From Chronological Calculus to Exponential Representations of ...

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 12, DECEMBER 2007

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From Chronological Calculus to Exponential Representations of Continuous- and Discrete-Time Dynamics: A Lie-Algebraic Approach Salvatore Monaco, Fellow, IEEE, Dorothée Normand-Cyrot, Fellow, IEEE, and Claudia Califano

Abstract—In this paper, formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived. It is shown that the chronological exponential admits an ordinary exponential representation, the exponent being given by an explicitly computable Lie series expansion. The results are then used to describe controlled dynamics, dynamics under sampling and forced discrete-time dynamics. The study emphasizes the role of Lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations, dynamics under sampling and forced discrete-time dynamics up to hybrid ones. Index Terms—Chronological calculus, discrete-time and sampled dynamics, nonautonomous differential equations, nonlinear systems.

I. INTRODUCTION

S

ERIES expansions of solutions to differential equations and controlled dynamics have been widely investigated in the literature. As is well known, in the simpler case of linear systems, they reduce to the Peano Baker series [1] making use of linear algebra calculus and exponentials of matrices only. The introduction of Chen’s formalism [2], exponential Lie series [3], and chronological calculus [4] is at the basis of the more recent developments in the nonlinear context. As a matter of fact, in the control theory literature these techniques have been used in various formats such as Chen-Fliess series by Fliess [5], chronological series by Agrachev-Gamkrelidze [4], and exponential series by Sussmann [6]. They have been developed following different points of view uniting control and combinatorics. In this context, the input-affine systems behaviors admit Volterra series expansions with computable kernels [7]–[10], or Chen-Fliess series expansions in the successive iterated integrals of the inputs [5], [11], or representations as infinite direct products of exponentials [12], [13]. By exploiting the combinatoric and Lie algebraic Manuscript received October 6, 2004; revised January 25, 2006 and November 4, 2006. Recommended by Associate Editor J. M. Berg. This work was supported in part by MIUR and by the Vinci Programme Galilée under the auspices of the University Franco/Italienne-Italo/Francese. S. Monaco and C. Califano are with the Dipartimento di Informatica e Sistemistica “Antonio Ruberti,” Università di Roma “La Sapienza,” 00184 Rome, Italy. D. Normand-Cyrot is with the Laboratoire des Signaux et Systèmes, CNRS, Supélec, University Paris-Sud, F-91192, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2007.902734

structures of these series solutions, a variety of analysis and design control problems have been investigated [7], [14]–[22]. In the present paper, starting from the chronological series expressing the solution to nonautonomous differential equations, we derive an ordinary exponential representation of the solution and explore its formal and combinatoric properties. This result represents by itself an original contribution. Its use in the area of systems and control gives two interesting outcomes. The first concerns its application to dynamics under sampling while the latter its application to discrete-time dynamics. Equivalent sampled models of continuous-time dynamics under higher orderholding devices and exponential representations of controlled discrete-time mappings are thus proposed. The conclusion is that, even if at a first glance continuous-time dynamics, dynamics under sampling or forced discrete-time dynamics may appear different, there are many connections and they all lead to essentially equivalent developments thus providing a unified framework for handling hybrid dynamics including jumps and resets [23]. Convergence issues are not explicitly addressed in this paper which essentially attempts to unite in an identical formal treatment continuous-time, discrete-time, and sampled dynamics. The application of these formal manipulations to solve numerical problems requires a better understanding of convergence performances. It must also be noted that approximated solutions based on series truncations are sufficiently accurate in practice and that several realistic classes of systems exhibit series solutions of finite order [24] (finite sampling). Some specific aspects related to the results of this work have been preliminarily discussed in [25]. In the sequel, an overview of the main results as well as the terminology and assumptions used in the paper are given. Section II is concerned with nonautonomous differential equations of the form

(1) is a smooth and complete vector field, analytically where parameterized by . According to [4], in the operator language of chronological calculus, the solution is obtained through the to the identity function and the application of the flow

0018-9286/$25.00 © 2007 IEEE

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result evaluated at . The flow is represented by the right chronological exponential series, i.e., for

(2)

The explicit asymptotic representation of the flow proposed in Section II exhibits the following properties: • it takes the form of an ordinary exponential (3) • its exponent admits a Lie series expansion in powers of with Lie polynomials of degree in the ( )s as coefficients (4) • each coefficient can be explicitly and iteratively computed. in (1), It is worth noting that in the autonomous case, and (3) reduces to

with . The formal expression (3) provides an explicit integrated form of (2) since it no longer requires the integration of the composed vector fields depending in (2). A different integrated form has been deon rived in [27] in terms of expansions in the state variables with time dependent coefficients. Our result relies on expansions in . This solution represents an improvement of the Lie series expansion introduced in [3], since it provides an explicit Taylor type expansion of the flow with respect to time. The formulas here given can find various uses in different contexts. We illustrate their usage in systems and control theory for representing discrete-time dynamics and computing, possibly through approximations, dynamics under sampling and digital controllers ([25], [26]). As far as their interest in numerical integration schemes is concerned, a deep understanding of the convergence properties is necessary for pursuing a comparative study with existing methods, e.g., the one proposed in [27]. However, this goes beyond the purpose of this paper. Section III studies dynamics under sampling, i.e., the evolubeing the sampling time, of the controlled tion at times dynamics (5)

