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Detectability and Invariance Properties for Set Dynamical Systems*
10th 10th IFAC IFAC Symposium Symposium on on Nonlinear Nonlinear Control Control Systems Systems August 23-25, 2016. California, USA 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, Monterey, California, USA 10th IFAC Symposium on Nonlinear Control Systems August 23-25, 2016. Monterey, California, USA online at www.sciencedirect.com Available August 23-25, 2016. Monterey, California, USA
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Detectability and Invariance Properties Detectability and Invariance Properties Detectability and Invariance Properties Detectability and Invariance Properties for Set Dynamical Systems for Set Dynamical Systems for Systems for Set Set Dynamical Dynamical Systems ∗ ∗∗
Nathalie Risso Risso ∗ Ricardo Ricardo G. G. Sanfelice Sanfelice ∗∗ Nathalie ∗ ∗∗ Nathalie Nathalie Risso Risso ∗ Ricardo Ricardo G. G. Sanfelice Sanfelice ∗∗ ∗ ∗ Department of Electrical and Computer Engineering, University of ∗ Department of Electrical and Computer Engineering, University of and Engineering, University ∗ Department of Electrical Arizona, AZ USA. Department of Electrical and Computer Computer Engineering, University of of Arizona, Tucson, Tucson, AZ 85719, 85719, USA. Arizona, Tucson, AZ 85719, USA. (e-mail: [email protected].) Arizona, Tucson, AZ 85719, USA. (e-mail: [email protected].) ∗∗ (e-mail: [email protected].) ∗∗ Department of Computer Engineering, University University of of California, California, (e-mail: [email protected].) Computer Engineering, ∗∗ Department of of Computer of ∗∗ Department Santa Cruz, CA 95064, USA. [email protected]) Department of CA Computer Engineering, University of California, California, Santa Cruz, 95064, Engineering, USA. (e-mail: (e-mail:University [email protected]) Santa Santa Cruz, Cruz, CA CA 95064, 95064, USA. USA. (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Abstract: Invariance Invariance properties properties and and convergence convergence of of solutions solutions of of set set dynamical dynamical systems systems are are Abstract: Invariance properties and convergence of solutions of set dynamical systems are studied. Using a framework for systems with set-valued states, notions of stability Abstract: Invariance properties convergence of solutions of setnotions dynamical systems and are studied. Using a framework for and systems with set-valued states, of stability and studied. Using a for with set-valued states, notions stability and detectability, similar to results classical dynamical systems, are defined and studied. Using a framework framework for systems systems with set-valued states, notions of stability and detectability, similar to the the existing existing results for for classical dynamical systems, are of defined and used used detectability, similar to the existing results for classical dynamical systems, are defined and used to obtain information about the convergence properties of solutions. In particular, it is shown detectability, similar toabout the existing results for properties classical dynamical systems, are defined used to obtain information the convergence of solutions. In particular, it and is shown to obtain about convergence properties solutions. particular, it that local stability, boundedness can convergence of to obtain information about the the and convergence properties ofcombined solutions.to Inconclude particular, it is is shown shown that local information stability, detectability, detectability, and boundedness can be beof combined toIn conclude convergence of that local stability, detectability, and boundedness can be combined to conclude convergence set-valued solutions. Under the assumption of bounded solutions and outer semicontinuity of that local stability, and boundedness can besolutions combinedand to conclude convergence of set-valued solutions.detectability, Under the assumption of bounded outer semicontinuity set-valued solutions. Under the assumption bounded solutions outer of the maps define system’s dynamics, properties for set-valued solutions. Under the the assumption of boundedinvariance solutions and and outer semicontinuity semicontinuity of the set-valued set-valued maps that that define the system’s of dynamics, invariance properties for set set dynamical dynamical the set-valued maps that define the system’s dynamics, invariance properties for set dynamical systems are also presented along with an invariance principle. The invariance principle involves the set-valued maps that define system’s dynamics, invariance set dynamical systems are also presented alongthe with an invariance principle. The properties invariance for principle involves systems are also along an principle. The involves the to invariant sets. illustrate the systems areLyapunov-like also presented presentedfunctions along with with an invariance invariance principle. The invariance invariance principle involves the use use of of Lyapunov-like functions to locate locate invariant sets. Examples Examples illustrateprinciple the results. results. the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results. the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Invariance Invariance principle, principle, set set dynamical dynamical systems, systems, stability stability of of nonlinear nonlinear systems, systems, Keywords: Invariance principle, set dynamical systems, stability of nonlinear Lyapunov methods, discrete event systems Keywords: Invariancediscrete principle, setsystems dynamical systems, stability of nonlinear systems, systems, Lyapunov methods, event Lyapunov Lyapunov methods, methods, discrete discrete event event systems systems 1. INTRODUCTION in Artstein Artstein and and Rakovi´ Rakovi´cc (2011). (2011). Even Even though though some some early early 1. INTRODUCTION in 1. in Artstein Rakovi´ cc (2011). though some predecessors exists, aa tool for dynamical that 1. INTRODUCTION INTRODUCTION in Artstein and and Rakovi´ (2011). Even thoughsystems some early early predecessors exists, tool for set set Even dynamical systems that predecessors exists, a tool for set dynamical systems that resembles the widely used invariance principle is currently Complex technological systems available nowadays require tool invariance for set dynamical that resembles theexists, widelyaused principlesystems is currently Complex technological systems available nowadays require predecessors resembles the widely used invariance principle is currently not available. Complex technological systems available nowadays require a suitable representation for realistic behavior analysis. the widely used invariance principle is currently not available. Complex systems availablebehavior nowadaysanalysis. require resembles a suitabletechnological representation for realistic not available. a suitable representation for behavior dynamics often restrictions that aSystem suitable representation for realistic realistic behavior analysis. analysis. System dynamics often incorporate incorporate restrictions that the the not Withavailable. the the objective objective of of developing developing an an invariance invariance principle principle System dynamics often incorporate restrictions that the state variables variables must satisfy, as well wellrestrictions as a a model modelthat of the With System dynamics often incorporate state must satisfy, as as of With the objective of developing an principle so as to characterize the behavior of systems setWith objective of the developing invariance principle so as the to characterize behavioranofinvariance systems with with setstate variables must satisfy, as well as a model of the disturbances. Methods to check properties of the behavior state variablesMethods must satisfy, well as aofmodel of the so disturbances. to checkasproperties the behavior as to characterize the behavior of systems with setvalued solutions, in this paper we study convergence of so as to characterize the behavior of systems with setvalued solutions, in this paper we study convergence of disturbances. Methods to check properties of the behavior of such systems, without the need of explicitly computing disturbances. Methods to the check properties of the behavior solutions of such systems, without need of explicitly computing valued solutions, in this paper we study convergence of and invariance properties of set dynamical sysvalued solutions, in this paper we study convergence of solutions and invariance properties of set dynamical sysof such systems, without the need of explicitly computing solutions are main used in analysis system of such systems, the need computing solutions are the thewithout main tools tools used of in explicitly analysis and and system solutions and invariance properties of set dynamical systems. Here, we consider systems with a set-valued state solutions andweinvariance of set dynamicalstate systems. Here, consider properties systems with a set-valued solutions are the main tools used in analysis and system design nowadays. For such purposes, formulations based solutions are the main toolspurposes, used in analysis and system design nowadays. For such formulations based tems. Here, with state that in discrete time to tems. Here, we we consider systems with aa set-valued set-valued state that evolves evolves in consider discrete systems time according according to aa set-valued set-valued design nowadays. For based on are suited design nowadays.frameworks For such such purposes, purposes, formulations based that on set-theoretic set-theoretic frameworks are well wellformulations suited for for design design evolves in discrete time according to a set-valued map and a constraint given in terms of a set, both in that evolves in discrete time according to a set-valued map and a constraint given in terms of a set, both in on set-theoretic frameworks well for problems involving constraints, uncertainties, opon set-theoretic frameworks are well suited suitedmultiple for design design problems involving constraints,are uncertainties, multiple op- map and a constraint given in terms of a set, both in Euclidean space. Solutions associated to such systems are map and a constraint given in terms of a set, both in Euclidean space. Solutions associated to such systems are problems involving constraints, uncertainties, multiple operation among others. problems involving constraints, eration points, points, among others. uncertainties, multiple op- Euclidean space. Solutions associated to such systems are given by sequences of sets, rather than sequences of points Euclidean space. Solutions associated such systems are given by sequences of sets, rather than to sequences of points eration points, among others. eration points, among others. given of than sequences points as the case dynamical in Most by sequences of sets, sets, rather rather than systems sequences ofdiscrete points as in in by thesequences case of of classical classical dynamical systems inof discrete Most popular popular methods methods to to study study the the behavior behavior of of dynamical dynamical given as in the case of classical dynamical systems in discrete time. In particular, we extend the framework introduced Most popular methods to study the behavior of dynamical systems consider a classical state representation, namely a as in the case of classical dynamical systems in discrete time. In particular, we extend the framework introduced Most popular methods to study the behavior of dynamical systems consider a classical state representation, namely a In the introduced in (2014) dynamical systems outputs systems consider a namely aa time. single-valued vector in space. On the time. In particular, particular, weset extend the framework framework introduced in Sanfelice Sanfelice (2014) to towe setextend dynamical systems with with outputs systems consider a classical classical state representation, namely single-valued vector in some somestate space.representation, On the the other other hand, hand, the in Sanfelice (2014) to set dynamical systems with outputs and formulate appropriate notions of detectability and stasingle-valued vector in some space. On the other hand, the generation of tools to study the properties of systems in (2014) to set dynamical with and outputs andSanfelice formulate appropriate notions of systems detectability stasingle-valued in some On the other hand, the generation of vector tools to studyspace. the properties of systems in in and formulate