MODERN technologies are increasingly demanding

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH n-bit Stabilization of n-dimensional Nonlinear Systems in Feedforward Form Claudio De Persis, Member, IEEE Abstract A methodology is presented which allows to design encoder, decoder controller for stabilizing a nonlinear system in feedforward form using saturated encoded state feedback basically under stard assumption, namely local Lipschitz property of the vector field defining the system. (respectively, +1) bits are used to encode the state information needed to the purpose of semiglobally (globally) stabilizing an -dimensional system. Minimality of the data rate is discussed. Index Terms Communication, decoders, encoders, Lebesgue sampling, nonlinear control systems, quantization, small inputs, smart actuators, smart sensors, stabilization. I. INTRODUCTION MODERN technologies are increasingly deming expertise in analyzing ultimately controlling large-scale complex systems comprised by a large number of sub-units which exchange information. This information typically travels through finite data rate communication channels. If the purpose is to control one or more of these sub-units, then a typical situation which may arise is that measurements taken by sensors in one place may be needed by a controlling device at a remote location (see [1] [18] the references therein). Furthermore, complex systems usually exhibit behaviors which are better described by nonlinear dynamics. Controlling or, more specifically, stabilizing a nonlinear system by feedback information which travels through finite data rate information channel is precisely the problem considered in this paper. For reasons which will be clear later on, we shall refer to the feedback information transmitted through a communication channel by the term encoded feedback [4]. We define now the main feature of the channel considered in this paper: It allows to send a packet of bits at times with a known constant. Moreover, in order to avoid cumbersome complexities at this stage, the channel will be assumed noise-less delay-free. Hence, any packet of bits which is transmitted at one end of the channel is immediately received at the other end without undergoing any modification. Note that the controller receives the feedback information at times which are not uniformly spaced, a situation which is clearly related to the concept of Lebesgue sampling discussed in [19]. Manuscript received June 8, 2004; revised November 11, Recommended by Associate Editor L. Magni. The author is with the Dipartimento di Informatica e Sistemistica, Università di Roma La Sapienza, Rome, Italy ( depersis@dis.uniroma1.it). Digital Object Identifier /TAC There are other papers which have approached the problem of stabilizing a nonlinear system by encoded feedback. In particular, Liberzon in a series of recent papers on the subject initiated with [6] continued in [17] [18] has thoroughly investigated the problem for those nonlinear systems which can be made input-to-state stable (ISS) with respect to measurement errors. Among other things, he provided a value for the data rate of the communication channel (or, equivalently, on the number of bits) needed to guarantee the solvability of the stabilization problem, found out that the value depends on the size of the subset of the state space to which the initial condition of the system belongs on the sampling period (assuming uniformly spaced sampling). Unfortunately, the hypothesis of input-to-state stability severely restricts the class of systems to which the result applies. To mitigate this limitation, the author Isidori have shown in [20] that, if the data rate is allowed to take a different value than that in [18], then any nonlinear control system which can be (semi-)globally asymptotically stabilized by stard (i.e., with no encoding) feedback can also be (semi-)globally asymptotically stabilized by encoded feedback. The issue of minimality of the data rate, on the other h, has been recently addressed in [21], where the authors have elegantly characterized the minimal data rate to achieve set invariance for topological dynamical systems. For discrete-time systems defined on the Euclidean space, they have shown the minimal data rate needed to guarantee local uniform asymptotic stabilizability. The aim of this paper is to show that, if the nonlinear system to control has a special upper triangular structure (that is, the system falls in the class of the celebrated feedforward systems of [22] [24]), then asymptotic stability by encoded feedback can be guaranteed while providing some additional interesting features. First of all, asymptotic stabilization can be achieved using small inputs. Moreover, the prescribed data rate is independent of the set of initial conditions of the system, of the time-varying sampling period (a property which makes the overall control system potentially robust with respect to phenomena such as loss of packets congestion in the channel), more specifically, it can be simply assessed from the dimension of the system (denoted by this dimension, bits are needed for global stabilization for semiglobal stabilization). Additionally, it is worth stressing that the data rate we employ to achieve our semiglobal result equals the data rate which one would obtain by using for the Euler discretization of (1) the method proposed in [21] to achieve local uniform stabilization, can be shown to yield an asymptotic average data rate arbitrarily close to the /$ IEEE