is a complete vector field, analytically paramewhere the control terized by . Over the time interval variable is given by a, possibly finite, expansion of the form which could be interpreted as the Taylor’s like expansion of a smooth control. A nonautonomous dynamics of the form (1), parameterized by the ( )s, is thus obtained. The results of Section II can be applied to characterize the asymptotic expansion of the multi-input mapping which describes the evolution under sampling

The representation (3) can thus be used to compute sampled equivalent models, possibly through finite order approximations. The computations are developed in Section III for input-affine dynamics. The obtained representation, which is no more than an integrated form of the Chen-Fliess series ([2], [15], [5]), is useful in the design of piecewise continuous control laws. More precisely, such a model can be used in the design of advanced digital controllers involving holding devices of different orders or for solving difficult control problems when standard smooth control solutions cannot be worked out. This is illustrated in Section III on the basis of an example. Nonlinear controlled discrete-time dynamics are discussed in Section IV. Even if difference equations, usually described by can be considered far away from mappings differential equations, the application to the discrete-time context of the results derived in Section II is straightforward by interchanging time and input dependencies. To this end, discrete-time dynamics are represented as coupled differential/difference equations as proposed in [20] and [28]. More precisely, and a complete vector field , analytically given a map parameterized by , a differential/difference single-input discrete-time dynamics will be defined over each time interval by (6) and (7) below. The differential equation (7) describes the variation with respect , while the difference equation (6) describes the free evolution and sets the initial condition, i.e. (6)

(7) Once again the results of Section II can be applied so obtaining that the flow associated with the differential equation (7), namely the chronological exponential

admits the standard exponential representation

MONACO et al.: CONTINUOUS AND DISCRETE-TIME DYNAMICS

We thus get the same representation as in (3)–(4) now parameterized by instead of . The evaluation of the flow at the initial condition (6) gives back a discrete-time dynamics in the usual . How the vector form of a map, i.e., defining in (7) enter in the geometric structural fields characterization of nonlinear discrete-time systems has been illustrated in several publications [16], [17], [20]. In the present context, a major advantage of the approach described herein is to allow us to represent compositions of nonlinear mappings over several time-steps or for different control values as compositions of exponential forms. As an example, this makes the computation of the kernels in the associated Volterra series expansions easier.

A. Terminology and Assumptions Even if not making strict use of the operator notations of chronological calculus, we work under the assumptions stated in [4] and we refer to the terminology in paragraph 2 of [18] for all the material not here specified (see also [21], [6], [12], [14], [13], [18], [19], [31], [22], [29], [27]). Throughout the paper, , an open set of which , a neighborhood of zero in . can be all are smooth—infinitely differenMaps and vector fields on denotes the commutative algebra over of tiable—. smooth real valued functions on . The nonautonomous vector (respectively, and ), defining field the ordinary differential equation (1) [respectively, (5) and (7)], is assumed to be analytically parameterized by (respectively, )—admitting convergent Taylor series expansions with respect to (respectively, ) for any fixed —, and complete—an abexists for all (respecsolutely continuous solution through tively, )—. Given a function and a vector field , their and , or evaluation at a point is denoted by and ; when performing derivatives with equivalently by respect to or , evaluation is also denoted by “ ”. The application of over gives and its eval. If is a -vector valued funcuation at is denoted by , for tion with each component acts componentwise. , denotes the canonical projection of over the identity map over and 1 of . the identity operator over the dual space The successive application of two vector fields and is deand their Lie bracket by noted by ( -times) is denoted by and ; , with . The Lie moreover, bracket defines a Lie algebra structure on the set of vector fields. We just recall that in this formalism, all the objects (points, maps and vector fields) are considered well defined members so that equalities involving such objects are well of defined. In this context, given any operator or composition of operators, say , its evaluation at , or the evaluation at of its application to the identity function, will be denoted equiva. Given a diffeomorphism over lently

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while . As an ex, the following property ample, defining of commutation (see [3]) holds true:

(8) All over the paper, identities must be interpreted in the , denotes sense of asymptotic series. Finally, the transport of along which satisfies at the equality so that . Iteratively, we and for so that define , defines the transport of along (composed -times). Given a finite set , an alphabet, with elements called letters (or equivalently given a set of indexed formal of length over variables), we define a word any finite sequence of length of elements in . The set of all words of length is denoted by and the set of all words is . , equipped with the usual associative concatenation product: , with the empty word, 1, as neutral element, has a monoid structure. indicates with real the linear space of all formal power series coefficients . can be written as where is a finite linear combination of words of length . If has zero constant term (the coefficient of the and empty word is zero), we can define with . One easily and (see [1], [30]). verifies that can be equipped with the shuffle product, denoted by “ ” (e.g., [30]), a commutative product defined in a recursive way and for on the words any two words and of length and , respectively

When is equipped with the Lie bracket: a Lie monomial is a Lie bracket of letters (respectively, formal variables); a Lie polynomial is a finite sum with real coefficients of Lie monomials; a Lie series is an infinite sum of Lie polynomials and a Lie element denotes any of them. When a degree is assigned to each letter (respectively, formal variable), the notions of degree for monomials and homogeneity for polynomials are straightand forward. Given two noncommuting formal variables we indicate by the Baker-Campbell-Hausdorff exponent, i.e.

