Lyapunov Design for EventTriggered Exponential Stabilization
Abstract.
Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS) programming for polynomial systems. Dealing with continuoustime systems, the CLFbased controller is also continuoustime, whereas practical implementation on a digital platform requires sampledtime control. In this paper, we show that if the continuoustime controller provides exponential stabilization, then an exponentially stabilizing eventtriggered control strategy exists with the convergence rate arbitrarily close to the rate of the continuoustime system.
1. Introduction
The idea to use Lyapunov functions as a control design tool (Kalman and Bertram, 1960) naturally leads to the method of Control Lyapunov Functions (CLF). Being a natural extension of the usual Lyapunov functions for controlled systems, a CLF is a function that becomes a Lyapunov function of the closedloop system under an appropriate choice of the controller. The existence of a CLF is necessary and sufficient for stabilization of a general nonlinear system, as implied by the fundamental Artstein’s theorem (Artstein, 1983). This theorem, however, provides no constructive way to design the stabilizing control, moreover, this control in general can be “relaxed” (randomized), mapping a system’s state into a probability measure. These limitations may be overcome in the case of affine systems. Sontag’s theorem (Sontag, 1989) gives an explicit formula for one stabilizing feedback, which appears to be continuous everywhere except for the equilibrium point.
Whereas to find a CLF for a given control system is a nontrivial problem, in some important situations it can be found in an explicit form. Examples include some homogeneous systems (Faubourg and Pomet, 2000), passive and feedbackpassive systems (Kokotović and Arcak, 2001; Khalil, 1996) and cascaded systems, for which both CLF and stabilizing controller can be delivered by the backstepping and forwarding procedures (Krstić et al., 1995; Sepulchre et al., 1997). CLFs can often be computed by using numerical tools, e.g. the Sum of Square (SOS) programming (Furqon et al., 2017) and Zubov’s method (Camilli et al., 2008).
Nowadays the method of CLF is recognized as a powerful tool in nonlinear control systems’ design (Krstić et al., 1995; Khalil, 1996; Sepulchre et al., 1997). A CLF gives a solution to the HamiltonJacobiBellman equation for an appropriate performance index, giving a solution to the inverse optimality problem (Freeman and Kokotović, 1996). The method of CLF has been extended to discretetime (Kellett and Teel, 2004), timedelay (Jancović, 2001) and hybrid systems (Sanfelice, 2013; Ames et al., 2014). Combining CLFs and Control Barrier Functions (CBFs), correctbydesign controllers for stabilization under safety constraints can be obtained (Romdlony and Jayawardhana, 2016; Ames et al., 2017), enabling to design safetycritical control systems, arising e.g. in automotive (Ames et al., 2017; Nilsson et al., 2016) and aerospace (Luo et al., 2005) applications.
Typically, CLFbased controllers are continuoustime. Their implementation on digital platforms requires to introduce time sampling. A straightforward approach, often used in engineering, is to emulate the continuoustime feedback by a discretetime control, sampled periodically at a high rate. However, rigorous techniques for nonlinear controllers’ discretization have appeared only recently (Nešić and Teel, 2004; Arcak and Nešić, 2004) and are highly nontrivial. As an alternative to these techniques, digital controllers based on eventtriggered sampling can be used. Eventtriggered sampling has a number of advantages over periodic (timetriggered) sampling, providing parsimonious use of communication and power (Aström and Bernhardsson, 2002; Tabuada, 2007; Borgers and Heemels, 2014; Araujo et al., 2014).
A natural question arises whether the existence of a (continuoustime) CLF enables one to design an eventtriggered stabilizing controller. Such controllers have been found for only a few special cases. The most studied is the case where the system admits a so called ISS Lyapunov function (Tabuada, 2007), being a special CLF that ensures a special inputtostate stability (ISS) (Sontag and Wang, 1995) property of the closedloop system. A more recent result from (Seuret et al., 2014) relaxes the ISS condition to a stronger version of usual asymptotic stability, however the control algorithm from (Seuret et al., 2014), in general, does not ensure positive dwell time between the consecutive events, nor even the absence of Zeno behaviors. Another approach, based on Sontag’s universal formula (Sontag, 1989) and inheriting its basic limitations has been proposed in (Marchand et al., 2013a; Marchand et al., 2013b). All of these results rely on restrictive assumptions, discussed in detail in Section 2, and do not allow to estimate the convergence rate efficiently. In many situations a CLF can be designed that provides exponential convergence rate (Ames et al., 2014) in continuous time. A natural question arises whether eventbased controllers can provide the same (or an arbitrarily close) convergence rate. In this paper, we give an affirmative answer to this fundamental question. Under natural assumptions, we design an eventtriggered controller, similar in structure to the one proposed in (Seuret et al., 2014), but retaining the exponential convergence and providing a positive dwell time between consecutive events.
