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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 612198, 17 pages
http://dx.doi.org/10.1155/2012/612198
Research Article

Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators

Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia), P.O. Box 644 de Bilbao, 48080 Bilbao, Spain

Received 13 August 2012; Accepted 17 October 2012

Academic Editor: Haydar Akca

Copyright © 2012 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The stabilization of dynamic switched control systems is focused on and based on an operator-based formulation. It is assumed that the controlled object and the controller are described by sequences of closed operator pairs on a Hilbert space of the input and output spaces and it is related to the existence of the inverse of the resulting input-output operator being admissible and bounded. The technical mechanism addressed to get the results is the appropriate use of the fact that closed operators being sufficiently close to bounded operators, in terms of the gap metric, are also bounded. That philosophy is followed for the operators describing the input-output relations in switched feedback control systems so as to guarantee the closed-loop stabilization.

1. Introduction

Control Theory is a relevant field from the mathematical theoretical point of view as well as in many applications. What is important, in particular, is the closed-loop stabilization of dynamic system under appropriate feedback control as a minimum requirement to design a well-posed feedback system. Concerning the stabilization, the stabilization accomplishing with the properties of absolute stability is a very important issue (stabilization for whole sets of families of nonlinear controlled systems subject to nonlinear controllers satisfying Lure’s-type or Popov-type inequalities) and hyperstability (the nonlinearity can be, in addition, time-varying) or its most general property of passivity. See, for instance, [115] and references therein. If the feed-forward controlled object is linear, then hyperstability of the whole closed-loop system requires, in addition, the positive realness of the feed-forward loop of the controlled system. See [49] and references therein. It is also important to maintain the stability properties with a certain tolerances to modelling errors to better describe real situations, that is, the achievement of closed-loop robust stabilization. See, for instance, [2, 16, 17] and references there in. On the other hand, the problems of closed-loop stabilization as oscillatory behaviour in switched and impulsive dynamic systems with several potential active parameterizations has been investigated in the last years with an important set of background results. See, for instance, [1827] and references there in. In particular, it can be said that if all the parameterizations are stable and linear and possess a common Lyapunov function then the closed-loop stabilization of the switched system is possible under arbitrary switching. However, in the general case, it is needed to maintain a minimum residence time at each active parameterization before next switching or, alternatively, at certain active parameterizations, being active after a bounded whole time from its last activation. Some formulations replace the minimum residence time required for stabilization of the switched system by a sufficiently large averaged time at each stable parameterization. See, [1825, 2837] and references there in for a background subject coverage. Extensions have been proposed for certain classes of hybrid systems and time-delay systems. See, for instance, [2127] and references there in. Generally speaking, most of the proposed results about the stabilization under switching rules for the various involved parameterizations have been formulated for feedback regulation controls, that is, for a closed-loop regulation system in the absence of an external reference signal.

This paper gives a formal framework for the case when both controlled system and controller are described by closed operators. In this way, the stability of the switched system is not mainly related to a feedback control law but to the switching law in-between parameterizations. The given formulation is based on the properties of the operators describing the input-output relations of the combined controlled object and its controller. In particular: (a) bounded operators are closed-operators while the converse is not true, in general, (b) linear closed operators being sufficiently close to bounded operators are also bounded, and (c) the input-output operator of a stable dynamic system has a bounded nonlinear inverse operator and vice versa for any admissible stabilizing controller, [38, 39]. The closeness between operator controlled object/controller -parameterized pairs associated with the given switching law is characterized in terms of “smallness” of the gap metric on the Hilbert space of inputs and outputs.

