n-Bit Stabilization of n-Dimensional Nonlinear Systems in Feedforward ...

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 3, MARCH 2005

299

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 and controller for stabilizing a nonlinear system in feedforward form using saturated encoded state feedback basically under standard assumption, namely local Lipschitz property of the ) bits are used vector field defining the system. (respectively, to encode the state information needed to the purpose of semiglobally (globally) stabilizing an -dimensional system. Minimality of the data rate is discussed.

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Index Terms—Communication, decoders, encoders, Lebesgue sampling, nonlinear control systems, quantization, small inputs, smart actuators, smart sensors, stabilization.

I. INTRODUCTION

M

ODERN technologies are increasingly demanding expertise in analyzing and 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] and 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

and a known constant. Moreover, in order to avoid cumbersome complexities at this stage, the channel will be assumed noise-less and 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, 2004. Recommended by Associate Editor L. Magni. The author is with the Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza,” 00184 Rome, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.843847

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] and continued in [17] and [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, and 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 and 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 and 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 “standard” (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 hand, 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 and congestion in the channel) and, more specifically, it can be simply assessed from the dimension of the system (denoted by this dimension, bits are needed for global stabilization and 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, and can be shown to yield an asymptotic average data rate arbitrarily close to the

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300


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]

.. .

where , state variables the ’s. Assumption

(1)

and having denoted by the set of . The following is assumed for

, 1: Functions , and are locally Lipschitz. Remark: As in [25], rather than considering the system described by

with -sub-

we are dropping for the dependence on . For those systems depends on , the results in this paper hold only for which locally and 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 , , , and 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 and a large amount of results is available, to our purpose we stability result established shall only make use of an -toin [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 is provided by the decoder, and the state we elaborate the results of [25] to tune the parameters , of a nested saturated controller (see Section IV) which stabilizes of (1), robustly with rethe components spect to the encoding (estimation) errors, so as to guarantee the , and we explicitly calcuboundedness of late 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 and the decoder which are able to produce an estimate of using only 1 bit. Pointing out that the encoder and the decoder can always provide a reliable estimate of the state, and iterating the procedure of the component outlined above times, the entire encoder, decoder and controller are obtained. Notice also that, if we were able to design the encoder and 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 and 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 and 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] and [20]), restrictive assumptions, and local results (as in [21]). In fact, our approach leads to an alternative way of designing encoders and 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] and [17]) which allows us to locate a region of the state bits. space where the state lies in finite time and using 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 and 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 and controller which semiglobally (respectively, globally) ) bits is asymptotically stabilize (1) using (respectively, obtained by combining the results of Sections III and 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])

DE PERSIS: -BIT STABILIZATION OF -DIMENSIONAL NONLINEAR SYSTEMS

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—and therefore this is surely known to the encoder—because of the physical separation between sensors and 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 is known to the loss of generality, a range vector belongs to the quantization redecoder for which the state and whose edges have gion whose centroid is the origin in . This implies that at time length given by the entries of no overflow is occurring. In the latter case, on the other can not be guaranteed hand, the existence of such a vector any longer and 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 bits are available for encoding. This is a preliminary that step toward the global asymptotic estimation of the state of the process from encoded feedback information and ultimately toward the global asymptotic stabilization by encoded feedback. Remark: We shall see in the next section that our design of encoders and decoders will result in no overflow for all the time, . Hence, provided that no overflow is occurring at time if the compact set of initial conditions is known to the decoder, overflow will never occur and bits suffice to encode the information regarding the location of the state within the quantization region (see Sections III and IV). On the other hand, if such a set is unknown, then overflow may occur and in this case an additional bit will be needed to distinguish the two situations of overflow and 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 and 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, and 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 and decoder—the latter upon reception of the “all-0” lies outside the quantization packet of bits—know that region which therefore must be expanded. To this purpose it is set . The following arguments are inspired by the well-known fact that system (1) with is forward comfor all the times and, therefore, plete. Now, , the th component of the there exist an update law for range vector, such as (2)

301

with

, and a nonnegative integer

(3)

such that for all and for all . At time , the encoder sets the th bit to is now available. The 1 to signal the decoder an estimate of decoder implements the same equation (2). Hence, when the dethe packet of bits where only the th coder receives at time is different from zero, it can promptly derive that for all and for all , . In the that is it can determine a bound on used by the ensequel, we will derive update laws for each coder. The decoder will implement exactly the same equations, so that, when a special sequence of bits is received, it will be . For the sake of conciseable to derive a bound on each ness, the equations for the decoder will not be explicitly given. can be computed as well, since we have An estimate of

for all

, where (4)

Note that this estimate is available only for , as only the number and the maximum in (4) can at time actually be computed. As a consequence, there exist an update , namely law for (5) and a nonnegative integer

(6)

such that for each and for . At time the encoder sets the th bit all has become available. A to 1 to signal that an estimate of straightforward exercise shows that this argument can be iterated. In particular, we have that, for each , , for all for each

302


having set (7) .. . which is available for law for

. Hence, there exist an update

(8) and a nonnegative integer (9) such that Hence, at time

for all , with

and for all

.

