A Closed Form Expression of Nonlinear Scalings for Lyapunov Functions of ISS Networks ⁎

11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems Systems Vienna, Austria, Sept. 4-6, 2019 11th Symposium on Control 11th IFAC Symposium on Nonlinear Control Systems Available online at www.sciencedirect.com Vienna, Austria, Sept. 2019 Vienna, Austria, Sept. 4-6, 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 544–549 Closed Form Expression of Nonlinear Scalings Closed Form Expression of Nonlinear Scalings ⋆⋆ Closed Form Expression of Nonlinear Scalings Closed Form Expression of Nonlinear Scalings for Lyapunov Functions of ISS Networks Closed Form Expression of Nonlinear Scalings for Lyapunov Functions of ISS Networks ⋆⋆ Closed Form Expression of Nonlinear Scalings for Lyapunov Functions of ISS Networks ⋆ for Lyapunov Functions of ISS Networks for Lyapunov Functions of ISS Networks ∗ Hiroshi Itoof for Lyapunov Functions ISS Networks ⋆ ∗

Hiroshi Ito ∗ ∗ Hiroshi Ito Hiroshi Ito ∗ ∗ Hiroshi Ito Systems, ∗ and Control ∗ Department of Intelligent Hiroshi Ito Systems, Department of Intelligent and Control ∗ Technology, Iizuka,ofFukuoka 820-8502, JapanSystems, (e-mail: ∗ Department Intelligent and Control Intelligent and Technology, Iizuka,of Fukuoka 820-8502, JapanSystems, (e-mail: ∗ Department Intelligent and Control Control ∗ Department Technology, Iizuka,ofFukuoka 820-8502, JapanSystems, (e-mail:

