Input-to-state stability and interconnections of discontinuousdynamical systemsCitation for published version (APA):Heemels, W. P. M. H., & Weiland, S. (2007). Input-to-state stability and interconnections of discontinuousdynamical systems. (DCT rapporten; Vol. 2007.104). Technische Universiteit Eindhoven.

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Download date: 18. Sep. 2020

DCT report 2007-104

Input-to-state stability and interconnections of

discontinuous dynamical systems

W.P.M.H. Heemels and S. WeilandEindhoven Univ. of Technology

P.O. Box 513, 5600 MB Eindhoven, The Netherlandsm.heemels@tue.nl and s.weiland@tue.nl

July 31, 2007

1

Abstract

In this paper we will extend the input-to-state stability (ISS) frame-work introduced by Sontag to continuous-time discontinuous dynamicalsystems adopting Filippov’s solution concept and using non-smooth ISSLyapunov functions. The main motivation for investigating non-smoothISS Lyapunov functions is the recent focus on “multiple Lyapunov func-tions” for which feasible computational schemes are available that haveproven useful in the stability theory for hybrid systems. This paper pro-poses an extension of the well-known Filippov’s solution concept, that isappropriate for ‘open’ systems so as to allow interconnections of hybridsystems. It is proven that the existence of a non-smooth ISS Lyapunovfunction for a discontinuous system implies ISS. In addition, a (smallgain) ISS interconnection theorem is derived for two discontinuous dy-namical systems that both admit a non-smooth ISS Lyapunov function.This result is constructive in the sense that an explicit ISS Lyapunovfunction for the interconnected system is given. It is shown how theseresults can be applied to piecewise linear (PWL) systems. In particu-lar, it is shown how piecewise quadratic ISS Lyapunov functions can beconstructed for PWL systems via linear matrix inequalities (LMIs). Thetheory will be illustrated by an example of an extended ‘flower’ system,which also demonstrates the computational machinery.

1 Introduction

The concept of input-to-state stability (ISS), see e.g. [24, 22, 25, 14, 13, 18], isinstrumental for the study of stability in various types of dynamical systems.Especially, for interconnected dynamical systems, input-to-state stability hasplayed an important role. Typically, the interconnection of two dynamical sys-tems which are ISS and satisfy a small gain condition, can be proven to beglobally asymptotically stable (GAS) or ISS with respect to the remaining ex-ternal signals (see e.g. [14, 13]). Stability results on interconnected systems areof key importance for understanding complexity issues in large-scale dynamicalsystems. Indeed, by decomposing a complex system into components with spe-cific (ISS) properties, one can infer properties of the complex system by usingqualities of the interconnection. Often, these results are to derive when the sys-tem would be studied in its complete form. Also in the context of observer-basedcontroller design, ISS interconnection results are very appealing as the closed-loop system is often considered as the interconnection of the estimation errordynamics on one hand and a state feedback controlled plant on the other. Theapproach of designing the observer and the state feedback controller separately(as in the linear case), and applying ISS-interconnection results to prove thestability of the closed loop system was used for Lipschitz continuous nonlinearsystems in e.g. [3, 2] and for continuous piecewise affine systems using smoothISS Lyapunov functions in [19].

Considering the recent attention for discontinuous and hybrid dynamical sys-tems, it is of interest to extend the ISS machinery to this class of systems. Forcontinuous-time systems that are in general discontinuous, like hybrid systems,

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the ISS interconnection theory of [14, 13] does not apply. The first reason thathampers the use of the results in [14, 13] is that discontinuous systems do nothave a Lipschitz continuous vector field. The second reason is that Lyapunovfunctions for hybrid systems are generally non-smooth, while the common ISSapproach (see e.g. [24, 22, 25, 13, 18]) considers smooth Lyapunov functions.Indeed, for hybrid systems the use of multiple Lyapunov functions is advocatedin e.g. [4] and [6] and many applications of multiple Lyapunov functions haveoriginated from these works. In particular for piecewise affine systems, piece-wise quadratic Lyapunov functions (see e.g. [15, 20, 7]) are popular as they canbe constructed from linear matrix inequalities (LMIs). The availability of abroad range of computational tools for the calculation of non-smooth Lyapunovfunctions motivates the development of an ISS theory for continuous-time dis-continuous systems by using non-smooth Lyapunov functions.

