Stabilizing Transmission Intervals for Nonlinear Delayed Networked Control Systems
[Extended Version]
Abstract
In this article, we consider a nonlinear process with delayed dynamics to be controlled over a communication network in the presence of disturbances and study robustness of the resulting closedloop system with respect to networkinduced phenomena such as sampled, distorted, delayed and lossy data as well as scheduling protocols. For given plantcontroller dynamics and communication network properties (e.g., propagation delays and scheduling protocols), we quantify the control performance level (in terms of gains) as the transmission interval varies. Maximally Allowable Transfer Interval (MATI) labels the greatest transmission interval for which a prescribed gain is attained. The proposed methodology combines impulsive delayed system modeling with LyapunovRazumikhin techniques to allow for MATIs that are smaller than the communication delays. Other salient features of our methodology are the consideration of variable delays, corrupted data and employment of modelbased estimators to prolong MATIs. The present stability results are provided for the class of Uniformly Globally Exponentially Stable (UGES) scheduling protocols. The wellknown Round Robin (RR) and TryOnceDiscard (TOD) protocols are examples of UGES protocols. Finally, two numerical examples are provided to demonstrate the benefits of the proposed approach.
1 Introduction
Networked Control Systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators and controllers is realized by a shared (wired or wireless) communication network [12]. NCSs offer several advantages, such as reduced installation and maintenance costs as well as greater flexibility, over conventional control systems in which parts of control loops exchange information via dedicated pointtopoint connections. At the same time, NCSs generate imperfections (such as sampled, corrupted, delayed and lossy data) that impair the control system performance and can even lead to instability. In order to reduce data loss (i.e., packet collisions) among uncoordinated NCS links, scheduling protocols are employed to govern the communication medium access. Since the aforementioned networkinduced phenomena occur simultaneously, the investigation of their cumulative adverse effects on the NCS performance is of particular interest. This investigation opens the door to various tradeoffs while designing NCSs. For instance, dynamic scheduling protocols (refer to [11] and [17]), modelbased estimators [8] or smaller transmission intervals can compensate for greater delays at the expense of increased implementation complexity/costs [5].
In this article, we consider a nonlinear delayed system to be controlled by a nonlinear delayed dynamic controller over a communication network in the presence of exogenous/modeling disturbances, scheduling protocols among lossy NCS links, timevarying signal delays, timevarying transmission intervals and distorted data. Notice that networked control is not the only source of delays and that delays might be present in the plant and controller dynamics as well. Therefore, we use the term delayed NCSs. The present article takes up the emulationbased approach from [27] for investigating the cumulative adverse effects in NCSs and extends it towards plants and controllers with delayed dynamics as well as towards nonuniform timevarying NCS link delays. In other words, different NCS links induce different and nonconstant delays. It is worth mentioning that [27] generalizes [11] towards corrupted data and the socalled large delays. Basically, we allow communication delays to be larger than the transmission intervals. To the best of our knowledge, the work presented herein is the most comprehensive study of the aforementioned cumulative effects as far as the actual plantcontroller dynamics (i.e., timevarying, nonlinear, delayed and with disturbances) and interconnection (i.e., output feedback) as well as the variety of scheduling protocols (i.e., UGES protocols) and other networkinduced phenomena are concerned (i.e., variable delays, lossy communication channels with distortions). For instance, [19] focuses on timevarying nonlinear control affine plants (i.e., no delayed dynamics in the plant nor controller) and state feedback with a constant delay whilst neither exogenous/modeling disturbances, distorted data nor scheduling protocols are taken into account. The authors in [15] and [33] consider linear control systems, impose ZeroOrderHold (ZOH) sampling and do not consider noisy data nor scheduling protocols. In addition, [15] does not take into account disturbances. Similar comparisons can be drawn with respect to other related works (see [32, 12, 11, 27, 19] and the references therein).
