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To cite this version: Rotella, Frédéric and Zambettakis, Irène An automatic control course without the Laplace transform. An operational point of view. (2008) e-STA, 5 (3). 18-28. ISSN 1954-3522
Any correspondence concerning this service should be sent to the repository administrator: [email protected] 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 [email protected]
Institut Universitaire de Technologie de Tarbes, Université Toulouse III Paul Sabatier, 1 rue Lautréamont, 65016 Tarbes CEDEX, France [email protected]
Abstract— The question is: Is the Laplace transform needed response signal of a system with respect to a given input in an Automatic Control course? The answer is : Obviously, signal. The Laplace transform is also important for the analysis not! Based on an operational standpoint the rst parts are and design of control systems [13]. This tool appears thus a about guidelines for a primer in automatic control. Beyond necessary and unavoidable burden for students participating in an undergraduate course, the two last parts, a little bit more automatic control courses. However, the transform has some technical, are devoted on the one hand to get the model of a skeletons-in-the-closet [27], [34]. In this article, we discuss computer controlled system and on the other hand to relate the an alternative approach to the use of Laplace transform in operational standpoint to usual tables used in some cases. automatic control courses. Résumé— Devant les inconvénients pédagogiques engendrés par l'utilisation de la transformée de Laplace en automatique Consider a signal x(t); dened for a positive time t and le développement de méthodes basées sur une présentation satisfying some appropriate growth conditions. The Laplace purement opérationnelle permet de focaliser les raisonnements transform of x(t) is sur l'aspect pratique de cette discipline. À partir d'un point 1 de vue développé par Heaviside, nous passons en revue, dans les X(s) = Lf x(t)g = x(t)e st dt; (1) premières parties, quelques résultats de base en automatique. Ces Z0 résultats sont bien sûr obtenus à l'aide de la notion de transfert where s is a complex variable. This denition can require ce qui nous permet d'en préciser l'interprétation et les différentes advanced mathematical machinery [34]. This machinery is acceptions. Les deux dernières parties, qui ne sont pas nécessaires very demanding, and usually beyond the skills of most under- au premier abord sont consacrées d'une part à la construction graduate students. This generates difculties that lead deser- du modèle d'un système commandé par calculateur et d'autre tion of students from automatic control classes. Equation (1) part à la liaison que l'on peut établir entre l'approche proposée assumes that all considerations, diagrams and developments et l'utilisation des tables. Cet article n'a pas pour objectif de are embedded in a space of transformed signals. Students changer l'automatique mais se propose de fournir les moyens ask frequently two questions in regards to the usefulness of d'en changer la présentation actuelle. equation (1). How can we experimentally exhibit or visualize Index Terms— Transfer operator, automatic control course, op- the transformed signals for example with an oscilloscope? Are erational calculus, Laplace transform, Carson transform, Heav- there some forbidden signals in automatic control methods? iside, Mikusi nski.´ For instance, exp( t2) has no Laplace transform [36]. Mots-clés— Opérateur de transfert, cours d'automatique, cal- cul opérationnel, transformée de Laplace, transformée de Carson, Some fundamental theorems of Laplace transform have also Heaviside, Mikusi nski.´ 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 = I. I NTRODUCTION sX (s) x(0) : When x(0) vanishes the complex variable s First level courses in automatic control or basic textbooks is considered as a time derivative operator. However, this can of this topic contain rst lessons of Laplace transforms. be considered in only the space of transformed signals. Still The Laplace transform approach leads to dene the transfer about this fundamental theorem, the lower limit of integration function of a system. It is used to get the corresponding 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) - F (s) - Y (s) extensively by Deakin [8], [9], was introduced rst in the form as equation (1) by Bateman in 1910 to solve the differential Fig. 1. Basic block diagram. The transformed output signal Y (s) is given equation x_(t) = x (t) where is a nonzero real number. 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 In the following, we describe guidelines for starting an notation between signals and systems. automatic control course without using the Laplace transform. The presentation is based on the use of a pure operational [39] to overcome problems encountered in some particular point of view that provides an opportunity to link methods applications. An attempt to solve this problem and to unify developed in automatic control to laboratory applications. The the Laplace formalisms is proposed in [32]. Nevertheless the essential proofs based on Laplace transform theorems can be use of this tool is still an unjustied hypothesis. For the read in standard textbooks of automatic control ( e.g. [20], 1 case, we are faced with the bilateral Laplace transform, [30], [22]). Concerning the notation, we consider signals as which is questionable also [34]. The purpose of this article elements belonging to the set C of integrable real valued is to show that the mathematical machinery required by functions f = ff(t)g ; supposed to be m times continuously the Laplace transform [47] can be avoided. Moreover, the differentiable on [0 ; 1) except at isolated points where it is pedagogic difculty can be cleared in a natural way. 