Let be a Banach space and let denote a nonempty set of linear bounded (then continuous) operators on with operator norms ; for all , where is an indicator set denoted by the notation . It will be said that a sequence of operators where is the corresponding indicator set. The notation for the indicator refers to a finite or infinite number of members of the operator set with all the subscripts from unit to being present in the set while the notation for the indicator (a finite or infinite numerable set of natural numbers) means they are not always consecutive natural numbers. Let us denote by the set of all convergent sequences of operators in which converge to some , where stands for the closure. Such a convergence, in principle, is open to happen in any well-posed sense as, for instance, weak convergence, strong convergence or uniform convergence in the sense that the convergence happens in the weak, strong, or uniform topologies. Sequences of operators where convergence properties are of interest are denoted simply by .

2. Preliminary Results on Closed and Bounded Sequences of Linear Operators

The subsequent result relies on the convergence of sequences of operators to limits so that we give the following simple result.

Theorem 2.1. ; for all .

Proof. as in the strong operator topology. Thus, for any such that, in the strong operator topology, as ranges over the unit ball in . Now, assume that in the uniform operator topology as . Thus, for any , such that ; for all . Since , , then if as in the uniform operator topology, for , , and . But the choice of is arbitrary and then the above constraint fails if for any given . Hence, is what contradicts the assumption as in the uniform operator topology so that there is some infinite subsequence of such that . Thus, either with improper set inclusion or . (Here, we are considering the sequences as sets what is trivially consistent). By reversing the roles of and one concludes that with improper set inclusion or . Then, .

Note that proof of the above result might be easily readdressed with the uniform operator topology as well with the replacements and for any . Note that is compatible with the existence of distinct convergent sequences to either or to which can contain common operators . The above result is linked with the so-called gap metric [1, 2], as follows. If are closed subspaces of then the directed gap from to is defined by where are the corresponding projection operators. Note that the directed gap is not symmetric, in general. The gap metric is defined by where the last identity holds (see, e.g., [38]). Note that the gap metric has the symmetry property so that it is a well-posed metric. Then, the metric space , with the gap metric , is a complete metric space which is also the Banach space , defined for the above norm, which induces the strong topology on . We can also use the above concept to define distances between closed operators via the gap metric. Since closed operators on of domain are bounded, the gap metric is also useful to quantify the “separation” between linear bounded (then continuous) operators whose domain is the whole vector space of the Banach space and which are not necessarily projection operators. If and are closed linear operators on , then the gap distance between them is , where is the graph of ; and we will denote such a gap distance by for the sake of simplicity. Note that the linear operator is closed if its graph is closed in the direct sum . The following result involves the proofs of some properties of convergent sequences on a Hilbert space under contractive conditions for the gap metric.

Theorem 2.2. Consider a sequence of linear closed operators on a Hilbert space , where is the set of nonnegative integers, such that is also bounded. Then, the following properties hold:(i)The operators in are also bounded if . Also, the sequence converges to a unique bounded operator on , which is bounded and unique and as , if the following constraints hold for some real sequence : (ii)Assume that is invertible with bounded inverse and . Then, the sequence exists consisting of bounded operators on , In addition, such a sequence of inverse operators converges to a unique bounded operator on , which is the bounded inverse operator of on in Theorem 2.2(i), if (2.5)-(2.6) hold.(iii)If the operators in are linear and closed and then the operators in are also densely defined. If, in addition, (2.5)-(2.6) hold then the densely defined operators of the sequence converge to a limit operator which is also densely defined, that is, their domains ; and are dense subsets of and their images ; and are contained in .

