Abstract

This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes.

1. Introduction

The study of switched and general hybrid dynamic systems has gained lots of interest in recent years (see, e.g., [1–11]). These mathematical abstractions are practically motivated by many significant engineering applications, in which digital devices interact with an analog environment. On the other side, the analytic tools and results related to the switched control constitute a valuable part of the modern systems theory. The control design technique based on the switched systems methodology is nowadays a mature adequate approach to the synthesis of controllers for several nonlinear interconnected dynamic systems. We refer to [2, 4, 5, 7, 10, 12–17] for the basic facts and some interesting real world applications of switched and hybrid dynamic models.

However, in the face of a recent progress in switched systems theory, there are numbers of fundamental properties of these systems that have not been investigated in sufficient detail. Among others, these include a formal description of the so-called Zeno executions (see [2, 4, 10, 18–22]). A behaviour of a system is called a Zeno dynamics, if it takes infinitely many discrete transitions on a finite time interval. The real world engineering and physical systems are, of course, non-Zeno, but a complex switched-type mathematical model of a physical process may be Zeno, due to the usual modelling overabstractions.

Since the above-mentioned abstraction provides an inevitable theoretic basis for a concrete control design procedure, understanding when it leads to a Zeno effect in a switched system is essential, for an adequate numerical implementation of the designed control algorithms. Let us also note that a Zeno-like switched dynamic behaviour can cause imprecise computer simulations and imply the corresponding calculating and modelling errors. It is explainable that the main computer tools and numerical packages developed for switched systems (see, e.g., [19, 22]) get stuck when a large number of discrete transitions take place within a short time interval. A possible improvement of the mentioned simulation technique can be realized on a basis of some adequate extensions of the available theory of the Zeno effect including some approximative approaches. Recall that the Zeno hybrid automata have been usually examined from the point of view of theoretical computer science [2, 23] such that the necessary investigations of the continuous part of a system under consideration and the discrete-continuous interplay are quite underrated. The general Zeno hybrid executions under some specific assumptions have been deeply investigated in [18, 20–22]. On the other hand, an analytic approach based on the modern variational analysis and consistent approximation techniques have not been sufficiently advanced to the Zeno switched systems setting and to the corresponding control design procedures.

Usually, one avoids the examination of a Zeno effect in a work on switched control design by some specific additional assumptions (see, e.g., [5, 12, 13, 17, 24, 25]). The aim of our contribution is to give a constructive characterization of the Zeno executions in a specific case of affine switched systems. We propose an analytic technique that eliminates the original Zeno effect from the consideration. These approximations are consistent in the sense of the resulting trajectories. Theoretic results obtained in our paper can be useful, for instance, when designing switched controllers and simulating the correspondingly sophisticated dynamic behaviour.

The remainder of the paper is organized as follows: Section 2 contains some necessary formal concepts and facts related to the class of systems under consideration. Section 3 includes the main analytic results and the corresponding formal proofs. We propose here a novel approximative approach to the Zeno dynamics. In Section 4 we discuss a specific case of a Zeno behaviour determined by a switched manifold (in the state space) and extend our approximating scheme to this common situation. In this connection we also mention shortly the sliding mode control processes. Section 5 is devoted to the novel variational description of the Zeno effect in affine switched systems. We establish a similarity between the approximative Zeno dynamics and the conventional optimal control methodology. Section 6 summarizes our paper.

2. Zeno Executions in Affine Switched Systems

This paper concerns affine switched control systems (affine with respect to the control inputs) described by the following concept (see also [13, 24, 25]).

Definition 1. An affine switched system (ASS) is 7-tuplewhere(i) is a finite set of indices;(ii), , is a family of state spaces such that ;(iii) is a set of admissible control input values (called control set);(iv), , are families of -functions (real analytic functions) determined on an open set ;(v) is the set of admissible control functions;(vi) is a transitions’ set.

We say that a location switching from to occurs at a switching time from the given (finite) time interval . Consider a specific ASS from Definition 1 with switching timesDenote for a given . In general, the above finite or infinite sequence of switching times is not an a priory given sequence. A switched control system remains in location for all , . We additionally assume that the class of admissible controls is the set of bounded measurable functionswhere is assumed to be compact. Here denotes the Lebesgue space of all -valued bounded measurable functions. A “continuous trajectory” of an ASS under consideration is determined as follows.