2. Preliminaries and problem setup
Given a map such that , we use to denote its Jacobian matrix.
2.1. Control Lyapunov functions in stabilization problems
In this paper, we deal only with CLFs for global asymptotic stabilization of general nonlinear systems of the form
(1) 
Here stands for the state vector and is the control input. Our goal is to find a controller , where is some causal operator, such that for any the closedloop system has a forward complete (existing up to ) solution, and all solutions converge to an equilibrium, assumed, without loss of generality, to be
(2) 
Following the definition from (Sontag, 1989), henceforth all CLFs are supposed to be radially unbounded, or proper (Sontag, 1989).
Definition 2.1 ().
(Sontag, 1989) A smooth function is called a control Lyapunov function (CLF) in the stabilization problem, if
(3)  
(4) 
The condition (4), obviously, can be reformulated as follows
(5) 
If is Lebesgue measurable (e.g., continuous), then the measurable selector theorem (Himmelberg, 1975, Theorem 5.2) implies that the function can be chosen measurable. This function is, in general, discontinuous, so that the closedloop system has no classical solutions. However, the existence of a CLF is necessary and sufficient (Artstein, 1983) for the existence of a relaxed stabilizing control , where is a probability distribution on .
The situation becomes much simpler in the case where the system (1) is affine: . Assuming that and are continuous and is convex, the existence of a CLF ensures the possibility to design a controller , where is continuous everywhere except for, possibly (the continuity at requires an additional assumption) (Artstein, 1983). Extending this result, Sontag (Sontag, 1989) has proposed an explicit universal formula, giving a broad class of stabilizing controllers. In the simplest case where (scalar control) and , Sontag’s formula gives the following controller
(6) 
Here the functions are defined as
and is a continuous function, . Controllers similar to (6) have been proposed for , being the Euclidean space with (Sontag, 1989) and a closed ball in (Lin and Sontag, 1991).
2.2. CLF and eventtriggered control
Dealing with continuoustime systems (1), Lyapunov controllers are also continuoustime, which makes it impossible to implement them directly on a digital platform. In reality, the control is always sampled time, that is, the control command is computed and sent to the plant only at discrete instants , remaining constant between them. The simplest time sampling is periodic . In spite of the belief that highfrequency periodic control (with small ) satisfactorily emulates the continuoustime controller, mathematically rigorous analysis of the resulting nonlinear sampledtime system appears to be nontrivial (Nešić and Teel, 2004; Arcak and Nešić, 2004). Alternatively, sampling can be triggered by some condition, or event, e.g., can be the first instant after when the absolute value of the “error” reaches a predefined threshold (Tabuada, 2007). This approach, known as eventbased or eventtriggered sampling has many advantages over periodic control, in particular, it uses communication and energy resources parsimoniously (Aström and Bernhardsson, 2002; Tabuada, 2007; Borgers and Heemels, 2014; Araujo et al., 2014).
A natural question arises whether a continuoustime CLF can be employed to design an eventtriggered stabilizing controller. Up to now, only a few results of this type have been reported in the literature. In the seminal work (Tabuada, 2007), an eventtriggered controller requires the existence of a special CLF, called ISS Lyapunov function, for which the conditions (3),(4) are replaced by the following
(7) 
Here () are functions^{1}^{1}1That is, are continuous, strictly increasing, and . and the mappings , , and are assumed to be locally Lipschitz. The continuoustime control not only stabilizes the system, but in fact also provides input to state stability (ISS) with respect to the measurement error . The eventtriggered controller, offered in (Tabuada, 2007), is as follows
(8) 
The controller (8) provides positive dwell time between consecutive events , which is uniformly positive for the solutions, starting in any compact set.