Stability theory for continuous-time hybrid systems using non-smooth Lya-punov functions is well developed (see e.g. [4, 6, 15, 20, 7] and the referencestherein). Typically, the solution trajectories are considered in the traditionalsense and not in a generalized sense such as the convex definition given in [8].We will adopt a framework for extended Filippov solutions in this paper. Tomotivate this choice, consider the discontinuous piecewise linear system

x(t) =

A1x(t), when x1(t) > 0A2x(t), when x1(t) 6 0

(1)

with

A1 =(−3 1−5 1

)and A2 =

(−3 −15 1

).

This system allows a continuous piecewise quadratic Lyapunov function of theform V (x) = x>P1x when x1 > 0 and V (x) = x>P2x when x1 6 0 with

P1 =(

3.9140 −2.0465−2.0465 1.5761

)and P2 =

(3.9140 2.04652.0465 1.5761

),

which is computed via the procedure outlined in [15]. The function V is con-tinuous over the switching plane x1 = 0. According to the results in [15] thisproves the exponential stability of the system along “ordinary” continuouslydifferentiable solutions (without sliding motions). In [15] it is already indicatedthat one has to take additional measures to include sliding modes (e.g. usingFilippov solutions) and this is actually demonstrated by system (1). The slidingmode dynamics at x1 = 0 is given by x2 = x2, which is unstable, in spite of thepresence of a piecewise quadratic Lyapunov function. This example indicatesthat generalized solutions require extensions of the standard stability condi-tions, as will be presented in the current paper. Especially, in the case of ISSthis is particularly important as external inputs can easily trigger a sliding mode(cf. the example of the extended flower system at the end of the paper). Theseminal book [8] and the paper [21] provide extensions of stability analysis fordiscontinuous dynamical systems using non-smooth Lipschitz continuous Lya-punov functions. We will extend this work towards ISS and ISS interconnection

3

results using a generalized Filippov’s solution concept. A first contribution ofthe present paper is to extend Filippov’s solution concept to open systems, i.e.,systems that allow for free input signals. This extension includes the classicalFilippov solution concept as a special case. In the next section we motivatewhy the classical Filippov solution concept is not adequate for interconnectionpurposes and we will prove several properties of the newly introduced solutionconcept.

Recently, the interest for the application of the concept of ISS within thecontext of hybrid systems is increasing, see e.g. [5, 27, 17] for continuous-timesystems. However, the emphasis is mainly on smooth Lyapunov functions, whichmight not have the computational advantages that non-smooth Lyapunov func-tions have. The only exception is formed by the work [27] that studies ISS forswitched systems using multiple Lyapunov functions and average dwell time as-sumptions. However, [27] differs from our work as we do not adopt an averagedwell-time assumption. Another motivation for the current paper is the paper[17], that actually advocates the application of the ISS framework for verifyingstability of hybrid systems. Our paper contributes to this research line andgeneralizes [17] by adopting the “multiple ISS Lyapunov” function approach.To summarize, this work extends [21] towards ISS and ISS interconnection the-ory for discontinuous dynamical systems, and expands the line of research in[5, 27, 17] towards the usage of non-smooth ISS Lyapunov functions (withoutdwell time assumptions).

Next to the introduction of extended Filippov solutions suitable for inter-connection purposes, the contributions of the paper are as follows. First, we willshow that the existence of a non-smooth (but continuous) ISS Lyapunov func-tion for a discontinuous system implies ISS. Secondly, we develop ISS theory forcontinuous-time discontinuous dynamical systems using non-smooth Lyapunovfunctions which leads to a general interconnection result for discontinuous dy-namical systems. Such a (small gain) interconnection result can be based en-tirely on the time-domain analysis of systems as in [14], which holds for generaldynamical systems (cf. [17]). We focus here on an alternative to this time-domain setting in that we explicitly construct a suitable ISS Lyapunov functionfor the interconnected system, from the ISS Lyapunov functions of the subsys-tems. This extends the counterpart for continuous systems as given in [13].As argued also in [17], having an (ISS) Lyapunov function provides additionalinsight into the behavior of a stable system and plays an important role in per-turbation analysis and estimating the region of attraction. This indicates, that aLyapunov formulation of a discontinuous small gain theorem is of independentinterest. As a third contribution, we will use these general results to obtaincomputational tools based on linear matrix inequalities (LMIs) to assess ISSand stability of PWA systems using Filippov solutions. This result generalizesthe well-known stability theorem for PWA systems in [15]. The effectivenessof the theory will be shown on an illustrative example obtained as a modifica-tion of the ‘flower system’ of [15]. Hereby, we also extend a preliminary workof the authors in [11] for which only autonomous interconnected systems wereconsidered with an observer-based control application using common quadratic

4

ISS Lyapunov functions.