2 300 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 infimum one (see the first remark after Proposition 3). Finally, if the set to which the initial condition of the system belongs is unknown, the special structure of the system allows to remedy this lack of information efficiently provided that a number of maximization problems can be solved (see Section II). The systems under consideration in this paper take the form [25]. (1) where, having denoted by the set of state variables. The following is assumed for the s. Assumption 1: Functions, with, are locally Lipschitz. Remark: As in [25], rather than considering the -subsystem described by we are dropping for the dependence on. For those systems for which depends on, the results in this paper hold only locally may employ a number of bits larger than, whose exact value can be determined by applying the results of [20] to the -subsystem. Remark: The models of many mechanical systems are of the form considered here. Furthermore, feedforward systems appear as sub-components of more complex objects. We also point out that the class of systems for which the results of this paper hold can actually be enlarged to include the class of the so-called block feedforward systems, for which with, for,,, are neutrally stable matrices of appropriate dimensions. However, for the sake of simplicity, this extension is not explicitly pursued here. Our paper is inspired by the idea of [11], where the (Jordan) structure of the system was exploited to better analyze the system. However, those works are mainly concerned with linear discrete-time systems, while in this paper we consider genuinely nonlinear continuous-time systems in feedforward form. Hence, the techniques used in our paper are completely different by those adopted in [11]. Although nonlinear feedforward systems represent a very active research field in the control community a large amount of results is available, to our purpose we shall only make use of an -to- stability result established in [25]. In a nutshell, our approach is as follows. The decoder reconstructs the state of the system to be used by the controller. starting from the information transmitted by the encoder. We assume that, for some index, an estimate of the components of the state is provided by the decoder, we elaborate the results of [25] to tune the parameters, of a nested saturated controller (see Section IV) which stabilizes the components of (1), robustly with respect to the encoding (estimation) errors, so as to guarantee the boundedness of, we explicitly calculate the bound. Exploiting the upper triangular structure of the system to control, such a bound is used to design the additional components of the encoder the decoder which are able to produce an estimate of using only 1 bit. Pointing out that the encoder the decoder can always provide a reliable estimate of the component of the state, iterating the procedure outlined above times, the entire encoder, decoder controller are obtained. Notice also that, if we were able to design the encoder the decoder without relying on the boundedness property for the state (as it is the case for systems with a globally Lipschitz vector field cf. comments at the end of Section III), then the stabilization result would be a straightforward application of the results in [22]. The need for the knowledge of the bounds on the components of the state forced us to conceive the iterated procedure in which the design of the encoder/decoder the design of the controller are closely intertwined. This iterative design of the overall scheme for stabilizing nonlinear systems under data rate constraints represents the main contribution of our paper allows us to overcome some of the drawbacks found in other approaches to the problem, such as the use of large (or not minimal) data rate (as in [18] [20]), restrictive assumptions, local results (as in [21]). In fact, our approach leads to an alternative way of designing encoders decoders which use a minimal data rate even for linear discrete-time systems with an upper-triangular structure with no need to determine the Jordan form [11], [26]. In Section II, we examine the case in which an estimate of the size of the set of initial conditions is unknown. We implement a localization procedure ( zooming out procedure in [6] [17]) which allows us to locate a region of the state space where the state lies in finite time using bits. The feedforward structure of the system makes it possible to give an explicit characterization of the time needed to locate such a region. In Section III, we illustrate the iterative design of the encoder the decoder under the assumption that finite bounds on the state components of (1) are known. These bounds can be explicitly computed by an appropriate design of the parameters of the nested saturated control. This is carried out in Section IV. Hence, the overall design of the encoder, decoder controller which semiglobally (respectively, globally) asymptotically stabilize (1) using (respectively, ) bits is obtained by combining the results of Sections III IV (respectively, Sections II IV). II. LOCALIZATION STAGE As we will see in the next section, all the methods which will be presented in this paper are based on the fact that at a certain time a parallelepiped (the quantization region see [11])