Whereas the condition (7) holds for linear systems (Tabuada, 2007) and some polynomial systems (Anta and Tabuada, 2010), in general it is restrictive and not easily verifiable. Another approach to CLFbased design of eventtriggered controllers have been proposed in (Marchand et al., 2013a; Marchand et al., 2013b). Discarding the ISS condition (7), this approach is based on Sontag’s theory (Sontag, 1989) and inherits its basic assumptions: first, the system has to be affine , where , second, the Sontag formula (Sontag, 1989) gives an admissible controller, that is, for any . The controllers from (Marchand et al., 2013a; Marchand et al., 2013b) also provide the positive dwelltime (or “minimal intersampling interval”, MSI (Marchand et al., 2013a)) property.
An alternative eventtriggered control algorithm, substantially relaxing the ISS condition (7) and applicable, unlike (Marchand et al., 2013a; Marchand et al., 2013b), to nonaffine systems, has been proposed in (Seuret et al., 2014). This approach requires the existence of a CLF that satisfies (7) with , i.e.
(9) 
The events are triggered in a way providing that strictly decreases along any nonequilibrium trajectory
(10) 
Here for any and is function. In general, this algorithm does not guarantee the dwell time positivity and may even lead to Zeno solutions (Seuret et al., 2014).
In this paper, we consider an algorithm similar in spirit to the algorithm from (Seuret et al., 2014). Unlike (Seuret et al., 2014), in this paper we confine ourselves to CLFs that give exponentially stabilizing continuoustime controllers, which requires to modify the condition (9). In the case where such a CLF exists, we prove (under some natural assumptions) that exponential convergence can also be provided by an eventtriggered controller. Furthermore, such a controller provides convergence rate arbitrary close to the rate of the continuoustime control and provides positive dwell time between consecutive switchings of the control input. Unlike (Tabuada, 2007; Anta and Tabuada, 2010), we do not assume that CLF satisfies the ISS condition (7). Unlike (Marchand et al., 2013a; Marchand et al., 2013b), the affinity of the system is not needed, and the convergence rate can be explicitly estimated. Unlike (Seuret et al., 2014), we prove the dwell time positivity (which, in particular, implies that all solutions are nonZeno).
2.3. Exponential stabilization. Problem setup.
Whereas the existence of CLF typically allows to find a stabilizing controller, it can potentially be unsatisfactory due to very slow convergence. Throughout this paper, we assume that the continuoustime CLFbased controller provides exponential convergence rate; such a CLF is also called exponentially stabilizing, or ESCLF (Ames et al., 2014). Although finding of ESCLF can be nontrivial, the inverse Lyapunov theorem (Khalil, 1996) implies that it usually exists in the vicinity of the equilibrium if the system can be exponentially stabilized.
Definition 2.2 ().
A function , satisfying (3), is said to be an ESCLF with exponent , if there exists a map , satisfying the conditions
(11) 
Note that the map is not assumed to be continuous, so that the controller can be “infeasible”, that is, for some initial conditions the closedloop system has no classical (Caratheódory’s) solutions. For forward complete solutions, (11) implies that
Note that, in general, need not be a monotone function of the norm , so (11) does not imply (9).
In this paper, we address the following fundamental question: does the existence of an ESCLF allow to design an eventtriggered mechanism, also providing exponential convergence? In fact, we seek for eventtriggered controllers whose convergence rates are arbitrarily close to the rate of the continuoustime controller.
Problem. Given an ESCLF with exponent and a constant , design an eventtriggered controller, providing the exponential convergence with exponent
(12) 
3. Eventtriggered Controller Design
Henceforth we suppose that an ESCLF and the corresponding feedback map from (11) are fixed. By definition, for any we have . To simplify notation, denote
(13) 
The design of our eventtriggered algorithm, to be discussed in what follows, provides that
(14) 
which evidently implies (12).
As usual in eventtriggered control, the input switches at sampling instants , whose sequence depends on the solution. At the initial instant , compute the control input and consider the solution corresponding to the input . If , then the system is already at the equilibrium and remains there due to the assumption . Otherwise, for sufficiently close to one has
(15) 
since and . The next sampling instant is the first time when (15) is violated; let if such an instant does not exist. If , we repeat the procedure and compute the new control input , which remains unchanged till the next sampling instant . If , then the system stays at the equilibrium under the control input and we put . Otherwise, for sufficiently close to the following inequality holds
(16) 
Let the next sampling instant be the first time when (16) is violated and if such an instant does not exist. Iterating this procedure, the sequence of instants is constructed in a way that the control for satisfies (14). If , is the first time when
(17) 
The sequence of sampling instants terminates if or (17) does not hold at any , in this case we formally define and the control remains constant for .
