ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 71

An Operational Standpoint in Electrical Engineering

Fred´ eric´ Rotella and Irene` Zambettakis

Abstract—In electrical engineering education exists a major is based on the use of a pure operational point of view difficulty for first level students, namely the Laplace transform. that provides an opportunity to link methods developed in The question is: does this ubiquitous tool is needed in an electrical electrical engineering with experiments and applications. The engineering course? Our answer is: Obviously, not. Based on an operational standpoint the paper describes some guidelines paper is organized as follows. In a first part we remind some and results for a primer on handling signals and linear systems well-known problems about the use of Laplace transform in without using the Laplace transform. The main advantage is operational calculus. Among engineers and mathematicians that the operational standpoint leads to simplified proofs for Heaviside appears, in the historical developments of opera- well-known results. tional calculus, as the focal point. His ideas on the use of the Index Terms—Transfer operator, operational calculus, Laplace differential operator and on the definition of the transfer (resis- transform, Carson transform, signal generator, Heaviside, tance) operator are the basis of guidelines for an engineering Mikusinski.´ course without the Laplace transform. A particular subsection Original Research Paper is devoted to Oliver Heaviside. The second part deals with DOI: 10.7251/ELS1317071R the transfer operator definition. Let us insist here that it must not be confused with the transfer function definition which I.INTRODUCTION can be related to the frequency response only. This point is detailed in the third part of the paper. The next part is devoted N signal processing, electrical engineering or automatic to some linear models analysis. For instance, we point out that control the Laplace transform is used as an ubiquitous poles and zeros meaning, DC gain calculus or error analysis tool.I First level courses in electrical engineering or basic are obvious within our approach. The final part talks about textbooks on these areas contain lessons on properties and a recent application of operational calculus, namely algebraic uses of Laplace transforms. For instance, in automatic control estimation. the Laplace transform approach leads to define the Laplace The essential proofs based on Laplace transform theorems transform of a signal or the transfer function of a system. can be read in standard textbooks on automatic control (e.g. On the one hand it is used to get the corresponding response [24], [28], [36]). Nevertheless, we will see that the operational signal of a system with respect to a given input signal. On standpoint leads to simplified proofs of well-known results. the other hand it is important for the analysis and design of Concerning the notation, we consider signals as elements control systems [14]. This tool appears thus as a necessary belonging to the set C of integrable real valued functions f = and unavoidable burden for students participating in electrical f(t) , supposed to be m times continuously differentiable engineering courses. Nevertheless, the induced mathematical { } on [0, ) except at isolated points where it is assumed that background leads to some problems for teachers from a pure ∞ both left limit and right limit exist. As f(t) denotes the educational standpoint [7], [27], [27] For instance, one of the { } signal f while f(t) stands for its value at time t, we write conclusions of [27] is “that the Laplace transform is one of the for two signals a and b in C: for all t 0, a(t) = b(t), or most difficult topics for learning engineering electric circuit ≥ a(t) = b(t) , or a = b. However, when no confusion is theory.”Roughly speaking, the difficulty arises with the con- { } { } possible the braces or “for all t 0” may be dropped. nection between mathematical framework and physical world. ≥ However, this transform has some skeletons-in-the-closet [33], II.OPERATIONAL CALCULUS [40]. In this article, we discuss an alternative approach to the use of Laplace transform. This approach is based on a pure Let us consider a signal x(t) defined for a positive time t and operational standpoint which has been proposed more than a satisfying some appropriate growth conditions. The Laplace century ago by Oliver Heaviside. Based on this standpoint an transform of x(t) is automatic control course has been detailed in [51], [52]. ∞ st X(s) = x(t) = x(t)e− dt, (1) In the following, we describe guidelines for starting a L{ } 0 course without using the Laplace transform. The presentation Z where s is a complex variable. This definition requires ad- Manuscript received 26 November 2013. Received in revised form 25 vanced mathematical machinery [40], [56] which is very de- December 2013. Accepted for publication 26 December 2013. manding, and usually, beyond the skills of most undergraduate F. Rotella is with the Ecole Nationale d’Ingenieur´ de Tarbes, 47 avenue students. This generates difficulties that lead desertion of d’Azereix, BP 1629, 65016 Tarbes CEDEX, France and Laboratoire de Genie´ de Production, 47 avenue d’Azereix, BP 1629, 65016 Tarbes CEDEX, France students from basic electrical engineering classes. Equation (1) (e-mail: [email protected]). assumes that all considerations, diagrams and developments I. Zambettakis is with the Universite´ Toulouse III Paul Sabatier, rue are embedded in a space of transformed signals. Students ask Lautreamont,´ 65016 Tarbes CEDEX, France and Laboratoire de Genie´ de Production, 47 avenue d’Azereix, BP 1629, 65016 Tarbes CEDEX, France frequently two questions in regards to the usefulness of equa- (e-mail: [email protected]). tion (1). How can we experimentally exhibit or visualize the 72 ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013