II. NONAUTONOMOUS DIFFERENTIAL EQUATIONS It is well known and not difficult to verify by making use of Taylor series arguments that the flow of an autonomous differenadmits the exponential representation tial equation

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. We may then explicitly characterize the underlying Lie group action or the additive one-parameter group: , we have for any

How to extend these results to a nonautonomous differential equation is the object of the present section. The main difficulty arises from the fact that the composition of vector fields do not commute. Consider again (1)

for , (analogously ). Its solution can be represented either by a series of multiple integrals, the right chronological exponential series [4] or by Taylor like expansions [3]. a bounded, analytic, uniRemark 1: Assuming as in [4], formly integrable vector field, or as in [21], analytic and comat , denoted by plete, the solution to (1) through , is well defined by the convergent series detailed in the sequel for sufficiently close to . As previously noted, convergence issues will not be addressed so that we make reference to asymptotic expansions of the solutions.

B. The Taylor’s Series Expansion On the other hand, the Taylor’s series expansion of the solution to (1) takes the form

with . Define now on the extended vector field . By interpreting time as a new dependent variable, we can set in the new independent variable . Consequently we can define, , so that (10) accordingly to [3], the Lie series operator can be rewritten as

(11)

with

A. The Chronological Exponential Series Consider as in [4], the integral form of (1) at

, i.e.

where , denotes the projection on and . Equating the solutions (10) and (11), we get

The following computations are instrumental to express each in (11) as an homogeneous polynomial of degree in the ( )s with real coefficients. Through multiple application of , we get for the first terms

(9) Substituting in (9), as follows:

with its integral form computed

we get

The next Proposition provides a first integrated form of is given. takes the form Proposition 2.1: The solution to (1) at

Substituting again the mapping with its integral form and iterating this procedure for the mappings for , we get the right chronological exponential which gives the asymptotic expansion of the solution

(10)

(12) where the asymptotic expansion in of

is given by

with (13)


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is an homogeneous polynomial of degree in the ), with real coeffi( )s (said by convention of degree and , we have cients. For

(14) Proof: It works by induction. Starting from so that , we set for

The corollary below describes an algorithm for iteratively com. puting the decomposition of each , we compute itCorollary 2.1: Starting from for according to eratively

where

and compute formally . By applying the derivation rule with respect to , to the composition of differential operators , we get

(15)

denotes a formal derivation acting over

as

with

when and for Proof: Taking into account that

.

and, thus the proof is an immediate consequence of (15). Working out the induction rule on the first terms, we get

We easily deduce that formula here holds true

and, for

, the recurrent

for C. The Exponential Representation Since, when

, the combinatoric equality

is verified then (14) follows. For the first terms, one has

In the sequel, we show that the flow admits an ordinary exponential representation whose exponent is described by with elements in , the Lie a formal power series in . The proof is based on algebra generated by the vector fields a result well known in different formats in the combinatoric literature (see, for example, [2], [30], [13]). with Theorem 2.1: A series is an exponential Lie series, that is is a Lie element, if and only if the coefficients satisfy the shuffle relations, i.e., where . In [25], we applied this result in the context of nonautonomous differential equations showing that, since the in (13) verify the coefficients

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shuffle relations, then the formal logarithmic expansion of the is a Lie element. More precisely flow

can be iteratively decomposed as a Lie • Each in the vector fields defining polynomial of degree the dynamics and it can be iteratively computed beginning according to the rule from (22)

where the coefficient is an homogeneous Lie polynomial of degree in its arguments. When no confusion is possible from the context, the dependency on its , of the , will be omitted. arguments, the operators Lemma 2.1 below reformulates this result. Slightly different formulations exist in the literature about chronological calculus (see, for example, [4], [13], [31], [22]). For completeness, combinatoric proofs of (17) and (18) are proposed in the Appendix. Lemma 2.1: Let the function be defined as

where the ( )s are the Bernoulli numbers ( for ) and denote by inverse, i.e.

its formal

Proof: Since by definition , (19) immediately follows from the formal inversion of the logarithmic function. According to these definitions, (21) is nothing else but a rewriting of (17) in Lemma 2.1. By definition and from (21) we compute

which proves that, by construction, is a Lie polynomial in the ( )s. Its decomposition as an homogeneous polynomial can be computed by equating term-by-term the series expansions in the left and right-hand sides of (21), i.e.

