Minimal data rate stabilization of nonlinear systems over networks with large delays
Control systems over networks with a finite data rate can be conveniently modeled as hybrid (impulsive) systems. For the class of nonlinear systems in feedfoward form, we design a hybrid controller which guarantees stability, in spite of the measurement noise due to the quantization, and of an arbitrarily large delay which affects the communication channel. The rate at which feedback packets are transmitted from the sensors to the actuators is shown to be arbitrarily close to the infimal one.
The problem of controlling systems under communication constraints
has attracted much interest in recent years.
many papers have investigated how to cope with the finite bandwidth of
the communication channel in the feedback loop.
For the case of linear systems (cf. [2, 4, 30, 7, 5, 23, 31, 3] to cite a few)
the problem has been very well
understood, and an elegant characterization of the minimal data rate
– that is the minimal rate at which the measured information
must be transmitted to the actuators –
above which stabilization is always possible is available.
Loosely speaking, the result shows that the minimal data rate
is proportional to the inverse of the product of the
unstable eigenvalues of the dynamic matrix of the system.
Controlling using the minimal data rate is
interesting not only from a theoretical point of view, but also from
a practical one, even in the presence of communication
channels with a large bandwidth. Indeed, having control techniques which
employ a small number of bits to encode the feedback information
implies for instance that the number of
different tasks which can be simultaneously carried out
is maximized, results in explicit procedures to convert the
analog information provided by the sensors into the digital form
which can be transmitted, and improves the performance
of the system (). We refer the reader to
 for an excellent survey on the topic of control
under data rate constraints.
The problem for nonlinear systems has been investigated as well (cf. [14, 16, 24, 29, 11, 27, 28]). In , the author extends the results of  on quantized control to nonlinear systems which are input-to-state stabilizable. For the same class, the paper  shows that the approach in  can be employed also for continuous-time nonlinear systems, although in  no attention is paid on the minimal data rate needed to achieve the result. In fact, if the requirement on the data rate is not strict, as it is implicitly assumed in , it is shown in  that the results of  actually hold for the much broader class of stabilizable systems. This observation is useful also to address the case in which only output feedback is transmitted through the channel. This is investigated in  for the class of uniformly completely observable systems. The paper  shows, among the other results, that a minimal data rate theorem for local stabilizability of nonlinear systems can be proven by focusing on linearized system. To the best of our knowledge, non local results for the problem of minimal data rate stabilization of nonlinear systems are basically missing. Nevertheless, the paper  has pointed out that, if one restricts the attention to the class of nonlinear feedforward systems, then it is possible to find the infimal data rate above which stabilizability is possible. We recall that feedforward systems represent a very important class of nonlinear systems, which has received much attention in recent years (see e.g. [33, 22, 10, 8, 18], to cite a few), in which many physical systems fall (), and for which it is possible to design stabilizing control laws in spite of saturation on the actuators. When no communication channel is present in the feedback loop, a recent paper (, see also ) has shown that any feedforward nonlinear system can be stabilized regardless of an arbitrarily large delay affecting the control action.
In this contribution, exploiting the results of , we show that the minimal data rate theorem of  holds when an arbitrarily large delay affects the channel (in , instantaneous delivery through the channel of the feedback packets was assumed). Note that the communication channel not only introduces a delay, but also a quantization error and an impulsive behavior, since the packets of bits containing the feedback information are sent only at discrete times. Hence, the methods of , which are studied for continuous-time delay systems, can not be directly used to deal with impulsive delay systems in the presence of measurement errors. On the other hand, the approach of  must be modified to take into account the presence of the delay. To do this, we use different proof techniques than in . Namely, while in the former paper the arguments were mainly trajectory-based, here we employ a mix of trajectory- and Lyapunov-based techniques. As a consequence, our result requires an appropriate redesign not only of the parameters in the feedback law of , but also of the encoder and the decoder of . Moreover, differently from , the parameters of the feedback control law are now explicitly computed (see Theorem 1 below). See  for another approach to control problems in the presence of delays and quantization.
In the next section, we present some preliminary notions useful to formulate the problem. The main contribution is stated in Section 3. Building on the coordinate transformations of [32, 20], we introduce in Section 4 a form for the closed-loop system which is convenient for the analysis discussed in Section 5. In the conclusions, it is emphasized how the proposed solution is also robust with respect to packet drop-out. The rest of the section summarizes the notation adopted throughout the paper.