transformed signals for example by mean of an oscilloscope? U(s) - F (s) - Y (s) Are some signals forbidden in controlled systems or linear electronics? For instance, exp(t2) has no Laplace transform Fig. 1. Basic block diagram. [43]. Some fundamental theorems linked to Laplace transform have also provided some misunderstandings about the actual III.HISTORICAL STANDPOINT meaning of s. For instance, consider the Laplace transform of a derivative function with the initial condition x(0), namely, As a matter of fact, the use of Laplace transforms is one x˙(t) = sX(s) x(0). When x(0) vanishes the complex method among many others [35], [39] to justify the Heaviside L{ } − variable s is considered as a time derivative operator. How- operational calculus (Figure 2). Note that the operational ever, it just stands in the space of transformed signals only. calculus is used to solve differential equations (in most cases Concerning definition (1), the lower limit of integration is linear) rather than automatic control problems. The history + of the Laplace transform has been studied extensively by often replaced by 0−, 0 , or [14], [36], [46] to overcome −∞ discontinuity problems arising in case of particular functions. Deakin [9], [10]. It has been first introduced in the form An attempt to solve this question and to unify the Laplace (1) by Bateman in 1910 to solve the differential equation x˙(t) = λx(t) λ formalisms is proposed in [38]. For the case, we are faced where is a nonzero real number. ∞ − with the bilateral Laplace transform, which is questionable Nevertheless and independently, Oliver Heaviside (1850- as well [40]. The purpose of this article is to show that the 1925), electrical engineer at the Great Northern Telegraph mathematical machinery required by the Laplace transform Company, has introduced a pure operational calculus. His [57] can be avoided. Moreover, the pedagogic difficulty can seminal works have been collected in two books : be cleared in a natural way. Electrical Papers, 1873 1891; • → Electromagnetic Theory, 3 volumes, 1891 1893 (ET1), When the transfer function of a linear system has to be • → 1894 1898 (ET2), 1900 1912. defined, Laplace transform is applicable. For a single-input → → single-output system the transfer function is defined as the During his life he brought great breakthroughs on different quotient of the Laplace transform of the output y(t) to the electrical or electromagnetic topics such as : Laplace transform of the input u(t) with the assumption of Maxwell’s field equations. He reworded them in terms of • zero initial conditions. In other words, the transfer function is electric and magnetic forces and energy flux. The use of defined by vector analysis to write them as a fourth-order system is y(t) Y (s) due to Heaviside. Consequently, the well-known Maxwell F (s) = L{ } = . u(t) U(s) equations should be known as the Maxwell-Heaviside L{ } equations; Although the name transfer function as a mathematical tool is atmospheric layers. He predicted the existence of ionized adequate for s = jω, where j2 = 1 and ω is the frequency • − layers by which radio signals are transmitted around the [3], [29], this ad-hoc definition generates some interesting Earth’s curvature. The existence of the ionosphere was questions. The Laplace transform of signals cannot be obtained confirmed in 1923 only. These layers are bearing the in practice, and sometimes we wonder how to determine the name KennellyHeaviside; transfer function of a system? For instance this question is transmission lines. He developed and patented (1880) the • avoided in identification procedures [37] which use ARMAX coaxial cable; models involving recurrence relationships instead of transfer fractional derivatives. He introduced a 1/2-order deriva- • functions. In several high quality textbooks on discrete-time tive operator to modelize a diffusive process; systems (e.g. [2]), the complex variable z of the Z-transform operational calculus. The major part of its ideas about • [32] and the shift-forward operator q are both used. However, operational calculus are gathered together in ET2. The the choice between z and q is not always argued. So this writing is oriented for practice : subtle differences, which is mysterious for students, is not “Of course, I do not write for rigourists but for really useful. According to our personal experience in teaching – a wider circle of readers who have fewer preju- automatic control, it is very important to be able to give an dices,”(ET2); experimental meaning of the transfer of a system, irrespective “There is, however, practicality in theory as well in of previous formal definitions. In practice students often forget – practice.”(ET1 ) to relate the transfer to the differential operation induced by the system. Indeed, the use of Laplace transforms leads to the while the proposed developments lead him to name his diagram depicted in Figure 1 which describes the relationship method “my operational method”. between the Laplace transforms of input and output signals and The interested reader about life and works of Oliver Heavi- the transfer function F (s) of the system. But this formalism side can see the following biographies : G. Lee, Oliver Heavi- hides the time variable and, there is no different notations side, 1947; H.J. Josephs, Oliver Heaviside ; a biography, 1963; between signals and systems. So, the essential meaning is G.F.C. Searle, Oliver Heaviside, the man, 1987; P.J. Nahin, lost. The reader can already notice that with the forthcoming Oliver Heaviside : sage in solitude, 1988; I. Yavetz, From developments we will not use anymore the term transfer obscurity to enigma : the work of Oliver Heaviside, 1995. Let function but just transfer for the transfer model of a system. us briefly, describe the Heaviside operational method. ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 73

Leibniz (1695) Euler (1730) Laplace (1812) Servois (1814)

Cauchy (1827) Gregory (1846) Boole (1859)

Heaviside (1884-1895) Kirchoff (1891)

Wagner (1915) Bromwich (1916) Carson (1917-1922) Smith (1925) Levy (1926) Dirac (1926) Jeffreys-March (1927) Van der Pol (1929) Doestch (1930)

Florin (1934) Schwartz (1945,1947) Mikusinski´ (1950)

Dimovski (1982)

Fig. 2. Genealogy of operational calculus. This diagram is detailed in (Rotella, Zambettakis, 2006). Date indicates publication year of a major contribution in operational calculus.