In order to account for large delays, our methodology employs impulsive delayed system modeling and LyapunovRazumikhin techniques when computing Maximally Allowable Transmission Intervals (MATIs) that provably stabilize NCSs for the class of Uniformly Globally Exponentially Stable (UGES) scheduling protocols (to be defined later on). Besides MATIs that merely stabilize NCSs, our methodology is also capable to design MATIs that yield a prespecified level of control system performance. As in [11], the performance level is quantified by means of gains. According to the batch reactor case study provided in [27], MATI conservativeness repercussions of our approach for the small delay case appear to be modest in comparison with [11]. This conservativeness emanates from the complexity of the tools for computing gains of delayed (impulsive) systems as pointed out in Section 5 and, among others, [3]. On the other hand, delayed system modeling (rather than ODE modeling as in [11]) allows for the employment of modelbased estimators, which in turn increases MATIs (see Section 5 for more). In addition, reallife applications are characterized by corrupted data due to, among others, measurement noise and communication channel distortions. In order to include distorted information (in addition to exogenous/modeling disturbances) into the stability analyses, we propose the notion of stability with bias.
The main contributions of this article are fourfold: a) the design of MATIs in nonlinear delayed NCSs with UGES protocols even for the socalled large delays; b) the LyapunovRazumikhinbased procedure for rendering stability of nonlinear impulsive delayed systems and computing the associated gains; c) the consideration of NCS links with nonidentical timedependent delays; and d) the inclusion of modelbased estimators. In contrast to our conference paper [27], this article incorporates variable delays, contains proofs, provides a nonlinear numerical example with delayed plant dynamics and designs modelbased estimation that prolongs MATIs [8]. Furthermore, this article accompanies [28].
The remainder of this article is organized as follows. Section 2 presents the utilized notation and stability notions regarding impulsive delayed systems. Section 3 states the problem of finding MATIs for nonlinear delayed NCSs with UGES protocols in the presence of nonuniform communication delays and exogenous/modeling disturbances. A methodology to solve the problem is presented in Section 4. Detailed numerical examples are provided in Section 5. Conclusions and future challenges are in Section 6. The proofs are provided in the Appendix.
2 Preliminaries
2.1 Notation
To simplify notation, we use . The dimension of a vector is denoted . Next, let be a Lebesgue measurable function on . We use
to denote the norm of when restricted to the interval . If the corresponding norm is finite, we write . In the above expression, refers to the Euclidean norm of a vector. If the argument of is a matrix , then it denotes the induced 2norm of . Furthermore, denotes the (scalar) absolute value function. The dimensional vector with all zero entries is denoted . Likewise, the by matrix with all zero entries is denoted . The identity matrix of dimension is denoted . In addition, denotes the nonnegative orthant. The natural numbers are denoted or when zero is included.
Lefthand and righthand limits are denoted and , respectively. Next, for a set , let for every exists in for all and for all but at most a finite number of points . Observe that denotes the family of rightcontinuous functions on with finite lefthand limits on contained in and whose discontinuities do not accumulate in finite time. Finally, let denote the zero element of .
2.2 Impulsive Delayed Systems
In this article, we consider nonlinear impulsive delayed systems
(1) 
where is the state, is the input and is the output. The functions and are regular enough to guarantee forward completeness of solutions which, given initial time and initial condition , where is the maximum value of all timevarying delay phenomena, are given by rightcontinuous functions . Furthermore, denotes the translation operator acting on the trajectory defined by for . In other words, is the restriction of trajectory to the interval and translated to . For , the norm of is defined by . Jumps of the state are denoted and occur at time instants , where , . The value of the state after a jump is given by for each . For a comprehensive discussion regarding the solutions to (1) considered herein, refer to [2, Chapter 2 & 3]. Even though the considered solutions to (1) allow for jumps at , we exclude such jumps in favor of notational convenience.
Definition 1 (Uniform Global Stability).
For , the system is said to be Uniformly Globally Stable (UGS) if for any there exists such that, for each and each satisfying , each solution to satisfies for all and can be chosen such that .
Definition 2 (Uniform Global Asymptotic Stability).