appropriate notions of detectability and stability. We follow similar ideas to those existing for classical generation of tools to study the properties of systems in which set-valued states represent variables or multivalued and formulate appropriate notions of detectability and stability. We follow similar ideas to those existing for classical generation of tools to study the properties systems in which set-valued states represent variables orofmultivalued bility. We follow similar ideas to those existing for classical dynamical systems as well as those in the literature of setwhich set-valued states represent variables or multivalued signals has experienced a more delayed development. Early bility. We follow similar ideas to those existing for classical dynamical systems as well as those in the literature of setwhich set-valued states represent variables or multivalued signals has experienced a more delayed development. Early dynamical systems as well as those in the literature of setvalued dynamical systems, such as Panasyuk (1986), Pelsignals has experienced a more delayed development. Early approaches in this direction can be found in works such systemssystems, as well as those the literature of Pelsetvalued dynamical such as in Panasyuk (1986), signals has experienced a morecan delayed development. approaches in this direction be found in worksEarly such dynamical valued dynamical systems, such as Panasyuk (1986), Pelczar (1994), and Artstein (1995). The proposed concepts approaches in this direction can be found in works such as Pelczar (1977), where a type of generalized systems dynamical systems,(1995). such asThe Panasyuk (1986), Pelczar (1994), and Artstein proposed concepts approaches in this direction can be in works such valued as Pelczar (1977), where a type of found generalized systems czar (1994), Artstein (1995). proposed concepts are then then usedand to obtain obtain information about the convergence convergence as Pelczar where a of systems is and are (1994), and Artstein (1995). The The proposed concepts are used to information about the as Pelczar (1977), (1977), wherestability a type type properties of generalized generalized systems czar is presented presented and basic basic stability properties are studied, studied, are then used to obtain information about the convergence properties of solutions and to state an invariance principle is presented and basic stability properties are studied, with follow-up developments for properties the study study of of limit sets are then used to obtainand information about the convergence properties of solutions to state an invariance principle is presented and basic stability arelimit studied, with follow-up developments for the sets properties of and to state principle systems. our with follow-up developments for limit in (1991). Notions for properties of solutions solutions and In to particular, state an an invariance invariance principle for set set dynamical dynamical systems. In particular, our contributions contributions with follow-up developments for the the study study ofgeneralized limit sets sets for in Pelczar Pelczar (1991). Notions of of reachability reachability forof generalized for set dynamical systems. In particular, our contributions include: in Pelczar (1991). Notions of reachability for generalized pseudo dynamical systems are formally studied in Pelczar set dynamical systems. In particular, our contributions include: in Pelczar (1991). systems Notions are of reachability for generalized pseudo dynamical formally studied in Pelczar for include: pseudo systems are studied Pelczar (1994). More properties of with setpseudo dynamical systems are formally formally studied in in Pelczar (1994). dynamical More recently, recently, properties of systems systems with set- include: •• Basic Basic assumptions assumptions on on the the system system data data for for key key strucstruc(1994). More recently, properties of systems with setvalued states states in continuous time are are studied inwith Artstein (1994). More in recently, properties of studied systemsin setvalued continuous time Artstein •• Basic assumptions on the system data for tural properties, namely, a semigroup property, a Basic assumptions on theasystem dataproperty, for key key strucstructural properties, namely, semigroup a sesevalued states in continuous time are studied in Artstein (1995) and a framework for the design of output feedback valued states in continuous time are studied in feedback Artstein (1995) and a framework for the design of output tural properties, namely, aa semigroup property, a sequential compactness property, and closedness of tural properties, namely, semigroup property, a the sequential compactness property, and closedness of the (1995) and a framework for the design of output feedback algorithms incorporating set-valued systems is developed (1995) and incorporating a framework for the design of output feedback algorithms set-valued systems is developed quential compactness property, and closedness of the set-valued states and outputs. quential compactness property, and closedness of the set-valued states and outputs. algorithms algorithms incorporating incorporating set-valued set-valued systems systems is is developed developed set-valued states and •• Relationships between local set-valued states and outputs. outputs. Relationships between local stability, stability, detectability, detectability, This research research has has been been partially partially supported supported by by the the National National Science Science This •• Relationships between local stability, detectability, and boundedness to conclude convergence of Relationships between local stability, detectability, and boundedness to conclude convergence of setsetFoundation under CAREER Grant no. ECCS-1450484 and Grant This research has been partially supported by the National Science Foundation under Grant no. ECCS-1450484 andScience Grant This research hasCAREER been partially supported by the National and boundedness to conclude convergence of valued solutions. and boundedness to conclude convergence of setsetvalued solutions. no. CNS-1544396, CNS-1544396, and by the the Air Air Force Office of Scientific Scientificand Research Foundation under and CAREER Grant no.Office ECCS-1450484 Grant no. by Force of Research Foundation under CAREER Grant no. ECCS-1450484 and Grant valued solutions. valued solutions. under Grant no. FA9550-16-1-0015. no. CNS-1544396, and by the Air Force Office of Scientific Research under Grant no. FA9550-16-1-0015. no. CNS-1544396, and by the Air Force Office of Scientific Research
under Grant no. FA9550-16-1-0015. under Grant no. FA9550-16-1-0015. Copyright © 2016 1048 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 1048Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2016 responsibility IFAC 1048Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 1048 10.1016/j.ifacol.2016.10.303
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• Results about convergence of set-valued solutions based on invariant sets involving the output map. • An invariance principle, similar to the one available for continuous-time and discrete-time systems; see, e.g., LaSalle (1967). Under the said basic assumptions, our invariance principle uses a real-valued Lyapunov-like function to locate invariant sets of bounded and complete solutions. Although results on convergence and stability properties for dynamical systems under a set-valued approach have been generated in the past, see Artstein and Rakovi´c (2011) for results concerning systems with output feedback, and results pertaining to invariance of sets are available (see Blanchini and Miani (2007), Artstein and Rakovi´c (2008)), our approach provides a tool to analyze convergence of solutions by using an invariance principle for set dynamical systems, which allows to use nonstrict Lyapunov functions for determining where set-valued solutions converge to. To the best of our knowledge, such tool is not available in the literature. 1 The reminder of this paper is organized as follows. After basic notation is introduced, Section 2 presents concepts associated to properties that are relevant for set dynamical systems behavior. The framework for set dynamical systems with outputs is defined in Section 3. The main results are presented in Section 4. More precisely, results relating detectability, stability, and convergence of solutions are described in Section 4.1. Invariance properties for set dynamical systems are presented in Section 4.2, where a formulation of an invariance principle for set-valued systems is proposed. Examples throughout the paper illustrate the ideas.
Definition 2.2. (inner and outer limit). For a sequence of n sets {Ti }∞ i=0 in R :
• The inner limit of the sequence {Ti }∞ i=0 , denoted by lim inf i→∞ Ti , is the set of all points x ∈ Rn for which there exist points xi ∈ Ti , i ∈ N, such that limi→∞ xi = x; • The outer limit of the sequence {Ti }∞ i=0 , denoted by lim supi→∞ Ti , is the set of all points x ∈ Rn for ∞ which there exist a subsequence {Tik }∞ k=0 of {Ti }i=0 and points xk ∈ Tik , for all k ∈ N, such that limk→∞ xk = x.
The limit of the sequence exists if the outer and the inner limit sets are equal; namely, lim Ti = lim inf Ti = lim sup Ti i→∞
Some basic definitions and properties that are used to characterize set dynamical systems are given in this section. We consider the standard topology in Rn . A set is closed if and only if it contains its boundary. A set is compact if bounded and closed. Definition 2.1. (distance between sets). The Hausdorff distance between two closed sets A1 , A2 ⊂ Rn is given by d(A1 , A2 ) = max sup |x|A2 , sup |z|A1 z∈A2
Note that Proposition 9.1 in Artstein and Rakovi´ c (2011) is a convergence result that requires the Lyapunov function to strictly decrease while the convergence criterion of sets in Proposition 4.5 in Artstein and Rakovi´ c (2008) requires a strict contraction property.
i→∞
Definition 2.3. (union of a collection of sets). Let C be a collection of sets. The union of C is C = {x : x ∈ X for some X ∈ C}. Definition 2.4. (Rockafellar and Wets, 2009, convergence of a sequence of sets) When the limit of the sequence n {Ti }∞ i=0 in R exists in the sense of Definition 2.2, and is equal to T , the sequence of sets {Ti }∞ i=0 is said to converge to the set T . 3. SET DYNAMICAL SYSTEMS WITH OUTPUTS We consider set dynamical systems defined by X+ = G(X) Y = H(X)
Notation: The following notation is used throughout this paper. N denotes the natural numbers including 0, i.e., N = {0, 1, . . .}. Rn denotes the n-dimensional Euclidean space. R≥0 denotes the nonnegative real numbers. B denotes the closed unit ball in Euclidean space. Given a set A, its closure is denoted by A. Given a map V , dom V denotes its domain of definition. Given x ∈ Rn , |x| denotes the Euclidean vector norm and |x|1 denotes the taxicab norm. For a closed set A ⊂ Rn and x ∈ Rn , we define the distance |x|A = inf y∈A |x − y|. Given a function V : dom V → R and a constant r ∈ R, its r-sublevel set is given by LV (r) := {x ∈ dom V : V (x) ≤ r }. For a map G, Gn denotes the n-th composition.
x∈A1
i→∞
The inner and outer limits of a sequence always exist and are closed 2 , although the limit itself might not exist.
2. PRELIMINARIES
1
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(1)
X⊂D where X is the set-valued state, Y is the system’s output, G : Rn ⇒ Rn and H : Rn ⇒ Rm are set-valued maps defining the right-hand side and the output map, respectively, and D ⊂ Rn defines a constraint that solutions to the system must satisfy. We say that (D, G, H) is the data of (1). A solution to the system in (1) is defined as the sequence of nonempty sets {Xj }Jj=0 and its associated output is defined by the sequence {Yj }Jj=0 , J ∈ N ∪ {∞}, satisfying Xj+1 = G(Xj ) Yj = H(Xj ) Xj ⊂ D
over the domain of definition of the solution {Xj }Jj=0 , which is given by the collection {0, 1, 2, . . . , J} ∩ N, and denoted by dom X. The first entry of the solution, X0 , is the initial set. We assume X0 to be compact. If a solution has J = 0 then we say that it is trivial, and if it has J > 0 we say that it is nontrivial. If it has J = ∞, we say that it is complete. A solution {Xj }Jj=0 is said to be maximal if it cannot be further extended. Given an initial set X0 ⊂ Rn , S(X0 ) denotes the set of maximal solutions 3 to (1) from X0 . 2 For a sequence of empty sets, the inner and outer limit is the empty set. 3 By Lemma 3.7, maximal solutions to (1) are unique.