Kyushu Institute of Kyushu Institute of [email protected]) Kyushu Institute of Kyushu Institute [email protected]) Kyushu Institute of of [email protected]) Department ofFukuoka Intelligent and Control Systems, Kyushu Institute of Technology, Technology, Iizuka, Iizuka, Fukuoka 820-8502, 820-8502, Japan Japan (e-mail: (e-mail: [email protected]) [email protected]) Technology, Iizuka, Fukuoka of 820-8502, Japan stable (e-mail: [email protected]) Abstract: For networks consisting input-to-state systems, it is well-known that the Abstract: For networks consisting of input-to-state stable systems, it is well-known that the popular Lyapunov function called the max-separable function can be constructed by solely Abstract: For networks consisting of input-to-state stable systems, it is well-known that the Abstract: For networks consisting of input-to-state systems, it is well-known that the popular Lyapunov function called the max-separablestable function can be constructed by solely Abstract: For networks consisting of input-to-state stable systems, it is well-known that the relying on component-wise inverse maps of one single path characterizing the monotonicity of popular Lyapunov function called the max-separable function can be constructed by solely Abstract: For networks consisting of input-to-state stable systems, it is well-known that the popular Lyapunov function called the max-separable function can be constructed by solely relying on component-wise inverse maps of one single path characterizing the monotonicity of popularon Lyapunov function called maps thesystems. max-separable function can behave constructed by solely dissipation inequalities of component Numerical algorithms been developed to relying component-wise inverse of one single path characterizing the monotonicity of popular Lyapunov function called maps thesystems. max-separable function can behave constructed by solely relying on component-wise inverse of one single path characterizing the monotonicity of dissipation inequalities of component Numerical algorithms been developed to relying on component-wise inverse maps of one single path characterizing the monotonicity of compute such a path. This paper proposes a useful closed-form expression for the inverse maps dissipation inequalities of component systems. Numerical algorithms have been developed to relying onsuch component-wise inverseproposes maps ofaone single pathalgorithms characterizing the monotonicity of dissipation inequalities of component systems. Numerical have been developed to compute a path. This paper useful closed-form expression for the inverse maps dissipation inequalities of component systems. Numerical algorithms have been developed to of a path and its extension to generate a sufficient variety of paths. The solution not only gives compute such a path. This paper proposes aa useful closed-form expression for the inverse maps dissipation inequalities of component systems. Numerical algorithms have been developed to compute such a path. This paper proposes useful closed-form expression for the inverse maps of a path and its extension to generate a sufficient variety of paths. The solution not only gives compute a path. This paper proposes a usefulfunction closed-form expression for substantiates thenot inverse nonlinear scalings of the max-separable Lyapunov butsolution also the of a a path path such and its extension to generate generate sufficient variety ofexplicitly, paths. The only maps gives compute such a path. This paper proposes a usefulvariety closed-form expression for substantiates thenot inverse maps of its to aaaLyapunov sufficient of paths. The gives nonlinear scalings of the max-separable butsolution also the of a path and and its extension extension to generate sufficient function variety ofexplicitly, paths. The solution not only only gives rounding-off technique to remove the non-differentiable nature from the max-separable function. nonlinear scalings of the max-separable Lyapunov function explicitly, but also substantiates the of a path and its extension to generate aLyapunov sufficient function variety ofexplicitly, paths. The solution not only gives nonlinear scalings of the max-separable but also substantiates the rounding-off technique to remove the non-differentiable nature from the max-separable function. nonlinear scalings of thetomax-separable Lyapunov function explicitly, butmax-separable also substantiates the rounding-off technique remove the non-differentiable nature from the function. Keywords: Nonlinear systems, Lyapunov methods; Input-to-state stability; Small gain function. nonlinear scalings of the max-separable Lyapunov function explicitly, butmax-separable also substantiates the rounding-off technique to remove the non-differentiable nature from the © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. rounding-offNonlinear techniquesystems, to remove the non-differentiable nature from the max-separable Keywords: Lyapunov methods; Input-to-state stability; Small gain function. theorem; Interconnected rounding-off techniquesystems, tosystems. remove the non-differentiable nature from the max-separable function. Keywords: Nonlinear Lyapunov methods; Input-to-state stability; Small gain Keywords:Interconnected Nonlinear systems, systems, Lyapunov methods; methods; Input-to-state Input-to-state stability; stability; Small Small gain gain theorem; systems. Keywords: Nonlinear Lyapunov theorem; Interconnected systems. Keywords: Nonlinear systems, Lyapunov methods; Input-to-state stability; Small gain theorem; Interconnected Interconnected systems. theorem; systems. 1. INTRODUCTION and Wirth (2012)). Numerical computation can be done theorem; Interconnected systems. 1. INTRODUCTION and Wirth (2012)). Numerical computation done only in a prescribed bounded region (R¨ uffercan andbe Wirth 1. INTRODUCTION and Wirth (2012)). Numerical computation can be done 1. INTRODUCTION only in a prescribed bounded region (R¨ u ffer and Wirth and Wirth (2012)). Numerical computation can be done 1. provide INTRODUCTION and Wirth Numerical computation can done It prescribed is(2012)). noteworthy that aregion particular path Pbe (v) can Lyapunov functions useful information for analysis (2011)). only in a bounded (R¨ u ffer and Wirth 1. INTRODUCTION and Wirth (2012)). Numerical computation can be done (2011)). It is noteworthy that a particular path P (v) can only in a prescribed bounded region (R¨ u ffer and Wirth Lyapunov functions provide useful information for analysis only in a prescribed bounded region (R¨ u ffer and Wirth be expressed analytically by using the result in Karafyllis and designfunctions of nonlinear control systems (see e.g.for Sepulchre (2011)). It is noteworthy that a particular path P (v) can Lyapunov provide useful information analysis only in a prescribed bounded region (R¨ u ffer and Wirth be expressed analytically by using the result in Karafyllis (2011)). It is noteworthy that a particular path P (v) and design of nonlinear control systems (see e.g. Sepulchre Lyapunov functions provide useful information for analysis is analytically noteworthy a particular path P (v) can can and JiangIt(2011). However,that it using does not warrant closed-form Lyapunov functions provide useful information for et al.design (1997)). The small-gain theorem in terms ofanalysis input- (2011)). be expressed by the result in Karafyllis and of nonlinear control systems (see e.g. Sepulchre (2011)). is analytically noteworthy that a particular path P (v) can and JiangIt(2011). However, it using does not warrant closed-form be expressed by the result in Karafyllis Lyapunov functions provide useful information et al.design (1997)). The small-gain theorem in terms ofanalysis inputand design of nonlinear nonlinear control systems (see e.g.for Sepulchre be expressed analytically by using the result in Karafyllis expressions of its component-wise inverses ρ used in (1). and of control systems (see e.g. Sepulchre to-state stability (ISS) is one of several popular and powi and Jiang (2011). However, it does not warrant closed-form et (1997)). The small-gain theorem in terms of be expressed byit thewarrant result Karafyllis expressions ofanalytically its However, component-wise inverses ρi in used in (1). and Jiang (2011). does closed-form and of nonlinear control systems (see e.g. Sepulchre to-state stability (ISS) is one of severalby popular and pow- Having et al. al.design (1997)). The small-gain theorem in terms of inputinputand Jiang (2011). However, it using does not warrant closed-form closed-form expressions of not the scaling functions ρi et al. (1997)). The theorem in terms of inputerful frameworks forsmall-gain building systems interconnections expressions of its component-wise inverses ρ used in (1). to-state stability (ISS) is one of several popular and powi functions and Jiang (2011). However, it does not warrant closed-form Having closed-form expressions of the scaling ρini expressions of its component-wise inverses ρ used in et al. (1997)). The small-gain theorem in terms of inputerful frameworks for building systems by interconnections to-state stability (ISS) is one of several popular and powi expressions of for its component-wise inverses ρi functions used in (1). (1). convenient further use ofofathe Lyapunov function to-state (ISS) is one severalbypopular and gives pow- is (Jiangframeworks etstability al. (1994)). The ISSof small-gain framework Having closed-form expressions scaling ρ erful for building systems interconnections i expressions of its component-wise inverses ρ used in (1). is convenient for further use of a Lyapunov function Having closed-form expressions of the scaling functions ρ to-state stability (ISS) is one of several popular and pow(Jiang et al. (1994)). The ISS small-gain framework gives erful frameworks for building systems by interconnections i i Having closed-form expressions ofincluding scaling functions ρin analysis and design of networks infinite dimenerful frameworks for building interconnections not only a stability test, butsystems also a by Lyapunov function i is convenient for further use of aathe Lyapunov function in (Jiang et al. (1994)). The ISS small-gain framework gives Having closed-form expressions of the scaling functions ρ analysis and design of networks including infinite dimenis convenient for further use of Lyapunov function in erful frameworks for building systems by interconnections not only a stability test, but also a Lyapunov function (Jiang et al. (1994)). The ISS small-gain framework gives i is convenient for further use of a Lyapunov function in sional systems (Dashkovskiy and Mironchenko (2013)). (Jiang et al. (1994)). The ISS small-gain framework gives analysis endorsing the stabilitytest, of the interconnected system (Jiang and design of networks including infinite dimennot only a stability but also a Lyapunov function is convenient for further use of a Lyapunov function in sional systems (Dashkovskiy and Mironchenko (2013)). analysis and design of networks including infinite dimen(Jiang et al. (1994)). The ISS small-gain framework gives endorsing the stability of the interconnected system (Jiang not only a stability test, but also a Lyapunov function analysis and design of networks including infinite dimennot athe stability test, but Lyapunov function et al.only (1996)). This framework isalso now aavailable for networks sional systems (Dashkovskiy and Mironchenko (2013)). endorsing stability of interconnected system (Jiang and design networks dimensional systems (Dashkovskiy Mironchenko (2013)). The of thisofpaper isand to including provide aninfinite useful closednot only athe stability test, but Lyapunov function et al. (1996)). This framework isalso now aavailable forinnetworks endorsing the stability of the the interconnected system (Jiang analysis sionalobjective systems (Dashkovskiy and Mironchenko (2013)). endorsing stability of the interconnected system (Jiang in which ISS components are interconnected general The objective of this paper is to provide an useful closedet al. (1996)). This framework is now available for networks sional systems (Dashkovskiy and Mironchenko (2013)). form expression for the scalings ρ . As in the standard endorsing the stability of the interconnected system (Jiang in which ISS components are interconnected in general et al. (1996)). This framework is now available for networks i The objective of this paper is to provide an useful closedet al. (1996)). This framework is now available for networks topology (Dashkovskiy et are al. interconnected (2007); Jiang and Wang The form expression for the scalings ρ . As in the standard objective of this paper is to provide an useful closedi in which ISS components in general The objective of this paper isare to described provide an useful closedsetup, component systems by dissipation et (1996)). This framework is interconnected now available topology (Dashkovskiy et al. (2007); Jiangfor and Wang in al. which ISS components are interconnected innetworks general form expression for the scalings ρ in the standard i .. As in which ISS components are in general The objective of this paper is to provide an useful closed(2008); Dashkovskiy et al. (2010); Karafyllis and Jiang setup, component systems are described by dissipation form expression for the scalings ρ As in the standard i . functions topology (Dashkovskiy et al. (2007); Jiang and Wang form expression for the scalings ρ As in the standard inequalities consisting of dissipation and supply in which ISS components are interconnected in general (2008); Dashkovskiy et al. (2010); Karafyllis and Jiang topology (Dashkovskiy et al. (2007); Jiang and Wang i setup, component systems are described by dissipation topology (Dashkovskiy et al. (2007); and Wang form expression for the scalings ρ . As in the standard (2011); Liu et al. (2014); Dashkovskiy et Jiang al. (2011)). When inequalities consisting of dissipation functions and supply setup, component systems are described by dissipation i (2008); Dashkovskiy et al. (2010); Karafyllis and Jiang setup, component systems are described by dissipation functions which determine nonlinear gains of component topology (Dashkovskiy et al. (2007); Jiang and Wang (2011); Liu et al. (2014); Dashkovskiy et al. (2011)). When (2008); Dashkovskiy et al. (2010); Karafyllis and Jiang inequalities consisting of dissipation functions and supply (2008); Dashkovskiy al. of(2010); Karafyllis and When Jiang setup, component systems are described by dissipation a network described byetISS n components is guaranteed functions which determine nonlinear gains of component inequalities consisting of dissipation functions and supply (2011); Liu et al. (2014); Dashkovskiy et al. (2011)). inequalities consisting of functions and supply with respect to dissipation connecting inputs. The goal is (2008); Dashkovskiy al. of(2010); Karafyllis and When Jiang systems a network described byetISS n components is guaranteed (2011); Liu et aal. al.Lyapunov (2014); Dashkovskiy etthe al. (2011)). (2011)). When functions which determine nonlinear gains of component (2011); Liu et (2014); Dashkovskiy et al. inequalities consisting of dissipation functions and supply to be stable, function of max-separable systems with respect to connecting inputs. The goal is functions which determine nonlinear gains of component ato network described by ISS of n components is guaranteed functions which determine nonlinear gains of component to express ρ s with those functions. The expression is (2011); Liu et al. (2014); Dashkovskiy et al. (2011)). When be stable, a Lyapunov function of the max-separable a network described by ISS of n components is guaranteed i systems with respect to connecting inputs. The