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 When the transfer function of a linear system has to be write for two signals a and b in C: for all t 0; a (t) = b(t), dened, Laplace transform is applicable. For a single-input or fa(t)g = fb(t)g, or a = b: However, when no confusion single-output system the transfer function is dened as the is possible the braces or “for all t 0” may be dropped. quotient of the Laplace transform of the output y(t) to the Nevertheless we do not deal with discrete-time systems except Laplace transform of the input u(t) with the assumption of to dene the transfer of a computer controlled system. zero initial conditions. In other words, the transfer function is dened by: II. T RANSFER OPERATOR Lf y(t)g Y (s) F (s) = = : We begin by considering a linearized system around an Lf u(t)g U(s) equilibrium point. We suppose this system can be described Although the name transfer function as a mathematical tool by the linear differential equation 2 is adequate for s = j!; where j = 1 and ! is the (n) (n 1) (n 2) y (t) + an 1y (t) + an 2y (t) + frequency [23], [3], this ad-hoc denition generates some (1) interesting questions. The Laplace transform of signals cannot +a1y (t) + a0y(t) (m) (m 1) (2) be obtained in practice, and sometimes we wonder how to = bmu (t) + bm 1u (t) + (1) determine the transfer function of a system? For instance this +b1u (t) + b0u(t); question is avoided in identication procedures [31] where where y(t) and u(t) stand for the differences of output and ARMAX models involving recurrence relationships instead of input signals with their setpoint values respectively and n transfer functions are used. In several high quality textbooks and m are two integers. In (2) the coefcients ai and bj are for discrete-time systems e.g. [2], the complex variable z of constant parameters. the Z-transform [26] and the shift-forward operator q are both used. However, the choice between z and q is not argued A. Coding in every case. These subtle differences, which are mysterious The aim is to provide a tool with which it is easy to for students, are useless and can be avoided. In view of our manipulate mathematical expressions, which may be used personal experience in teaching automatic control, one desire instead of linear differential equations, to separate input and is to give an experimental meaning of the transfer of a system, output variables from the system. Following Heaviside [24], irrespective of previous formal denitions. Rarely, in real [40] or Carson [5], we introduce the derivative operator occasions students may relate the transfer to the differential d operation induced by the system. Indeed the use of the Laplace p ; transform leads to the diagram depicted in Figure 1. This gure , dt describes the relationship between the Laplace transforms of which upon acting on x(t) in C gives the codings external signals and the transfer of the system. So, the essential 2 (n) n meaning is lost. It can be noticed here that we do not use the x_(t) = px (t); x•(t) = p x(t);:::; x (t) = p x(t);::: (3) term transfer function, although we call it transfer as can be In view of these codings and using the distributivity property seen in the sequel. for every real numbers and ; As a matter of fact, the use of Laplace transforms is one [ p n + p m]x(t) = x (n)(t) + x (m)(t); method among many others [33], [29] to justify the Heaviside operational calculus (Figure 2). Note that the operational equation (2) becomes n n 1 n 2 calculus is used to solve differential equations (in most cases p + an 1p + an 2p + + a1p + a0 y(t) = m m 1 linear) rather than automatic control problems. From an histor- bmp + bm 1p + + b1p + b0 u(t): ical perspective, the Laplace transform, which has been studied (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)
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) Mikusi nski´ (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. Among engineers and mathematicians, Heaviside appears as the focal point. His ideas on the use of the differential operator and on the denition of the transfer (resistance) operator are the basis of guidelines for an automatic control course without the Laplace transform.