Given an integer , the vector
will be succinctly denoted by the
corresponding uppercase letter with index , i.e. . For , we will
equivalently use the symbol or simply . denotes the
identity matrix. (respectively, )
an matrix whose entries are all (respectively, ).
When only one index is present, it is meant that the matrix is a
(row or column) vector.
If is a vector, denotes the standard Euclidean norm, i.e. , while denotes the infinity norm . The vector will be simply denoted as . (respectively, ) is the set of nonnegative integers (real numbers), is the positive orthant of . A matrix is said to be Schur stable if all its eigenvalues are strictly inside the unit circle. The symbol denotes the induced matrix 2-norm.
The symbol , with a scalar variable, denotes the sign function which is equal to if , if , and equal to otherwise. If is an -dimensional vector, then is an -dimensional vector whose th component is given by . Moreover, is an diagonal matrix whose element is .
Given a vector-valued function of time , the symbol denotes the supremum norm . Moreover, represents the right limit . In the paper, two time scales are used, one denoted by the variable in which the delay is , and the other one denoted by , in which the delay is . Depending on the time scale, the following two norms are used: , .
The saturation function  is an odd function such that for all , for all , and for all . Furthermore, , with a positive real number. The functions are those functions such that [20, 32]
with , .
where , is the vector of state variables , , each function is , and there exists a positive real number such that for all , if , then
We additionally assume that a bound on the compact set of initial conditions is available, namely a vector is known for which
We investigate the problem of stabilizing the system above, when the measurements of the state variables travel through a communication channel. There are several ways to model the effect of the channel. In the present setting, we assume that there exists a sequence of strictly increasing transmission times , satisfying
for some positive and known constants , at which a packet of bits, encoding the feedback information, is transmitted. The packet is received at the other end of the channel units of time later, namely at the times . In problems of control under communication constraints, it is interesting to characterize how often the sensed information is transmitted to the actuators. In this contribution, as a measure of the data rate employed by the communication scheme we adopt the average data rate  defined as
where is the total number of bits transmitted during the time interval . An encoder carries out the conversion of the state variables into packets of bits. At each time , the encoder first samples the state vector to obtain , and then determines a vector of symbols which can be transmitted through the channel. We recall below the encoder which has been proposed in , inspired by [31, 16], and then propose a modification to handle the presence of the delay.
2.1 Encoder in the delay-free case
The encoder in  is as follows:
where is the encoder state, is the feedback information transmitted
through the channel, is a
Schur stable matrix, and
rationale behind the encoder (7) is easily
explained (we refer the reader to [31, 16, 27] for more details).
During the time at which no new feedback information is
encoded, that is for , the encoder tracks the state of the
system. This explains the first equation of
(7), which is a copy of the system (2).
The (positive) values which appear in the entries of are
the lengths of the edges of the
quantization region, which is a cuboid with center .
This vector does not change during continuous flow.
At , when a new feedback information must be encoded and
transmitted, the encoder splits the cuboid into
subregions of equal volume, and select the subregion where the state
lies. The center of this subregion is taken as the new
state of the encoder (see the third equation in
(7)), and communicates this choice of the new
state to the decoder through the symbol . Note that each
component of takes value in , therefore can
be transmitted as a packet of bits of finite length. In
particular, if is on the left of then is
transmitted, if it is on the right, then is transmitted.
Finally, the size of the quantization region is updated, as it is
shown by the fourth equation in (7).
The system (7) is an impulsive system ([1, 12, 26]) and its behavior is as follows. At , given an initial condition , the updates of the encoder state and of the output are obtained. The former update serves as initial condition for the continuous-time dynamics, and the state is computed over the interval . At the endpoint of the interval, a new update is obtained and the procedure can be iterated an infinite number of times to compute the solution for all .
At the other end of the channel lies a decoder, which receives the packets , and reconstructs the state of the system. The decoder is very similar to the encoder. In fact, we have:
with . The control law is the well-known nested saturated feedback controller
the functions are defined in (1), and the parameters and the saturation levels of are to be designed (see Theorem 1). Note that we assume the decoder and the actuator to be co-located, and hence this control law is feasible. If the encoder and the decoder agree to set their initial conditions to the same value, then it is not hard to see () that and for all . Moreover, one additionally proves that is an asymptotically correct estimate of , and the latter converges to zero .