Notation. R+ denotes all nonnegative real numbers. For a set Ω ⊆ Rn,cl Ω denotes its closure, intΩ denotes its interior and coΩ its closed convex hull.For two sets Ω1, Ω2, we define the set difference Ω1 \ Ω2 as x ∈ Ω1 | x 6∈ Ω2.A set Ω ⊆ Rn is called a polyhedron, if it is the intersection of a finite numberof open or closed half spaces. For a matrix A ∈ Rn×m we denote by A>

its transpose. For a positive semi-definite matrix A ∈ Rn×n, λmin(A) andλmax(A) will denote its minimal and maximal eigenvalue. The operator col(·, ·)stacks subsequent arguments into a column vector, e.g. for a ∈ Rn and b ∈ Rm

col(a, b) = (a>, b>)> ∈ Rn+m. A function u : R+ → Rn is piecewise continuous,if on every bounded interval the function has only a finite number of points atwhich it is discontinuous. Without loss of generality we will assume that everypiecewise continuous function u is right continuous, i.e. limt↓τ u(t) = u(τ) forall τ ∈ R+. A function x : [a, b] → Rn is called absolutely continuous, if x iscontinuous and there exists a function x in L1[a, b], the set of integrable functionson [a, b], such that x(t) = x(a) +

∫ t

ax(τ)dτ for all t ∈ [a, b]. The function x is

called the derivative of x on [a, b]. Note that an absolutely continuous function xis almost everywhere differentiable. With | · | we will denote the usual Euclideannorm for vectors in Rn, and ‖ · ‖ denotes the L∞ norm for time functions,i.e. ‖u‖ = supt∈R+

|u(t)| for a time function u : R+ → Rn. For two functionsf and g we denote by f g the composition (f g)(x) = f(g(x)). A functionγ : R+ → R+ is of class K if it is continuous, strictly increasing and γ(0) = 0.It is of class K∞ if, in addition, it is unbounded, i.e. γ(s) → ∞ as s → ∞. Afunction β : R+×R+ → R+ is of class KL if, for each fixed t ∈ R+, the functionβ(·, t) is of class K and for each fixed s ∈ R+, the function β(s, ·) is decreasingand tends to zero at infinity. A function γ : R+ → R+ is called positive definite,if γ(s) > 0, when s > 0. For a real-valued, differentiable function V : Rn 7→ R,∇V denotes its gradient. For a set-valued function F from Rn to Rm, we use thenotation F : Rn → Rm to indicate that F(x) is a subset of Rm for all x ∈ Rn.

2 Solution concepts and interconnections of dis-continuous dynamical systems

Consider the differential equation with discontinuous right-hand side of the form

x(t) = f(x(t), u(t)) (2)

with x(t) ∈ Rn and u(t) ∈ Rm, the state and control input at time t ∈ R+,respectively. The vector field f is assumed to be a piecewise continuous functionfrom Rn × Rm to Rn in the sense that

f(x, u) = fi(x, u) when col(x, u) ∈ Ωi, i = 1, 2, . . . , N. (3)

Here, Ω1, . . . , ΩN are closed subsets of Rn × Rm that form a partitioning ofthe space Rn × Rm in the sense that intΩi ∩ intΩj = ∅, when i 6= j and

5

⋃Ni=1 Ωi = Rn×Rm. The functions fi : Ωi → Rn are locally Lipschitz continuous

on their domains Ωi (this means including the boundary).

