Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets

Proceedings, 6th IFAC Conference on Analysis and Design of Hybrid Systems Proceedings, 6th IFAC Conference on Analysis and Design of Hybrid Systems Proceedings, 6th 11-13, IFAC Conference on Analysis and Design of Oxford, UK, July 2018 Hybrid Systems Available online at www.sciencedirect.com Oxford, UK, July 11-13, 2018 Hybrid Systems Proceedings, 6th IFAC Conference on Analysis and Design of Oxford, UK, July 11-13, 2018 Oxford,Systems UK, July 11-13, 2018 Hybrid Oxford, UK, July 11-13, 2018

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IFAC PapersOnLine 51-16 (2018) 115–120 Compositional Analysis of Hybrid Systems Compositional Analysis of Hybrid Systems Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets Compositional Analysis of Hybrid Systems Defined Over Finite Alphabets Defined ∗Over Finite Alphabets ∗ ∗ Murat Defined Cubuktepe Mohamadreza Ahmadi ∗ Ufuk Topcu ∗ Finite Alphabets Murat Cubuktepe ∗∗Over Mohamadreza Ahmadi ∗∗ ∗ Ufuk Topcu ∗ Brandon Hencey

Murat Mohamadreza Ahmadi ∗ Ufuk Topcu ∗ ∗∗ Murat Cubuktepe Cubuktepe ∗ Brandon Mohamadreza Ahmadi Ufuk Topcu Hencey ∗∗ Hencey ∗∗ ∗ Brandon ∗ ∗ Hencey Mohamadreza Ahmadi Ufuk Topcu ∗ Murat Cubuktepe Brandon at Austin, Austin, ∗∗ Texas 78712 USA (e-mail: ∗ The University of TexasBrandon The University of Texas at Austin,Hencey Austin, [email protected]) 78712 USA (e-mail: ∗ [email protected], [email protected], Austin, Austin, Texas 78712 USA (e-mail: ∗ The University of Texas at The University of Texas at Austin,Dayton, Austin, Ohio [email protected]) 78712USA USA(e-mail: (e-mail: [email protected], [email protected], ∗∗ Air Force Research Laboratory, 45433 [email protected], [email protected], [email protected]) ∗ ∗∗ [email protected], [email protected], [email protected]) Air Force Research Laboratory, Dayton, Ohio 45433 USA (e-mail: The University of Texas at Austin, Austin, Texas 78712 USA (e-mail: ∗∗ Laboratory, Dayton, 45433 ∗∗ Air Force [email protected]) Air Force [email protected]) Laboratory, Dayton, Ohio Ohio [email protected]) 45433 USA USA (e-mail: (e-mail: [email protected], [email protected], [email protected]) ∗∗ Air Force [email protected]) Laboratory, Dayton, Ohio 45433 USA (e-mail: Abstract: We consider the stability and the input-output analysis problems of a class of [email protected]) Abstract: We consider the stabilityofand the input-output analysis problems of adynamics class of large-scale hybrid systemsthe composed continuous dynamics analysis coupled with discrete Abstract: We consider stability and the input-output problems of aa class of Abstract: We consider the stability and the input-output analysis problems of class of large-scale hybrid systems composed of continuous dynamics coupled with discrete dynamics defined over finitesystems alphabets, e.g., deterministic finite state coupled machineswith (DFSMs). This class large-scale hybrid composed of continuous dynamics discrete dynamics large-scale hybrid systems composed ofand continuous dynamics coupled discrete defined over finite alphabets, e.g., deterministic finite statecontrolled machineswith (DFSMs). class Abstract: We consider the stability the input-output analysis problems of This adynamics class of of hybrid systems can be used to model physical systems by software. For such defined over finite alphabets, e.g., deterministic finite state machines (DFSMs). This class defined over finite alphabets, e.g., deterministic finite state machines (DFSMs). This class of hybrid systems can be used to model physical systems controlled by software. For such large-scale hybrid systems composed of continuous dynamics coupled with discrete dynamics classes of systems, we use aused method basedphysical on dissipativity theory for compositional analysis of hybrid systems can be to systems controlled by software. For such of hybrid systems be stability, to model model systems controlled by software. For such classes ofover systems, we use aused method basedphysical on and dissipativity for compositional analysis defined finite alphabets, e.g., deterministic finite statetheory machines (DFSMs). class that allows us to can study passivity input-output norms. We showThis that the classes of systems, we use a method based on dissipativity theory for compositional analysis classes of systems, we use a method based on dissipativity theory for compositional analysis that allows us to study stability, passivity and input-output norms. We show that the of hybrid systems can be used to model physical systems controlled by software. For such certificates ofusthetomethod based on dissipativity theory can be computed by solving a set of that allows study stability, passivity and input-output norms. We show that the that allows study stability, passivity input-output norms. We show that certificates ofus thetomethod based on dissipativity theory canon besemi-definite computed byprograms solving abecome setthe of classes of systems, we use a method based on and dissipativity theory for compositional analysis semi-definite programs. Nonetheless, the formulation based certificates of the method based on dissipativity theory can be computed by solving a set of certificates of the method based on dissipativity theory can be computed by solving a set of semi-definite programs. Nonetheless, the formulation based on semi-definite programs become that allows us to study stability, passivity and input-output norms. We show that the computationally intractable for relatively large number of discrete and continuous states. We semi-definite programs. Nonetheless, the based on programs become semi-definite programs. Nonetheless, the formulation formulation based onbesemi-definite semi-definite programs computationally forwith relatively large number of discrete andof continuous states. certificates theintractable method based on large dissipativity theory can computed solving abecome setWe of demonstrateofthat, for systems number of states consisting anby interconnection of computationally intractable for relatively large number of discrete and continuous states. We computationally intractable for relatively large number of discrete and continuous states. We demonstrate that, for systems with large number of states consisting of an interconnection of semi-definite programs. Nonetheless, the formulation based on semi-definite programs become smaller hybrid systems, accelerated alternating method of multipliers can beinterconnection used to carry out demonstrate that, for systems with large of consisting of an of demonstrate that, systems with large number number of states states consisting of continuous an of smaller hybrid systems, accelerated alternating method of can beinterconnection used is tostates. carry out computationally intractable forand relatively large number of multipliers discrete and We the computations infor a scalable distributed manner. The proposed methodology illustrated smaller hybrid systems, accelerated alternating method of multipliers can be used to carry out smaller hybrid systems, accelerated alternating method of multipliers can be used to carry out the computations in a scalable and distributed manner. The proposed methodology is illustrated demonstrate that, for systems with large number of states consisting of an interconnection of by an example of in a system with 60distributed continuousmanner. states and 18proposed discrete methodology states. the computations a and The is illustrated the computations a scalable scalable and The proposed by an example of in a system with 60distributed continuous states and discrete methodology states. smaller hybrid systems, accelerated alternatingmanner. method of18 multipliers can be used is toillustrated carry out by an example of a with 60 continuous states and 18 discrete states. by an example of in a system system with 60distributed continuous states and 18proposed discrete states. the computations a scalable and The methodology is illustrated © 2018, IFAC (International Federation of Automaticmanner. Control) Hosting by Elsevier All rights reserved.the actuation 1. INTRODUCTION framework is natural, forLtd. example, whenever by an 1. example of a system with 60 continuous states and 18 discrete states. INTRODUCTION framework is natural, for example, whenever the switch actuation the form of anforon/off or awhenever multi-level or 1. INTRODUCTION INTRODUCTION framework is natural, example, the actuation Over the past decades, we have witnessed a dramatic takes 1. framework is natural, for example, whenever the