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Robustness of recurrence for a class of stochastic hybrid systems*
Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Preprints, 5th IFAC Conference on Analysis and Design of Hybrid Systems Preprints, Preprints, 5th 5th IFAC IFAC Conference Conference on on Analysis Analysis and and Design Design of of Hybrid Hybrid Systems Preprints, 5th IFAC Conference on Analysis and Design of Hybrid October USAonline at www.sciencedirect.com Systems14-16, 2015. Georgia Tech, Atlanta, Available Systems October 14-16, 2015. Georgia Tech, Atlanta, USA Systems October 14-16, 14-16, 2015. Georgia Georgia Tech, Atlanta, Atlanta, USA October October 14-16, 2015. 2015. Georgia Tech, Tech, Atlanta, USA USA
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Robustness of recurrence for a class Robustness of recurrence for a class Robustness of recurrence for a Robustness of recurrence for a class class stochastic hybrid systems stochastic hybrid systems stochastic hybrid systems
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Anantharaman Subbaraman, Andrew R. Teel Anantharaman Subbaraman, Andrew R. Teel Anantharaman Subbaraman, Andrew R. Anantharaman Teel Anantharaman Subbaraman, Subbaraman, Andrew Andrew R. R. Teel Teel Department of of Electrical Electrical and and Computer Computer Engineering Engineering Department Department Electrical and Engineering University of of California, Santa Barbara 93106-9560. 93106-9560. Department of Electrical and Computer Engineering Department of Electrical Santa and Computer Computer Engineering University of California, Barbara University of California, Santa Barbara 93106-9560. [email protected], [email protected] University of California, Santa Barbara 93106-9560. University of California, Santa Barbara 93106-9560. [email protected], [email protected] [email protected], [email protected], [email protected] [email protected], [email protected] [email protected] Abstract: We study stability property recurrence for aa class hybrid Abstract: We study a a weak weak stability property called called recurrence for class of of stochastic stochastic hybrid hybrid Abstract: We weak stability called recurrence for systems. Robustness the recurrence property to various state-dependent perturbations is Abstract: We study weak stability property called recurrence for class of stochastic hybrid Abstract: We study study aa aof weak stability property property called recurrence for aaa class class of of stochastic stochastic hybrid systems. Robustness of the recurrence property to various state-dependent perturbations is systems. Robustness of the recurrence property to various state-dependent perturbations is established under mild regularity conditions for the stochastic hybrid system. A potential systems. Robustness of the recurrence property to various state-dependent perturbations is systems. Robustness of the recurrence property to various state-dependent perturbations is established under mild regularity conditions for the stochastic hybrid system. A potential established under mild regularity conditions for the stochastic hybrid system. A potential application of the robustness results in the development of converse Lyapunov theorems is established under mild regularity conditions for the stochastic hybrid system. A potential established under mild regularity conditions for the stochastic hybridLyapunov system. A potential application of the robustness results in the development of converse theorems is application of also outlined. application of the robustness results in the development of converse Lyapunov theorems is application of the the robustness robustness results results in in the the development development of of converse converse Lyapunov Lyapunov theorems theorems is is also outlined. also outlined. also outlined. also outlined. © 2015, IFAC (International Federation of Automatichybrid Control)systems Hosting by Elsevier Ltd. All rights reserved. Keywords: Recurrence, Robustness, Stochastic Keywords: Recurrence, Robustness, Stochastic hybrid systems Keywords: Recurrence, Robustness, Stochastic hybrid systems Keywords: Keywords: Recurrence, Recurrence, Robustness, Robustness, Stochastic Stochastic hybrid hybrid systems systems 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Hybrid systems are a of dynamical systems that Hybrid systems are aa class class of dynamical systems that Hybrid systems are class of dynamical systems that involve both continuous-time evolution and discrete-time Hybrid systems are a class of dynamical systems that Hybrid both systems are a class of dynamical systems that involve continuous-time evolution and discrete-time involve both continuous-time evolution and discrete-time events. Stochastic hybrid systems (SHS) this involve continuous-time evolution and discrete-time involve both both continuous-time evolution andgeneralize discrete-time events. Stochastic hybrid systems (SHS) generalize this events. Stochastic hybrid systems (SHS) generalize this class by adding randomness. In stochastic hybrid sysevents. Stochastic hybrid systems (SHS) generalize this events. Stochastic hybrid systems (SHS) generalize this class by adding randomness. In stochastic hybrid sysclass by adding randomness. In stochastic hybrid systems randomness can affect the continuous-time dynamics, class by adding randomness. In stochastic hybrid sysclass randomness by adding can randomness. In stochastic hybrid systems affect the continuous-time dynamics, tems randomness can affect the continuous-time dynamics, discrete-time dynamics and also the transition between tems can continuous-time tems randomness randomness can affect affect the continuous-time dynamics, discrete-time dynamics andthe also the transition transitiondynamics, between discrete-time dynamics and also the between them. Frameworks for modeling SHS are in Yin and Zhu discrete-time dynamics and also the transition between discrete-time dynamics and alsoSHS the are transition between them. Frameworks for modeling in Yin and Zhu them. Frameworks for modeling SHS are in Yin and Zhu (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros them. Frameworks for modeling SHS are in Yin and Zhu them. Frameworks for modeling SHS are in Yin and Zhu (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2006) and Teel et al. (2014b). (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2006) and Teel et al. (2014b). (2006) and Teel et al. (2014b). (2006) Teel (2014b). (2006) and andframeworks Teel et et al. al. for (2014b). Modeling stochastic hybrid systems seldom Modeling frameworks for stochastic hybrid systems seldom seldom Modeling frameworks for stochastic hybrid systems account for systems that generate non-unique solutions. Modeling frameworks for stochastic hybrid systems seldom Modelingfor frameworks for stochastic hybrid systems seldom account systems that generate non-unique solutions. account for systems that generate non-unique solutions. In Teel (2013) and Teel a generalized account for that generate non-unique solutions. account for systems systems that(2014b) generate non-uniqueframework solutions. In Teel (2013) and Teel (2014b) a generalized framework In Teel (2013) and Teel (2014b) generalized framework for modeling solutions proposed. In (2013) and Teel (2014b) In Teel Teel (2013)SHS andwith Teel non-unique (2014b) aaa generalized generalized framework for modeling SHS with non-unique solutions is isframework proposed. for modeling SHS with non-unique solutions is proposed. The need to study stochastic systems with non-unique for modeling SHS with non-unique solutions is proposed. for modeling SHS with non-unique solutions is proposed. The need to study stochastic systems with non-unique The need to study stochastic systems with non-unique solutions Firstly, analyzing such systems is The to study systems non-unique The need need is totwofold. study stochastic stochastic systems with with non-unique solutions is twofold. Firstly, analyzing such systems is solutions is twofold. Firstly, analyzing such systems is crucial to developing a robust stability theory. Robust solutions is twofold. Firstly, analyzing such systems is solutionsto isdeveloping twofold. Firstly, analyzing such systems is crucial a robust stability theory. Robust crucial to developing robust stability theory. Robust stability non-stochastic hybrid systems is studied in crucial developing aaa robust stability theory. Robust crucial to tofor developing robust stability theory. Robust stability for non-stochastic hybrid systems is studied in stability for non-stochastic hybrid systems is studied in Goebel et al. (2012). Secondly, such system models allow stability for non-stochastic hybrid systems is studied in stability for non-stochastic hybrid systems is studied in Goebel et al. (2012). Secondly, such system models allow Goebel et al. (2012). Secondly, such system models allow flexibility in control design applications. See Subbaraman Goebel et al. (2012). Secondly, such system models allow Goebel et in al.control (2012).design Secondly, such system models allow flexibility applications. See Subbaraman flexibility in control design applications. See Subbaraman et al. (2013), Venkateswaran et al. (2013) and et al. flexibility in design See Subbaraman flexibility in control control design applications. applications. See Poveda Subbaraman et al. (2013), Venkateswaran et al. (2013) and Poveda et al. et al. (2013), Venkateswaran et al. (2013) and Poveda et (2015). et al. (2013), Venkateswaran et al. (2013) and Poveda et al. et al. (2013), Venkateswaran et al. (2013) and Poveda et al. al. (2015). (2015). (2015). (2015). The stochastic stability property of interest in this paper The stochastic stability property of interest in this paper The stochastic stability property of interest in this paper is called recurrence. Loosely speaking, recurrence of an The stochastic stability property of in The stochastic stability property of interest interest in this this paper paper is called recurrence. Loosely speaking, recurrence of an is called recurrence. Loosely speaking, recurrence of an open set implies that solutions visit the set infinitely is recurrence. Loosely speaking, recurrence of is called called recurrence. Loosely speaking, recurrence of an an open set implies that solutions visit the set infinitely open set implies that solutions visit the set infinitely often with probability one. The recurrence property is open set implies that solutions visit the set infinitely open set implies that one. solutions visit the set infinitely often with probability The recurrence property is often with probability one. The recurrence property is frequently studied in the literature. See Yin and Zhu often with probability one. The recurrence property is often with studied probability one.literature. The recurrence property is frequently in the See Yin and Zhu frequently studied in the literature. See Yin and Zhu (2009) and Meyn and Tweedie (2009). Recurrence is aa frequently studied in literature. See Yin frequently studied in the the literature. SeeRecurrence Yin and and Zhu Zhu (2009) and Meyn and Tweedie (2009). is (2009) and Meyn and Tweedie (2009). Recurrence is weaker notion of stability compared to more commonly (2009) and Meyn and Tweedie (2009). Recurrence is a (2009) and Meyn and Tweedie (2009). Recurrence is aa weaker notion of stability compared to more commonly weaker notion of stability compared to more commonly studied notions of mean square asymptotic stability and weaker notion of stability compared to more commonly weaker notion stability compared to more commonly studied notionsofof mean square asymptotic stability and studied notions of mean square asymptotic stability and asymptotic stability in probability. Recurrence of an open studied notions of mean square asymptotic stability and studied notions of mean square asymptotic stability and asymptotic stability in probability. Recurrence of an open asymptotic stability in probability. Recurrence of an open bounded set need not imply any stability-like property for asymptotic stability in probability. Recurrence of asymptotic stability inimply probability. Recurrenceproperty of an an open open bounded set need not any stability-like for bounded set need not imply any stability-like property for the set. Recurrence of a set does not imply any invariancebounded set need not imply any stability-like property for bounded set need not imply any not stability-like property for the set. Recurrence of a set does imply any invariancethe set. Recurrence of a set does not imply any invariancelike property for the set and finally, recurrence of an open, the set. Recurrence of a set does not imply any invariancethe set. Recurrence ofset a set does notrecurrence imply any of invariancelike property for the and finally, an open, like property for the set and finally, recurrence of an open, bounded set does not imply solutions stay bounded in like for set and finally, recurrence of like property property for the the set andthat finally, recurrence of an an open, open, bounded set does not imply that solutions stay bounded in bounded set does not imply that solutions stay bounded in a probabilistic sense. Recurrence is still a useful property bounded set does not imply that solutions stay bounded in bounded set does not imply that solutions stay bounded in aa probabilistic sense. Recurrence is still a useful property probabilistic sense. Recurrence is still a useful property study since it provides an alternative when stronger ato probabilistic sense. Recurrence is still a useful property a probabilistic sense. Recurrence is still a useful property to study since it provides an alternative when stronger to study since it provides an alternative when stronger to to study study since since it it provides provides an an alternative alternative when when stronger stronger Research supported in part by AFOSR FA9550-12-1-0127,