goal is atype network described by ISS function of n components is guaranteed functions which determine nonlinear gains of component to express ρ s with those functions. The expression systems with respect to connecting inputs. The goal is i to be stable, a Lyapunov of the max-separable systems with respect to connecting inputs. The goal is sought whenever the ISS small-gain condition is satisfied. a network described by ISS of n components is guaranteed type to be stable, a Lyapunov function of the max-separable to express ρi ssrespect with those functions. The expression expression is to be stable, aVLyapunov function the max-separable with to small-gain connecting inputs. The goal is sought whenever the ISS condition isexpression satisfied. to express ρ those functions. The type (x) = max ρi (Vof (x (1) systems express ρii s with with those functions. The expression is i )) max-separable The development directly gives a closed-loop to be stable, aVILyapunov function the type sought whenever the ISS small-gain condition is satisfied. max ρi (Vofii (x (1) to type to express ρ s with those functions. The expression is I (x) = i∈{1,2...,n} i )) The development directly gives a closed-loop expression sought whenever the ISS small-gain condition is satisfied. i sought whenever the ISS small-gain condition is satisfied. of the max-separable function for ISS networks. Another type i∈{1,2...,n} V (x) = max ρ (V (x )) (1) i (Vi (xi )) The development directly gives aa ISS closed-loop expression VII (x) = max ρ (1) sought whenever the ISS small-gain condition is satisfied. i (Vi (xi )) of the max-separable function for networks. Another The development directly gives closed-loop expression i∈{1,2...,n} V (x) = max ρ (1) can be constructed. The max-separable Lyapunov funci i The development a ISS closed-loop expression contribution of thisdirectly paper isgives to for generalize the closed-form i∈{1,2...,n} the max-separable function networks. Another VI = ρii (Vthe (1) of can constructed. The max-separable Lyapunov i∈{1,2...,n} i (xi )) The a ISS closed-loop expression contribution ofproduce thisdirectly paper isgives to for generalize the closed-form of the max-separable function networks. Another tion be VI (x) is asI (x) simple asmax dividing state spacefuncinto of thedevelopment max-separable function for ISS networks. Another i∈{1,2...,n} can be constructed. The max-separable Lyapunov funcexpression to a useful variety of admissible scalcontribution of this paper is to generalize the closed-form tion V (x) is as simple as dividing the state space into can be constructed. The max-separable Lyapunov funcI of the max-separable function for ISS networks. Another expression to produce a useful variety of admissible scalcontribution of this paper is to generalize the closed-form can be constructed. The max-separable Lyapunov funcregions to which Lyapunov functions of individual compocontribution of this paper is to generalize the closed-form tion V (x) is as simple as dividing the state space into ings that increase the flexibility in constructing Lyapunov I expression to produce a useful variety of admissible scalcan be constructed. The max-separable Lyapunov funcregions to which Lyapunov functions of individual compotion V (x) is as simple as dividing the state space into I contribution of this paper is to generalize the closed-form ings that increase the flexibility in constructing Lyapunov expression to produce a useful variety of admissible scaltion V (x) is as simple as dividing the state space into nents are assigned exclusively. In each region, a Lyapunov I expression to produce a useful variety admissible scalregions to which Lyapunov functions of individual compofunctions. This paper demonstrates howofthe generalization ings that increase the flexibility in constructing Lyapunov tion V (x) is as simple as dividing the state space into nents are assigned exclusively. In each region, a Lyapunov regions to which Lyapunov functions of individual compoI expression to produce a useful variety of admissible scalfunctions. This paper demonstrates how the generalization ings that increase the flexibility in constructing Lyapunov regionsare toVassigned which Lyapunov ofregion, individual compofunction a singlefunctions component system serves as gives i (xi ) of ings that increase the flexibility in constructing Lyapunov nents exclusively. In each a Lyapunov a solution on which the rounded Lyapunov funcfunctions. This paper demonstrates how the generalization regions to which Lyapunov functions of individual compofunction V (x ) of a single component system serves as nents are assigned exclusively. In each region, a Lyapunov i i function ings that increase the flexibility in constructing Lyapunov gives a solution on which the rounded Lyapunov funcfunctions. This paper demonstrates how the generalization nents are assigned exclusively. In each region, a Lyapunov the Lyapunov of the entire network. The difunctions. This paper demonstrates how the generalization function of a single component system serves as tion constructed. the corners of the funcmaxi (x i) givesis solution on Rounding which theoff rounded Lyapunov nents are V In entire each region, a Lyapunov the Lyapunov network. The difunction of a component system as i (x i ) function This paper demonstrates how the generalization tion isaaaconstructed. Rounding off the corners of the funcmaxgives solution on which the rounded Lyapunov function Vassigned of exclusively. a single single component system serves as functions. vision is V determined by of thethe scaling functions ρserves whose i (xi ) function iThe gives solution on which the rounded Lyapunov functhe Lyapunov of the entire network. diseparable Lyapunov function is an approach proposed very tion is constructed. Rounding off the corners of the maxfunction V (x ) of a single component system serves as vision is determined by the scaling functions ρ whose the Lyapunov function of the entire network. The dii i i gives a solution on which the rounded Lyapunov funcseparable Lyapunov function is an approach proposed very tion is constructed. Rounding off the corners of the maxthe Lyapunov of the entire network. The di- recently inverses form a function path connecting the point at infinity and tion is constructed. Rounding off the corners of thediffermaxvision is determined by the scaling functions ρ whose to geometrically construct a continuously iThe separable Lyapunov function is an approach proposed very the Lyapunov function of the entire network. diinverses form a path connecting the point at infinity and vision is determined by the scaling functions ρ whose i whose tion is constructed. Rounding off the corners of the maxrecently to geometrically construct a continuously differseparable Lyapunov function is an approach proposed very vision is determined by the scaling functions ρ the origin in the non-negative space of the vector z = i separable function is an approach proposed very inverses form a path connecting the point at infinity and entiable function for interconnected ISS system recently Lyapunov toLyapunov geometrically construct a continuously continuously differvision is determined by the scaling functions ρ whose the origin in the non-negative space of the vector z = inverses form a path connecting the point at infinity and i separable Lyapunov function is an approach proposed very entiable Lyapunov function for interconnected ISS system recently to geometrically construct a differinverses form a path connecting the point at infinity and [V (x ), V (x ), ..., V (x )] listing the values of Lyapunov 1 1 2 2 n n recently to geometrically construct a continuously differthe origin in the non-negative space of the vector z = (Ito (2018)) and ISS networks (Ito and R¨ u ffer (2019)). As entiable Lyapunov function for interconnected ISS system inverses form a path connecting the point at infinity and [V (x ), V (x ), ..., V (x )] listing the values of Lyapunov the origin in the non-negative space of the vector z = 1 1 2 2 n n recently to geometrically construct a continuously differ(Ito (2018)) and ISS networks (Ito and R¨ u ffer (2019)). As entiable Lyapunov function for interconnected ISS system the origin in the non-negative space of the vector z = functions of component systems. In Fig.1, it of is illustrated entiable Lyapunov function for(Ito interconnected ISSfunction system [V V (x ..., V )] listing the values Lyapunov illustrated inand Fig.1, level sets of the Lyapunov 1 (x 1 ), 2of 2 ), n (x nsystems. (Ito (2018)) ISS networks and R¨ u ffer (2019)). As the origin in the non-negative space of the vector z = functions component In Fig.1, it is illustrated [V (x ), V (x ), ..., V (x )] listing the values of Lyapunov 1 1 2 2 n n entiable Lyapunov function for interconnected ISS system illustrated in Fig.1, level sets of the Lyapunov function (Ito (2018)) and ISS networks (Ito and R¨ u ffer (2019)). As [V (x ), V (x ), ..., V (x )] listing the values of Lyapunov by the rectangles and the path P (v) for n = 2. The con1 1 2of component 2 n nsystems. In Fig.1, it is illustrated (Itorounded (2018))inand ISS networks and R¨ uffer (2019)). As functions are cuboids, whichsets are(Ito often convenient estimates illustrated Fig.1, level of the Lyapunov function [V (x ), V (x ), ..., V (x )] listing the values of Lyapunov by the rectangles and the path P (v) for n = 2. The confunctions of component systems. In Fig.1, it is illustrated 1 1 2 2 n n (Ito (2018)) and ISS networks (Ito and R¨ u ffer (2019)). As are rounded cuboids, which are often convenient estimates illustrated in Fig.1, level sets of the Lyapunov function functions of the component systems. In Fig.1, it=is2.illustrated struction of popular max-separable Lyapunov function illustrated in Fig.1, level sets of the Lyapunov function by the rectangles and the path P (v) for n The conof forward invariant sets, compared with unexpectedly are rounded cuboids, which are often convenient estimates functions of component systems. In Fig.1, it is illustrated struction of the popular max-separable Lyapunov function by the rectangles and the path P (v) for n = 2. The conillustrated in Fig.1, level sets of the Lyapunov function of forward invariant sets, compared with unexpectedly are rounded cuboids, which are often convenient estimates by the rectangles and the path P (v) for n = 2. The con(1) relies ofonthe the knowledge of a pathLyapunov P (v). Numerical are rounded cuboids, which often convenient struction popular max-separable function sets produced by are other methods. Theestimates rounded of forward invariant sets, compared with unexpectedly by the rectangles and the path Pa(v) for = 2.Numerical The con- distorted (1) relies the knowledge of discussed path Pnin (v). struction of the popular max-separable Lyapunov function are rounded cuboids, which are often convenient estimates distorted sets produced byfrom other methods. The rounded of forward invariant sets, compared with unexpectedly struction ofon the popular max-separable Lyapunov function construction of such a path is Dashkovskiy of forward invariant sets, compared with unexpectedly (1) relies on the knowledge of a path P (v). Numerical Lyapunov function is free discontinuous changes in distorted sets produced by other methods. The rounded struction of the popular max-separable Lyapunov function construction such a path discussed (1) relies on the knowledge of a P (v). Numerical forwardsets invariant sets, compared with unexpectedly Lyapunov function is action free from discontinuous changes in distorted produced by other methods. The rounded (1) relies on of the knowledge ofextensive a path path study Pin (v).Dashkovskiy Numerical et al. (2010). There has beenis on the de- of distorted sets produced by other methods. The rounded construction of such a path is discussed in Dashkovskiy estimate and control which can be caused by the Lyapunov function is free from discontinuous changes in (1) relies on the knowledge of a path P (v). Numerical et al. (2010). There has been extensive study on the deconstruction of such a path is discussed in Dashkovskiy distorted sets produced by other methods. The rounded estimate and control action which can be caused by the Lyapunov function is free from discontinuous changes in construction of such a path is discussed in Dashkovskiy velopment of efficient numerical algorithms for the path Lyapunov function free from discontinuous changes in et al. (2010). There has been extensive study on the deswitching of maximization inwhich (1). The switching implies estimate and and controlis action can be be caused by the the construction of such a path is discussed in Dashkovskiy velopment of efficient numerical algorithms for the path et al. (2010). There has been extensive study on the deLyapunov function is free from discontinuous changes in switching of maximization in (1). The switching implies estimate control action which can caused by et al. (2010). There has been extensive study on the decomputation (e.g., Dashkovskiy et al. (2010); Geiselhart estimate and control action which can be caused by the velopment of (e.g., efficient numerical algorithms foron the path that the max-separable Lyapunov function is only implies locally switching of maximization in (1). The switching et al. (2010). There has been extensive study the decomputation Dashkovskiy et al. (2010); Geiselhart velopment of efficient numerical algorithms for the path estimate and control action which can be caused by the that the max-separable Lyapunov function is only locally switching of maximization in (1). The switching implies velopment of (e.g., efficient numerical et algorithms forGeiselhart the path Lipschitz of that maximization in (1). The switching implies computation Dashkovskiy al. (2010); so methods from non-smooth analysis had ⋆ that the max-separable Lyapunov function is only locally velopment efficient numerical algorithms forGeiselhart the Grant path switching computation (e.g., Dashkovskiy al. (2010); The work of was supported in part et by JSPS KAKENHI switching of that maximization in (1). The switching implies Lipschitz so methods from non-smooth analysis had that the max-separable Lyapunov function is only locally ⋆ computation (e.g., Dashkovskiy (2010); Geiselhart The work was supported in part et by al. JSPS KAKENHI Grant that the max-separable Lyapunov function is only locally to be used to avoid them (Jiang et al. (1996)). This paper Lipschitz so that methods from non-smooth analysis had computation (e.g., Dashkovskiy et al. (2010); Geiselhart Number 17K06499. ⋆ that max-separable function isanalysis only to bethe used tothat avoid themLyapunov (Jiang et al. (1996)). Thislocally paper Lipschitz so methods from non-smooth had The work was supported in part by JSPS KAKENHI Grant ⋆ Number 17K06499. The work was supported in part by JSPS KAKENHI Grant Lipschitz so that methods from non-smooth analysis had ⋆ The work was supported in part by JSPS KAKENHI Grant to be used to avoid them (Jiang et al. (1996)). This paper Lipschitz so that methods from non-smooth analysis had Number 17K06499. to be used to avoid them (Jiang et al. (1996)). This paper ⋆ Number 17K06499. The work was supported in part by JSPS KAKENHI Grant to be used to avoid them (Jiang et al. (1996)). This paper Number 17K06499. Copyright © 2019 IFAC 931 to be used to avoid them (Jiang et al. (1996)). This paper Number 17K06499. 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 931 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 931 Peer review© responsibility of International Federation of Automatic Copyright © under 2019 IFAC IFAC 931 Control. Copyright © 2019 IFAC 931 10.1016/j.ifacol.2019.12.018 Copyright © 2019 IFAC 931

