Lyapunov Event-triggered Stabilizationwith a Known Convergence Rate

Lyapunov Event-triggered Stabilization
with a Known Convergence Rate

Anton V. Proskurnikov,  and Manuel Mazo Jr.,  The authors are with Delft Center for Systems and Control, Delft University of Technology, The Netherlands. E-mail: anton.p.1982@ieee.org; m.mazo@tudelft.nlThe work is supported by NWO Domain TTW, Netherlands, under the project TTW#13712 “From Individual Automated Vehicles to Cooperative Traffic Management – predicting the benefits of automated driving through on-road human behavior assessment and traffic flow models” (IAVTRM).A special case of Theorem 1 (dealing with exponentially stabilizing CLF) is reported in the paper [1] to be presented on ACM Conference on Hybrid Systems Computation and Control, Porto, Portugal, April 11-13, 2018.
Abstract

A constructive tool of nonlinear control systems design, the method of Control Lyapunov Functions (CLF) has found numerous applications in stabilization problems for continuous-time, discrete-time and hybrid systems. In this paper, we address the fundamental question: given a CLF, corresponding to the continuous-time controller with some predefined (e.g. exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provide positive dwell-times between the consecutive events. Furthermore, we prove the existence of self-triggered and periodic event-triggered controllers, providing stabilization with a known convergence rate.

{keywords}

Control Lyapunov Function, Event-triggered Control, Stabilization, Nonlinear Systems

I Introduction

The seminal idea to use the second Lyapunov method as a tool of control design [2] has naturally lead to the idea of control Lyapunov Function (CLF). The concept of CLF is a natural extension of the usual Lyapunov functions for controlled systems. Namely, a CLF is a function that becomes a Lyapunov function of the closed-loop system under an appropriate (usually, non-unique) choice of the controller. The fundamental Artstein’s theorem [3] states that the existence of a CLF is necessary and sufficient for stabilization of a general nonlinear system by a “relaxed” controller, mapping the system’s state into a probability measure. For an affine unconstrained system, a usual static stabilizing controller can always be found, as shown in the seminal work [4].

In general, to find a CLF for a given control system is a non-trivial problem since the set of CLFs may have a very sophisticated structure, being non-convex and even disconnected [5]. However, in some important situations a CLF can be explicitly found. Examples include, but are not limited to, some homogeneous systems [6], feedback-linearizable, strictly passive or feedback-passive systems [7, 8] and cascaded systems [9], for which both CLFs and stabilizing controllers can be delivered by the backstepping and forwarding procedures [10, 11]. The CLF method has recently been empowered by the development of algorithms and software for convex optimization [12, 13] and genetic programming [14]. Another numerical method to compute CLFs [15] employs the so-called Zubov equation; a similar in flavor technique has been proposed in [16].

Nowadays the method of CLF is recognized as a powerful tool in nonlinear control systems design [10, 8, 11]. A CLF gives a solution to the Hamilton-Jacobi-Bellman equation for an appropriate performance index, giving a solution to the inverse optimality problem [17]. The method of CLF has been extended to uncertain [17, 18], discrete-time [19], time-delay [20] and hybrid systems [21, 22]. Combining CLFs and Control Barrier Functions (CBFs), correct-by-design controllers for stabilization under safety constraints can be obtained [23, 24]; this makes the CLF method important in designing safety-critical control systems, arising e.g. in automotive and aerospace applications [24, 25, 26].

For continuous-time systems, the CLF-based controllers are also continuous-time, and their implementation on digital platforms requires to introduce time sampling. The simplest approach is based on emulation of the continuous-time feedback by a discrete-time control, sampled at a high rate. In spite of its simplicity, rigorous stability analysis of the resulting sampled-time systems is non-trivial; we refer the reader to [27] for a detailed survey of the existing methods. A more general framework to sampled-time control design, based on a direct discretization of the nonlinear control system and approximating it by a nonlinear discrete-time inclusion, has been developed in [28, 29, 30]. This method allows to design controllers that cannot be directly redesigned from continuous-time feedback policies, however, the relevant design procedures and stability analysis are in general very sophisticated.