Definition 2. An admissible trajectory associated with an ASS is an absolutely continuous function such that (i) is an absolutely continuous function on with continuously prolongable to , , ;(ii) for almost all times , where is a restriction of an admissible control function on the time interval and .

Note that under the above assumptions (smoothness of the families , and boundedness of the admissible controls) the trajectories from Definition 2 have bounded derivatives almost everywhere. We refer to [3, 4, 6, 10, 17] for some alternative concepts of switched, hybrid, and general interconnected dynamic systems. Evidently, every from Definition 2 determines a finite or infinite partitioning of . The adjoint time intervals and the corresponding locations from are assumed to be a priori unknown. Let be a trajectory of an ASS associated with an admissible for a given (finite) number . This trajectory is an absolutely continuous solution of the following initial value problem:where is a characteristic function of the interval for and . Note that, in contrast to the general hybrid systems, the switching times from a sequence do not depend on the state vector. The complete (switched-type) control input associated with the ASS under consideration can now be expressed as follows:where , and the set of all admissible sequences is characterized by the generic conditionsLet us note that , where is the standard Lebesgue space of all scalar absolutely integrable functions.

Definition 3. For an admissible control input , wherean execution of the ASS is defined as a collection , where and is a solution to (5). Here is the corresponding sequence of switching times.

Let us assume that every admissible control generates a unique execution of the given ASS. Following [18, 20–22] we now introduce a formal concept of a Zeno execution.

Definition 4. An execution of an ASS is called Zeno execution if the corresponding switching sequence is an infinite sequence such thatAn admissible input , wherewhich generates the Zeno execution is called a Zeno input.

Due to the specific character of the characteristic functions (elements of the sequence ) and taking into account Definition 2, we can express as a solution to the initial value problem (5) with . By we denote here a set of the “conventional” control inputs that imply a Zeno execution in the given ASS.

The existence of the Zeno dynamics for various classes of switched and hybrid systems is deeply discussed in [22]. Note that the Zeno behaviour is a typical effect that occurs in a switched and hybrid dynamic system. It appears in mathematical models of many real engineering systems, such as water tank model and mechanics of a bouncing ball, in aircraft conflict resolution model and in the dynamic models for safety control. We refer to [2, 10, 18–22] for some further switched-type mathematical models that admit Zeno executions. Let us also mention the celebrated Fuller’s problem that constitutes a generic example of the Zeno behaviour (see [26]).

3. Consistent Approximations of the Zeno Dynamics

Let us introduce some necessary mathematical concepts and facts. We next use the notation for the Banach space of all -valued continuous functions on equipped with the usual maximum-norm .

Definition 5. Consider an ASS given by system (5) and assume that all the conditions of Section 2 are satisfied. One says that this ASS possesses a strong approximability property (with respect to the admissible control inputs) if there exists a -weakly convergent sequenceof functions , such that the initial value problem (5) with the control input has a unique solution for every and is a -convergent (uniformly convergent) sequence.

We now formulate the main analytic result that guarantees the strong approximability property for the ASS given affine systems (5) (see [13, 24, 25] for the additional mathematical details).

Theorem 6. Let all the assumptions from Section 2 be satisfied and letbe a Zeno execution generated by a Zeno input . Consider the initial value problem (5) associated with a -weakly convergent sequence of control inputs , whereand is a -weak limit of . Then, for all the corresponding initial value problem (5) has a unique absolutely continuous solution determined on and

Proof. -weak convergence implies the -wea convergence (see, e.g., [27]). Therefore, also converges -weakl to the Zeno input . From the equivalent definition of the -wea convergence (see [27]) we next deduce the following:Here is the corresponding “conventional” part of the given Zeno input . Using the basic properties of the Lebesgue spaces (-spaces) on the finite-measure sets, we obtainBy we denote here the topologically dual space to . Consequently, (15) is also true for all from . This fact implies the -weak convergence of the given sequence of control inputsto the same functionThe expected result now follows from the main result of [24, Theorem , p. 5]. The proof is completed.

Theorem 6 provides an analytic basis for the constructive approximations of the Zeno executions. This approximation tools can not only be useful for a possible simplified analysis of the Zeno involved ASSs but also be used on the systems modelling phase. Note that the real world engineering processes do not present an exact Zeno behaviour. On the other side the corresponding mathematical models of some engineering interconnected systems may be Zeno due to the associated formal abstraction. This situation evidently implies a specific modelling conflict. The proposed approximations of the Zeno dynamics in ASSs can help to restore the adequateness of the abstract dynamic model under consideration.