actuation takes the form of an on/off or a multi-level switch or Over the past decades, we have witnessed a dramatic when the output is a binary or a quantized signal. This takes form of an on/off or a multi-level switch or increase research on hybrid cyber-physical sys- takes Over the theinpast past decades, we have haveand witnessed a dramatic dramatic the form of an on/off or a multi-level switch or 1. INTRODUCTION when output is a binary or a quantized signal. This framework is natural, for example, whenever the actuation Over decades, we witnessed a increase in research on hybrid and cyber-physical sysclass of systems is referred to as systems defined over when the output a binary or a quantized signal. This tems (Ahmadi et al. on (2014); Kamgarpour et al. (2017)). increase in research hybrid and cyber-physical syswhen the output a binary or a quantized signal. This class of systems is referred to as systems defined over takes form of an on/off or a multi-level switch or increase on hybrid and cyber-physical sys- finite tems (Ahmadi et al.systems (2014); Kamgarpour et al.be (2017)). alphabets. Tarraf et al.to(2008) proposed a robust Over theinpast decades, we in have witnessed dramatic class of systems is referred as systems defined over Examples ofresearch such real-world canaal. found tems (Ahmadi et al. (2014); Kamgarpour et (2017)). class of systems is referred to as systems defined over finite alphabets. Tarraf et al. (2008) proposed a robust when the output a binary or a quantized signal. This tems (Ahmadi et al.systems (2014); et Examples such inKamgarpour real-world canal.be(2017)). found input-output analysis framework for proposed such systems and increase inofresearch on hybrid and cyber-physical sys- finite alphabets. Tarraf et (2008) a in robotics and Gregg (2017)), biological netExamples of(Hamed such systems systems in real-world real-world can be be found found finite alphabets. Tarraf et al. al. proposed a robust robust input-output analysis framework for such for systems and class of systems is applied referred to(2008) as systems defined over Examples of such in can in robotics (Hamed and Gregg (2017)), biological netTarraf (2011, 2014) this framework controller tems (Ahmadi et al. (2014); Kamgarpour et al. (2017)). input-output analysis framework for such systems and works (Lincoln and Tiwari (2004)), and biological in power netsys- input-output in robotics (Hamed and Gregg Gregg (2017)), analysis framework for such systems and Tarraf (2011, 2014) applied this framework for controller finite alphabets. Tarraf et al. (2008) proposed a robust in robotics (Hamed and (2017)), biological networks (Lincoln and Tiwari (2004)), and in power sysExamples ofet such systems in real-world can be found synthesis. Tarraf (2011, 2014) applied this framework for controller tems (Totu al. (2017)). works (Lincoln and Tiwari (2004)), and in power sysTarraf (2011, 2014) applied this framework for controller synthesis. input-output analysis framework for such systems and works (Lincoln Tiwari (2004)), and biological in power netsys- synthesis. tems (Totu et al.and (2017)). in robotics (Hamed and Gregg (2017)), tems (Totu et al. (2017)). The contributions of this paper are threefold. First, we prosynthesis. Tarraf (2011, 2014) applied this framework for controller The literature is and rich in analysis and verification methods tems (Totu et al. (2017)). works (Lincoln Tiwari (2004)), and in power sys- The contributions of this paper are threefold. First, we proThehybrid literature is rich (Zhao in analysis and (2008)). verification methods a stability and an input-output analysis framework synthesis. The contributions of paper threefold. First, we for and Hill Despite the pose tems (Totu systems et al. (2017)). The literature is rich in analysis and verification methods The contributions of this this paper are are threefold. First, we propropose aclass stability and ansystems input-output analysis framework The literature is rich in analysis and verification methods for hybrid systems (Zhao and Hill (2008)). Despite the for a of hybrid with discrete components pose a stability and an input-output analysis framework available tools for the analysis and verification, scalability for hybrid systems (Zhao and Hill (2008)). Despite the pose a stability and an input-output analysis framework for a class of hybrid systems with discrete components The contributions of this paper are threefold. First, we profor hybrid andand Hill (2008)). Despite the defined available tools forrich the(Zhao analysis verification, scalability overofahybrid finite alphabet baseddiscrete on Lyapunov and The literature is in analysis and verification methods for aa class systems with components still poses asystems challenge. Therefore, there has been a surge available tools for the analysis and verification, scalability for class of hybrid systems with discrete components defined over a finite alphabet based on Lyapunov and pose a stability and an input-output analysis framework available tools for the analysis and verification, scalability still poses a challenge. Therefore, there has been a surge dissipativity analysis. In particular, the discrete dynamics for hybrid systems (Zhao and Hill (2008)). Despite the defined over a finite alphabet based on Lyapunov and in analysis techniques. methods, in for stillcompositional poses aa challenge. challenge. Therefore, thereThese has been been a surge surge defined over aform finite alphabet based on Lyapunov and dissipativity In particular, the discrete dynamics abeclass ofanalysis. hybrid systems with discrete components still poses Therefore, there has a in compositional analysis techniques. These methods, in can in the of deterministic finite state machines, available tools for the analysis and verification, scalability dissipativity analysis. In particular, the discrete dynamics general, decompose the analysis problem of a large-scale in compositional analysis techniques. These methods, in dissipativity analysis. In particular, the discrete dynamics can be in the form of deterministic finite state machines, defined over a finite alphabet based on Lyapunov and in compositional techniques. in which general, decompose theTherefore, analysis problem of been amethods, large-scale can be form used oftodeterministic model software. Second, we genstill poses a challenge. thereThese has areduce surge can be in the finite state machines, hybrid system intoanalysis smaller sub-problems, which can general, decompose the analysis problem of a large-scale can be in the form of deterministic finite state machines, which can be used to model software. Second, we gendissipativity analysis. In particular, the discrete dynamics general, decompose the analysis problem of amethods, large-scale hybrid system intoanalysis smaller sub-problems, which can reduce eralize can the methodology proposed in Arcak et al.we(2016) in compositional techniques. These in which be used to model software. Second, genthe computational burden significantly. Itwhich was shown that hybrid system into into smaller smaller sub-problems, can reduce reduce which can beet used todeterministic model software. Second, we(2016) generalize the methodology proposed infinite Arcak et al. can be in the form of state machines, hybrid system sub-problems, which can the computational burden significantly. It was shown that and Meissen al. (2015), wherein a dissipativity-based general, decompose the analysis problem of a large-scale eralize the methodology proposed in Arcak et al. (2016) dissipativity theoryburden can besignificantly. used as a tool forshown decompothe computational It was that eralize the methodology proposed inproposed etforal.continuand Meissen al. (2015), wherein aArcak dissipativity-based which can beetanalysis used to method model software. Second, we(2016) genthe computational burden significantly. It was shown that dissipativity theory can be used as a tool for decompocompositional was hybrid system into smaller sub-problems, which can reduce Meissen et al. (2015), wherein aa dissipativity-based sitional stability andcan detectability analysis (Teel (2010)). and dissipativity theory be used as a tool for decompoand Meissen et al. (2015), wherein dissipativity-based compositional analysis method was proposed for continueralize the methodology proposed in Arcak et al. (2016) dissipativity theory can besignificantly. used as a tool for decompositional stability and detectability analysis (Teel (2010)). systems, toanalysis hybrid method systems was defined over finite