2.1 Filippov’s convex solution concept

Since the system (2) has a discontinuous right-hand side, a generalized solutionconcept will be used. The most commonly used solution concept for (2) is Fil-ippov’s convex definition [8, p. 50]. However, as Filippov’s solution is intendedfor ‘closed’ systems (without external inputs) it is not immediately suitablefor interconnection purposes. Indeed, Filippov considered systems of the formx = f(x, t), and defined their solutions as solutions of the differential inclusion

x(t) ∈ Ff (x(t), t) :=⋂ε>0

⋂

M⊆Rn,µ(M)=0

co f(Bε(x(t)) \M, t), (4)

where Bε(x) is the open ball of radius ε around x and⋂M⊆Rn,µ(M)=0 indicates

the intersection over all sets M of Lebesgue measure 0. Loosely speaking, thismeans that the set Ff (x, t) is defined as the convex hull of all limit pointslimk→∞ f(xk, t) for sequences xkk∈N with xk → x when k →∞ and (xk, t) 6∈D, where D is the set of discontinuity points of f . Applied to (2), this meansthat for any fixed input u : R+ → Rm a solution of (2) is a solution of

x(t) ∈ Ff (x(t), u(t)) (5)

withFf (x, u) :=

⋂ε>0

⋂

M⊆Rn,µ(M)=0

co f(Bε(x) \M, u). (6)

The mapping f 7→ Ff can be seen as an operator that maps functions f :Rn × Rm → Rn into set-valued functions Ff : Rn × Rm → Rn.

If we interconnect the system (2) with a second system (a controller) thatgenerates the input signal u in (2), then this second system has the explicitpurpose to restrict the solution set of state trajectories of the open system (2)to a specific subset. Interconnection of systems is then merely synonymousto intersection of solution sets, a point of view that has been advocated inthe behavioral framework for several years. Nevertheless, the Filippov solutionconcept for the interconnection of open systems of the form (2) refrains fromhaving this property, as shown in the following example.

Example 2.1 Consider a scalar-input scalar-state system consisting of fourdifferent mode dynamics xa(t) = fa(xa(t), u(t)) = fa

i (xa(t), u(t)) with corre-sponding closed regions Ωi ⊂ R2, i = 1, 2, 3, 4 as depicted in Figure 1. ApplyingFilippov’s convex definition yields the differential inclusion (5) with Ffa(0, 0) =cofa

2 (0, 0), fa4 (0, 0).

Suppose that the input u is generated by the system xb = f b(xa, xb) := xa

and u = xb (i.e., an integrator). Then the interconnection is an autonomous

6

¡¡

¡¡

¡¡

¡¡¡

@@

@@

@@

@@@

Ω3

Ω4

Ω1

Ω2

u

x

Figure 1: Partitioning of state-input space

system with state variable x = col(xa, xb). Applying Filippov’s solution conceptto the interconnected system

(xa

xb

)= f(xa, xb) :=

(fa(xa, xb)f b(xa, xb)

)(7)

yields col(xa, xb) ∈ Ff (xa, xb) with

Ff (0, 0) 6⊂ Ffa(0, 0)×Ffb(0, 0)

as Ffb(0, 0) = 0 and

Ff (0, 0) = cofa1 (0, 0), fa

2 (0, 0), fa3 (0, 0), fa

4 (0, 0) × 0.

Hence, the dynamics of the xa-variable depends on all 4 original modes in thecontrolled system, whereas it only depends on the modes 2 and 4 in the uncon-trolled system. In particular, the vector field (set) in the origin of the controlledsystem is not necessarily a restriction of the vector field at the same point inthe uncontrolled system.

This example shows that Filippov’s convex definition for systems with in-puts can be unsatisfactory if no prior knowledge is assumed on input variables.Since we aim at deriving general interconnection conditions based on local (ISS)properties of the individual systems, we will need a generalization of Filippov’ssolution concept.

2.2 Interconnecting discontinuous dynamical systems

In this paper, we focus on interconnections of dynamical systems as in (7) ofExample 2.1. More generally, we consider interconnected systems of the type

Σa : xa = fa(xa, xb, ua) = faia

(xa, xb, ua)

when col(xa, xb, ua) ∈ Ωaia

for ia = 1, . . . , Na (8a)

Σb : xb = fb(xa, xb, ub) = fbib

(xa, xb, ub)

when col(xa, xb, ub) ∈ Ωbib

for ib = 1, . . . , Nb (8b)

7

with xa(t) ∈ Rna and xb(t) ∈ Rnb the state at time t of subsystem Σa andΣb, respectively, and where col(xb(t), ua(t)) ∈ Rnb+ma and col(xa(t), ub(t)) ∈Rna+mb are the external inputs at time t for subsystem Σa and Σb, respectively.The collections Ωa

1 , . . . , ΩaNa and Ωb

1, . . . , ΩbNb consist of closed sets that

form partitionings of Rna+nb+ma and Rna+nb+mb , respectively, as in (2).

