ابزارهای پیشرفته برای تجزیه و تحلیل و کنترل سیستم های نمونه برداری شده-داده غیر خطی

It is shown that the formalism of asymptotic series expansions recently developed by the authors for computing the solutions of non autonomous differential equations, can be profitably employed to obtain the equivalent model to a nonlinear continuous system under generalized sampling procedures. Tools and insights for the design of sampled-data control systems are derived.

Several reasons motivate the renewed interest in sampled-data systems, among them:  the highly demanding performances for computer controlled systems;  the remarkable technological advances which push forward the limits of applications and make real theoretical research transfer;  the fact that sampled-data systems are inherent components of hybrid processes and embedded systems: pervasive technologies which are present in the most challenging research programmes of the international scientific community. In the attempt to stress a rather schematic classification of the methodologies, even though too simplistic for specialists, let us introduce the different ways to attack sampled-data control problems. These comments, valid in general, will essentially focus on nonlinear systems. The continual evolution of digital technologies in terms of speed, costs and precision is at the basis of the increasing diffusion of digital control. A control system is, in this context, characterized by the digital computation of a control law from sampled measures performed on the given continuous plant. Two kinds of phenomena are therefore present : sampling of the measures and restoring of the continuous-time inputs of the plant; analog-to-digital and digital-to-analog conversions of the signals involved in the computation of the control law. The designer of a digital control scheme must therefore take into account, from the very beginning, problems linked to sampling, continuous- time restoring and quantization. To study the effects of different sampling devices over the analysis and control properties of a given continuous-time dynamics is thus of major interest. The questions are: what about the properties of a given, open loop or closed loop, continuous-time system under sampling? Is it possible to preserve/to improve the performances of the control system? What about the behaviours in the inter-sample? Digital control design should answer these purposes.The difficulties, already present in the linear context, give rise to increasing complexity in a nonlinear setting. To cite a few we note the appearance of critical zero dynamics and the lost of control properties like non-interaction or feedback linearization. As rather usual in the digital control literature, this paper is concerned only with sampling and continuoustime restoring. Quantization effects, often represented by an additive noise with statistical characteristics depending on the device, raise up nowadays a renewed interest in the context of hybrid systems. These aspects, investigated in [2], will not be discussed in this paper which is focused on sampled-data control systems. Sampling and holding are assumed to be synchronized and the sampled equivalent model of the plant represents the link between the input to the holder device and the state or output sampled values of the plant. As far as the continuous-time reconstruction of the signals is concerned, we make use of zero-order holding devices ‘usual sampling’ or higher-order and/or multirate holding devices, ‘generalized sampling’. As well known, the following three approaches, extensively developed in the linear context starting from the early 1950s, see [6], can be pursued for the design of digital controllers.  Continuous-time design (CTD) The continuoustime controller is designed and then implemented in discrete time. When dealing with static feedback on a nonlinear plant, the most common approach consists in the direct implementation through zero-order holder of the continuous control computed at the sampling instants; this procedure is denoted as ‘emulation’. To compensate the effects of sampling and holding devices, a modified continuous model plant or a modified design procedure can be used, ‘indirect digital control’ or ‘redesign methods’. First results in the nonlinear context in these lines are in [11,24, 25,27,35,42] where some new insights are given. Emulation and redesign techniques have been extensively employed in the more recent nonlinear literature. The investigations are usually confined to understand up to what extent the continuous-time performances are maintained by the resulting sampleddata control scheme and to pursue a quantitative analysis for guaranteeing a safe sampling range. Referring to the sampling period length, the question is: up to what extent is it possible to guarantee, through emulated control, the performances of a continuous-time control design? The most accurate quantitative analysis on the limits of emulation make use of tools and methods developed in the area of robust control Lyapunov’s type techniques in the wide framework of input-to-state-stability [23,36,53,54,56,61].  