3 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 301 is known to which the state of the system belongs. Even though we are assuming that sensors are providing measurements of the state of the system therefore this is surely known to the encoder because of the physical separation between sensors controller the state is in general unknown to the decoder. Two cases are possible: The decoder knows a compact subset of the state space to which the initial state of the system belongs or it does not. In the former case, we can assume, without loss of generality, a range vector is known to the decoder for which the state belongs to the quantization region whose centroid is the origin in whose edges have length given by the entries of. This implies that at time no overflow is occurring. In the latter case, on the other h, the existence of such a vector can not be guaranteed any longer overflow becomes possible. The aim of this section is to show that, nevertheless, there exists a way to encode the information so that the decoder can determine in a finite time a quantization region to which the state belongs, provided that bits are available for encoding. This is a preliminary step toward the global asymptotic estimation of the state of the process from encoded feedback information ultimately toward the global asymptotic stabilization by encoded feedback. Remark: We shall see in the next section that our design of encoders decoders will result in no overflow for all the time, provided that no overflow is occurring at time. Hence, if the compact set of initial conditions is known to the decoder, overflow will never occur bits suffice to encode the information regarding the location of the state within the quantization region (see Sections III IV). On the other h, if such a set is unknown, then overflow may occur in this case an additional bit will be needed to distinguish the two situations of overflow no overflow (see later). The program of determining in finite time a quantization region to which the state belongs may be pursued by exploiting the feedforward form of the system. Since no a priori information is available to the decoder concerning the initial state, the encoder at time takes the centroid equal to the origin the range vector equal to some vector [9], [11], [12]. The same does the decoder. If the state happens to belong to the quantization region just defined, then our search ends with. In particular, the encoder sets the th bit to 1 to signal the decoder that no overflow is occurring, it uses the remaining bits to specify which sub-region of the quantization region the state belongs to. The specific method adopted by the encoder to code this information will be explained in the next section. Otherwise, the encoder transmits bits all set to 0, thus denoting overflow. At this point, both encoder decoder the latter upon reception of the all-0 packet of bits know that lies outside the quantization region which therefore must be exped. To this purpose it is set. The following arguments are inspired by the well-known fact that system (1) with is forward complete. Now, for all the times, therefore, there exist an update law for, the th component of the range vector, such as with, a nonnegative integer such that for all for all. At time, the encoder sets the th bit to 1 to signal the decoder an estimate of is now available. The decoder implements the same equation (2). Hence, when the decoder receives at time the packet of bits where only the th is different from zero, it can promptly derive that for all for all, that is it can determine a bound on. In the sequel, we will derive update laws for each used by the encoder. The decoder will implement exactly the same equations, so that, when a special sequence of bits is received, it will be able to derive a bound on each. For the sake of conciseness, the equations for the decoder will not be explicitly given. An estimate of can be computed as well, since we have for all, where Note that this estimate is available only for, as only at time the number the maximum in (4) can actually be computed. As a consequence, there exist an update law for, namely a nonnegative integer such that for each for all. At time the encoder sets the th bit to 1 to signal that an estimate of has become available. A straightforward exercise shows that this argument can be iterated. In particular, we have that, for each, for each, for all (3) (4) (5) (6) (2)

4 302 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 having set which is available for law for. a nonnegative integer (7). Hence, there exist an update such that for all for all. Hence, at time, with (8) (9) that the state belongs to the quantization region with centroid the origin in range vector. In fact, we have already discussed that this can happen because either a compact set to which the initial condition belongs is known (in which case is the vector whose entries equal the length of the edges of the parallelepiped centered around the origin containing the set to which the initial condition belongs) or such a compact set is unknown but, by the localization procedure examined in Section II, have been determined. The former case is explicitly dealt with in this section in the next one, the semiglobal results we obtain are intended to hold under the hypothesis that bits are available for encoding. The latter case, which is not explicitly dealt with for the sake of concision, can be analogously tackled (by keeping in mind the results of Section II easily adapting the arguments adopted for the former case) to obtain global results, provided that bits are used for encoding. We now describe in more detail the functioning of the encoder the decoder from time on. In the sequel, it will be convenient to consider at each time step not only the entire quantization region but also its projections onto the subspaces (10) a fixed real number, the state lies within the quantization region. At this point, the encoder sets the th bit to 1 to signal that no overflow is occurring any longer it uses the remaining bits to specify