The procedure just described can be written mathematically as
(18) 
(for brevity, we assume that ).
Note that implementation of Algorithm (18) assumes implicitly that a constructive procedure (e.g. optimizationbased) is available to compute at each sampling instant, however, it does not require any closedform analytic expression for .
To assure the practical applicability of the algorithm (18), one has to prove that the solution of the closedloop system is unique and forward complete, addressing the following two problems. First, one has to establish the solution’s existence and uniqueness between two sampling instants. In particular, one has to show that the event (17) is detected earlier than the solution runs away to infinity. Second, one has to show the absence of Zeno trajectories, for which the sequence converges to a finite limit.
Definition 3.1 ().
Notice that even for nonZeno trajectories it may happen that as , which makes it problematic to implement the algorithm on a realtime platform. Thus we are primarily interested in the more restrictive condition of the dwelltime positivity.
Definition 3.2 ().
The value is called the dwelltime of the solution. Algorithm (18) provides positive dwell time if . We say that the algorithm provides locally uniformly positive dwell time if the function is uniformly positive on any compact set , e.g. .
In this paper, we establish criteria for local uniform (called sometimes “semiuniform” (Marchand et al., 2013a)) positivity of the dwell time .
3.1. The intersampling behavior of solutions
To examine the solution’s behavior between two sampling instants, consider the following Cauchy problem
(19) 
where . To provide the unique solvability of (19), henceforth the following nonrestrictive assumption is adopted. {assumption} For , the map is locally Lipschitz, and hence the function is continuous.
Proposition 3.3 ().
Proof.
The first statement follows from the PicardLindelöf theorem (Khalil, 1996). It remains to prove that the solution cannot grow infinite while . Indeed, the latter condition implies that , and thus . Recalling that is proper, one obtains boundedness of . ∎
Corollary 3.4 ().
Applying Corollary 3.4 to , and , one shows that the sequence of sampling instants in (18) is well defined, and the instant depends only on and .
Corollary 3.5 ().
By construction of the sampling instants, the inequality (14) holds between them and, in particular, the CLF is nonincreasing along each trajectory.
3.2. Dwell time positivity
In this subsection, we formulate our main result, namely, the criterion of dwell time positivity in Algorithm (18). This criterion relies on additional assumptions.
For any and , denote
(22) 
The set is bounded for any bounded set since is supposed to be continuous and radially unbounded
Accordingly to Assumption 3.1, the following supremum is finite
(23) 
for any (in the case where and , let ). We adopt a stronger version of Assumption 3.1. {assumption} The Lipschitz constant in (23) is a locally bounded function of (that is, is bounded on any compact).
Assumption 3.2 holds, for instance, if the mapping is locally bounded and the derivative is continuous in and . The next assumption is a stronger version of CLF’s smoothness. {assumption} The function is locally Lipschitz.
Along with , we introduce the Lipschitz constant of the gradient on the compact set as follows
(24) 
Since for any compact the set is bounded, one has
Assumption 3.2 thus implies that from (24) is locally bounded.
Finally, we adopt an assumption that allows to establish the relation between the convergence rates of the ESCLF under the continuoustime control and the solution . Notice that (11) gives no information about the speed of the solution’s convergence since depends only on the velocity’s projection on the gradient vector , whereas the transversal components can be arbitrary. These transversal dynamics can potentially lead to very slow and “nonsmooth” convergence, e.g., the velocity can be unbounded as . Our final assumption excludes this pathological behavior. For brevity, let
The ESCLF and the corresponding controller satisfy the following properties:
(25) 
Here stands for the angle between the vectors and (Fig. 1), is locally bounded, and is locally strictly positive^{2}^{2}2In other words, on any compact set the function is bounded and the function is uniformly strictly positive..
The inequalities (25) imply that the solution does not oscillate near the equilibrium since as , and the angle between the vectors^{3}^{3}3The inequality (11) implies that both vectors are nonzero unless and remains strictly obtuse as , i.e. the flow is not transversal to the CLF’s gradient.
Assumption 3.2 can be reformulated as follows.
Lemma 3.6 ().
For an ESCLF , Assumption 3.2 holds if and only if a locally bounded function exists such that
(26) 
Proof.