alphathe computational burden It was shown that ous compositional proposed for This result was further extended inanalysis (Naldi and Sanfelice sitional stability and detectability detectability (Teel (2010)). compositional method was proposed for continucontinuous systems, to hybrid systems defined over finite alphaand Meissen etanalysis al.decomposes (2015), wherein a dissipativity-based sitional stability and analysis (Teel (2010)). This result was further extended in (Naldi and Sanfelice bets. Our method the analysis problem of the dissipativity theory can be used as a tool for decompoous systems, to hybrid systems defined over finite alpha(2014)) to present sufficient conditions for passivity and compositional This result was further extended in (Naldi and Sanfelice ous systems, to hybrid systems defined over finite alphabets. Our method decomposes the analysis problem of the analysis method was proposed for continuThis result was further extended in (Naldi and Sanfelice (2014)) to present sufficient conditions for passivity and overall interconnected hybrid system defined over finite sitional stability and detectability analysis (Teel (2010)). bets. Our method decomposes the analysis problem of the stability analysis a class ofconditions interconnected hybrid sys(2014)) to to presentofsufficient sufficient for passivity passivity and ous bets. Our method decomposes the analysis problem of the overall interconnected hybrid system defined overalphafinite systems, tosmaller hybrid systems defined over finite (2014)) present conditions for and stability analysis of a class of interconnected hybrid sysalphabet into local sub-problems for subsystems This result was further extended in (Naldi and Sanfelice overall interconnected hybrid system defined over finite tems with sums of storage functions. Some well-posedness stability analysis of a class of interconnected hybrid sysoverall interconnected hybrid system defined over finite alphabet into smaller local sub-problems for subsystems bets. Our method decomposes the analysis problem of stability analysis a class ofconditions interconnected hybrid systems with sums ofofstorage functions. Some takes into advantage oflocal a global storage function, whichthe is (2014)) to present sufficient forwell-posedness passivity and and alphabet smaller sub-problems for subsystems issues and input-output notions for interconnected hybrid tems with sums of storage functions. Some well-posedness alphabet into smaller local sub-problems for subsystems and takes advantage of a global storage function, which is overall interconnected hybrid system defined over finite tems with sums of storage functions. Some well-posedness issues and input-output notions for interconnected hybrid the sum of local storage functions. Thirdly, to carry out stability analysis of a class of interconnected hybrid sysand takes advantage of a global storage function, which is systems were discussed in (Sanfelice (2011)). Nonetheless, issues and and input-output notions for interconnected interconnected hybrid alphabet and takes advantage of a global storage function, which is the sum of local storage functions. Thirdly, to carry out into smaller local sub-problems for subsystems issues input-output notions for hybrid systems were discussed in (Sanfelice (2011)). Nonetheless, the computation in a distributed manner, we use accelertems with sums of storage functions. Some well-posedness sum of local storage functions. Thirdly, to carry out one issuewere withdiscussed the aboveincompositional methods for hybrid and systems (Sanfelice (2011)). Nonetheless, the sum ofadvantage local storage functions. Thirdly, carry out computation in aofdistributed manner, weto use accelertakes aet global storage function, which is systems were discussed in (Sanfelice (2011)). Nonetheless, one issue with the above compositional methods for hybrid ated ADMM (Goldstein al. (2014)). To use accelerated issues and input-output notions for interconnected the computation in aa distributed manner, we use accelersystems iswith thatthe they often do not provide a computational one issue above compositional methods for hybrid computation in distributed manner, we use accelerated ADMM (Goldstein et al. (2014)). To use accelerated the sum of local storage functions. Thirdly, to carry out one issueis with the above methods for hybrid that they often do not provide a computational ADMM, which has a faster convergence rate compared to systems were incompositional (Sanfelice Nonetheless, ADMM (Goldstein et (2014)). To use accelerated framework to discussed find theoften certificates and(2011)). rely ad-hoc ana- ated systems is that that they do not not provide provide a on computational ated ADMM (Goldstein et al. al. (2014)). To use accelerated ADMM, which has aa faster convergence rate to the computation insmoothing distributed manner, wecompared use accelersystems is they often do a computational framework to find the certificates and rely on ad-hoc anaADMM, we utilize techniques (Nesterov (2005); one issue with the above compositional methods for hybrid which has a faster convergence rate compared to lytical techniques the overall system. framework to find find to theanalyze certificates and rely rely on ad-hoc anaADMM, which has a faster convergence rate compared to we utilize smoothing techniques (Nesterov (2005); ated ADMM (Goldstein et al. (2014)). To use accelerated framework to the certificates and on ad-hoc analytical techniques to often analyze etweal. (2011)), which techniques has been (Nesterov used to improve systems is that they do the not overall providesystem. a computational Becker ADMM, utilize smoothing (2005); lytical techniques to analyze the overall system. we utilize smoothing techniques (Nesterov (2005); Becker et al. (2011)), which has been used to improve ADMM, which has a faster convergence rate compared to A largetechniques class of discrete systems possess inputs and/or lytical to the overall system. convergence rate ofwhich similar first order methods. We framework to find theanalyze certificates and rely on ad-hoc ana- the Becker et al. (2011)), has been used to improve A large class of discrete systems possess inputs and/or Becker et al. (2011)), which has been used to improve the convergence rate of similar first order methods. We ADMM, we utilize smoothing techniques (Nesterov (2005); outputs that take inthe finite sets.system. This modeling also convergence discuss the effects ofsimilar restarting accelerated ADMM, lytical tovalues analyze overall A largetechniques class of discrete discrete systems possess inputs and/or rate of first order methods. We A large class of inputsmodeling and/or the outputs that take values systems in finitepossess sets. This the convergence rate ofwhich first order We also discuss effects ofsimilar restarting accelerated Becker et shown al.the (2011)), hasconvergence been usedmethods. to ADMM, improve which has to improve the rate of simi1 outputs that take values in finite sets. This modeling also discuss the effects of restarting accelerated ADMM, Partly funded by the awards AFRL UTC 17-S8401-10-C1, and outputs that take values inAFRL finiteUTC sets. This A large funded class of discrete systems possess inputsmodeling and/or also discuss thealgorithms effects restarting accelerated which has shown to improve the convergence rateADMM, of et simithe convergence rate ofofsimilar first (2013); order methods. We 1 Partly by the awards 17-S8401-10-C1, and lar accelerated (Nesterov Becker al. which has to the rate of simiONR N000141613165. 1 Partly funded by thevalues awards AFRL 17-S8401-10-C1, and outputs that take finiteUTC sets. This modeling which has shown shown to improve improve the convergence convergence rateADMM, of et simi1 lar accelerated (Nesterov (2013); Becker al. also discuss thealgorithms effects of restarting accelerated ONR N000141613165. Partly funded by the awardsin AFRL UTC 17-S8401-10-C1, and lar accelerated algorithms (Nesterov (2013); Becker et al. ONR N000141613165. lar accelerated algorithms (Nesterov (2013); Becker et al. which has shown to improve the convergence rate of simi1 ONR N000141613165. Partly funded by the awards AFRL UTC 17-S8401-10-C1, and Copyright © 2018 IFAC 115 lar accelerated algorithms (Nesterov (2013); Becker et al. ONR N000141613165. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Copyright 2018 IFAC 115 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 115Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 115 10.1016/j.ifacol.2018.08.020 Copyright © 2018 IFAC 115