Σa Σb- ¾ ¾¾

xa

ubxb

xa

- -

xbua

Figure 2: System interconnections

The interconnection of Σa and Σb is illustrated in Figure 2. The statevariable xa of Σa is input to subsystem Σb and the state variable xb of Σb is inputto subsystem Σa. The signals ua and ub remain external inputs with respect tothe interconnected system, which we denote by Σ. The overall system in thecombined state variable x = col(xa, xb) ∈ Rn with n = na + nb and externalsignal u = col(ua, ub) ∈ Rm with m = ma + mb is then given by

x = f(x, u) = f(ia,ib)(x, u) =

= col(faia

(xa, xb, ua), f bib

(xa, xb, ub))when col(x, u) ∈ Ωia,ib

, (9)

where

Ωia,ib:= col(x, u) ∈ Rn+m | col(xa, xb, ua) ∈ Ωa

ia

and col(xa, xb, ub) ∈ Ωbib (10)

for each pair (ia, ib) ∈ 1, . . . , Na × 1, . . . , N b. Hence, we have at mostN := NaNb regions Ωia,ib

in Rn+m. Observe that the sets Ωia,ib, ia = 1, . . . , Na,

ib = 1, . . . , Nb are closed, satisfy intΩi1a,i1b∩ intΩi2a,i2b

= ∅ when (i1a, i1b) 6=(i2a, i2b) and that their union is equal to Rn+m. Hence, the sets Ωia,ib

, ia =1, . . . , Na, ib = 1, . . . , Nb form a partitioning of the combined state/input spaceRn+m and, as such, the interconnected system falls within the class of systemscorresponding to (2).

Example 2.1 illustrated that Ff (x, u) ⊆ Ffa(xa, xb, ua)×Ffb(xa, xb, ub) does

not hold in general for Filippov’s solution concept. Such an interconnectionrelation for the solution concept is desirable as it enables the derivation ofproperties of the interconnection Σ from properties of the subsystems Σa andΣb. Therefore, we propose an extended solution concept1 that generalizes the

1In literature also other proposals for solution concepts of discontinuous dynamical systemswere made, for instance the control equivalent definition in [26] and the general definition in[1]. These definitions were used for interconnections of smooth systems with controllers that

8

well-known Filippov solutions and satisfies this interconnection relation. Thenew solution concept will replace the differential equation (2) by the differentialinclusion

x(t) ∈ Ef (x(t), u(t)) (11)

whereEf (x, u) :=

⋂ε>0

⋂

M⊆Rn+m,µ(M)=0

co f(Bε(x, u) \ M). (12)

Here, Bε(x, u) is a ball of radius ε around col(x, u) in Rn+m and the set valuedmap f(B) needs to be read as f(B) = f(x, u) | (x, u) ∈ B for any B ⊂Rn+m. Loosely speaking, the set Ef (x, u) is therefore defined as the convexhull of all limit points limk→∞ f(xk, uk) for all sequences col(xk, uk)k∈N withcol(xk, uk) → col(x, u) when k → ∞ and (xk, uk) 6∈ D, where D is the set ofdiscontinuity points of f . We can make the latter more precise by introducingthe mapping f 7→ Cf where Cf : Rn × Rm → Rn is the set valued map

Cf (x, u) := cofi(x, u) | i ∈ I(x, u) (13)

and whereI(x, u) := i ∈ 1, . . . , N | col(x, u) ∈ Ωi (14)

is the index set indicating the regions Ωi to which the state-input vector col(x, u)belongs. It is then immediate (from the closedness of the regions) that Ef (x, u) ⊆Cf (x, u) for all (x, u) ∈ Rn+m. We emphasize that in general Ef (x, u) 6⊇ Cf (x, u).If there exist points (x, u) for which Ef (x, u) 6= Cf (x, u) then this is due to thefact that (x, u) belongs to a region Ωi ⊂ Rn+m of Lebesgue measure 0. Asufficient condition for the absence of such points is that Ωi is not (includedin) a lower-dimensional manifold of Lebesgue measure 0 and hence, fi is notdiscarded in the extended Filippov solution by removing sets M of measurezero in (12). We will say that the system (2) has non-degenerate regions in thatcase and will assume this in the sequel.

