Response impulse

If the impulse response function g(x) of a system is known, the output signal y(x) of the system is given for any input signal u(x). The integral equation, which is called superposition integral. 

The function g(x) is named impulse response of the system, because it is the response to an unit pulse 5(x) applied at =0 [2]. This unit impulse 5(x), also called Dirac impulse or delta-function, is defined as... 

Eq.(2) describes an impulse with the area of 1 [1-3]. Fig. 1 (left) shows such an unit impulse S(x) and an example for an impulse response g(x) at the output of the system. 

Often an unit impulse is not available as a signal to get the impulse response function g(x). Therefore an other characteristic signal, the unit step, is be used. 

The step response function h(x) is also determined by the integral equation (1). The relationship between step response h(x) and the impulse response g(x) is represented by... 

The superposition integral (1) corresponds to a division of the input signal u(x) into a lot of Dirac impulses 5 x). which are scaled to the belonging value of the input. The output of each impulse 5fx) is known as the impulse response g(x). That means, the output y(x) is got by addition of a lot of local shifted and scaled impulse responses. 

Fig. 1 (right) shows upside an example of an input. The marked points are some of the scaled Dirac impulses. The belonging scaled impulse responses are shown downside. 

Unit impulse response Unit step response responses for input example... 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

Chapter 4.3. discusses the explained theory for choosed examples. For several cracks the output is pre-calculated by using the impulse response and compared with measurement data. 

All described sensor probes scan an edge of the same material to get the characteristic step response of each system. The derivation of this curve (see eq.(4) ) causes the impulse responses. The measurement frequency is 100 kHz, the distance between sensor and structure 0. Chapter 4.2.1. and 4.2.2. compare several sensors and measurement methods and show the importance of the impulse response for the comparison. 

The first example presents the importance of the impulse response function for the comparison of several sensors with the same arrangement from chapter 3.1.. 

Figures Impulse responses of different coils on a material edge...

It should be a symmetrical form in the impulse response of a linear system. 

The following examples represent the importance of the impulse response for the comparison of different magnetic field sensors. For presentation in this paper only one data curve per method is selected and compared. The determined signals and the path x are related in the same way like in the previous chapter. 

Figure 7 Impulse responses of different sensor systems...

This is visible in the behaviour of the impulse responses as well (fig, 7), There the amplitude of curve (1) (gWng, )) is the highest, of curve (2) ig(L,)) the lowest, but the maxima are not located at the same place. 

The difference in widths of the impulse responses are small. Especially visible the pulse response of the inductive sensors, curves (1) and... 

For calculation the known data are the. .input signal", cracks of different widths, and the impulse response. The material of the crack model is assigned to the value 0, the air to 1. 

The determined eddy-eurrent parameter is the inductance of the eomplex impedance measured by impedance analyzer at j=100 kHz. Therefore the impulse response function from chapter 4.2.1. is used for calculation. The depth of the cracks is big in comparison to coil size. For presentation the measured and pre-calculated data are related to their maxima (in air). The path X is related to the winding diameter dy of the coil. 

Methods from the theory of LTI-systems are practicable for eddy-current material testing problems. The special role of the impulse response as a characteristic function of the system sensor-material is presented in the theory and for several examples. 

So, a comparison of different types of magnetic field sensors is possible by using the impulse response function. High amplitude and small width of this bell-formed function represent a high local resolution and a high signal-to-noise-characteristic of a sensor system. On the other hand the impulse response can be used for calculation of an unknown output. In a next step it will be shown a solution of an inverse eddy-current testing problem. 

The function h(t) to be restored is the impulse response of the medium x(t) is the transmitted pulse measured by reflection on a perfect plane reflector, for example the interface between air and water and y(t) is the observed signal. 

The method proposed by Papoulis [7] to determine h(t) as a function of its Fourier transform within a band, is a non-linear adaptive modification of a extrapolation method.[8] It takes advantage of the finite width of impulse responses in both time and frequency. 

The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical hnear models expressed in either step-response or impulse-response form. For simphcity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipiilated variable u can be expressed as... 

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. 

The step-response model in Eq. (8-63) is equivalent to the following impulse response model ... 

Implementation Issues A critical factor in the successful application of any model-based technique is the availability of a suitaole dynamic model. In typical MPC applications, an empirical model is identified from data acquired during extensive plant tests. The experiments generally consist of a series of bump tests in the manipulated variables. Typically, the manipulated variables are adjusted one at a time and the plant tests require a period of one to three weeks. The step or impulse response coefficients are then calculated using linear-regression techniques such as least-sqiiares methods. However, details concerning the procedures utihzed in the plant tests and subsequent model identification are considered to be proprietary information. The scaling and conditioning of plant data for use in model identification and control calculations can be key factors in the success of the apphcation. 

The impulse response funetion, equation (3.26) is shown in Figure 3.11. 

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