(16) , one has

If

(17) and reciprocally (18) The main Theorem can now be stated. Theorem 2.2: The flow associated with the nonautonomous differential equation (1) admits the exponential representation

with

so that

(19) • The exponent series series expansion

is defined by the Lie

(20) which satisfies the equality of formal series

(21)

Equating the terms of the same power in , we conclude the proof.


Equations (20) and (22) explicitly describe the series exponent. For the first terms, we get

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describes the one parameter Lie group structure associated with the nonautonomous vector field . Given and , the composition of flows reads (23) with equality of operators

and

. Rewriting (23) as the

with one deduces the group law “#” over the exponent so that # for

(24)

From (24), it follows that by applying the Baker-CampbellHausdorff formula ([1], [2], [15], [30], [29]) to

Expanding the computations, we get

and

with

Remark 2: It is immediately verified that when the fields commute, which is equivalent to , then and thus . provides an asymptotic Remark 3: For fixed and expansion of the vector field which defines the averaged dy. namics ([31], [32]) associated with over Remark 4: The role of higher order approximations in the study of stability by means of averaging techniques when commutation does not hold has been put in light in [31]. Our result can be considered instrumental in that context. Remark 5: (18) can be written as

so expressing the dynamics in terms of the series exponent . Remark 6: By performing the formal logarithmic expansion of (2) under the chronological product (see, for with example, in [22], ), one gets a series of Lie brackets of iterated integrals [14], [12], [27]. provides an integrated form of such a series with respect to time. Remark 7: The exponential mapping

, one has

Example 1: Let us consider the simple example of the one chained dynamics [32] (25) Setting in (25), values

and

with constant

, we get

which takes the form of a nonautonomous dynamics (26) with . By applying (20)–(21), the solution to (26) can be computed in terms of the exponential representation of its flow. At time we get (27) with and all the other Lie brackets of vector fields equal to zero. Expanding (27) in powers of , we immediately get

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(28)

for terval, (29) can be rewritten as

which in fact corresponds to the integrated form of (26) easily computable in this simple case.

. On this time in-

(33)

III. CONTROLLED DYNAMICS UNDER SAMPLING We are now in a position to investigate the effects of the derived expansions on analysis and design under sampling. To this end, with reference to an input-affine continuous-time dynamics (29) (30) we describe the sampled equivalent models under zero-order or higher order holding devices and give an insight about the possible use and potential utility of the proposed approach.

Remark 8: It is worth noting that, assuming , (33) describes the dynamics (29) under smooth control, preliminarily expanded according to its successive time-derivatives. replaced by Noticing that (33) is in the form of (1) with by for and by , Theorem 2.2 , initialized directly applies. The solution of (33) at time at , is given by (omitting the -dependency) (34) with

A. Piecewise Constant Control Consider first the simpler case of an input signal constant . Denote by the over small time intervals of length control value over the interval and by , the at time . As well known, the solution at value of of (29), starting from , takes the form time

(31) (31) defines the so-called sampled equivalent to (29). The exponential operator describes the flow , associated with the vector field over the in. Given an input sequence of length terval , the input-state behavior over time-steps takes the form of a composition of exponentials

(32) Equivalently, due to (8), we get for the input-output behavior

These formulas are useful to compute the expansions of the evolutions in powers of the control, possibly by means of symbolic manipulation tools. Volterra expansions and approximated control schemes are just two examples of application of these results from the theoretical and practical points of view, respectively. What is interesting to note is that, thanks to the introduced formalism, similar series representations can be obtained in the more general case of higher order control. B. Higher Order Control Let, over the time interval , the control signal take the form of a summation (possibly finite),

(35) The sampled equivalent to (33) is thus defined by the exponential form (34) and (35), so generalizing (31) to the present context of higher order control. with in (20), we compute Substituting the first terms in (35), i.e.


so that, restricting to

the exponent expansion, we get

Expanding the exponential (or integrating the chronological , we get series) with truncations in

Let us stress that the exponent series evidentiates in the equiva, lent sampled dynamics the vector fields the so-called controllability Lie brackets. Remark 9: Chronological series solutions are currently compared with Chen-Fliess series [2], [15], [5], which concern the chronological series associated with controlled input-affine dynamics expanded according to the successive iterated integrals in the control; our solution provides its integrated form with respect to time. C. An Insight About Control Design As well known, sampled equivalent models are used in the design of digital controllers [33]. The control law is computed on each time interval from measurements at the sampling instants for achieving prefixed performances. An improvement of the controller capability can be achieved by making use of multirate techniques or higher order devices. Multirate techniques consist in using different rates for the controller and the measurements; higher frequency on the controller results in a piecewise constant control with multiple changes over each sampling interval. Multirate and higher order control schemes are both characterized by the fact that, in practice, the number of control variables is increased with respect to usual schemes where the control variable is the constant input only. Such techniques, which complicate the design, have been recently developed and successfully applied in the nonlinear context too [34], [26]. The formulas here proposed can be profitably employed in the design of nonlinear controllers in these contexts. • Multirate digital control. Given (29), the th-order multirate sampled equivalent (the control changes -times during the sampling interval), takes the form of a map

