An automatic control course without the Laplace transform

An operational point of view 10/12/2016
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An automatic control course without the Laplace transform


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            <date dateType="Submitted">Mon 18 Feb 2019</date>
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1 An automatic control course without the Laplace transform An operational point of view F. ROTELLA , I. ZAMBETTAKIS Ecole Nationale d'Ingénieur de Tarbes, 47 avenue d'Azereix, BP 1629, 65016 Tarbes CEDEX, France Institut Universitaire de Technologie de Tarbes, Université Toulouse III Paul Sabatier, 1 rue Lautréamont, 65016 Tarbes CEDEX, France Abstract— The question is: Is the Laplace transform needed in an Automatic Control course? The answer is : Obviously, not! Based on an operational standpoint the rst parts are about guidelines for a primer in automatic control. Beyond an undergraduate course, the two last parts, a little bit more technical, are devoted on the one hand to get the model of a computer controlled system and on the other hand to relate the operational standpoint to usual tables used in some cases. Résumé— Devant les inconvénients pédagogiques engendrés par l'utilisation de la transformée de Laplace en automatique le développement de méthodes basées sur une présentation purement opérationnelle permet de focaliser les raisonnements sur l'aspect pratique de cette discipline. À partir d'un point de vue développé par Heaviside, nous passons en revue, dans les premières parties, quelques résultats de base en automatique. Ces résultats sont bien sûr obtenus à l'aide de la notion de transfert ce qui nous permet d'en préciser l'interprétation et les différentes acceptions. Les deux dernières parties, qui ne sont pas nécessaires au premier abord sont consacrées d'une part à la construction du modèle d'un système commandé par calculateur et d'autre part à la liaison que l'on peut établir entre l'approche proposée et l'utilisation des tables. Cet article n'a pas pour objectif de changer l'automatique mais se propose de fournir les moyens d'en changer la présentation actuelle. Index Terms—Transfer operator, automatic control course, op- erational calculus, Laplace transform, Carson transform, Heav- iside, Mikusi´nski. Mots-clés—Opérateur de transfert, cours d'automatique, cal- cul opérationnel, transformée de Laplace, transformée de Carson, Heaviside, Mikusi´nski. I. INTRODUCTION First level courses in automatic control or basic textbooks of this topic contain rst lessons of Laplace transforms. The Laplace transform approach leads to de ne the transfer function of a system. It is used to get the corresponding response signal of a system with respect to a given input signal. The Laplace transform is also important for the analysis and design of control systems [13]. This tool appears thus a necessary and unavoidable burden for students participating in automatic control courses. However, the transform has some skeletons-in-the-closet [27], [34]. In this article, we discuss an alternative approach to the use of Laplace transform in automatic control courses. Consider a signal x(t); de ned for a positive time t and satisfying some appropriate growth conditions. The Laplace transform of x(t) is X(s) = Lfx(t)g = Z 1 0 x(t)e st dt; (1) where s is a complex variable. This de nition can require advanced mathematical machinery [34]. This machinery is very demanding, and usually beyond the skills of most under- graduate students. This generates dif culties that lead deser- tion of students from automatic control classes. Equation (1) assumes that all considerations, diagrams and developments are embedded in a space of transformed signals. Students ask frequently two questions in regards to the usefulness of equation (1). How can we experimentally exhibit or visualize the transformed signals for example with an oscilloscope? Are there some forbidden signals in automatic control methods? For instance, exp(t2 ) has no Laplace transform [36]. Some fundamental theorems of Laplace transform have also provided some misunderstandings about the actual meaning of s. For instance, consider the Laplace transform of a derivative function with the initial