the sub-region to which the state belongs. As mentioned above, the precise way in which the encoder performs this will be explained in the next section. We summarize with the following. Lemma 1: Under Assumption 1, if bits are available to encode information, then there exists a finite time, with given by (10), at which the state of process (1) is enclosed in the parallelepiped with centroid the origin edges given by the range vector (11) where the s are defined through (2) (5), (7), (8). Remark: From the arguments preceding the lemma, it can be seen that belongs to the parallelepiped with centroid the origin range vector for all. Remark: By (10), it is seen that in the case in which lies within the quantization region whose centroid is the origin the edges are given by the range vector. Remark: Despite of what happens for a general nonlinear system (cf., e.g., [18]), the feedforward form of the system allows to find an explicit characterization of the time required to determine the quantization region in which the state of the process belongs. Of course, the applicability of the method proposed here relies on the possibility to efficiently solve the maximization problems. III. ITERATIVE DESIGN OF ENCODERS AND DECODERS We have seen that there is no loss of generality in assuming the existence of a finite time a vector such We also set. Let be an index belonging to the set. At each time, for, the encoder constructs the region defined as the Cartesian product of segments, with, i.e., where having denoted by the center of the segment by its length. The region is defined as the quantization region at time, whereas the regions, for, are its projections onto. Remark: The reason for us to introduce, not only the quantization region, but also its projections onto the subspaces, lies in our iterative approach to the design of the encoder/decoder the controller. In fact, the functioning of the encoder obeys a set of differential/difference equations we start designing the th differential/difference equation [see (13), with, (14)]. This differential/difference equation lives in the subspace define the region in such a way that, no matter what the control input is, the component of the state is guaranteed to always belong to an asymptotic estimate of is provided by the th differential/difference equation we implemented. The results of [25] are then elaborated in Section IV to show the existence of a controller fed by which guarantees boundedness of. On the basis of this result, we can proceed to: Design the th differential/difference equation of the encoder which, together with the th equation of the encoder, lives in the subspace define the region ; Prove that the sub-component of the state always

5 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 303 belongs to, hence, prove that an asymptotic estimate of is generated by the th differential/difference equation we implemented. Again, an appropriate design of the controller, fed by, guarantees boundedness of. This procedure is iterated until each equation of the encoder/decoder the entire controller are designed (see Propositions 1 2). Having constructed, each, for, is divided into two parts [16] which results in a partition of the region into sub-regions. Assume that at all the times, with, the state of the system belongs to (Lemma 3 Proposition 1 will show that this results in no loss of generality). By knowing, the encoder can determine which one of the sub-regions in belongs to therefore the centroid of the sub-region. In particular, the encoder calculates [16] the numbers 1 sign with, resulting in for each, if if (12) is updated in the fol- First of all, the length of the segment lowing way: (14) where is the th entry of the vector introduced at the beginning of the section. Now, let, be suitable positive constants (the values for these constants will be made explicit in Section IV). Having specified the update law for, bearing in mind Assumption 1, we can set introduce the constant for which for each pair for each satisfying Recursively, for laws (15), we introduce the update The packet of bits, which is the binary representation of, is taken equal to the following sequence of 0 s 1 s: where, for if if. (16) where is the th entry of the vector introduced at the beginning of the section, set We explicitly notice that only the symbol is actually sent through the channel. We are left with specifying the equations which define the center the range of each segment hence define the region. Henceforth, these equations introduced later are intended to be defined over their maximal interval of existence. For each, the center of is taken equal to [18], where is derived from the solution of the equation define the constant for each pair satisfying 2 for which for each (13) with initial condition. The equations for the range of the segments are introduced in a recursive fashion as follows. 1 Setting b (t )=1if x (t ) 0 C (t )=0is arbitrary. One might as well decide to set b (t )=0. The choice only affects the values taken by the vector ^X (t ) by the symbol s (t ) defined below, but it does not affect the results stated in the remaining of this paper. Remark: If the bound is not available, one can replace it with the difference, with, the passages below will still be valid. Of course, this requires encoders decoders to be endowed with a timer, as it was assumed in the previous section. We observe that the parameters used to introduce the Lipschitz constants are welldefined. This is a consequence of the following straightforward 2 We observe that constants F, with j =1; 2;...;n0 i 0 1, will depend on ` ;...;`, Z ;...;Z, F ;...;F, U T.