tn 1) Coding. Introducing the derivative operator and because p nH(t) = he gets − n! d n , D (before 1886) , p (after 1886), t dt C = a ; n n! n 0 he codes a linear differential equation (1884) for an X≥ electrical circuit or every system that can be, by analogy, In 1895, he deals with the sinusoidal input E = reduced to an electrical circuit. In this framework, he sin(nt) = (exp(nt)). Using of the Euler shifting = points out the resistance operator, namely the trans- theorem fer operator, Z(p) to defines the operational solution E at at C = where C et E are the output and input ϕ(p)e f(t) = e ϕ(p + a)f(t), Z(p) voltage respectively. he proposes to solve the differential equation by succes- 2) Algebrization. When E is fixed at the outset he proposes d sively replacing p with ni and, then i with . different methods to get the solution of the differential d(nt) equation. For example, let E be the Heaviside function Let us consider for example The RL circuit with Z(p) = or step signal H(t) defined by H(t) = 1 for t 0 and 0 R+Lp. For the input E = H(t), the expansion theorem yields ≥ elsewhere. Using the expansion theorem (1886) he gets R E eλkt Z(0) = R, k = 1, λk = ,Z0(λk) = L, C = + E , − L Z(0) λkZ (λk) k 0 thus X E Rt where Z(λ ) = 0. Later on, using the series expansions C = 1 e− L . k R − with respect to p (1888)   1 The series expansion with respect to p− leads to n C = αnp H(t), E 1 n 0 C = , X≥ Lp R n 1 + Lp and supposing p H(t) = 0 he gets C = α0. Then, using 1 E R R2 the series expansions with respect to p− (1892) = (1 + ), Lp − Lp L2p2 − · · · n C = anp− H(t), E Rt = 1 e− L . n 0 X≥ R −   74 ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013

For the input E = sin(nt), the successive replacements So, equation (2) becomes method induces n n 1 n 2 p + an 1p − + an 2p − + + a1p + a0 y(t) = 1 R Lni − m − m 1 ··· C = sin(nt) = − sin(nt), bmp + bm 1p − + + b1p + b0 u(t). R + Lp R2 + n2L2  − ···  (4) R Ln d To separate input and output signals from the system we divide = 2 2 2 sin(nt) 2 2 2 sin(nt). n n 1 R + n L − R + n L d(nt) equation (4) by the polynomial factor p + an 1p − + − ··· Despite the fact that Oliver Heaviside was disapproved by +a0, which yields to code the input-output relationship as mathematicians, at the begining of the twentieth century the y(t) = F (p)u(t) with operational calculus became a high challenge. For instance, in m m 1 bmp + bm 1p − + + b1p + b0 − 1928 E.T. Whittaker quotes F (p) = n n 1 n ···2 . (5) p + an 1p − + an 2p − + + a1p + a0 “We should now place the operational calculus with − − ··· Poincare’s´ discovery of automorphic functions and Ricci’s We must insist here that y(t) = F (p)u(t) cannot be consid- discovery of the tensor calculus as the three most important ered as the solution of the differential equation (2). Indeed, the mathematical advances of the last quarter of the nineteenth initial conditions are not known. y(t) is determined with this century. Applications, extensions and justifications of it consti- writing as with the writing (2). In equation (5), F (p) stands tute a considerable part of the mathematical activity of today.” for the transfer operator. In essence, it is the transfer, which Justifications of the operational calculus (see [39], [47] and represents the operation induced by the system to transform the references therein) can be gathered together in two different input signal into the output signal. The operational approach kinds of methods : on the one hand the integral transforms provides an opportunity to relate the transfer operator (5) to such as the Laplace transform and, on the other hand the the differential equation (2). The diagram, depicted in figure algebraic methods based on the Paul Levy´ standpoint linked 3, corresponds to an experimental situation. Notice that, in to the integral operator. These last ones, which to our point of this figure, t denotes the time variable and p the derivative view are the only ones to preserve the practical meaning have operator. The difference between signals and system is due to lead to the Mikusinski´ operational calculus. In the sequel we the use of these notations. The action performed by a system carry on with the way paved by Heaviside approach, keeping on an input signal is retained. The essential meaning of the in mind that the mathematical background must not cover up transfer F (p) is the linear differential equation that links the the practical meaning. output signal to the input signal.

IV. TRANSFER OPERATOR - - We begin by considering a linearized system around an u(t) F (p) y(t) equilibrium point. We suppose this system can be described by the linear differential equation Fig. 3. Operational block diagram. The output signal y(t) is obtained by F (p)u(t) where u(t) is the input signal. (n) (n 1) (n 2) y (t) + an 1y − (t) + an 2y − (t) + (1) − − ··· +a1y (t) + a0y(t) = (m) (m 1) (2) bmu (t) + bm 1u − (t) + − ··· +b u(1)(t) + b u(t), 1 0 B. Operational Calculus As Polynomial Calculus where y(t) and u(t) stand for the differences of output and Operational calculus is understood as algebraic methods for input signals with their setpoint values respectively and n solving differential or recurrence equations, specifically in a and m are two integers. In (2) the coefficients ai and bj are linear time-invariant framework. In our point of view solving constant parameters. a differential equation for a given input is a mathematical exercise only [40]. In electrical engineering or, more generally, A. Coding in automatic control operational calculus means rules for The aim is to provide a tool making easy the manipulation transfer connections or decompositions through polynomial of linear time-invariant differential equations and, which al- calculus. Thus the coding (4) is useless when we are not lows to separate input and output variables from the system. allowed to associate transfer operators. From the operational Following Heaviside [30], [47] or Carson [5], we introduce standpoint the connection rules can be demonstrated through the derivative operator the following steps. The transfers of the connected system d provide differential equations. The connections and the elim- p , , dt ination of intermediate signals lead to a differential equation between the output and input signals. The encoding of this which applied on x(t) in C gives the codings differential equation with p ensures the results. Although x˙(t) = px(t), x¨(t) = p2x(t), . . . , x(n)(t) = pnx(t),... (3) we can use this procedure in every case, it is sufficient to exemplify it with respect to series or parallel connections for In view of these codings and using the distributivity property two first-order systems. we get, for every real numbers α and β, Let us consider such two linear systems described by n m (n) (m) [αp + βp ]x(t) = αx (t) + βx (t). y1(t) = F1(p)u1(t) and y2(t) = F2(p)u2(t) where u1(t) and ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 75