(10) a fixed real number, the state lies within the and th quantization region. At this point, the encoder sets the bit to 1 to signal that no overflow is occurring any longer and 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. bits are available to Lemma 1: Under Assumption 1, if , with encode information, then there exists a finite time given by (10), at which the state of process (1) is enclosed in the parallelepiped with centroid the origin and edges given by the range vector (11) ’s are defined through (2)–(5), (7), and (8). where the Remark: From the arguments preceding the lemma, it can belongs to the parallelepiped with centroid the be seen that . origin and 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 and 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 and a vector such the existence of a finite time

belongs to the quantization region with centhat the state and range vector . In fact, we have altroid the origin in ready discussed that this can happen because either a compact set to which the initial condition belongs is known (in which and is the vector whose entries equal the length case of the edges of the parallelepiped centered around the origin and containing the set to which the initial condition belongs) or such a compact set is unknown but, by the localization proceand have been determined. dure examined in Section II, The former case is explicitly dealt with in this section and in the next one, and 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 and 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 enon. In the sequel, it will coder and the decoder from time be convenient to consider at each time step not only the entire quantization region but also its projections onto the subspaces

We also set

. Let be an index belonging to the set . At each time , for , the encoder constructs defined as the Cartesian product of the region segments , with , i.e.,

where and

having denoted by the center of the segment and by its length. The region is defined as the , for quantization region at time , whereas the regions , 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 and the controller. In fact, the functioning of the encoder obeys differential/difference equations and we start dea set of signing the th differential/difference equation [see (13), with , and (14)]. This differential/difference equation lives in and define the region in such a way the subspace is, the component that, no matter what the control input of the state is guaranteed to always belong to and of is provided by the th an asymptotic estimate differential/difference equation we implemented. The results of [25] are then elaborated in Section IV to show the existence which guarantees boundedness of of a controller fed by . On the basis of this result, we can proceed to: Design th differential/difference equation of the encoder the which, together with the th equation of the encoder, lives in the subspace and define the region ; Prove of the state always that the sub-component


belongs to , and hence, prove that an asymptotic of is generated by the th estimate differential/difference equation we implemented. Again, an , appropriate design of the controller, fed by . This procedure guarantees boundedness of is iterated until each equation of the encoder/decoder and the entire controller are designed (see Propositions 1 and 2). Having constructed , each , for , is divided into two into parts [16] which results in a partition of the region sub-regions. Assume that at all the times , with , the state of the system belongs to (Lemma 3 and Proposition 1 will show that this results in no loss of generality). , the encoder can determine which one of the By knowing sub-regions in belongs to and therefore the centroid of the sub-region. In particular, the encoder calculates [16] the numbers1 sign

with

if if , resulting in

303

First of all, the length of the segment lowing way:

is updated in the fol-

(14) where is the th entry of the vector introduced at the bebe suitginning of the section. Now, let , able positive constants (the values for these constants will be made explicit in Section IV). Having specified the update law , and bearing in mind Assumption 1, we can set for

and introduce the constant

for which

for each pair

and for each

satisfying

for each

(15)

, and (12)

Recursively, for laws

, we introduce the update

The packet of bits , which is the binary representation of , is taken equal to the following sequence of 0’s and 1’s: (16) where, for

where is the th entry of the vector introduced at the beginning of the section, set if if

.

We explicitly notice that only the symbol is actually sent through the channel. We are left with specifying the equations which define the and hence decenter and the range of each segment . Henceforth, these equations—introduced fine the region later—are intended to be defined over their maximal interval of , the center of existence. For each is taken equal to [18] , where is derived from the solution of the equation

(13) . The equations for the range with initial condition of the segments are introduced in a recursive fashion as follows.

0

1Setting b (t ) = 1 if x (t ) C (t ) = 0 is arbitrary. One might as well decide to set b (t ) = 0. The choice only affects the values taken by the vector ^ (t ) and by the symbol s (t ) defined below, but it does not affect the results X stated in the remaining of this paper.

and define the constant

for which

for each pair satisfying2

and for each

Remark: If the bound is not available, one can replace , with , and the passages it with the difference below will still be valid. Of course, this requires encoders and decoders to be endowed with a timer, as it was assumed in the previous section. used to inWe observe that the parameters troduce the Lipschitz constants are welldefined. This is a consequence of the following straightforward 2We observe that constants F ;...;` , Z pend on `

;

0 0 1, will de-

, with j = 1; 2; . . . ; n ...;Z , F ;...;F

i

, U and T .

