## 1. Introduction

Impulse response functions (IRFs) have been largely used in experimental modal analysis in order to extract the modal parameters (natural frequencies, damping factors and modal forms) in different areas. IRFs occupy a prominent place in applications of aeronautical, machinery and automobile industries, mainly when the system has coupled modes. Additionally, IRFs have practical advantages for use in control theory for many reasons, e.g.:

For very complex systems, they can be determined by experimental tests, or using data of input and output measured by load cells or accelerometers, or directly with an impact hammer.

The identified model is essentially nonparametric.

Normally a finite impulse response (FIR) model of the structure is employed. Thus, the stability can be warranted a priori. Additionally, the many adaptive controlers are based on an FIR structure and it is easy to perform a recursive estimation.

In general, IRFs can be identified by impact tests with an instrumented hammer or by using numerical algorithms implemented in commercial software. IRFs can be determined with those algorithms through different methods, e. g., the covariance method based on the sum of convolutions of the measured input forces. However, there is an over parametrization that is a drawback when the lag memory is high. Fortunately, an expansion of the IRFs into orthonormal basis functions can enhance the procedure of reducing the number of parameters [15]. For describing mechanical vibrating systems, Kautz filters are interesting orthogonal functions set in Hilbert space [21] that include a priori knowledge about the dominant poles. The eigenvalues associated to vibrating mechanical systems are conjugated complex poles, so, the IRFs can be expanded in orthonormal basis functions with those conditions. Kautz filters are orthogonal funcions that can be used for this purpose. These filters can decrease the computational cost and accelerate the convergence rate providing a good estimate of the IRFs [14].

Kautz filters have found several applications, e.g., acoustic and audio [20], circuit theory [17], experimental modal analysis in mechanical systems [2, 12, 13, 14], vibration control [6], model reduction [4], robust control [18], predictive control [19], general system identification [5, 6, 22, 25], non-linear system identification with Volterra models [7, 8, 9, 10, 11], etc. Although it may seem that the mathematical and theoretical aspects of Kautz filters are more interesting for academic purposed, some practical applications can be found in the literature. For example, the flight testing certification of aircrafts for aeroelastic stability was completely charecterized through a series connection of Kautz filters in [1]. The application used a simulated nonlinear prototypical two-dimensional wing section and F/A-18 active aeroelastic wing ground vibration test data.

In specific control applications with Kautz filters, the strategies are, normally, based on active noise control using feedforward compensation, e. g. as performed in [26]. It is well-known that Wiener theory can be used to describe internal model control to change the control architecture from feedforward to feedback [3]. However, feedback compensation can also be directly implemented. Thus, the goal of the present chapter is to apply Kautz filters for active vibration control. The main steps and characteristics involved in this procedure are described. Specifically, this chapter emphasises the following:

Feedback control, considering dynamic canceling.

It is not necessary to have a complete mathematical model and the controller is designed directly in the digital domain for fast practical implementation.

The control method is based on experimental IRFs (nonparametric) and in orthonormal basis functions. Thus, the method is grey-box because prior knowledge of the mechanical vibrating system treated is assumed (poles of Kautz filter to represent the system). Additionally, complex vibration system can be controlled.

An example of a single-degree of freedom mechanical model is used to illustrate the main steps.

Additionally, an experimental example by using a clamped beam with PZT actuator and PVDF sensor is presented.

The chapter is organized as follows. First, the IRF identification and covariance method is reviewed briefly, followed by the Kautz filter with multiple poles for expansion of impulse response. After, a vibration control strategy is described and example applications involving single-input-single-output vibrating systems are used to illustrate the approach. Finally, the results are discussed and suggestions for a non-linear identification procedure are proposed.

## 2. Impulse response function

The output of a linear discrete-time and invariant system can be written as:

where the sequences

Normally, to obtain the IRF, eq. (1), is truncated in

The approach in eq. (2) changes an infinite impulse response model (IIR) into a finite impulse response model (FIR). The most common method to identify the

where the correlation function

the stability of the identified model is guaranteed a priori, since the model is FIR.

the model is assumed to be described only for arbitrary zeros and poles at the origin of the complex plane.

the model is linear in the parameters, hence the LS approach can be performed.

However, this identification technique often leads to conservative results because a common vibration system is hardly ever represented by a FIR model. Thus, the practical drawback is that a large number of parameters

## 3. Covariance method expanded in orthonormal basis functions

The IRF

where

The convergence of

By incorporating the eq. (4) into Wiener-Hopf equation, eq. (3), one can obtain:

where

Eq. (8) is basically a filtering of the input signal

The effectiveness of the model is limited by the choice of the filters

## 4. Kautz filter

The Kautz filters can be given by [16, 22, 24]:

where the constants

A sequence of filters is utilized with different poles in each section describing the modal behavior in the frequency range of interest. A question is relative for choosing the poles and the IRFs iteratively based on application of eq. (2) and output experimental signal

where

The optimization problem can be described by objective function that employs an Euclidean norm and the Kautz poles are functions of the frequencies and damping factors that are the optimization parameters. These parameters can be restricted in a range searching. This optimization problem can be solved by several classical approaches. A detailed explanation in this point can be found in [12].