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To make notation easier to follow, at times, the sequence (B1) For any solution X to (1) and any ˆj ∈ dom X, we of sets {Xj }Jj=0 is represented as Xj (or even just X). We ˆ given by X ˆ j = X ˆ for each j ∈ dom X ˆ= have that X j+j make the same notational simplification when referring to j : j + ˆj ∈ dom X is a solution to (1). the output Y. The term solution-output pair {X, Y} is used to represent a solution X and its associated output (B2) Suppose the data (D, G) of (1) satisfies (A0) and Y = H(X). Notation {Xij }∞ i=0 refers to the sequence of (A1) of Assumption 3.4. Let {Xi0 }∞ i=0 be an eventually solutions Xij indexed by i, where j is the associated discrete bounded (with respect to Rn ) sequence of sets convergtime. ing to a compact set X0 and suppose {Xi }∞ i=0 is such that Xi ∈ S (Xi0 ). Then, there exists a subsequence of {Xi }∞ 3.1 Definitions and Assumptions i=0 converging to some X ∈ S (X0 ). (B3) Suppose the data (D, G, H) of (1) satisfies Assumpi ∞ tion 3.4. Let {Xij }∞ i=0 and {Yj }i=0 be a solutionDefinition 3.1. (stability of a set). The set A is stable if output pair to (1), such that {Yji }∞ for each > 0 there exists δ > 0 such that each solution i=0 is bounded. Then, there exists a subsequence of {Yji }∞ Xj to (1) with d(X0 , A) ≤ δ satisfies d(Xj , A) ≤ for all i=0 that converges to a closed set. j ∈ dom Xj . For classical systems, detectability means that when the output remains at zero, the norm of solutions converges to zero as time goes to infinity. A related concept is defined for set dynamical systems next. Definition 3.2. (detectability). Let the sets A ⊂ Rn , N ⊂ Rm , and M ⊂ Rn be given, where A is closed. For the system in (1), the distance to A is detectable relative to N on M if for each complete solution X ∈ S(X0 ), with Xj closed for each j ∈ dom X, and associated output Y X⊂M =⇒ lim d(Xj , A) = 0 Y⊂N j→∞ In the next sections, results characterizing key dynamical properties of a set dynamical system as in (1) are proposed. Those results require the data of (1) to satisfy certain mild conditions, which are stated in the following assumption. Definition 3.3. (outer semicontinuity). The set-valued map G : Rn ⇒ Rn is outer semicontinuous at x ∈ Rn if for n each sequence {xi }∞ i=0 converging to a point x ∈ R and ∞ each sequence {yi }i=0 such that yi ∈ G(xi ) for each i, converging to a point y, it holds that y ∈ G(x). It is outer semicontinuous if G(x) is outer semicontinuous at each x ∈ Rn . Assumption 3.4. The set dynamical system defined in (1), with data (D, G, H) satisfies the following properties: (A0) The set-valued map G : Rn ⇒ Rn is outer semicontinuous, locally bounded 4 , and, for each x ∈ D, G(x) is a nonempty subset of Rn . (A1) The set D ⊂ Rn is closed. (A2) The set-valued map H : Rn ⇒ Rn is outer semicontinuous, locally bounded, and, for each x ∈ D, H(x) is a nonempty subset of Rn . 3.2 Preliminary results The following result establishes a structural property of the state and output trajectory of (1). A similar result appeared in (Sanfelice, 2014, Proposition 4.6). Proposition 3.5. (basic properties of solutions) The following properties hold for system (1): The map G is locally bounded if for each compact set K ⊂ Rn there exists a compact set K ⊂ Rn such that G(K) ⊂ K . 4
Theorem 3.6. (Rockafellar and Wets, 2009, Theorem 4.18) n Every sequence of nonempty sets {Ti }∞ either i=0 in R escapes to the horizon or has a subsequence converging to a nonempty set T ⊂ Rn , i.e., there exists a subsequence ∞ {Tik }∞ k=0 of {Ti }i=0 such that limk→∞ Tik = T . Lemma 3.7. Given a solution {Xj }Jj=0 to (1) and any subsequence {ji }Ii=0 of {j}Jj=0 , we have
∀i ∈ {0, 1, . . . , I} ∩ N Xji = Gji (X0 ) Moreover, every maximal solution to (1) is unique.
(2)
Now, we define the notion of ω-limit set. This notion is used later to establish convergence properties of solutions. Definition 3.8. (ω-limit set). The ω-limit set of a solution {Xj }Jj=0 to (1) is given by ω (Xj ) = {Y ⊂ Rn : ∃{ji }∞ i=0 , lim ji = ∞, Y = lim Xji } i→∞
i→∞
Note that ω (Xj ) is a collection of sets.
(3)
Theorem 3.9. (Sanfelice, 2014, Theorem 4.13) Suppose the data (D, G) of (1) satisfies Assumption 3.4 and that V : Rn → R is a Lyapunov-like function. Then, every solution {Xj }Jj=0 , J ∈ N ∪ {∞} to (1) from X0 ⊂ D satisfies 0, sup V (x) ⊂ 0, sup V (x) x∈Xj+1
(4)
x∈Xj
for all j ∈ {0, 1, . . . , J − 1} ∩ N. Moreover, if J = ∞ then (5) 0, sup V (x) lim 0, sup V (x) = j→∞
x∈Xj
j∈N
x∈Xj
and if, furthermore, {Xj }∞ j=0 is bounded then there exists r ∈ R≥0 such that V ( ω (Xj )) ⊂ [0, r] (6) 4. DETECTABILITY AND AN INVARIANCE PRINCIPLE This section pertains to the study of convergence and invariance properties of set dynamical systems. Notions and results on detectability and convergence are presented in Section 4.1. In Section 4.2, invariance properties of set
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dynamical systems are characterized and then used to prove an invariance principle, which corresponds to the main result in this paper. The proofs of some of the results borrow ideas from variational analysis in Rockafellar and Wets (2009) and the results for discrete-time systems in LaSalle (1967) and Teel (2006).
X3
X2
4.1 Detectability and Convergence X1
Structural properties of the solutions to set dynamical systems are used in this section, along with detectability, to conclude convergence of solutions to compact sets. Theorem 4.1. Let A and M be compact subsets of Rn . Suppose system (1) satisfies Assumption 3.4, the set A is stable, and the distance to A is detectable relative to N . Let X ∈ S(X0 ) be complete and Y be the associated output. If X ⊂ M and limj→∞ Yj = N then lim d(Xj , A) = 0 j→∞
Example 4.2. (illustration of Theorem 4.1) Consider the set dynamical system in (1) with G(X) = {g(x) : x ∈ X}, and g(x) = H(X) = {h(x) : x ∈ X} where 1 x21 , and D = {x ∈ R2 : , h(x) = 0.5 − x1 x2 1 + e−x1 |x|1 ≤ cD }, with cD > 0. This system satisfies Assumption 3.4 since both g and h are continuous and D is compact. Consider the compact sets M = {x ∈ R2 : |x|1 ≤ cM } with cM ≥ 0, A = {(0, 0)} and N = {0}. Let Xj be a complete solution from X0 = {x ∈ R2 : |x|1 ≤ c0 } with c0 < 1 and let Yj be its associated output. For cM < 1 and c0 ≤ cM , Xj ⊂ M for all j ∈ dom Xj since g is contractive on M in the sense that |g1 (x)| < |x1 | < 1 and |g2 (x)| < 1 for each x ∈ M. To establish stability of A, let > 0. For δ = , the solution Xj from X0 is such that d(Xj−1 , A) ≤ d(Xj , A) ≤ δ < 1, since g is contractive on X ⊂ M, so d(Xj , A) ≤ for all j ∈ dom Xj . We have that Y = N if and only if X ⊂ {x ∈ R2 : x1 = 0}. So any solution with X ⊂ M and Y = N is such that Yj = {0} for all j ∈ dom Xj , which implies that X0 ⊂ {x ∈ R2 : x1 = 0} and X1 = G(X0 ) = {(0, 0)}. Then, limj→∞ d(Xj , A) = 0. Thus, the distance to A is detectable relative to N on M. The assumptions in Theorem 4.1 are satisfied, which implies that the set-valued solution converges to A. A solution from the initial set X0 with c0 = 0.6, cM = 0.8, and cD = 0.9 is presented in Figure 1, where convergence to A can be observed. Theorem 4.3. Let A and M be compact subsets of R . Suppose system (1) satisfies Assumption 3.4, the set A is stable for system (1), and A is detectable relative to N on M. Then, for each > 0, there exists j ∗ ∈ N such that, for each solution-output pair (X, Y) with X ∈ S(X0 ), X⊂M =⇒ d(Xj , A) ≤ ∀j ≥ j ∗ , j ∈ dom X Y=N n
4.2 Invariance Properties In this section we formulate a convergence result based on invariant sets involving the output map. We introduce the concept of Lyapunov-like functions in the sense of Sanfelice (2014), and use such functions to locate invariant sets for
j X0
x1
x2
Fig. 1. Solution to the set dynamical system in Example 4.2 from initial set X0 = {x : |x|1 ≤ 0.6} (blue). As j increases, the solution converges to A = {(0, 0)}. solutions to set dynamical systems. We start by defining the concept of invariance for set dynamical systems. Definition 4.4. (forward and backward invariance). A set M ⊂ Rn is said to be forward invariant for (1) if for every set T ⊂ M ∩ D we have G(T ) ⊂ M ∩ D. A set M ⊂ Rn is said to be backward invariant for (1) if for every set T ⊂ M ∩ D for which there exists a set T with the property T = G(T ), we have T ⊂ M ∩ D for every such set T . A set M ⊂ Rn is said to be invariant if it is both forward and backward invariant. Recall that, according to the definition of solution, we have that all maximal solutions to (1) are unique. The following result from Sanfelice (2014) characterizes properties of the omega limit sets of solutions to set dynamical systems. Proposition 4.5. (Sanfelice, 2014, Proposition 4.9) Suppose the data (D, G) in (1) satisfies (A0) and (A1) in Assumption 3.4. Let Xj be a bounded and complete solution to (1). Then, ω (Xj ) is nonempty, compact, and invariant. Given a set dynamical system as in (1) and a set N , we define Y ∗ := {x ∈ D : H(x) ⊂ N }
which will be used in the formulation of the next results. Theorem 4.6. Let M be compact and let X ∈ S(X0 ) be a complete solution to the system in (1) satisfying Assumption 3.4. If X ⊂ M and its associated output Y satisfies limj→∞ Yj = N , then the solution X converges to the largest invariant set contained in M ∩ Y ∗ ∩ D. Proof Sketch: Since X remains in M for all j ∈ dom X, the solution is bounded and by Proposition 4.5, the omega limit set is nonempty, compact, and invariant. Since the system satisfies Assumption 3.4, from the definition of Yj we have that lim Yj = lim H(Xj ) = N . If X∗ is an j→∞
j→∞
element of ω (X), there is a subsequence of Xj indexed by ji such that lim H(Xji ) = H(X∗ ) = N . Moreover, for i→∞
(X), X∗ ⊂ {x ∈ D : H(x) ⊂ N } = Y ∗ . each X∗ ∈ ω
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Theorem 4.6 is later used in proving our invariance principle, which is in Theorem 4.9. Before that, we make a link between detectability and stability. Theorem 4.7. Let A and M be compact subsets of Rn and suppose, for the system described in (1), that the set A is stable. Then, the following statements are equivalent:
Remark 4.10. If, for the set dynamical system in (1), we have that G = g, with g : Rn → Rn , then Theorem 4.9 reduces to the original invariance principle for discretetime systems given in LaSalle (1967), with a sublevel set of V instead of a level set as stated in the original principle.