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which can be written as the composition of exponentials as (32) with

Such a formula can be used for computing, possibly by means of formal manipulations, approximated sampled models. Example 1. (Continued)—Multirate Digital Control Over : Consider again the dynamics (25). We recall that such an example, even simple, is of peculiar interest because it cannot be steered under continuous-time smooth control. We illustrate hereafter that digital control designs, based either on multirate or higher order techniques, yield to exact state steering. The exact sampled equivalent to (25), under piecewise constant controls over time intervals of length , is described by the nonlinear difference equations

where indicates the constant value of the th component over the interval . On these bases, the second-order multirate sampled equivalent dynamics, over the holding the control constant and equal to first interval and to over the second interval, with , takes the form

An an example, to exactly steer to , we directly compute, assuming

in one step of length

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• Higher order control. A -order holding device generates, over each interval of length , a control of the form

(36) The equivalent sampled dynamics can be written as the exponential (34) and computed according to (35). Example 1. (Continued)—First-Order Holding Digital Conand in trol Over : Set is given by (28) (25). The sampled equivalent over with

s to be constant. It is also intuitively assumption on the clear that the direct design of a constant digital control on (38), which results in a discontinuous control of fixed order in of the form (36) on (29), should improve the overall performances of the control scheme since the dynamics (38), expanded as in (35), is explicitly described in terms of the controllability directions. The problem of investigating the links between (33) and (38) is thus set. How do the properties of (38) reflect in that of (33)? How do the performances of a control designed on (38) reflect on (33)? How to set a design procedure on (38) from requirements on (33)? In some sense, referring to Remark 2, the idea sketched here can be put in parallel to a control design based on averaged dynamics [32], [31]. IV. CONTROLLED DISCRETE-TIME DYNAMICS

Assuming

The purpose of this section is to put in light the profitable usage of the expansions derived in Section II in the study of nonlinear forced discrete-time dynamics. Such a parallel between two contexts apparently quite different is based on a new paradigm proposed in [20] for modeling, representing and thus defining, discrete-time dynamics in state-space form as coupled nonlinear differential/difference equations.

, the controller

A. The Differential/Difference Representation

achieves the same objective as before, steering the dynamics to . Let us underline that in one step of length from and . We note that in practice, approximated models of finite degree in ([35], [34], [26], [36]) and feedback laws designed over a finite number of additional control terms are sufficient to ensure significant improvements. We conclude this section by noticing that the exponential and , can be interpreted as the flow form (3), with associated with the dynamics (37) For each fixed , it is an autonomous differential equation satwhenever isfying the property that, at time . Moreover, in the context of higher order sampling devices, we deduce from (34)–(35) that the differential equation (37) takes the form (38) It results that under the higher order control the dynamics (29) (equivalently the time varying dynamics (33)), is equivalent under sampling to (38) over the sampling interval; under the constant control then at . This comprovided ment suggests to design the control on (38), even relaxing the

Let us first recall what we will refer to as the differential/ difference representation of a controlled discrete-time dynamics (DDR). Definition 4.1: Given a map and a vector field , analytically parameterized by around , a differential/difference single-input discrete-time dynamics is defined by (39) (40) can be expanded in powers of The vector field (or ), so getting

around

(41) with

for (analogously and ). Consider now the nonlinear difference equation described in the form of a map as it is usual in the literature (42)

The existence of an equivalent DDR follows from the existence on , parameterized by , satisfying the of a vector field equality (43) . and setting Reciprocally, starting from a discrete-time dynamics described by (39)–(40), a difference equation in the form of a map


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(42) can be recovered through the integration of (40) between and with initial condition set according to (39), i.e.

Remark 10: Given (42), the invertibility of (equivalently the existence of such that is invertible) as well as the existence of is sufficient to guarantee which results to be locally and uniquely defined as . Remark 11: Given a sequence of inputs , the state behavior over several time steps is obtained by the multiple application of the difference equation (39), which fixes the initial condition at each time instant, followed by the integration . More in details, starting of (40) over the control interval and the evalfrom , the integration of (40) between 0 and leads to the computation uation of the result at ; then, the integration of (40) between 0 and , of and the evaluation of the result at leads to ; iteratively the result folthe computation of lows.

is a Lie polynomial of degree in the • Each , from ( )s which can be iteratively computed for , according to

• Moreover, the solution to (39)–(40) admits the exponential representation

(47) Proof: The proof is the same as for Theorem 2.2 with the with (equivalently, with and substitution of with for ) and of with . For the first terms of the exponent, we find similarly