Assumption 3.2 restricts the solution to approach the equilibrium “smoothly” in the sense that the state cannot change much faster than the CLF decreases along it. Note that the definition of ESCLF (11) implies the following “relaxed” version of this assumption. First, implies that , that is, and thus , in other words, at any . Second, the angle between and has to be obtuse unless . It is convenient to verify Assumption 3.2 and the condition (11) simultaneously since both of these conditions involve and .
We now formulate a key technical lemma which allows to establish the criterion of dwell time positivity in Algorithm (18). This lemma, proved in Appendix, entails that the time elapsed between two consecutive events cannot be smaller than , where is a locally uniformly positive function (depending only on and ). Consider again the solution to the system (19) with and .
Lemma 3.7 ().
Let the system (1) and the ESCLF satisfy Assumptions 3.23.2. Then for any there exists a function , featured by the following properties:

is uniformly strictly positive on any compact set;

for any the function is welldefined on and, furthermore,
(27)
If the functions , and ) are globally bounded, then is uniformly strictly positive on .
The proof of Lemma 3.7 will be given in Appendix. Note that Algorithm (18) does not employ the functions , and in any way; they influence only the dwell time estimate . The explicit formula for , given in Appendix, shows that is nonincreasing in , being maximal for and vanishing as . Recalling that regulates the convergence speed of the algorithm, one can notice that the price paid for the fast convergence is the small dwell time between the consecutive events (or, equivalently, large number of events per unit of time). Our main result is the following criterion of dwell time positivity.
Theorem 3.8 ().
Proof.
Notice first that the function from (28) is uniformly strictly positive on any compact set since
is bounded and thus is strictly positive on . Remark (3.1) implies that each set is forward invariant, in particular, the solution starting at remains in . If an event occurs at , then Lemma 3.7 applied to entails that the next event cannot occur earlier than at , that is, for any . ∎
3.3. Extensions
We now consider two important extensions of the main result, dealing with nonexponential stability and safetycritical systems.
3.3.1. Nonexponential convergence
Our algorithm (18) can be easily modified to cope with many CLFs that do not provide exponential convergence. For instance, replacing (11) by the inequality
(29) 
with and , one has as since
The arbitrarily close convergence rate is provided by the modification of algorithm (18), where (17) is replaced by
Instead of (14), such an algorithm provides the condition
giving an explicit estimate of the convergence rate
(30) 
A closer analysis of the proofs reveals that all statements from Subsect. 3.2, including the dwell time positivity criterion from Theorem 3.8, retain their validity for such a modified algorithm.
3.3.2. Safetycritical control
For many safetycritical systems, such as e.g. autonomous robots, smart factories and power grids, safety has to be provided by the control design. Often the requirement of safety can be mathematically described as avoiding of some “dangerous” set by the solution . As has been demonstrated in (Romdlony and Jayawardhana, 2016), in many situation the stabilization problem with this additional restriction can be solved by using control LyapunovBarrier functions (CLBF). We do not consider here the general definition of CLBF from (Romdlony and Jayawardhana, 2016) and only formulate a simple result, concerned with safe stabilization. As usual, denotes the interior of the set , and stands for its boundary.
Lemma 3.9 ().
Let stand for the closed set of “dangerous” states, we assume that . Suppose that a CLF also serves as a barrier certificate in the sense that for any one has . Then for any such that , the eventtriggered algorithm (18) provides safety .
Proof.
Indeed, the design of the algorithm provides that (14) holds along any solution, in particular, . Therefore, the solution cannot cross the boundary of the set . ∎
4. Examples
We illustrate algorithm (18) by considering two examples.
4.1. Example 1. Eventtriggered backstepping
Eventtriggered control proves to be an important tool in design of cooperative control algorithms for automated driving, where communication between the vehicles is seriously restricted by the wireless network bandwidth (Dolk et al., 2017a, b). In this subsection, we consider a simplified problem of two vehicle platoons merging (Chien et al., 1995). Assume that the lead platoon (Fig. 2) travels at constant speed , an algorithm is wanted allowing the follower (trail) platoon to merge safely with it. Denoting the velocity of the trail platoon’s leader by and its distance to the lead platoon (Fig. 2) by , the merging goal can be formulated as follows (Chien et al., 1995)
(31) 
where is the desired safe intervehicle distance. In general, more complicated speed control policies are required (Alvarez and Horowitz, 1999), ensuring safety in the case where the lead platoon applies emergency braking. Such merging algorithms are beyond the scope of this paper.