(1) The distance to A is detectable relative to N on M. (2) The largest invariant set contained in M ∩ Y ∗ is a subset of A.
Example 4.11. (illustration of Theorem 4.9) Consider the set dynamical as defined in (1) in R2 with G(X) = αx 2 1 + x21 {g(x) : x ∈ X } and g(x) = βx1 , The following definition introduces the notion of Lyapunovlike functions for set dynamical systems. 1 + x22 Definition 4.8. (Lyapunov-like function). A continuous func- D = [0, p] × [0, q], with p, q real positive numbers and tion V : Rn → R is called a Lyapunov-like function for α, β ≥ 0. The set dynamical system data satisfies Assumption 3.4 since g is continuous and D is compact. Consider system (1) if it satisfies as the Lyapunov-like function V (x) = x21 + x22 , which, for V (x) ≥ 0 ∀x ∈ D ∪ G(D) each x ∈ D and η = G(x), leads to
V (η) − V (x) ≤ 0 ∀x ∈ D, η ∈ G(x) α2 β2 2 − 1 x + − 1 x22 V (η) − V (x) = 1 (1 + x22 )2 (1 + x21 )2 Now we are ready to present our main result. ≤ (β 2 − 1)x21 + (α2 − 1)x22 Theorem 4.9. (invariance principle) Suppose system (1) For α2 < 1 and β 2 < 1, V (η)−V (x) ≤ 0 for all x ∈ X, with satisfies Assumption 3.4. Let V be a Lyapunov-like func- X ⊂ D, η ∈ G(X), and V is a valid Lyapunov-like function tion for (1). Let Xj be a bounded and complete solution for the set dynamical system. For each x ∈ D ∪ G(D), we to (1). Then, there is a number r ∈ R≥0 such that Xj have V (x) ≥ 0. Moreover, let Xj be a complete solution converges to the largest invariant set contained in from X0 ⊂ D. Since D is compact, X is bounded. The assumptions in Theorem 4.9 are satisfied E ∩ LV (r) ∩ D and the solution converges to the largest invariant set in E ∩LV (r)∩D for where some r ≥ 0. Since supη∈G(X) V (η) = supx∈X V (x) only for X = {(0, 0)}, the set E contains only the element {(0, 0)}, E := X ⊂ D : sup V (η) = sup V (x) x∈X η∈G(X) and the largest invariant set is the origin. A solution from the initial set X0 = [0.5, 2] × [0.5, 2], with p = 5, q = 4, and LV (r) := x ∈ D : V (x) ≤ r . α = β = 0.75 is presented in Figure 2. (Xj ) Proof Sketch: Since Xj is bounded and complete, ω is nonempty, compact, and invariant. Using Theorem 3.9, since Assumption 3.4 holds for system (1), V ( ω (Xj )) ⊂ [0, r] for some r ∈ R≥0 . Consider the set dynamical system in (1) with Y = (Y1 , Y2 ) = H(X) = (H1 (X), H2 (X)), where
H1 (X) = 0, sup V (x) − sup V (η)
x∈X
H2 (X) = 0, sup V (x) x∈X
η∈G(X)
to which the given Xj is a solution. We can apply Theorem 4.6, defining the sets N = N1 × N2 , N1 = {0}, N2 = {x ∈ D : V (x) ≤ r} and M = {x ∈ D : V (η) ≤ V (x)}. We have Xj ⊂ M for all j ∈ dom X. Using Theorem 3.9 we have that supx∈G(Xj ) V (x) ≤ supx∈G(Xj−1 ) V (x) for each j ∈ dom Xj \ {0} and the sequence of points {supx∈Xj V (x)}∞ j=0 converges to a real number b ∈ [0, r], with b = limj→∞ supx∈Xj V (x). Furthermore, we have that limj→∞ supx∈G(Xj ) V (x) − supx∈G(Xj−1 ) V (x) = 0. Then, for the first component of the output, Y1 , limj→∞ Y1j = N1 . The second output Y2 satisfies limj→∞ Y2j = N2 . Then, since limj→∞ Yj = N , using Theorem 4.6 for this system, we have that the solution Xj converges to thelargest invariant set in M∩Y ∗ ∩D, which corresponds to E ∩ LV (r) ∩ D.
For α2 = β 2 = 1, condition V (η)−V (x) ≤ 0 still holds, for all x ∈ X, with X ⊂ D, η ∈ G(X), so V is a Lyapunovlike function for the set dynamical system. In this case, condition supη∈G(X) V (η) = supx∈X V (x) is satisfied for X with elements x such
that
1 1 2 − 1 x1 + − 1 x22 = 0 (1 + x22 )2 (1 + x21 )2 which holds for each x in X = {x ∈ D : x1 = 0, x2 ≥ 0} ∪ {x ∈ D : x2 = 0, x1 ≥ 0}. Since this set of points belongs to the boundary of D and is invariant for G, we have that it defines the set E as well, namely, E = {x ∈ D : x1 = 0, x2 ≥ 0} ∪ {x ∈ D : x2 = 0, x1 ≥ 0}. Since the assumptions in Theorem 4.9 are satisfied, the set-valued solution converges to the largest invariant set in E ∩ LV (r) ∩ D, in this case E ∩ LV (r) (7) for some r ≥ 0. For a solution from the initial set X0 = [0, 1] × [0, 1], with p = 5, q = 4, the set-valued state converges to {x ∈ R2 : x1 = 0, x2 ≤ 1} ∪ {x ∈ R2 : x2 = 0, x1 ≤ 1} which is the set in (7) with r = maxx∈X0 V (x) = 1. Figure 3 shows the set-valued solution up to j = 6. 5. CONCLUSION
Convergence and invariance properties for set dynamical systems with outputs are studied in this paper. The
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(a) Set-valued state evolution.
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(a) Set-valued state evolution.
(b) Set-valued state convergence in the plane x1 − x2 .
(b) Set-valued state convergence in the plane x1 − x2 .
Fig. 2. Solution to the set dynamical system in Example 4.11, with α = β = 0.75 and X0 = [0.5, 2] × [0.5, 2], for j = 0 to 6.
Fig. 3. Solution to the set dynamical system in Example 4.11, with α = β = 1 and X0 = [0, 1] × [0, 1], for j = 0 to 6.
mathematical framework in Sanfelice (2014) was extended to systems with outputs and notions of stability and detectability for these systems were defined. Then, under the assumption of bounded solutions and outer semicontinuity, results relating stability, detectability and state trajectory convergence were developed. Invariance properties for set dynamical systems were characterized and an invariance principle, involving the use of Lyapunov-like functions to locate invariant sets, under assumption of bounded solutions and outer semicontinuity of the set-valued map that defines the system dynamics, was presented.
LaSalle, J.P. (1967). An invariance principle in the theory of stability. Differential Equations and Dynamical Systems (JP La Salle and JK Hale editors), Acad. Press, NY, 277–286. Panasyuk, A.I. (1986). Dynamics of sets defined by differential inclusions. Siberian Mathematical Journal, 27(5), 757–765. Pelczar, A. (1977). Stability in generalized pseudodynamical systems. Zeszyty Naukowe UJ, Prace Matematyczne, 19, 33–38. Pelczar, A. (1991). Remarks on limit sets in generalized semi-systems. Univ. Iagel. Acta Math, 28, 29–39. Pelczar, A. (1994). On modified prolongational limit sets and prolongations in dynamical systems on metric spaces. Univ. Iagel. Acta Math, 31, 175–202. Rockafellar, R.T. and Wets, R.J.B. (2009). Variational analysis, volume 317. Springer Science & Business Media. Sanfelice, R.G. (2014). Asymptotic properties of solutions to set dynamical systems. In 2014 IEEE 53rd Annual Conference on Decision and Control, 2287–2292. Teel, A.R. (2006). Class notes on nonlinear discrete-time systems.
