Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations
Introduction
Differential equations [1] are the essence of modern control theory. Solution of differential equations of widely varying types have attracted many researchers since the early days of mathematics. The main tool for tackling differential equations in the modern age is the numerical analysis, and to be explicit, numerical integration.
The work by Gear [2] discusses numerical initial value problems related to ODE’s and Hairer presents a useful study of non-stiff ODE’s. The work by Butcher [3] gives an impressive overview of numerical methods for ordinary differential equations. The age old 4th order Runge–Kutta method has undergone so many improvements and modifications and the book by Butcher [4] provides an exhaustive study on the subject. In a recent work Cools et al. [5] discuss numerical integration with special emphasis on oscillatory integrals. Simos’s [6] work on modified Runge–Kutta methods for the numerical solution of ODEs with oscillating solutions tackles simultaneous first order ODE’s very much similar to state space equations in control theory.
In control theory, usually any one works with the mathematical model of any system whose behavior is of interest. And such task mostly revolves round the solution of one or many differential equations of different types and different orders. However, in the well-known state variable method of linear control theory, one deals with n number of first order differential equations in vector matrix form.
For more than three decades, solution of differential equations as well as integral equations related to fields of engineering including control theory was also attempted by researchers employing piecewise constant basis functions (PCBF) like Walsh functions, block pulse functions etc.
It was in 1910, when Haar functions [7] 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. The first ever attempt to solve differential and integral equations using Walsh functions was made by Corrington [8]. Later, block pulse function (BPF) [9], [10] set and its variants [11] proved to be the most efficient because of their simplicity and versatility in analysis [12] as well as synthesis [11], [13] of control systems in state space.
In 1998, an orthogonal set of sample-and-hold functions [14] were introduced by Deb et al. and the same was applied to solve problems related to discrete time systems with zero order hold. Like block pulse functions, the set of sample-and-hold functions approximate any square integrable function of Lebesgue measure in a piecewise constant manner. But this set proved to be tailor made for solving problems associated to sample-and-hold systems.
In 2003, orthogonal triangular functions [15] were introduced by Deb et al. which could approximate any square integrable function in a piecewise linear manner. The same function set was applied to control system related problems including analysis and system identification on state space platform.
In this paper, a new set of orthogonal hybrid functions (HF), which is a combination of sample-and-hold function and triangular function, is presented. This new function set is utilized for:
- (i)
Function approximation.
- (ii)
Computation of the operational matrix for integration in hybrid function (HF) domain.
- (iii)
Integration of time functions using the operational matrix for integration.
- (iv)
Computation of the operational matrix for differentiation.
- (v)
Differentiation of time functions using the operational matrix for differentiation.
- (vi)
Solution of differential equations and state equations using the operational matrices for differentiation and integration.
The results obtained for the cases (i) and (iii) are compared with the exact solutions. The results obtained for item nos. (iv)–(vi) above are compared with the results computed via exact analysis as well as other methods discussed by Simos [6].
Section snippets
Brief review of block pulse functions (BPF) [9]
The ith member of a set of BPF, Ψ(m)(t), comprised of m component functions, is defined aswhere 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 [9], [10] into an m-term BPF series in t ∈ [0, T) aswhere [⋯]T denotes transpose and the ith BPF
Brief review of sample-and-hold functions (SHF) [14]
Any square integrable function f(t) may be represented by a sample-and-hold function set in the semi-open interval [0, T) by consideringwhere 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 [14] are similar
Brief review of triangular functions (TF) [16,17]
A block pulse function can be dissected along two diagonals to generate two triangular functions. That is, when we add two component triangular functions, we get back the original block pulse function. This dissection process is shown in Fig. 2, where the first member ψ0(t) is resolved into two component triangular functions T10(t) and T20(t).
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
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 ith member Hi(t) of the m-set hybrid function H(m)(t) in 0 ⩽ t < T aswhere i = 0, 1, 2, … , (m − 1), ai and bi are scaling constants. For convenience, in the following, we write T instead of T2.
While the block pulse function set provides us with a staircase solution, the hybrid
Function approximation by hybrid functions
Consider a function f(t) in an interval t ∈ [0, T). If we take (m + 1) equidistant samples c0, c1, c2, … , cm of the function with a sampling period h (i. e., T = mh), we can write Illustrative Example 1 Take up the function f1(t) = sin(πt) and express it via a hybrid function set for m = 8 and T = 1 s. Following Eq. (12), the result is
Operational matrices 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 matrices for integration in hybrid function domain, we proceed in a familiar manner [9], [18] 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 matrices
Let f(t) be a square integrable function which can be expanded in hybrid function domain from Eq. (12) as
Integrating Eq. (28) with respect to t, we get
Now we use (29) to perform integration of a few simple square integrable functions. Illustrative Example 2 Integrate the function f2(t) = t via hybrid function
Differentiation matrices in HF domain
Let a square integrable function f(t) of Lebesgue measure be expressed in HF domain, for m = 4, as
When a function f(t) is expressed in HF domain, it is converted to a piecewise linear function in [0, T). If this converted function is differentiated, the result will be a staircase function. For such a function, any attempt to compute the higher derivatives will give rise to delta as well as double delta functions.
To avoid this
Differentiation of a function in HF domain
Illustrative Example 3 Let us consider a function . Expanding it in HF domain, for m = 10 and T = 1 s, we have
Now we differentiate the function given in (40) using the matrices of Eqs. (38), (39) for m = 10. The result of differentiation in HF domain is obtained as
Solution of differential equations using differential operational matrices
Now solution of differential equations is attempted using both differentiation and integration operational matrices.
If we try to solve any differential equation with the operational matrices DS and DT of Eqs. (38), (39), the attempt is met with a permanent difficulty: samples of the unknown function, say x(t), are required as elements of both the differential matrices. Obviously, any such attempt is certain to fail, because these samples of x(t) are yet to be derived as the solution of the
Solution of a set of simultaneous differential equation
A set of simultaneous differential equations is no different from the vector matrix equation known as state equation in the area of control theory.
A homogeneous state equation is given bywhere A is an (n × n) system matrix,
x is the state vector given bywith the initial conditions
Integrating (72), we get,
Using (12) and expanding x and x(0) via an m-set hybrid function, we getwhere
Conclusion
In this paper, a new set of orthogonal functions, comprised of a 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 one example 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
Acknowledgement
The authors are indebted to unnamed Reviewers and Professor Binayak Samaddar Choudhury of Department of Mathematics, Bengal Engineering and Science University for their valuable suggestions which led to significant upgradation of the work.
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