The necessity to use communication, computational and power resources parsimoniously has motivated to study digital controllers that are based on event-triggered sampling, which has a number of advantages over classical time-triggered control [31, 32, 33, 34, 35]. Event-triggered control strategies can be efficiently analyzed by using the theories of hybrid systems [36, 35, 37], switching systems [38], delayed systems [39, 40] and impulsive systems [41]. It should be noticed that the event-triggered sampling is aperiodic and, unlike the classical time-triggered designs, the inter-sampling interval need not necessarily be sufficiently small: the control can be frozen for a long time, provided that the behavior of the system is satisfactory and requires no intervention. On the other hand, with event-triggered sampling one has to prove the existence of positive dwell time between consecutive events: even though mathematically any non-Zeno trajectory is admissible, in real-time control systems the sampling rate is always limited.

A natural question arises whether the existence of a CLF makes it possible to design an event-triggered controller. In a few situations, the answer is known to be affirmative. The most studied is the case where the CLF appears to be a so called ISS Lyapunov function [32, 35] and allows to prove the input-to-state stability (ISS) of the closed-loop system with respect to measurement errors. A more recent result from [42] relaxes the ISS condition to a stronger version of usual asymptotic stability, however the control algorithm from [42], in general, does not ensure the absence of Zeno solutions. Another approach, based on Sontag’s universal formula [4] has been proposed in [43, 44]. All of these results impose limitations, discussed in detail in Section II. In particular, the estimation of the convergence rate for the methods proposed in [42, 43, 44] is a non-trivial problem. The method from [32] allows to estimate the convergence rate, but relies on the restrictive ISS condition, which is often violated or cannot be efficiently tested. In many situations a CLF can be designed that provides some known convergence rate (e.g. exponentially stabilizing CLFs [22]) in continuous time. A natural question arises whether event-based 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 event-triggered controller, providing a known convergence rate and a positive dwell time between consecutive events. In the special case of exponentially stabilizing CLF, this criterion has been reported in the conference paper [1]. Furthermore, we design self-triggered and periodic event-triggered controllers that simplify real-time task scheduling. We also extend our results to the problem of robust stabilization in presence of uncertain disturbances.

The paper is organized as follows. Section II gives the definition of CLF and related concepts and sets up the problem of event-triggered stabilization with a predefined convergence rate. The solution to this problem, being the main result of the paper, is offered in Section III, where event-triggered, self-triggered and periodic event-triggered stabilizing controllers are designed. In Section IV, extensions of the main result are discussed, dealing with safety-critical control, set stabilization and robust stabilization. In Section V, the main results are illustrated by numerical examples. Section VI concludes the paper. In Appendix, some technical proofs are given.

Ii Preliminaries and problem setup

Henceforth stands for the set of real matrices, . Given a function that maps into , we use to denote its Jacobian matrix.

Ii-a Control Lyapunov functions in stabilization problems

To simplify matters, henceforth we deal with the problem of global asymptotic stabilization; some extensions are discussed in Section IV. Consider the following control system

(1)

where stands for the state vector and is the control input (the case corresponds to the absence of input constraints). Our goal is to find a controller , where is some causal (non-anticipating) operator, such that for any the solution to the closed-loop system is forward complete (exists up to ) and converges to the unique equilibrium

(2)

We now give the definition of CLF; following [4], CLFs are henceforth assumed to be proper (radially unbounded).

Definition 1

[4] A -smooth function is called a control Lyapunov function (CLF)

(3)
(4)

The condition (4), obviously, can be reformulated as follows

(5)

If is Lebesgue measurable (e.g., continuous), then the set is also measurable and the Aumann measurable selector theorem [45, Theorem 5.2] implies that the function can be chosen measurable; however, it can be discontinuous and infeasible (the closed-loop system has no solution for some initial condition). Some systems (1) with continuous right-hand sides cannot be stabilized by usual controllers in spite of the existence of a CLF, however, they can be stabilized by a “relaxed” control [3] , where is a probability distribution on .