304


result concerning the asymptotic behavior of vector . solution of (14)–(16) globally expoLemma 2: Vector nentially converges to zero. Proof: It is enough to notice that system (14), (16) admits an upper triangular representation

. This implies that by the decoder from the symbol and therefore for all . This ends the proof of the claim. As for the at each time step. encoder, (19)–(21) define the region , point Indeed, for each and range define the segment . The Cartesian product

(17) where is an are defined as follows: For for

matrix whose entries , , and

(18) . This is a linear time-invariant discrete-time system whose dynamic matrix has all the eigenvalues contained in the unitary globally exponentially disc of the complex plane. Hence, . converges to zero as Remark: It should be kept in mind that the exponential conis preserved even in the case in vergence to the origin of is replaced by , with , provided that which for each . Suppose now that the set of bits is available at the other , recalling (12) end of the channel. For each it is seen that can be exactly and that provided that reconstructed by the decoder from and are known to the decoder. This is actually the case provided that the decoder implements the equations

(19) with initial condition

, and the equations

denotes the region . We introduce now the following assumption. for which Assumption 2: There exists a constant , 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 and 2 hold, then, for each , we have for all (22) where is generated by the decoder (19) with and encoded by encoder (13) with (20), starting from state and (14). Furthermore, for all (23) where , . Proof: By hypothesis,3 Over the interval the difference

. satisfies

and, hence, , which, keeping , can be rewritten as in mind the update law (20) for . We now proceed by induction. , , Assume that for some for all . As before, we have for ,

(20) and, for which proves the first part of the thesis. As far as the second part is concerned, we observe that

(21) where and 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, and and, hence, can be exactly reconstructed by the decoder. This implies that and, therefore, over the interval . . Suppose Then, we are guaranteed that , we have and now that, for some . Then, can be exactly reconstructed

From this, the thesis is immediately inferred. Proposition 1: Let be an index belonging to the set4 . Let Assumptions 1 and 2 hold and suppose that, for each

Then, for each , for all , we have

, for each

(24) 3Hereafter, we are using the standard argument (see, e.g., [11], [16], [18]) that , x (t ) y (t ) L(t)=2, then x(t) y (t) L(t)=4. if, for some t 4The case in which i = n has been already dealt with in Lemma 3.

2

j

0

j

j

0

j


where each is generated by the decoder (19), (20), encoded by encoder (13), (14), (21) starting from state , for each , (16). Furthermore, for all

305

where we have exploited the definition of and in deriving the latter inequality. The use of the inductive hypothesis yields

for some . Proof: First of all, observe that, because of Assumption 2, and, therefore, the state of the decoder , exists the state . Recall that, by construction, the state lies for all . By Lemma 3 within the quantization region

for all evolution of integer

. We now proceed backward and consider the over the interval for . To this purpose, we assume that, for some , for each , for all

having exploited the definition (21) for , for each that, for all

. We conclude

To proceed further, we adopt again an inductive argument and , for all , we start by assuming that, for some for each and consider the evolution of over the same . As a consequence of the latter inequality and of interval the hypothesis , we have

(26) and prove that for all

(25) for all

, and each lying in the region

. Observe that

By (26), we have in particular

yields (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), and the definition of , integration of the previous equality yields

At this point, we start again with the th component of the difand then proceed backward. As before, we ference have for

which proves the thesis for the the component of Assume now that, for some integer , for all

. , for each

As a consequence, (25) holds for all , and . The study of the evolution of for each

306


over the interval can be carried out analogously to the investigation conducted over the in. In particular, the use of the inductive hypothesis terval 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. IV. ASYMPTOTIC STABILIZATION BY ENCODED FEEDBACK

The first part of the thesis is, therefore, proven. To prove the second part of the statement we note that, in view , for all , for of Lemma 2 and (24), for each each

where

and

are real numbers for which

The thesis follows immediately with and . 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 . If this is the case, then hypothesis on the function is no longer needed, and, denoted boundedness of the Lipschitz constants of functions by , respectively, the and are equations which describe . All the results presented above defined for each 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 and landing (VTOL) aircraft

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 and the controller. are To this purpose, additional properties of each function to be introduced. Assumption 3: Each function is zero at the origin and is such that the linearization of (1) at the origin exists and is stabifor which lizable; There exist class- 5 functions