2.2 Encoders for delayed channels
When a delay affects the channel, both the encoder and the decoder, as well as the parameters of the controller, must be modified. As in [20, 32], we shall adopt a linear change of coordinates in which the control system takes a special form convenient for the analysis. Differently from , this change of coordinates plays a role also in the encoding/decoding procedure. Denoted by the nonsingular matrix which defines the change of coordinates, and which we specify in detail in Section 4, the functions , which appear in the encoder and, respectively, the decoder are modified as
Now, observe that the decoder does not know the first state sample throughout the interval , and hence it can not provide any feedback control action. The control is therefore set to zero. As the successive samples are all received at times , the decoder becomes aware of the value of units of time later. Hence, the best one can expect is to reconstruct the value of (see Lemma 1 below), and to this purpose the following decoder is proposed:
Observe that, as , and bearing in mind that
reconstructs , the state
is driven by
a control law which is a function of . This
justifies the appearance of , rather than ,
in the first equation of the decoder above.
We also need to modify the encoder. Indeed, as mentioned in the case with no delay, for the encoder to work correctly, the control law (9), and hence , must be available to the encoder. To reconstruct this quantity, the following equations are added to the encoder (7):
The initial conditions of the encoder and decoder are set as
and, finally, the vector which is transmitted through the channel takes the expression
Overall, the equations which describe the encoder are:
The following can be easily proven:
In the above setting, we have: (i) for all , (ii) and for all .
By definition of , after the reset the initial conditions become
As for the right-hand side of (13) is equal to zero, surely we have that over such time interval provided that no state reset occurs. Even when a state reset occurs (this is the case for instance if the delay is larger than the sampling interval) we can draw the same conclusion. In fact, before the reset takes place, say at time , for what we have just observed. But then, exactly as in (14), immediately after the reset the two variables are equal, and they continue to be the same up to time because the growth of their difference is identically zero and other possible resets produce no effect as before. Iterating these arguments shows (i). For the second part of the thesis, observe that, by definition, , and that implies exactly as in (14). Moreover, at , , for depends on , while depends on , and we have already proven that for all . The second part of (ii) is trivially true.
As anticipated, the encoder and decoder we introduced above are such that the internal state of the former is exactly reconstructed from the internal state of the latter. Hence, in the analysis to come it is enough to focus on the equations describing the process and the decoder only.
3 Main result
The problem we tackle in this paper is, given any value of the delay , find the control (9) and the matrices in (12) and (10) which guarantee the state of the entire closed-loop system to converge to the origin. As recalled in the previous section, at times , the measured state is sampled, packed into a sequence of bits, and fed back to the controller. In other words, the information flows from the sensors to the actuator with an average rate given by (6). In this setting, it is therefore meaningful to formulate the problem of stabilizing the system while transmitting the minimal amount of feedback information per unit of time, that is using the minimal average data rate. The problem can be formally cast as follows:
System (2) is
semi-globally asymptotically and locally exponentially stabilizable
an average data rate arbitrarily close to the infimal one if,
for any , , , a controller (9),
(12), a decoder (10),
and initial conditions (4),
exist such that for the closed-loop system with state
(i) There exist a compact set containing the origin, and , such that for all ;
(ii) For all , for some positive real numbers , ;
Remark. In the definition above we are slightly abusing the terminology, since for the sake of simplicity no requirement on Lyapunov (simple) stability is added. Nevertheless, it is not difficult to prove that the origin is a stable equilibrium point, in the sense that, for each , there exists such that, implies for all .
Remark. Item (iii) explains what is meant by stabilizability using an average data rate arbitrarily close to the infimal one. As a matter of fact, (iii) requires that the average data rate can be made arbitrarily close to zero, which of course is the infimal data rate. It is “infimal” rather than “minimal”, because we could never stabilize an open-loop unstable system such as (2) with a zero data rate (no feedback).
Compared with the papers [33, 22, 10, 18], concerned with the stabilization problem of nonlinear feedforward systems, the novelty here is due to the presence of impulses, quantization noise which affects the measurements and delays which affect the control action (on the other hand, we neglect parametric uncertainty, considered in ). In , it was shown robustness with respect to measurement errors for non-impulsive systems with no delay. In , the input is delayed, but neither impulses nor measurement errors are present. Impulses and measurement errors are considered in , where the minimal data rate stabilization problem is solved, but instantaneous delivery of the packets is assumed.