For , the system is said to be Uniformly Globally Asymptotically Stable (UGAS) if it is UGS and uniformly globally attractive, i.e., for each there exists such that for every and every .
Definition 3 (Uniform Global Exponential Stability).
For , the system is said to be Uniformly Globally Exponentially Stable (UGES) if there exist positive constants and such that, for each and each , each solution to satisfies for each .
Definition 4 (Stability with Bias ).
Let . The system is stable with bias from to with (linear) gain if there exists such that, for each and each , each solution to from satisfies for each .
Definition 5 (Detectability).
Let . The state of is detectable from with (linear) gain if there exists such that, for each and each , each solution to from satisfies for each .
3 Problem Formulation
Consider a nonlinear control system consisting of a plant with delayed dynamics
(2) 
and a controller with delayed dynamics
(3) 
where and are the states, and are the outputs, and and are the inputs of the plant and controller, respectively, where and are external disturbances to (and/or modeling uncertainties of) the plant and controller, respectively. The translation operators and are defined in Section 2.2 while the corresponding plant and controller delays are and , respectively. For notational convenience, constant plant and controller delays are considered.
Let us now model the communication network between the plant and controller over which intermittent and realistic exchange of information takes place (see Figure 1). The value of computed by the controller that arrives at the plant is denoted . Similarly, the values of that the controller actually receives are denoted . Consequently, we have
(4) 
on the right hand sides of (2) and (3). In our setting, the quantity is the delayed and distorted input fed to the plant (2) while the quantity is the delayed and distorted version of received by the controller (3). We proceed further by defining the error vector
(5) 
where and are translation operators and the maximal networkinduced delay (e.g., propagation delays and/or delays arising from protocol arbitration). The operator in (5) delays each component of for the respective delay. Essentially, if the component of , that is , is transmitted with delay , then the component of , that is , is in fact . Accordingly, .
Due to intermittent transmissions of the components of and , the respective components of and are updated at time instants , i.e.,
(6) 
where and model measurement noise, channel distortion and the underlying scheduling protocol. The role of and is as follows. Suppose that the NCS has links. Accordingly, the error vector can be partitioned as . In order to avoid cumbersome indices, let us assume that each NCS link is characterized by its own delay. Hence, there are merely (rather than ) different delays in (5). Besides the already introduced upper bound on ’s, we assume that ’s are differentiable with bounded . As orchestrated by (6), if the NCS link is granted access to the communication medium at some , the corresponding components of jump to the received values. It is to be noted that all other components of remain unaltered. Consequently, the related components of reset to the noise present in the received data, i.e.,
(7) 
and we assume that
Noise , which is embedded in and , models any discrepancy between the received values and their actual values at time (when the NCS link of was sampled). As already indicated, this discrepancy can be a consequence of measurement noise and channel distortion. We point out that has nothing to do with nor . Observe that outoforder packet arrivals, as a consequence of the timevarying delays, are allowed for.
In between transmissions, the values of and need not to be constant as in [11], but can be estimated in order to extend transmission intervals (consult [8] for more). In other words, for each we have
(8) 
where the translation operators and are with delay . The commonly used ZOH strategy is characterized by and .
Definition 6.
Consider the noisefree setting, i.e., . The protocol given by is UGES if there exists a function such that is locally Lipschitz (and hence almost everywhere differentiable) for every , and if there exist positive constants , and such that

, and

,
for all .
Notice that, even though the delays could result from protocol arbitration, the delays are not a part of the UGES protocol definition [20, 11]. In addition, is not a part of the protocol, but rather a consequence, as it is yet to be designed. Commonly used UGES protocols are the Round Robin (RR) and TryOnceDiscard protocol (TOD) (consult [20, 11, 5]). The corresponding constants are , , for RR and , for TOD. Explicit expressions of the noisefree for RR and TOD are provided in [20], but are not needed in the context of this article.
The properties imposed on the NCS in Figure 1 are summarized in the following standing assumption.
Assumption 1.