Our goal is to design the algorithm for the leading vehicle of the trail platoon, providing the control goal (31). Unlike (Chien et al., 1995), dealing with highly nonlinear controllers for the throttle and braking systems of the vehicle, we suppose that the vehicle’s longitudinal dynamics can be approximated (Stotsky et al., 1995) by the equation
(32) 
Here is the leading vehicle’s actual acceleration, whereas can be treated as the desired acceleration command. Note that, in general, the system (32) is nonlinear due to the presence of function , depending on the dynamics of the servoloop and characterizing time lag between the commanded and actual accelerations. We suppose that is known, the trail platoon’s leader measures and is aware of the lead platoon’s speed .
To design an ESCLF for this stabilization problem, we use the wellknown backstepping procedure (Krstić et al., 1995; Khalil, 1996). Choosing a parameter , we introduce the new state variables as follows
By noticing that and , the equations (32) are rewritten as follows
(33)  
The backstepping procedure implies that is the ESCLF for the system (33), associated with the controller
A straightforward computation shows that
entailing (11) with . It can be easily shown that all assumptions of Theorem 3.8 (in particular, (26)) hold. The algorithm (18) is an eventtriggered controller for platoons’ merging.
In Fig. 3, we simulate the behavior of the algorithm (18) with , choosing and . The initial condition corresponds to the situation where , , . In other words, at the initial time the trail platoon has the same speed as the lead platoon , , whereas the distance to the lead platoon exceeds the desired reference value by 10m. One may notice that the maneuver of the trail platoon’s leader includes a short period of “harsh” braking, which causes discomfort for human occupants of the vehicle. Vehicle platooning under realistic safety and comfort constraints is a nontrivial problem, which is beyond the scope of this paper.
4.2. Example 2. Nonexponential stabilization
Our second example is borrowed from (Anta and Tabuada, 2008) and deals with a twodimensional homogeneous system
(34) 
The quadratic form is not an ESCLF, however, it satisfies (29) with , where since
According to (30), the eventtriggered algorithm (18) provides the stabilization with the convergence rate
(35) 
To compare our algorithm with the one reported in (Marchand et al., 2013a), we simulate the behavior of the system for , choosing . The results of numerical simulation (Fig. 4) are very similar to those presented in (Marchand et al., 2013a). Although the convergence of the solution is slow (, and hence ), its second component (and thus also the control input) converges very fast. During the first s, only two events are detected at times and , after the second event the control input is . Unlike the controller from (Marchand et al., 2013a), based on the Sontag formula (Sontag, 1989), our algorithm provides the explicit estimate of the solution’s convergence rate (30).
5. Conclusion
In this paper, we address the following fundamental question: let a nonlinear system admit a control Lyapunov function (CLF), corresponding to a continuoustime stabilizing controller. Does this imply the existence of an eventtriggered controller, also providing exponential convergence? Under certain natural assumptions, we give an affirmative answer and show that such a controller in fact also provides the positive dwell time between consecutive events, and the convergence rate of the closedloop system can be arbitrarily close to the continuous time system’s rate. The results remain valid for nonexponentially stabilizing CLFs, that provide polynomial convergence rate. Two examples are considered, illustrating application of the proposed method to nonlinear systems.
Although the existence of CLFs often can be derived from the inverse Lyapunov theorems, to find a CLF satisfying Assumptions 3.23.2 is a nontrivial problem; computational approaches to cope with it are subject of ongoing research. Finally, it should be noted that the CLF method is not the only approach to eventtriggered control of nonlinear system, e.g. in (Liu et al., 2014) an impulsive eventtriggered controller, exponentially stabilizing a nonlinear system, has been proposed. Unlike our controller, this controller leads to discontinuous trajectories (at each sampling instant, the continuous dynamics is stopped and restarted in another point) and imposes some other limitations, e.g. the system has to be fully actuated with globally Lipschitz righthand side.
References
 (1)
 Alvarez and Horowitz (1999) L. Alvarez and R. Horowitz. 1999. Safe Platooning in Automated Highway Systems. Part II: Velocity Tracking Controller. Vehicle System Dynamics 32 (1999), 57–84.
 Ames et al. (2014) A.D. Ames, K. Galloway, K. Sreenath, and J.W. Grizzle. 2014. Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics. IEEE Trans. Autom. Control 59, 4 (2014), 876–891.