properties like convergence or asymptotic stability are properties like convergence or asymptotic stability are properties like convergence or asymptotic stability are impossible to establish. This is particularly true in the properties like convergence or asymptotic stability are properties like convergence oris asymptotic stability are impossible to establish. This particularly true in the impossible to establish. This is particularly true in the case of systems affected by persistent disturbances. We impossible to establish. This is particularly true in the impossible to establish. This is particularly true in We the case of systems affected by persistent disturbances. case of systems affected by disturbances. We refer the reader to (Subbaraman and Teel, 2015, Example case of systems affected by persistent disturbances. We case of systems affected by persistent persistent disturbances. We refer the reader to (Subbaraman and Teel, 2015, Example refer the reader to (Subbaraman and Teel, 2015, Example 1) and Isaacs et al. (2014) for more details. refer the reader to (Subbaraman and Teel, 2015, Example refer the reader to (Subbaraman and Teel, 2015, Example 1) and Isaacs et al. (2014) for more details. 1) and Isaacs et al. (2014) for more details. 1) and al. for more details. 1) and Isaacs Isaacsofet etstability al. (2014) (2014) for more details. Robustness can be loosely defined as the staRobustness of stability can be loosely defined as the staRobustness of stability can be loosely defined as the stability property being preserved when the nominal system Robustness of stability can be loosely defined as the staRobustness of stability can be loosely defined as the stability property being preserved when the nominal system bility property being preserved when the nominal system is affected by sufficiently small perturbations. The robility property being preserved when the nominal system bility property being preserved when the nominalThe system is affected by sufficiently small perturbations. rois affected by sufficiently small perturbations. The robustness results in this paper exploit the good regularity is affected by sufficiently small perturbations. The is affected by sufficiently small perturbations. The rorobustness results in this paper exploit the good regularity bustness results in this paper exploit the good regularity properties of the nominal system. The results generated bustness results in this paper exploit the good regularity bustness results innominal this paper exploit theresults good regularity properties of the system. The generated properties of nominal system. The generated in this work have potential application in establishing properties of the nominal system. The results generated properties of the the nominal system. The results results generated in this work have potential application in establishing in this work have potential application in establishing novel converse Lyapunov theorems for stochastic hybrid in this work have potential application in establishing in thisconverse work have potential application in establishing novel Lyapunov theorems for stochastic hybrid novel converse Lyapunov theorems for stochastic hybrid systems. For non-stochastic hybrid systems, results on novel converse Lyapunov theorems for stochastic hybrid novel converse Lyapunov theorems for stochastic hybrid systems. For non-stochastic hybrid systems, results on systems. For non-stochastic hybrid systems, results on robustness of the recurrence property and the relation systems. For non-stochastic hybrid systems, results on systems. For non-stochastic hybrid systems, results on robustness of the recurrence property and the relation robustness of the recurrence property and the relation between recurrence and uniform ultimate boundedness of robustness of the recurrence property and the relation robustness of the recurrence property and the relation between recurrence and uniform ultimate boundedness of between recurrence and uniform ultimate boundedness of solutions are presented in Subbaraman and Teel (2015). between recurrence and uniform ultimate boundedness of between recurrence andinuniform ultimate boundedness of solutions are presented Subbaraman and Teel (2015). solutions are presented in Subbaraman and Teel (2015). solutions are presented in Subbaraman and Teel (2015). solutions are presented in organized Subbaraman and TeelSection (2015). 2 The rest of the paper is as follows. The rest of the paper is organized as follows. Section 2 The rest of the paper is organized as follows. Section 2 presents basic notation and definitions to be used in The of paper is as Section 2 The rest rest the of the the paper is organized organized as follows. follows. Section 2 presents the basic notation and definitions to be used in presents the basic notation and definitions to be used in the paper. Section 3 introduces the SHS framework that presents the basic notation and definitions to be used in presents theSection basic notation and the definitions to be used in the paper. 3 introduces SHS framework that the paper. Section 333 the introduces the SHS that will be considered rest of the The recurrence the paper. Section introduces the SHS framework that the paper. Sectionin introduces thepaper. SHS framework framework that will be considered in the rest of the paper. The recurrence will be considered in the rest of the paper. The recurrence property is explained in Section 4. Section 5 introduces will be in rest of The recurrence will be considered considered in the the rest of the the4.paper. paper. The recurrence property is explained in Section Section 5 introduces property is explained in Section 4. Section introduces viability and reachability probabilities which be used property is in 4. 555can introduces property is explained explained in Section Section 4. Section Section introduces viability and reachability probabilities which can be used viability and reachability probabilities which can be to prove the main results of the paper. Section 6 establishes viability and reachability probabilities which can be used viability andmain reachability probabilities which 6can be used used to prove the results of the paper. Section establishes to prove the main results of the paper. Section 666 establishes some basic bounds related to these probabilities. The main to prove the main results of the paper. Section establishes to prove the main results of the paper. Section establishes some basic bounds related to these probabilities. The main some basic bounds related to these probabilities. The main results are presented in Section Section 88 establishes some basic bounds related to probabilities. The some basic bounds related to these these7. probabilities. The main main results are presented in Section 7. Section establishes results are presented in Section 7. Section 8 establishes the link between the main results and the development results are presented in Section 7. Section 8 establishes results are presented in Section 7. and Section 8development establishes the link between the main results the the link the results and development of converse Lyapunov theorems. Section 99 presents some the link between the main results and the development the link between between the main main results and the the development of converse Lyapunov theorems. Section presents some of converse Lyapunov theorems. Section 9 presents some concluding comments and future work. The proofs of the of converse Lyapunov theorems. Section 9 presents some of converse Lyapunov theorems. Section 9 presents some concluding comments and future work. The proofs of the concluding comments and future work. The proofs of the main results are not presented due to space restrictions. concluding comments and future work. The proofs of concluding comments and future work. The restrictions. proofs of the the main results are not presented due to space main results are not presented due to space restrictions. main results are not presented due to space restrictions. main results are not presented due to space restrictions. 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND DEFINITIONS DEFINITIONS n n For a closed set S ⊂ R n and x ∈ Rn , |x|S := inf y∈S |x − For a closed set S ⊂ R and x ∈ R inf n n S := y∈S |x o− n n ,, |x| For aaa the closed set ∈ R inf y| is Euclidean distance B (resp., n and S := y∈S |x For closed set S ⊂ R and x ∈ R ,, |x| |x| := inf |xB − o− S y∈S For closed set S S ⊂ ⊂R R and x xof ∈x Rnto |x|S. inf −) y| is the Euclidean distance of x to S. B (resp., B S := y∈S |x o n o) y| is the Euclidean distance of x to S. B (resp., B denotes the closed (resp., open) unit ball in R . Given o y| is the Euclidean distance of x to S. B (resp., B n y| is thethe Euclidean distance of unit x to ball S. B (resp., B ))) denotes closed open) in R n . Given n(resp., n denotes the closed (resp., open) unit ball in R Given a closed set S ⊂ R and > 0, S + B represents set n .. the denotes the closed (resp., open) unit ball in R Given n denotes closed open) unit ball in R . the Given aa closed set S ⊂ R and > 0, S + B represents set n(resp., nthe n closed set S ⊂ R and > 0, S + B represents the set ∈ R : |x| ≤ }. R denotes the non-negative real n aa{x closed set S ⊂ R and > 0, S + B represents the set n S ≥0 closed set S ⊂ R and > 0, S + B represents the set {x ∈ R : |x| ≤ }. R denotes the non-negative real n S ≥0 n {x ∈ R : |x| ≤ }. R denotes the non-negative real numbers; Z denotes the non-negative integers. Let T be n : ≥0 S ≤ ≥0 denotes {x ∈ R |x| }. R the non-negative real S ≥0 {x ∈ R : |x| ≤ }. R denotes the non-negative real numbers; Z Sdenotes the ≥0non-negative integers. Let T be ≥0 numbers; Z denotes the non-negative integers. Let T be aanumbers; topological space. A function Ψ : T → R is upper ≥0 Z denotes the non-negative integers. Let T ≥0 Let numbers; Z≥0 denotesAthe non-negative integers. T be be topological function Ψ :: T → R upper ≥0 space. ≥0 is topological space. A function Ψ T → R is upper if for every converging sequence {t } → t, ≥0 aaasemicontinuous topological space. A function Ψ : T → R is upper i topological space. Aevery function Ψ : Tsequence → R≥0 isi }upper semicontinuous if for converging → t, ≥0 {t semicontinuous if for every converging sequence {t } → i semicontinuous if for every converging sequence {t } → t, i semicontinuous if for every converging sequence {ti } → t, t,
Research supported in part by AFOSR FA9550-12-1-0127, FA9550-15-1-0155 and NSF Research supported supported in ECCS-1232035. part by by AFOSR FA9550-12-1-0127, FA9550-12-1-0127, Research in part FA9550-15-1-0155 and NSF Research supported in ECCS-1232035. part by AFOSR AFOSR FA9550-12-1-0127, FA9550-15-1-0155 and NSF ECCS-1232035. FA9550-15-1-0155 and NSF ECCS-1232035. FA9550-15-1-0155 and NSF ECCS-1232035. Copyright © 2015, IFAC 2015 304 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 304 Copyright ©under IFAC responsibility 2015 304Control. Peer review of International Federation of Automatic Copyright © IFAC 2015 304 Copyright © IFAC 2015 304 10.1016/j.ifacol.2015.11.192
2015 IFAC ADHS October 14-16, 2015. Atlanta, USA Anantharaman Subbaraman et al. / IFAC-PapersOnLine 48-27 (2015) 304–309
lim supi→∞ Ψ(ti ) ≤ Ψ(t). For S ⊂ Rn , the symbol IS denotes the indicator function of S i.e., IS (x) = 1 for x ∈ S and IS (x) = 0 otherwise. Following Teel (2013), we define for sets S1 , S2 ⊂ Rp , I⊂S1 (S2 ) = 1−maxx∈S2 IRn \S1 (x) and I∩S1 (S2 ) = maxx∈S2 IS1 (x) with the convention that the maxima are zero when S2 = ∅. For τ ≥ 0, we define the set Γ≤τ := (s, t) ∈ R2 : s+t ≤ τ . (The sets Γ<τ , Γ≥τ are defined similarly). A set-valued mapping M : Rp ⇒ Rn is outer semicontinuous if, for each (xi , yi ) → (x, y) ∈ Rp × Rn satisfying yi ∈ M (xi ) for all i ∈ Z≥0 , y ∈ M (x). A mapping M is locally bounded if, for each bounded set K ⊂ Rp , M (K) := x∈K M (x) is bounded. B(Rm ) denotes the Borel σ-field. A set F ⊂ Rm is measurable if F ∈ B(Rm ). For a measurable space (Ω, F), a mapping M : Ω ⇒ Rn is measurable (Rockafellar and Wets, 1998, Def. 14.1) if, for each open set O ⊂ Rn , the set M −1 (O) := {ω ∈ Ω : M (ω) ∩ O = ∅} ∈ F. The functions πi : R≥0 × R≥0 × Rn → R≥0 are such that πi (t1 , t2 , z) = ti for each i ∈ {1, 2}. 3. PRELIMINARIES ON STOCHASTIC HYBRID SYSTEMS We consider a class of SHS with randomness restricted to the discrete-time dynamics introduced in Teel (2013). Let the state x ∈ Rn and random input v ∈ Rm . Then, the SHS is written formally as x∈C
x∈D
x˙ ∈ F (x) +
(1) +
x ∈ G(x, v )
(2)
µ(·) (3) where C, D ⊂ Rn represent the flow and jump sets respectively and F, G represent the flow and jump maps respectively. So, the continuous-time dynamics is modeled by a differential inclusion and the discrete-time dynamics is modeled by a stochastic difference inclusion. The distribution function µ is derived from the probability space (Ω, F, P) and a sequence of independent, identically distributed (i.i.d.) input random variables vi : Ω → Rm defined on (Ω, F, P) for i ∈ Z≥1 . Then µ is defined as µ(A) = P(ω ∈ Ω : vi (ω) ∈ A) for every A ∈ B(Rm ). We denote by Fi the collection of sets {ω : (v1 (ω), ..., vi (ω)) ∈ A}, A ∈ B((Rm )i ) which are the sub-σ fields of F that form the natural filtration of v = {vi }∞ i=1 . For simplicity we will refer to the data of the stochastic hybrid system through its data as H = (C, F, D, G, µ). We now define the notion of random solution to H under the following basic assumptions on the system data which is imposed throughout the rest of the paper. Standing Assumption 1. The data of the SHS H satisfies the following conditions: (1) The sets C, D ⊂ Rn are closed; (2) The mapping F : Rn ⇒ Rn is outer-semicontinuous, locally bounded with nonempty convex values on C; (3) The mapping G : Rn × Rm ⇒ Rn is locally bounded and the mapping v → graph(G(·, v)) := {(x, y) ∈ R2n : y ∈ G(x, v)} is measurable with closed values.