Discrete-time design (DTD) The controller is directly designed on the exact or approximate equivalent sampled model of the plant. This appears to be the natural framework for studying digital control but, in the nonlinear context, difficult problems must be faced : in general the usual sampled model does not admit a closed form representation ([41,62]), and the problem is still more complicated if not standard holding devices are used; properties of the plant which are relevant for the controller design may be lost under sampling; only a few number of design procedures involving rather complex computational aspects are available in discrete time. To quote a few, among the most traditional drawbacks of sampled models, put in light in a linear context too, let us recall that : minimum phase property is lost under usual sampling with the appearance of critical sampling zeroes [5,44,65]; the sampling period length is a crucial parameter of sampled-data systems [57,60]; structural or control properties can be lost depending on the sampling procedure and/or the order of approximation performed in the computation [7]. All these aspects request ad hoc analysis and control methodologies as well as non standard sampling procedures and devices [11,20,45,46,48,50]. Once a sampled equivalent model is assumed available, its practical computation being discussed lateron, direct digital and discrete-time control can be merged. Notable progresses have been done over the last years about nonlinear discrete-time control theory pursuing various different approaches either algebraic or geometric ones in the lines of consolidated mathematical frameworks set in the continuous-time theory. Results regarding controllability and observability, invariance, decoupling, regulation and observer design, are proposed and confronted with realistic examples. Nowadays, we can say that the available knowledge about discrete-time control theory in a nonlinear context covers a large variety of problems satisfactorily worked out on examples (see [3,4,7,8,12,16,19,29,33,34,36–39,43,47,64]). To conclude about this aspect let us say that: significant progresses have been made regarding nonlinear discrete-time control theory over the last years so providing techniques and tools for solving nonlinear discrete-time control problems; important problems remain the choice of the sampling procedure, the computation of the sampled equivalent model and/or the accuracy ot the approximated sampled model also in relation with the properties that must be maintained. In spite of this, exact sampled models are still difficult to compute so that the Euler approximation is the most popular sampled model overwhich digital control is designed. For such an approximate model, quantitative analysis regarding the sampling period length are available. However, as control design over Euler approximations restitutes emulated control, this restricts its interest as a discrete-time design approach.  Sampled data design (SDD) The controller is still designed on the discrete-time model of the plant, but now strictly taking into account two major aspects: the discrete-time model is issued from sampling and the variables under control are continuous-time ones. These aspects reflect in setting suitable performances on the behaviours not only at the sampling instants [9,10,17,20]. As a matter of fact, discrete-time dynamics issued from sampling do exhibit specific properties which can be used to get more accurate results. Two general questions which can be posed in this context are: up to what extent is it possible to reproduce a given continuous-time controlled behaviour? Up to what extent is it possible to guarantee prefixed performances on the continuous-time output? The first question let one understand the slight difference between SDD and CTD approaches; such a difference remaining confined to the use of the sampled model of the plant or not. On the other side, the second question suggests the possibility to achieve request which cannot be set in continuous time as for example the equivalent of the well known flat response in a linear digital control context. Arguing so, digital control solutions are designed around the nominal continuous-time solution, if it exists, or are described as series expansions in powers of the sampling period when a smooth control solution does not exist. Several contributions attack the problem (see [50] and the references therein). Even if, pursuing a formal analysis, the existence of such solutions can be proven and a procedure to iteratively compute the successive terms of their expansions is available, let us say that in practice, approximate solutions containing just a few, of course finite, number of terms, is implemented. Satisfactorily results are obtained and illustrated for different control requests on several case studies up to real implementations (e.g. [12,14]). The design of sampled-data controller achieving input-state matching [22] can be assumed belonging to such a context. Inspite of all this, a parodox stands in the fact that, while the wide theoretical research effort in nonlinear control during the early 1980s, [28], stimulated to work out adequate digital solutions, the increasing performances of computers and interface devices in terms of computational power, speed and precision, suggested, at the opposite, to make use of emulation with appropriate sampling frequency. This induced to think that there were no real need for ad hoc sampleddata design methods: fast sampling and emulated design suffice. Major objections to such a point of view can be raised:  fast sampling interacts with quantization and induces complex nefast effects which degradate the performances;  solutions based on redesign, DTD and SDD, are more efficient;  sampled-data controllers can also be used to solve continuous-time control problems which do not admit standard solutions. The contributions in the Minitutorial: ‘Advances in Nonlinear Sampled-Data Systems’, cover some important differents aspects recalled in this introduction. The effects of quantization are addressed by P. Albertos, M. Valle´s and A. Crespo in [2]; the lost of the minimum phase property under sampling