Remark: For , function depends on only. In the sequel, function denotes a saturation function, i.e., a function which is differentiable at the origin and , such that, for all , for which there exist i) ; . ii) The design of the controller rests on the following notion from [25]. of system Definition: The state (29) is said to satisfy the induction hypothesis if there exist numbers , , , and classfunctions , , , such that, for all and satisfying and6 , all , all , the response of satisfies (29) with

with , , the horizontal position, the vertical position and, respectively, the roll angle of the aircraft, and where is a parameter which is assumed known. The equations of the centroid of the encoder are immediately obtained. Assuming for all , we have (30)

and, therefore, the equations which define the range vector are

We rephrase [25, Lemma 3.2] in the following way. Lemma 4: Consider the locally Lipschitz control system

(31) where is a 6 6 matrix whose entries are obtained from (18), , and , , equal to the values with given previously. The encoder (and the decoder) just designed provides an and can be estimate which exponentially converges to 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 are not the more general case, in which functions

where i) ii)

is stabilizable and is critically stable; the state satisfies the induction hypothesis with and ;

K

5Classfunctions are nonnegative, continuous and nondecreasing functions. 6The following notation, adopted from [25], is in use: (v; a) = v a (v=a) and (v; 0) = v .

0


iii)

, with a class-

function;

iv)

307

functions , , , , for which the response of (1) in closed-loop with controller

.

for which Assume there exists a function . Then, there exist and a vector such that, for , the state of system (31) with control each

(32)

satisfies the induction hypothesis with being

and

, (33) , and where, for by the decoder (19), (20), (21) starting from state by encoder (13), (14), (16), satisfies

Remark: In [25], the author considers a system of the form (31), and shows that, if the -subsystem satisfies the induction hypothesis with as the fictitious disturbance and as the actual disturbance (and under suitable additional assumptions), such that then there exists a feedback control law the overall closed-loop system satisfies the induction hypothesis with as the fictitious disturbance and as the actual disturbance. In Lemma 4, we aim to show that, under basically the same assumptions of [25, Lemma 3.2], the perturbed feedis able to guarantee that the back control law overall closed-loop control system satisfies the induction hypothesis with as the fictitious disturbance (as in [25]), and with the actual disturbance given by plus . This amounts to show that, under the conditions of [25, Lemma 3.2], the re. In sult holds not with respect to , but with respect to particular, most of the proof is concerned with pointing out satisfies the induction hypothesis with that if fictitious disturbance and actual disturbance , then system satisfies the induction hypoth. esis with fictitious disturbance and actual disturbance Proof: See Appendix I. As in [25], Theorem 2.2, in the proof of the next result we iteratively apply Proposition 1 and Lemma 4 to design the encoder/decoder and the nested saturated controller and 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 and the controller to stabilize nonlinear feedforward 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, which feeds the controller, and tune obtain the estimation is the parameters , in such a way that boundedness of guaranteed by Lemma 4 and the constant is determined. Proposition 2: Let Assumptions 1 and 3 hold. There exist , positive numbers and positive numbers , for , and classvectors and, respectively, , for

is generated encoded

(34) with

and Furthermore, vector ular it satisfies

belongs to

(35) as well, and in partic-

(36) for suitable , and for all . Proof: The proof is by induction. Rewrite the system of (1) as follows:

-sub-

(37) Denoted (37) with control

(38) where

and (39)

308


by subsystem

As a consequence, one first shows by standard arguments [25] that system (40)

we have the following. Claim: There exists such that, for each subsystem (40) satisfies the induction hypothesis with and . Furthermore, , with7

, satisfies requirements i)–iv) of Lemma 4. By (46) and applying Proposition 1, one concludes that there exists a function for which , namely (41)

and

satisfies By definition of function by

and signal

, the latter given

(42) Proof: See Appendix II. Now, for some system in closed-loop with controller

it is seen that , consider the

satisfies

-sub(47) no matter what is. Lemma 4 can then be applied to infer the and of , with existence of such that , and the fulfillment of the induction and by hypothesis, with system (44), (45) and

(48)

(43) for each

. This, in particular, implies

and succinctly denote it by

Rewrite it as (44) where denotes the response of the closed-loop with control (43), namely

-subsystem in (49) (45)

Assume that, for , there exists vectors and positive numbers such that, for each , system (45) satisfies the induction hypothesis with and . Furthermore, assume that , with

for all and , , and all . As the previous inequalities also hold over each finite interval with truncated norms (see [25]), and one can assume without loss of generality , the following bound on holds:

and (46)

7Henceforth, the fact that the inequalities hold over the interval [t explicitly denoted unless needed for the sake of clarity.