u2(t) are the input signals, F1(p) and F2(p) the transfers of the V. SYSTEM RESPONSES systems, and y1(t) and y2(t) the corresponding output signals. System analysis is often the study of some particular re- In this regard, we have sponses of the system and, specially, the step and frequency

b1p + b0 β1p + β0 responses. F1(p) = and F2(p) = , a1p + a0 α1p + α0 A. Step Response where a0, a1, b0, b1, α0, α1, β0, and β1 are constant parameters. The series connection is defined by u2(t) = y1(t), The step response of a system is the solution of the differ- u(t) = u1(t), and y(t) = y2(t). The application of the ential equation of the system to a step input signal with zero procedure yields initial conditions. With operational calculus, we can expand transfer operator as a linear combination of simple transfers β b p2 + (β b + β b )p + β b y(t) = 1 1 0 1 1 0 0 0 u(t), an 1 α a p2 + (α a + α a )p + α a or , 1 1 0 1 1 0 0 0 (p + a)n pn where we recognize the product F1(p)F2(p). The parallel where a stands for a nonzero complex number and n for an connection is defined by u2(t) = u1(t) = u(t) and y(t) = integer. The step response of a multiple integrator with the y (t) + y (t), which yields 1 tn 1 2 transfer is . Let us consider the step response s (t) of pn n! n 2 n ((a1β1 + α1b1)p + (a1β0 + b1α0 + a0β1 a the Strejc´ system [48] with the transfer . For n = 1 + b0α1)p + (a0β0 + b0α0)) (p + a)n y(t) = 2 u(t), a α1a1p + (α0a1 + α1a0)p + α0a0 the associated differential equation to the transfer is p + a where we recognize the sum F1(p)+F2(p). These results can ay(t)+y ˙(t) = au(t) and we obtain by usual methods [61] the be extended to any order by induction, thus the transfer of the corresponding response to a given input u(t) with the initial series connection of two systems is the product of their transfer condition y(0) and the transfer of the parallel connection of two systems is t at aν their sum. y(t) = e− y(0) + a e u(ν)dν . Parallel and series rules give a meaning to the decomposi-  Z0  tions and the handling of transfer operators with polynomial For u(t) = 1 and zero initial conditions the step response calculus. These operations on transfer operators are the basis becomes at of operational calculus in automatic control. We can apply s (t) = 1 e− . 1 − usual techniques as Mason’s rule associated to the signal- a flow graphs [41]. This operational calculus can be applied also For n = 2 we have s2(t) = s1(t) from which it follows p + a for multiple-input multiple-output systems with the difference t that commutativity does not occur anymore. Let us note that at aν at s2(t) = ae− (e 1) dν = 1 e− (1 + at). (6) the polynomial formalism is used in several textbooks on 0 − − multivariable systems [33], [34] with no need of the Laplace Z For the general case n 1, let us suppose transform. ≥ n 1 at − i s (t) = 1 e− c t , n − i,n i=0 ! C. The Delay Operator X and A pure time delay of T between input and output signals n at i induces y(t) = u(t T ). This particular linear system cannot s (t) = 1 e− c t , − n+1 − i,n+1 be associated to a differential equation as (2). A special i=0 ! treatment must be used for delay equations. Following an idea X where the coefficients c and c are constant parameters. of Euler [15], the Taylor expansion of u(t T ) yields i,n i,n+1 − These signals are linked by the differential equation 2 n T (n) ( T ) u(t T ) = u(t) u˙(t)T +¨u(t) +u (t) − + , s˙n+1(t) + asn+1(t) = asn(t), − − 2 −· · · n! ··· which leads to the the relationships c = 1 and, for i = 1 which is encoded to give 0,n+1 to n, i T 2 ( T )n a a y(t) = u(t) pT u(t) + p2 u(t) + pn − u(t) + , ci,n+1 = ci 1,n = . − 2 − · · · n! ··· i − i! n We deduce the well known result that, for the Strejc´ model, ( pT ) an = − u(t) = (exp( pT )) u(t). , the step response is  n!  − n n 0 (p + a) X≥   n 1 i i We obtained the transfer operator for the time delay T as at − a t pT sn(t) = 1 e− . F (p) = e− . − i! i=0 ! X 76 ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013