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Hiroshi Ito / IFAC PapersOnLine 52-16 (2019) 544–549

provides scalings whose knowledge has to be assumed in the rounding-off construction (Ito and R¨ uffer (2019)). In this paper, for x, y ∈ Rn := (−∞, ∞)n we write x ≤ y if y − x ∈ Rn+ := [0, ∞)n . We write x < y if x ≤ y and x ̸= y. The symbol x ≪ y is used if y − x is in the interior of Rn+ . One writes ξ ∈ P for a continuous function ξ : R+ → R+ if ξ(s) > 0 for all s ∈ R+ \ {0}, and ξ(0) = 0. A class P function ξ is said to be of class K if it is strictly increasing. It is of class K∞ if, in addition, lims→∞ ξ(s) = ∞. A continuous function β : R+ ×R+ → R+ is said to belong to KL if, for each fixed t ≥ 0, β(·, t) is of class K and, for each fixed s > 0, β(s, ·) is decreasing and limt→∞ β(s, t) = 0. Composition of ξ1 , ξ2 : R+ → R+ is written ⊙n as ξ1 ◦ ξ2 . For ordered repeated composition, we use i=1 ξi = ξ1 ◦ ξ2 ◦ · · · ◦ ξn . The symbol Id denotes the identity function. The zero function on the space of interest is denoted by 0. The symbol | · | denotes the Euclidean norm. For a function f : Rm → R, its positive part (f (·))+ : Rm → R+ is defined by (f (x))+ = max{f (x), 0}. The power set of a set A is denoted by 2A . For a subset A of a set B, 1A : B → {0, 1} denotes the indicator function. That is, 1A (i) = 1 when i ∈ A. Otherwise, 1A (i) = 0. This paper employs the convention ∞ ≥ ∞. 2. NETWORK OF ISS SYSTEMS Consider a network consisting of n systems described by x˙ i (t) = fi (x(t), ri (t)), i ∈ n (2)