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(2011)). We illustrate the proposed method by a numerical example. This paper is structured as follows. In the following section, we present the problem formulation. In Section 3, we define notions of hybrid Lyapunov and storage functions for hybrid systems defined over finite alphabets. In Section 4, we propose a method based on dissipativity for compositional analysis of the class of hybrid systems under study. The proposed methodology is illustrated by an example in Section 6. Finally, Section 7 concludes the paper and gives directions for future research. Notation: R≥0 denotes the set [0, ∞).  ·  denotes the Euclidean vector norm on Rn . The set of integers are denoted by Z. For a function f : A → B, f ∈ Lp (A, B), 1  |f (t)|p dt p < ∞ and 1 ≤ p < ∞, implies that A supt∈A |f (t)| < ∞ for p = ∞. Equivalently, for a discrete signal s : A → B, s ∈ lp (A, B), 1 ≤ p < ∞, implies that 1  ( A |s(n)|p ) p < ∞ and supn∈A |s(n)| < ∞ for p = ∞. For symmetric matrices A1 , . . . , An , diag(A1 , . . . , An ) denotes the diagonalized matrix   A1 0 0   diag(A1 , . . . , An ) =  0 . . . 0  . 0 0 An

For a vector s ∈ Rns , s ≡ 0 denotes the element-wise equality to zero. If x ∈ X, then f ∈ C 1 (X) implies that the function f is continuously differentiable in x. 2. PROBLEM FORMULATION