 Ames et al. (2017) A.D. Ames, X. Xu, J.W. Grizzle, and P. Tabuada. 2017. Control Barrier Function Based Quadratic Programs for Safety Critical Systems. IEEE Trans. Autom. Control (2017). publ. online.
 Anta and Tabuada (2008) A. Anta and P. Tabuada. 2008. Selftriggered stabilization of homogeneous control systems. In Proc. American Control Conf. 4129–4134.
 Anta and Tabuada (2010) Adolfo Anta and Paulo Tabuada. 2010. To Sample or not to Sample: SelfTriggered Control for Nonlinear Systems. IEEE Trans. Autom. Control 55, 9 (2010), 2030–2042.
 Araujo et al. (2014) J. Araujo, M. Mazo, A. Anta, P. Tabuada, and K.H. Johansson. 2014. System Architectures, Protocols and Algorithms for Aperiodic Wireless Control Systems. IEEE Trans. Ind. Inform. 10, 1 (2014), 175–184.
 Arcak and Nešić (2004) M. Arcak and D. Nešić. 2004. A framework for nonlinear sampleddata observer design via approximate discretetime models and emulation. Automatica 40, 11 (2004), 1931–1938.
 Artstein (1983) Z. Artstein. 1983. Stabilization with relaxed control. Nonlinear Analysis. Theory, Methods & Applications 7, 11 (1983), 1163–1173.
 Aström and Bernhardsson (2002) K.J. Aström and B.M. Bernhardsson. 2002. Comparison of Riemann and Lebesgue Sampling for First Order Stochastic Systems. In Proc. of IEEE Conf. Decision and Control. Las Vegas, 2011–2016.
 Borgers and Heemels (2014) D.P. Borgers and W.P.M.H. Heemels. 2014. EventSeparation Properties of EventTriggered Control Systems. IEEE Trans. Autom. Control 59, 10 (2014), 2644–2656.
 Camilli et al. (2008) F. Camilli, L. Grüne, and F. Wirth. 2008. Control Lyapunov functions and Zubov’s method. SIAM J. Control Optim. 47, 1 (2008), 301–326.
 Chien et al. (1995) C.C. Chien, Y. Zhang, and M. Lai. 1995. Regulation Layer Controller Design for Automated Highway Systems. Math. Comput. Modeling 22, 47 (1995), 305–327.
 Dolk et al. (2017a) V.S. Dolk, D.P. Borgers, and W.P.M.H. Heemels. 2017a. Outputbased and decentralized dynamic eventtriggered control with guaranteed gain performance and Zenofreeness. IEEE Trans. Autom. Control 62, 1 (2017), 34–39.
 Dolk et al. (2017b) V.S. Dolk, J. Ploeg, and W.P.M.H. Heemels. 2017b. EventTriggered Control for StringStable Vehicle Platooning. IEEE Trans. Intelligent Transportation Syst. 18, 12 (2017), 3486–3500.
 Faubourg and Pomet (2000) L. Faubourg and J.B. Pomet. 2000. Control Lyapunov functions for homogeneous JurdjevicQuinn systems. ESAIM: Control, Optim., Calculus Variations 5 (2000), 293–311.
 Freeman and Kokotović (1996) R.A. Freeman and P.V. Kokotović. 1996. Inverse optimality in robust stabilization. SIAM J. Control Optim. 34 (1996), 1365–1391.
 Furqon et al. (2017) Radian Furqon, YingJen Chen, Motoyasu Tanaka, Kazuo Tanaka, and Hua Wang. 2017. An SOSbased Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems. IEEE Trans. Fuzzy Syst. (2017). publ. online.
 Himmelberg (1975) C. Himmelberg. 1975. Measurable relations. Fundamenta Mathematicae 87, 1 (1975), 53–72.
 Jancović (2001) M. Jancović. 2001. Control LyapunovRazumikhin Functions and Robust Stabilization of Time Delay Systems. IEEE Trans. Autom. Control 46, 7 (2001), 1048–1060.
 Kalman and Bertram (1960) R.E. Kalman and J.E. Bertram. 1960. Control System Analysis and Design Via the “Second Method” of Lyapunov. I. ContinuousTime Systems. Journal of Basic Engineering 32 (1960), 371–393.
 Kellett and Teel (2004) C.M. Kellett and A.R. Teel. 2004. Discretetime asymptotic controllability implies smooth control Lyapunov function. Syst. Control Lett. 52 (2004), 349–59.