u(t) - F (p) - y(t) B. Operational calculus as polynomial calculus Operational calculus is understood as algebraic methods Fig. 3. Operational block diagram. The output signal y(t) is obtained by for solving differential or recurrence equations, specically F (p)u(t) where u(t) is the input signal. Notice that t denotes the time in a linear time-invariant framework. In our point of view 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 solving a differential equation for a given input, is not the on an input signal is retained. The essential meaning of the transfer F (p) aim of automatic control but is a mathematical exercise [34]. In is the linear differential equation that links the output signal and the input automatic control operational calculus means rules for transfer signal. 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 To separate input and output signals from the system we divide the connection rules can be demonstrated through the fol- equation (4) by the polynomial factor pn+ a pn 1+ n 1 lowing steps. The transfers of the connected system provide +a ; which yields to code the input-output relationship as 0 differential equations. The connections and the elimination of y(t) = F (p)u(t) with intermediate signals lead to a differential equation between the output and input signals. The encoding of this differential b pm + b pm 1 + + b p + b F (p) = m m 1 1 0 : (5) equation with p ensures the results. Although we can use pn a pn 1 a pn 2 a p a + n 1 + n 2 + + 1 + 0 this procedure in every case it is sufcient to exemplify with respect to series or parallel connections for two rst-order We must insist here that y(t) = F (p)u(t) cannot be consid- systems. ered as the solution of the differential equation (2). Indeed, the initial conditions are not known. is determined with this y(t) Let us consider two linear systems described by y1(t) = writing as with the writing (2). In equation (5), stands F (p) F1(p)u1(t) and y2(t) = F2(p)u2(t) where u1(t) and u2(t) are for the transfer operator. In essence it is the transfer, which the input signals, F1(p) and F2(p) the transfers of the systems, represents the operation induced by the system to transform the and y1(t) and y2(t) are the corresponding output signals. In input signal into the output signal. The operational approach this regard we have provides an opportunity to relate the transfer operator (5) to b p + b p + the differential equation (2). The diagram, depicted in Figure F (p) = 1 0 and F (p) = 1 0 ; 1 a p + a 2 p + 3 corresponds to an experimental situation. 1 0 1 0 where a0; a 1; b 0; b 1; 0; 1; 0, and 1 are constant e-STA copyright 2008 by SEE parameters. The series connection is dened as u2(t) = y1(t); Volume 5, N°3, pp 18-28 4
u(t) = u1(t); and y(t) = y2(t): The application of the A. Step response procedure yields Let us consider the Heaviside or step signal fH(t)g dened b p2 + ( b + b )p + b by H(t) = 1 for t 0 and 0 elsewhere. The step response y(t) = 1 1 0 1 1 0 0 0 u(t); of a system is the solution of the differential equation of the a p2 + ( a + a )p + a 1 1 0 1 1 0 0 0 system to a step input signal with zero initial conditions. With
where we recognize the product F1(p)F2(p): The parallel operational calculus, we can expand transfer operator as a connection is dened as u2(t) = u1(t) = u(t) and y(t) = linear combination of simple transfers n y1(t) + y2(t); which yields to a 1 n or n ; 1 (p + a) p y t ( ) = 2 where a stands for a nonzero complex number and n for an 1a1p + ( 0a1 + 1a0)p + 0a0 2 integer. The step response of a multiple integrator with the (( a1 1 + 1b1)p + ( a1 0 + b1 0 1 tn transfer is : Let us consider the step response sn(t) of +a0 1 + b0 1)p + ( a0 0 + b0 0))] u(t); pn n! an the Strej c´ system [41] with the transfer : For n = 1 where we recognize the sum F1(p)+ F2(p): These results can (p + a)n be extended to any order by induction. It can be seen that the a the associated differential equation to the transfer is transfer of the series connection of two systems is the product p + a of their transfer and the transfer of the parallel connection of ay (t)+y _ (t) = au (t) and we obtain by usual methods [51] the two systems is the sum. corresponding response to a given input u(t) with the initial condition y(0) The parallel and series rules give a meaning to the decompo- t y(t) = e at y(0) + a ea u()d : sitions and the handling of transfer operators with polynomial 0 calculus. These operations on transfer operators are the basis Z For u(t) = 1 and zero initial conditions the step response of operational calculus in automatic control. We can apply becomes usual techniques as Mason's rule associated to the signal- s (t) = 1 e at : ow graph [35]. This operational calculus can be applied also 1 a for multiple-input multiple-ouput systems with the difference For n = 2 we have s2(t) = s1(t) from which follows that commutativity does not occur anymore. Let us note that p + a the polynomial formalism is used in several textbooks on t s (t) = ae at (ea 1) d = 1 e at (1 + at ): (6) multivariable systems [27], [28] with no need of the Laplace 2 Z0 transform. For the general case n 1; let us suppose n 1 at i C. The delay operator sn(t) = 1 e ci;n t ; i=0 ! A pure time delay of T between input and output signals X and induces y(t) = u(t T ): This particular linear system cannot n at i be associated to a differential equation as (2). A special sn+1 (t) = 1 e ci;n +1 t ; treatment must be used for delay equations. Following an idea i=0 ! X of Euler [14], the Taylor expansion of u(t T ) yields where the coefcients ci;n and ci;n +1 are constant parameters. These signals are linked by the differential equation T 2 ( T )n u(t T ) = u(t) u_(t)T +•u(t) +u(n)(t) + ; 2 n! s_n+1 (t) + as n+1 (t) = as n(t); which is encoded to give which to the the relationships