condition x(0). Namely, Lf _x(t)g = sX(s) x(0): When x(0) vanishes the complex variable s is considered as a time derivative operator. However, this can be considered in only the space of transformed signals. Still about this fundamental theorem, the lower limit of integration in equation (1) is often replaced by 0 , 0+ , or 1 [13], [30], e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 2 - -U(s) Y (s)F(s) Fig. 1. Basic block diagram. The transformed output signal Y (s) is given by F(s)U(s) where U(s) is the transformed input signal. As s is a complex variable the time is a hidden variable while there is no difference in the notation between signals and systems. [39] to overcome problems encountered in some particular applications. An attempt to solve this problem and to unify the Laplace formalisms is proposed in [32]. Nevertheless the use of this tool is still an unjusti ed hypothesis. For the 1 case, we are faced with the bilateral Laplace transform, which is questionable also [34]. The purpose of this article is to show that the mathematical machinery required by the Laplace transform [47] can be avoided. Moreover, the pedagogic dif culty can be cleared in a natural way. When the transfer function of a linear system has to be de ned, Laplace transform is applicable. For a single-input single-output system the transfer function is de ned as the quotient of the Laplace transform of the output y(t) to the Laplace transform of the input u(t) with the assumption of zero initial conditions. In other words, the transfer function is de ned by: F(s) = Lfy(t)g Lfu(t)g = Y (s) U(s) : Although the name transfer function as a mathematical tool is adequate for s = j!; where j2 = 1 and ! is the frequency [23], [3], this ad-hoc de nition generates some interesting questions. The Laplace transform of signals cannot be obtained in practice, and sometimes we wonder how to determine the transfer function of a system? For instance this question is avoided in identi cation procedures [31] where ARMAX models involving recurrence relationships instead of transfer functions are used. In several high quality textbooks for discrete-time systems e.g. [2], the complex variable z of the Z-transform [26] and the shift-forward operator q are both used. However, the choice between z and q is not argued in every case. These subtle differences, which are mysterious for students, are useless and can be avoided. In view of our personal experience in teaching automatic control, one desire is to give an experimental meaning of the transfer of a system, irrespective of previous formal de nitions. Rarely, in real occasions students may relate the transfer to the differential operation induced by the system. Indeed the use of the Laplace transform leads to the diagram depicted in Figure 1. This gure describes the relationship between the Laplace transforms of external signals and the transfer of the system. So, the essential meaning is lost. It can be noticed here that we do not use the term transfer function, although we call it transfer as can be seen in the sequel. As a matter of fact, the use of Laplace transforms is one method among many others [33], [29] to justify the Heaviside operational calculus (Figure 2). Note that the operational calculus is used to solve differential equations (in most cases linear) rather than automatic control problems. From an histor- ical perspective, the Laplace transform, which has been studied extensively by Deakin [8], [9], was introduced rst in the form as equation (1) by Bateman in 1910 to solve the differential equation _x(t) = x(t) where is a nonzero real number. In the following, we describe guidelines for starting an automatic control course without using the Laplace transform. The presentation is based on the use of a pure operational point of view that provides an opportunity to link methods developed in automatic control to laboratory applications. The essential proofs based on Laplace transform theorems can be read in standard textbooks of automatic control (e.g. [20], [30], [22]). Concerning the notation, we consider signals as elements belonging to the set C of integrable real valued functions f = ff(t)g ; supposed to be m times continuously differentiable on [0; 1) except at isolated points where it