ABSTRACT
We propose a novel model for discrete linear periodic time varying (LPTV) systems using wavelets. The new model is compared with the ‘raised model’, which is commonly used for modeling LPTV systems. In fact, it turns out that the new model can be viewed as a generalization of the raised model. The wavelets model will be shown to be particularly suitable for adaptive identification of LPTV systems. It orders a compromise between time- and frequency-based algorithms. Time resolution is needed for modeling reasons and minimizing processing delay. Frequency resolution enables faster convergence of adaptive algorithms in general and the least mean square algorithm used here, in particular. Simulations show that for a colored input using the new model results not only in faster convergence compared to the raised model based algorithm, but also produce a lower steady-state error. This is at no significant additional cost in numerical complexity.
INTRODUCTION
On-line identification of general linear time variant (LTV) systems, although very important for a variety of applications, is still a relatively open issue. If the changes are sufficiently slow a linear time invariant(LTI) model can be used and the changes are traced by an adaptive algorithm However, when these changes are fast, the time variations need to be modeled. If there are some underlying constant parameters which model the system, they can be Estimate using adaptive or online algorithms. Furthermore, even if these parameters vary slowly an adaptive algorithm can track these variations. So, for the identification of LTV systems, a priori knowledge about the variations in time is assumed. Usually, one assumes that the changes are characterized by a finite set of functions However, all these cases refer only to limited classes of LTV systems. One important class of LTV systems includes the linear periodically time varying (LPTV) systems For quite a number of applications periodicity in the time variation can be observed A number of physical phenomena have periodic characteristics and using a LPTV model is very reasonable. Orbital motion, AC motors rhythm
the heart, or periodic disturbances (vibrations) in helicopters phenomena. Design of optimum periodic time varying filters for applications in diagnostics of combustion engines appears in [21] and periodic optimal control is discussed in Note that if a system is LTV ,but its time variations consist of a fast mode which is periodic and a slow mode, an LPTV model can be used combined with an adaptive algorithm. The LPTV model then handles the fast changes and the adaptive algorithm the slow changes. A common way to model LPTV systems is via the raising method described a new approach for modeling discrete LPTV systems with finite impulse responses using wavelets. We assume that the period is known a priori and, for simplicity, that it is a power of two (when the sampling time is under the user’s control, he can choose it accordingly). In fact, we show that using wavelets can be viewed as a generalization of the raised model. Using wavelets in modeling LTV systems is discussed by Doroslovacki and Fan After introducing the model we investigate its use for adaptive identification with the LMS algorithm. This algorithm is simple, has reasonable tracking abilities and low computational complexity. Its main disadvantage, as is well known, is its slow convergence. This disadvantage becomes more acute here since time scale for any adaptive algorithm is slowed down with LPTV systems, by a factor equal to their period. The convergence rate disadvantage has motivated us to consider using wavelets in adaptive identification of linear systems. Wavelets have been used for adaptive identification of LTI systems. The convergence rate of the LMS algorithm is shown to be higher when wavelets are used The motivation behind these algorithms is separation of the parameters, similar to frequency domain least mean square (FDLMS) The FDLMS uses the fast Fourier transform (FFT) for reasons of lower complexity and faster convergence rate. Convergence rate is also a motivation for sub-band approaches Frequency resolution scheme senile separation and therefore faster convergence, while time resolution is needed for minimizing the delay and for accurate modeling of the unknown system’s impulse response. Wavelets provide Mixable trade-as between time and frequency resolutions. Sub band approaches are used in applications such as echo cancellation and equalizers. Splitting a stationary input signal into different frequency bands and decimations, result in a lower spread of eigenvalues. A number of independent adaptive filters are updated at a reduced rate. The price is a larger steady-state error because of insufficient order estimation and aliasing. In some of those approaches an order is made to minimize the aliasing error caused by non-ideal filters. It is shown in that decimated based models used for LTI systems are in fact LPTV models. Moreover, these approaches are shown to be special cases of the new model we present here, which is more general for modeling both LPTV and LTI systems
PROCESS OF IDENTIFICATION APPROACH
a. Experiment design:
Its purpose is to obtain good experimental data, and it includes the choice of measured variables and of the character of input signals
b. Selection of model structure:
A suitable model structure is chosen using prior knowledge and trial and error
c. Choice of the criterion to fit:
A suitable cost function is chosen, which reflects how well the model fits the experimental data
d. Parameter estimation :
An optimization problem is solved to obtain the numerical values of the model parameters and system stability is identified
e. Model validation
The model is tested in order to reveal any inadequacies in system like its order of system, stability and dynamic nature of system parameter WAVELET “The Wavelet transform is a tool that cuts up data , functions or operators into different frequency components ,and then studies each component with resolution matched to its scale”.