Proof. Let be an abbreviated notation for where the graph of , is the range of the operator defined on , that is, of the operator defined on . Proceed by complete induction by taking any and assuming that is bounded, such that and take Then, there exists such that . Define so that by Schwarz’s inequality. Since one gets from (2.8)-(2.9) that so that provided that is what guarantees that is bounded since is bounded. The above inequality is homogeneous in then it is true for all . Thus, is bounded so that it is . As a result, if is a sequence of closed operators with being bounded then is also bounded since is bounded since . Thus, if is a sequence of linear closed operators with bounded then is also a sequence of linear bounded operators. Now, note from (2.11) that as , then there is a unique (uniqueness follows by construction), under , if is what holds if Thus, note that the constraint (2.5) is a sufficient condition for the necessary to hold to guarantee a well-posed (2.11). Also, since converges to , as and there is a sufficiently large finite such that ; for all , it is proven that then is bounded since is bounded. Assume not, then there exists and such that and for all and some nonzero for any given . Then, and, one gets by taking , that leading to ; for all which leads to the contradiction . Thus, is bounded, and then closed since linear, and since and are linear, closed, bounded, and then continuous operators on . Property (i) has been proven.
To prove Property (ii), note that if exists then provided that exists and is bounded and . Thus, it suffices to prove that is invertible. Assume not and proceed by contradiction by assuming that . By linearity of the operator , it always exists with such that , and with so that: Then and is invertible. By complete induction it follows that if for all , and is invertible with bounded inverse then exists consisting of linear bounded operators. If, in addition, (2.5)-(2.6) hold then the sequences of inverse operators which is bounded and consists of sufficiently close linear operators converges to the bounded inverse linear operator .
To proof Property (iii), consider the inequality for any given . Therefore, , where stands for the closure, for the norm-induced metric on and, since , this leads to the homogeneous inequality for any . Then, . If consists of closed operators and is a sequence of densely defined operators then is a densely defined sequence of operators. Proceeding by complete induction, it follows that the sequence is densely defined and is closed by an analogous reasoning that the corresponding one used in the proof of Property (i). Furthermore, one has for large enough that so that and is densely defined. The proof is complete.

Remark 2.3. Note that if Theorem 2.2(i) holds then ; for all since the sequence of linear closed operators and its linear closed limit operator are bounded.

The next result extends for a sequence of convergent operators that a sequence of closed operators is also a linear sequence of operators if each of its elements are sufficiently close in terms of difference of norms, or in terms of the gap metric, to some linear operator and the above theorem holds.

Theorem 2.4. is a sequence of linear operators on if it is a sequence of closed operators on is linear operator, and The sequence has a linear limit operator if (2.18) is replaced by the stronger condition: for some given real sequence with satisfying ; for all for which a sufficient condition is

Proof. The first part related to (2.18) follows from Theorem 2.2 for arbitrarily close to , being either bounded or densely defined if is a linear closed operator on for any given . Then, either or so that is, furthermore, linear and closed since is closed and the property that bounded or densely defined linear operators are closed. Then, if is linear, then it follows by complete induction that is a sequence of linear operators. The second part of the theorem is proven by first noting that (2.19) guarantees (2.18) so that is still a sequence of linear operators which are also either bounded or densely defined. Furthermore, it is guaranteed from Theorem 2.2 that converges to a limit linear operator either bounded or densely defined provided that (2.19) holds under the necessary condition: namely, if which is guaranteed if , that is, if ; for all .

3. Stability of Dynamic Systems with Eventual Switches

We now describe a closed-loop (or feedback) linear dynamic system of a separable Hilbert space by the operator pair , formally identifying the physical closed-loop system, where and are operators on describing the input-output relationships of the controlled system (sometimes, simply referred to as the “plant” to be controlled) and its controller, respectively, as follows: where and and and are, respectively, externally applied (reference and noise) inputs and inputs to the controlled system and its controller, respectively. The separable Hilbert space is assumed over vector fields or for the input to the controlled system, of state dimension , and over vector fields or for the input to the controller, of state dimension . The closed-loop dynamic system operator pair is defined by operator , where demotes the -identity matrix, defined on the direct sum of the extended space of the Hilbert space with itself, where and , subject to if , is a projection operator which defines the seminorm on for and . The subscript denoting the orders of identity matrices will be omitted in the following when no confusion is expected. The family of seminorms defines the resolution topology on , since is a resolution of the identity with being a truncation of ; for all , the separation property for , such that and the convergence in this topology is defined as follows: converges to if as ; for all . It is said that the closed-loop system is well-posed if the internal input is a causal function of the external input . This is equivalent to the operator to be causally invertible. If is a discrete set starting at , all invertible operators are bounded and causally invertible. The closed-loop system is said to be stable if has a bounded causal inverse defined on such that where is a matrix of -order composed of block matrices of which the first one is the -identity matrix and the remaining ones are zero, so as to collect the current components of in a vector of components. is an operator defining the response to initial conditions from to if the operators and represent linear dynamic systems subject to initial conditions , , , ,. If is a set of consecutive nonnegative integers, then (3.3) describes a causally invertible linear controlled discrete system. It can also describe a linear continuous dynamic system if is formed for nonnegative real intervals of the form . The stability of the linear dynamic system is associated with the existence of a causal inverse of as follows, [38].