B. The Exponential Representation According to Definition 4.1, discrete-time dynamics can be thus seen as nonautonomous differential equations with respect to the input variable with jumps piloted by the drift term. The results stated in Theorem 2.2 can be applied by formally interchanging the role of and . Recalling that

the following results, which mimic those in Theorem 2.2, can be stated. Theorem 4.1: The flow associated with (40) admits the exponential representation (44) • The exponent series expansion

is defined by the Lie series

(45)

with

Remark 12: As in Remark 5, reversing (46), we find out the equality of series below

The next corollary specifies the results at and Corollary 4.1: For a given get

with and

for

. , we

.

which satisfies the equality C. Volterra Series Expansions

(46)

The introduction of the differential/difference representation simplifies the computation of the Volterra kernels characterizing the input-output expansions [9]–[11]. By expressing the output

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mapping as a functional depending on multiple products of inputs, we define the discrete-time Volterra series expansions (see, for example, [37] and [38], for details), i.e.

As an example, consider the DDR characterized by only one control vector field , i.e.

The associated integrated form is then

The functions describe the successive around . Thanks to the Volterra kernels of order DDR, the output mapping admits an exponential representation we iteratively too. In fact, from (47), starting from compute

In this case, the input-output mapping reduces to

with the Volterra kernels

where, thanks to the employed formalism

with by definition

Iteratively, after -time steps, we get

Remark 13: With respect to continuous-time dynamics, we note that discrete-time input-state and input-output mappings exhibit both multiple products of inputs either at different time instants or at the same time instant; consequently the Volterra kernels must be differentiated. This difficulty occurs even if the DDR is described by only one vector field (equivalently the differential equation (40) is not -depending) as illustrated above. In spite of this, thanks to the DDR formalism, we have shown in previous works (see [20]) that these dynamics are comparable to input-affine continuous-time ones if one refers to their structural and control properties. D. Sampled Systems as Discrete-Time Ones

These expressions can be expanded in the multiple products of inputs to describe the successive Volterra kernels so obtaining

Let us go back to Section III. It is interesting to note that the sampled equivalents (31) and (34) do admit representations of the form (39)–(40) since the drift term is al. ways invertible for sufficiently small , i.e., Hereafter, Theorem 4.1 is applied to compute the DDR of (31) and (34), respectively. 1) Piecewise Constant Control: Theorem 4.2: For a given sampling period , the sampled equivalent to (31) under piecewise constant control, denoted by , admits the differential/difference representation (48) (49)


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with

and keeping in mind that, according to (16)

(50) are described in terms of the vector fields The vector fields as follows:

(50) follows from (54) and (53). The expression of thus obtained by performing derivatives of (50); i.e.

can be

taking into account that

provides an expoRemark 14: The expression around the free evonential representation of the flow lution which can be used to compute approximated models in . Remark 15: Regarding the input-state or input-output behaviors over several time steps and the Volterra kernels computation, the former expressions can be used setting

.. .

. 2) Higher Order Control: Let us describe the DDR of (34). Theorem 4.3: For a fixed sampling period , the sampled s over a equivalent to (33) under constant values of the , adtime interval of length , denoted by mits the differential/difference representation (55)

Corollary 4.2: The integration of (49) between 0 and with the initial condition given by (48) gives back a discretetime dynamics in the form of a map over intervals of length . We have

(56)

(51) with

with

(52) (53) (57) Proof: Once (50) is proven, (51) and (52) follow directly. A proof of (53) is given in the Appendix. The vector field satisfying is uniquely defined by the equality (54)

given in (35). and We note that in this case, interpreted as a multi-input case [28], the DDR is described by the set of partial derivatives equations (56). The expressions (57) describe the controlled vector , as Lie polynomials in the s. fields

14


V. CONCLUSION It has been shown that the flow associated with the solution to a nonautonomous differential equation admits an ordinary exponential form with an iteratively computable Lie series exponent. The results have been applied to the discrete-time and sampled control contexts. The study has been presently detailed for single-input discrete-time dynamics to simplify the notations but it can be generalized to the multi-input case. The fact that the solutions to nonlinear continuous-time, discrete-time and sampled dynamics admit a unified formal exponential representation clarifies their analogies. The study provides an efficient and elegant tool for representing the evolutions of nonlinear hybrid dynamics mixing up continuous-time behaviors with jumps or switches. In practice, the approach, based on series expansions, makes the computation of approximated solutions possible. Along the same lines, it could be possible to set the study in the context of parameter dependent dynamics to better understand the behavior of singularly or regularly perturbed dynamics versus chaotic ones. From the design point of view, the exponential form of the exact or approximated sampled model of a controlled dynamics suggests to investigate new digital design procedures. APPENDIX Lemma 2.1: The following identities hold true

(18)

(19) Corollary 4.2: The following identity holds true: (63) be the formal algebra of all formal Proof: Let power series in the non commuting formal operators and equipped with the formal product “ .” Let be the operator over defined by , respectively, so that . Define by