6 304 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 result concerning the asymptotic behavior of vector. Lemma 2: Vector solution of (14) (16) globally exponentially converges to zero. Proof: It is enough to notice that system (14), (16) admits an upper triangular representation (17) where is an matrix whose entries are defined as follows: For,, for (18). This is a linear time-invariant discrete-time system whose dynamic matrix has all the eigenvalues contained in the unitary disc of the complex plane. Hence, globally exponentially converges to zero as. Remark: It should be kept in mind that the exponential convergence to the origin of is preserved even in the case in which is replaced by, with, provided that for each. Suppose now that the set of bits is available at the other end of the channel. For each, recalling (12) that it is seen that can be exactly reconstructed by the decoder from provided that are known to the decoder. This is actually the case provided that the decoder implements the equations by the decoder from the symbol. This implies that therefore for all. This ends the proof of the claim. As for the encoder, (19) (21) define the region at each time step. Indeed, for each, point range define the segment. The Cartesian product denotes the region. We introduce now the following assumption. Assumption 2: There exists a constant for which, for all. Remark: Under this assumption, all the equations defined previously hold over the interval. The following two statements can be proven. Lemma 3: If Assumptions 1 2 hold, then, for each, for all,wehave (22) where is generated by the decoder (19) with (20), starting from state encoded by encoder (13) with (14). Furthermore, for all (23) where,. Proof: By hypothesis, 3. Over the interval the difference satisfies with initial condition, the equations (19), hence,, which, keeping in mind the update law (20) for, can be rewritten as. We now proceed by induction. Assume that for some,, for all. As before, we have for, (20), for which proves the first part of the thesis. As far as the second part is concerned, we observe that (21) where are the same as in the equations for the range vector of the encoder. As a consequence, for all. We claim now that for all as well. First of all, by definition,, hence, can be exactly reconstructed by the decoder. This implies that, therefore, over the interval. Then, we are guaranteed that. Suppose now that, for some, wehave. Then, can be exactly reconstructed From this, the thesis is immediately inferred. Proposition 1: Let be an index belonging to the set 4. Let Assumptions 1 2 hold suppose that, for each Then, for each, for all, for each,wehave (24) 3 Hereafter, we are using the stard argument (see, e.g., [11], [16], [18]) that if, for some t 2, jx(t )0y(t )j L(t)=2, then jx(t)0y(t)j L(t)=4. 4 The case in which i = n has been already dealt with in Lemma 3.

7 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 305 where each is generated by the decoder (19), (20), (21) starting from state encoded by encoder (13), (14), (16). Furthermore, for all, for each, where we have exploited the definition of in deriving the latter inequality. The use of the inductive hypothesis yields for some. Proof: First of all, observe that, because of Assumption 2, the state, therefore, the state of the decoder, exists for all. Recall that, by construction, the state lies within the quantization region. By Lemma 3 for all. We now proceed backward consider the evolution of over the interval for. To this purpose, we assume that, for some integer, for each, for all having exploited the definition (21) for that, for all, for each. We conclude To proceed further, we adopt again an inductive argument we start by assuming that, for some, for all, for each consider the evolution of over the same interval. As a consequence of the latter inequality of the hypothesis,wehave prove that for all (26) (25) for all, each. Observe that lying in the region yields By (26), we have in particular (27) Now, for, the following equation holds: which implies state at time to lie within the segment. Therefore, by construction (28) In view of Assumption 2, inequality (25), the definition of, integration of the previous equality yields At this point, we start again with the th component of the difference then proceed backward. As before, we have for which proves the thesis for the the component of. Assume now that, for some integer, for each, for all As a consequence, (25) holds for all, for each. The study of the evolution of