;

1 is not )

Asymptotic stability is proven promptly.


Proposition 3: Let Assumptions 1 and 3 hold. Then, the control law

309

with , and let 3, we conclude that the estimate , for each

. By Lemma satisfies, for all

(50) is given by (33), is generated by the dewhere coder (19), (20), (21), with starting from state en, guarantees the coded by encoder (13), (14), (16) with of the closed-loop system (1), (50) to satisfy the response following properties. i)

all ii)

, there exists implies

For each

such that for

(52) with . Now, according to Proposition 1, in order , boundedness of is required. Let us deto estimate termine the constant for which . Introduce function , and let be the classfunction for which the solution of the scalar difthe classferential equation

. .

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 —can from above the (nonuniform) sampling intervals be chosen as large as desired, the average data rate—which or —can be made arbitrarily close equals 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

satisfies . By Lemma 4 and [25, Lemma 3.2], one can show that the controller guarantees the fulfillment of the induction and . 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

.

Remark: Expression (50) for the controller explains why asavailable results in no loss of suming the encoder to have may generality. As a matter of fact, even though vector not be accessible by the encoder, we have already observed that for all . This implies that the encoder can via the knowledge of exactly reconstruct the control input and function . Proof: By (34) and (35)

, and , . Hence, for this example, we can take with

Because of the latter inequality and the bounds on , we clearly have that for all . Note also that, for each

and

(51)

, and choose . This choice and (51) imply , that is property (i) is fulfilled. Property (ii) amounts to show asymptotic convergence. But by Proposition 2, we have and [25]. We conclude that asymptotically converges to zero by Barbalat’s lemma. Example: Consider the system Fix

and, hence, keeping in mind that we have, for all , for each

Letting

and assume that lies within the square with centroid the . Let origin and range vector with

, and

,

,

310


we conclude by Proposition 1 that, for each each

, for

By definition, Fact 1], we have

. Hence, recalling [25,

(53) Keeping in mind that, by definition, that

By Proposition 2, there exists

such that the overall closed-loop system satisfies the induction and . The asymptotic stahypothesis with . bility result then follows provided that , this is actually the case. By (52), (53) and the definition of

, we also note

As a consequence

V. CONCLUSION This paper proposes a solution to the problem of global and 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 and 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 and 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. APPENDIX I PROOF OF LEMMA 4

(54) where

Now, observe that

By definition of , definition of the saturation function and assumption on , we see that . Let be such that . Then, we have

(55)

First of all, note that the -subsystem with control (32) can be rewritten as

where

Inequalities (54) and (55) yield the thesis. having set . The following inequality is then straightforward in view of iii):

where is a classfunction. Bearing in mind [25, Lemma 3.2], we are left with proving that the state of

satisfies the induction hypothesis with By hypothesis, we have

and

.

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 and there is no “disturbance” nor -subsystem. To this end, notice that requirements i) and ii) of the lemma are trivially satisfied. As far as requirement iii) is concerned, we observe that even this requirement is satisfied, since

Finally, requirement iv) is fulfilled by construction. Therefore, there exists such that, for each , system (37) with control (38) satisfies the induction hypothesis with and , provided that there exists a law for which belongs to . Such a law actually exists. In fact, as , Assumption the 2 is fulfilled and—under the assumption that at time state lies within the region with centroid the origin and range


vector —Lemma 3 applies. Then, as Assumption 1 holds, in and the signal , view of the definition of the function it is seen that (56) -signal whose -norm depends and as such it is an , namely (41) holds. Finally, assuming on without loss of generality that , we have

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Claudio De Persis (M’03) received the Laurea degree in electrical engineering and the Ph.D. degree in systems engineering, both from the University of Rome “La Sapienza,” Rome, Italy, in 1996 and 1999, respectively. He was a Research Associate at Washington University, St. Louis, MO, from 1999 to 2001, and at Yale University, New Haven, CT, from 2001 to 2002. Since November 2002, he has been with the Department of Computer and Systems Science at the University of Rome “La Sapienza” as an Assistant Professor. He has held visiting positions at Texas Tech University, Lubbock, TX, and the University of California, Davis (1998–1999). In 2004–2005, he was Visiting Professor at the Center for Embedded Software Systems, Aalborg University, Aalborg, Denmark. His current research interests include observation and control with limited information, hybrid systems, monitoring in large-scale systems, complex systems, networks, modern communication, and post-genomic biology. Dr. De Persis served on the IEEE Control Systems Society Conference Editorial Board in 2004–2005.