Formally, we consider the following class of hybrid systems    x(t) ˙ = f (x(t), w(t); p(t))   C:   y(t) = h (x(t); p(t))   + G: q(t ) = g (q(t), u(t), x(t))  D:   p(t) = l (q(t), u(t))    q(t0 ) = q0 , x(t0 ) = x0 .

(1)

where x ∈ Rn and q ∈ Q ⊂ N represent continuous and discrete states. In the continuous module C, f (·, ·; p) : Rn × Rm → Rn , f (0, w; p) ≡ 0, ∀(w, p) ∈ W × P, is a family of mappings with index p ∈ P ⊂ Z and similarly h(·; p) : Rn → Rny is a family of output mappings. y ∈ Rny and w ∈ Rm are the continuous outputs and inputs, respectively. In the discrete module D, g : Q×U ×Rn → Q and l : Q × U → Q. p ∈ P and u ∈ U ⊂ Znu are the discrete outputs and inputs, respectively. The sets associated with the discrete module Q, P and U are assumed to be finite. In the sequel, we abuse the notation and use q + to represent q(t+ ). The discrete module D can characterize a rich class of systems defined over finite alphabets (Tarraf et al. (2008)) and can be used to model systems ranging from quantizers to deterministic finite state machines. The hybrid system (1) can also be studied in the context of hybrid automata (Henzinger (1996)). However, note that for hybrid automata, analysis tools such as Lyapunov functions or storage functions are not available in general. 116

3. STABILITY AND DISSIPATIVITY ANALYSIS We can study the input-output and stability properties of system (1) by using a dissipativity-type and Lyapunovtype argument, respectively. To this end, we use the notion of hybrid Lyapunov or storage function, which is described as follows. Definition 1. (Hybrid Lyapunov Function). A function V : Rn × Q → R≥0 such that V (0, q) = 0, ∀q ∈ Q, and V ∈ C 1 (Rn ) is called a hybrid Lyapunov function for system (1) with u ≡ 0 and w ≡ 0, if it satisfies the following inequalities V (x, q) > 0, ∀x ∈ Rn \ {0}, ∀q ∈ Q, (2)  T ∂V (x, q) f (x, 0; p) < 0, ∀x ∈ Rn , ∀q ∈ Q, ∀p ∈ P, ∂x (3) and V (x, q + ) − V (x, q) ≤ 0, ∀x ∈ Rn , ∀q ∈ Q. (4) Theorem 1. The hybrid system (1) is asymptotically stable, i.e., limt→∞ x(t) = 0, ∀q ∈ Q, if there exists a hybrid Lyapunov function. Proof. See Appendix A in (Cubuktepe et al. (2018)). Definition 2. (Hybrid Storage Function). A function V : Rn × Q → R≥0 such that V ∈ C 1 (Rn ) is called a hybrid storage function for system (1), if it satisfies the following inequalities V (x, q) ≥ 0, ∀x ∈ Rn , ∀q ∈ Q, (5)  T ∂V (x, q) f (x, w; p) ≤ Wc (w, y), ∂x ∀x ∈ Rn , ∀q ∈ Q, ∀p ∈ P (6) and V (x, q + ) − V (x, q) ≤ Wd (u, p), ∀x ∈ Rn , ∀q ∈ Q (7) where the integrable functions Wc : Rm × Rny → R and Wd : Q × U → R are the continuous and the discrete supply rates, respectively. Theorem 2. The hybrid system (1) is dissipative with respect to the supply rates Wc and Wd , if there exists a hybrid storage function. Proof. The dissipativity of the continuous dynamics is standard and follows from integrating (6). The dissipativity of the discrete dynamics follows from Theorem 3 and Lemma 2 in Tarraf et al. (2008). G1

..