Assumption 2.2 The system (2) has non-degenerate regions in the sense thatthe regions Ω1, . . . , ΩN form a partitioning of Rn × Rm that consists of closedsubsets of Rn ×Rm with the property that cl(int(Ωi)) = Ωi for all i = 1, . . . , N .

As noted, when this assumption does not hold, we only have Ef (x, u) ⊆Cf (x, u) instead of equality. However, by removing the measure zero (parts of)regions one can obtain a partitioning that is non-degenerate so that the systemstill allows the same extended Filippov solutions. Hence, under Assumption 2.2we can consider, instead of (11), the differential inclusion

x(t) ∈ Cf (x(t), u(t)). (15)

are switched static state feedbacks and as such do not fit the general framework consideredhere. Also for a subclass of discontinuous dynamical systems consisting of piecewise linearsystems an extended Caratheodory solution was proposed in [12], which does not allow forsliding modes. Hence, also this proposal is not general enough for our purposes. See [9] formore details on solution concepts.

9

Definition 2.3 A function x : [a, b] 7→ Rn is an extended Filippov solution to(2) for the piecewise continuous input function u : [a, b] 7→ Rm, if it is a solutionto (15) for the input u, in the sense that x is absolutely continuous and satisfiesx(t) ∈ Cf (x(t), u(t)) for almost all t ∈ [a, b].

Under the conditions given here, it can be shown (by using § 2.6 (page 69)in [8]) that the mappings (x, u) 7→ Ef (x, u) and (x, u) 7→ Cf (x, u) are uppersemicontinuous on Rn × Rm. As such, the multi-valued mappings that assignEf (x, u(t)) or Cf (x, u(t)) to a point (t, x) as in (12) for any bounded piecewisecontinuous function u is upper semicontinuous in (t, x) on T × Rn, where Tindicates any interval where u is continuous. As Ef (x, u) and Cf (x, u) are non-empty, bounded, convex and closed for any col(x, u) ∈ Rn+m, local existence ofsolutions to (2) (interpreted either via (11) or (15)) given an initial conditionx(t0) = x0 and a piecewise continuous input function is guaranteed from Theo-rem 1, page 77 in [8]. To obtain also uniqueness, additional conditions have tobe imposed on (2), see e.g. § 10 in [8].

We present a number of properties of the vector field sets introduced so farin the next two theorems. The proofs can be found in the appendix.

Theorem 2.4 Consider system (2).

1. Ff (x, u) ⊆ Cf (x, u) and Ef (x, u) ⊆ Cf (x, u) for all x ∈ Rn and u ∈ Rm.

2. In case the system (2) has non-degenerate regions, i.e. Assumption 2.2holds, then Ff (x, u) ⊆ Ef (x, u) = Cf (x, u) for all x ∈ Rn and u ∈ Rm.

3. If the regions Ωi, i = 1, . . . , N of system (2) are non-degenerate and onlystate dependent, i.e. Ωi = Ωx

i × Rm, i = 1, . . . , N with Ωxi ⊆ Rn, then

Ff (x, u) = Ef (x, u) = Cf (x, u) for all x ∈ Rn and u ∈ Rm.

Hence, in general only the inclusions Ff (x, u) ⊆ Cf (x, u) and Ef (x, u) ⊆Cf (x, u) hold, while under the non-degeneracy assumption we also obtain theadditional inclusions Ff (x, u) ⊆ Ef (x, u) and Cf (x, u) ⊆ Ef (x, u). In case thenon-degeneracy assumption does not hold, it is not difficult to find counterex-amples for the latter two inclusions (take regions depending on the input uonly and include a region of measure zero). Even under the non-degeneracyassumption, the inclusions Ef (x, u) ⊆ Ff (x, u) and Cf (x, u) ⊆ Ff (x, u) do notgenerally hold, as shown by Example 2.1. However, if the switching is onlystate-dependent, then all vector field sets coincide.

Theorem 2.5 Suppose that the component systems (8a) and (8b) satisfy As-sumption 2.2.

1. The interconnection (9) of (8a) and (8b) satisfies

Cf (x, u) = Cfa(xa, xb, ua)× Cfb(xa, xb, ub) (16)

for all points col(x, u) ∈ Rn+m.