The need for the regularity properties listed in Standing Assumption 1 are twofold. Firstly the robustness results established in this paper exploit the system properties 305
305
in Standing Assumption 1. Secondly, it guarantees the existence of random solutions for the SHS. See Teel (2013) for more details. We first explain the notion of a solution to non-stochastic hybrid systems studied in Goebel et al. (2009) and Goebel et al. (2012). A compact hybrid time domain is a subset of R≥0 × Z≥0 of the form ∪Jj=0 ([tj , tj+1 ] × {j}) for some J ∈ Z≥0 and real numbers 0 = t0 ≤ t1 .... ≤ tJ+1 . A hybrid time domain is a set E ⊂ R≥0 × Z≥0 such that for each T, J, the set E ∩ ([0, T ] × {0, ..., J}) is a compact hybrid time domain. A hybrid arc is a mapping φ : E → Rn such that E is a hybrid time domain and for each j ∈ Z≥0 , the mapping t → φ(t, j) is locally absolutely continuous.
Let (Ω, F) be a measurable space. A stochastic hybrid arc is a mapping x defined on Ω such that x(ω) is a hybrid arc for each ω ∈ Ω and the set-valued mapping from Ω to Rn+2 defined by ω → graph(x(ω)) :=
{(t, j, z) : φ = x(ω), (t, j) ∈ dom(φ), z = φ(t, j)} is F-measurable with closed values. Define graph(x(ω))≤j := graph(x(ω)) ∩ (R≥0 × {0, ..., j} × Rn ). An {Fj }∞ j=0 adapted stochastic hybrid arc is a stochastic hybrid arc x such that the mapping ω → graph(x(ω))≤j is Fj measurable for each j ∈ Z≥0 . An adapted stochastic hybrid arc x is a solution starting from x denoted x ∈ Sr (x) if x(ω) is a solution to H with inputs {vi (ω)}∞ i=1 ; that is with φω := x(ω) we have • φω (0, 0) = x; • if (t1 , j), (t2 , j) ∈ dom(φω ) with t1 < t2 then, for almost every t ∈ [t1 , t2 ], φω (t, j) ∈ C and φ˙ ω (t, j) ∈ F (φω (t, j)); • if (t, j), (t, j + 1) ∈ dom(φω ) then φω (t, j) ∈ D and φω (t, j + 1) ∈ G(φω (t, j), vj+1 (ω)).
We refer the reader to Teel (2013) for more details on the solution concept to SHS. In this paper, we will sometimes focus on solutions that are maximal (See (Goebel et al., 2012, Def 2.7)). 4. RECURRENCE AND UNIFORM RECURRENCE In this section we define the notion of recurrence for open sets. An open, bounded set O ⊂ Rn is globally recurrent for H if there are no finite escape times for (1) and for each x ∈ Rn and x ∈ Sr (x), lim P graph(x) ⊂ (Γ<τ × Rn ) τ →∞ ∨ graph(x) ∩ (Γ≤τ × O) = 1.
Loosely speaking, the above condition insists that almost surely the sample paths of the random solution x are either not complete or hit the set O. We refer the reader to Teel (2013) for more details. An open, bounded set O ⊂ Rn is uniformly globally recurrent for H if there are no finite escape times for (1) and for each ρ > 0 and R > 0 there exists τ ≥ 0 such that for ξ ∈ RB and x ∈ Sr (ξ), n P graph(x) ⊂ (Γ<τ × R ) ∨ graph(x) ∩ (Γ≤τ × O)
2015 IFAC ADHS 306 October 14-16, 2015. Atlanta, USA Anantharaman Subbaraman et al. / IFAC-PapersOnLine 48-27 (2015) 304–309
≥ 1 − ρ. The following result establishes equivalence between uniform and non-uniform recurrence. We refer the reader to (Teel, 2014a, Thm 6) for a proof. Proposition 1. An open, bounded set O is globally recurrent for H if and only if it is uniformly globally recurrent for H.
Proposition 2. Let O ⊂ Rn be an open, bounded set. The following statements are equivalent: (1) O is globally recurrent. (2) limτ →∞ m⊂Rn \O (τ, x) = 0 for all x ∈ Rn . (3) For every compact set K ⊂ Rn , and ρ > 0, there exists τ ≥ 0 such that supx∈K m⊂Rn \O (τ, x) ≤ ρ. We also utilize reachability probabilities studied in (Teel, 2013, Section 8). For x ∈ Rn ,τ ≥ 0 and closed set S ⊂ Rn , we define m∩S (τ, x) := sup P graph(x) ∩ (Γ≤τ × S) = ∅ .
The equivalence between recurrence and uniform recurrence need not hold in the absence of Standing Assumption 1. We refer the reader to (Subbaraman and Teel, 2015, Example 2) for more details. The following example illustrates that recurrence of a set does not imply any stabilitylike properties for the stochastic system. Consider the simple discrete-time stochastic system x+ = max{0, x+v}, x ∈ Z≥0 , v ∈ {−1, 1} with µ(−1) = 0.6 and µ(1) = 0.4. Then, for the open set O = (−1, 1), V (x) = |x| is a Lyapunov function that guarantees global recurrence from (Subbaraman and Teel, 2013, Thm 1) as
The weak reachability probability m∩S (τ, x) is related to the worst case probability that random solutions starting from x reach the set S within hybrid time less than or equal to τ . It can be established similar to (Teel, 2013, Prop 10.2) that the supremum in the above definition is achieved for some random solution.
E[V (x+ )] = V (x) − 0.2, x ∈ Z≥0 \O.
6. PRELIMINARY BOUNDS ON VIABILITY AND REACHABILITY PROBABILITIES
It follows from (Meyn and Tweedie, 2009, Thm 8.1.2) that every set of the form Or = (r, r + 2) for every r ∈ Z≥0 is globally recurrent. This implies that no compact set for this system is stable in a probabilistic sense as solutions can go arbitrarily far away with probability one because of recurrence of sets Or = (r, r + 2) for every r ∈ Z≥0 . We can also easily observe that the recurrent set does not have any invariance-like property and solutions need not be bounded in a probabilistic sense due to recurrence of sets arbitrarily far from the origin. 5. VIABILITY AND REACHABILITY PROBABILITIES Since the recurrence property needs to hold for every random solution from an initial condition, it is useful to work with worst case probabilities related to recurrence. Motivated by Teel et al. (2014a) and Teel (2013) we characterize the recurrence property in terms of viability probabilities defined below (See (Teel, 2013, Section 9) for more details). For x ∈ Rn , τ ≥ 0 and closed set S ⊂ Rn , we define m⊂S (τ, x) := sup P graph(x) ∩ (Γ≥τ × Rn ) = ∅ x∈Sr (x)
∨ graph(x) ∩ (Γ≤τ
x∈Sr (x)
˜ be any SHS satisfying the conditions of Standing Let H Assumption 1 for which every maximal random solution x ˜ has almost surely complete sample paths. generated by H The probability bounds in this section are generated for ˜ This assumption will be satisfied for the the system H. SHS used in the robustness results in Section 7 and it also simplifies some of the proofs. We now present a series of bounds related to viability and reachability probabilities in this section. The following result when applied with the set S = Rn \O gives an alternative characterization of recurrence of the set O similar to (Subbaraman and Teel, 2013, Lemma 3). Proposition 3. Let S ⊂ Rn be closed. If there exists γ < 1 such that supx∈S m ⊂S (x) ≤ γ, then m ⊂S (x) = 0 for all x ∈ S.
The next result is motivated by the result in (Teel et al., 2014a, Lemma 3) and is similar in nature to the semi-group property for non-stochastic systems. Proposition 4. For closed sets S0 , S1 ⊂ Rn and (k1 , k2 , x) ∈ Z≥0 × Z≥0 × Rn , m⊂S0 (k1 + k2 , x) ≤ m⊂S1 (k1 , x) +
× Rn ) ⊂ R2 × S .
The viability probability m⊂S (τ, x) is related to the worst case probability that random solutions starting from x stay in the set S for hybrid time less than or equal to τ and not stop before that time. This probability condition is complementary to the condition for recurrence when the set S = Rn \O and when τ → ∞. It is established in (Teel, 2013, Prop 10.2, Prop 9.1) that the supremum in the above definition is achieved for some random solution and the mapping (τ, x) → m⊂S (τ, x) is upper semicontinuous. Define m ⊂S (x) := limτ →∞ m⊂S (τ, x). The limit is well defined due to the mapping τ → m⊂S (τ, x) being nonincreasing for every x. 306
sup ξ∈Rn \S1
m⊂S0 (k2 , ξ).