The situation becomes much simpler in the case of affine system (1) with . 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  [3]. While the original proof from [3] was not fully constructive, Sontag [4] has proposed an explicit universal formula, giving a broad class of stabilizing controllers. Assuming that , let

Then (4) means that whenever and . In the scalar case (), Sontag’s controller is

(6)

Here is a continuous function, . It is shown [4] that the control (6) is continuous at any , moreover, if , and are -smooth (respectively, real analytic), the same holds for in the domain . The global continuity requires an addition “small control” property [4]. A similar controller exists [4] for . In the subsequent paper [46] explicit stabilizing controllers have been found for the case where is a closed ball in .

Ii-B CLF and event-triggered control

Dealing with continuous-time systems (1), the CLF-based controller is also continuous-time, and its implementation on digital platforms requires time-sampling. Formally, the control command is computed and sent to the plant at time instants and remain constant for . The approach broadly used in engineering is to emulate the continuous-time feedback by sufficiently fast periodic or aperiodic sampling (the intervals are small). We refer the reader to [27] for the survey of existing results on stability under sampled-time control.

As an alternative to periodic sampling, methods of non-uniform event-based sampling have been proposed [31, 32]. With these methods, the next sampling instant instant is triggered by some event, depending on the previous instant and the system’s trajectory for . Special cases are self-triggered controllers [47, 48], where is determined by and , and there is no need to check triggering conditions, and periodic even-triggered control [49], which requires to check the triggering condition only periodically at times . The advantages of event-triggered control over traditional periodic control, in particular the economy of communication and energy resources, have been discussed in the recent papers [31, 32, 33, 34]. Event-triggered control algorithms are widespread in biology, e.g. oscillator networks [50].

A natural question arises whether a continuous-time CLF can be employed to design an event-triggered stabilizing controller. Up to now, only a few results of this type have been reported in the literature. In [32], an event-triggered controller requires the existence of a so-called ISS Lyapunov function and a controller , satisfying the conditions

(7)

Here () are -functions111A function belongs to the class if it is continuous and strictly increasing with and . and the mappings , , and are assumed to be locally Lipschitz. Subsituting into (7), one easily shows that the ISS Lyapunov function satisfies (4), being thus a special case of CLF; the corresponding feedback not only stabilizes the system, but in fact also provides its input to state stability (ISS) with respect to the measurement error . The event-triggered controller, offered in [32], is as follows

(8)

The controller (8) guarantees a positive dwell time between consecutive events , which is uniformly positive for the solutions, starting in a compact set.

Whereas the condition (7) holds for linear systems [32] and some polynomial systems [47], in general it is restrictive and not easily verifiable. Another approach to CLF-based design of event-triggered controllers has been proposed in [43, 44]. Discarding the ISS condition (7), this approach is based on Sontag’s theory [4] and inherits its basic assumptions: first, the system has to be affine , where , second, Sontag’s controller is admissible ( for any ). The controllers from [43, 44] also provide positivity of the dwell time (“minimal inter-sampling interval”).

An alternative event-triggered control algorithm, substantially relaxing the ISS condition (7) and applicable to non-affine systems, has been proposed in [42]. This approach requires the existence of a CLF that satisfies (7) with

(9)

The events are triggered in a way providing that strictly decreases along any non-equilibrium trajectory

(10)

Here for any and is -function. As noticed in [42], this algorithm in general does not provide dwell time positivity, and may even lead to Zeno solutions.

As will be discussed below, the conditions (9) can be considered as the estimate for the CLF’s convergence rate. In this paper, we assume that the CLF satisfies a more general convergence rate condition, and design an event-triggered controller, which is similar to (10), however, provides positive dwell time between consecutive switchings and an arbitrarily close convergence rate of the CLF. Moreover, we show that for each bounded region of the state space, self-triggered and periodic event-triggered controllers exist that stabilize any solution that starts in this region. Our approach substantially differs from the previous works [32, 47, 43, 44, 42]. Unlike [32, 47], we do not assume that CLF satisfies the ISS condition (7). Unlike [43, 44], the affinity of the system is not needed, and the solution’s convergence rate can be explicitly estimated. Unlike [42], we prove the dwell time positivity (in particular, absence of Zeno solutions).