Theorem 3.1. The closed-loop system is stable if and only if equivalently (in geometric terms), if and only if the orthogonal projection of onto is an invertible operator, where is the graph of defined on and is the inverse graph of , being the subspace of , whose graph is defined on .

Note that if the closed-loop system is stable then and are bounded operators. Assume not bounded. Then, one can take and with such that is unbounded, thus is not stable. Thus, is bounded. Assume that is not bounded. Then, there is some bounded such that is unbounded from (3.3), since is bounded so it is , so that is unbounded. But then the external input is unbounded from (3.1), here a contradiction, so that the operator is bounded. The stability of the controlled system is now discussed under eventual switching in the parameterizations in both controlled object and its controller. To establish the particular stability properties, Theorem 3.1 is addressed together with the relevant results of Section 2. In the following, we adopt the convention that the projector is defined for all with ; for all so as to facilitate the formal presentation of some of the subsequent equations. The subsequent result, supported by Theorem 2.2, relies on the stability of a switched system with switches between several possible stable parameterizations provided that there is a convergence to one of them either in finite time (i.e., the switching process ends in finite time) or asymptotically.

Theorem 3.2. Assume that there is a finite or infinite switching set of strictly ordered time instants with for any and some , where . Consider the sequence of linear closed operators on the Hilbert space , where such that is also bounded and invertible with bounded inverse , and where . Then, the following properties hold:(i)The sequence exists and it consists of bounded operators on . (ii)In addition, such a sequence of inverse operators converges to a unique bounded operator on , which is the bounded inverse operator of on in Theorem 2.2(i), if hold with the replacement . Furthermore, the switched closed-loop system of sequence pairs of operators , each of them being stable, is stable.

Proof. One gets from (3.3) that if , where is an infinite cardinal number for numerable sets, and if provided that the above inverse operators exist, where with a number of parameterization switches being ; for all for some two consecutive switching time instants such that . One has under zero initial conditions that provided that exists. Note that if the inverses exist, where , for all , and it is bounded if , for all , for all from Theorem 2.2 with the replacements being what is guaranteed if ; for all , for all , since Thus, is a sequence of sufficiently close operators in terms of the gap metric so that they are bounded closed operators from Theorem 2.2((i)-(ii)) with the replacements following complete induction. The remaining of the proof follows also by complete induction concerning the convergence of the sequence to a bounded closed operator follows from the convergence conditions (2.5)-(2.6) of Theorem 2.2 to get conditions (3.7)-(3.8). Thus, the operator sequence of elements defined in (3.11) is also closed with an associate bounded existing sequence of bounded inverse operators which has a bounded invertible limit if converges under the conditions (3.7)-(3.8). Note also from (3.11) that if exists and for all if . Thus, is bounded with a an existing finite sequence of bounded and closed inverse operators converging to a bounded limit if with having also a bounded inverse; for all for and for all if . Then (3.4) holds and the switched system is stable from Theorem 3.1 since: since exists consisting of bounded operators on , and by construction on for all . In addition, such a sequence of inverse operators converges to a unique bounded operator on so that (3.16) implies (3.4) and, equivalently, (3.5) in Theorem 3.1. Then, the switched closed-loop system defined by the convergent sequence of operator pairs is stable.

