Transfer function identification from impulse response via a new set of orthogonal hybrid functions (HF)

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Abstract

The present work proposes a new set of hybrid functions (HF) which evolved from the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). Traditional block pulse functions (BPF) still continue to be attractive to many researchers in the arena of control theory. Block pulse functions also gave birth to a few useful variants. Two such variants are SHF and TF. The former is efficient for analyzing sample-and-hold control systems, while triangular functions established their superiority in obtaining piecewise linear solution of various control problems. After developing the basic theory of HF, a few square integrable functions are approximated via this set in a piecewise linear manner. For such approximation, it is shown, the mean integral square error (MISE) is much less than block pulse function based approximation. The operational matrices for integration in HF domain are also derived. Finally, this new set is employed for solving identification problem from impulse response data. The results are compared with the solutions obtained via BPF, SHF, TF, etc. and many illustrations are presented.

Introduction

For more than three decades different piecewise constant basis functions (PCBF) have been employed to solve problems in different fields of engineering including control theory. It was in 1910, when Haar functions [1] appeared as the first set of the PCBF family. As far as shapes were concerned, this function set was entirely different from the ‘orthodox’ sine–cosine functions and was the genesis of a new class of orthogonal functions. Piecewise constant nature of this ‘new’ class of functions attracted many researchers to explore its appropriate application areas. Of this class, the block pulse function (BPF) [2], [3] set and its variants [4] proved to be the most efficient because of their simplicity and versatility in analysis [5] as well as synthesis [4], [6] of control systems.

In 1998, an orthogonal set of sample-and-hold functions [7] was introduced by Deb et al. and the same was applied to solve problems related to discrete time systems with zero order hold. The set of sample-and-hold functions approximates any square integrable function of Lebesgue measure in a piecewise constant manner and it proved to be more convenient for solving problems associated to sample-and-hold control systems.

In 2003, orthogonal triangular functions [8] were introduced by Deb et al. and the same were applied to control system related problems including analysis and system identification. The set of triangular functions approximates any square integrable function in a piecewise linear manner. From this basic property relevant theories were developed to make this set powerful enough to deal with control system related problems [9], [10].

In this paper, a new set of orthogonal hybrid functions (HF), which is a combination of sample-and-hold functions and triangular functions, is presented.

This article is organized as follows:

  • (i)

    Function approximation.

  • (ii)

    Computation of operational matrices for integration in hybrid function (HF) domain.

  • (iii)

    Integration of time functions using the operational matrices for integration in hybrid function (HF) domain.

  • (iv)

    Transfer function approximation from impulse response data.

The results obtained for the cases (i) and (iii) are compared with the exact solutions along with the results obtained via block pulse functions [2], [3]. The results obtained for Item No. (iv) above are compared with the results computed via traditional block pulse functions, non-optimal block pulse functions, sample-and-hold functions [7] and triangular function [10] based analysis.

Section snippets

Brief review of block pulse functions (BPF) [2]

The ith member of a set of BPF, Ψ(m)(t), comprised of m component functions, is defined as:ψi(t)=1,foriht<(i+1)h,0,elsewhere,where i = 0, 1, 2,  , (m  1).

Fig. 1 shows a set of eight block pulse functions.

The BPF set is orthogonal as well as complete in t  [0, T).

A square integrable time function f(t) of Lebesgue measure may be expanded into an m-term BPF series in t  [0, T) as:f(t)i=0m-1fiψi(t)=f0f1f2fif(m-1)Ψ(m)(t)fori=0,1,2,,(m-1)F(m)TΨ(m)(t),where, [  ]T denotes transpose and the (i + 1)th BPF

Brief review of sample-and-hold functions (SHF) [7]

Any square integrable function f(t) may be represented by a sample-and-hold function set in the semi-open interval [0, T) by considering:fi(t)f(ih),i=0,1,2,,(m-1),where, h is the sampling period (=T/m), fi(t) is the amplitude of the function f(t) at time ih and f(ih) is the first term of the Taylor series expansion of the function f(t) around the point t = ih, because, for a zero order hold (ZOH) the amplitude of the function f(t) at t = ih is held constant for the duration h.