for xi (t) ∈ RNi , ri (t) ∈ RPi and n := {1, 2, ..., n}. Defining the state x(t) = [x1 (t)T , x2 (t)T , . . . , xn (t)T ]T ∈ RN and the external signal r(t) = [r1 (t)T , r2 (t)T , . . . , rn (t)T ]T ∈ RP allows the network (2) to be expressed as x(t) ˙ = f (x(t), r(t)). Assume that f : RN × RP → RN is locally Lipschitz, and r : R+ → RP is locally essentially bounded. Following the standard definition in Sontag (1989), the network (2) is said to be ISS if for any x(0) ∈ RN and any locally essentially bounded r : R+ → RP , a unique solution x(t) to (2) exists for all t ∈ R+ , and satisfies ( ) |x(t)| ≤ β(|x(0)|, t) + γ ess sup |r(τ )| , τ ∈[0,t]

∀t ∈ R+ (3)

for some β ∈ KL and γ ∈ K. ISS implies global asymptotic stability of the origin x = 0 for r(t) ≡ 0, which is called 0-GAS of the network (2). If there exist a continuously differentiable, positive definite and radially unbounded function V : RN → R+ , and α ∈ K, σ ∈ K such that ∂V (x)f (x, r) ≤ −α(V (x)) + σ(|r|), ∀x ∈ RN , r ∈ RP , ∂x (4) lim α(s) ≥ lim σ(s) (5) s→∞

s→∞

are satisfied, then the network (2) is ISS (Sontag and Wang (1995); Ito (2006)). In this paper, a function V fulfilling the above requirement is referred to as an ISS Lyapunov function. If a function V satisfies (4) with α ∈ P for r = 0, it is called a 0-GAS Lyapunov function. This paper addresses the problem of constructing an 0GAS/ISS Lyapunov function of (2) under the following. Assumption 1. There exist continuously differentiable, positive definite and radially unbounded functions Vi : RNi → R+ , and αi ∈ K, σi,j ∈ K ∪ {0}, κi ∈ K, i, j = 1, 2, . . . , n (i ̸= j), such that 932

545

∂Vi (xi )fi (x, ri ) ≤ −αi (Vi (xi )) + max σi,j (Vj (xj )) ∂xi j∈n\{i} + κi (|ri |), ∀x ∈ RN, ri ∈ RPi, (6) lim αi (s) = ∞ or ∀j ∈ n \ {i} lim αi (s) > lim σi,j (s) s→∞

s→∞

s→∞

(7)

are satisfied for all i ∈ n. The pair of (6) and (7) makes sure that each xi -system is ISS with respect to the connecting input xj individually for all j ∈ n \ {0}. The non-strict inequality lims→∞ αi (s) ≥ lims→∞ σi,j (s) better fits (5) than the strict one in (7). However, eventually, the small-gain condition guaranteeing 0-GAS of the network (2) requires the strict margin as in (7) (e.g. Jiang et al. (1994); Dashkovskiy et al. (2010); Liu et al. (2014)) unless one formulates an essentially harder problem by invoking integral input-to-state stability, which is a broader notion than ISS (e.g. Sontag (1998); Ito (2006); Ito et al. (2013); Angeli and Astolfi (2007)). 3. CLOSED FORM EXPRESSION FOR PATHS This section presents results on the scaling functions ρi which are central to the construction of the Lyapunov function of the form (1). To find closed form expressions of the functions ρi , we define, for s ∈ R+ and z ∈ Rn+ , (8) gi,j (s) := pi,j ◦ σi,j (s), i ∈ n, j ∈ n \ {i},   max g1,j (zj )   α1 (z1 )   j∈n\{1}  max g2,j (zj )   α2 (z2 )     , G(z) =  j∈n\{2} A(z) =   , (9) .. ..     . .   αn (zn ) max gn,j (zj ) j∈n\{n}

L(z) = −A(z) + G(z), (10) n Ω = {z ∈ R+ : L(z) ≪ 0}. (11) The functions pi,j ∈ K ∪ {0} have yet to be determined. Define a directed graph G(V, E) from (6) by the vertex set V and the edge set E as follows. Let V = {1, 2, . . . , n}. The ordered pair (i, j) is an directed edge from vertex j to vertex i unless σi,j = 0. Let P(i, j; G) denote the set of all paths from vertex j to vertex i of the directed graph G. As the main answer in this section, define n ∑ ζj (s) = Ki θi,j (s) (12) i=1

with respect to given constants Ki > 0 for i ∈ n, where (13) θi,j = αi−1 ◦ gi,j , j ∈ n \ {i},  s, if j = i   |u| ⊙ θi,j (s) = (14) θu(k),u(k+1) (s), otherwise.   max u∈P(i,j;G)

k=1

Here, | · | denotes the length. The well-posedness of αi−1 in (13) is addressed whenever it appears in this paper. The next theorem shows that the vector Z(s) = [ζ1−1 (s), ζ2−1 (s), . . . , ζn−1 (s)]T (15) parametrized by s ∈ R+ has a desired property. Theorem 2. Suppose that (16) θi,j ∈ K ∪ {0} is satisfied with pi,j ∈ K ∪ {0} for i ∈ n and j ∈ n \ {i}. Let the constants Ki > 0 be fixed arbitrarily. If (17) L(z) ̸≥ 0, ∀z ∈ Rn+ \ {0}

2019 IFAC NOLCOS 546 Vienna, Austria, Sept. 4-6, 2019

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holds, then θi,j ∈ K ∪ {0} ζi ∈ K ∞

θi,j (ζj−1 (s)) < ζi−1 (s), ∀s ∈ (0, ∞) L(Z(s)) ≪ 0, ∀s ∈ (0, ∞) hold for all i ∈ n and j ∈ n \ {i}.