is assumed that both left limit and right limit exist. As ff(t)g denotes the signal f while f(t) stands for its value at time t we write for two signals a and b in C: for all t 0; a(t) = b(t), or fa(t)g = fb(t)g, or a = b: However, when no confusion is possible the braces or “for all t 0” may be dropped. Nevertheless we do not deal with discrete-time systems except to de ne the transfer of a computer controlled system. II. TRANSFER OPERATOR We begin by considering a linearized system around an equilibrium point. We suppose this system can be described by the linear differential equation y(n) (t) + an 1y(n 1) (t) + an 2y(n 2) (t) + +a1y(1) (t) + a0y(t) = bmu(m) (t) + bm 1u(m 1) (t) + +b1u(1) (t) + b0u(t); (2) where y(t) and u(t) stand for the differences of output and input signals with their setpoint values respectively and n and m are two integers. In (2) the coef cients ai and bj are constant parameters. A. Coding The aim is to provide a tool with which it is easy to manipulate mathematical expressions, which may be used instead of linear differential equations, to separate input and output variables from the system. Following Heaviside [24], [40] or Carson [5], we introduce the derivative operator p , d dt ; which upon acting on x(t) in C gives the codings _x(t) = px(t); •x(t) = p2 x(t); : : : ; x(n) (t) = pn x(t); : : : (3) In view of these codings and using the distributivity property for every real numbers and ; [ pn + pm ]x(t) = x(n) (t) + x(m) (t); equation (2) becomes pn + an 1pn 1 + an 2pn 2 + + a1p + a0 y(t) = bmpm + bm 1pm 1 + + b1p + b0 u(t): (4) e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 3 Leibniz (1695) Euler (1730) Laplace (1812) Servois (1814) Cauchy (1827) Gregory (1846) Boole (1859) Kirchoff (1891) Wagner (1915) Bromwich (1916) Carson (1917-1922) Jeffreys-March (1927) Van der Pol (1929) Doestch (1930) Mikusi´nski (1950) Dimovski (1982) Schwartz (1945,1947) Florin (1934) Levy (1926) Dirac (1926) Smith (1925) Heaviside (1884-1895) 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. Among engineers and mathematicians, Heaviside appears as the focal point. His ideas on the use of the differential operator and on the de nition of the transfer (resistance) operator are the basis of guidelines for an automatic control course without the Laplace transform. - -u(t) y(t)F(p) Fig. 3. Operational block diagram. The output signal y(t) is obtained by F(p)u(t) where u(t) is the input signal. Notice that t denotes the time variable and p the derivative operator. The difference between signals and system is due to the use of these notations. The action performed by a system on an input signal is retained. The essential meaning of the transfer F(p) is the linear differential equation that links the output signal and the input signal. To separate input and output signals from the system we divide equation (4) by the polynomial factor pn + an 1pn 1 + +a0; which yields to code the input-output relationship as y(t) = F(p)u(t) with F(p) = bmpm + bm 1pm 1 + + b1p + b0 pn + an 1pn 1 + an 2pn 2 + + a1p + a0 : (5) We must insist here that y(t) = F(p)u(t) cannot be consid- ered as the solution of the differential equation (2). Indeed, the initial conditions are not known. y(t) is determined with this writing as with the writing (2). In equation (5), F(p) stands for the transfer operator. In essence it is the transfer, which represents the operation induced by the system to transform the input signal into the output signal. The operational approach provides an opportunity to relate the transfer operator (5) to the differential equation (2). The diagram, depicted in Figure 3 corresponds to an experimental situation. B. Operational calculus as polynomial calculus Operational calculus is understood as algebraic methods for solving differential or recurrence equations, speci cally in a linear time-invariant framework. In our point of view solving a differential equation for a given input, is not the aim of automatic control but is a mathematical exercise [34]. In automatic control operational calculus means rules for transfer connections or decompositions through polynomial calculus. Thus the coding (4) is useless when we are not allowed to associate transfer operators. From the operational standpoint the connection