So, the step response of a given system defined by a transfer the frequency is the only actual transfer function. Graphic operator can be calculated using polynomial calculus. representations such as Bode, Black-Nichols, or Nyquist loci Stability analysis does not use the Laplace transform. How- may be used to analyze the frequency response [36]. For ever, we may insist again on the link between the transfer unstable systems the loci are valid as calculated representations operator, the differential equation and the transient behavior. only. While for stable systems experiments can allow to get an Let us consider the transfer operator where a and n the frequency response as well. (p + a)n have the same meaning as before. From the previous paragraph we can see that the step response is composed of a constant VI.ANALYSIS term and a time-dependent term. The first term is the forced A. Poles and Zeros response and the second term is the transient behavior. The transient behavior tends asymptotically to zero if and only if The names of poles and zeros come from the interpretation the real part of a is strictly negative. More generally, consider of a transfer operator F (p) as a function of a complex variable the n-th order transfer p. This interpretation is a consequence of the formulation of b pm + + b p + b Laplace transform and it misunderstands the physical meaning F (p) = m ··· 1 0 , ρ1 ρ2 ρr of these notions. The consideration of a transfer operator as (p p1) (p p2) (p pr) − − ··· − a coding of a differential equation provides an immediate where p1, p2, . . . , pr are the r complex poles of F (p) and ρi r physical interpretation. Namely, let us consider the transfer are the respective multiplicities with n = i=1 ρi. The poles operator p t p t p t generate terms associated with the signals e 1 , e 2 , . . . , e r p + a weighted by time polynomials of order ρ P1 respectively. So, F (p) = , (7) i − p + b when all the poles have strictly negative real part, the transient behavior tends asymptotically to zero. Namely, the system is where a and b are constant parameters. In the operational asymptotically stable. standpoint the transfer (7) corresponds to the input-output differential equation y˙(t)+by(t) =u ˙(t)+au(t) where y(t) and u(t) are the output and input signals. First, consider u(t) = 0 B. Frequency Response for t > 0 and a nonzero initial condition y(0) we then get For asymptotically stable systems the frequency response bt y(t) = y(0)e− for t > 0. Second, consider a zero initial is deduced from the steady-state output response to a given at condition for the output and the input signal u(t) = e− for jωt sinusoidal input signal u(t) = e where ω is the frequency t > 0 we obtain u˙(t) + au(t) = 0 so y(t) = 0 for t > 0. 2 and j = 1. Consider a system defined by the transfer More generally, poles correspond to signals generated by the − B(p) operator F (p) = where A(p) and B(p) are two system with zero input. Zeros correspond to signals absorbed A(p) polynomials. The operational approach leads to the encoded or blocked by the system. Let us write the transfer operator input-output differential equation as A(p)y(t) = B(p)u(t). (5) as jωt ν1 ν2 νd For the sinusoidal input u(t) = e , we obtain (p z1) (p z2) (p zd) F (p) = k − ρ − ρ ··· − ρ , B(p)u(t) = B(jω) ej(ωt+arg(B(jω))), (p p1) 1 (p p2) 2 (p pr) r | | − − ··· − where B(jω) and arg(B(jω)) denote the module and the where k = bm, zi, i = 1, . . . , d and pi, i = 1, . . . , r are | | argument of the complex number B(jω) respectively. The complex numbers, and νi, i = 1, . . . , d and ρi, i = 1, . . . , r p t output y(t) is the sum of a particular solution of the differential are integers. For i = 1, . . . , r, e i is solution of the coded equation and the general solution of the differential equation differential equation without second member. The general solution characterizes (p p )ρ1 (p p )ρ2 (p p )ρr y(t) = 0, the transient response that vanishes in case of asymptotically − 1 − 2 ··· − r stable systems. For a particular solution, we look for the and for i = 1, . . . , d, ezit is solution of the coded differential y(t) = Y ej(ωt+ϕ) steady-state behavior as the form where equation Y and ϕ are constant parameters. Replacing this expression for y(t) in the differential equation yields (p z )ν1 (p z )ν2 (p z )νd u(t) = 0. − 1 − 2 ··· − d B(jω) On the one hand we can use the correspondence between Y = | | = F (jω) , pit A(jω) | | e and the transfer denominator roots pi that characterizes | | the transient rate in the linear constant parameter framework ϕ = arg(B(jω)) arg(A(jω)) = arg(F (jω)). − only. The same remark can be said about the correspondence zit The frequency response is defined by the evolution of between e and the transfer numerator roots zi. On the other ( F (jω) , arg(F (jω))) as the frequency ω varies from 0 to hand this signal approach for the pole and zeros meaning can | | + . We can notice that F (jω) is the transfer function of be extended to time-varying or nonlinear multivariable systems ∞ the system such as Harris defined it [29]. In our standpoint, with an algebraic standpoint [16], [17]. this transfer function must not be confused with the transfer In order to underline and to exemplify the important operator F (p). Nevertheless, F (jω) such as a function of problem of pole/zero cancellation let us consider the series ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 77 connection with the systems : input-error transfer is 1 F (p), we obtain a steady-state error − 1 of zero order when the input-error DC gain is zero. This is a y(t) = u(t), fundamental remark for the following. p 1 p − 1 Let us notice that r1(t) is the integral of r0(t). Namely, z(t) = y(t). 