SHFs [7] are similar

Brief review of triangular functions (TF) [10]

From a set of block pulse function, Ψ(m)(t), we can generate two sets of orthogonal triangular functions (TF), namely T1(m)(t) and T2(m)(t) such that:Ψ(m)(t)=T1(m)(t)+T2(m)(t),

Fig. 3a, Fig. 3b show the orthogonal triangular function sets, T1(m)(t) and T2(m)(t), where m has been chosen arbitrarily as 8. These two TF sets are complementary to each other. For convenience, we call T1(m)(t) the left handed triangular function (LHTF) vector and T2(m)(t) the right handed triangular function (RHTF)

Hybrid Function (HF): A Combination of SHF and TF

We can use a set of sample-and-hold functions and the RHTF set of triangular functions to form a hybrid function set, which we name a ‘Hybrid Function’ set.

To define a hybrid function (HF) set, we express the (i + 1)th member Hi(t) of the m-set hybrid function H(m)(t) as:Hi(t)=aiSi(t)+biT2i(t),where, i = 0, 1, 2,  , (m  1), ai and bi are scaling constants, 0  t < T. For convenience, in the following, we write T instead of T2.

While the block pulse function set provides us with a staircase solution, the

Method of function approximation by hybrid functions

Consider a function f(t) in an interval t  [0, T). If we consider (m + 1) equidistant samples c0, c1, c2,  , cm of the function with a sampling period h (i.e., T = mh), we can write:f(t)c0c1c2cicmS(m)(t)+[(c1-c0)(c2-c1)(c3-c2)(ci-ci-1)(cm-c(m-1))]T(m)(t)n=0m-1cSnSn(t)+n=0m-1cTnTn(t)CSTS(m)(t)+CTTT(m)(t).

Algorithm of function approximation by hybrid function

Table 1 presents basic properties of the basic functions BPF, SHF, TF, NOBPF and HF.

Example 1

Let us expand the function f1(t) = t in hybrid function domain taking m = 8 and T = 1 s. Following the method

Operational matrix for integration

A hybrid function set is a combination of a sample-and-hold function set and a triangular function set (RHTF). In order to derive the operational matrix for integration in hybrid function domain, we proceed in a familiar manner [2], [12] adopted for Walsh and block pulse functions. Here, for hybrid function set, we consider both the component function sets separately.

Integration of functions using hybrid function domain operational matrix

Let f(t) be a square integrable function which can be expanded in hybrid function domain as:f(t)c0c1c2c(m-1)S(m)(t)+(c1-c0)(c2-c1)(c3-c2)cm-c(m-1)T(m)(t)=CSTS(m)(t)+CTTT(m)(t),where, [  ]T denotes transpose.

Integrating Eq. (30) with respect to t, we get:.

Now we use (31) to perform integration of a few simple square integrable functions.

Example 5

Integrate the function f1(t) = t via hybrid function method taking T = 1 s, m = 8 and h = T/m = 0.125 s.

The integration of the given function is t22 and direct expansion

Error analysis

The representational error for equal width block pulse function expansion of any square integrable function of Lebesgue measure has been investigated by Rao and Sivakumar [13], while the error analysis for pulse-width modulated generalized block pulse function (PWM-GBPF) expansion has been carried out by Deb et al. [4].

Hybrid function approximation has two components: sample-and-hold function component and triangular function component. To present an error analysis in hybrid function domain, it

Parameter estimation of a linear system

Now we employ the set of hybrid functions to estimate the parameters of a linear system from its impulse response [14] data.

Let H(t) be the p × r impulse response matrix of a linear time-invariant multivariable system where H(t) is specified graphically or analytically. Also, let H(s)=L[H(t)] be the transfer function matrix of the system.

Now consider the rational function matrix G(S) of the form:G(s)=Bpsp+B(p-1)sp-1++B1s+B0sq+a(q-1)sq-1++a1s+a0,p<q.

Expansion of H(s) and G(s) in power series is

Conclusion

In this paper, a new set of orthogonal functions, comprised of triangular functions and sample-and-hold functions, termed as hybrid functions (HF), has been proposed. The theoretical aspects of function approximation via this new function set have been established with two examples treated to illustrate its potential. As in case of Walsh and block pulse functions, the operational matrix for integration in hybrid function domain, namely a combination of the rectangular matrices R1(m×2m) and R2(m×

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