REFERENCES Artstein, Z. (1995). A calculus for set-valued maps and setvalued evolution equations. Set-Valued Analysis, 3(3), 213–261. Artstein, Z. and Rakovi´c, S.V. (2011). Set invariance under output feedback: a set-dynamics approach. International Journal of Systems Science, 42(4), 539–555. Artstein, Z. and Rakovi´c, S.V. (2008). Feedback and invariance under uncertainty via set-iterates. Automatica, 44(2), 520–525. Blanchini, F. and Miani, S. (2007). Set-theoretic methods in control. Springer Science & Business Media.
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Detectability and Invariance Properties Detectability and Invariance Properties Detectability and Invariance Properties Detectability and Invariance Properties for Set Dynamical Systems for Set Dynamical Systems for Systems for Set Set Dynamical Dynamical Systems ∗ ∗∗
Nathalie Risso Risso ∗ Ricardo Ricardo G. G. Sanfelice Sanfelice ∗∗ Nathalie ∗ ∗∗ Nathalie Nathalie Risso Risso ∗ Ricardo Ricardo G. G. Sanfelice Sanfelice ∗∗ ∗ ∗ Department of Electrical and Computer Engineering, University of ∗ Department of Electrical and Computer Engineering, University of and Engineering, University ∗ Department of Electrical Arizona, AZ USA. Department of Electrical and Computer Computer Engineering, University of of Arizona, Tucson, Tucson, AZ 85719, 85719, USA. Arizona, Tucson, AZ 85719, USA. (e-mail: [email protected].) Arizona, Tucson, AZ 85719, USA. (e-mail: [email protected].) ∗∗ (e-mail: [email protected].) ∗∗ Department of Computer Engineering, University University of of California, California, (e-mail: [email protected].) Computer Engineering, ∗∗ Department of of Computer of ∗∗ Department Santa Cruz, CA 95064, USA. [email protected]) Department of CA Computer Engineering, University of California, California, Santa Cruz, 95064, Engineering, USA. (e-mail: (e-mail:University [email protected]) Santa Santa Cruz, Cruz, CA CA 95064, 95064, USA. USA. (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Abstract: Invariance Invariance properties properties and and convergence convergence of of solutions solutions of of set set dynamical dynamical systems systems are are Abstract: Invariance properties and convergence of solutions of set dynamical systems are studied. Using a framework for systems with set-valued states, notions of stability Abstract: Invariance properties convergence of solutions of setnotions dynamical systems and are studied. Using a framework for and systems with set-valued states, of stability and studied. Using a for with set-valued states, notions stability and detectability, similar to results classical dynamical systems, are defined and studied. Using a framework framework for systems systems with set-valued states, notions of stability and detectability, similar to the the existing existing results for for classical dynamical systems, are of defined and used used detectability, similar to the existing results for classical dynamical systems, are defined and used to obtain information about the convergence properties of solutions. In particular, it is shown detectability, similar toabout the existing results for properties classical dynamical systems, are defined used to obtain information the convergence of solutions. In particular, it and is shown to obtain about convergence properties solutions. particular, it that local stability, boundedness can convergence of to obtain information about the the and convergence properties ofcombined solutions.to Inconclude particular, it is is shown shown that local information stability, detectability, detectability, and boundedness can be beof combined toIn conclude convergence of that local stability, detectability, and boundedness can be combined to conclude convergence set-valued solutions. Under the assumption of bounded solutions and outer semicontinuity of that local stability, and boundedness can besolutions combinedand to conclude convergence of set-valued solutions.detectability, Under the assumption of bounded outer semicontinuity set-valued solutions. Under the assumption bounded solutions outer of the maps define system’s dynamics, properties for set-valued solutions. Under the the assumption of boundedinvariance solutions and and outer semicontinuity semicontinuity of the set-valued set-valued maps that that define the system’s of dynamics, invariance properties for set set dynamical dynamical the set-valued maps that define the system’s dynamics, invariance properties for set dynamical systems are also presented along with an invariance principle. The invariance principle involves the set-valued maps that define system’s dynamics, invariance set dynamical systems are also presented alongthe with an invariance principle. The properties invariance for principle involves systems are also along an principle. The involves the to invariant sets. illustrate the systems areLyapunov-like also presented presentedfunctions along with with an invariance invariance principle. The invariance invariance principle involves the use use of of Lyapunov-like functions to locate locate invariant sets. Examples Examples illustrateprinciple the results. results. the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results. the use of Lyapunov-like functions to locate invariant sets. Examples illustrate the results. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Invariance Invariance principle, principle, set set dynamical dynamical systems, systems, stability stability of of nonlinear nonlinear systems, systems, Keywords: Invariance principle, set dynamical systems, stability of nonlinear Lyapunov methods, discrete event systems Keywords: Invariancediscrete principle, setsystems dynamical systems, stability of nonlinear systems, systems, Lyapunov methods, event Lyapunov Lyapunov methods, methods, discrete discrete event event systems systems 1. INTRODUCTION in Artstein Artstein and and Rakovi´ Rakovi´cc (2011). (2011). Even Even though though some some early early 1. INTRODUCTION in 1. in Artstein Rakovi´ cc (2011). though some predecessors exists, aa tool for dynamical that 1. INTRODUCTION INTRODUCTION in Artstein and and Rakovi´ (2011). Even thoughsystems some early early predecessors exists, tool for set set Even dynamical systems that predecessors exists, a tool for set dynamical systems that resembles the widely used invariance principle is currently Complex technological systems available nowadays require tool invariance for set dynamical that resembles theexists, widelyaused principlesystems is currently Complex technological systems available nowadays require predecessors resembles the widely used invariance principle is currently not available. Complex technological systems available nowadays require a suitable representation for realistic behavior analysis. the widely used invariance principle is currently not available. Complex systems availablebehavior nowadaysanalysis. require resembles a suitabletechnological representation for realistic not available. a suitable representation for behavior dynamics often restrictions that aSystem suitable representation for realistic realistic behavior analysis. analysis. System dynamics often incorporate incorporate restrictions that the the not Withavailable. the the objective objective of of developing developing an an invariance invariance principle principle System dynamics often incorporate restrictions that the state variables variables must satisfy, as well wellrestrictions as a a model modelthat of the With System dynamics often incorporate state must satisfy, as as of With the objective of developing an principle so as to characterize the behavior of systems setWith objective of the developing invariance principle so as the to characterize behavioranofinvariance systems with with setstate variables must satisfy, as well as a model of the disturbances. Methods to check properties of the behavior state variablesMethods must satisfy, well as aofmodel of the so disturbances. to checkasproperties the behavior as to characterize the behavior of systems with setvalued solutions, in this paper we study convergence of so as to characterize the behavior of systems with setvalued solutions, in this paper we study convergence of disturbances. Methods to check properties of the behavior of such systems, without the need of explicitly computing disturbances. Methods to the check properties of the behavior solutions of such systems, without need of explicitly computing valued solutions, in this paper we study convergence of and invariance properties of set dynamical sysvalued solutions, in this paper we study convergence of solutions and invariance properties of set dynamical sysof such systems, without the need of explicitly computing solutions are main used in analysis system of such systems, the need computing solutions are the thewithout main tools tools used of in explicitly analysis and and system solutions and invariance properties of set dynamical systems. Here, we consider systems with a set-valued state solutions andweinvariance of set dynamicalstate systems. Here, consider properties systems with a set-valued solutions are the main tools used in analysis and system design nowadays. For such purposes, formulations based solutions are the main toolspurposes, used in analysis and system design nowadays. For such formulations based tems. Here, with state that in discrete time to tems. Here, we we consider systems with aa set-valued set-valued state that evolves evolves in consider discrete systems time according according to aa set-valued set-valued design nowadays. For based on are suited design nowadays.frameworks For such such purposes, purposes, formulations based that on set-theoretic set-theoretic frameworks are well wellformulations suited for for design design evolves in discrete time according to a set-valued map and a constraint given in terms of a set, both in that evolves in discrete time according to a set-valued map and a constraint given in terms of a set, both in on set-theoretic frameworks well for problems involving constraints, uncertainties, opon set-theoretic frameworks are well suited suitedmultiple for design design problems involving constraints,are uncertainties, multiple op- map and a constraint given in terms of a set, both in Euclidean space. Solutions associated to such systems are map and a constraint given in terms of a set, both in Euclidean space. Solutions associated to such systems are problems involving constraints, uncertainties, multiple operation among others. problems involving constraints, eration points, points, among others. uncertainties, multiple op- Euclidean space. Solutions associated to such systems are given by sequences of sets, rather than sequences of points Euclidean space. Solutions associated such systems are given by sequences of sets, rather than to sequences of points eration points, among others. eration points, among others. given of than sequences points as the case dynamical in Most by sequences of sets, sets, rather rather than systems sequences ofdiscrete points as in in by thesequences case of of classical classical dynamical systems inof discrete Most popular popular methods methods to to study study the the behavior behavior of of dynamical dynamical given as in the case of classical dynamical systems in discrete time. In particular, we extend the framework introduced Most popular methods to study the behavior of dynamical systems consider a classical state representation, namely a as in the case of classical dynamical systems in discrete time. In particular, we extend the framework introduced Most popular methods to study the behavior of dynamical systems consider a classical state representation, namely a In the introduced in (2014) dynamical systems outputs systems consider a namely aa time. single-valued vector in space. On the time. In particular, particular, weset extend the framework framework introduced in Sanfelice Sanfelice (2014) to towe setextend dynamical systems with with outputs systems consider a classical classical state representation, namely single-valued vector in some somestate space.representation, On the the other other hand, hand, the in Sanfelice (2014) to set dynamical systems with outputs and formulate appropriate notions of detectability and stasingle-valued vector in some space. On the other hand, the generation of tools to study the properties of systems in (2014) to set dynamical with and outputs