8 306 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 over the interval can be carried out analogously to the investigation conducted over the interval. In particular, the use of the inductive hypothesis yields, for all globally Lipschitz, then the boundedness requirement becomes essential. For the latter to be fulfilled, a more careful use of the results of [25] must be carried out. This is pursued in the next section. The first part of the thesis is, therefore, proven. To prove the second part of the statement we note that, in view of Lemma 2 (24), for each, for all, for each where are real numbers for which IV. ASYMPTOTIC STABILIZATION BY ENCODED FEEDBACK The attention will now be turned to the fulfillment of the boundedness conditions under which the results of the previous section hold. As a result of this investigation, we obtain an iterative procedure, summarized in Proposition 2, for the concurrent design of the encoder/decoder the controller. To this purpose, additional properties of each function are to be introduced. Assumption 3: Each function is zero at the origin is such that the linearization of (1) at the origin exists is stabilizable; There exist class- 5 functions for which The thesis follows immediately with. Before ending this section, we observe that the arguments shown previously become more straightforward if Assumption 1 is replaced by a global Lipschitz condition for each function. If this is the case, then hypothesis on the boundedness of is no longer needed,, denoted by the Lipschitz constants of functions, respectively, the equations which describe are defined for each. All the results presented above continue to hold provided that the local Lipschitz constants are replaced by the global ones. For instance, consider the equations of the vertical take off ling (VTOL) aircraft with,, the horizontal position, the vertical position, respectively, the roll angle of the aircraft, where is a parameter which is assumed known. The equations of the centroid of the encoder are immediately obtained. Assuming for all,wehave, therefore, the equations which define the range vector are Remark: For, function depends on only. In the sequel, function denotes a saturation function, i.e., a function which is differentiable at the origin for which there exist, such that, for all, i) ; ii). The design of the controller rests on the following notion from [25]. Definition: The state of system (29) is said to satisfy the induction hypothesis if there exist numbers,,, class- functions,,, such that, for all satisfying 6, all, all, the response of (29) with satisfies We rephrase [25, Lemma 3.2] in the following way. Lemma 4: Consider the locally Lipschitz control system (30) where is a 6 6 matrix whose entries are obtained from (18), with,,, equal to the values given previously. The encoder ( the decoder) just designed provides an estimate which exponentially converges to can be straightforwardly used to devise a control action, designed following [25], which asymptotically steers to the origin the state of system (1), provided that Assumption 3 holds. In the more general case, in which functions are not where i) is stabilizable is critically stable; ii) the state satisfies the induction hypothesis with ; (31) 5 Class-K functions are nonnegative, continuous nondecreasing functions. 6 The following notation, adopted from [25], is in use: (v; a)=v0a(v=a) (v;0) = v.

9 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 307 iii) a class- function; iv)., with forward systems via encoded feedback using bits. In fact, at each iteration when following [25] the controller parameters, are to be tuned, we first design the th component of the encoder/decoder by relying on the results of Section III, obtain the estimation which feeds the controller, tune the parameters, in such a way that boundedness of is guaranteed by Lemma 4 the constant is determined. Proposition 2: Let Assumptions 1 3 hold. There exist positive numbers, for, positive numbers vectors, respectively,, for, classfunctions,,,, for which the response of (1) in closed-loop with controller Assume there exists a function for which. Then, there exist a vector such that, for each, the state of system (31) with control (32) satisfies the induction hypothesis with, being (33) where, for, is generated by the decoder (19), (20), (21) starting from state encoded by encoder (13), (14), (16), satisfies Remark: In [25], the author considers a system of the form (31), shows that, if the -subsystem satisfies the induction hypothesis with as the fictitious disturbance as the actual disturbance ( under suitable additional assumptions), then there exists a feedback control law such that the overall closed-loop system satisfies the induction hypothesis with as the fictitious disturbance as the actual disturbance. In Lemma 4, we aim to show that, under basically the same assumptions of [25, Lemma 3.2], the perturbed feedback control law is able to guarantee that the overall closed-loop control system satisfies the induction hypothesis with as the fictitious disturbance (as in [25]), with the actual disturbance given by plus. This amounts to show that, under the conditions of [25, Lemma 3.2], the result holds not with respect to, but with respect to.in particular, most of the proof is concerned with pointing out that if satisfies the induction hypothesis with fictitious disturbance actual disturbance, then system