Starting with a brief review of block pulse functions (BPF), sample-and-hold functions (SHF) and triangular functions (TF), this chapter presents the genesis of hybrid functions (HF) mathematically. Then different elementary properties and operational rules of HF are discussed. The chapter ends with a qualitative comparison of BPF, SHF, TF and HF.

The Hybrid Function (HF) and Its Properties

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control

I Piecewise constant orthogonal basis functions.- II Operations on square integrable functions in terms of PCBF spectra.- III Analysis of lumped continuous linear systems.- IV Analysis of time delay systems.- V Solution of functional differential equations.- VI Analysis of non-linear and time-varying systems.- VII Optimal control of linear lag-free systems.- VIII Optimal control of time-lag systems.- IX Solution of partial differential equations (PDE) [W55].- X Identification of continuous lumped parameter systems.- XI Parameter identification in distributed systems.

Piecewise Constant Orthogonal Functions and Their Application to Systems and Control

The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.

A new set of orthogonal functions and its application to the analysis of dynamic systems

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems

The present work searches for a suitable set of orthogonal functions for the analysis of control systems with sample-and-hold ( S/H ). The search starts with the applicability of the well known block pulse function (BPF) set and uses an operational technique by defining a block pulse operational transfer function ( BPOTF ) to analyse a few control systems. The results obtained are found to be fairly accurate. But this method failed to distinguish between an input sampled system and an error sampled system. To remove these limitations, another improved approach was followed using a sample-and-hold operational matrix, but it also failed to come up with accurate results. Further, the method needed a large number of component block pulse functions leading to a much larger amount of storage as well as computational time. To search for a more efficient technique, a new set of piecewise constant orthogonal functions, termed sample-and-hold functions (SHF), is introduced. The analysis, based upon a similar operational technique, in the SHF domain results in the same accuracy as the conventional z -transform analysis. Here, the input signal is expressed as a linear combination of sample-and-hold functions; the plant having a Laplace transfer function G(s) is represented by an equivalent sample-and-hold operational transfer function ( SHOTF ), and the output in the SHF domain is obtained by means of simple matrix multiplication. This technique is able to do away with the laborious algebraic manipulations associated with the z -transform technique without sacrificing accuracy. Also, the accuracy does not depend upon m and the presented method does not need any kind of inverse transformation. A few linear sample-and-hold SISO control systems, open loop as well as closed loop, are analysed as illustrative examples. The results are found to match exactly with the z -transform solutions. Finally, an error analysis has been carried out to estimate the upper bound of the mean integral squared error (m.i.s.e.) of the SHF approximation of a function f(t) of Lebesgue measure.

The Hybrid Function (HF) and Its Properties

Citations

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control

References

Piecewise Constant Orthogonal Functions and Their Application to Systems and Control

A new set of orthogonal functions and its application to the analysis of dynamic systems

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems

A new set of piecewise constant orthogonal functions for the analysis of linear siso systems with sample-and-hold

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control

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