The focus of this thesis is on the identification of systems based on the concept of Linear Time-varying (LTV) models. Unlike in the LTI case, here the impulse response function – hence the dynamics – of a linear time-varying system can be changed at any time.
Many authors refer to LTV systems as a special class of Linear Parameter-Varying (LPV) systems [39], [40], [41].
DEFINITION 2.14A discrete, linear parameter-varying system can be defined by its two dimensional impulse response function (denoted by ℎ [𝑡, 𝑝 ]) as follows:
𝑦[𝑡] = ∑ ℎ [𝜏, 𝑝 ]𝑢[𝑡 − 𝜏]
∑ ℎ [𝑡 − 𝜏, 𝑝 ]𝑢[𝑡] (2.3)
where the parameter 𝑝 is the scheduling variable.
Chapter 2- Systems
It is obvious when the variable 𝑝 represents the time, then the LPV model will depend on the time as well, which leads us to the definition of the LTV systems [42].
DEFINITION 2.15The 𝐺 discrete, linear time dependent system can be defined by its two dimensional impulse response function (denoted by ℎ [𝑡, 𝜏]) as follows:
G {u[𝑡]} = 𝑦[𝑡] =
∑ ℎ [𝑡, 𝜏]𝑢[𝑡 − 𝜏] = ∑ ℎ [𝑡, 𝑡 − 𝜏]𝑢[𝜏] (2.4) where the parameter 𝑡 is the global time and 𝜏 is the system time [22].
It is also important to remark that some authors refer to ℎ [𝑡, 𝜏] as the time-varying kernel. The main concept of this thesis is based on this impulse response notion.
This means that the response of the system to an impulse depends on the time on at which the excitation is applied. An illustration is shown in Figure 2.1. Observe that the responses to two impulses applied at different time instants are different. In this case ℎ [𝑡, 𝜏] can be seen as snapshots of the instantaneous dynamic behavior at these two time instants of the impulses. Next, the linearity and stability will be defined.
DEFINITION 2.16𝐺 is linear because the superposition holds, i.e. with c1, c2 arbitrary constant values:
y = G {c 𝑢 [𝑡] + c 𝑢 [𝑡]} =
c G {𝑢 [𝑡]} + c G {𝑢 [𝑡]} (2.5)
DEFINITION 2.17 𝐺 discrete, linear time dependent system is causal when the following is true:
ℎ [𝑡, 𝜏] = 0, when 𝜏 < 0 (2.6)
Note that this definition differs from the LTI case where the constraint is on the – global – time variable 𝑡. In Chapter 6 this will be explained in details. Last but not least, the stability can be defined in the same way as for the LTI case [43].
DEFINITION 2.18 An LTV system is stable if, for every bounded input, the output is bounded and finite at any global time of 𝑡 [44].
Figure 2.1: An example of an LTV system.
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Before we go any further, a brief overview about different approaches used for – different types of – LTV systems will be given here.
Although LTV systems can be described – in theory – uniquely in the time domain with a two dimensional impulse response function, the related description in the frequency domain is not so trivial. It sounds contradictory but the – classical – transfer function does not exist for LTV systems. A transfer function describes the amplitude and phase relation between input and output sinusoidal signals of an LTI system [24]. However in this case this definition does not hold because the response of an LTV system to a sinusoidal is not sinusoidal anymore. It is due to the fact that the poles and zeros of an LTV system are not fixed and they can move continuously [43].
Therefore a generalization of the transfer function is needed. One possibility is proposed in [42], [44] and applied to practical problems for instance in [45]. This is the system function, also known as instantaneous transfer function (ITF). This is well studied and discussed in [40], [41], [45], [46], [47]. It is possible to interpret this notion as a snapshot of the instantaneous dynamic behavior – it is analogous to ℎ [𝑡, 𝜏] in the time domain.