satisfies the induction hypothesis with fictitious disturbance actual disturbance. Proof: See Appendix I. As in [25], Theorem 2.2, in the proof of the next result we iteratively apply Proposition 1 Lemma 4 to design the encoder/decoder the nested saturated controller explicitly take care of the additional disturbance we are injecting into the system at each iteration. The proof is a comprehensive illustration of how the results of this paper concur to the design of the encoder/decoder the controller to stabilize nonlinear feed- with (34) (35) Furthermore, vector belongs to as well, in particular it satisfies -sub- for suitable, for all. Proof: The proof is by induction. Rewrite the system of (1) as follows: Denoted (37) with control where (36) (37) (38) (39)

10 308 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 by subsystem (40) As a consequence, one first shows by stard arguments [25] that system we have the following. Claim: There exists such that, for each, subsystem (40) satisfies the induction hypothesis with. Furthermore,, with 7 (41) satisfies requirements i) iv) of Lemma 4. By (46) applying Proposition 1, one concludes that there exists a function for which, namely satisfies By definition of function signal, the latter given by (42) Proof: See Appendix II. Now, for some, consider the -subsystem in closed-loop with controller it is seen that satisfies (47) no matter what is. Lemma 4 can then be applied to infer the existence of of, with such that, the fulfillment of the induction hypothesis, with by system (44), (45) succinctly denote it by (43) for each. This, in particular, implies (48) Rewrite it as (44) where denotes the response of the -subsystem in closed-loop with control (43), namely (45) Assume that, for, there exists vectors positive numbers such that, for each, system (45) satisfies the induction hypothesis with. Furthermore, assume that, with (49) for all,, all. As the previous inequalities also hold over each finite interval with truncated norms (see [25]), one can assume without loss of generality, the following bound on holds: (46) 7 Henceforth, the fact that the inequalities hold over the interval [t ;1) is not explicitly denoted unless needed for the sake of clarity. Asymptotic stability is proven promptly.

11 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 309 Proposition 3: Let Assumptions 1 3 hold. Then, the control law (50) where is given by (33), is generated by the decoder (19), (20), (21), with starting from state encoded by encoder (13), (14), (16) with, guarantees the response of the closed-loop system (1), (50) to satisfy the following properties. i) For each, there exists such that implies for all. ii). Remark: Phrased in a different way, the result states that asymptotic stabilization is achieved by transmitting or bits every units of time. As the constant which bounds from above the (nonuniform) sampling intervals can be chosen as large as desired, the average data rate which equals or can be made arbitrarily close to zero. Remark: Recalling the definition of in Section II, in the case in which at time the size of the set of initial conditions is unknown, it is seen that the stabilizing control law is given by for for. Remark: Expression (50) for the controller explains why assuming the encoder to have available results in no loss of generality. As a matter of fact, even though vector may not be accessible by the encoder, we have already observed that for all. This implies that the encoder can exactly reconstruct the control input via the knowledge of function. Proof: By (34) (35) with, let. By Lemma 3, we conclude that the estimate satisfies, for all, for each (52) with. Now, according to Proposition 1, in order to estimate, boundedness of is required. Let us determine the constant for which. Introduce the class- function, let be the class- function for which the solution of the scalar differential equation satisfies. By Lemma 4 [25, Lemma 3.2], one can show that the controller guarantees the fulfillment of the induction hypothesis for the -subsystem with. In particular, it is possible to show that the solution of the -subsystem in closed-loop with the aforementioned controller satisfies where:,, with. Hence, for this example, we can take with Because of the latter inequality the bounds on, we clearly have that for all. Note also that, for each (51) Fix, choose. This choice (51) imply, that is property (i) is fulfilled. Property (ii) amounts to show asymptotic convergence. But by Proposition 2, we have [25]. We conclude that asymptotically converges to zero by Barbalat s lemma. Example: Consider the system, hence, keeping in mind that,, we have, for all, for each Letting assume that lies within the square with centroid the origin range vector. Let with,