Ii-C CLF with known convergence rate

Whereas the existence of CLF typically allows to find a stabilizing controller, it can potentially be unsatisfactory due to very slow convergence. Throughout the paper, we assume that a CLF gives a controller with known convergence rate.

Definition 2

Consider a continuous function , such that (and hence ). A function , satisfying (3), is said to be a -stabilizing CLF, if there exists a map , satisfying the conditions

(11)
Remark 1

The condition (9), as well as the stronger ISS condition (7), imply that is -stabilizing CLF with ( is continuous since are -functions). In general, neither -CLF is a monotone function of the norm , nor is monotone. Hence (11) is more general than (9). Note that may be discontinuous and “infeasible” (the closed-loop system may have no solutions).

To examine the behavior of solutions of the closed-loop system, we introduce the following function

(12)

The definition (12) implies that is positive when and negative for . Since, , is increasing and hence the limits (possibly, infinite) exist

The inverse is increasing and -smooth. If , we define for .

To understand the meaning of the function , consider now a special situation, where the equality in (11) is achieved

(13)

The CLF can be treated as some “energy”, stored in the system at time , whereas can be treated as the energy dissipation rate or “power” consumed by the closed-loop system (“work” done by the system per unit of time) with feedback . By noticing that , the function may be considered as the “energy-time characteristics” of the system: it takes the system time to move from the energy level to the energy level .

In general, the function gives an upper bound for the value of CLF along a solution.

Proposition 1

Let the system (1) have a -stabilizing CLF , corresponding to the controller . Let be a solution to

(14)

Then on the interval of the solution’s existence the function satisfies the following inequality

(15)
{proof}

If at any time when the solution exists, then and

(16)

which implies (15) since is increasing. Suppose now that vanishes at some , and let be the first such instant. By definition, for one has , which entails (16) and (15). Since is non-increasing, for , and thus (15) holds also for .

Corollary 1

If , then the solution of (14) converges to in finite time (provided that it exists on . If and is a forward complete solution to (14), then .

Depending on the finiteness of , Proposition (1) explicitly estimates either time or rate of the CLF’s convergence to .

Example 1. Let , where is a constant. In this case , , , . The -stabilizing CLF provides exponential stabilization (being an ES-CLF [22]). The inequality (15) reduces to

(17)

Example 2. Let with . We have , , , , and (15) boils down to

(18)

Example 3. Let with . Similar to the case , one has and , however, . The condition (15) again leads to (18), however, the right-hand side vanishes for , e.g. the solution converges in finite time .

Ii-D Problem setup

In this paper, we address the following fundamental question: does the existence of a continuous-time -stabilizing CLF allow to design an event-triggered mechanism, providing the same convergence rate as the continuous-time control ? Relaxing the latter requirement, we seek for event-triggered controllers whose convergence rates are arbitrarily close to the convergence rate of the continuous-time controller.

Problem. Assume that is a -stabilizing CLF, where is a known function, and is a fixed constant. Design an event-triggered controller, providing the following condition

(19)

Applying Proposition 1 to (which corresponds to ), it is shown that (19) entails that

(20)

For instance, in the Example 1 considered above (20) implies exponential convergence with exponent (that is, ) (versus the rate in continuous time). When , (20) implies finite-time convergence in no longer than units of time.

Iii Event-triggered, Self-Triggered and Periodic Event-Triggered Controller Designs

Henceforth we suppose that a continuous strictly positive function , a -stabilizing CLF and the corresponding feedback map are fixed. All algorithms considered in this paper provide that ; without loss of generality, we assume that . We are going to design an event-triggered algorithm that ensures (19). The input switches at sampling instants , where and the next instants depends on the solution, remaining constant on each sampling interval .