T 2 ( T )n c0;n +1 = 1 ; y(t) = u(t) pT u (t) + p2 u(t) + pn u(t) + ; 2 n! a ai for i = 1 to n : c = c = : i;n +1 i i 1;n i ( pT )n ! = u(t) = (exp( pT )) u(t): We deduce the well known result that, for the Strej c´ model 0 n! 1 n0 X an @ A ; We obtained the transfer operator for the time delay T as (p + a)n F (p) = e pT : the step response sn(t) is
n 1 i i III. S YSTEM RESPONSES a t s (t) = 1 e at : n i! System analysis is often the study of some particular re- i=0 ! X sponses of the system. Of special interest are the step and So the step response of a given system dened by a transfer frequency responses. operator can be calculated using polynomial calculus. e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 5
B. Stability the frequency is the only actual transfer function. Graphic Due to the fact that stability analysis does not use the representations such as Bode, Black-Nichols, or Nyquist loci Laplace transform there is no novelty in this paragraph. may be used to analyze the frequency response [30]. For However, 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. But for stable systems, experiments cannot allow to an obtain the frequency response. Let us consider the transfer operator where a and n (p + a)n have the same meaning as before. From the previous paragraph IV. A NALYSIS we can see that the step response is composed of a constant term and a time-dependent term. The rst term is the forced A. Poles and zeros response and the second term is the transient behavior. The The names of poles and zeros come from the interpretation transient behavior tends asymptotically to zero if and only if of a transfer operator F (p) as a function of a complex variable the real part of a is strictly negative : More generally, consider p: This interpretation is a consequence of the formulation of the n-th order transfer Laplace transform and it misunderstands the physical meaning m bmp + + b1p + b0 of these notions. The consideration of a transfer operator as F (p) = ; (p p1) 1 (p p2) 2 (p pr) r a coding of a differential equation provides an immediate physical interpretation. Namely, let us consider the transfer where p1; p 2;:::; p r are the r complex poles of F (p) and are the respective multiplicities with n = r . For the operator i i=1 i p + a transient response the poles generate terms associated with the F (p) = ; (7) signals ep1t; e p2t;:::; e pr t weighted by timeP polynomials of p + b order i 1 respectively. If all the poles have strictly negative where a and b are constant parameters. In the operational real part, the transient behavior tends asymptotically to zero. standpoint the transfer (7) corresponds to the input-output Namely, the system is asymptotically stable. 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 C. Frequency response 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 For asymptotically stable systems the frequency response is at deduced from the steady-state output response corresponding condition for the output and the input signal u(t) = e for t > we obtain u t au t so y t for t > : to a given sinusoidal input signal u(t) = ej!t where ! is 0 _( ) + ( ) = 0 ( ) = 0 0 the frequency and j2 = 1. Consider a system dened by B(p) In general poles correspond to signals generated by the the transfer operator F (p) = where A(p) and B(p) A(p) system with zero input. Zeros correspond to signals absorbed are two polynomials. From the operational approach we have or blocked by the system. Let us write the transfer operator the encoded input-output differential equation as A(p)y(t) = (5) as B(p)u(t). For the sinusoidal input u(t) = ej!t ; we obtain 1 2 d (p z1) (p z2) (p zd) j(!t +arg( B(j! ))) F (p) = k ; B(p)u(t) = jB(j! )j e ; (p p1) 1 (p p2) 2 (p pr) r
where jB(j! )j and arg( B(j! )) denote the module and the where k = bm; z i; i = 1 ;:::;d and pi; i = 1 ;:::;r are argument of the complex number B(j! ) respectively. The complex numbers, and i; i = 1 ;:::;d and i; i = 1 ;:::;r output y(t) is the sum of a particular solution of the differential are integers. For i = 1 ;:::;r , epit is solution of the coded equation and the general solution of the differential equation differential equation without second member. The general solution characterizes 1 2 r the transient response that vanishes in case of asymptotically (p p1) (p p2) (p pr) y(t) = 0 ; stable systems. For a particular solution, we look for the and for i = 1 ;:::;d , ezit is solution of the coded differential steady-state behavior as the form y(t) = Y e j(!t +') where equation Y and ' are constant parameters. Replacing this expression for y(t) in the differential equation yields 1 2 d (p z1) (p z2) (p zd) u(t) = 0 :