Remark 3.3. Note that the above result also holds if a finite time interval is removed from the analysis, that is, if there is a finite number of switches between a set of parameterizations not all being stable and after such a finite time interval the hypotheses hold. Note that (3.6) in Theorem 3.2 guarantees the existence of the inverse operator and its boundedness since they are closed and also sufficiently close (in terms of the gap metric) to each next consecutive element within such a sequence. On the other hand, the constraint (3.8) in Theorem 3.2 guarantees the convergence of the sequence of existing inverse operators to a closed operator which is also bounded so that Theorem 3.1 holds.

Remark 3.4. Note that Theorem 3.2 guarantees the stability of a switched system whose sequence of parameterizations converges to a stable configuration while all such parameterizations are stable. However, it is easy to generalize the result to two weaker conditions as follows:(1)Not all the parameterizations ; are stable but Theorem 3.2 conditions are fulfilled for , where and , with are two consecutive marked elements of which are not necessarily consecutive in such that is stable.(2)Theorem 3.2 is fulfilled only for the subset obtained by removing a finite set from .

The fact that the convergence of the sets of parameterizations to a stable one is not required for stabilization purposes is now discussed while it is sufficient that the switched parameterized sequence has consecutive stable parameterizations of sufficiently large norms.

Theorem 3.5. Assume the following.(1)There is a switching set of infinite cardinal of strictly ordered time instants with , for any and some , where .(2)The sequence of linear closed operators on the Hilbert space , where , is subject to being bounded and invertible with a bounded inverse , and where ; for all , for all . (3) where for some nonnegative strictly decreasing real sequence and some bounded nonnegative real sequence which depends on the active parameterization within which is parameterized by a bounded function of parameters .
Then, the switched closed-loop system of sequence pairs of operators , each of them being stable, is stable.

Proof. From the first part of Theorem 3.2, one deduces that the sequence exists and it consists of bounded operators on . The assumption implies as since , with as , according to (3.18). Thus, from (3.15) in the proof of Theorem 3.2, is bounded, the sequence exists and consists of bounded and closed operators while it converges to a bounded closed operator on and then (3.16)-(3.17) hold and the switched closed-loop system is stable.

The condition of (3.18) with in Theorem 3.5 implies that the operators describing the controlled object and controller both converge. It is not required for the parameterization, whose worst-case contribution to the norm , given by , to converge. In real situations, the switching process can activate stable parameterizations without convergence to a particular one provided that a sufficiently large residence time , such that , is respected at each active parameterization so that as in (3.18).

Theorem 3.5 is extended as follows by addressing the existence of the inverses of the relevant operators for finite strips of composite operators rather that for each individual operator.

Theorem 3.6. Assume the following.(1)There is a switching set of infinite cardinal of strictly ordered time instants with , for any and some , where .(2)There is a switching set of infinite cardinal of marked strictly ordered time instants such the composite operator sequence is a composite operator defined for and for some , , and where the operator consists of finite strips of linear closed operators on such that is bounded and invertible of bounded inverse satisfying , and for some nonnegative strictly decreasing real sequence and some bounded nonnegative real sequence which depends on the active parameterization within which is parameterized by a bounded function of parameters .
Then, the switched closed-loop system of sequence pairs of composite operators , each of them being stable, is stable.

Proof. Linked to (3.13), let us now consider (3.19)-(3.20). The theorem hypothesis guarantee the existence of the inverse operator being closed and bounded. From (3.21), the finite operators ; for all , for all converge to a closed invertible operator of bounded inverse.

Acknowledgments

The author thanks MEC, Basque Government, and University of Basque Country UPV/EHU for their supports through Grants DPI2012-30651, IT378-10, and UFI 11/07. The author also thanks the referees for their useful suggestions.

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