We now present a result that relates the viability and reachability probabilities. A similar result for discrete-time stochastic systems is in (Teel et al., 2014a, Lemma 1). Proposition 5. For closed sets S, S1 , S2 ⊂ Rn such that S ⊂ S1 ∪ S2 and for each x ∈ Rn and τ ≥ 0, m⊂S (τ, x) ≤ m⊂S1 (τ, x) + m∩S2 (τ, x). The final result is similar to (Teel et al., 2014a, Lemma 4) and establishes that the reachability probabilities m∩S (τ, x) can be made arbitrarily small for a fixed τ ≥ 0, for initial conditions x in a compact set, when the set S = Rn \RBo by choosing R > 0 sufficiently large. The
2015 IFAC ADHS October 14-16, 2015. Atlanta, USA Anantharaman Subbaraman et al. / IFAC-PapersOnLine 48-27 (2015) 304–309
307
proof follows along the same lines as (Teel et al., 2014a, Lemma 4) using the fact that the sample paths of random solutions are almost surely complete, the reachable set from a compact set of initial conditions for finite time is bounded for x ∈ C, x˙ ∈ F (x) using (Teel, 2013, Lemma 6.16), the local boundedness G and the dynamic programming methods in (Teel, 2013, Section 8.1). Proposition 6. For each k ∈ Z≥0 , ε > 0 and r > 0 there exists R > 0 such that, with S = Rn \RBo , m∩S (k, x) ≤ ε for all x ∈ rB.
The next result establishes that finite time viability probabilities related to a perturbation of a set S from a compact set of initial conditions can be made arbitrarily close to worst case probabilities related to the original set S provided the perturbation is small enough. The proof follows directly from the sequential compactness results established in Teel (2014a). Proposition 9. Let S ⊂ Rn be closed. For each (, ρ) ∈ Z≥0 × R>0 and K ⊂ Rn compact there exists a ε > 0 such that, for every x ∈ K,
7. MAIN RESULTS: ROBUSTNESS OF RECURRENCE
m⊂S+εB (, x) ≤ max m ˜ ⊂S (, ξ) + ρ. ξ∈K
In this section we discuss robustness of the recurrence property to perturbations. The proofs are motivated by the results in Subbaraman and Teel (2013) and Subbaraman and Teel (2015). Firstly, we establish that recurrence of an open bounded set implies recurrence of a smaller open set within the original set. This type of result can be viewed as robustness to perturbations in the set. Secondly, we prove recurrence is preserved when the data of the SHS is modified to slow down recurrence. Finally, we show that by perturbing the system data in a sufficiently small manner we preserve recurrence. We will work with SHS for which the maximal random solutions have almost surely complete sample paths. This modification will preserve recurrence and will play an important role in developing converse Lyapunov theorems. If the open, bounded set O is globally recurrent for H, := (C, F, Rn , G, µ) where consider the inflated system H v) = G1 (x, v) ∪ G2 (x) with G1 (x, v) = G(x, v) for G(x, x ∈ D and G1 (x, v) = ∅ for x ∈ / D, and G2 (x) = x∗ ∗ for some x ∈ O and for all x ∈ Rn . From the data of , recurrence of the set O for H and the hybrid system H solutions of (1) not exhibiting finite escape times it follows that is maximal, the that for every random solution of H sample paths are almost surely complete. The proof of the next result follows directly using (Teel et al., 2014a, Prop 2) and (Rockafellar and Wets, 1998, Prop 14.11 b). satisfies Standing Lemma 7. The data of the SHS H Assumption 1.
The next result establishes that the recurrence property is preserved for the modified system H. Lemma 8. If the open, bounded set O is globally recurrent for H, then O is globally recurrent for H. We can also observe that since the solutions of H are a if any set O is globally recurrent subset of solutions of H, then O is also globally recurrent for H. for H, 7.1 Robustness to perturbations of the set
The probabilities used in this subsection are generated for which the random solutions have using the system H almost surely complete sample paths. We define
Theorem 10. If an open bounded set O ⊂ Rn is globally then there exists an open bounded set O recurrent for H and ε > 0 such that O+εB ⊂ O and O is globally recurrent for H.
One of the main motivations for establishing the existence of smaller recurrent sets within a given recurrent set is that Theorem 10 plays an important role in developing converse Lyapunov theorems. In particular, it will be required in smoothing Lyapunov functions. We refer the reader to the converse theorems in Subbaraman and Teel (2015) and Subbaraman and Teel (2013) and Section 8 for more details. 7.2 Robustness to slowing down recurrence results in preserving The next inflation of the data of H recurrence while making certain quantities related to average value of worst case first hitting time for solutions to the set O grow unbounded in the distance of the state to the set O. The result is important in the context of developing converse Lyapunov theorems with radially unbounded Lyapunov functions. Since the result might have other applications we include it here.
For ν ∈ K∞ , define the continuous set-valued mapping Mν (x) := {x∗ } + ν(|x − x∗ |)B for x∗ ∈ Rn . Consider the ν := (C, F, Rn , G ν , µ) where G(x) inflated mapping H = G1 (x) ∪ Mν (x). The proof of the next result is very similar to Lemma 7. ν Lemma 11. For every ν ∈ K∞ , the data of the SHS H satisfies Standing Assumption 1.
The next theorem claims the existence of a ν ∈ K∞ small enough to preserve recurrence of the set O for the inflated ν if O is globally recurrent for H and x∗ ∈ O. system H A similar result is established for stochastic difference inclusions in (Subbaraman and Teel, 2013, Theorem 4) and the proof differs only in the construction of the function ν. Theorem 12. If the open, bounded set O ⊂ Rn is globally then for any x∗ ∈ O, there exists ν ∈ K∞ recurrent for H ν . such that O is globally recurrent for H
Consider the hybrid system with state x ∈ R2 , C = {x : |x| ≤ r}, D = {x : |x| ≥ r} for some r > 1, the flow map f (x) = 1 and the jump map g(x, v) = 0. It can be o m ˜ ⊂S (, ξ) := sup P(graph(x) ∩ (Γ< × Rn ) ⊂ (R2 × S)). observed that the set O = B is globally recurrent. For this system, the first time for solutions to hit the set O x∈Sr (ξ) 307
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does not increase with the distance of the state to the set O, as from large initial conditions the solutions reach the set O in one jump. Hence, with the modification D = Rn and the augmentation of the mapping Mν (x) = ν(|x|)B with x∗ = {0} and ν(s) = s/2, it can be observed that the recurrence property is preserved but the worst-case first time for solutions to reach the set O increases as the state grows larger. In essence, the modification slows down the recurrence property while preserving it.
there exists a continuous function ρ : Rn → R>0 and µ > 0 such that
∇V (x), f ≤ −ρ(x) + µIO (x), ∀x ∈ C, f ∈ F (x)
Rm
7.3 Robustness to perturbations of system data Finally, we analyze the robustness of recurrence to sufficiently small state dependent perturbations. For a continuous, positive-valued function δ : Rn → R>0 , we denote the δ , µ) by H δ with data (Cδ , Fδ , Rn , G perturbed version of H defined as Cδ := {x ∈ Rn : (x + δ(x)B) ∩ C = ∅}
Fδ (x) := conF (x + δ(x)B) ∩ C) + δ(x)B + δ(x)B, v)} Gδ (x, v) := {w ∈ Rn : w ∈ g + δ(g)B, g ∈ G(x where con refers to the closed convex hull. The next result follows from (Goebel et al., 2012, Proposition 6.28) and (Teel et al., 2014a, Prop 8). Lemma 13. For every continuous δ : Rn → R>0 , the data δ satisfies Standing Assumption 1. of the hybrid system H
The next result establishes closeness of probabilities between the perturbed and unperturbed SHS. For constant perturbations we use δ(x) ≡ δ for all x ∈ Rn . In this subsection we denote the probabilities generated by the δ with the subscript δ. system H Proposition 14. Let S ⊂ Rn be closed. For each (, ρ) ∈ Z≥0 × R>0 and K ⊂ Rn compact there exists a δ > 0 such that, for every x ∈ K, mδ,⊂S (, x) ≤ max m ˜ ⊂S (, ξ) + ρ. ξ∈K
The next result establishes that recurrence of the set open, bounded O set can be preserved when the state dependent perturbations are sufficiently small. The proof follows along the same lines as (Subbaraman and Teel, 2013, Thm 5). Theorem 15. If the open bounded set O ⊂ Rn is globally then there exists a continuous function recurrent for H, n δ . δ : R → R>0 such that O is globally recurrent for H
The proof of Theorem 15 relies primarily on the regularity properties of the SHS. In (Subbaraman and Teel, 2015, Example 4) it is established that the result of Theorem 15 need not hold even for the case of non-stochastic systems without the regularity properties in Standing Assumption 1. 8. APPLICATION TO CONVERSE THEOREMS
We now present a Lyapunov function based characterization of recurrence for open, bounded sets. A smooth function V : Rn → R≥0 is a Lyapunov function with respect to the set O for H if it is radially unbounded and 308
max V (g)µ(dv) ≤ V (x) − ρ(x) + µIO (x), ∀x ∈ D.
g∈G(x,v)
It is established in Teel (2013) that the existence of V satisfying the above conditions implies O is globally recurrent. Now we outline how the robustness results established in this paper will apply to the development of converse Lyapunov theorems. The outline is similar to that of Subbaraman and Teel (2015), Subbaraman and Teel (2013), Teel et al. (2014a). We start from the assumption that O is recurrent for H. The converse theorem will now need to establish the existence of a Lyapunov function satisfying the above decrease conditions. by Lemma Since O is recurrent for H, it is recurrent for H and > 0 8. Then, Theorem 10 implies that there exists O o + B ⊂ O and O is recurrent for H. Theorem such that O is 12 establishes that there exists ν ∈ K∞ such that O recurrent for Hν . Finally, Theorem 15 proves that there is recurrent for H ν,δ . The exists δ : Rn → R>0 such that O goal now is to construct a preliminary Lyapuov function candidate V0 : Rn → R≥0 (possibly non-smooth) which is radially unbounded and satisfies
Rm
max
g∈G(x,v)\O
DV0 (x) ≤ −ρ0 (x), ∀x ∈ C\O,
V0 (g)µ(dv) ≤ V0 (x) − ρ0 (x), ∀x ∈ D\O,
where D refers to the Dini derivative and ρ0 is a positive function characterizing decrease properties. Slowing down recurrence using Theorem 12 will help to guarantee that V0 is radially unbounded, particularly when V0 is related to the expected value of the worst first hitting time for See Subbaraman and Teel (2015) solutions to the set O. and Subbaraman and Teel (2013) for details. Then, we is smooth the Lyapunov function V0 that certifies O recurrent for Hν,δ to get a smooth function Vs (x) := V (x+σ(x)η)ψ(η)dη for appropriately chosen functions Rn 0 σ, ψ. Then Vs would be a smooth Lyapunov function with respect to O for the system H. The structure of Vs is motivated by Subbaraman and Teel (2013) and Teel et al. (2014a). 9. CONCLUSION We have presented a series of robustness results for a weak stability property called recurrence. We have outlined how the results can lead to the development of converse Lyapunov theorems for SHS. The robustness results are a result of the good regularity properties for the nominal SHS. Future work will focus on constructing the Lyapunov function V0 outlined in Section 8 and establish a converse Lyapunov theorem. REFERENCES Bujorianu, M.L. and Lygeros, J. (2006). Toward a general theory of stochastic hybrid systems. In Stochastic Hybrid Systems, 3–30. Springer.