andSanfelice formulate appropriate notions of systems detectability stasingle-valued in some On the other hand, the generation of vector tools to studyspace. the properties of systems in in and formulate appropriate notions of detectability and stability. We follow similar ideas to those existing for classical generation of tools to study the properties of systems in which set-valued states represent variables or multivalued and formulate appropriate notions of detectability and stability. We follow similar ideas to those existing for classical generation of tools to study the properties systems in which set-valued states represent variables orofmultivalued bility. We follow similar ideas to those existing for classical dynamical systems as well as those in the literature of setwhich set-valued states represent variables or multivalued signals has experienced a more delayed development. Early bility. We follow similar ideas to those existing for classical dynamical systems as well as those in the literature of setwhich set-valued states represent variables or multivalued signals has experienced a more delayed development. Early dynamical systems as well as those in the literature of setvalued dynamical systems, such as Panasyuk (1986), Pelsignals has experienced a more delayed development. Early approaches in this direction can be found in works such systemssystems, as well as those the literature of Pelsetvalued dynamical such as in Panasyuk (1986), signals has experienced a morecan delayed development. approaches in this direction be found in worksEarly such dynamical valued dynamical systems, such as Panasyuk (1986), Pelczar (1994), and Artstein (1995). The proposed concepts approaches in this direction can be found in works such as Pelczar (1977), where a type of generalized systems dynamical systems,(1995). such asThe Panasyuk (1986), Pelczar (1994), and Artstein proposed concepts approaches in this direction can be in works such valued as Pelczar (1977), where a type of found generalized systems czar (1994), Artstein (1995). proposed concepts are then then usedand to obtain obtain information about the convergence convergence as Pelczar where a of systems is and are (1994), and Artstein (1995). The The proposed concepts are used to information about the as Pelczar (1977), (1977), wherestability a type type properties of generalized generalized systems czar is presented presented and basic basic stability properties are studied, studied, are then used to obtain information about the convergence properties of solutions and to state an invariance principle is presented and basic stability properties are studied, with follow-up developments for properties the study study of of limit sets are then used to obtainand information about the convergence properties of solutions to state an invariance principle is presented and basic stability arelimit studied, with follow-up developments for the sets properties of and to state principle systems. our with follow-up developments for limit in (1991). Notions for properties of solutions solutions and In to particular, state an an invariance invariance principle for set set dynamical dynamical systems. In particular, our contributions contributions with follow-up developments for the the study study ofgeneralized limit sets sets for in Pelczar Pelczar (1991). Notions of of reachability reachability forof generalized for set dynamical systems. In particular, our contributions include: in Pelczar (1991). Notions of reachability for generalized pseudo dynamical systems are formally studied in Pelczar set dynamical systems. In particular, our contributions include: in Pelczar (1991). systems Notions are of reachability for generalized pseudo dynamical formally studied in Pelczar for include: pseudo systems are studied Pelczar (1994). More properties of with setpseudo dynamical systems are formally formally studied in in Pelczar (1994). dynamical More recently, recently, properties of systems systems with set- include: •• Basic Basic assumptions assumptions on on the the system system data data for for key key strucstruc(1994). More recently, properties of systems with setvalued states states in continuous time are are studied inwith Artstein (1994). More in recently, properties of studied systemsin setvalued continuous time Artstein •• Basic assumptions on the system data for tural properties, namely, a semigroup property, a Basic assumptions on theasystem dataproperty, for key key strucstructural properties, namely, semigroup a sesevalued states in continuous time are studied in Artstein (1995) and a framework for the design of output feedback valued states in continuous time are studied in feedback Artstein (1995) and a framework for the design of output tural properties, namely, aa semigroup property, a sequential compactness property, and closedness of tural properties, namely, semigroup property, a the sequential compactness property, and closedness of the (1995) and a framework for the design of output feedback algorithms incorporating set-valued systems is developed (1995) and incorporating a framework for the design of output feedback algorithms set-valued systems is developed quential compactness property, and closedness of the set-valued states and outputs. quential compactness property, and closedness of the set-valued states and outputs. algorithms algorithms incorporating incorporating set-valued set-valued systems systems is is developed developed set-valued states and •• Relationships between local set-valued states and outputs. outputs. Relationships between local stability, stability, detectability, detectability, This research research has has been been partially partially supported supported by by the the National National Science Science This •• Relationships between local stability, detectability, and boundedness to conclude convergence of Relationships between local stability, detectability, and boundedness to conclude convergence of setsetFoundation under CAREER Grant no. ECCS-1450484 and Grant This research has been partially supported by the National Science Foundation under Grant no. ECCS-1450484 andScience Grant This research hasCAREER been partially supported by the National and boundedness to conclude convergence of valued solutions. and boundedness to conclude convergence of setsetvalued solutions. no. CNS-1544396, CNS-1544396, and by the the Air Air Force Office of Scientific Scientificand Research Foundation under and CAREER Grant no.Office ECCS-1450484 Grant no. by Force of Research Foundation under CAREER Grant no. ECCS-1450484 and Grant valued solutions. valued solutions. under Grant no. FA9550-16-1-0015. no. CNS-1544396, and by the Air Force Office of Scientific Research under Grant no. FA9550-16-1-0015. no. CNS-1544396, and by the Air Force Office of Scientific Research
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• Results about convergence of set-valued solutions based on invariant sets involving the output map. • An invariance principle, similar to the one available for continuous-time and discrete-time systems; see, e.g., LaSalle (1967). Under the said basic assumptions, our invariance principle uses a real-valued Lyapunov-like function to locate invariant sets of bounded and complete solutions. Although results on convergence and stability properties for dynamical systems under a set-valued approach have been generated in the past, see Artstein and Rakovi´c (2011) for results concerning systems with output feedback, and results pertaining to invariance of sets are available (see Blanchini and Miani (2007), Artstein and Rakovi´c (2008)), our approach provides a tool to analyze convergence of solutions by using an invariance principle for set dynamical systems, which allows to use nonstrict Lyapunov functions for determining where set-valued solutions converge to. To the best of our knowledge, such tool is not available in the literature. 1 The reminder of this paper is organized as follows. After basic notation is introduced, Section 2 presents concepts associated to properties that are relevant for set dynamical systems behavior. The framework for set dynamical systems with outputs is defined in Section 3. The main results are presented in Section 4. More precisely, results relating detectability, stability, and convergence of solutions are described in Section 4.1. Invariance properties for set dynamical systems are presented in Section 4.2, where a formulation of an invariance principle for set-valued systems is proposed. Examples throughout the paper illustrate the ideas.
Definition 2.2. (inner and outer limit). For a sequence of n sets {Ti }∞ i=0 in R :
• The inner limit of the sequence {Ti }∞ i=0 , denoted by lim inf i→∞ Ti , is the set of all points x ∈ Rn for which there exist points xi ∈ Ti , i ∈ N, such that limi→∞ xi = x; • The outer limit of the sequence {Ti }∞ i=0 , denoted by lim supi→∞ Ti , is the set of all points x ∈ Rn for ∞ which there exist a subsequence {Tik }∞ k=0 of {Ti }i=0 and points xk ∈ Tik , for all k ∈ N, such that limk→∞ xk = x.
The limit of the sequence exists if the outer and the inner limit sets are equal; namely, lim Ti = lim inf Ti = lim sup Ti i→∞
Some basic definitions and properties that are used to characterize set dynamical systems are given in this section. We consider the standard topology in Rn . A set is closed if and only if it contains its boundary. A set is compact if bounded and closed. Definition 2.1. (distance between sets). The Hausdorff distance between two closed sets A1 , A2 ⊂ Rn is given by d(A1 , A2 ) = max sup |x|A2 , sup |z|A1 z∈A2
Note that Proposition 9.1 in Artstein and Rakovi´ c (2011) is a convergence result that requires the Lyapunov function to strictly decrease while the convergence criterion of sets in Proposition 4.5 in Artstein and Rakovi´ c (2008) requires a strict contraction property.
i→∞
Definition 2.3. (union of a collection of sets). Let C be a collection of sets. The union of C is C = {x : x ∈ X for some X ∈ C}. Definition 2.4. (Rockafellar and Wets, 2009, convergence of a sequence of sets) When the limit of the sequence n {Ti }∞ i=0 in R exists in the sense of Definition 2.2, and is equal to T , the sequence of sets {Ti }∞ i=0 is said to converge to the set T . 3. SET DYNAMICAL SYSTEMS WITH OUTPUTS We consider set dynamical systems defined by X+ = G(X) Y = H(X)
Notation: The following notation is used throughout this paper. N denotes the natural numbers including 0, i.e., N = {0, 1, . . .}. Rn denotes the n-dimensional Euclidean space. R≥0 denotes the nonnegative real numbers. B denotes the closed unit ball in Euclidean space. Given a set A, its closure is denoted by A. Given a map V , dom V denotes its domain of definition. Given x ∈ Rn , |x| denotes the Euclidean vector norm and |x|1 denotes the taxicab norm. For a closed set A ⊂ Rn and x ∈ Rn , we define the distance |x|A = inf y∈A |x − y|. Given a function V : dom V → R and a constant r ∈ R, its r-sublevel set is given by LV (r) := {x ∈ dom V : V (x) ≤ r }. For a map G, Gn denotes the n-th composition.
x∈A1
i→∞
The inner and outer limits of a sequence always exist and are closed 2 , although the limit itself might not exist.
2. PRELIMINARIES
1
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(1)
X⊂D where X is the set-valued state, Y is the system’s output, G : Rn ⇒ Rn and H : Rn ⇒ Rm are set-valued maps defining the right-hand side and the output map, respectively, and D ⊂ Rn defines a constraint that solutions to the system must satisfy. We say that (D, G, H) is the data of (1). A solution to the system in (1) is defined as the sequence of nonempty sets {Xj }Jj=0 and its associated output is defined by the sequence {Yj }Jj=0 , J ∈ N ∪ {∞}, satisfying Xj+1 = G(Xj ) Yj = H(Xj ) Xj ⊂ D
over the domain of definition of the solution {Xj }Jj=0 , which is given by the collection {0, 1, 2, . . . , J} ∩ N, and denoted by dom X. The first entry of the solution, X0 , is the initial set. We assume X0 to be compact. If a solution has J = 0 then we say that it is trivial, and if it has J > 0 we say that it is nontrivial. If it has J = ∞, we say that it is complete. A solution {Xj }Jj=0 is said to be maximal if it cannot be further extended. Given an initial set X0 ⊂ Rn , S(X0 ) denotes the set of maximal solutions 3 to (1) from X0 . 2 For a sequence of empty sets, the inner and outer limit is the empty set. 3 By Lemma 3.7, maximal solutions to (1) are unique.