(18) (19)

ρ−1 2 (v2 )

(20) (21)

ρ−1 2 (v1 )

Proof: Applying (16) to the arguments in (Ito et al., 2013, Proof of Lemma 1) proves that (17) ensure (18). θi,i (s) = s and Ki > 0 in (12) yield (19). Property (17) also implies θi,j ◦ θj,i (s) < s, ∀s ∈ (0, ∞) for all i ∈ n and j ∈ n \ {i}, and θi,k ◦ θk,j (s) ≤ θi,j (s), ∀s ∈ R+ for all i ∈ n, j ∈ n \ {i} and k ∈ n \ {i, j}. Hence, due to (12) and (14), for all i ∈ n and j ∈ n \ {i}, ζj ◦ θj,i (s) ≤ ζj ◦ θj,i (s) ∑ = Kj θj,i + Kk θk,j ◦ θj,i (s) k∈n\{j}

≤ Ki θi,j ◦ θj,i (s) + Kj θj,i +

∑

Kk θk,i (s)

k∈n\{j,i}

(22) < ζi (s), ∀s ∈ (0, ∞). Using (13), θi,j ∈ K ∪ {0} and ζi ∈ K∞ we obtain gj,i ◦ ζi−1 (s) < αj ◦ ζj−1 (s), ∀s ∈ (0, ∞) for all i ∈ n and j ∈ n\{i}. We arrive at (21). Finally, (20) is obtained from ζj ◦ θj,i (s) < ζi (s) proved in (22).  Property (20) hints the idea of defining θi,j , while (21) is the targeted inequality which qualifies (1) with ρi = ζi to be a Lyapunov function of (2). It is worth mentioning that Karafyllis and Jiang (2011) gave a different set of ζi−1 s achieving L(Z(s)) ≤ 0 instead of (21), although the inverses ζi−1 do not warrant a closed-form expression of (1). The parametrized vector Z(s) can be seen as a path of infinite length in Rn+ . The constants Ki in the proposed solution ζi in (12) provide the path Z(s) a degree of freedom in achieving the targeted inequality (21). It is expected that the “thickness” of the set Ω increases the freedom. For instance, when all θi,j s are of class K∞ , the path Z(s) gets closer to the curve of [ −1 ]T −1 −1 θi,1 (s), θi,2 (s), . . . , θi,n (s) as Ki becomes larger than the others Kj , j ∈ n \ {i} while (21) is maintained. Since a change in Ki alters all components of Z(s), this flexibility offered by Ki is limited.

Our next goal is to increase the flexibility of solutions to (21), so that some components in Z(s) can be modified individually, while the remaining components are fixed. For such flexibility that is more than Ki s, replace Z(s) by ζˆi = ζi + rˆi (23) ˆ Z(s) = [ζˆ1−1 (s), ζˆ2−1 (s), . . . , ζˆn−1 (s)]T . (24) The following theorem presents a condition on rˆi ∈ K∪{0} to guarantee that this modification of Z(s) retains (21). Theorem 3. Suppose that (16) and (17) are satisfied with pi,j ∈ K ∪ {0} for i ∈ n and j ∈ n \ {i}. Assume that µi,j ∈ K satisfies (Id + µi,j ) ◦ θj,i ◦ θi,j (s) < s,

∀s ∈ (0, ∞)

z2 

(25)

933

P (v)

r(v2 )  r(v1 )

 z1 −1 ρ−1 1 (v1 ) ρ1 (v2 ) Fig. 1. The rounded rectangles are level sets of W (z) defined with (32) for two levels 0 < v1 < v2 , while the rectangles are of (1). 0

for i ∈ n and j ∈ n \ {i}. Let Kj > 0 be fixed arbitrarily for i ∈ n. If the functions rˆi ∈ K ∪ {0} satisfy rˆi ◦ θi,j (s) ≤ Kj µi,j ◦ θj,i ◦ θi,j (s), ∀s ∈ R+ , j ∈ n \ {i} (26) for all i ∈ n, then ζˆi ∈ K∞ , ˆ L(Z(s)) ≪ 0,

(27) ∀s ∈ (0, ∞)

(28)

hold for all i ∈ n and j ∈ n \ {i}. Proof: Property (27) follows from (23) and rˆi ∈ K ∪ {0} since ζi ∈ K∞ . Applying (25) to an argument used in the proof of Theorem 2 gives ζi ◦ θi.j (s) ≤ ζi ◦ θi,j (s)

≤ Kj θj,i ◦ θi,j (s) + Ki θi,j +

∑

Kk θk,j (s)

k∈n\{i,j}

< ζj (s) − Kj µi,j ◦ θj,i ◦ θi,j (s),

∀s ∈ (0, ∞)

for all i ∈ n and j ∈ n \ {i}. Hence, due to (23), property (26) yields ζˆi ◦ θi.j (s) < ζj (s), ∀s ∈ (0, ∞) for all i ∈ n and j ∈ n \ {i}. Since ζj (s) ≤ ζˆj (s) holds for all s ∈ R+ , the claim (28) is established by (24).  The functions µi,j ∈ K in (25) describe the margin in condition (17), which is interpreted as the thickness of the set Ω. Theorem 3 swings the path Z(s) within the margin. Remark 4. By definition, the function ζi is continuously differentiable if the functions θi,j are continuously differentiable. Under the continuous differentiability of θi,j , the function θi,j is only locally Lipschitz due to the maximization in (14). However, the dents caused by the maximization can be smoothed from above by picking a slightly larger and continuously differentiable function θi,j . The margin in (17) can allow this modification. Importantly, condition (17) does not have to be checked again after smoothing θi,j . Instead, one can check the targeted inequality (21) directly. The smoothing is valid as long as (21) holds. Verifying (21) defined for s ∈ R+ is computationally less expensive than (17) defined for z ∈ Rn+ . Thus, starting by assuming continuously differentiable θi,j is not unreasonable. The function ζˆi is continuously differentiable if rˆi is selected as a continuously differentiable function in addition to continuous differentiable ζi,j s.

2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019

Hiroshi Ito / IFAC PapersOnLine 52-16 (2019) 544–549

4. ROUNDED LYAPUNOV FUNCTIONS 4.1 Rounding-off the max-separable function The network (2) is guaranteed to be ISS if gain functions of component systems satisfy a small-gain condition (Dashkovskiy et al. (2007); Jiang and Wang (2008); Dashkovskiy et al. (2010); Liu et al. (2014)). An expression of the small-gain condition is the existence of ωi ∈ K∞ (i ∈ n) satisfying D ◦ Σ(z) ̸≥ A(z), ∀z ∈ Rn+ \ {0}, (29) where   max σ1,j (zj ) j∈n\{1}    max σ2,j (zj )   j∈n\{2}  Σ(z) =  (30) , ..   .   max σn,j (zj ) j∈n\{n}

D(z) = [z1 + ω1 (z1 ), z2 + ω2 (z2 ), . . . , zn + ωn (zn )]T. (31)

The vector form condition (29) is known to be equivalent to the cyclic small-gain condition of Jiang and Wang (2008); Liu et al. (2014). It is known that under the satisfaction of the ISS small-gain condition (29), an ISS Lyapunov function of the max-separable form (1) can be constructed for the network (2). To get rid of the nondifferentiability of (1), recently, a rounding-off approach has been proposed in Ito (2018); Ito and R¨ uffer (2019). The idea is to round off the corners of the (hyper-)rectangles which are level sets of the max-separable Lyapunov function (1). The rounding-off is represented by n [ ∑ ( ) ]2 J(v, z) = zi −ρ−1 (v)+r(v) − r(v)2 (32) i + i=1

for v ∈ R+ and z = [z1 , z2 , ..., zn ]T ∈ Rn+ , where ρi ∈ K∞ and r ∈ K ∪ {0}. It can be verified that if J(W (z), z) = 0 admit a solution W : Rn+ → R+ for all z ∈ Rn+ , the level set of W (z) is a rounded cuboid whose side lengths are ρ−1 i (v) and the radius of the rounding-off is r as depicted in Fig.1 for n = 2. Define T