1 − r (t) = r (t), thus we have p + 1 1 p 0 The pole 1 induces, in the transient behavior or in the initial t ε (t) = r (t) y (t), conditions effect for the first system, an e signal. This signal 1 1 − 1 is blocked by the second system, which has 1 as zero. As 1 1 t = r0(t) F (p) r0(t), limt e = , this fact forbids such a connection. Indeed, p − p →∞ ∞ while the input and output signals are zero, there exists in the 1 F (p) = − r (t). system a non observed and non controlled unbounded signal. p 0 The conclusion is different if we consider the series connection with the systems : Clearly, from the previous result for ε0( ), ε1( ) vanishes ∞ ∞ 1 F (p) 1 if and only if the DC gain of the transfer operator − y(t) = u(t), p p + 1 is equal to zero. Since p + 1 z(t) = y(t). 2 p 1 1 F (p) (a0 b0) + (a1 b1)p + (a2 b2)p + − = − − 2 − n ··· − p p(a0 + a1p + a2p + + anp ) t Due to the pole/zero cancellation at 1, y(t) has an e− ··· − component that vanishes at . Except during the transient we obtain ε1( ) = 0 if and only if a0 = b0 and a1 = b1. It ∞ ∞ behavior, the pole/zero cancellation is acceptable for asymp- can be seen that : totically stable canceled zeros. when a = b , we have ε ( ) = 0 and ε ( ) = • 0 0 0 1 a06 b0 ∞ 6 ∞ limp 0 − = ; B. DC Gain → pa0 ±∞ when a = b , we obtain ε ( ) = 0 and ε ( ) = • 0 0 0 1 Let us keep in mind that the transfer operator F (p) in a1 b1 ∞ ∞ − . Moreover, ε1( ) = 0 when a1 = b1. equation (5) is just a coding of the differential equation (2). a0 ∞ In the case of an asymptotically stable system, with a constant In the same way we can show by induction that the system value U as input, the step response analysis indicates that the has a steady-state error of order N if and only if its transfer output tends, as t goes to + , to a constant value Y given by F (p) in equation (5) is such that, for i = 0 to N, ai = bi. ∞ Y the relationship a Y = b U. The ratio defines the DC gain The steady-state error of order N + 1 is then 0 0 U of the system G . The stability condition implies a = 0, DC 0 6 aN+1 bN+1 and from (5) we obtain GDC = F (0). εN+1( ) = − , ∞ a0 and the next ones have an infinite module. Thus, the degree C. Steady-State Error Analysis of the steady-state error can be obtained just by a visual In all this section systems are supposed to be asymptotically inspection of the transfer operator of the system. stable, namely the transient behavior vanishes and only the permanent behavior remains. For the reference inputs ri(t) ti VII.ALGEBRAIC ESTIMATION defined by, for t > 0, r (t) = , and for t < 0, r (t) = 0, i i! i the corresponding outputs are yi(t) = F (p)ri(t). The input- Recently, M. Fliess and his co-workers [21], [22], [42] have output error ε (t) = r (t) y (t) is called the system error of proposed new methods to estimate the parameters of a system i i − i order i. Two notions can be pointed out here. First a norm of or the first derivatives of a measured signal. For instance, the instantaneous system error εi(t) characterizes the system the last point is implemented in a model-free control of a performance. Secondly, the value εi( ) = limt εi(t) system [19], [20]. These methods are based on operational ∞ →∞ during the permanent behavior characterizes the steady-state calculus. According to the presented operational framework error. In a basic lecture of automatic control this last notion is of our paper, let us describe the used skills for the particular usually considered. We detail it according to our formulation, case of derivative estimation. namely without the use of the final value theorem. Firstly, the signal s(t) is approximated on a short-time A steady-state error of order N is ensured if ε ( ) = 0, window [ T, 0] with the polynomial : i ∞ − for i = 0 to N, and ε ( ) = 0. Consider the transfer N+1 ∞ 6 2 operator F (p) of an asymptotically stable system defined in s(t) = s0 + s1t + s2t , (8) (5). The corresponding permanent step response value is given s s = b0 where 1 stands for the estimation of the first derivative, 1 by the DC gain , so ε0( ) = 0 if and only if b0 = a0. Thus s˙(0). a0 ∞ we conclude that a steady-state error of zero order is fulfilled Secondly, the time functions 1, t and t2 are associated to whether the DC gain is equal to 1. In other words since the operationald forms, namely, their signal generators. 78 ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013