. GN

  z ζ

  w u

M

  y p

  d µ

Fig. 1. Interconnected system with hybrid inputs (d, µ)T and hybrid outputs (z, ζ)T .

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

4. INTERCONNECTION OF HYBRID SYSTEMS DEFINED OVER FINITE ALPHABETS We consider interconnected systems as illustrated in Fig. 1, where the subsystems {Gi }N i=1 are known and have dynamics in the form of (1). We associate each subsystem with a set of functions {fi , hi , gi , li } and xi ∈ Rni , qi ∈ Qi , i i wi ∈ Rnw , ui ∈ Ui , yi ∈ Rny and pi ∈ Pi . The static interconnection is characterized by a matrix M where N i N i N n = i=1 ni , nw = i=1 nw and ny = i=1 ny . That is, M satisfies     w y z  d (8)  u  = M p , ζ µ

where d ∈ Rnd and z ∈ Rnz are the continuous exogenous inputs and outputs, respectively. Similarly, µ ∈ M ⊂ Znµ and ζ ∈ Z ⊂ Znζ are the discrete exogenous inputs and outputs, respectively. We assume this interconnection is well-posed, i.e., for all d ∈ L2e and initial condition x(0) ∈ Rn , there exist unique z, w, y ∈ L2e that causally depend on d for all p ∈ P ⊂ Znp . Furthermore, we define   Mc , M= Md

where Mc ∈ Rnw +ny ×Rnw +ny and Md ∈ Znµ +nζ ×Znµ +nζ . The continuous local and global supply rates, Wci (wi , yi ) and Wc (d, z), respectively, are defined by quadratic functions. That is,  T   d d Wc (d, z) = S . (9) z z where S ∈ Rnd +nz × Rnd +nz .

Analogously, for discrete subsystems, we have Wdi (pi , ui ) and Wd (p, u) given by quadratic functions  T   u u R . (10) Wd (p, u) = p p where R ∈ Znu +np × Znu +np is a symmetric matrix.

We next show that certifying the dissipativity of an overall interconnected system can be concluded, if each of the subsystems satisfy the local dissipativity property. Let  Li = (Si , Ri ) | Gi is dissipative w.r.t. 

  T   wi ui u Ri i , , and yi pi pi

(11)

     T Mc Mc T < 0 , S, {Si }N | P Q P c c i=1 c 1n y 1ny

(12)

  T    Md Md N T P d Q d Pd <0 , Ld = R, {Ri }i=1 | 1np 1np

(13)

Lc = and

wi yi

T

Si



117

wherein Qc = diag(S1 , . . . , SN , −S), Qd = diag(R1 , . . . , RN , −R), and Pc and Pd are permutation matrices defined by 117

   w1 u1  y1   p       .   .1  w u  .   .   .   .  z ζ    = Pc    (14)  y  , and uN  = Pd  p  .  wN  y  p   N  N d µ  d  µ z ζ Proposition 3. Consider the interconnection of N subsystems as given in (8) with the global supply rates (9) and N (10). If there exists {Si }N i=1 and {Ri }i=1 satisfying (Si , Ri ) ∈ Li , i = 1, . . . , N, (15) and (S1 , . . . , SN , −S) ∈ Lc , (16) (R1 , . . . , RN , −R) ∈ Ld , (17) then the interconnected system is dissipative with respect to the global supply rates Wd and Wc . A storage function N certifying global dissipativity is V (x, q) = i=1 Vi (xi , qi ), where Vi is the storage function certifying dissipativity of subsystem i as in Li . Proof. See Appendix B in (Cubuktepe et al. (2018)). Proposition 3 provides the means to decompose the analysis of interconnected subsystems to smaller problems that are computationally more amenable. However, even the above discussed decompositional analysis method can be computationally involved for large-scale hybrid systems. In the next section, we propose a method based on accelerated ADMM to carry out such computations in a distributed manner. It is worth mentioning Oehlerking and Theel (2009) proposed a compositional method based on graph-based reasoning rather than dissipativity for hybrid systems that are not subject to inputs and outputs and are described by a hybrid automaton. Whereas, our formulation allows for (both discrete and continuous) inputs and outputs and is pertained to hybrid systems composed of continuous dynamics and a discrete subsystem defined over finite alphabets. In addition, Oehlerking and Theel (2009) did not bring forward a method based on distributed optimization to address the computations in the case of largescale systems. Such distributed computational method is discussed in the next section. 5. COMPUTATIONAL FORMULATION USING ACCELERATED ADMM For small-scale systems, we can solve the optimization problem outlined in Proposition 3 using publicly available SDP solvers like MOSEK (Andersen and Andersen (2012)). But, these SDP solvers do not scale well for larger problems, as they use interior point methods, which requires solving a system of equations in each iteration. However, the structure in our problem allows us to decompose the constraints in (5), (6) and (7), leading to a distributed algorithm. Specifically, the ADMM (Boyd et al. (2011)) approach decomposes the convex optimization problems into a set of smaller problems. A generic convex optimization problem minimize F (b) subject to b ∈ C , (18)

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where b ∈ Rn , F is a convex function, and C is a convex set, can be written in ADMM form as

the best of our knowledge, these methods have not been applied in compositional analysis. Consider the following perturbation of the problem in (20)

minimize F (b) + G(v) subject to b = v, where G is the indicator function of C .