is addressed by G.C. Goodwin, J.I. Yuz and M.E. Salgado in [18]; the state reconstruction problem is addressed for nonlinear discrete-time systems by Mi. Xiao, N. Kazantzis and C. Kravaris in [63] and in the sampled-data context by S. Diop in [15]. Coming back to our contribution, it should be clear from the previous discussion, that digital control methods, from emulation of continuous controllers to sampled data design, require the knowledge of the discrete-time model of the plant. The paper goal is to show how series expansions associated to formal integration can be profitably used to handle generalized sampling procedures. The proposed analysis is formal and convergence issues are not explicitely addressed. Eventhough the control solutions are described by their asymptotic expansions in powers of the sampling period, it is often sufficient in practice to consider only a finite number of terms to get efficient solutions. The method is constructive. Being the approach essentially based on formal series manipulations, combinatoric properties and tools, it is interesting to interpret sampling and integration as two formal inverse procedures. At the same time, sampling and inverse sampling are shown to be instrumental for reproducing under sampled-data control the behaviours of a continuous-time controller and vice versa, under continuous-time control the performances of a digital one. In-between, we can propose digital solutions when smooth continuous solutions do not exist. To conclude, we note the possible usefulness of this unified formalism in the context of hybrid behaviours mixing up continuous-time and discrete evolutions with switchs and resets. The paper is organized as follows. The introduction ends with some notations. Section 2 is devoted to generalized sampling and inverse sampling. The asymptotic expansion characterizing the flow associated with a nonautonomous differential equation representing a nonlinear system under higherorder holder device is described. Vice versa, the inverse sampling problem is formally solved for a class of parameterized maps. In both cases, an iterative procedure to compute the series expansions are described so providing approximate solutions which can be used in practice. This Section is essentially referred to the results proposed in [51]. Section 3 illustrates the use of these results regarding the problem of input-state matching of a given continuoustime dynamics by a sampled-data control and vice versa (a preliminary work [52]). A procedure for computing the control laws, possibly through finite order approximations, is given. Section 4 extends the results of Section 3 to multirate techniques. Notations- Throughout the paper x 2 Rn, t 2 R, u 2 U, a neighborhood of 0 in R. All the objects, maps, vector fields, control systems are analytic on their domains of definition, that is infinitely differentiable and admitting convergent Taylor series expansions. The vector fields are complete, that is the associated flow is defined at any time and for any initial condition. All over the paper, identities must be interpreted in the sense of asymptotic series. Given a generic map its evaluation at a point x is denoted either by ‘(x)’ or ‘j x’. Given a function Rn ! R and a vector field  on Rn, the differential operator L acts on  as L:¼ @ @x  so expressing the Lie derivative of  along . The repeated use of this Lie derivative gives for p>0, Lp :¼ @Lp1   @x  with Lp :¼ L      L; p– times and L0 ¼ 1, the identity operator, while Id indicates the identity map on Rn:eL ¼ 1 þ P p1 Lp  p! denotes the exponential series associated with L denoted as e for notational convenience. When no confusion is possible, the composition of Lie derivatives Li1      Lil associated with the vector fields i is denoted by i1      il . Given another vector field , [,] denotes the usual Lie bracket of vector fields; the following equality holds L½,  ¼ ðL  L L  LÞ; we note ad0  ðÞ:¼  and for p>0, adpþ1  ðÞ:¼ ½, adp  ðÞ. The evaluation at a point x of an operator or a composition of operators applied to the identity function, LðIdÞ  x is denoted by (x) when no confusion is possible; identically LðIdÞj eðxÞ:¼ ðeðxÞÞ. Given a set of formal variables, a Lie monomial is a Lie bracket of them; i.e. a monomial in these variables with respect to the Lie bracket operation. Analogously, 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 formal variable, the notions of degree for monomials and homogeneity for polynomials are straightforward.

Starting from the affirmation that digital control methods from emulation to sampled-data design require the accurate knowledge of discrete-time models, we clarified how some recent results developed in the context of formal integration, are at the basis of the sampling process and are instrumental for the design of advanced digital control procedures. Eventhough the control solutions are described by their asymptotic expansions in powers of the sampling period, the method is constructive and it is often sufficient in practice to take into account only a finite number of terms to get efficient solutions. The problems of reproducing under digital control the behaviour of a continuous-time controller and vice versa under continuous-time control, the performances of a digital one, have been respectively investigated. On these bases and arguing as in [50], it is possible to make use of sampled-data control design techniques to set discontinuous controllers when smooth solutions do not exist. Work is progressing in this direction. Finally, we note the usefulness of such a formulation in continuous-time, discrete-time and sampled data contexts versus the hybrid one.