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To make notation easier to follow, at times, the sequence (B1) For any solution X to (1) and any ˆj ∈ dom X, we of sets {Xj }Jj=0 is represented as Xj (or even just X). We ˆ given by X ˆ j = X ˆ for each j ∈ dom X ˆ= have that X j+j make the same notational simplification when referring to j : j + ˆj ∈ dom X is a solution to (1). the output Y. The term solution-output pair {X, Y} is used to represent a solution X and its associated output (B2) Suppose the data (D, G) of (1) satisfies (A0) and Y = H(X). Notation {Xij }∞ i=0 refers to the sequence of (A1) of Assumption 3.4. Let {Xi0 }∞ i=0 be an eventually solutions Xij indexed by i, where j is the associated discrete bounded (with respect to Rn ) sequence of sets convergtime. ing to a compact set X0 and suppose {Xi }∞ i=0 is such that Xi ∈ S (Xi0 ). Then, there exists a subsequence of {Xi }∞ 3.1 Definitions and Assumptions i=0 converging to some X ∈ S (X0 ). (B3) Suppose the data (D, G, H) of (1) satisfies Assumpi ∞ tion 3.4. Let {Xij }∞ i=0 and {Yj }i=0 be a solutionDefinition 3.1. (stability of a set). The set A is stable if output pair to (1), such that {Yji }∞ for each > 0 there exists δ > 0 such that each solution i=0 is bounded. Then, there exists a subsequence of {Yji }∞ Xj to (1) with d(X0 , A) ≤ δ satisfies d(Xj , A) ≤ for all i=0 that converges to a closed set. j ∈ dom Xj . For classical systems, detectability means that when the output remains at zero, the norm of solutions converges to zero as time goes to infinity. A related concept is defined for set dynamical systems next. Definition 3.2. (detectability). Let the sets A ⊂ Rn , N ⊂ Rm , and M ⊂ Rn be given, where A is closed. For the system in (1), the distance to A is detectable relative to N on M if for each complete solution X ∈ S(X0 ), with Xj closed for each j ∈ dom X, and associated output Y X⊂M =⇒ lim d(Xj , A) = 0 Y⊂N j→∞ In the next sections, results characterizing key dynamical properties of a set dynamical system as in (1) are proposed. Those results require the data of (1) to satisfy certain mild conditions, which are stated in the following assumption. Definition 3.3. (outer semicontinuity). The set-valued map G : Rn ⇒ Rn is outer semicontinuous at x ∈ Rn if for n each sequence {xi }∞ i=0 converging to a point x ∈ R and ∞ each sequence {yi }i=0 such that yi ∈ G(xi ) for each i, converging to a point y, it holds that y ∈ G(x). It is outer semicontinuous if G(x) is outer semicontinuous at each x ∈ Rn . Assumption 3.4. The set dynamical system defined in (1), with data (D, G, H) satisfies the following properties: (A0) The set-valued map G : Rn ⇒ Rn is outer semicontinuous, locally bounded 4 , and, for each x ∈ D, G(x) is a nonempty subset of Rn . (A1) The set D ⊂ Rn is closed. (A2) The set-valued map H : Rn ⇒ Rn is outer semicontinuous, locally bounded, and, for each x ∈ D, H(x) is a nonempty subset of Rn . 3.2 Preliminary results The following result establishes a structural property of the state and output trajectory of (1). A similar result appeared in (Sanfelice, 2014, Proposition 4.6). Proposition 3.5. (basic properties of solutions) The following properties hold for system (1): The map G is locally bounded if for each compact set K ⊂ Rn there exists a compact set K ⊂ Rn such that G(K) ⊂ K . 4
Theorem 3.6. (Rockafellar and Wets, 2009, Theorem 4.18) n Every sequence of nonempty sets {Ti }∞ either i=0 in R escapes to the horizon or has a subsequence converging to a nonempty set T ⊂ Rn , i.e., there exists a subsequence ∞ {Tik }∞ k=0 of {Ti }i=0 such that limk→∞ Tik = T . Lemma 3.7. Given a solution {Xj }Jj=0 to (1) and any subsequence {ji }Ii=0 of {j}Jj=0 , we have
∀i ∈ {0, 1, . . . , I} ∩ N Xji = Gji (X0 ) Moreover, every maximal solution to (1) is unique.
(2)
Now, we define the notion of ω-limit set. This notion is used later to establish convergence properties of solutions. Definition 3.8. (ω-limit set). The ω-limit set of a solution {Xj }Jj=0 to (1) is given by ω (Xj ) = {Y ⊂ Rn : ∃{ji }∞ i=0 , lim ji = ∞, Y = lim Xji } i→∞
i→∞
Note that ω (Xj ) is a collection of sets.
(3)
Theorem 3.9. (Sanfelice, 2014, Theorem 4.13) Suppose the data (D, G) of (1) satisfies Assumption 3.4 and that V : Rn → R is a Lyapunov-like function. Then, every solution {Xj }Jj=0 , J ∈ N ∪ {∞} to (1) from X0 ⊂ D satisfies 0, sup V (x) ⊂ 0, sup V (x) x∈Xj+1
(4)
x∈Xj
for all j ∈ {0, 1, . . . , J − 1} ∩ N. Moreover, if J = ∞ then (5) 0, sup V (x) lim 0, sup V (x) = j→∞
x∈Xj
j∈N
x∈Xj
and if, furthermore, {Xj }∞ j=0 is bounded then there exists r ∈ R≥0 such that V ( ω (Xj )) ⊂ [0, r] (6) 4. DETECTABILITY AND AN INVARIANCE PRINCIPLE This section pertains to the study of convergence and invariance properties of set dynamical systems. Notions and results on detectability and convergence are presented in Section 4.1. In Section 4.2, invariance properties of set
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dynamical systems are characterized and then used to prove an invariance principle, which corresponds to the main result in this paper. The proofs of some of the results borrow ideas from variational analysis in Rockafellar and Wets (2009) and the results for discrete-time systems in LaSalle (1967) and Teel (2006).
X3
X2
4.1 Detectability and Convergence X1
Structural properties of the solutions to set dynamical systems are used in this section, along with detectability, to conclude convergence of solutions to compact sets. Theorem 4.1. Let A and M be compact subsets of Rn . Suppose system (1) satisfies Assumption 3.4, the set A is stable, and the distance to A is detectable relative to N . Let X ∈ S(X0 ) be complete and Y be the associated output. If X ⊂ M and limj→∞ Yj = N then lim d(Xj , A) = 0 j→∞
Example 4.2. (illustration of Theorem 4.1) Consider the set dynamical system in (1) with G(X) = {g(x) : x ∈ X}, and g(x) = H(X) = {h(x) : x ∈ X} where 1 x21 , and D = {x ∈ R2 : , h(x) = 0.5 − x1 x2 1 + e−x1 |x|1 ≤ cD }, with cD > 0. This system satisfies Assumption 3.4 since both g and h are continuous and D is compact. Consider the compact sets M = {x ∈ R2 : |x|1 ≤ cM } with cM ≥ 0, A = {(0, 0)} and N = {0}. Let Xj be a complete solution from X0 = {x ∈ R2 : |x|1 ≤ c0 } with c0 < 1 and let Yj be its associated output. For cM < 1 and c0 ≤ cM , Xj ⊂ M for all j ∈ dom Xj since g is contractive on M in the sense that |g1 (x)| < |x1 | < 1 and |g2 (x)| < 1 for each x ∈ M. To establish stability of A, let > 0. For δ = , the solution Xj from X0 is such that d(Xj−1 , A) ≤ d(Xj , A) ≤ δ < 1, since g is contractive on X ⊂ M, so d(Xj , A) ≤ for all j ∈ dom Xj . We have that Y = N if and only if X ⊂ {x ∈ R2 : x1 = 0}. So any solution with X ⊂ M and Y = N is such that Yj = {0} for all j ∈ dom Xj , which implies that X0 ⊂ {x ∈ R2 : x1 = 0} and X1 = G(X0 ) = {(0, 0)}. Then, limj→∞ d(Xj , A) = 0. Thus, the distance to A is detectable relative to N on M. The assumptions in Theorem 4.1 are satisfied, which implies that the set-valued solution converges to A. A solution from the initial set X0 with c0 = 0.6, cM = 0.8, and cD = 0.9 is presented in Figure 1, where convergence to A can be observed. Theorem 4.3. Let A and M be compact subsets of R . Suppose system (1) satisfies Assumption 3.4, the set A is stable for system (1), and A is detectable relative to N on M. Then, for each > 0, there exists j ∗ ∈ N such that, for each solution-output pair (X, Y) with X ∈ S(X0 ), X⊂M =⇒ d(Xj , A) ≤ ∀j ≥ j ∗ , j ∈ dom X Y=N n
4.2 Invariance Properties In this section we formulate a convergence result based on invariant sets involving the output map. We introduce the concept of Lyapunov-like functions in the sense of Sanfelice (2014), and use such functions to locate invariant sets for
j X0
x1
x2
Fig. 1. Solution to the set dynamical system in Example 4.2 from initial set X0 = {x : |x|1 ≤ 0.6} (blue). As j increases, the solution converges to A = {(0, 0)}. solutions to set dynamical systems. We start by defining the concept of invariance for set dynamical systems. Definition 4.4. (forward and backward invariance). A set M ⊂ Rn is said to be forward invariant for (1) if for every set T ⊂ M ∩ D we have G(T ) ⊂ M ∩ D. A set M ⊂ Rn is said to be backward invariant for (1) if for every set T ⊂ M ∩ D for which there exists a set T with the property T = G(T ), we have T ⊂ M ∩ D for every such set T . A set M ⊂ Rn is said to be invariant if it is both forward and backward invariant. Recall that, according to the definition of solution, we have that all maximal solutions to (1) are unique. The following result from Sanfelice (2014) characterizes properties of the omega limit sets of solutions to set dynamical systems. Proposition 4.5. (Sanfelice, 2014, Proposition 4.9) Suppose the data (D, G) in (1) satisfies (A0) and (A1) in Assumption 3.4. Let Xj be a bounded and complete solution to (1). Then, ω (Xj ) is nonempty, compact, and invariant. Given a set dynamical system as in (1) and a set N , we define Y ∗ := {x ∈ D : H(x) ⊂ N }
which will be used in the formulation of the next results. Theorem 4.6. Let M be compact and let X ∈ S(X0 ) be a complete solution to the system in (1) satisfying Assumption 3.4. If X ⊂ M and its associated output Y satisfies limj→∞ Yj = N , then the solution X converges to the largest invariant set contained in M ∩ Y ∗ ∩ D. Proof Sketch: Since X remains in M for all j ∈ dom X, the solution is bounded and by Proposition 4.5, the omega limit set is nonempty, compact, and invariant. Since the system satisfies Assumption 3.4, from the definition of Yj we have that lim Yj = lim H(Xj ) = N . If X∗ is an j→∞
j→∞
element of ω (X), there is a subsequence of Xj indexed by ji such that lim H(Xji ) = H(X∗ ) = N . Moreover, for i→∞
(X), X∗ ⊂ {x ∈ D : H(x) ⊂ N } = Y ∗ . each X∗ ∈ ω
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Theorem 4.6 is later used in proving our invariance principle, which is in Theorem 4.9. Before that, we make a link between detectability and stability. Theorem 4.7. Let A and M be compact subsets of Rn and suppose, for the system described in (1), that the set A is stable. Then, the following statements are equivalent:
Remark 4.10. If, for the set dynamical system in (1), we have that G = g, with g : Rn → Rn , then Theorem 4.9 reduces to the original invariance principle for discretetime systems given in LaSalle (1967), with a sublevel set of V instead of a level set as stated in the original principle.