(19)

minimize

η

subject to (5),(6),(7), (16), and (17)

(20)

which is outlined in Proposition 3. For example, η can be the upper-bound on the continuous ζ z induced norm dLL2 or the discrete induced norm µll2 2 2 that we wish to minimize. Using the above form, the problem in (20) can be written in ADMM form with F (b) is defined as sum of η and the indicator function of (5),(6) and (7), and G(v) is defined as the indicator function of (16) and (17). Then, the scaled form of ADMM algorithm for problem in (19) is b

k+1

k

= arg min F (b) + (ρ/2)||b − v + b

η +  Di (Si , Ri , Vi , )

subject to (5),(6),(7), (16), and (17)

The problem we want to find a compositional formulation can be given as {Si }N ,{Ri }N ,{Vi }N i=1 i=1 i=1

minimize

{Si }N ,{Ri }N ,{Vi }N i=1 i=1 i=1

(21)

for some fixed smoothing parameter  > 0 and a strongly convex function D that satisfies 1 (22) D(b) ≥ D(b0 ) + ||b − b0 ||22 2 for some point b0 . Specifically, we choose Di = Si F + Vi F + Ri F , where  · F is the Frobenius norm. For some problems, it is shown that for small enough , the approximate problem (21) is equivalent to the original problem Becker et al. (2011). When F (b) and G(v) are strongly convex, the ADMM algorithm can be modified with an acceleration step to achieve O( k12 ) convergence after k iterations Goldstein et al. (2014). Then, the accelerated ADMM algorithm is bki = arg min Fi (b) + (ρ/2)||bi − v¯ik + s¯ki ||22 ,

sk ||22 ,

bi

k

v = arg min G(v) + (ρ/2)||bk − v + s¯k ||22 ,

v k+1 = arg min G(v) + (ρ/2)||bk+1 − v + sk ||22 ,

v

v

sk+1 = sk + bk+1 − v k+1 , where b and v are the vectorized form of the matrices N N {Si }N i=1 , {Vi }i=1 , {Ri }i=1 , z is the scaled dual variable and ρ > 0 is the penalty parameter, which is a penalty for primal infeasibility, i.e penalty of not satisfying the constraint b = v. As F (b) is separable for each subsystem, the ADMM algorithm can be parallelized as follows: = arg min Fi (b) + (ρ/2)||bi − vik + ski ||22 , bk+1 i bi

v k+1 = arg min G(v) + (ρ/2)||bk+1 − v + sk ||22 , v

sk+1 = sk + bk+1 − v k+1 , Under mild assumptions, the ADMM algorithm converges Boyd et al. (2011), but the convergence is only asymptotic in general, therefore it may require many iterations to achieve sufficient accuracy. 5.1 Accelerated ADMM Several algorithms (Nesterov (2013); Chen and Ozdaglar (2012)) shows that acceleration schemes can improve the performance significantly. These methods achieve O( k12 ) convergence after k iterations, which is shown to be optimal for a first order method Nesterov (1983). However, they usually require the function F (b) to be differentiable with a known Lipschitz constant on the ∇F (b), which does not exist when the problem is constrained. For the case when F (b) or G(v) is not strongly convex or smooth, smoothing approaches have been used Nesterov (2005); Becker et al. (2011) to improve convergence. However, to 118

sk = s¯k + bk − v k ,  1 + 1 + 4αk2 αk+1 = 2 αk − 1 k k+1 k =v + (v − v k−1 ) v¯ αk+1 αk − 1 k s¯k+1 = sk + (s − sk−1 ), αk+1 where ρ is a positive constant that satisfies ρ ≤  to make 1 sure the accelerated ADMM converges with O( 2 ) rate, k and α1 = 1. Note that α update can be carried out in parallel while achieving O( k12 ) convergence, which cannot be achieved by the standard ADMM or accelerated proximal methods if there are constraints in the problem. In general, we do not have access to the Lipschitz constant or strongly convexity parameter in the feasible region of the subproblems because of the constraints, which may reduce the performance of the accelerated method Becker et al. (2011). One approach to deal with the case of unknown Lipschitz constant or strongly convexity parameter is so-called restart method, which is used in Nesterov (2013); Becker et al. (2011), and it is shown that restart methods can improve the convergence rate significantly. To apply the method, we restart the algorithm, i.e, we set the acceleration parameter αk = 1 after a certain number of iterations while using the point in iteration k as the starting point for the restart, which resets the acceleration parameter, and reruns the accelerated ADMM algorithm from the next starting point. Examples in Becker et al. (2011) show that the restarting methods can greatly improve performance, but they note that the restart method