WAVELET
“The Wavelet transform is a tool that cuts updata , functions or operators into different frequency components,and then studies each component with resolution matched to its scale”.
RAISED MODEL
This is commonly referred to as the raised model of the LPTV system. Clearly, the raised form , is a MIMO LTI system. Depending on the properties of the matrix sequence {H[k]} one can talk about an FIR or IIR system with the corresponding transfer function matrix,
H(z) =Z_ H[k]Zk.
This sequence contains all the information of the LPTV system and identifying the system is identifying this sequence (or the corresponding transfer function matrix).The raising concept appears in the ‘filter bank’ literature where it is referred to as ‘poly phase transform. The main observation we make in this model is that any LPTV system can be modeled as consisting of three parts: a (generic system independent) linear periodic transform 2 (the poly phase transform in the raised model case) and its inverse and in between a LTI system which captures the particular system parameters. This is demonstrated in Fig. 2. With this in mind we proceed to present our approach next. Final Interpretation of Raised Model .The final interpretation of Raised Model technique is given by following matrix which gives the diagonal zing matrix which is the specified need for the finding out the programming output and also block diagram shows specified general block diagram of raised model which is then compared with HAAR Model with general equation of it
y n =Σh[ n ,l] x[ l]
where k is an N times slower time scale than n
Next : LPTV MODELLING APPROACH
still updating
We propose a novel model for discrete linear periodic time varying (LPTV) systems using wavelets. The new model is compared with the ‘raised model’, which is commonly used for modeling LPTV systems. In fact, it turns out that the new model can be viewed as a generalization of the raised model. The wavelets model will be shown to be particularly suitable for adaptive identification of LPTV systems. It orders a compromise between time- and frequency-based algorithms. Time resolution is needed for modeling reasons and minimizing processing delay. Frequency resolution enables faster convergence of adaptive algorithms in general and the least mean square algorithm used here, in particular. Simulations show that for a colored input using the new model results not only in faster convergence compared to the raised model based algorithm, but also produce a lower steady-state error. This is at no significant additional cost in numerical complexity.
INTRODUCTION
On-line identification of general linear time variant (LTV) systems, although very important for a variety of applications, is still a relatively open issue. If the changes are sufficiently slow a linear time invariant(LTI) model can be used and the changes are traced by an adaptive algorithm However, when these changes are fast, the time variations need to be modeled. If there are some underlying constant parameters which model the system, they can be Estimate using adaptive or online algorithms. Furthermore, even if these parameters vary slowly an adaptive algorithm can track these variations. So, for the identification of LTV systems, a priori knowledge about the variations in time is assumed. Usually, one assumes that the changes are characterized by a finite set of functions However, all these cases refer only to limited classes of LTV systems. One important class of LTV systems includes the linear periodically time varying (LPTV) systems For quite a number of applications periodicity in the time variation can be observed A number of physical phenomena have periodic characteristics and using a LPTV model is very reasonable. Orbital motion, AC motors rhythm
the heart, or periodic disturbances (vibrations) in helicopters phenomena. Design of optimum periodic time varying filters for applications in diagnostics of combustion engines appears in [21] and periodic optimal control is discussed in Note that if a system is LTV ,but its time variations consist of a fast mode which is periodic and a slow mode, an LPTV model can be used combined with an adaptive algorithm. The LPTV model then handles the fast changes and the adaptive algorithm the slow changes. A common way to model LPTV systems is via the raising method described a new approach for modeling discrete LPTV systems with finite impulse responses using wavelets. We assume that the period is known a priori and, for simplicity, that it is a power of two (when the sampling time is under the user’s control, he can choose it accordingly). In fact, we show that using wavelets can be viewed as a