V (x) = [V1 (x1 ), V1 (x1 ), . . . , Vn (xn )] for the functions Vi defined in Assumption 1. This section demonstrates that the functions developed in the previous section qualify VI (x) = W (V (x)) to serve as a 0-GAS/ISS Lyapunov function of the network (2). 4.2 0-GAS The following proposition is a refined presentation of a result proved in Ito and R¨ uffer (2019). Proposition 5. Assume that ρi ∈ K∞ (i ∈ n) and r ∈ K ∪ {0} are continuously differentiable on R+ and satisfy ρ′i (s) > 0, ∀s ∈ (0, ∞), i ∈ n, (33) ′ r′ (s) < min(ρ−1 i ) (s), i∈n

∀s ∈ (0, ∞),

i ∈ n,

(34)

−1 [ρ−1 1 (s) − 1B (1)r(s), ρ2 (s) − 1B (2)r(s),

T · · · , ρ−1 n (s) − 1B (n)r(s)] ∈ Ω,

∀s ∈ (0, ∞), ∀B ⊆ 2n \ n

for Ω defined in (11) with pi,j = Id, i ∈ n, j ∈ n \ {i}.

(35) (36) 934

547

Then there exists a unique C1 function W : Rn+ → R+ such that J(W (z), z) = 0 holds for all z ∈ Rn+ . Moreover, VI (x) = W (V (x)) is a 0-GAS Lyapunov function of the network (2), and it is continuously differentiable everywhere if r ∈ K. When B = ∅ is taken, it is well-known that the existence of ωi ∈ K∞ (i ∈ n) satisfying (29) guarantees the existence of ρi ∈ K∞ (i ∈ n) achieving (35) (see e.g. Dashkovskiy et al. (2010)). With the help of re-parametrization of ρi , continuity arguments would lead to the existence of a continuously differentiable ρi satisfying (33) and (35), and even allow a non-zero r in (34) and (35). However, the existence alone is not satisfactory. Proposition 5 requires ρi and r. The next theorem demonstrates that Theorem 3 in the previous section gives desired ρi ∈ K∞ and r ∈ K∪{0}. Theorem 6. Suppose that ωi ∈ K∞ (i ∈ n) satisfy (29). Let pi,j be defined as in (36) Assume that the functions θi,j are continuously differentiable on R+ for i ∈ n and j ∈ n \ {i}. Let Ki > 0 be arbitrary. Then for i ∈ n and j ∈ n \ {i}, there exists µi,j ∈ K satisfying (25). Moreover, ρi = ζ i (37) defined with (12) achieves ρi ∈ K∞ and (33), In addition, for any rˆi ∈ K ∪ {0} (i ∈ n) which are continuously differentiable on R+ and satisfy (26), the function ζi−1 − ζˆi−1 defined with (23) is continuously differentiable on R+ and satisfies (38) ζi−1 − ζˆi−1 ∈ K ∪ {0}. Furthermore, properties (34) and (35) are achieved for any r ∈ K ∪ {0} satisfying r(s) ≤ min{ζ −1 (s) − ζˆ−1 (s)}, ∀s ∈ R+ , (39) i∈n

i

r′ (s) < min(ζi−1 )′ (s), i∈n

i

∀s ∈ (0, ∞).

(40)

Proof: Let pi,j = Id for i ∈ n and j ∈ n \ {i}. Property (29) yields (17) and the existence of µi,j ∈ K satisfying (25). The assumption (7) guarantees (16). Hence, one can invoke Theorem 3 to prove the claims. For the choice (37), property (27) confirms ρ ∈ K∞ . Since θi,j is continuously differentiable on R+ and belong to K ∪ {0}, we have ′ θi,j (s) ≥ 0 for all s ∈ R+ . Thereby, the functions ζi ∈ K∞ given by (12) are continuously differentiable on R+ . Due to Kk > 0 for all k ∈ n, the linear term in (12) implies ζi′ (s) > 0 for all s ∈ R+ . Thus, (33) is obtained from (37). In addition, the continuous differentiability of ζi−1 ∈ K∞ on R+ follows. The continuous differentiability of rˆi ∈ K ∪ {0} gives the continuous differentiability of ζi + rˆi ∈ K∞ on R+ . Since ζi′ (s) > 0 and rˆi′ (s) ≥ 0 hold for all s ∈ R+ , we have ζi′ (s) + rˆi′ (s) > 0 for all s ∈ R+ . Therefore, the continuous differentiability of ζˆi−1 = (ζi + rˆi )−1 ∈ K∞ and ζi−1 ensures the continuous differentiability of ζi−1 − ζˆi−1 on R+ . Property rˆi ∈ K ∪ {0} yields (38). In fact, we have ζi−1 − ζˆi−1 = qi ◦ (ζi + rˆi )−1 for qi := ζi−1 ◦ (ζi + rˆi ) − Id ∈ K ∪ {0} and (ζi + rˆi )−1 ∈ K∞ . The pair (37) and (40) ensure (34) directly. From (37) and (39) we obtain ˆ−1 ρ−1 i (s) − r(s) ≥ ζi (s) for i ∈ n. Hence, for (11), property (35) follows from (28).  Due to (38) and ζi ∈ K∞ defined by (12), there always exists r ∈ K satisfying (39) and (40) in Theorem 6, although r = 0 is a trivial solution. As stated in Proposition 5. VI (x) = W (V (x)) is continuously differentiable

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everywhere if r ̸= 0 is chosen. When r = 0 is chosen, the function W achieving J(W (z), z) = 0 can be expressed as (41) W (z) = max ρi (z). i∈n

This function leading to (1) is not differentiable. The function r is the radius of the arc rounding off the corner of the max-separable function (41) to render VI (x) = W (V (x)) continuously differentiable. This paper provides not only a closed form expression of ρi , but also an appropriate rounding-off radius r. 4.3 ISS The following proposition shows that ISS can be extracted from pi,j ̸= Id, which extends a result in Ito and R¨ uffer (2019) by replacing lims→∞ αi (s) = ∞ with (7). Proposition 7. Let hi ∈ (0, 1) and ωi ∈ K∞ (i ∈ n) be such that (42) lim αi (s) ≥ lim (Id + hi ωi ) ◦ σi,j (s) s→∞

s→∞

hold for all j ∈ n \ {i}. Assume that ρi ∈ K∞ (i ∈ n) and r ∈ K ∪ {0} are continuously differentiable on R+ and satisfy (33) (34) and (35) for Ω defined in (11) with (43) pi,j = Id + hi ωi , i ∈ n, j ∈ n \ {i}. 1 n Then there exists a unique C function W : R+ → R+ such that J(W (z), z) = 0 holds for all z ∈ Rn+ . Moreover, if ( ( ) ) 1 −1 −1 lim inf min ωi ◦ (Id+hi ωi ) ◦αi ρi (s)− 1− √ r(s) s→∞ i∈n n −1 ′ mink∈n ρk ◦ ρk (s) =∞ (44) × maxj∈n ρ′j ◦ ρ−1 j (s) is satisfied, VI (x) = W (V (x)) is an ISS Lyapunov function of the network (2), and it is continuously differentiable everywhere if r ∈ K. Proof: The arguments in Ito and R¨ uffer (2019) lead us to ∂VI f ≤ −α ˆ (V (x(t))) + ηˆ(V (x(t)))κ(|r(t)|). ∂x along the trajectories of (2), where ( ( ) ) 1 −1 √ α ˜ i ρi (v) − 1− r(s) n α ˆ (s) = min , √ n ∑ i∈n n k=1

Therefore, by virtue of (45), we have ∂VI f ≤ −α ˆ (V ) + ηˆ(V )κ(|r|) ≤ −α ˜ (V ) + σ(|r|) ∂x with α ˜ := δ α ˆ ∈ K, σ := [Y −1 ◦ κ]κ + Eκ ∈ K ∪ {0}.