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Davis, M.H. (1984). Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models. Journal of the Royal Statistical Society. Series B (Methodological), 353–388. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2009). Hybrid dynamical systems. IEEE Control Systems Magazine, 29(2), 28–93. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2012). Hybrid dynamical systems: modeling, stability, and robustness. Princeton University Press. Isaacs, J.T., Magee, C., Subbaraman, A., Quitin, F., Fregene, K., Teel, A.R., Madhow, U., and Hespanha, J.P. (2014). GPS-optimal micro air vehicle navigation in degraded environments. In American Control Conference (ACC), 2014, 2581–2587. IEEE. Meyn, S.P. and Tweedie, R.L. (2009). Markov chains and stochastic stability. Cambridge University Press. Poveda, J., Teel, A., and Nesic, D. (2015). Flexible nash seeking using stochastic difference inclusions. In Proceedings of the 2015 American Control Conference. Rockafellar, R.T. and Wets, R.J.B. (1998). Variational analysis, volume 317. Springer. Subbaraman, A., Hartman, M., and Teel, A.R. (2013). A stochastic hybrid algorithm for robust global almost sure synchronization on the circle: all-to-all communication. In Proceedings of the 52nd IEEE Conference on Decision and Control, 600–605. Subbaraman, A. and Teel, A.R. (2013). A converse Lyapunov theorem for strong global recurrence. Automat-
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Robustness of recurrence for a class Robustness of recurrence for a class Robustness of recurrence for a Robustness of recurrence for a class class stochastic hybrid systems stochastic hybrid systems stochastic hybrid systems
of of of of
Anantharaman Subbaraman, Andrew R. Teel Anantharaman Subbaraman, Andrew R. Teel Anantharaman Subbaraman, Andrew R. Anantharaman Teel Anantharaman Subbaraman, Subbaraman, Andrew Andrew R. R. Teel Teel Department of of Electrical Electrical and and Computer Computer Engineering Engineering Department Department Electrical and Engineering University of of California, Santa Barbara 93106-9560. 93106-9560. Department of Electrical and Computer Engineering Department of Electrical Santa and Computer Computer Engineering University of California, Barbara University of California, Santa Barbara 93106-9560. [email protected], [email protected] University of California, Santa Barbara 93106-9560. University of California, Santa Barbara 93106-9560. [email protected], [email protected] [email protected], [email protected], [email protected] [email protected], [email protected] [email protected] Abstract: We study stability property recurrence for aa class hybrid Abstract: We study a a weak weak stability property called called recurrence for class of of stochastic stochastic hybrid hybrid Abstract: We weak stability called recurrence for systems. Robustness the recurrence property to various state-dependent perturbations is Abstract: We study weak stability property called recurrence for class of stochastic hybrid Abstract: We study study aa aof weak stability property property called recurrence for aaa class class of of stochastic stochastic hybrid systems. Robustness of the recurrence property to various state-dependent perturbations is systems. Robustness of the recurrence property to various state-dependent perturbations is established under mild regularity conditions for the stochastic hybrid system. A potential systems. Robustness of the recurrence property to various state-dependent perturbations is systems. Robustness of the recurrence property to various state-dependent perturbations is established under mild regularity conditions for the stochastic hybrid system. A potential established under mild regularity conditions for the stochastic hybrid system. A potential application of the robustness results in the development of converse Lyapunov theorems is established under mild regularity conditions for the stochastic hybrid system. A potential established under mild regularity conditions for the stochastic hybridLyapunov system. A potential application of the robustness results in the development of converse theorems is application of also outlined. application of the robustness results in the development of converse Lyapunov theorems is application of the the robustness robustness results results in in the the development development of of converse converse Lyapunov Lyapunov theorems theorems is is also outlined. also outlined. also outlined. also outlined. © 2015, IFAC (International Federation of Automatichybrid Control)systems Hosting by Elsevier Ltd. All rights reserved. Keywords: Recurrence, Robustness, Stochastic Keywords: Recurrence, Robustness, Stochastic hybrid systems Keywords: Recurrence, Robustness, Stochastic hybrid systems Keywords: Keywords: Recurrence, Recurrence, Robustness, Robustness, Stochastic Stochastic hybrid hybrid systems systems 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION Hybrid systems are a of dynamical systems that Hybrid systems are aa class class of dynamical systems that Hybrid systems are class of dynamical systems that involve both continuous-time evolution and discrete-time Hybrid systems are a class of dynamical systems that Hybrid both systems are a class of dynamical systems that involve continuous-time evolution and discrete-time involve both continuous-time evolution and discrete-time events. Stochastic hybrid systems (SHS) this involve continuous-time evolution and discrete-time involve both both continuous-time evolution andgeneralize discrete-time events. Stochastic hybrid systems (SHS) generalize this events. Stochastic hybrid systems (SHS) generalize this class by adding randomness. In stochastic hybrid sysevents. Stochastic hybrid systems (SHS) generalize this events. Stochastic hybrid systems (SHS) generalize this class by adding randomness. In stochastic hybrid sysclass by adding randomness. In stochastic hybrid systems randomness can affect the continuous-time dynamics, class by adding randomness. In stochastic hybrid sysclass randomness by adding can randomness. In stochastic hybrid systems affect the continuous-time dynamics, tems randomness can affect the continuous-time dynamics, discrete-time dynamics and also the transition between tems can continuous-time tems randomness randomness can affect affect the continuous-time dynamics, discrete-time dynamics andthe also the transition transitiondynamics, between discrete-time dynamics and also the between them. Frameworks for modeling SHS are in Yin and Zhu discrete-time dynamics and also the transition between discrete-time dynamics and alsoSHS the are transition between them. Frameworks for modeling in Yin and Zhu them. Frameworks for modeling SHS are in Yin and Zhu (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros them. Frameworks for modeling SHS are in Yin and Zhu them. Frameworks for modeling SHS are in Yin and Zhu (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2006) and Teel et al. (2014b). (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2009),Teel (2013), Davis (1984), Bujorianu and Lygeros (2006) and Teel et al. (2014b). (2006) and Teel et al. (2014b). (2006) Teel (2014b). (2006) and andframeworks Teel et et al. al. for (2014b). Modeling stochastic hybrid systems seldom Modeling frameworks for stochastic hybrid systems seldom seldom Modeling frameworks for stochastic hybrid systems account for systems that generate non-unique solutions. Modeling frameworks for stochastic hybrid systems seldom Modelingfor frameworks for stochastic hybrid systems seldom account systems that generate non-unique solutions. account for systems that generate non-unique solutions. In Teel (2013) and Teel a generalized account for that generate non-unique solutions. account for systems systems that(2014b) generate non-uniqueframework solutions. In Teel (2013) and Teel (2014b) a generalized framework In Teel (2013) and Teel (2014b) generalized framework for modeling solutions proposed. In (2013) and Teel (2014b) In Teel Teel (2013)SHS andwith Teel non-unique (2014b) aaa generalized generalized framework for modeling SHS with non-unique solutions is isframework proposed. for modeling SHS with non-unique solutions is proposed. The need to study stochastic systems with non-unique for modeling SHS with non-unique solutions is proposed. for modeling SHS with non-unique solutions is proposed. The need to study stochastic systems with non-unique The need to study stochastic systems with non-unique solutions Firstly, analyzing such systems is The to study systems non-unique The need need is totwofold. study stochastic stochastic systems with with non-unique solutions is twofold. Firstly, analyzing such systems is solutions is twofold. Firstly, analyzing such systems is crucial to developing a robust stability theory. Robust solutions is twofold. Firstly, analyzing such systems is solutionsto isdeveloping twofold. Firstly, analyzing such systems is crucial a robust stability theory. Robust crucial to developing robust stability theory. Robust stability non-stochastic hybrid systems is studied in crucial developing aaa robust stability theory. Robust crucial to tofor developing robust stability theory. Robust stability for non-stochastic hybrid systems is studied in stability for non-stochastic hybrid systems is studied in Goebel et al. (2012). Secondly, such system models allow stability for non-stochastic hybrid systems is studied in stability for non-stochastic hybrid systems is studied in Goebel et al. (2012). Secondly, such system models allow Goebel et al. (2012). Secondly, such system models allow flexibility in control design applications. See Subbaraman Goebel et al. (2012). Secondly, such system models allow Goebel et in al.control (2012).design Secondly, such system models allow flexibility applications. See Subbaraman flexibility in control design applications. See Subbaraman et al. (2013), Venkateswaran et al. (2013) and et al. flexibility in design See Subbaraman flexibility in control control design applications. applications. See Poveda Subbaraman et al. (2013), Venkateswaran et al. (2013) and Poveda et al. et al. (2013), Venkateswaran et al. (2013) and Poveda et (2015). et al. (2013), Venkateswaran et al. (2013) and Poveda et al. et al. (2013), Venkateswaran et al. (2013) and Poveda et al. al. (2015). (2015). (2015). (2015). The stochastic stability property of interest in this paper The stochastic stability property of interest in this paper The stochastic stability property of interest in this paper is called recurrence. Loosely speaking, recurrence of an The stochastic stability property of in The stochastic stability property of interest interest in this this paper paper is called recurrence. Loosely speaking, recurrence of an is called recurrence. Loosely speaking, recurrence of an open set implies that solutions visit the set infinitely is recurrence. Loosely speaking, recurrence of is called called recurrence. Loosely speaking, recurrence of an an open set implies that solutions visit the set infinitely open set implies that solutions visit the set infinitely often with probability one. The recurrence property is open set implies that solutions visit the set infinitely open set implies that one. solutions visit the set infinitely often with probability The recurrence property is often with probability one. The recurrence property is frequently studied in the literature. See Yin and Zhu often with probability one. The recurrence property is often with studied probability one.literature. The recurrence property is frequently in the See Yin and Zhu frequently studied in the literature. See Yin and Zhu (2009) and Meyn and Tweedie (2009). Recurrence is aa frequently studied in literature. See Yin frequently studied in the the literature. SeeRecurrence Yin and and Zhu Zhu (2009) and Meyn and Tweedie (2009). is (2009) and Meyn and Tweedie (2009). Recurrence is weaker notion of stability compared to more commonly (2009) and Meyn and Tweedie (2009). Recurrence is a (2009) and Meyn and Tweedie (2009). Recurrence is aa weaker notion of stability compared to more commonly weaker notion of stability compared to more commonly studied notions of mean square asymptotic stability and weaker notion of stability compared to more commonly weaker notion stability compared to more commonly studied notionsofof mean square asymptotic stability and studied notions of mean square asymptotic stability and asymptotic stability in probability. Recurrence of an open studied notions of mean square asymptotic stability and studied notions of mean square asymptotic stability and asymptotic stability in probability. Recurrence of an open asymptotic stability in probability. Recurrence of an open bounded set need not imply any stability-like property for asymptotic stability in probability. Recurrence of asymptotic stability inimply probability. Recurrenceproperty of an an open open bounded set need not any stability-like for bounded set need not imply any stability-like property for the set. Recurrence of a set does not imply any invariancebounded set need not imply any stability-like property for bounded set need not imply any not stability-like property for the set. Recurrence of a set does imply any invariancethe set. Recurrence of a set does not imply any invariancelike property for the set and finally, recurrence of an open, the set. Recurrence of a set does not imply any invariancethe set. Recurrence ofset a set does notrecurrence imply any of invariancelike property for the and finally, an open, like property for the set and finally, recurrence of an open, bounded set does not imply solutions stay bounded in like for set and finally, recurrence of like property property for the the set andthat finally, recurrence of an an open, open, bounded set does not imply that solutions stay bounded in bounded set does not imply that solutions stay bounded in a probabilistic sense. Recurrence is still a useful property bounded set does not imply that solutions stay bounded in bounded set does not imply that solutions stay bounded in aa probabilistic sense. Recurrence is still a useful property probabilistic sense. Recurrence is still a useful property study since it provides an alternative when stronger ato probabilistic sense. Recurrence is still a useful property a probabilistic sense. Recurrence is still a useful property to study since it provides an alternative when stronger to study since it provides an alternative when stronger to to study study since since it it provides provides an an alternative alternative when when stronger stronger Research supported in part by AFOSR FA9550-12-1-0127,
