX model structure. By averaging in the ridge direction the dimensionality of the problem can be reduced to 2 dimensions (with the third being time). If there is only one ridge displayed by the system then no in-

formation is lost in the dimensionality reduction. The averaging can be performed via the equations

   H ω1+ω2+...+ωj n (ω, t)   = Z ω1+ω2+...+ωj=ω 1 Nω |Hn(ω1, . . . , ωn, t)|dω (5.4) and φ H_nω1+ω2+...+ωj(ω, t)  = Z ω1+ω2+...+ωj=ω 1 Nω φ(Hn(ω₁, . . . , ωn, t))dω, (5.5)

for the gain and phases of the system respectively, where φ(_·)denotes the phase and N_f is the number of samples along the direction of the ridge at each frequency

ω =ω₁+ω₂+. . .+ω_j. The variable j depends on the models structure.

The TV-GFRF analysis scheme [49] for a TV nonlinear system that can be de- scribed by a fixed model structure with a TV parameter vector follows as

Step 1: Calculate the system TV-GFRFs, Hn(ω₁, . . . , ωn, t), via Equation (5.2)

up to the required order, N∗

m.

Step 2: Evaluate the TV-GFRFs at each sample time t=kTs. Step 3: Determine the GFRF ridge directions.

Step 4: Average the evaluated TV-GFRFs via equations (5.4) and (5.5). This approach hence allows the higher order GFRFs of a TV non-linear system to be visualised in a lower dimensionality providing insight into how the input frequencies combine to generate complex non-linear effects. However, there are disadvantages to this scheme in terms of the computational cost involved in eval- uating the GFRF due to their high dimensionality. This is further exacerbated in calculating the symmetric GFRF because the number of permutations across all the input frequencies also increases rapidly with the GFRF order. This calculation must then be performed at each time step.

More importantly, from the TV-GFRF alone it is not possible to find exactly how the system dynamics behave in generating the system output for a given input excitation. NOFRFs, an extension of the GFRF framework, were introduced in the previous chapter and are used in the next section to address the time varying case.

5.1.3 Time varying non-linear output frequency response functions