rules can be demonstrated through the fol- lowing steps. The transfers of the connected system provide differential equations. The connections and the elimination of intermediate signals lead to a differential equation between the output and input signals. The encoding of this differential equation with p ensures the results. Although we can use this procedure in every case it is suf cient to exemplify with respect to series or parallel connections for two rst-order systems. Let us consider two linear systems described by y1(t) = F1(p)u1(t) and y2(t) = F2(p)u2(t) where u1(t) and u2(t) are the input signals, F1(p) and F2(p) the transfers of the systems, and y1(t) and y2(t) are the corresponding output signals. In this regard we have F1(p) = b1p + b0 a1p + a0 and F2(p) = 1p + 0 1p + 0 ; where a0; a1; b0; b1; 0; 1; 0, and 1 are constant parameters. The series connection is de ned as u2(t) = y1(t);e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 4 u(t) = u1(t); and y(t) = y2(t): The application of the procedure yields y(t) = 1b1p2 + ( 0b1 + 1b0)p + 0b0 1a1p2 + ( 0a1 + 1a0)p + 0a0 u(t); where we recognize the product F1(p)F2(p): The parallel connection is de ned as u2(t) = u1(t) = u(t) and y(t) = y1(t) + y2(t); which yields to y(t) = 1 1a1p2 + ( 0a1 + 1a0)p + 0a0 ((a1 1 + 1b1)p2 + (a1 0 + b1 0 +a0 1 + b0 1)p + (a0 0 + b0 0))] u(t); where we recognize the sum F1(p)+F2(p): These results can be extended to any order by induction. It can be seen that the transfer of the series connection of two systems is the product of their transfer and the transfer of the parallel connection of two systems is the sum. The parallel and series rules give a meaning to the decompo- sitions and the handling of transfer operators with polynomial calculus. These operations on transfer operators are the basis of operational calculus in automatic control. We can apply usual techniques as Mason's rule associated to the signal- ow graph [35]. This operational calculus can be applied also for multiple-input multiple-ouput systems with the difference that commutativity does not occur anymore. Let us note that the polynomial formalism is used in several textbooks on multivariable systems [27], [28] with no need of the Laplace transform. C. The delay operator A pure time delay of T between input and output signals induces y(t) = u(t T): This particular linear system cannot be associated to a differential equation as (2). A special treatment must be used for delay equations. Following an idea of Euler [14], the Taylor expansion of u(t T) yields u(t T) = u(t) _u(t)T +•u(t) T2 2 +u(n) (t) ( T)n n! + ; which is encoded to give y(t) = u(t) pTu(t) + p2 T2 2 u(t) + pn ( T)n n! u(t) + ; = 0 @ X n 0 ( pT)n n! 1 A u(t) = (exp( pT)) u(t): We obtained the transfer operator for the time delay T as F(p) = e pT : III. SYSTEM RESPONSES System analysis is often the study of some particular re- sponses of the system. Of special interest are the step and frequency responses. A. Step response Let us consider the Heaviside or step signal fH(t)g de ned by H(t) = 1 for t 0 and 0 elsewhere. The step response of a system is the solution of the differential equation of the system to a step input signal with zero initial conditions. With operational calculus, we can expand transfer operator as a linear combination of simple transfers an (p + a)n or 1 pn ; where a stands for a nonzero complex number and n for an integer. The step response of a multiple integrator with the transfer 1 pn is tn n! : Let us consider the step response sn(t) of the Strej´c system [41] with the transfer an (p + a)n : For n = 1 the associated differential equation to the transfer a p + a is ay(t)+ _y(t) = au(t) and we obtain by usual methods [51] the corresponding response to a given input u(t) with the initial condition y(0) y(t) = e at y(0) + a Z t 0 ea u( )d : For u(t) = 1 and zero initial conditions the step response becomes s1(t) = 1 e at : For n = 2 we have s2(t) = a p + a s1(t) from which follows s2(t) = ae at Z t 0 (ea 1) d = 1 e at (1 + at): (6) For the general case n 1; let us suppose sn(t) = 1 e at n 1X i=0 ci;nti ! ; and sn+1(t) = 1 e at nX i=0 ci;n+1ti ! ; where the coef cients ci;n and ci;n+1 are constant parameters. These signals are linked by the differential equation _sn+1(t) + asn+1(t) = asn(t); which to the the relationships c0;n+1 = 1; for i = 1 to n : ci;n+1 = a i ci 1;n = ai i! : We deduce the well known result that, for the Strej´c model an (p + a)n ; the step response sn(t) is sn(t) = 1 e at n 1X i=0 ai ti i! ! : So the step response of a given system de ned by a transfer operator can be calculated using polynomial calculus. e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 5 B. Stability Due to the fact that stability analysis does not use the Laplace transform there is no novelty in this paragraph. However, we may insist again on the link between the transfer operator, the differential equation and the transient behavior. Let us consider the transfer operator an (p + a)n where a and n have the same meaning as before. From the previous paragraph we can see that the step response is composed of a constant term and a time-dependent term. The rst term is the forced response and the second term is the transient behavior. The transient behavior tends asymptotically to zero if and only if the real part of a is strictly negative: More generally, consider the n-th order transfer F(p) = bmpm + + b1p + b0 (p p1) 1 (p p2) 2 (p pr) r ; where p1; p2; : : : ; pr are the r complex poles of F(p) and i are the respective multiplicities with n = Pr i=1 i. For the transient response the poles generate terms associated with the signals ep1t ; ep2t ; : : : ; eprt weighted by time polynomials of order i 1 respectively. If all the poles have strictly negative real part, the transient behavior tends asymptotically to zero. Namely, the system is asymptotically stable. C. Frequency response For asymptotically stable systems the frequency response is deduced from the steady-state output response corresponding to a given sinusoidal input signal u(t) = ej!t where ! is the frequency and j2 = 1. Consider a system de ned by the transfer operator F(p) = B(p) A(p) where A(p) and B(p) are two polynomials. From the operational approach we have the encoded input-output differential equation as A(p)y(t) = B(p)u(t). For the sinusoidal input u(t) = ej!t ; we obtain B(p)u(t) = jB(j!)j ej(!t+arg(B(j!))) ; where jB(j!)j and arg(B(j!)) denote the module and the argument of the complex number B(j!) respectively. The output y(t) is the sum of a particular solution of the differential equation and the general solution of the differential equation without second member. The general solution characterizes the transient response that vanishes in case of asymptotically stable systems. For a particular solution, we look for the steady-state behavior as the form y(t) = Y ej(!t+') where Y and ' are constant parameters. Replacing this expression for y(t) in the differential equation yields Y = jB(j!)j jA(j!)j = jF(j!)j ; ' = arg(B(j!)) arg(A(j!)) = arg(F(j!)): The frequency response is de ned by the evolution of (jF(j!)j ; arg(F(j!))) as the frequency ! varies from 0 to +1: We can notice that F(j!) is the transfer function of the system such as Harris de ned it [23]. In our standpoint, this transfer function must not be confused with the transfer operator F(p). Nevertheless, F(j!) such as a function of the frequency is the only actual transfer function. Graphic representations such as Bode, Black-Nichols, or Nyquist loci may be used to analyze the frequency response [30]. For unstable systems the loci are valid as calculated representations only. But for stable systems, experiments cannot allow to obtain the frequency response. IV. ANALYSIS A. Poles and zeros The names of poles and zeros come from the interpretation of a transfer operator F(p) as a function of a complex variable p: This interpretation is a consequence of the formulation of Laplace transform and it misunderstands the physical meaning of these notions. The consideration of a transfer operator as a coding of a differential equation provides an immediate physical interpretation. Namely, let us consider the transfer operator F(p) = p + a p + b ; (7) where a and b are constant parameters. In the operational 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 for t > 0 and a nonzero initial condition y(0) we obtain y(t) = y(0)e bt for t > 0: Second consider a zero initial condition for the output and the input signal u(t) = e at for t > 0 we obtain _u(t) + au(t) = 0 so