jB(j! )j On the one hand we can use the correspondence between Y = = jF (j! )j ; pit jA(j! )j e and the transfer denominator roots pi that characterizes the transient rate in the linear constant parameter framework ' = arg( B(j! )) arg( A(j! )) = arg( F (j! )) : only. The same remark can be said about the correspondence zit The frequency response is dened by the evolution of between e and the transfer numerator roots zi. On the other (jF (j! )j ; arg( F (j! ))) as the frequency ! varies from 0 to hand this signal approach for the pole and zeros meaning can +1: We can notice that F (j! ) is the transfer function of be extended to time-varying or nonlinear multivariable systems the system such as Harris dened it [23]. In our standpoint, with an algebraic standpoint [15], [16]. this transfer function must not be confused with the transfer In order to underline and to exemplify the important operator F (p). Nevertheless, F (j! ) such as a function of 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 : order is fullled whether the DC gain is equal to 1. In other 1 words since the input-error transfer is 1 F (p); we obtain a y(t) = u(t); steady-state error of zero order when the input-error DC gain p 1 is zero. This is a fundamental remark for the following. p 1 z(t) = y(t): p + 1 If we notice that r1(t) is the integral of r0(t); namely The pole 1 induces, in the transient behavior or in the initial 1 r1(t) = r0(t); we have conditions effect for the rst system, an et signal. This signal p is blocked by the second system, which has 1 as zero. As "1(t) = r1(t) y1(t); t lim t!1 e = 1; this fact forbides such a connection. Indeed, 1 1 while the input and output signals are zero, there exists in the = r0(t) F (p) r0(t); p p system a non observed and non controlled unbounded signal. 1 F (p) The conclusion is different if we consider the series connection = r0(t): with the systems : p Clearly, from the previous result for "0(1); " 1(1) vanishes 1 1 F (p) y(t) = u(t); if and only if the DC gain of the transfer operator p + 1 p p + 1 z(t) = y(t): is equal to zero. Since p 1 2 1 F (p) (a0 b0) + ( a1 b1)p + ( a2 b2)p + t = 2 n Due to the pole/zero cancellation at 1, y(t) has an e p p(a0 + a1p + a2p + + anp ) component that vanishes at 1: Except during the transient we obtain " (1) = 0 if and only if a = b and a = b : It behavior, the pole/zero cancellation is acceptable for asymp- 1 0 0 1 1 can be seen that totically stable cancelled zeros. when a0 6= b0; we have "0(1) 6= 0 and with "1(1) = a0 b0 lim p!0 = 1 ; B. DC gain pa 0 when a = b ; we obtain " (1) = 0 and with " (1) = Let us keep in mind that the transfer operator F (p) in 0 0 0 1 a1 b1 equation (5) is just a coding of the differential equation (2). : Moreover "1(1) = 0 when a1 = b1. a0 In the case of an asymptotically stable system, with the input In the same way we can show by recurrence that the system taking a constant value U; the step response analysis indicates can have a steady-state error of order N if and only if its that the output tends to a constant value Y as t goes to +1 transfer F p in equation (5) is such that, for i to N; Y ( ) = 0 given by the relationship a Y = b U: The ratio denes the a = b . In this case the steady-state error of order N + 1 is 0 0 U i i DC gain of the system GDC : The stability condition implies aN+1 bN+1 a 6= 0 ; and from (5) we obtain G = F (0) . "N+1 (1) = ; 0 DC a0 and the next ones have an innite module. The degree of the C. Steady-state error analysis steady-state error can be obtained by just a visual inspection In all this part systems are supposed to be asymptotically of the transfer operator of the system. stable, namely the transient behavior vanishes and only the V. C OMPUTERCONTROLLEDSYSTEMS permanent behavior remains. For the reference inputs ri(t) ti The last question we wish to deal with concerns the con- dened as, for t > 0; r (t) = ; and for t < 0; r (t) = 0 ; i i! i struction of the model of a linear system controlled by a the corresponding outputs are yi(t) = F (p)ri(t): The input- numerical algorithm ( e.g. [2], [19]). The problem is to nd the output error "i(t) = ri(t) yi(t) is called the system error of discrete-time model corresponding to the structure presented order i: Two notions can be pointed out here. First a norm of in Figure 4 where DAC denotes a digital-analog converter the instantaneous system error "i(t) characterizes the system and it is usually modelled as the series connection of an performance. Second the value "i(1) = lim t!1 "i(t) during ideal sampler and a zero-order hold. ADC denotes an analog- the permanent behavior characterizes the steady-state error. In digital converter usually modelled by an ideal sampler and basic lecture of automatic control this last notion is usually the discrete output signal of the ADC device is yk = y(kT s) considered. We detail it according to our formulation, namely for k in N. Both are supposed to be synchronized with the without the use of the nal value theorem. sampling period Ts: In this section we use the notations fxkg for the discrete-time real valued signal dened for k in N 1 1 A steady-state error of order N is ensured if "i(1) = 0 ; for and q for the delay operator q fxkg = fxk 1g. All the i = 0 to N; and "N+1 (1) 6= 0 : Consider the transfer operator considered discrete-time signals are supposed to be zero for F (p) in equation (5) of an asymptotically stable system. The negative values of k: For the discrete input signal fukg the corresponding permanent step response value is given by the output of the DAC device is b0 DC gain : It can be seen that "0(1) = 0 if and only if a0 u(t) = uk (H(t kT s) H(t (k + 1) Ts) ; (8) b0 = a0: We can conclude that a steady-state error of zero k0 e-STA copyright 2008 by SEE X Volume 5, N°3, pp 18-28 7