The jump times of the NCS links at the controller and plant end obey the underlying UGES scheduling protocol (characterized through ) and occur at transmission instants belonging to , where for each with arbitrarily small. The received data is corrupted by measurement noise and/or channel distortion (characterized through as well). In addition, each NCS link is characterized by the networkinduced delay , .
The existence of a strictly positive , and therefore the existence of , is demonstrated in Remark 3.
A typical closedloop system (2)(8) with continuous (yet delayed) information flows in all NCS links might be robustly stable (in the sense according to (15)) only for some sets of , . We refer to the family of such delay sets as the family of admissible delays and denote it . Next, given some admissible delays , , the maximal which renders stability (with a desired gain) of the closedloop system (2)(8) is called MATI and is denoted . We are now ready to state the main problem studied herein.
Problem 1.
Remark 1.
Even though our intuition (together with the case studies provided herein and in [27]) suggests that merely “small enough" delays (including the zero delay) are admissible because the control performance impairs (i.e., the corresponding gain increases) with increasing delays, this observation does not hold in general [10],[21, Chapter 1.],[22]. In fact, “small" delays may destabilize some systems while “large" delays might destabilize others. In addition, even a second order system with a single discrete delay might toggle between stability and instability as this delay is being decreased. Clearly, the family needs to be specified on a casebycase basis. Hence, despite the fact that the case studies presented herein and in [27] yield MATIs that hold for all smaller timeinvariant delays (including the zero delay) than the delays for which these MATIs are computed for, it would be erroneous to infer that this property holds in general.
4 Methodology
Along the lines of [20], we rewrite the closedloop system (2)(8) in the following form amenable for smallgain theorem (see [14, Chapter 5]) analyses:
(9a)  
(9b) 
where , , and functions , and are given by (10) and (11). We assume enough regularity on and to guarantee existence of the solutions on the interval of interest [2, Chapter 3]. Observe that differentiability of ’s and boundedness of play an important role in attaining regularity of . For the sake of simplicity, our notation does not explicitly distinguish between translation operators with delays , , or in (10), (11) and in what follows. In this regard, we point out that the operators and are with delays and , respectively, the operators and within and are with delay while all other operators are with delay . In what follows we also use , which is the maximum value of all delay phenomena in (11).
(10)  
(11) 
For future reference, the delayed dynamics
(12a)  
(12b) 
are termed the nominal system , and the impulsive delayed dynamics
(13a)  
(13b) 
are termed the error system . Observe that contains delays, but does not depend on nor as seen from (12). Instead, and constitute the error subsystem as seen from (13).
The remainder of our methodology interconnects and using appropriate outputs. Basically, from Definition 6 is the output of while the output of , denoted , is obtained from and as specified in Section 4.2. Notice that the outputs and are auxiliary signals used to interconnect and and solve Problem 1, but do not exist physically. Subsequently, the smallgain theorem is employed to infer stability with bias. Proofs of the upcoming results are in the Appendix.
4.1 Stability with Bias of Impulsive Delayed LTI Systems
Before invoking the smallgain theorem in the upcoming subsection, let us establish conditions on the transmission interval and delay that yield stability with bias for a class of impulsive delayed LTI systems. Clearly, the results of this subsection are later on applied towards achieving stability with bias and an appropriate gain of .
Consider the following class of impulsive delayed LTI system
(14a)  
(14b) 
where and , initialized with some . In addition, is a continuous function upper bounded by while denote external inputs and .
Lemma 1.
Assume , and consider a positive constant . In addition, let , and for or merely for . If there exist constants , such that the conditions

, and

hold, then the system (14) is UGES and for all .
The previous lemma, combined with the work presented in [1], results in the following theorem.
4.2 Obtaining MATIs via the SmallGain Theorem
We are now ready to state and prove the main result of this article. Essentially, we interconnect and via suitable outputs (i.e., and , respectively), impose the smallgain condition and invoke the smallgain theorem.
Theorem 2.