(1) The distance to A is detectable relative to N on M. (2) The largest invariant set contained in M ∩ Y ∗ is a subset of A.
Example 4.11. (illustration of Theorem 4.9) Consider the set dynamical as defined in (1) in R2 with G(X) = αx 2 1 + x21 {g(x) : x ∈ X } and g(x) = βx1 , The following definition introduces the notion of Lyapunovlike functions for set dynamical systems. 1 + x22 Definition 4.8. (Lyapunov-like function). A continuous func- D = [0, p] × [0, q], with p, q real positive numbers and tion V : Rn → R is called a Lyapunov-like function for α, β ≥ 0. The set dynamical system data satisfies Assumption 3.4 since g is continuous and D is compact. Consider system (1) if it satisfies as the Lyapunov-like function V (x) = x21 + x22 , which, for V (x) ≥ 0 ∀x ∈ D ∪ G(D) each x ∈ D and η = G(x), leads to
V (η) − V (x) ≤ 0 ∀x ∈ D, η ∈ G(x) α2 β2 2 − 1 x + − 1 x22 V (η) − V (x) = 1 (1 + x22 )2 (1 + x21 )2 Now we are ready to present our main result. ≤ (β 2 − 1)x21 + (α2 − 1)x22 Theorem 4.9. (invariance principle) Suppose system (1) For α2 < 1 and β 2 < 1, V (η)−V (x) ≤ 0 for all x ∈ X, with satisfies Assumption 3.4. Let V be a Lyapunov-like func- X ⊂ D, η ∈ G(X), and V is a valid Lyapunov-like function tion for (1). Let Xj be a bounded and complete solution for the set dynamical system. For each x ∈ D ∪ G(D), we to (1). Then, there is a number r ∈ R≥0 such that Xj have V (x) ≥ 0. Moreover, let Xj be a complete solution converges to the largest invariant set contained in from X0 ⊂ D. Since D is compact, X is bounded. The assumptions in Theorem 4.9 are satisfied E ∩ LV (r) ∩ D and the solution converges to the largest invariant set in E ∩LV (r)∩D for where some r ≥ 0. Since supη∈G(X) V (η) = supx∈X V (x) only for X = {(0, 0)}, the set E contains only the element {(0, 0)}, E := X ⊂ D : sup V (η) = sup V (x) x∈X η∈G(X) and the largest invariant set is the origin. A solution from the initial set X0 = [0.5, 2] × [0.5, 2], with p = 5, q = 4, and LV (r) := x ∈ D : V (x) ≤ r . α = β = 0.75 is presented in Figure 2. (Xj ) Proof Sketch: Since Xj is bounded and complete, ω is nonempty, compact, and invariant. Using Theorem 3.9, since Assumption 3.4 holds for system (1), V ( ω (Xj )) ⊂ [0, r] for some r ∈ R≥0 . Consider the set dynamical system in (1) with Y = (Y1 , Y2 ) = H(X) = (H1 (X), H2 (X)), where
H1 (X) = 0, sup V (x) − sup V (η)
x∈X
H2 (X) = 0, sup V (x) x∈X
η∈G(X)
to which the given Xj is a solution. We can apply Theorem 4.6, defining the sets N = N1 × N2 , N1 = {0}, N2 = {x ∈ D : V (x) ≤ r} and M = {x ∈ D : V (η) ≤ V (x)}. We have Xj ⊂ M for all j ∈ dom X. Using Theorem 3.9 we have that supx∈G(Xj ) V (x) ≤ supx∈G(Xj−1 ) V (x) for each j ∈ dom Xj \ {0} and the sequence of points {supx∈Xj V (x)}∞ j=0 converges to a real number b ∈ [0, r], with b = limj→∞ supx∈Xj V (x). Furthermore, we have that limj→∞ supx∈G(Xj ) V (x) − supx∈G(Xj−1 ) V (x) = 0. Then, for the first component of the output, Y1 , limj→∞ Y1j = N1 . The second output Y2 satisfies limj→∞ Y2j = N2 . Then, since limj→∞ Yj = N , using Theorem 4.6 for this system, we have that the solution Xj converges to thelargest invariant set in M∩Y ∗ ∩D, which corresponds to E ∩ LV (r) ∩ D.
For α2 = β 2 = 1, condition V (η)−V (x) ≤ 0 still holds, for all x ∈ X, with X ⊂ D, η ∈ G(X), so V is a Lyapunovlike function for the set dynamical system. In this case, condition supη∈G(X) V (η) = supx∈X V (x) is satisfied for X with elements x such
that
1 1 2 − 1 x1 + − 1 x22 = 0 (1 + x22 )2 (1 + x21 )2 which holds for each x in X = {x ∈ D : x1 = 0, x2 ≥ 0} ∪ {x ∈ D : x2 = 0, x1 ≥ 0}. Since this set of points belongs to the boundary of D and is invariant for G, we have that it defines the set E as well, namely, E = {x ∈ D : x1 = 0, x2 ≥ 0} ∪ {x ∈ D : x2 = 0, x1 ≥ 0}. Since the assumptions in Theorem 4.9 are satisfied, the set-valued solution converges to the largest invariant set in E ∩ LV (r) ∩ D, in this case E ∩ LV (r) (7) for some r ≥ 0. For a solution from the initial set X0 = [0, 1] × [0, 1], with p = 5, q = 4, the set-valued state converges to {x ∈ R2 : x1 = 0, x2 ≤ 1} ∪ {x ∈ R2 : x2 = 0, x1 ≤ 1} which is the set in (7) with r = maxx∈X0 V (x) = 1. Figure 3 shows the set-valued solution up to j = 6. 5. CONCLUSION
Convergence and invariance properties for set dynamical systems with outputs are studied in this paper. The
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(a) Set-valued state evolution.
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(a) Set-valued state evolution.
(b) Set-valued state convergence in the plane x1 − x2 .
(b) Set-valued state convergence in the plane x1 − x2 .
Fig. 2. Solution to the set dynamical system in Example 4.11, with α = β = 0.75 and X0 = [0.5, 2] × [0.5, 2], for j = 0 to 6.
Fig. 3. Solution to the set dynamical system in Example 4.11, with α = β = 1 and X0 = [0, 1] × [0, 1], for j = 0 to 6.
mathematical framework in Sanfelice (2014) was extended to systems with outputs and notions of stability and detectability for these systems were defined. Then, under the assumption of bounded solutions and outer semicontinuity, results relating stability, detectability and state trajectory convergence were developed. Invariance properties for set dynamical systems were characterized and an invariance principle, involving the use of Lyapunov-like functions to locate invariant sets, under assumption of bounded solutions and outer semicontinuity of the set-valued map that defines the system dynamics, was presented.
LaSalle, J.P. (1967). An invariance principle in the theory of stability. Differential Equations and Dynamical Systems (JP La Salle and JK Hale editors), Acad. Press, NY, 277–286. Panasyuk, A.I. (1986). Dynamics of sets defined by differential inclusions. Siberian Mathematical Journal, 27(5), 757–765. Pelczar, A. (1977). Stability in generalized pseudodynamical systems. Zeszyty Naukowe UJ, Prace Matematyczne, 19, 33–38. Pelczar, A. (1991). Remarks on limit sets in generalized semi-systems. Univ. Iagel. Acta Math, 28, 29–39. Pelczar, A. (1994). On modified prolongational limit sets and prolongations in dynamical systems on metric spaces. Univ. Iagel. Acta Math, 31, 175–202. Rockafellar, R.T. and Wets, R.J.B. (2009). Variational analysis, volume 317. Springer Science & Business Media. Sanfelice, R.G. (2014). Asymptotic properties of solutions to set dynamical systems. In 2014 IEEE 53rd Annual Conference on Decision and Control, 2287–2292. Teel, A.R. (2006). Class notes on nonlinear discrete-time systems.
REFERENCES Artstein, Z. (1995). A calculus for set-valued maps and setvalued evolution equations. Set-Valued Analysis, 3(3), 213–261. Artstein, Z. and Rakovi´c, S.V. (2011). Set invariance under output feedback: a set-dynamics approach. International Journal of Systems Science, 42(4), 539–555. Artstein, Z. and Rakovi´c, S.V. (2008). Feedback and invariance under uncertainty via set-iterates. Automatica, 44(2), 520–525. Blanchini, F. and Miani, S. (2007). Set-theoretic methods in control. Springer Science & Business Media.
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