## 5. Active vibration control strategy

If an IRF is well identified through covariance method expanded with Kautz filters, a model in

Considering ℎ(𝑘) is a causal sequence.

:A controller can be inserted in the direct branch of the control loop to try to reject the disturbance. This controller

where

It is worth to point out that one consider only the control of stable systems described by

Two examples are used to show the approach proposed. The first one is a single-degree-of-freedom model that is a simple and easy example for the interested reader reproduce it. The second one is based on active vibration control in a smart structure with PZT actuator and PVDF sensor for presenting its use employing experimental data.

The results are illustrated in a single-degree-of-freedom model given by:

where

An important step to identify the IRFs is the choice of Kautz poles that need to reflect adequately the dominant dynamics of the vibrating systems. In real-world application the choice of the poles is a complicated problem. However, a simple power spectral density of the output signal (in our example the displacement) can give an orientation to help in the selection. If the system is more complicated, an optimization procedure could be used [12]. Figure (2) shows the power spectral density of the displacement. Clearly, it seems a peak value close to

Based on the frequency of

The impulse response of the two sections of the Kautz filter are used to process the correlation function of the input signal

Once the IRF is identified, an experimental FIR model representative of the system is now known. This

where

where

Finally, the feedback transfer function

that corresponds to:

Clearly the effectiveness of the controller depends on the correct identification of the

Figure (6) shows the output displacement without and with control. The disturbance force is considered with the same level and type of the tests used in the uncontrolled condition.

A cantilever aluminium beam with a PZT actuator patch and a piezoelectric sensor (PVDF) symmetrically bonded to both sides of the beam is used to illustrate the process of IRF identification and design of a digital controller for active vibration reduction. The PZT and PVDF are bonded attached collocated near to the clamped end of the beam, as seen in fig. (7). The PZT patch is the model QP10N from ACX with size of

A white noise signal is generated in the computer, converted to analogic domain with a D/A converter and pre-processed by a voltage amplifier with gain of * dSPACE* 1104 acquisition board with a sample rate of

The first step in this approach is the choosing an adequate set of poles for the Kautz Filters. As the mathematical model is unknown, one needs to start by availing the power spectral density of the PVDF sensor (output) as suggested in the first example. Figure (10) presents the power spectral density of the output signal (PVDF) estimated using Welch method with Hanning window,

Based on the spectral analysis one must choose the continuous poles candidates given by

Once the fourth and fifth modes are apparently well damped by analysing the frequency response the correspond poles are also considered well damped (not dominants). The Kautz filter is described in the discrete-time domain. So, it is necessary to convert to

The cantilever beam is a SISO system, but with apparent five modes in the frequency range computed of interest. So, they are used

Figure (12) shows the comparison between the IFFT of the FRF from

Although, it seems that are not a complete visual agreement between the curves, the FRF seen in figure (13) presents a good agreement. It is worth to comments that with the same experimental data, [23] identified a state-space model through Eigensystem Realization Algorithms (ERA) combined with Observer/Kalman filter Identification (OKID). The results presented with Kautz filter allowed a better identification in this frequency range comparing than with ERA/OKID.

Figure (14) shows the output response of th PVDF estimated by a convolution between the IRF identified by Kautz filter with the input excitation from PZT actuator. The estimated output can be compared with the experimental measured response (see fig. 8).

The controller is designed based on the inverse of the identified system described by eq. (16), called by

where

It is important to observe that the three compensators,

It was decided to control only the

The fourth and fifth modes are not dominant.

Additionally, these modes are not well identified by the Kautz filter. One included the damping factor in these modes with these values shown above in order to correct identification the anti-ressonance region.

Figure (15) shows the FRF comparison between the uncontrolled (estimated by Kautz filter) and controlled system where is possible to observe the reduction in the resonance peak caused by the controller implemented.

Another advantage of this procedure face to state-feedback approaches is relative to the controlability and observability conditions. If one use procedures identification for obtaining a state-space realization, e. g. ERA/OKID as made by [23], is necessary to verify a prior the observability and controlability conditions. In some situations some modes are not controllable and observable adequately with a specific realization. Once the technique used in this chapter is not described in state-space variables and it is based on input/output variables with non-parametric IRF model, these kinds of drawbacks are avoided.

This chapter has described a procedure for non-parametric system identification of an impulse response function (IRF) based on input and output experimental data. Orthogonal functions are used to reduce the number of samples to be identified. A simple active vibration control procedure with a digital compensator that seeks to cancel the plant dynamic is also described. Once the IRF in the uncontrolled condition is well estimated by Kautz filters, the control strategy presented can increase the damping in a satisfactory level with low actuator requirements. Single-input-single-output vibrating systems have been used to illustrate the performance and the main aspects for practical implementation. This procedure can also be extended for nonlinear systems using Hammerstein or Wiener block-oriented models.

## Acknowledgement

The authors are thankful for the financial support provided by National Council for Scientific and Technological Development (CNPq/Brasil), INCT and São Paulo Research Foundation (FAPESP). The authors acknowledge the helpful suggestions of the Editor. The authors also are thankful the help of Prof. Dr. Gustavo LuizChagasManhN´ es de Abreu and Sanderson Manoel da Conceição for providing the experimental data in the clamped beam.

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## Notes

- Considering ℎ(𝑘) is a causal sequence.