 Khalil (1996) H.M. Khalil. 1996. Nonlinear systems. PrenticeHall, Englewood Cliffs, NJ.
 Kokotović and Arcak (2001) P. V. Kokotović and M. Arcak. 2001. Constructive nonlinear control: A historical perspective. Automatica 37, 5 (2001), 637–662.
 Krstić et al. (1995) M. Krstić, I. Kanellakopoulos, and P.V. Kokotović. 1995. Nonlinear and adaptive control design. Wiley.
 Lin and Sontag (1991) Y. Lin and E.D. Sontag. 1991. A universal formula for stabilization with bounded controls. Syst. Control Lett. 16 (1991), 393–397.
 Liu et al. (2014) B. Liu, D.N. Liu, and C.X. Dou. 2014. Exponential stability via eventtriggered impulsive control for continuoustime dynamical systems. In Proc. Chinese Control Conf. 4346–4350.
 Luo et al. (2005) W. Luo, Y.C. Chu, and K.V. Ling. 2005. Inverse Optimal Adaptive Control for Attitude Tracking of Spacecraft. IEEE Trans. Autom. Control 50, 11 (2005), 1639–1654.
 Marchand et al. (2013a) N. Marchand, S. Durand, and J.F.G. Castellanos. 2013a. A General Formula for EventBased Stabilization of Nonlinear Systems. IEEE Trans. Autom. Control 58, 5 (2013), 1332–1337.
 Marchand et al. (2013b) N. Marchand, J.J. Martinez, S. Durand, and J.F. GuerreroCastellanos. 2013b. Lyapunov eventtriggered control: a new event strategy based on the control. IFAC Proceed. Volum. 46, 23 (2013), 324–328.
 Nešić and Teel (2004) D. Nešić and A.R. Teel. 2004. A Framework for Stabilization of Nonlinear SampledData Systems Based on Their Approximate DiscreteTime Models. IEEE Trans. Autom. Control 49, 7 (2004), 1103–1122.
 Nilsson et al. (2016) Petter Nilsson, Omar Hussien, Ayca Balkan, Yuxiao Chen, A.D. Ames, J.W. Grizzle, N. Ozay, H. Peng, and P. Tabuada. 2016. CorrectbyConstruction Adaptive Cruise Control: Two Approaches. IEEE Trans. Control Syst. Tech. 24, 4 (2016), 1294–1307.
 Romdlony and Jayawardhana (2016) M.Z. Romdlony and B. Jayawardhana. 2016. Stabilization with guaranteed safety using Control Lyapunov–Barrier Function. Automatica 66 (2016), 39–47.
 Sanfelice (2013) R. Sanfelice. 2013. On the Existence of Control Lyapunov Functions and StateFeedback Laws for Hybrid Systems. IEEE Trans. Autom. Control 58, 12 (2013), 3242–3248.
 Sepulchre et al. (1997) R. Sepulchre, M. Janković, and P.V. Kokotović. 1997. Robust Nonlinear Control Design. StateSpace and Lyapunov Techniques. Springer London.
 Seuret et al. (2014) A. Seuret, C. Prieur, and N. Marchand. 2014. Stability of nonlinear systems by means of eventtriggered sampling algorithms. IMA Journal of Mathematical Control and Information 31 (2014), 415–433.
 Sontag (1989) E.D. Sontag. 1989. A “universal” construction of Artstein’s Theorem on nonlinear stabilization. Syst. Control Lett. 13 (1989), 117–123.
 Sontag and Wang (1995) E.D. Sontag and Y. Wang. 1995. On characterizations of the inputtostate stability property. Syst. Control Lett. 24, 5 (1995), 351–359.
 Stotsky et al. (1995) A. Stotsky, C.C. Chien, and P. Ioannou. 1995. Robust PlatoonStable Controller Design for Autonomous Intelligent Vehicles. Math. Comput. Modeling 22, 47 (1995), 287–303.
 Tabuada (2007) P. Tabuada. 2007. EventTriggered RealTime Scheduling of Stabilizing Control Tasks. IEEE Trans. Autom. Control 52, 9 (2007), 1680–1685.
Appendix A Proof of Lemma 3.12
We start with several technical propositions, establishing some useful properties of the solution to a general Cauchy problem (19). Throughout this subsection, we assume that Assumptions 3.23.2 hold. Henceforth we always assume that in (19) and . Denoting the (unique) solution to (19) by , let stand for the first instant when and