properties like convergence or asymptotic stability are properties like convergence or asymptotic stability are properties like convergence or asymptotic stability are impossible to establish. This is particularly true in the properties like convergence or asymptotic stability are properties like convergence oris asymptotic stability are impossible to establish. This particularly true in the impossible to establish. This is particularly true in the case of systems affected by persistent disturbances. We impossible to establish. This is particularly true in the impossible to establish. This is particularly true in We the case of systems affected by persistent disturbances. case of systems affected by disturbances. We refer the reader to (Subbaraman and Teel, 2015, Example case of systems affected by persistent disturbances. We case of systems affected by persistent persistent disturbances. We refer the reader to (Subbaraman and Teel, 2015, Example refer the reader to (Subbaraman and Teel, 2015, Example 1) and Isaacs et al. (2014) for more details. refer the reader to (Subbaraman and Teel, 2015, Example refer the reader to (Subbaraman and Teel, 2015, Example 1) and Isaacs et al. (2014) for more details. 1) and Isaacs et al. (2014) for more details. 1) and al. for more details. 1) and Isaacs Isaacsofet etstability al. (2014) (2014) for more details. Robustness can be loosely defined as the staRobustness of stability can be loosely defined as the staRobustness of stability can be loosely defined as the stability property being preserved when the nominal system Robustness of stability can be loosely defined as the staRobustness of stability can be loosely defined as the stability property being preserved when the nominal system bility property being preserved when the nominal system is affected by sufficiently small perturbations. The robility property being preserved when the nominal system bility property being preserved when the nominalThe system is affected by sufficiently small perturbations. rois affected by sufficiently small perturbations. The robustness results in this paper exploit the good regularity is affected by sufficiently small perturbations. The is affected by sufficiently small perturbations. The rorobustness results in this paper exploit the good regularity bustness results in this paper exploit the good regularity properties of the nominal system. The results generated bustness results in this paper exploit the good regularity bustness results innominal this paper exploit theresults good regularity properties of the system. The generated properties of nominal system. The generated in this work have potential application in establishing properties of the nominal system. The results generated properties of the the nominal system. The results results generated in this work have potential application in establishing in this work have potential application in establishing novel converse Lyapunov theorems for stochastic hybrid in this work have potential application in establishing in thisconverse work have potential application in establishing novel Lyapunov theorems for stochastic hybrid novel converse Lyapunov theorems for stochastic hybrid systems. For non-stochastic hybrid systems, results on novel converse Lyapunov theorems for stochastic hybrid novel converse Lyapunov theorems for stochastic hybrid systems. For non-stochastic hybrid systems, results on systems. For non-stochastic hybrid systems, results on robustness of the recurrence property and the relation systems. For non-stochastic hybrid systems, results on systems. For non-stochastic hybrid systems, results on robustness of the recurrence property and the relation robustness of the recurrence property and the relation between recurrence and uniform ultimate boundedness of robustness of the recurrence property and the relation robustness of the recurrence property and the relation between recurrence and uniform ultimate boundedness of between recurrence and uniform ultimate boundedness of solutions are presented in Subbaraman and Teel (2015). between recurrence and uniform ultimate boundedness of between recurrence andinuniform ultimate boundedness of solutions are presented Subbaraman and Teel (2015). solutions are presented in Subbaraman and Teel (2015). solutions are presented in Subbaraman and Teel (2015). solutions are presented in organized Subbaraman and TeelSection (2015). 2 The rest of the paper is as follows. The rest of the paper is organized as follows. Section 2 The rest of the paper is organized as follows. Section 2 presents basic notation and definitions to be used in The of paper is as Section 2 The rest rest the of the the paper is organized organized as follows. follows. Section 2 presents the basic notation and definitions to be used in presents the basic notation and definitions to be used in the paper. Section 3 introduces the SHS framework that presents the basic notation and definitions to be used in presents theSection basic notation and the definitions to be used in the paper. 3 introduces SHS framework that the paper. Section 333 the introduces the SHS that will be considered rest of the The recurrence the paper. Section introduces the SHS framework that the paper. Sectionin introduces thepaper. SHS framework framework that will be considered in the rest of the paper. The recurrence will be considered in the rest of the paper. The recurrence property is explained in Section 4. Section 5 introduces will be in rest of The recurrence will be considered considered in the the rest of the the4.paper. paper. The recurrence property is explained in Section Section 5 introduces property is explained in Section 4. Section introduces viability and reachability probabilities which be used property is in 4. 555can introduces property is explained explained in Section Section 4. Section Section introduces viability and reachability probabilities which can be used viability and reachability probabilities which can be to prove the main results of the paper. Section 6 establishes viability and reachability probabilities which can be used viability andmain reachability probabilities which 6can be used used to prove the results of the paper. Section establishes to prove the main results of the paper. Section 666 establishes some basic bounds related to these probabilities. The main to prove the main results of the paper. Section establishes to prove the main results of the paper. Section establishes some basic bounds related to these probabilities. The main some basic bounds related to these probabilities. The main results are presented in Section Section 88 establishes some basic bounds related to probabilities. The some basic bounds related to these these7. probabilities. The main main results are presented in Section 7. Section establishes results are presented in Section 7. Section 8 establishes the link between the main results and the development results are presented in Section 7. Section 8 establishes results are presented in Section 7. and Section 8development establishes the link between the main results the the link the results and development of converse Lyapunov theorems. Section 99 presents some the link between the main results and the development the link between between the main main results and the the development of converse Lyapunov theorems. Section presents some of converse Lyapunov theorems. Section 9 presents some concluding comments and future work. The proofs of the of converse Lyapunov theorems. Section 9 presents some of converse Lyapunov theorems. Section 9 presents some concluding comments and future work. The proofs of the concluding comments and future work. The proofs of the main results are not presented due to space restrictions. concluding comments and future work. The proofs of concluding comments and future work. The restrictions. proofs of the the main results are not presented due to space main results are not presented due to space restrictions. main results are not presented due to space restrictions. main results are not presented due to space restrictions. 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND 2. BASIC NOTATION AND DEFINITIONS 2. BASIC NOTATION AND DEFINITIONS DEFINITIONS n n For a closed set S ⊂ R n and x ∈ Rn , |x|S := inf y∈S |x − For a closed set S ⊂ R and x ∈ R inf n n S := y∈S |x o− n n ,, |x| For aaa the closed set ∈ R inf y| is Euclidean distance B (resp., n and S := y∈S |x For closed set S ⊂ R and x ∈ R ,, |x| |x| := inf |xB − o− S y∈S For closed set S S ⊂ ⊂R R and x xof ∈x Rnto |x|S. inf −) y| is the Euclidean distance of x to S. B (resp., B S := y∈S |x o n o) y| is the Euclidean distance of x to S. B (resp., B denotes the closed (resp., open) unit ball in R . Given o y| is the Euclidean distance of x to S. B (resp., B n y| is thethe Euclidean distance of unit x to ball S. B (resp., B ))) denotes closed open) in R n . Given n(resp., n denotes the closed (resp., open) unit ball in R Given a closed set S ⊂ R and > 0, S + B represents set n .. the denotes the closed (resp., open) unit ball in R Given n denotes closed open) unit ball in R . the Given aa closed set S ⊂ R and > 0, S + B represents set n(resp., nthe n closed set S ⊂ R and > 0, S + B represents the set ∈ R : |x| ≤ }. R denotes the non-negative real n aa{x closed set S ⊂ R and > 0, S + B represents the set n S ≥0 closed set S ⊂ R and > 0, S + B represents the set {x ∈ R : |x| ≤ }. R denotes the non-negative real n S ≥0 n {x ∈ R : |x| ≤ }. R denotes the non-negative real numbers; Z denotes the non-negative integers. Let T be n : ≥0 S ≤ ≥0 denotes {x ∈ R |x| }. R the non-negative real S ≥0 {x ∈ R : |x| ≤ }. R denotes the non-negative real numbers; Z Sdenotes the ≥0non-negative integers. Let T be ≥0 numbers; Z denotes the non-negative integers. Let T be aanumbers; topological space. A function Ψ : T → R is upper ≥0 Z denotes the non-negative integers. Let T ≥0 Let numbers; Z≥0 denotesAthe non-negative integers. T be be topological function Ψ :: T → R upper ≥0 space. ≥0 is topological space. A function Ψ T → R is upper if for every converging sequence {t } → t, ≥0 aaasemicontinuous topological space. A function Ψ : T → R is upper i topological space. Aevery function Ψ : Tsequence → R≥0 isi }upper semicontinuous if for converging → t, ≥0 {t semicontinuous if for every converging sequence {t } → i semicontinuous if for every converging sequence {t } → t, i semicontinuous if for every converging sequence {ti } → t, t,
Research supported in part by AFOSR FA9550-12-1-0127, FA9550-15-1-0155 and NSF Research supported supported in ECCS-1232035. part by by AFOSR FA9550-12-1-0127, FA9550-12-1-0127, Research in part FA9550-15-1-0155 and NSF Research supported in ECCS-1232035. part by AFOSR AFOSR FA9550-12-1-0127, FA9550-15-1-0155 and NSF ECCS-1232035. FA9550-15-1-0155 and NSF ECCS-1232035. FA9550-15-1-0155 and NSF ECCS-1232035. Copyright © 2015, IFAC 2015 304 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 304 Copyright ©under IFAC responsibility 2015 304Control. Peer review of International Federation of Automatic Copyright © IFAC 2015 304 Copyright © IFAC 2015 304 10.1016/j.ifacol.2015.11.192
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lim supi→∞ Ψ(ti ) ≤ Ψ(t). For S ⊂ Rn , the symbol IS denotes the indicator function of S i.e., IS (x) = 1 for x ∈ S and IS (x) = 0 otherwise. Following Teel (2013), we define for sets S1 , S2 ⊂ Rp , I⊂S1 (S2 ) = 1−maxx∈S2 IRn \S1 (x) and I∩S1 (S2 ) = maxx∈S2 IS1 (x) with the convention that the maxima are zero when S2 = ∅. For τ ≥ 0, we define the set Γ≤τ := (s, t) ∈ R2 : s+t ≤ τ . (The sets Γ<τ , Γ≥τ are defined similarly). A set-valued mapping M : Rp ⇒ Rn is outer semicontinuous if, for each (xi , yi ) → (x, y) ∈ Rp × Rn satisfying yi ∈ M (xi ) for all i ∈ Z≥0 , y ∈ M (x). A mapping M is locally bounded if, for each bounded set K ⊂ Rp , M (K) := x∈K M (x) is bounded. B(Rm ) denotes the Borel σ-field. A set F ⊂ Rm is measurable if F ∈ B(Rm ). For a measurable space (Ω, F), a mapping M : Ω ⇒ Rn is measurable (Rockafellar and Wets, 1998, Def. 14.1) if, for each open set O ⊂ Rn , the set M −1 (O) := {ω ∈ Ω : M (ω) ∩ O = ∅} ∈ F. The functions πi : R≥0 × R≥0 × Rn → R≥0 are such that πi (t1 , t2 , z) = ti for each i ∈ {1, 2}. 3. PRELIMINARIES ON STOCHASTIC HYBRID SYSTEMS We consider a class of SHS with randomness restricted to the discrete-time dynamics introduced in Teel (2013). Let the state x ∈ Rn and random input v ∈ Rm . Then, the SHS is written formally as x∈C
x∈D
x˙ ∈ F (x) +
(1) +
x ∈ G(x, v )
(2)
µ(·) (3) where C, D ⊂ Rn represent the flow and jump sets respectively and F, G represent the flow and jump maps respectively. So, the continuous-time dynamics is modeled by a differential inclusion and the discrete-time dynamics is modeled by a stochastic difference inclusion. The distribution function µ is derived from the probability space (Ω, F, P) and a sequence of independent, identically distributed (i.i.d.) input random variables vi : Ω → Rm defined on (Ω, F, P) for i ∈ Z≥1 . Then µ is defined as µ(A) = P(ω ∈ Ω : vi (ω) ∈ A) for every A ∈ B(Rm ). We denote by Fi the collection of sets {ω : (v1 (ω), ..., vi (ω)) ∈ A}, A ∈ B((Rm )i ) which are the sub-σ fields of F that form the natural filtration of v = {vi }∞ i=1 . For simplicity we will refer to the data of the stochastic hybrid system through its data as H = (C, F, D, G, µ). We now define the notion of random solution to H under the following basic assumptions on the system data which is imposed throughout the rest of the paper. Standing Assumption 1. The data of the SHS H satisfies the following conditions: (1) The sets C, D ⊂ Rn are closed; (2) The mapping F : Rn ⇒ Rn is outer-semicontinuous, locally bounded with nonempty convex values on C; (3) The mapping G : Rn × Rm ⇒ Rn is locally bounded and the mapping v → graph(G(·, v)) := {(x, y) ∈ R2n : y ∈ G(x, v)} is measurable with closed values.