- u(-t) y(-t) - 1 uk DAC F (p) ADC yk where gl; l 0; are real numbers. With fvkg = G(q ) fukg we obtain Fig. 4. Linear computer controlled system. Equipped with digital-analog (DAC) and analog-digital (ADC) converters, a continuous-time model ( F p ) ( ) fv g = g u = u S : leads to a discrete-time model. The discrete-time model is a coding of the l 8 k l k9 8 k l k9 u k0 k0 recurrence equation between the discrete input signal k and the sampled Although the Z-transform doesn't suffer the same drawbacks k as the Laplace transform, for instance discrete impulse is really gk = K(1 D ); a discrete signal, the Laplace transform appears once more Ts time here. To carry on we obtain the discrete-time transfer where D = e : We deduce operator F (q 1) in the operational framework. G(q 1) = K (1 Dk)q k; Denoting by S(t) the step response of F (p); the response k X0 of F (p) to uk(H(t kT s) H(t (k + 1) Ts) is uk(S(t K(1 D)q 1 kT ) S(t (k + 1) T ): The sampling of this response at the = ; s s (1 q 1)(1 Dq 1) time period Ts leads to a value for t = lT s; l in N; given by uk (Sl k Sl k 1) where Sk stands for S(kT s): Denoting so the discrete-time transfer of (10) is l the independent integer variable the corresponding discrete (1 D)q 1 signal is fu (S S )g = (1 q 1) fu S g for F (q 1) = K : k l k l k 1 k l k (1 Dq 1) a given k: Because of linearity we deduce from (8) that the sampled response corresponding to the input signal fukg is VI. W HAT 'S ABOUT TABLES ? The main reason of using the Laplace transform is the tables fy g = (1 q 1) u S : l 8 k l k9 we have at our disposal. First they contain information to k0 1 n0 to link input u(t) and output y(t) signals of a linear system (1 + d1q + + dn0 q ) fvkg = 1 m0 by a differential equation coded as y(t) = F (p)u(t): Until (n + n q + + nm q ) fukg ; 0 1 0 now we obtained the step response by solving this differential which leads to the discrete transfer operator equation when initial conditions are all zero. In case of no n + n q 1 + + n q m0 input and non zero initial conditions such a transfer operator G(q 1) = 0 1 m0 : 1 n0 produces an output signal solution of the associated homo- 1 + d1q + + dn0 q 1 geneous differential equation. The coding of this differential Let us denote the numerator and the denominator of G(q ) equation with the p operator denes then the generator of by N(q 1) and D(q 1) respectively. The division of N(q 1) 1 this signal. Two ways can be considered to take into account by D(q ) leads to initial conditions in this coding. Namely, on the one hand the 1 l G(q ) = glq ; Mikusi nski´ operational calculus and on the other hand the l0 integral form of a differential equation. e-STA copyright 2008 by SEEX Volume 5, N°3, pp 18-28 8 (n 1) A. The Mikusi´nski operational calculus with initial conditions f(0) = f0; f_(0) = f1;:::; f (0) = f ; where n is an integer and the are real numbers. With All the previous developments can be rigorously proved n 1 i (11) the coding of (12) leads to by means of operational calculus of Mikusi nski´ [37], which n is based on convolution algebra of operators. Let us briey i describe this operational calculus whereas keeping in mind ip f(t) PIC (p; f 0;:::; f n 1) = 0 ; "i=0 # that the considerations below are not needed in a rst level X course. Convolution product is a fundamental tool in dynamic where PIC (p; f 0;:::; f n 1) is a polynomial in p that depends systems eld [45], [46], specically in case of linear systems on the initial conditions and the coefcients i: We obtain [21], [10]. This tool is dened by then the generator of ff(t)g t PIC (p; f 0;:::; f n 1) ff(t)g = n i : (13) (f; g ) 7! gf = f( )g(t )d ; [ i=0 ip ] Z0 The second theorem indicates thatP when the one-sided Laplace while the Heaviside function H = f1g is of great importance transform of a signal exists, its expression is identical to the due to the fact that we have for every f in the set of integrable generator of the signal, the complex variable s of Laplace function C transform being changed into the derivative operator p (to t avoid any confusion). A major consequence is that the tables Hf = f(x)dx : 1 0 [12], [48], can be used. Since H = p we remark that the Z generator can be written indifferently with the operators H or Consequently, H appears as the integration operator. The suc- p. cessive powers of H with respect to the convolution product are For example when we look for a generator, say for sin( !t ); tn 1 we observe that sin( !t ) is the solution of the differential for all n in N; n 1; H n = : (n 1)! equation 2 To distinguish a constant signal f g with the operator x•(t) + ! x(t) = 0 ; x (0) = 0 ; x_(0) = 1 : (14) dened by the constant gain we denote it by [ ] : For all f Using (11) the coded form is obtained as in C p2x(t) 1 + !2x(t) = 0 ; t f g f = f( )d and [ ] f = f f (t)g : which leads to the generator of sin( !t ) Z0 1 fsin( !t )g = ; [1] is the unit element for the convolution and we can give a M p2 + !2 meaning to H0 as H0 = [1] . We dene the derivative operator where the symbol “M” denotes “in the Mikusi nski´ sense”. as the solution of the convolutional equation pH = [1] ; and Indeed, we see in the next section an alternative to the then we write p = H 1: With the understanding p0 = H 0 = Mikusi nski´ approach. The Mikusi nski´ operational calculus has n n [1] , we have p = H for n in N: been extended recently by the convolutional calculus [11]. Mikusi nski´ [37] has proved the two results below, which B. The integral form are essential for our purpose. A second point of view can