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In this section, we illustrate the proposed distributed analysis method with a large scale example, where we compare the convergence rate of ADMM with accelerated ADMM and several restart methods. We implemented both standard ADMM and accelerated ADMM algorithms in MATLAB using the CVX toolbox (Grant and Boyd (2014)) and MOSEK (Andersen and Andersen (2012)) to solve SDP problems. 6.1 Example In order to test and compare the different forms of ADMM methods discussed in Section 5, we randomly generated N = 6 subsystems with linear continuous dynamics, each with 10 continuous states, 2 continuous inputs and outputs, and with 3 discrete modules. The continuous dynamics of each subsystem is characterized by the equations (1), with the maximum real part for an eigenvalue of Ai is normalized to −2, and all of the subsystems are connected to 3 other subsystems, which forms the interconnection matrix M . In total, the system we consider has 60 continuous states with 3 different dynamics depending on the discrete module, which makes the centralized approaches impractical as the number of semidefinite variables and constraints grow large. We note that, MOSEK run into numerical problems while solving the analysis problem with a centralized approach, therefore we only include comparisons between different compositional methods. The discrete module Di is shown in Figure 2 with three states q1 , q2 , and q3 , inputs U = {0, 1} and outputs P = {0, 1, 2}. The output function p is defined as  for i = 1, u, p(qi , u) : 1 − u, for i = 2,  0, for i = 3. We apply the compositional approach underlined by the problem in (21) to find the minimum induced L2 -norm between the output and input of randomly chosen systems. Since the continuous dynamics are linear, we consider quadratic Lyapunov functions for subsystems. For each subsystem, let  T    Pi ri xi xi . V (xi ) = qi qi riT λi The iterative methods were initialized using Vi0 = Si0 = Ri0 = Ui0 = I. For each method, we plot the norm of primal u=0 q1

u = 0, 1

q3

u=1

u=1

q2

u=0

Fig. 2. Di in the Example. 119

Primal residual versus number of iterations 100 10−1 Primal Residual

6. NUMERICAL EXPERIMENTS

residual in Figure 3, which is defined as rk = bk − v k , and it is the residual for primal feasibility. Also, we show the norm of the dual residual sk = ρ(v k − v k−1 ) in Figure 4, which can be viewed as a residual for the dual feasibility condition.

10−2 10−3 10−4

ADMM Accelerated ADMM Acc ADMM, restart every 10 Acc ADMM, restart every 20 Acc ADMM, restart every 50

10−5 10−6 10−7

0

50

100

150

200

Number of iterations Fig. 3. Norm of primal residual versus number of iterations for the decentralized synthesis problem with standard and accelerated ADMM with various restart methods. Dual residual versus number of iterations 10−1 Dual Residual

requires tuning for different problems to optimize the performance, i.e, the performance can vary significantly with different restart schemes.

119

10−3 10−5

ADMM Accelerated ADMM Acc ADMM, restart every 10 Acc ADMM, restart every 20 Acc ADMM, restart every 50

10−7 0

50

100

150

200

Number of iterations Fig. 4. Norm of dual residual versus number of iterations for the decentralized synthesis problem with standard and accelerated ADMM with various restart methods. We can see that accelerated ADMM achieves superior convergence in primal and dual residuals compared to ADMM. In Fig. 3, we can see that ADMM can only achieve accuracy up to 10−1 after 200 iterations, which is not a sufficient accuracy. However, all of the accelerated ADMM methods outperform ADMM in convergence of primal residual. We can see in Fig. 4 that the convergence rate of dual residual is better compared to convergence rate of primal residual for ADMM, notably achieving accuracy up to 10−3 after 200 iterations. However, like in convergence of primal residual, all accelerated ADMM methods outperform ADMM.

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We note that restarting the algorithm in every 50 iterations makes the convergence of the primal residual and dual residual significantly faster compared to ADMM and other accelerated ADMM with different restart schemes. Also, we can see that restarting every 10 or 20 iterations makes the convergence rate of primal residual slower compared to accelerated ADMM without restarting the algorithm in Fig. 3. These two restart schemes does not have a significant effect of convergence rate of dual residual compared to accelerated ADMM without restart. Even though some of the restart methods does not perform as well as restarting in every 50 iterations, all of the accelerated variants outperforms ADMM. After 200 iterations with the accelerated ADMM, the minimum (upper-bound on) induced L2 -norm from the input d to the output output y is 0.29. 7. CONCLUSIONS We proposed a method for compositional analysis of large scale hybrid systems defined over finite alphabets, for which an underlying interconnection topology is given. For such systems, we decompose the global analysis problem into a number of smaller local analysis problems using dissipativity theory. Furthermore, we proposed a distributed optimization method with smoothing techniques, which enables to employ accelerated ADMM. Numerical results show that the accelerated ADMM method with different restart methods significantly improves the convergence rate compared to standard ADMM. REFERENCES Ahmadi, M., Mojallali, H., and Wisniewski, R. (2014). Guaranteed cost H∞ controller synthesis for switched systems defined on semi-algebraic sets. Nonlinear Analysis: Hybrid Systems, 11, 37 – 56. Andersen, E.D. and Andersen, K.D. (2012). The mosek optimization software. EKA Consulting ApS, Denmark. Arcak, M., Meissen, C., and Packard, A. (2016). Networks of Dissipative Systems. SpringerBriefs in Control, Automation and Robotics. Springer New York. Becker, S.R., Cand`es, E.J., and Grant, M.C. (2011). Templates for convex cone problems with applications to sparse signal recovery. Mathematical programming computation, 3(3), 165–218. Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. R in Machine Learning, 3(1), Foundations and Trends 1–122. Chen, A.I. and Ozdaglar, A. (2012). A fast distributed proximal-gradient method. In Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, 601–608. IEEE. Cubuktepe, M., Ahmadi, M., Topcu, U., and Hencey, B. (2018). Compositional analysis of hybrid systems defined over finite alphabets. arXiv preprint arXiv:1803.00622. Goldstein, T., O’Donoghue, B., Setzer, S., and Baraniuk, R. (2014). Fast alternating direction optimization methods. SIAM Journal on Imaging Sciences, 7(3), 1588– 1623. 120

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