Assume that all the conditions of Theorem 6 are fulfilled. Since is bounded and is weakly convergent, is a uniformly integrable sequence. This is a simple consequence of the celebrated Dunford-Pettis Theorem (see, e.g., [28]). Therefore one can choose an (existing) subsequence of the approximating sequence such thatfor every measurable set with measure , . For example, can be chosen as a sequence of step-functions. A concrete approximation , , , of elements of a Zeno input can be found from a suitable class of numerically tractable approximations. In the context of Theorem 6 one can consider, for example, the Fourier series:The Fourier series converges (as ) in the sense of -norm to a characteristic function of the chosen time interval . This fact is a simple consequence of the Parseval equality (see, e.g., [27, 29]). The given -convergence implies -convergence of (20). Let us now assume that as (the simple point wise convergence). This construction implies -convergence of the sequence to (see Figure 1).

Figure 1 

Approximation of an element of a Zeno input by Fourier series.

Evidently, the component of a Zeno input takes its values in the set of vertices of the following -simplex:-weakly convergent component of an approximating (Zeno input) sequence can also be defined as a function that takes its values from the above simplex . In that case possesses the additional useful properties summarized in the following theorem.

Theorem 7. Assume that all the conditions of Theorem 6 are satisfied. Let be -weakly convergent component of the approximating sequence and let for almost all . Then also converges strongly to .

Proof. Since is a convex polyhedron, the tangent cone to this simplex at every vertex is a pointed cone (does not contain a complete line). Therefore, for an element there exists a number and a unit vector such that for all we haveBy and we denote here the norm and the scalar product in . The above estimation and -weak convergence of imply the following:where . This completes the proof.

Evidently, Theorem 7 brings out the best choice of the element of an approximating sequence associated with a Zeno execution.

4. The Zeno-Type Dynamics Determined by a Switching Manifold

Let us consider a smooth manifoldwithHere , , is a continuously differentiable function. Following [30, 31] we call an invariant manifold associated with a dynamic system of the formif for all . Here is an appropriate smooth function. The next abstract result gives a general invariance criterion.

Theorem 8. A smooth manifold is invariant for system (26) iff belongs to the tangent space of for all .

The proof of Theorem 8 is based on the extended Lyapunov-type technique (see [30] for details). Recall that the tangent space of takes the formwhere denotes a derivative of .

We now consider a specific case of ASSs such that the switching times , , are determined by a smooth manifold :where is a solution of (5). This modelling approach corresponds to some particular mathematical models of hybrid control processes. We refer to [20] for relevant examples and further formal details.

Theorem 9. Assume that all the conditions of Theorem 6 are satisfied and the switching times of an ASS are determined by (28), where is a smooth manifold. Let be a Zeno execution generated by a Zeno input . Let be an approximating sequence from Theorem 6. Then there exist projected dynamic processes given by the following initial value problem:where is a strong limit of a subsequence of and is a projection of a vector on the manifold . Moreover, is an invariant manifold for (29) and

Proof. From the assumptions of Section 2 (smoothness of the families , and boundedness of the admissible controls) it follows that the trajectories of the ASS under consideration have bounded derivatives for almost all . This fact and the basic Definition 4 imply where denotes the Euclidean distance between a vector and . Consider -weakly convergent sequence of controls from Theorem 6 such thatwhere is a sequence of solutions to (5) generated by . From (31) and (32) we next deduce Here is a projection on the tangent space . Moreover, the above relations (33) implySince the mappings and are continuous, we next obtainwhere is -limit of if ,The continuity and linearity of the projection operator on the subspace implyfor almost all . The obtained relation can be rewritten as follows:The smoothness of the families , from Definition 1 implies the projected differential equationdetermined on the tangent space . We refer to [32] for the necessary concepts and facts from the theory of projected dynamic systems defined by ordinary differential equations. Here is -limit (strong limit) of a subsequence of . Note that every -weakly convergent sequence has a strongly convergent subsequence. The inclusion is a consequence of the closeness of the set of admissible controls. Using the basic property (idempotency ) of the projection operator, we finally obtainand . By construction, is an invariant manifold for system (39). The proof is completed.