be used to introduce initial con- Theorem 1 For every continuous function f in C, ditions in differential equations. In particular, let us consider f (1) (t) = pf [f(0)] : More generally, for every integer the result below [44] k fx_(t) = f(t); x (0) = x0g k 1 (k) k (i) k i 1 if and only if (15) f (t) = p f f (0) p : (11) t x(t) = x + f( )d : i=0 0 0 n o X h i Denoting the integral operator byR H, the derivative operator 1 tp Theorem 2 For every f in C such that 0 e f(t)dt exists by p; and using the fact that for zero initial conditions, pH = Hp = I where I stands for the identity operator, we can 1 R f = e tp f(t)dt: code the differential equation (15) by x(t) x0 = Hf (t) or 0 px (t) px (0) = f(t) =x _ (t): In general, from the differential Z equation n The rst theorem allows to write the generator of a signal x( )(t) = f(t); ff(t)g when is known the differential equation whose this with the initial conditions x(0) = x0; x_(0) = x1;:::; signal is solution. Indeed, let us suppose that this differential (n 1) x (0) = xn 1; we obtain equation can be written n 1 tk t n x(t) = xk + f( )d : (16) (i) k! f (t) = 0 ; (12) k 0 i X=0 Z Z i=0 n times e-STA copyright 2008X by SEE Volume 5, N°3, pp 18-28 | {z } 9 In particular for ff(t)g = 0 and the initial conditions x0 = we must keep the coherence in using tables. For instance, when x1 = = xk 1 = xk+1 = xn 1 = 0 and xk = 1 we want to know the beginning of the response we can develop tk y(t) in power of p 1. This procedure was used by Heaviside we obtain x(t) = : As the corresponding differential k! [24]. However, different functions may be associated to p k equation can be coded pkx(t) = 1 we get according to the adopted generator sense. Second, we can ob- tain the discrete-time model of a computer controlled system. tk 1 for every integer k; = : Since we see it above, this model uses the step response S(t) k! pk of a system dened by the transfer operator F (p): Incidentally, Moreover for zero initial conditions and for all integer k we we can remark that the step generator in the Mikusi nski´ sense have pkHk = Hkpk = I: These remarks induce for (16) the F (p) for S(t) is . In this framework we obtain the formula coding p n 1 (9) for the discrete-time model. Whatever the used generator (n) n n k x (t) = p x(t) xkp : (17) sense the tables can be used also. k=0 Moreover, in order to see the importance of the generator X The generator of a signal ff(t)g can be obtained in this for operational calculus, let us consider the following example framework by coding with (17) the differential equation (12) where two signals y1 and y2 are dened by the differential whose this signal is solution. We obtain the coding equations : n (p 1) y1(t) = u(t); (19) pi f(t) P (p; f ;:::; f ) = 0 ; i IC 0 n 1 (p 1) y2(t) = u(t); (20) "i=0 # X 0 0 and the initial conditions y1 and y2 respectively. If we consider where PIC (p; f 0;:::; f n 1) is a polynomial in p that depends on only the initial conditions and the coefcients : We get the operational calculus, the parallel connection y(t) = y1(t) i y t yields to : the generator of ff(t)g as 2( ) 1 1 PIC (p; f 0;:::; f n 1) y(t) = u(t) u(t) = 0 : ff(t)g = n i : (18) p 1 p 1 [ i=0 ip ] This conclusion is obviously wrong. Indeed our setting and the However, if for a given function f t we compare the P f ( )g suggested proof of the operational calculus indicate that we generators (13) and (18) it can be seen that P p; f ;:::; IC ( 0 consider formal differential equations, namely without initial f pP p; f ;:::; f . We have the following n 1) = IC ( 0 n 1) conditions. When we write : relationship between the generators : the second one is the rst 1 one multiplied by p: The generator (18) is the generator of f(t) y(t) = u(t); in the Carson sense. The Carson transform was introduced in p 1 1926 [6] and it differs from the Laplace transform by a factor it is just a coding of the differential equations (19) and (20) p: The Carson tables can be used in this framework. only. The initial conditions can be taken into account by mean of the generator notion. We can write (19) and (20) For example when we evaluate this generator for sin( !t ); as, respectively : we obtain from the differential equation (14) the coding 0 1 y1 2 2 y1(t) = u(t) + ; p x(t) p + ! x(t) = 0 ; M p 1 p 1 0 which leads to the generator of sin( !t ) 1 y2 y2(t) = u(t) + ; p M p 1 p 1 sin( !t ) = ; C p2 + !2 in the Mikusi nski's´ generator sense. It yields for y(t) = where the symbol “C” denotes “in the Carson sense”. It is y1(t) y2(t) the generator : obvious that the integral form approach appears more intuitive y0 y0 than the Mikusi nski´ approach but the results are quite similar. y(t) = 1 2 : M p 1 C. Consequences This point indicates that y(t) is solution of the differential equation : The generator of a signal allows to perform both of the 0 0 two points below. First, for a system dened by the transfer (p 1) y(t) = 0 ; y (0) = y1 y2; operator F (p) we can calculate the response y(t) to an input or, equivalently, y(t) = ( y0 y0)et: u(t) by using Carson or Laplace transform tables. Indeed when 1 2 This standpoint can also be explained in a more algebraic U(p) is a generator of u(t) we obtain the generator of the framework as the Fliess'module-theoretic approach [17]. Nev- output ertheless, let us remind a sentence of