The need for the regularity properties listed in Standing Assumption 1 are twofold. Firstly the robustness results established in this paper exploit the system properties 305
305
in Standing Assumption 1. Secondly, it guarantees the existence of random solutions for the SHS. See Teel (2013) for more details. We first explain the notion of a solution to non-stochastic hybrid systems studied in Goebel et al. (2009) and Goebel et al. (2012). A compact hybrid time domain is a subset of R≥0 × Z≥0 of the form ∪Jj=0 ([tj , tj+1 ] × {j}) for some J ∈ Z≥0 and real numbers 0 = t0 ≤ t1 .... ≤ tJ+1 . A hybrid time domain is a set E ⊂ R≥0 × Z≥0 such that for each T, J, the set E ∩ ([0, T ] × {0, ..., J}) is a compact hybrid time domain. A hybrid arc is a mapping φ : E → Rn such that E is a hybrid time domain and for each j ∈ Z≥0 , the mapping t → φ(t, j) is locally absolutely continuous.
Let (Ω, F) be a measurable space. A stochastic hybrid arc is a mapping x defined on Ω such that x(ω) is a hybrid arc for each ω ∈ Ω and the set-valued mapping from Ω to Rn+2 defined by ω → graph(x(ω)) :=
{(t, j, z) : φ = x(ω), (t, j) ∈ dom(φ), z = φ(t, j)} is F-measurable with closed values. Define graph(x(ω))≤j := graph(x(ω)) ∩ (R≥0 × {0, ..., j} × Rn ). An {Fj }∞ j=0 adapted stochastic hybrid arc is a stochastic hybrid arc x such that the mapping ω → graph(x(ω))≤j is Fj measurable for each j ∈ Z≥0 . An adapted stochastic hybrid arc x is a solution starting from x denoted x ∈ Sr (x) if x(ω) is a solution to H with inputs {vi (ω)}∞ i=1 ; that is with φω := x(ω) we have • φω (0, 0) = x; • if (t1 , j), (t2 , j) ∈ dom(φω ) with t1 < t2 then, for almost every t ∈ [t1 , t2 ], φω (t, j) ∈ C and φ˙ ω (t, j) ∈ F (φω (t, j)); • if (t, j), (t, j + 1) ∈ dom(φω ) then φω (t, j) ∈ D and φω (t, j + 1) ∈ G(φω (t, j), vj+1 (ω)).
We refer the reader to Teel (2013) for more details on the solution concept to SHS. In this paper, we will sometimes focus on solutions that are maximal (See (Goebel et al., 2012, Def 2.7)). 4. RECURRENCE AND UNIFORM RECURRENCE In this section we define the notion of recurrence for open sets. An open, bounded set O ⊂ Rn is globally recurrent for H if there are no finite escape times for (1) and for each x ∈ Rn and x ∈ Sr (x), lim P graph(x) ⊂ (Γ<τ × Rn ) τ →∞ ∨ graph(x) ∩ (Γ≤τ × O) = 1.
Loosely speaking, the above condition insists that almost surely the sample paths of the random solution x are either not complete or hit the set O. We refer the reader to Teel (2013) for more details. An open, bounded set O ⊂ Rn is uniformly globally recurrent for H if there are no finite escape times for (1) and for each ρ > 0 and R > 0 there exists τ ≥ 0 such that for ξ ∈ RB and x ∈ Sr (ξ), n P graph(x) ⊂ (Γ<τ × R ) ∨ graph(x) ∩ (Γ≤τ × O)
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≥ 1 − ρ. The following result establishes equivalence between uniform and non-uniform recurrence. We refer the reader to (Teel, 2014a, Thm 6) for a proof. Proposition 1. An open, bounded set O is globally recurrent for H if and only if it is uniformly globally recurrent for H.
Proposition 2. Let O ⊂ Rn be an open, bounded set. The following statements are equivalent: (1) O is globally recurrent. (2) limτ →∞ m⊂Rn \O (τ, x) = 0 for all x ∈ Rn . (3) For every compact set K ⊂ Rn , and ρ > 0, there exists τ ≥ 0 such that supx∈K m⊂Rn \O (τ, x) ≤ ρ. We also utilize reachability probabilities studied in (Teel, 2013, Section 8). For x ∈ Rn ,τ ≥ 0 and closed set S ⊂ Rn , we define m∩S (τ, x) := sup P graph(x) ∩ (Γ≤τ × S) = ∅ .
The equivalence between recurrence and uniform recurrence need not hold in the absence of Standing Assumption 1. We refer the reader to (Subbaraman and Teel, 2015, Example 2) for more details. The following example illustrates that recurrence of a set does not imply any stabilitylike properties for the stochastic system. Consider the simple discrete-time stochastic system x+ = max{0, x+v}, x ∈ Z≥0 , v ∈ {−1, 1} with µ(−1) = 0.6 and µ(1) = 0.4. Then, for the open set O = (−1, 1), V (x) = |x| is a Lyapunov function that guarantees global recurrence from (Subbaraman and Teel, 2013, Thm 1) as
The weak reachability probability m∩S (τ, x) is related to the worst case probability that random solutions starting from x reach the set S within hybrid time less than or equal to τ . It can be established similar to (Teel, 2013, Prop 10.2) that the supremum in the above definition is achieved for some random solution.
E[V (x+ )] = V (x) − 0.2, x ∈ Z≥0 \O.
6. PRELIMINARY BOUNDS ON VIABILITY AND REACHABILITY PROBABILITIES
It follows from (Meyn and Tweedie, 2009, Thm 8.1.2) that every set of the form Or = (r, r + 2) for every r ∈ Z≥0 is globally recurrent. This implies that no compact set for this system is stable in a probabilistic sense as solutions can go arbitrarily far away with probability one because of recurrence of sets Or = (r, r + 2) for every r ∈ Z≥0 . We can also easily observe that the recurrent set does not have any invariance-like property and solutions need not be bounded in a probabilistic sense due to recurrence of sets arbitrarily far from the origin. 5. VIABILITY AND REACHABILITY PROBABILITIES Since the recurrence property needs to hold for every random solution from an initial condition, it is useful to work with worst case probabilities related to recurrence. Motivated by Teel et al. (2014a) and Teel (2013) we characterize the recurrence property in terms of viability probabilities defined below (See (Teel, 2013, Section 9) for more details). For x ∈ Rn , τ ≥ 0 and closed set S ⊂ Rn , we define m⊂S (τ, x) := sup P graph(x) ∩ (Γ≥τ × Rn ) = ∅ x∈Sr (x)
∨ graph(x) ∩ (Γ≤τ
x∈Sr (x)
˜ be any SHS satisfying the conditions of Standing Let H Assumption 1 for which every maximal random solution x ˜ has almost surely complete sample paths. generated by H The probability bounds in this section are generated for ˜ This assumption will be satisfied for the the system H. SHS used in the robustness results in Section 7 and it also simplifies some of the proofs. We now present a series of bounds related to viability and reachability probabilities in this section. The following result when applied with the set S = Rn \O gives an alternative characterization of recurrence of the set O similar to (Subbaraman and Teel, 2013, Lemma 3). Proposition 3. Let S ⊂ Rn be closed. If there exists γ < 1 such that supx∈S m ⊂S (x) ≤ γ, then m ⊂S (x) = 0 for all x ∈ S.
The next result is motivated by the result in (Teel et al., 2014a, Lemma 3) and is similar in nature to the semi-group property for non-stochastic systems. Proposition 4. For closed sets S0 , S1 ⊂ Rn and (k1 , k2 , x) ∈ Z≥0 × Z≥0 × Rn , m⊂S0 (k1 + k2 , x) ≤ m⊂S1 (k1 , x) +
× Rn ) ⊂ R2 × S .
The viability probability m⊂S (τ, x) is related to the worst case probability that random solutions starting from x stay in the set S for hybrid time less than or equal to τ and not stop before that time. This probability condition is complementary to the condition for recurrence when the set S = Rn \O and when τ → ∞. It is established in (Teel, 2013, Prop 10.2, Prop 9.1) that the supremum in the above definition is achieved for some random solution and the mapping (τ, x) → m⊂S (τ, x) is upper semicontinuous. Define m ⊂S (x) := limτ →∞ m⊂S (τ, x). The limit is well defined due to the mapping τ → m⊂S (τ, x) being nonincreasing for every x. 306
sup ξ∈Rn \S1
m⊂S0 (k2 , ξ).