Suppose the underlying UGES protocol, and are given. In addition, assume that

there exists a continuous function such that the system given by (12) is stable from to for some , i.e., there exist such that
(15) for all , and

there exists and , , such that for almost all , almost all and for all it holds that
(16)
Then, the NCS (9) is stable with bias from to for each for which there exist and satisfying (I), (II) and with parameters and .
Remark 2.
According to Problem 1, condition (a) requires the underlying delays to be admissible, i.e., . Condition (a) implies that the nominal system (i.e., the closedloop system) is robust with respect to intermittent information and disturbances. Besides stability, typical robustness requirements encountered in the literature include InputtoState Stability (ISS) and passivity [30]. Condition (b) relates the current growth rate of with its past values. As shown in Section 5, all recommendations and suggestions from [20] and [11] regarding how to obtain a suitable readily apply because characterizes the underlying UGES protocol (and not the plantcontroller dynamics).
Remark 3 (Zenofreeness).
The lefthand sides of conditions (I) and (II) from Lemma 1 are nonnegative continuous functions of and approach as . Also, these lefthand sides equal zero for . Note that both sides of (I) and (II) are continuous in , , , and . Hence, for every , , , and there exists such that (I) and (II) are satisfied. Finally, since is continuous in and , we infer that for every finite there exists such that . In other words, for each admissible , , the unwanted Zeno behavior is avoided and the proposed methodology does not yield continuous feedback that might be impossible to implement. Notice that each yielding is a candidate for . Depending on , , and , the maximal such is in fact MATI .
Remark 4.
The right hand side of (16) might not be descriptive enough for many problems of interest. In general, (16) should be sought in the form , where and . As this general form leads to tedious computations (as evident from the proof of Lemma 1 in the Appendix), we postpone its consideration for the future. For the time being, one can intentionally delay the communicated signals in order to achieve a single discrete delay in (16). This idea is often found in the literature and can be accomplished via the Controller Area Network (CAN) protocol, timestamping of data and introduction of buffers at receiver ends (refer to [12] and references therein).
Remark 5.
Noisy measurements can be a consequence of quantization errors. According to [18], feedback control prone to quantization errors cannot yield closedloop systems with linear gains. Hence, the bias term in the linear gain stability with bias result of Theorem 2 cannot be removed without contradicting the points in [18]. Further investigations of quantized feedback are high on our future research agenda.
Remark 6.
Let us consider the case of lossy communication channels. If there is an upper bound on the maximum number of successive dropouts, say , simply use as the transmission interval in order for Theorem 2 to hold. Moreover, the transmission instants among NCS links need not to be (and often cannot be) synchronized. In this case, each NCS must transmit at a rate smaller than (instead of ), where is the MATI obtained for the RR protocol, in order to meet the prespecified performance requirements. Observe that this leads to asynchronous transmission protocols, which in turn increases the likelihood of packet collisions [17].
Corollary 1.
In the following proposition, we provide conditions that yield UGS and GAS of the interconnection and . Recall that and are the disturbance and noise settings, respectively, corresponding to UGS and GAS.
5 Numerical Examples
5.1 Constant Delays
The following example is motivated by [31, Example 2.2.] and all the results are provided for . Consider the following nonlinear delayed plant (compare with (2))
controlled with (compare with (3))
As this controller is without internal dynamics, therefore . Additionally, .
Let us consider the NCS setting in which noisy information regarding and are transmitted over a communication network while the control signal is not transmitted over a communication network nor distorted (i.e., ). In addition, consider that the information regarding arrives at the controller with delay while information regarding arrives in timely manner. For the sake of simplicity, let us take . Apparently, the output of the plant is and there are two NCS links so that . Namely, is transmitted through one NCS link while is transmitted through the second NCS link. The repercussions of these two NCS links are modeled via the following error vector (compare with (5))
The expressions (10) and (11) for this example become:
(17) 
(18) 
(19) 
where and .
According to [20] and [11], we select and , where is a diagonal matrix whose diagonal elements are lower bounded by and upper bounded by . Next, we determine , , and from Theorem 2 for the ZOH strategy (i.e., ) obtaining (20) and (21).
(20)  