12 310 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005 we conclude by Proposition 1 that, for each each, for By definition,. Hence, recalling [25, Fact 1], we have By Proposition 2, there exists (53) Keeping in mind that, by definition, that, we also note such that the overall closed-loop system satisfies the induction hypothesis with. The asymptotic stability result then follows provided that. By (52), (53) the definition of, this is actually the case. As a consequence V. CONCLUSION This paper proposes a solution to the problem of global semiglobal stabilization of nonlinear systems in feedforward form via encoded feedback. By exploiting the structure of the process to control, it is shown that a number of bits equal to the dimension of the process suffices to the purpose of encoding the feedback for semiglobal stabilization of the process (if a global stabilization result is desired, then an additional bit must be employed). The solution is constructive in the sense that encoder/decoder controller are iteratively designed. The proposed solution presents interesting features, among which it is worth pointing out the required bit rate being dependent on the dimension of the system to control only the capability of working nicely even in the situation in which the packets of bits do not arrive at uniformly spaced sampling times. A discussion on the minimality of the proposed data rate has also been added. where Now, observe that (54) By definition of,definition of the saturation function assumption on, we see that. Let be such that. Then, we have APPENDIX I PROOF OF LEMMA 4 First of all, note that the -subsystem with control (32) can be rewritten as where Inequalities (54) (55) yield the thesis. (55) having set. The following inequality is then straightforward in view of iii): where is a class- function. Bearing in mind [25, Lemma 3.2], we are left with proving that the state of APPENDIX II PROOF OF THE CLAIM As in [25], we apply Lemma 4 to (37), by identifying the -subsystem with the -subsystem of the Lemma, where now there is no disturbance nor -subsystem. To this end, notice that requirements i) ii) of the lemma are trivially satisfied. As far as requirement iii) is concerned, we observe that even this requirement is satisfied, since satisfies the induction hypothesis with. By hypothesis, we have Finally, requirement iv) is fulfilled by construction. Therefore, there exists such that, for each, system (37) with control (38) satisfies the induction hypothesis with, provided that there exists a law for which belongs to. Such a law actually exists. In fact, as, Assumption 2 is fulfilled under the assumption that at time the state lies within the region with centroid the origin range

13 DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS 311 vector Lemma 3 applies. Then, as Assumption 1 holds, in view of the definition of the function the signal, it is seen that (56) as such it is an -signal whose -norm depends on, namely (41) holds. Finally, assuming without loss of generality that,wehave ACKNOWLEDGMENT The author would like to thank G. Nair, P. Rinaldi, the anonymous reviewers for useful comments on this paper. He would also like to thank S. Tatikonda for kindly providing him with preprints of his work. REFERENCES [1] D. Delchamps, Extracting state information from a quantized output record, Syst. Control Lett., vol. 13, pp , [2] X. Feng K. Loparo, Active probing for information in control systems with quantized state measurements: a minimum entropy approach, IEEE Trans. Autom. Control, vol. 42, no. 2, pp , Feb [3] W. Wong R. Brockett, Systems with finite communication bwidth constraints 1: State estimation problems, IEEE Trans. Autom. 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Control, vol. 41, no. 11, pp , Nov [24] M. Jankovic, R. Sepulchre, P. Kokotovic, Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE Trans. Autom. Control, vol. 41, no. 12, pp l735, Dec [25] A. Teel, On L performance induced by feedbacks with multiple saturation, ESAIM: Control, Optim., Calc. Var., vol. 1, pp , [26] G. N. Nair R. J. Evans, Exponential stabilisability of finite-dimensional linear systems with limited data rates, Automatica, vol. 39, pp , Claudio De Persis (M 03) received the Laurea degree in electrical engineering the Ph.D. degree in systems engineering, both from the University of Rome La Sapienza, Rome, Italy, in , respectively. He was a Research Associate at Washington University, St. Louis, MO, from 1999 to 2001, at Yale University, New Haven, CT, from 2001 to Since November 2002, he has been with the Department of Computer Systems Science at the University of Rome La Sapienza as an Assistant Professor. He has held visiting positions at Texas Tech University, Lubbock, TX, the University of California, Davis ( ). In , he was Visiting Professor at the Center for Embedded Software Systems, Aalborg University, Aalborg, Denmark. His current research interests include observation control with limited information, hybrid systems, monitoring in large-scale systems, complex systems, networks, modern communication, post-genomic biology. Dr. De Persis served on the IEEE Control Systems Society Conference Editorial Board in