DEFINITION 2.19 For a causal LTV system, the system function is defined at any time instant 𝑡 with the ℒ transform (Laplace transform) as follows:
𝐻 (𝑡, 𝑠) = ℒ{ℎ (𝑡, 𝜏)} = ∫ ℎ (𝑡, 𝜏)𝑒 𝑑𝜏 (2.7) It can be proven that the system function has similar properties as the (LTI) transfer function [44], [48], [49], i.e. in steady-state (zero initial conditions):
i. the output of the system can be computed as 𝑦(𝑡) = ℒ {𝐻 (𝑡, 𝑠)𝑈(𝑠)}
where 𝑈(𝑠) = ℒ{𝑢(𝑡)} and ℒ is the inverse Laplace transform,
ii. the response to a sine wave excitation 𝑢(𝑡) = sin (𝜔𝑡) is given by the modulated output as follows: 𝑦(𝑡) = |𝐻 (𝑡, 𝑗𝜔𝑡)|sin (𝜔𝑡 + ∠𝐻 (𝑡, 𝜔𝑡)).
The discrete system function can be obtained by using the 𝒵-transform as follows:
DEFINITION 2.20 For a causal discrete LTV system, the discrete system function is defined at time instant 𝑡 with the 𝒵 transform (discrete Laplace transform) as follows:
𝐻 [𝑡, 𝑧] = 𝒵{ℎ [𝑡, 𝜏]} = ∑ ℎ [𝑡, 𝜏]𝑧 (2.8)
In this thesis, the proposed time domain methods will be compared to a special implementation of the above-mentioned concept [40].
Chapter 2- Systems
There is another similar approach to define somehow a transfer function. When the time variations are slow, during a short observation time the system can be well approximated by a time invariant model. Then it is a common practice to describe the system as a series of LTI systems. In these techniques, at each measurement time a
“frozen” LTI model is built. These LTI systems are called the frozen instantaneous systems. These models can describe the time-varying behavior quite well [39], [44]
[50], [51], [52].
The drawback of these methods is that there can be a quite significant time variation during a single experiment, such that an LTI model is not sufficient to describe the system’s behavior.
DEFINITION 2.21The transfer functions obtained by using “frozen” coefficients are the frozen transfer function (FTF) [41].
A longer description about the instantaneous and frozen approaches can be found in Chapter 10.
There is also a recently published similar approach in [53]. This method uses a – one dimensional – regularization technique (see Chapter 5). In the referred work a sliding window is used over different moments. The estimation is done by an extended kernel function (see Chapter 5). This is quite similar to the above-mentioned frozen LTI approaches.
It is also possible to use different recursion techniques to track the changes of the parameters such as in [54], [55], [56]. These techniques typically use a kind of sliding window with the assumption that the system is time invariant inside that window.
The most common techniques use time-varying ARX, ARMAX [57] parametric models.
Related to the parametric representations, some authors expand the time-varying coefficient onto a finite set of basis sequences, wavelets [58], [59]. There are some interesting wavelets techniques which provide (directly) a good estimation of the impulse responses [60], [61].
There are some distinguished methods where they build a model from the complete measured time window and considered frequency band at once using difference or differential equations such as in [62], [63], [64], [65]. The basic idea of this nonparametric estimation (see Section 3.1.2) is that the parameters need to be estimated at once which is similar to the main concept of this thesis. Based on this concept two proposed methods (using regularization and B-splines) will be shown in this thesis.
There are some alternatives in control and automation, where they prefer to use a state space representation instead. These studies basically describe the effect of the
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varying subspace parameters as a function of 𝑡. The structural properties of such an LTV system are analyzed using the solutions of the system of differential (difference) equations [65], [66], [67]. Some other authors try to identify state space models using wavelets [68], [69]. This kind of representation is out of the scope of the thesis. Last but not least, there is an important subclass of LTV systems: the periodically time-varying systems (LTP) which are studied for instance in [41], [70]. In this case, to estimate model parameters more advanced statistical methods can be used.
Related to the instantaneous and frozen as well as to the periodically varying systems approaches, there is a very important the concept that needs to be mentioned.
It is the best linear time-invariant (BLTI) approximation of an LTP system [71]. It tells us what we would obtain, if we would simply use an LTI model instead an LTV model.