We now present a result that relates the viability and reachability probabilities. A similar result for discrete-time stochastic systems is in (Teel et al., 2014a, Lemma 1). Proposition 5. For closed sets S, S1 , S2 ⊂ Rn such that S ⊂ S1 ∪ S2 and for each x ∈ Rn and τ ≥ 0, m⊂S (τ, x) ≤ m⊂S1 (τ, x) + m∩S2 (τ, x). The final result is similar to (Teel et al., 2014a, Lemma 4) and establishes that the reachability probabilities m∩S (τ, x) can be made arbitrarily small for a fixed τ ≥ 0, for initial conditions x in a compact set, when the set S = Rn \RBo by choosing R > 0 sufficiently large. The
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307
proof follows along the same lines as (Teel et al., 2014a, Lemma 4) using the fact that the sample paths of random solutions are almost surely complete, the reachable set from a compact set of initial conditions for finite time is bounded for x ∈ C, x˙ ∈ F (x) using (Teel, 2013, Lemma 6.16), the local boundedness G and the dynamic programming methods in (Teel, 2013, Section 8.1). Proposition 6. For each k ∈ Z≥0 , ε > 0 and r > 0 there exists R > 0 such that, with S = Rn \RBo , m∩S (k, x) ≤ ε for all x ∈ rB.
The next result establishes that finite time viability probabilities related to a perturbation of a set S from a compact set of initial conditions can be made arbitrarily close to worst case probabilities related to the original set S provided the perturbation is small enough. The proof follows directly from the sequential compactness results established in Teel (2014a). Proposition 9. Let S ⊂ Rn be closed. For each (, ρ) ∈ Z≥0 × R>0 and K ⊂ Rn compact there exists a ε > 0 such that, for every x ∈ K,
7. MAIN RESULTS: ROBUSTNESS OF RECURRENCE
m⊂S+εB (, x) ≤ max m ˜ ⊂S (, ξ) + ρ. ξ∈K
In this section we discuss robustness of the recurrence property to perturbations. The proofs are motivated by the results in Subbaraman and Teel (2013) and Subbaraman and Teel (2015). Firstly, we establish that recurrence of an open bounded set implies recurrence of a smaller open set within the original set. This type of result can be viewed as robustness to perturbations in the set. Secondly, we prove recurrence is preserved when the data of the SHS is modified to slow down recurrence. Finally, we show that by perturbing the system data in a sufficiently small manner we preserve recurrence. We will work with SHS for which the maximal random solutions have almost surely complete sample paths. This modification will preserve recurrence and will play an important role in developing converse Lyapunov theorems. If the open, bounded set O is globally recurrent for H, := (C, F, Rn , G, µ) where consider the inflated system H v) = G1 (x, v) ∪ G2 (x) with G1 (x, v) = G(x, v) for G(x, x ∈ D and G1 (x, v) = ∅ for x ∈ / D, and G2 (x) = x∗ ∗ for some x ∈ O and for all x ∈ Rn . From the data of , recurrence of the set O for H and the hybrid system H solutions of (1) not exhibiting finite escape times it follows that is maximal, the that for every random solution of H sample paths are almost surely complete. The proof of the next result follows directly using (Teel et al., 2014a, Prop 2) and (Rockafellar and Wets, 1998, Prop 14.11 b). satisfies Standing Lemma 7. The data of the SHS H Assumption 1.
The next result establishes that the recurrence property is preserved for the modified system H. Lemma 8. If the open, bounded set O is globally recurrent for H, then O is globally recurrent for H. We can also observe that since the solutions of H are a if any set O is globally recurrent subset of solutions of H, then O is also globally recurrent for H. for H, 7.1 Robustness to perturbations of the set
The probabilities used in this subsection are generated for which the random solutions have using the system H almost surely complete sample paths. We define
Theorem 10. If an open bounded set O ⊂ Rn is globally then there exists an open bounded set O recurrent for H and ε > 0 such that O+εB ⊂ O and O is globally recurrent for H.
One of the main motivations for establishing the existence of smaller recurrent sets within a given recurrent set is that Theorem 10 plays an important role in developing converse Lyapunov theorems. In particular, it will be required in smoothing Lyapunov functions. We refer the reader to the converse theorems in Subbaraman and Teel (2015) and Subbaraman and Teel (2013) and Section 8 for more details. 7.2 Robustness to slowing down recurrence results in preserving The next inflation of the data of H recurrence while making certain quantities related to average value of worst case first hitting time for solutions to the set O grow unbounded in the distance of the state to the set O. The result is important in the context of developing converse Lyapunov theorems with radially unbounded Lyapunov functions. Since the result might have other applications we include it here.
For ν ∈ K∞ , define the continuous set-valued mapping Mν (x) := {x∗ } + ν(|x − x∗ |)B for x∗ ∈ Rn . Consider the ν := (C, F, Rn , G ν , µ) where G(x) inflated mapping H = G1 (x) ∪ Mν (x). The proof of the next result is very similar to Lemma 7. ν Lemma 11. For every ν ∈ K∞ , the data of the SHS H satisfies Standing Assumption 1.
The next theorem claims the existence of a ν ∈ K∞ small enough to preserve recurrence of the set O for the inflated ν if O is globally recurrent for H and x∗ ∈ O. system H A similar result is established for stochastic difference inclusions in (Subbaraman and Teel, 2013, Theorem 4) and the proof differs only in the construction of the function ν. Theorem 12. If the open, bounded set O ⊂ Rn is globally then for any x∗ ∈ O, there exists ν ∈ K∞ recurrent for H ν . such that O is globally recurrent for H
Consider the hybrid system with state x ∈ R2 , C = {x : |x| ≤ r}, D = {x : |x| ≥ r} for some r > 1, the flow map f (x) = 1 and the jump map g(x, v) = 0. It can be o m ˜ ⊂S (, ξ) := sup P(graph(x) ∩ (Γ< × Rn ) ⊂ (R2 × S)). observed that the set O = B is globally recurrent. For this system, the first time for solutions to hit the set O x∈Sr (ξ) 307
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does not increase with the distance of the state to the set O, as from large initial conditions the solutions reach the set O in one jump. Hence, with the modification D = Rn and the augmentation of the mapping Mν (x) = ν(|x|)B with x∗ = {0} and ν(s) = s/2, it can be observed that the recurrence property is preserved but the worst-case first time for solutions to reach the set O increases as the state grows larger. In essence, the modification slows down the recurrence property while preserving it.
there exists a continuous function ρ : Rn → R>0 and µ > 0 such that
∇V (x), f ≤ −ρ(x) + µIO (x), ∀x ∈ C, f ∈ F (x)
Rm
7.3 Robustness to perturbations of system data Finally, we analyze the robustness of recurrence to sufficiently small state dependent perturbations. For a continuous, positive-valued function δ : Rn → R>0 , we denote the δ , µ) by H δ with data (Cδ , Fδ , Rn , G perturbed version of H defined as Cδ := {x ∈ Rn : (x + δ(x)B) ∩ C = ∅}
Fδ (x) := conF (x + δ(x)B) ∩ C) + δ(x)B + δ(x)B, v)} Gδ (x, v) := {w ∈ Rn : w ∈ g + δ(g)B, g ∈ G(x where con refers to the closed convex hull. The next result follows from (Goebel et al., 2012, Proposition 6.28) and (Teel et al., 2014a, Prop 8). Lemma 13. For every continuous δ : Rn → R>0 , the data δ satisfies Standing Assumption 1. of the hybrid system H
The next result establishes closeness of probabilities between the perturbed and unperturbed SHS. For constant perturbations we use δ(x) ≡ δ for all x ∈ Rn . In this subsection we denote the probabilities generated by the δ with the subscript δ. system H Proposition 14. Let S ⊂ Rn be closed. For each (, ρ) ∈ Z≥0 × R>0 and K ⊂ Rn compact there exists a δ > 0 such that, for every x ∈ K, mδ,⊂S (, x) ≤ max m ˜ ⊂S (, ξ) + ρ. ξ∈K
The next result establishes that recurrence of the set open, bounded O set can be preserved when the state dependent perturbations are sufficiently small. The proof follows along the same lines as (Subbaraman and Teel, 2013, Thm 5). Theorem 15. If the open bounded set O ⊂ Rn is globally then there exists a continuous function recurrent for H, n δ . δ : R → R>0 such that O is globally recurrent for H
The proof of Theorem 15 relies primarily on the regularity properties of the SHS. In (Subbaraman and Teel, 2015, Example 4) it is established that the result of Theorem 15 need not hold even for the case of non-stochastic systems without the regularity properties in Standing Assumption 1. 8. APPLICATION TO CONVERSE THEOREMS
We now present a Lyapunov function based characterization of recurrence for open, bounded sets. A smooth function V : Rn → R≥0 is a Lyapunov function with respect to the set O for H if it is radially unbounded and 308
max V (g)µ(dv) ≤ V (x) − ρ(x) + µIO (x), ∀x ∈ D.
g∈G(x,v)
It is established in Teel (2013) that the existence of V satisfying the above conditions implies O is globally recurrent. Now we outline how the robustness results established in this paper will apply to the development of converse Lyapunov theorems. The outline is similar to that of Subbaraman and Teel (2015), Subbaraman and Teel (2013), Teel et al. (2014a). We start from the assumption that O is recurrent for H. The converse theorem will now need to establish the existence of a Lyapunov function satisfying the above decrease conditions. by Lemma Since O is recurrent for H, it is recurrent for H and > 0 8. Then, Theorem 10 implies that there exists O o + B ⊂ O and O is recurrent for H. Theorem such that O is 12 establishes that there exists ν ∈ K∞ such that O recurrent for Hν . Finally, Theorem 15 proves that there is recurrent for H ν,δ . The exists δ : Rn → R>0 such that O goal now is to construct a preliminary Lyapuov function candidate V0 : Rn → R≥0 (possibly non-smooth) which is radially unbounded and satisfies
Rm
max
g∈G(x,v)\O
DV0 (x) ≤ −ρ0 (x), ∀x ∈ C\O,
V0 (g)µ(dv) ≤ V0 (x) − ρ0 (x), ∀x ∈ D\O,
where D refers to the Dini derivative and ρ0 is a positive function characterizing decrease properties. Slowing down recurrence using Theorem 12 will help to guarantee that V0 is radially unbounded, particularly when V0 is related to the expected value of the worst first hitting time for See Subbaraman and Teel (2015) solutions to the set O. and Subbaraman and Teel (2013) for details. Then, we is smooth the Lyapunov function V0 that certifies O recurrent for Hν,δ to get a smooth function Vs (x) := V (x+σ(x)η)ψ(η)dη for appropriately chosen functions Rn 0 σ, ψ. Then Vs would be a smooth Lyapunov function with respect to O for the system H. The structure of Vs is motivated by Subbaraman and Teel (2013) and Teel et al. (2014a). 9. CONCLUSION We have presented a series of robustness results for a weak stability property called recurrence. We have outlined how the results can lead to the development of converse Lyapunov theorems for SHS. The robustness results are a result of the good regularity properties for the nominal SHS. Future work will focus on constructing the Lyapunov function V0 outlined in Section 8 and establish a converse Lyapunov theorem. REFERENCES Bujorianu, M.L. and Lygeros, J. (2006). Toward a general theory of stochastic hybrid systems. In Stochastic Hybrid Systems, 3–30. Springer.
2015 IFAC ADHS October 14-16, 2015. Atlanta, USA Anantharaman Subbaraman et al. / IFAC-PapersOnLine 48-27 (2015) 304–309
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