Property (44) implies α ˜ ∈ K∞ .

(49) (50) (51)



Due to (7), there always exist hi ∈ (0, 1) and ωi ∈ K∞ (i ∈ n) such that (42) holds, The only difference between Proposition 5 and Proposition 7 is pi,j in (36) and (43). The difference brings in (42). The maps θi,j , θi,j , L and the set Ω are defined for given pi,j . Therefore, replacing (36) with (43) in Theorem 6 qualities ρi and r proposed by (37) and (39) to be solutions to (33), (34) and (35) required in Theorem 7. The following confirms this point. Theorem 8. Suppose that there exist ωi ∈ K∞ (i ∈ n) such that (29) and (52) lim αi (s) ≥ lim (Id + ωi ) ◦ σi,j (s) s→∞

s→∞

hold for all j ∈ n\{i}, Let hi ∈ (0, 1) be arbitrarily fixed for all i ∈ n. Then (42) is satisfied. Moreover, all the assertions in Theorem 6 holds true for pi,j defined in (43). Proof: Obtaining (42) is straightforward from (52) and hi ∈ (0, 1) . The achievement of (29) implies ˆ ◦ G(z) ̸≥ A(z), ∀z ∈ Rn+ \ {0} D ˆ ˆ 1 (z1 ), z2 + ω ˆ 2 (z2 ), . . . , zn + ω ˆ n (zn )]T D(z) = [z1 + ω where ω ˆ i := (Id+hi ωi )−1 ◦(Id+ωi )−Id = (Id+hi ωi )−1 ◦ (1 − hi )ωi . Here, ω ˆ i is of class K∞ since ωi ∈ K∞ and h ∈ (0, 1). As in the case of Theorem 6, the existence of ωi ∈ K∞ (i ∈ n) satisfying (29) guarantees the existence of ρi ∈ K∞ (i ∈ n) and r ∈ K satisfying (33), (34) and (35) for Ω defined with (43) for given hi ∈ (0, 1). 

ρ′k ◦ ρ−1 k (s)

i∈n

Let ψ ∈ K be such that (45) α ˆ (s) ≥ ηˆ(s)ψ(s), ∀s ∈ R+ , α ˆ (s) lim inf = lim ψ(s). (46) s→∞ η s→∞ ˆ(s) Since ηˆ(s) > 0 holds for all s ∈ (0, ∞), there exists η ∈ P such that ηˆ(s) ≥ η(s), ∀s ∈ R+ , E := sup ηˆ(s) − η(s) < ∞. s∈R+

Indeed, E = ηˆ(0) can be achieved. Pick δ ∈ (0, 1), and let Y ∈ K such that (47) (1 − δ)ˆ η (s)ψ(s) ≥ η(s)Y (η(s)), ∀s ∈ R+ , (48) lim Y (s) ≥ lim κ(s). s→∞

η(V )κ(|r|) ≤ η(V )Y (η(V )) + Y −1 (κ(|r|))κ(|r|), it holds that (1 − δ)ˆ η (V )ψ(V ) + [η(V ) + E]κ(|r|) ≤ Y −1 (κ(|r|))κ(|r|) + Eκ(|r|).

for

α ˜ i (s) = hi ωi ◦ (Id + hi ωi )−1 ◦ αi (s), √ ∑ ′ ηˆ(s) = n ρi ◦ ρ−1 κ(s) = max κi (s). i (s), i∈n

ˆ (s)/ˆ η (s) = ∞, property Since (44) implies lim inf s→∞ α (46) guarantees lims→∞ ψ(s) = ∞. Hence, there exists Y ∈ K∞ achieving (47) and (48). Due to

s→∞

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Note that if there exist ωi ∈ K∞ (i ∈ n) such that (29) hold, property (7) guarantees the existence of another set of ωi ∈ K∞ (i ∈ n) satisfying (29) and (52) simultaneously. The functions pi,j are used to place the path Z(s) closer to the “center” of the set Ω, while rˆi s swing Z(s) in Ω. Remark 9. In principle, we can relax (44) further in Proposition 7. For this purpose, one can simply check if lim α ˜ (s) ≥ lim σ(s) (53) s→∞

s→∞

holds for the functions defined by (50) and (51) in the proof of Proposition 7. An explicit expression of (53) is more complex than (44). 5. AN EXAMPLE Consider the network described by

2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019

Hiroshi Ito / IFAC PapersOnLine 52-16 (2019) 544–549

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REFERENCES

Fig. 2. The level set W (V ) = 2 (the light blue cube with rounded edges) computed via J(2, V ) = 0 with (32), and the path Z(s) (red curve) for (54); The other three surface plots are gain bounds of two component systems. 5s6 α1 (s) = 3s6 , σ12 (s) = s3 , σ13 (s) = (54a) 1 + s3 α2 (s) = 4s2 , σ21 (s) = s4 , σ23 (s) = 2s2 (54b) 4 s , σ32 (s) = s2 . (54c) α3 (s) = 4(s2 + s4 ), σ31 (s) = 5 + s2 The small-gain condition (29) holds with ω1 (s) = ω2 (s) = ω3 (s) = s/2. Thereby, Theorem 6 gives a path Z(s) = [ζ1−1 , ζ2−1 , ζ3−1 ] as in (12) with (36). Pick K1 = K2 = K3 = 1. The inequality (26) is satisfied for s2 4s , rˆ3 (s) = s. (55) rˆ1 (s) = , rˆ2 (s) = 2 5(s + 1) A radius function r fulfilling (39) and (40) is s2 . (56) r(s) = 2(s2 + 10) Using ρi = ζi in (32), Proposition 5 qualifies the function VI (x) = W (V (x)) as a Lyapunov function establishing 0GAS of (2) specified by (54). In Fig.2, the level set of the constructed Lyapunov function is plotted on the V -plane for the level v = 2. Figure 2 illustrates that the path Z(s) lies inside the tube Ω surrounded by the three manifolds specified by the three rows of the equation L(V ) = 0 with respect to (36). The path penetrates the rounded-off corner of a cube in which r is the radius of the rounding-off arc. 6. CONCLUSIONS This paper has developed closed form expressions for nonlinear scalings which directly compose the max-separable Lyapunov function for ISS networks. In contrast to the popularity of the max-separable Lyapunov function, its closed form expression remained challenging. This paper has shown that a simple yet judicious combination of gain functions of component systems gives the desired scalings explicitly. The scalings in closed form can be tuned to swing the corresponding path within an appropriate range. It has been shown that the tuning in the scalings can be expressed explicity again with gain functions and smallgain margins. This paper has demonstrated that the tuning leads to a solution which is simply assumed in Ito and R¨ uffer (2019) to round off the corners of the max-separable Lyapunov functions for getting rid of the inherent nondifferentiability of switching the scalings. 936

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