Iii-a The event-triggered control algorithm design

The condition (19) can be rewritten as , where the function is defined by

(21)

At the initial instant , calculate the control input . If , then the system starts at the equilibrium point and stays there under the control input . Otherwise, due to (11), and hence for sufficiently close to one has The next sampling instant is the first time when

we formally define if such an instant does not exist. If , we repeat the procedure, calculating the new control input . If , then the system has arrived at the equilibrium, and stays there under the control input . Otherwise, Otherwise, . Hence for close to one has The next sampling instant is the first time when , we define if such an instant does not exist. Iterating this procedure, the sequence of instants sampling is constructed in a way that the control for satisfies (11). If , is the first time when

(22)

The sequence of sampling instants terminates if or (22) does not hold at any , in this case we formally define and the control is frozen .

The procedure just described can be written as follows

(23)

(where ), or in the following “pseudocode form”.

; ; ;
while  do
     repeat
         ; is the current time
     until ;
     ; ; ;
end while;
freeze ; stay in the equilibrium
Algorithm (23) in the pseudocode form
Remark 2

Implementation of Algorithm (23) does not require any closed-form analytic expression for ; if suffices to have some numerical algorithm for computation of the value at a specific point .

Remark 3

Triggering condition (22) is similar to the condition (10), employed by the algorithm from [42], however, as explained in Remark 1, in general the conditions adopted in [42] do not hold. Furthermore, unlike [42], we give conditions for the positivity of dwell time (to be defined below) and explicitly estimate the convergence rate of the algorithm.

To assure the practical applicability of the algorithm (23), one has to prove that the solution of the closed-loop system is forward complete, addressing thus two problems. The first problem, addressed in Subsection III-B, is the solution existence between two sampling instants: to show that the event (22) is detected earlier than the solution to the following equation “explodes” (escapes from any compact)

(24)

The second problem, addressed in Subsection III-C, is to show the impossibility of Zeno solutions.

Definition 3

A solution to the closed-loop system (1),(23) is said to be Zeno, or exhibit Zeno behavior if the sequence of sampling instants is infinite and has a limit ; otherwise, the trajectory is said to be non-Zeno.

Although mathematically it can be possible to prolong the solution beyond the time  [51], the practical implementation of algorithm (23) with Zeno trajectories is problematic. Moreover, any real-time implementation of the algorithm imposes an implicit restriction on the minimal time between two consecutive events, referred to as the solution’s dwell-time. Since the control commands cannot be computed arbitrarily fast, in practice the solutions with zero dwell-time are also undesirable, even if they are forward complete.

Definition 4

The value is called the dwell-time or the minimal inter-sampling interval (MSI) [43] of the solution. Algorithm (23) provides locally uniformly positive dwell-time if is uniformly positive over all solutions, starting in a compact set : .

The proof of dwell-time positivity allows to design self-triggered and periodic event-triggered modifications of (23) that are discussed in Subsections III-D,E.

Iii-B The inter-sampling behavior of solutions

To examine the solutions’ behavior between two sampling instants, we introduce the auxiliary Cauchy problem

(25)

where . To provide the unique solvability of (25), henceforth the following non-restrictive assumption is adopted.

Assumption 1

For , the map is locally Lipschitz; in particular, is continuous222Recall that by definition of the CLF.

Proposition 2

Under Assumption 1, the Cauchy problem (25) has the unique solution , which satisfies at least one of the following two conditions holds

  1. for some ;

  2. the solution is bounded and forward complete.

{proof}

The first statement follows from the Picard-Lindelöf existence theorem [8]. Assume that on the interval of the solution’s existence we have (the first condition does not hold). Then , and hence also remains bounded on its interval of existence, and hence is forward complete.

Corollary 2

Under Assumption 1, is the only solution to the following Cauchy problem

(26)

where . If and , then .

Corollary 2 allows to show that the solution to the closed-loop system (1),(23) exists and unique for any initial condition. One can show via induction on that the sequence is uniquely defined by by noticing that is uniquely defined and if , then the next instant depends only on . If , then algorithms stops and . In view of Proposition 2, either event (22) occurs at some time (the first such instant is ), or the solution is well defined on and satisfies (19) (in which case ). In both situations, the solution is well defined on the th sampling interval .