A. Signal Generators Mikusinski´ [44] has proved the two results below, which The main reason of using the Laplace transform is the tables are essential to our purpose. we have at our disposal. Firstly, they contain information to Theorem 1 For every continuous function f in C, (1) determine the response of a system with respect to a given f (t) = pf [f(0)] . More generally, for every integer k − input signal. Secondly, they allow to get the discrete transfer  k 1 operator of a computer controlled system with a formula given (k) k − (i) k i 1 f (t) = p f f (0) p − − . (9) − in [52]. Although transforms are not used in our presentation, i=0 we show that these tables can be used without any change. To n o X h i do that, let us introduce the notion of generator of a continuous ∞ tp Theorem 2 For every f in C such that 0 e− f(t)dt exists signal, which consists in writing the time expression of this signal by means of the operator p. ∞ tp R f = e− f(t)dt. The previous parts show that the transfer operator allows to 0 Z link input u(t) and output y(t) signals of a linear system by a differential equation coded as y(t) = F (p)u(t). Until now, we Theorem 1 allows to write the generator of a signal f(t) { } get the step response by solving this differential equation when when the differential equation whose this signal is solution is initial conditions are all zero. In case of no input and non zero known. Indeed, let us suppose that this differential equation is initial conditions, such a transfer operator produces an output n signal solution of the associated homogeneous differential (i) αif (t) = 0, (10) equation. The coding of this differential equation with the i=0 p operator defines then the generator of this signal. Two X ˙ (n 1) ways can be considered to take into account initial conditions with initial conditions f(0) = f0, f(0) = f1, . . . , f − (0) = in this coding. Namely, on the one hand the Mikusinski´ fn 1, where n is an integer and the αi are real numbers. With − operational calculus and on the other hand the integral form (9) the coding of (10) leads to of a differential equation. n i Indeed, all the previous developments can be rigorously αip f(t) PIC(p, f0, . . . , fn 1) = 0, − − proved by means of operational calculus of Mikusinski´ [44], "i=0 # which is based on convolution algebra of operators. Let us X where PIC(p, f0, . . . , fn 1) is a polynomial in p that depends briefly describe this operational calculus whereas keeping in − on the initial conditions and the coefficients α . We obtain mind that the considerations below are not needed in a first i then the generator of f(t) level course. Convolution product is a fundamental tool in { } dynamic systems field [54], [55], specifically in case of linear PIC(p, f0, . . . , fn 1) f(t) = − . (11) systems [11], [25]. This tool is defined by n i { } [ i=0 αip ] t (f, g) gf = f(τ)g(t τ)dτ, Theorem 2 indicates that whenP the one-sided Laplace trans- 7→ − form of a signal exists, its expression is identical to the Z0 while the Heaviside function H = 1 is of great importance generator of the signal, the complex variable s of Laplace { } due to the fact that we have for every f in the set of integrable transform being changed into the derivative operator p (to avoid any confusion). A major consequence is that the tables function C 1 t [13], [58], can be used. Since H = p− we remark that the Hf = f(x)dx . generator can be written indifferently with the operators H or 0 Z  p. Consequently, H appears as the integration operator. The suc- For example, when we look for the generator of sin(ωt), cessive powers of H with respect to the convolution product we have just to observe that sin(ωt) is the solution of the are, for all n in N, n 1, ≥ differential equation tn 1 Hn = − . x¨(t) + ω2x(t) = 0, x(0) = 0, x˙(0) = 1. (12) (n 1)!  −  Using (9) the coded form is then To distinguish a constant signal α with the operator { } defined by the constant gain α we denote it [α] . For all f p2x(t) 1 + ω2x(t) = 0, in C − t which leads to the generator of sin(ωt) α f = α f(τ)dτ and [α] f = αf(t) . { } { } 1  Z0  sin(ωt) = , { } M p2 + ω2 The unit element for the convolution is [1] and we can give a meaning to H0 as H0 = [1] . We define then the derivative where the symbol “M” denotes “in the Mikusinski´ sense”. operator as the solution of the convolution equation pH = [1] , Indeed, this definition for the generator of a signal is not 1 0 0 and we write p = H− . With the understanding p = H− = unique because it depends on the used integral transformation. n n [1], we have p = H− for n in N. For instance, the reader can find in [52] another way to handle ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 79 a differential equation which leads to the signal generator of or, equivalently, y(t) = (y0 y0)et. 1 − 2 f(t) in the Carson sense. It is based on the result below [53] This standpoint can also be explained in a more algebraic framework as the Fliess’module-theoretic approach [18]. Nev- x˙(t) = f(t), x(0) = x0 if and only if { } t ertheless, let us quote a sentence of a recent paper [22] where (13) this point of view is used for the design of an algebraic x(t) = x0 + f(τ)dτ, Z0 identification procedure : “Let us add we tried to write the but for shortness sake we don’t develop this point here. The examples in such a way that they might be grasped without the Carson transform was introduced in 1926 [6] and it differs necessity of reading the sections on the algebraic background. from the Laplace transform by a factor p. The Carson tables Our standpoint on parametric identification should therefore can be used in this framework as well. be accessible to most engineers.” Let us mention that the Mikusinski´ operational calculus has been extended recently by the convolutional calculus [12]. B. Derivative Estimation For a system defined by the transfer operator F (p) we can tn calculate the response y(t) to an input u(t) by using Carson Let us associate to the signal , n 0, its signal n! ≥ or Laplace transform tables. Indeed when U(p) is a generator 1 generator in the Mikusinski´ sense . The polynomial of u(t) the generator of the output y(t) is F (p)E(p). We p=n + 1 must remark here that the generators can be obtained in any approximation (8) of a signal s(t) can then be written sense as defined (Mikusinski´ or Carson). However, we must 1 1 1 keep the consistency in using tables. For example, following s(t) = s0 + s1 + s2 . p p2 p3 the Heaviside series expansion, when we want to know the beginning of the response we can write the power series with To obtain s1 let us follow the steps : 1 3 respect to p− of the generator of y(t). However, different 1) Multiply with p k functions may be associated to p− according to the adopted 3 2 generator sense. p s(t) = p s0 + ps1 + s2. Moreover, in order to see the importance of the generator 2) Derivate with respect to p for operational calculus, let us consider the following example 2 3 where two signals y1 and y2 are defined by the differential 3p s(t) + p s0(t) = 2ps0 + s1, equations ds(t) (p 1)y1(t) = u(t), (14) where s (t) stands for . − 0 dp (p 1)y (t) = u(t), (15) 3) Divide with p − 2 0 0 2 s1 and the initial conditions y1 and y2 respectively. Let us 3ps(t) + p s0(t) = 2s0 + . consider the parallel connection y(t) = y (t) y (t). It yields p 1 − 2 1 1 4) Derivate with respect to p y(t) = u(t) u(t) = 0. p 1 − p 1 2 s1 − − 3s(t) + 5ps0(t) + p s00(t) = 2 . This conclusion is obviously wrong. Indeed, our setting indi- −p cates that we consider formal differential equations, namely, As p2 stands for a double time-derivative it cannot be imple- without initial conditions. When we write mented. Thus, the following step consists in a division with 1 3 y(t) = u(t), p . So, we obtain the operational estimation p 1 3 1 1 5 − 3p− s(t) + 5p− s0(t) + p− s00(t) = p− s . it is just a coding of differential equations (14) and (15). The − 1 initial conditions can be taken into account by means of the Notice that the use of a sufficient number of time-integrals generator notion. We can write (14) and (15) as, respectively, induces noise filtering. 0 In order to implement the derivative estimator we have then 1 y1 y1(t) = u(t) + , to come back in the time domain. There is no difficulty to M p 1 p 1 − − translate the time-integrals on [ T, 0] 1 y0 − y (t) = u(t) + 2 , 4 2 5 T M p 1 p 1 p− s = s , − − − 1 − 24 1 in the Mikusinski’s´ generator sense. It yields for y(t) = 2 y1(t) y2(t) the generator 1 (t τ) − 3 s(t) = s(τ)dτ = − s(τ)dτ. y0 y0 p 2 y(t) = 1 − 2 . ZZZ Z M p 1 For the other terms we must prove in a operational way the − This result indicates that y(t) is solution of the differential following well known result k equation d k 0 0 k (p 1)y(t) = 0, y(0) = y y , k s(t) = ( 1) t s(t). (16) − 1 − 2 dp − 80 ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013