generalization of the raised model. Using wavelets in modeling LTV systems is discussed by Doroslovacki and Fan After introducing the model we investigate its use for adaptive identification with the LMS algorithm. This algorithm is simple, has reasonable tracking abilities and low computational complexity. Its main disadvantage, as is well known, is its slow convergence. This disadvantage becomes more acute here since time scale for any adaptive algorithm is slowed down with LPTV systems, by a factor equal to their period. The convergence rate disadvantage has motivated us to consider using wavelets in adaptive identification of linear systems. Wavelets have been used for adaptive identification of LTI systems. The convergence rate of the LMS algorithm is shown to be higher when wavelets are used The motivation behind these algorithms is separation of the parameters, similar to frequency domain least mean square (FDLMS) The FDLMS uses the fast Fourier transform (FFT) for reasons of lower complexity and faster convergence rate. Convergence rate is also a motivation for sub-band approaches Frequency resolution scheme senile separation and therefore faster convergence, while time resolution is needed for minimizing the delay and for accurate modeling of the unknown system’s impulse response. Wavelets provide Mixable trade-as between time and frequency resolutions. Sub band approaches are used in applications such as echo cancellation and equalizers. Splitting a stationary input signal into different frequency bands and decimations, result in a lower spread of eigenvalues. A number of independent adaptive filters are updated at a reduced rate. The price is a larger steady-state error because of insufficient order estimation and aliasing. In some of those approaches an order is made to minimize the aliasing error caused by non-ideal filters. It is shown in that decimated based models used for LTI systems are in fact LPTV models. Moreover, these approaches are shown to be special cases of the new model we present here, which is more general for modeling both LPTV and LTI systems
PROCESS OF IDENTIFICATION APPROACH
a. Experiment design:
Its purpose is to obtain good experimental data, and it includes the choice of measured variables and of the character of input signals
b. Selection of model structure:
A suitable model structure is chosen using prior knowledge and trial and error
c. Choice of the criterion to fit:
A suitable cost function is chosen, which reflects how well the model fits the experimental data
d. Parameter estimation :
An optimization problem is solved to obtain the numerical values of the model parameters and system stability is identified
e. Model validation
The model is tested in order to reveal any inadequacies in system like its order of system, stability and dynamic nature of system parameter WAVELET “The Wavelet transform is a tool that cuts up data , functions or operators into different frequency components ,and then studies each component with resolution matched to its scale”.
WAVELET
“The Wavelet transform is a tool that cuts updata , functions or operators into different frequency components,and then studies each component with resolution matched to its scale”.
RAISED MODEL
This is commonly referred to as the raised model of the LPTV system. Clearly, the raised form , is a MIMO LTI system. Depending on the properties of the matrix sequence {H[k]} one can talk about an FIR or IIR system with the corresponding transfer function matrix,
H(z) =Z_ H[k]Zk.
This sequence contains all the information of the LPTV system and identifying the system is identifying this sequence (or the corresponding transfer function matrix).The raising concept appears in the ‘filter bank’ literature where it is referred to as ‘poly phase transform. The main observation we make in this model is that any LPTV system can be modeled as consisting of three parts: a (generic system independent) linear periodic transform 2 (the poly phase transform in the raised model case) and its inverse and in between a LTI system which captures the particular system parameters. This is demonstrated in Fig. 2. With this in mind we proceed to present our approach next. Final Interpretation of Raised Model .The final interpretation of Raised Model technique is given by following matrix which gives the diagonal zing matrix which is the specified need for the finding out the programming output and also block diagram shows specified general block diagram of raised model which is then compared with HAAR Model with general equation of it
y n =Σh[ n ,l] x[ l]
Next : LPTV MODELLING APPROACH
still updating
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