10

2. The interconnection (9) of (8a) and (8b) satisfies

Ef (x, u) ⊆ Efa(xa, xb, ua)× Efb(xa, xb, ub) (17)

for all points col(x, u) ∈ Rn+m.

3. If the interconnection (9) is autonomous, then

Ff (xa, xb) = Efa(xa, xb)× Efb(xa, xb)= Cfa(xa, xb)× Cfb(xa, xb). (18)

In view of Example 2.1, statements (1) and (2) of the above theorem es-tablish the desirable interconnection relation that at any point the vector fieldset of an interconnection is a subset of the product of the vector field sets ofthe component systems, a relation that does not hold for Filippov solutions asshown by Example 2.1. Statement (3) expresses that the proposed extensionof Filippov solutions is the smallest possible that contains all ordinary Filip-pov solutions of the interconnection. This is also illustrated by Example 2.1as Cfa(0, 0) = cofa

1 (0, 0), fa2 (0, 0), fa

3 (0, 0), fa4 (0, 0). Statement (1) of Theo-

rem 2.4 shows that an ordinary Filippov solution is also an extended Filippovsolution.

The regions of the interconnected system may be degenerate even when theregions of the component systems (8a) and (8b) are non-degenerate. Hence,Ff (x, u) ⊆ Ef (x, u) may not be true for some points col(x, u). Therefore weprefer to use x ∈ Cf (x, u) as Theorem 2.4, statement (1) shows that we still havethat Ff (x, u) ⊆ Cf (x, u) and Ef (x, u) ⊆ Cf (x, u). This means that propertiesproven for extended Filippov solutions in the sense of (15) transfer automaticallyto all ordinary Filippov solutions (solutions to (5)) and all solutions to (11) forthe interconnected system (9).

3 ISS for discontinuous dynamical systems

Given the possible non-uniqueness of Filippov solution trajectories, we definethe concept of input-to-state stability [22, 14, 13] for (2) as follows.

Definition 3.1 The system (2) is said to be input-to-state stable (ISS) if thereexist a function β of class KL and a function γ of class K such that for eachinitial condition x(0) = x0 and each piecewise continuous bounded input functionu defined on [0,∞),

• all corresponding Filippov solutions x of the system (2) exist on [0,∞)and,

• all corresponding Filippov solutions satisfy

|x(t)| 6 β(|x0|, t) + γ(‖u‖), ∀t > 0. (19)

11

In the study of hybrid systems often non-smooth or multiple Lyapunov func-tions are employed, see for instance [6, 4, 15, 20]. As such, we will considercontinuous Lyapunov functions that are composed of “multiple” Lyapunov func-tions Vj as

V (x) = Vj(x) when x ∈ Γj , j = 1, . . . , M, (20)

where Γ1, . . . , ΓM are closed subsets of Rn that form a partitioning of the spaceRn, i.e. int Γi ∩ int Γj = ∅, when i 6= j and

⋃Ni=1 Γi = Rn. For each j we

assume that Vj is a continuously differentiable function on some open domaincontaining Γj . Continuity of V implies that Vi(x) = Vj(x) when x ∈ Γi ∩ Γj .The continuity of the Lyapunov function is a typical condition used in the studyof stability for piecewise affine systems [15] in continuous-time. In discrete-timethe assumption on continuity of V can be dropped as proven in [7, 16]. Similarly,as in (14), we define the index set J(x) as

J(x) := j ∈ 1, . . . , M | x ∈ Γj. (21)

Definition 3.2 A function V of the form (20) is said to be an ISS-Lyapunovfunction for the system (2) if:

• V is continuous,

• there exist functions ψ1, ψ2 of class K∞ such that:

ψ1(|x|) 6 V (x) 6 ψ2(|x|), ∀x ∈ Rn, (22)

• there exist functions χ of class K and α positive definite and continuoussuch that for all x ∈ Rn and u ∈ Rm the implication

|x| > χ(|u|) ⇒ for all i ∈ I(x, u), j ∈ J(x)∇Vj(x)fi(x, u) 6 −α(V (x)) (23)

holds, or stated differently, for all x ∈ Rn and u ∈ Rm

|x| > χ(|u|), col(x, u) ∈ Ωi and x ∈ Γj ⇒∇Vj(x)fi(x, u) 6 −α(V (x)). (24)