a recent paper [18] y(t) = F (p)E(p): where is used this point of view for the design of a new We must remark here that the generators can be obtained in identication procedure : “ Let us add we tried to write the any sense as dened above (Mikusi nski´ or Carson). However, examples in such a way that they might be grasped without the e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28e-STA copyright 2008 by SEE Volume 5, N°3, pp 18-28 10 necessity of reading the sections on the algebraic background. [4] H. Blomberg, R. Ylinen, Algebraic Theory for Multivariable Linear Our standpoint on parametric identication should therefore Systems , Academic Press, 1983. [5] J.R. Carson, On a general expansion theorem for the transient oscilla- be accessible to most engineers. " 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 , VII. C ONCLUSION McGraw-Hill, 1926. We show in this short survey that from the use of the [7] D. Cochrane, G.H. Orcutt, Application of least squares regression to relationships containing autocorrelated error terms, J. Amer. Statist. differential operator we obtain all the usual results derived by Assoc. , vol. 44, pp. 32–61, 1949. means of the Laplace formulations. To teach a basic lecture [8] M.A.B. Deakin, The development of Laplace transform, 1737–1937 : in automatic control using the operational method offers some I. From Euler to Spitzer, 1737–1880, vol. 25, n. 4, pp. 343–90, Archive for History of Exact Science , 1981. advantages. The integral or derivative operators allow to link II. From Poincaré to Doetsch, 1880–1937, vol. 26, n. 4, pp. 351–81, every notion to its physical meaning. We keep in mind that Archive for History of Exact Science , 1982. a transfer operator is always related to only a differential [9] M.A.B. Deakin, The ascendancy of the Laplace transform and how it came about, vol. 44, n.3, pp. 265–86. Archive for History of Exact equation or a difference equation. Mathematical background Science , 1992. is minimized, however, when a rigorous justication is needed [10] C.A. Desoer, M. Vidyasagar, Feedback Systems : Input-Output Proper- the Mikusi nski´ operational calculus may be used. This opera- ties , Academic Press, 1975. [11] I.H. Dimovski, Convolutional Calculus , Kluwer Academic Publishers, tional calculus is based on the convolution operator, which is 1990. a natural tool for linear equations. [12] G. Doetsch, Theorie und Anwendung der Laplace-Transformation , Springer, 1937. [13] R.C. Dorf, R.H. Bishop, Modern Control Systems , Prentice Hall, 2001. Moreover, we meet here through a pedagogical step the [14] S. Dugowson, Les Différentielles Métaphysiques , Thèse de Doctorat, operational standpoints adopted directly in some advanced Université Paris XIII, 1994. textbooks to modelize the input-output relationship induced by [15] H. Bourlès, M. Fliess, Finite poles and zeros of linear systems : an intrinsic approach, Int. J. Control , vol. 68, pp. 897–822, 1997. a linear system. For instance, the discrete-time autoregressive [16] M. Fliess, S.T. Glad, An algebraic approach to linear and nonlinear con- 1 1 1 moving-average model A(q )yk = B(q )uk + C(q )k trol, Essays on Control : Perspectives in the Theory and its Applications , where A(q 1); B (q 1); and C(q 1) are polynomials in the H. Trentelman and J.C. Willems (eds.), pp. 223–267, 1993. delay operator and a noise signal is used in [7], [1] [17] M. Fliess, Une interprétation algébrique de la transformation de Laplace f kg et des matrices de transfert, Linear Algebra and its applications , vol. for identication purposes to describe the difference equation 203–204, pp. 429–442, 1994. between the input uk and the output signals yk of a given [18] M. Fliess, H. Sira-Ramirez, Closed-loop parametric identication system. For multivariable continuous-time linear systems [42], for continuous-time linear systems via new algebraic techniques, in Continuous-time model identication from sampled data , H. Garnier, [50], [49] introduce the model L. Wang (eds), Springer, 2007. [19] G.F. Franklin, J.D. Powell, L. Workman, Digital Control of Dynamic P (p)(t) = Q(p)u(t); Systems , Addison Wesley, 1990. [20] G.F. Franklin, J.D. Powell, A. Emami-Naemi, Feedback Control of y(t) = R(p)(t) + W (p)u(t); Dynamics Systems , Addison-Wesley, 1994. [21] I.C. Gohberg, M.G. Kre n, Theory and Applications of Volterra Opera- where P (p); Q (p); R (p); and W (p) are matric polynomials tors in Hilbert Space , American Mathematical Society, 1970. in the differential operator and (t) is a vector-valued function [22] G.C. Goodwin, S.F. Graebe, M.E. Salgado, Control System Design , of time called the partial state. More recently, [4] dene Prentice Hall, 2001. [23] H. Jr Harris, The frequency response of automatic control systems, the generator of a multivariable system as the polynomial Trans. on Electrical Engineering , vol. 65, pp. 539–46, 1946. matrix M(p) in the derivative operator ; which allows to [24] O. Heaviside, Electromagnetic Theory , vol. 2, The Electrician Printing write the relationship between input and output signals as and Publishing Company, London, 1899, rep. Elibron Classics, 2001. y(t) [25] J. Hladik, La Transformation de Laplace , Masson, 1969. M(p) = 0 . The generator in the sense dened in [4] [26] E.I. 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