Then,

and

Since follows that

commute and direct computations give

consists in all the formal series in

only, it

or, equivalently, tively. Setting

and

, respec-

we get the result expressed in (18) taking into account that [respectively, we get the result expressed in (19)]. Setting now, , we get the result expressed in . (63), that is ACKNOWLEDGMENT The authors would like to acknowledge the anonymous reviewers for their accurate reading of the draft and useful suggestions. REFERENCES [1] W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure and Appl. Math., vol. 7, pp. 649–673, 1954. [2] K. T. Chen, “Formal differential equations,” Annals of Math., vol. 73, pp. 110–133, 1961. [3] W. Gröbner, “Serie di Lie e Loro Applicazioni,” in Poliedro, Cremonese, Italy, 1973. [4] A. A. Agrachev and R. V. Gamkrelidze, “Exponential representations of flows and chronological calculus,” Math. USSR Sbornik, vol. 35, pp. 727–785, 1978. [5] M. Fliess, “Fonctionnelles causales non linéaires et indéterminées non commutatives,” Bull. Soc. Math. France, vol. 109, pp. 3–40, 1981. [6] H. J. Sussmann, “A Lie Volterra expansion for nonlinear systems,” Lecture Notes Cont. Inf. Sci., vol. 58, pp. 822–828, 1984. [7] R. W. Brockett, “Volterra series and geometric control,” Automatica, vol. 12, pp. 167–176, 1976. [8] K. Lesiak and A. J. Krener, “The existence and uniqueness of Volterra series for nonlinear systems,” IEEE Trans. Autom. Control, vol. 23, pp. 1090–1095, 1978. [9] W. J. Rugh, Nonlinear System Theory, the Volterra/Wiener Approach. Baltimore, MD: The Johns Hopkins Univ. Press, 1980, vol. 7. [10] I. W. Sandberg, “Volterra-like expansions for solutions of nonlinear integral equations and nonlinear differential equations,” IEEE Trans. Autom. Control, vol. 30, pp. 68–77, 1983. [11] A. Isidori, Nonlinear Control Systems. Berlin, Germany: SpringerVerlag, 1989. [12] H. J. Sussmann, “A product expansion of the Chen series,” in Theory and Applications of Nonlinear Control Systems, C. I. Byrnes and A. Lindquist, Eds. North Holland,: Elsevier, 1986, pp. 323–335. [13] M. Kawski and H. J. Sussmann, “Noncommutative power series and formal Lie algebraic techniques in control theory,” in Systems and Linear Algebra, U. Helmke, D. P. Wolters, and E. Zertz, Eds. Studgart, Germany: Springer, 1997, pp. 111–128. [14] A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, “Local invariants of smooth control systems,” Acta Applicandae Mathematicae, vol. 14, pp. 191–237, 1989. [15] M. Fliess and D. Normand-Cyrot, “Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intègrales itérées de K.T. Chen,” in Sèminaire de Probabilitès, J. Azema and M. Yor, Eds. : SpringerVerlag, 1980–1981, vol. 16, pp. 257–267, Lecture Notes Math.. [16] B. Jakubczyk and D. Normand-Cyrot, “Orbites de pseudo groupes de difféomorphismes et commandabilité des systèmes non linéaires en temps discret,” in C.R. Acad. Sci., Paris, France, 1984, vol. 298-I-11, pp. 257–260. [17] B. Jakubczyk and E. D. Sontag, “Controllability of nonlinear discrete time systems: A Lie algebraic approach,” SIAM J. Contr. Optim., vol. 28, pp. 1–33, 1990. [18] A. I. Tretyak, “Chronological calculus, high-order necessary conditions for optimality, and perturbation methods,” J. Dyn. Control Syst., vol. 4, pp. 77–126, 1998.


