Nonparametric time-variant frequency response function estimates using arbitrary excitations

TLDR: This paper proposes a method for estimating nonparametrically the dynamic part of the TV-FRF from known input, noisy output observations, and this method is applicable to arbitrary inputs.

About: This article is published in Automatica.The article was published on 2015-01-01 and is currently open access. It has received 23 citations till now. The article focuses on the topics: Legendre polynomials & Frequency response.

Figures

Figure 3: Second order bandpass filter with time-varying resonance frequency and damping ratio. The electronic circuit with input u(t) and output y(t) consists of a high gain operational amplifier (CA741CE), a JFET transistor (BF245B) with gate voltage p(t), three resistors (R1 = R2 = 10 kΩ and R3 = 470 kΩ), and two capacitors (C1 = C2 = 10 nF).

Figure 1: Block diagram of the slowly time-varying system (6) – direct model.

Figure 4: Estimated indirect model (8) of the electronic circuit (see also Fig. 2): FRFs Hr(jωk) (black) and their variance (grey).

Figure 5: Estimated direct model (7) of the electronic circuit (see also Fig. 1): FRFs Gr(jωk) (black) and their variance (grey).

Figure 2: Block diagram of a slowly time-varying system (6) with polynomial basis functions fr(t) – indirect model. This block diagram is a multiple-input u(t), u1(t), . . . , uNb (t), single-output y0(t) LTI equivalent of the single-input u(t), single-output y0(t) LTV system shown in Fig. 1.

Figure 6: Estimated TV-FRF G(jωk, t) of the electronic circuit. Top left: amplitude (red) and variance (blue) ofG(jωk, t); top right: phase of G(jωk, t); bottom left: time-frequency plot of |G(jωk, t)|; and bottom right: gate voltage p(t).

Frequently asked questions

Q1. What are the contributions in "Nonparametric time-variant frequency response function estimates using arbitrary excitations" ?

This paper proposes a method for estimating nonparametrically the dynamic part of the TV-FRF from known input, noisy output observations. 

Q2. What are the disadvantages of parametrising the system dynamics w.r.t.?

The three disadvantages of parametrising the system dynamics w.r.t. to a nonparametric representation are the following: (i) the type of dynamic model must be chosen: differential equation (s-domain), difference equation (z-domain), fractional differential equation (e.g., √ s-domain), or partial differential equation; (ii) the dynamic model order must be chosen (orders time-domain derivatives or time-domain shifts of the input and output signals); and (iii) estimating the model parameters mostly involves a nonlinear minimisation. 

Q3. What is the constant state response to sin(0t)?

The time-variant FRF (1) has the following properties:1. The steady state response to sin(ω0t) equals|G(jω0, t)|sin(ω0t+ ∠G(jω0, t)) (2)which is an amplitude and phase modulated sine wave. 

Q4. What are examples of class B dynamics?

Examples of class B dynamics are, regime switching in power electronics (Aguilera et al., 2014), econometrics (Hamilton, 1990), and control applications (Yin et al., 2009); and more general, hybrid systems (see Paoletti et al., 2007 and the references therein). 

Q5. What are examples of class A dynamics?

Examples of class A dynamics are, thermal drift in power electronics (Chen and Yuang, 2011); fatigue, aging and mortification in biomedical measurements (Aerts and Dirckx, 2010); pit corrosion of metals (Van Ingelgem et al., 2008); control of crane dynamics (Abdel-Rahman et al., 2003); airplane dynamics during take off and landing (Dimitriadis and Cooper, 2001); and impedance measurements for determining the state-of-charge of batteries (Rodrigues et al., 2000; Pop et al., 2005). 

Q6. What is the amplitude of the u(t)?

The amplitudes Ak, k = k1, k1 + 1, . . . , k2, are constant and chosen such that rms value of u(t) is 93 mV, while the phases φk, k = k1, k1 + 1, . . . , k2, are randomly selected according to auniform [0, 2π) distribution. 

Q7. What is the basic assumption made for class A dynamics?

The basic assumption made is that the system is time-invariant within the short sliding time window: see, for example, Spiridonakos and Fassois (2009) for noise power spectra and Sanchez et al. (2013) for FRFs. 

Q8. What is the simplest method for estimating the TV-FRF?

This paper has presented an indirect method for estimating nonparametrically the dynamics of the TV-FRF of linear time-variant systems, where the arbitrary timevariation is modelled by Legendre polynomials. 

Q9. What is the significance of the nonparametric estimation of the FRFs?

The nonparametric estimation of the FRFs Hr(jω), r = 0, 1, . . . , Nb, however, imposes additional conditions on the excitation u(t): in the frequency band of interest the input discrete Fourier transform (DFT) spectrum U(k) of u(t)U (k) = DFT (u (t)) = 1√ N N−1∑ n=0 u (nTs) e −j2πkn/N (11)with 

Q10. What are the disadvantages of parametrising the system dynamics?

Compared with the algorithms for model class 3, the methods developed for model class 4 have the disadvantage that they require a trade-off between accurate tracking of the time-variation (the sliding time window should be as small as possible) and sufficiently large frequency resolution of the estimated dynamics (the sliding time window should be as large as possible). 

Q11. What is the simplest way to estimate the TV-FRF?

Eq. (6) motivates the following assumption:Assumption 1. (Slow time-variation) The TV-FRF (1) of the linear time-variant system can be written as (6), where fr(t) , r = 0, 1, . . . , Nb, are polynomials of order r satisfying (5).