For simplicity sake, let us consider the signal meaning. We keep in mind that a transfer operator is always related to a differential equation. Mathematical background is ∞ tν x(t) = xν (0) . minimized, however, when a rigorous justification is needed ν! ν=0 the Mikusinski´ operational calculus may be used. This opera- X tν 1 tional calculus is based on the convolution operator, which is With the signal generators = , we get a natural tool for linear equations. ν! pν+1 Moreover, we meet here, through a pedagogical step, the ∞ 1 operational standpoints adopted directly in some advanced x(t) = xν (0) , pν+1 textbooks to modelize the input-output relationship induced by ν=0 X a linear system. For instance, the discrete-time autoregressive 1 1 1 which leads to moving-average model A(q− )yk = B(q− )uk + C(q− )k 1 1 1 d ∞ 1 where A(q− ),B(q− ), and C(q− ) are polynomials in the x(t) = (ν + 1)xν (0) , dp − pν+2 delay operator, k a noise signal, is used in [1], [8] for ν=0 { } X identification purposes to describe the difference equation ∞ tν+1 between the sampled input u and the sampled output signals = (ν + 1)xν (0) , k − (ν + 1)! yk of a given system. For multivariable continuous-time linear ν=0 X systems the following model is introduced in [49], [59], [60] ∞ tν+1 = xν (0) , − ν! P (p)ξ(t) = Q(p)u(t), ν=0 X y(t) = R(p)ξ(t) + W (p)u(t), ∞ tν = t xν (0) . − ν! ν=0 where P (p),Q(p),R(p), and W (p) are matrix polynomials X in the differential operator and ξ(t) is a vector-valued function d Thus x(t) = tx(t). It is easy to state the announced result of time named the partial state. More recently, the generator dp − (16) by induction. of a multivariable system is defined in [4] as the polynomial M(p) , We can then state the final form of the derivative estimation matrix in the derivative operator which allows to as write the relationship between input and output signals as y(t) 24 0 3 13 27 M(p) = 0. The generator in the sense defined in [4] \(1) 2 2 u(t) s (0) = 4 T T t + t s(t)dt.   −T T 4 − 2 4 must not be confused with signal generators. The interested Z−   reader can see the quoted literature. An example of a real-time application is given in figure 4 where we show the signal corrupted with noise and the real- time estimation of its derivative. REFERENCES [1] K.J. Astr˚ om,¨ T. Bohlin, Numerical identification of linear dynamic systems from normal operating records, in Theory of self-adaptative control systems, pp. 96–111, Plenum, 1965. [2] K.J. Astr˚ om,¨ B. Wittenmark, Computer Controlled Systems, Prentice Hall, 1994. [3] S. Bennet, History of Control Engineering, Peter Peregrinus, 1800–1930, 1978; 1930–1955, 1993. [4] H. Blomberg, R. Ylinen, Algebraic Theory for Multivariable Linear Systems, Academic Press, 1983. [5] J.R. Carson, On a general expansion theorem for the transient oscilla- tions of a connected system, The Physical Review, ser. 2, vol. X, n. 3, pp. 217–25, 1917. [6] J.R. Carson, Electric Circuit Theory and the Operational Calculus, McGraw-Hill, 1926. [7] A.-K. Carstensen, J. Bernhard, To learn a complex concept is to keep more than one concept in focal awareness simultaneously - an example from electrical engineering, 41st SEFI Conference, 16-20 September 2013, Leuven, Belgium, [8] D. Cochrane, G.H. Orcutt, Application of least squares regression to Fig. 4. Algebraic derivative estimation. relationships containing autocorrelated error terms, J. Amer. Statist. Assoc., vol. 44, pp. 32–61, 1949. [9] M.A.B. Deakin, The development of Laplace transform, 1737–1937 : I. From Euler to Spitzer, 1737–1880, vol. 25, n. 4, pp. 343–90, Archive for History of Exact Science, 1981. VIII.CONCLUSION II. From Poincare´ to Doetsch, 1880–1937, vol. 26, n. 4, pp. 351–81, We show in this short survey that from the use of the Archive for History of Exact Science, 1982. [10] M.A.B. Deakin, The ascendancy of the Laplace transform and how it differential operator we obtain all the usual results derived by came about, vol. 44, n.3, pp. 265–86. Archive for History of Exact means of the Laplace formulations. The teaching of a basic Science, 1992. lecture in electrical engineering and in automatic control using [11] C.A. Desoer, M. Vidyasagar, Feedback Systems : Input-Output Proper- ties, Academic Press, 1975. the operational method offers some advantages. The integral or [12] I.H. Dimovski, Convolutional Calculus, Kluwer Academic Publishers, derivative operators allow to link every notion to its physical 1990. ELECTRONICS, VOL. 17, NO. 2, DECEMBER 2013 81

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