[19] M. Kawski, “Controllability via chronological calculus,” Proc. IEEECDC, vol. 3, pp. 2920–25, 1999. [20] S. Monaco and D. Normand-Cyrot, “Geometric properties of a class of nonlinear discrete-time dynamics,” in Proc. ECC-97, Brussels, Belgium, 1997. [21] R. V. Gamkrelidze, “Exponential representations solutions of ordinary differential equations,” Lecture Notes Math., vol. 703, 1979. [22] A. V. Sarychev, “Lie- and chronologico-algebraic tools for studying stability of time-varying systems,” Syst. Control Lett., vol. 43, pp. 59–76, 2001. [23] S. Monaco, D. Normand-Cyrot, and C. Califano, “Discrete-time versus hybrid systems,” in Proc. IEEE-CDC-03, Maui, HI, 2003, vol. 5, pp. 5203–5208. [24] P. Di Giamberardino, S. Monaco, and D. Normand-Cyrot, “A hybrid control scheme for maneuvering space multibody systems,” J. Guid. Control Dyn., vol. 23, no. 2, pp. 231–240, 2000. [25] S. Monaco and D. Normand-Cyrot, “A unifying representation for nonlinear discrete-time and sampled dynamics,” J. Math. Syst., Est. Cont., vol. 7, pp. 477–503, 1997, (1995), pp. 101–103 (summary). [26] S. Monaco and D. Normand-Cyrot, “Issues on nonlinear digital systems,” Europ. J. Contr., vol. 7, pp. 160–178, 2001. [27] J. Stefanovski and K. Trencevski, “Analytic solutions of systems,” J. Dyn. Control Syst., vol. 8, pp. 463–486, 2002. [28] S. Monaco, D. Normand-Cyrot, and C. Califano, “Exponential representations of multi-input nonlinear discrete-time dynamics,” in Proc. ACC-04, Boston, MA, 2004, pp. 4998–5003. [29] L. Saenz and R. Suarez, “A combinatorial approach to the generalized Baker Campbell Hausdorff Dynkin formula,” Syst. Control Lett., vol. 45, pp. 357–370, 2002. [30] R. Ree, “Lie elements and an algebra associated with shuffles,” Ann. Math., vol. 68, pp. 210–220, 1958. [31] A. V. Sarychev, “Stability criteria for time-periodic systems via highorder averaging techniques,” in Nonlinear Control in the Year 2000, Lecture Notes in Contr. Inf. Sci.. Berlin, Germany: Springer-Verlag, 2001, vol. 259-2, pp. 365–377. [32] S. S. Sastry, Nonlinear Systems, Analysis, Stability and Control. New York: Springer, 1999. [33] K. Aström and Wittenmark, Computer Controlled Systems. Englewood Cliffs, NJ: Prentice-Hall, 1984. [34] J.-P. Barbot, S. Monaco, and D. Normand-Cyrot, “A sampled normal form for feedback linearization,” Math. Control Signal Syst., vol. 9, pp. 162–188, 1996. [35] J. W. Grizzle and P. V. Kokotovic, “Feedback linearization of sampleddata systems,” IEEE Trans. Autom. Control, vol. 33, pp. 857–859, 1988. [36] D. Nesic and A. R. Teel, “A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,” IEEE Trans. Autom. Control, vol. 49, pp. 1103–1034, 2004. [37] S. Monaco and D. Normand-Cyrot, “Finite Volterra series realizations and input output approximations of nonlinear discrete-time systems,” Int. J. Control, vol. 45-5, pp. 1771–1787, 1987. [38] S. Monaco and D. Normand-Cyrot, “Functional expansions for nonlinear discrete-time systems,” Math. Syst. Theory, vol. 21, pp. 235–254, 1989.

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Salvatore Monaco (SM’96–F’02) received the M.S. degree in electrical engineering in 1974 from he University of Rome “La Sapienza,” Italy. He was a Research Fellow in 1976, an Associate Professor in 1983, and now a Full Professor in Systems Theory, since 1986, with the University of Rome “La Sapienza,” Italy. Since 1981, he held several visiting positions with the Centre National de la Recherche Scientifique and University of Paris-Sud, France. His research activity is in the field of nonlinear systems and control theory, discrete-time and sampled-data systems, and applied research in aerospace and spacecraft control. Professor Monaco received the Outstanding Paper Award in the IEEE TRANSACTIONS ON AUTOMATIC CONTROL (1981–1982). He served as Member of the Councils of several Research Institutions, such as the University of l’Aquila, the Italian Space Agency (ASI), the European Union Control Association (EUCA), the Joint Research Center of the European Union, and the “Università Italo-Francese.”

Dorothée Normand-Cyrot (SM’96–F’05) received the Ph.D. degree in mathematics from the University of Paris VII, France, in 1978, and the Thèse d’Etat es-Sciences in physics from the University Paris-Sud in 1983. She was a Researcher with the Department of Electricité de France company from 1978 to 1980. In 1981, she joined the Laboratoire des Signaux et Systèmes of the Centre National de la Recherche Scientifique (CNRS), where she is currently a Senior Researcher since 1991. She has held Visiting Professor positions with the University of Rome “La Sapienza” and l’Aquila, Italy, in 1986. Her research activity is in the field of nonlinear systems and control theory, discrete-time and sampled data systems, digital control and its applications in electrical drives, aeronautics and spacecraft control, and robotics. She is an author and coauthor of about 125 refereed publications. Dr. Normand-Cyrot has served the European Union Control Association (EUCA), since 1991, as well as other control societies and journals in several capacities. She is currently Editor at Large of the European Journal of Control(EJC) and a Member of the Executive Council of the EUCA.

Claudia Califano received the Ph.D. degree in systems engineering from the University of Rome “La Sapienza,” Rome, Italy, in 1998. In 1999, she was awarded a CNR-NATO Fellowship and a CNR grant. Between 1999 and 2000 , she held a Postdoctoral position with the CNRS Laboratoire des Signaux et Systèmes, Gif sur Yvette, France. Since 2000, she has been an Assistant Professor with the Department of Computer and System Science (DIS), University of Rome “La Sapienza.” Her main research interests concern nonlinear systems in the discrete-time and continuous-time domain.