We state the main result of the paper:
Remark. The parameters of the nested saturated controller are very similar to those in , although and , ’s are, respectively, larger and smaller than the corresponding values of , to accommodate the presence of the quantization error. Moreover, in the proof, the values of the matrices are explicitly determined as well.
Remark. This result can be viewed as a nonlinear generalization of the well-known data rate theorem for linear systems. Indeed, the linearization of the feedforward system at the origin is a chain of integrators, for which the minimal data rate theorem for linear continuous-time systems states that stabilizability is possible using an average data rate arbitrarily close to zero.
4 Change of coordinates
Building on the coordinate transformations in [20, 32], we put the system composed of the process and the decoder in a special form (see (23) below). Before doing this, we recall that for feedforward systems encoders, decoders and controllers are designed in a recursive way [32, 33, 22, 10, 20, 27]. In particular, at each step , one focuses on the last equations of system (2), design the last equations of the encoder and the decoder, the first terms of the nested saturated controller, and then proceed to the next step, where the last equations of (2) are considered. To this end, it is useful to introduce additional notation to denote these subsystems. In particular, for , we denote the last equations of (2) by
with , while for the last equations of the decoder (10) we adopt the notation
where denotes the components from to of . Moreover, for given positive constants , , with defined in (3), we define the non singular positive matrices111The matrix will be simply referred to as . as:
where the functions are those defined in (1). Finally, let us also introduce the change of time scale
and the input coordinate change
Then we have:
In the previous sections, we have introduced the encoder, the decoder and the controller. In this section, in order to show the stability property, we carry out a step-by-step analysis, where at each step , we consider the subsystem (23) in closed-loop with (25). We first recall two lemmas which are at the basis of the iterative construction. The first one, which, in a different form, was basically given in , shows that the decoder asymptotically tracks the state of the process under a boundedness assumption.
Suppose (4) is true. If for some there exists a positive real number such that 222The conditions are void for .
and, for all ,
with333In the statement, the continuous dynamics of the impulsive systems are trivial – the associated vector fields are identically zero – and hence omitted.
and a Schur stable matrix, then for all ,
with , for , if , and
if , where is a row vector depending on , , and .
Proof. See . For the convenience of the Readers, we are adding the proof in the Appendix.
The following observation is useful later on. As a consequence of the mean value theorem, it is not difficult to realize (see also ) that, if , for some , then and in (23) (with ) obey the equations444Again, we adopt the symbol rather than .
and where the off-diagonal components of , rather than as functions of , are viewed as bounded (unknown) functions of , whose absolute value can be assumed without loss of generality to be upper bounded by a positive constant depending on , and . Now, concisely rewrite the system (27) as ()
with , . The following straightforward result shows that for the system above an exponential Lyapunov function exists (this fact was not pointed out in , where the proofs were not Lyapunov-based):
There exists a function such that, for all and for all for which , satisfies
for some positive constants , .
Proof. By Lemma 3, for any , , where is the index such that . Since the matrix is Schur stable, there exist positive constants and such that
with . We conclude that
that is the solution is exponentially stable. Then, we can apply a standard converse Lyapunov theorem for impulsive systems, such as Theorem 15.2 in , and infer the thesis.
The next statement, based on Lemma 10 in , shows that a controller exists which guarantees the boundedness of the state variables, a property required in the latter result. For the proof, observe that the arguments of  hold even in the presence of a “measurement” disturbance induced by the quantization, which can be possibly large during the transient but it is decaying to zero asymptotically.
Consider the system
where , is a positive real number, and additionally:
and are continuous functions for which positive real numbers and exist such that, respectively, , , for all .
is a piecewise-continuous function for which a positive time and a positive number exist such that , for all .
then there exist positive real numbers and such that , and for all ,
Proof. See .
To illustrate the iterative analysis in a concise manner, the following is very useful (cf. the analogous inductive hypothesis in ):
Inductive Hypothesis There exists such that . Moreover, for each , , for all , and there exists such that for all ,
Initial step () The initial step is trivially true, provided that , and .
Inductive step The inductive step is summarized in the following result:
If the inductive hypothesis is true for some , then it is also true for .