y(t) = 0 for t > 0: In general poles correspond to signals generated by the system with zero input. Zeros correspond to signals absorbed or blocked by the system. Let us write the transfer operator (5) as F(p) = k (p z1) 1 (p z2) 2 (p zd) d (p p1) 1 (p p2) 2 (p pr) r ; where k = bm; zi; i = 1; : : : ; d and pi; i = 1; : : : ; r are complex numbers, and i; i = 1; : : : ; d and i; i = 1; : : : ; r are integers. For i = 1; : : : ; r, epit is solution of the coded differential equation (p p1) 1 (p p2) 2 (p pr) r y(t) = 0; and for i = 1; : : : ; d, ezit is solution of the coded differential equation (p z1) 1 (p z2) 2 (p zd) d u(t) = 0: On the one hand we can use the correspondence between epit and the transfer denominator roots pi that characterizes the transient rate in the linear constant parameter framework only. The same remark can be said about the correspondence between ezit and the transfer numerator roots zi. On the other hand this signal approach for the pole and zeros meaning can be extended to time-varying or nonlinear multivariable systems with an algebraic standpoint [15], [16]. In order to underline and to exemplify the important problem of pole/zero cancellation let us consider the series e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 6 connection with the systems : y(t) = 1 p 1 u(t); z(t) = p 1 p + 1 y(t): The pole 1 induces, in the transient behavior or in the initial conditions effect for the rst system, an et signal. This signal is blocked by the second system, which has 1 as zero. As limt!1 et = 1; this fact forbides such a connection. Indeed, while the input and output signals are zero, there exists in the system a non observed and non controlled unbounded signal. The conclusion is different if we consider the series connection with the systems : y(t) = 1 p + 1 u(t); z(t) = p + 1 p 1 y(t): Due to the pole/zero cancellation at 1, y(t) has an e t component that vanishes at 1: Except during the transient behavior, the pole/zero cancellation is acceptable for asymp- totically stable cancelled zeros. B. DC gain Let us keep in mind that the transfer operator F(p) in equation (5) is just a coding of the differential equation (2). In the case of an asymptotically stable system, with the input taking a constant value U; the step response analysis indicates that the output tends to a constant value Y as t goes to +1 given by the relationship a0Y = b0U: The ratio Y U de nes the DC gain of the system GDC : The stability condition implies a0 6= 0; and from (5) we obtain GDC = F(0). C. Steady-state error analysis In all this part systems are supposed to be asymptotically stable, namely the transient behavior vanishes and only the permanent behavior remains. For the reference inputs ri(t) de ned as, for t > 0; ri(t) = ti i! ; and for t < 0; ri(t) = 0; the corresponding outputs are yi(t) = F(p)ri(t): The input- output error "i(t) = ri(t) yi(t) is called the system error of order i: Two notions can be pointed out here. First a norm of the instantaneous system error "i(t) characterizes the system performance. Second the value "i(1) = limt!1 "i(t) during the permanent behavior characterizes the steady-state error. In basic lecture of automatic control this last notion is usually considered. We detail it according to our formulation, namely without the use of the nal value theorem. A steady-state error of order N is ensured if "i(1) = 0; for i = 0 to N; and "N+1(1) 6= 0: Consider the transfer operator F(p) in equation (5) of an asymptotically stable system. The corresponding permanent step response value is given by the DC gain b0 a0 : It can be seen that "0(1) = 0 if and only if b0 = a0: We can conclude that a steady-state error of zero order is ful lled whether the DC gain is equal to 1. In other words since the input-error transfer is 1 F(p); we obtain a steady-state error of zero order when the input-error DC gain is zero. This is a fundamental remark for the following. If we notice that r1(t) is the integral of r0(t); namely r1(t) = 1 p r0(t); we have "1(t) = r1(t) y1(t); = 1 p r0(t) F(p) 1 p r0(t); = 1 F(p) p r0(t): Clearly, from the previous result for "0(1); "1(1) vanishes if and only if the DC gain of the transfer operator 1 F(p) p is equal to zero. Since 1 F(p) p = (a0 b0) + (a1 b1)p + (a2 b2)p2 + p(a0 + a1p + a2p2 + + anpn) we obtain "1(1) = 0 if and only if a0 = b0 and a1 = b1: It can be seen that when a0 6= b0; we have "0(1) 6= 0 and with "1(1) = limp!0 a0 b0 pa0 = 1; when a0 = b0; we obtain "0(1) = 0 and with "1(1) = a1 b1 a0 : Moreover "1(1) = 0 when a1 = b1. In the same way we can show by recurrence that the system can have a steady-state error of order N if and only if its transfer F(p) in equation (5) is such that, for i = 0 to N; ai = bi. In this case the steady-state error of order N + 1 is "N+1(1) = aN+1 bN+1 a0 ; and the next ones have an in nite module. The degree of the steady-state error can be obtained by just a visual inspection of the transfer operator of the system. V. COMPUTER CONTROLLED SYSTEMS The last question we wish to deal with concerns the con- struction of the model of a linear system controlled by a numerical algorithm (e.g. [2], [19]). The problem is to nd the discrete-time model corresponding to the structure presented in Figure 4 where DAC denotes a digital-analog converter and it is usually modelled as the series connection of an ideal sampler and a zero-order hold. ADC denotes an analog- digital converter usually modelled by an ideal sampler and the discrete output signal of the ADC device is yk = y(kTs) for k in N. Both are supposed to be synchronized with the sampling period Ts: In this section we use the notations fxkg for the discrete-time real valued signal de ned for k in N and q 1 for the delay operator q 1 fxkg = fxk 1g. All the considered discrete-time signals are supposed to be zero for negative values of k: For the discrete input signal fukg the output of the DAC device is u(t) = X k 0 uk (H(t kTs) H(t (k + 1)Ts) ; (8) e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 7 DAC ADCF(p)- - - -uk yk u(t) y(t) Fig. 4. Linear computer controlled system. Equipped with digital-analog (DAC) and analog-digital (ADC) converters, a continuous-time model (F(p)) leads to a discrete-time model. The discrete-time model is a coding of the recurrence equation between the discrete input signal uk and the sampled output yk = y(tk) where tk are the sampling times. The puzzle is to obtain this discrete-time model with the operational standpoint. where H(t kTs) stands for the delayed step signal : The discrete-time transfer of the system (Fig. 4) is obtained through the formula F(q 1 ) = (1 q 1 )Z L 1 F(p) p ; (9) where L 1 stands for the inverse Laplace transform and Z f:g stands for the Z-transform [26]. The expression in the square bracket denotes the sampling of the signal with a period Ts. Although the Z-transform doesn't suffer the same drawbacks as the Laplace transform, for instance discrete impulse is really a discrete signal, the Laplace transform appears once more time here. To carry on we obtain the discrete-time transfer operator F(q 1 ) in the operational framework. Denoting by S(t) the step response of F(p); the response of F(p) to uk(H(t kTs) H(t (k + 1)Ts) is uk(S(t kTs) S(t (k +1)Ts): The sampling of this response at the time period Ts leads to a value for t = lTs; l in N; given by uk (Sl k Sl k 1) where Sk stands for S(kTs): Denoting l the independent integer variable the corresponding discrete signal is fuk (Sl k Sl k 1)g = (1 q 1 ) fukSl kg for a given k: Because of linearity we deduce from (8) that the sampled response corresponding to the input signal fukg is fylg = (1 q 1 ) 8 < : X k 0 ukSl k 9 = ; : Denoting nP k 0 ukSl k o = fvlg ; fukg and fvkg are linked by a discrete convolution operation. Consequently [30], they are linked by a difference equation such as vk + d1vk 1 + + dn0 vk n0 = n0uk + n1uk 1 + + nm0 uk m0 ; where the di and the ni are real numbers and n0 and m0 are integers. In the same idea that for continuous systems, this equation can be coded by means of the delay operator q 1 (1 + d1q 1 + + dn0 q n0 ) fvkg = (n0 + n1q 1 + + nm0 q m0 ) fukg ; which leads to the discrete transfer operator G(q 1 ) = n0 + n1q 1 + + nm0 q m0 1 + d1q 1 + + dn0 q n0 : Let us denote the numerator and the denominator of G(q 1 ) by N(q 1 ) and D(q 1 ) respectively. The division of N(q 1 ) by D(q 1 ) leads to G(q 1 ) = X l 0 glq l ; where gl; l 0; are real numbers. With fvkg = G(q 1 ) fukg we obtain fvlg = 8 < : X k 0 gkul k 9 = ; = 8 < : X k 0 ukSl k 9 = ; : Since for k < 0; uk