Corollary 3

Let Assumption 1 hold. Then the sequence of sampling instants in the algorithm (23) is uniquely defined by the initial condition , and the solution between them is uniquely defined by the formula

(27)

where stands for the solution to (25).

Notice that the solution is automatically forward complete in the case where the sequence terminates (for some , we have ). This however is not guaranteed for the case where infinitely many events occur. To exclude the possibility of Zeno behavior, additional assumptions are required.

Iii-C Dwell time positivity

In this subsection, we formulate our first main result, namely, the criterion of dwell time positivity in Algorithm (23). This criterion relies on several additional assumptions.

For any and , denote

(28)

Algorithm (23) implies that is non-increasing due to (11), and hence for . In particular, sets are forward invariant along the solutions of (1),(23). For any bounded set , is also bounded since

Accordingly to Assumption 1, the following supremum is finite

(29)

for any (in the case where and , let ). We adopt a stronger version of Assumption 1.

Assumption 2

The Lipschitz constant in (29) is a locally bounded function of .

Assumption 2 holds, for instance, if the mapping is locally bounded and exists and is continuous in and .

Assumption 3

The gradient is locally Lipschitz.

Assumption 3 is a stronger version of CLF’s smoothness and holds e.g. when . Similar to (29), we introduce the Lipschitz constant of on the compact set :

(30)

Assumption 3 implies that is locally bounded since for any compact the set is bounded and

Finally, we adopt an assumption that allows to establish the relation between the convergence rates of the -CLF under the continuous-time 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 its transversal component can be arbitrary. These transversal dynamics can potentially lead to very slow and “non-smooth” convergence, e.g., the velocity can be unbounded as . Denoting

the definition of -CLF (11) implies that

Our final assumption requires these conditions to hold uniformly in the vicinity of in the following sense.

Assumption 4

The -CLF and the corresponding controller satisfy the following properties:

(31)

Here is the angle between and (Fig. 1), is locally bounded, and is strictly positive on any compact.

Fig. 1: Illustration to Assumption 4: the angle

The inequalities (31) imply that the solution does not oscillate near the equilibrium since as , and the angle between the vectors333The inequality (11) implies that both vectors are non-zero unless . and remains strictly obtuse as , i.e. the flow is not transversal to the CLF’s gradient. Assumption 4 can be reformulated as follows.

Lemma 1

For a -CLF , Assumption 4 holds if and only if a locally bounded function exists such that

(32)
{proof}

To prove the “only if” part, notice that . Therefore and (32) holds for . To prove “if” part, note that (32) and (11) imply that

and . Thus (31) holds with and .

We not turn to the key problem of dwell time estimation for Algorithm (23). In view of (27), to estimate of the time elapsed between consecutive events , it suffices to study the behavior of the solution to the Cauchy problem (25) with and , namely, to find the first instant such that . The following lemma implies that , where the function is locally uniformly positive on any compact.

Lemma 2

Let Assumptions 1-4 hold and be either non-decreasing or . Then a function exists, depending on , that satisfies two conditions:

  1. is uniformly strictly positive on any compact set;

  2. for any , the solution is well-defined on the closed interval and

    (33)

Moreover, if the functions , , and globally bounded, and , then .

The proof of Lemma 2 will be given in Appendix; in this proof the exact expression for will be found, which involves the functions . Note that Algorithm (23) does not employ , which is needed to estimate the dwell time. Notice that for a fixed , the value may be considered as a function of the parameter from (19). It can be shown that as . In other words, if the event-triggered algorithm provides the same convergence rate as the continuous-time control, the dwell time between consecutive events vanishes. Lemma 2 implies our main result.

Theorem 1

Let the assumptions of Lemma 2 hold. Then the following estimate for the dwell-time in (23) holds

(34)

where stands for the function from Lemma 2. The dwell-time is uniformly positive on any compact. Moreover, if the functions , , are globally bounded, and , then is uniformly strictly positive on .

{proof}

Notice first that the function from (34) is locally uniformly positive on any compact set since

due to the boundedness of the set and local uniform positivity of . Applying Lemma 2 to and using (<