Definition 3.2 is similar to the one proposed in [22, 14, 13], the only differencebeing that here we use non-smooth Lyapunov functions and that it is usedfor systems (2) in which the vector field might be discontinuous. To give aninterpretation of the condition (24) consider the case of an autonomous system(2) with state-dependent switching only, i.e. Ωi = Ωx

i × Rm, i = 1, . . . , N withΩx

i ⊆ Rn. In this case it is common practice to select the regions of the ISSLyapunov function Γj equal to the regions of the system Ωx

j , j = 1, . . . , M andM = N . In this case (24) becomes

|x| > χ(|u|), x ∈ Ωxi ∩ Ωx

j ⇒∇Vj(x)fi(x, u) 6 −α(V (x)). (25)

12

In this case, we observe that for i = j we have the common condition that the(ISS) Lyapunov condition Vi should decrease in the region where it is applied.In the case of absence of external inputs u, the conditions for i = j, togetherwith continuity of the Lyapunov functions over the boundary, are sufficient forguaranteeing global asymptotic stability (GAS), see e.g. [4, 15] along “ordinary”piecewise continuously differentiable solutions (without sliding modes). Here,we extend the framework in [4, 15] in two ways, (i) we consider ISS insteadof GAS and (ii) we considering extended Filippov solutions. The conditions(25) for i 6= j will be needed to accommodate for possible sliding modes. Theconditions for i = j were satisfied by the example (1) and the indicated piecewisequadratic Lyapunov function in the introduction. However, the conditions fori 6= j are violated, causing unstable sliding mode behavior.

To prove that the conditions in (24) can guarantee ISS under extended Fil-ippov solutions, we first derive conditions on the time derivative of an ISS Lya-punov V of the form (20) along extended Filippov solutions x of system (2)provided dV

dt (x(t)) and x(t) exist at time t. The complications are that a solu-tion trajectory might go along a surface on which ∇V does not exist and thatsolutions are of an extended Filippov type.

Theorem 3.3 If there exists an ISS Lyapunov function V of the form (20) forsystem (2) in the sense of Definition 3.2, then for any piecewise continuousbounded input function u : R+ → Rm and corresponding extended Filippovsolutions it holds that

d

dtV (x(t)) 6 −α(V (x(t)) (26)

at times t, where both dVdt (x(t)) and x(t) exist and |x(t)| > χ(|u(t)|).

Proof. We start by observing that since V is continuous and is com-posed of continuously differentiable functions (which are consequently, all lo-cally Lipschitz continuous), it follows that V is a locally Lipschitz continuousfunction. Suppose now that at time t both x(t) and dV (x(t))

dt exist and thatx(t) = y ∈ Cf (x(t), u(t)). We then obtain that

d

dtV (x(t)) = lim

h→0

V (x(t + h))− V (x(t))

h=

= limh↓0

V (x(t) + hy + g(h))− V (x(t))

h=

= limh↓0

V (x(t) + hy)− V (x(t))

h+

+ limh↓0

V (x(t) + hy + g(h))− V (x(t) + hy)

h=

= limh↓0

V (x(t) + hy)− V (x(t))

h(27)

where g : R 7→ Rn is a function that satisfies limh→0|g(h)|

h = 0. Note that inthe 4th equality we used

limh→0

V (x(t) + hy + g(h))− V (x(t) + hy)h

= 0

13

due to local Lipschitz continuity of V and limh→0|g(h)|

h = 0. Due to (14), y canbe written as

y =∑

i∈I(x(t),u(t))

αifi(x(t), u(t)) (28)

for some αi > 0, i ∈ I(x(t), u(t)) and∑

i∈I(x(t),u(t)) αi = 1. As x(t) ∈ Γj

iff j ∈ J(x(t)) and V is continuous we have that V (x(t)) = Vj(x(t)) for allj ∈ J(x(t)). To evaluate the right-hand side of (27), we have to realize thatV (x(t)+hy) = Vj(x(t)+hy) for all j ∈ J(x(t)+hy) in which the set J(x(t)+hy)depends on h. Since dV

dt (x(t)) exists, this means that

d

dtV (x(t)) = lim

h↓0Vj(x(t) + hy)− Vj(x(t))

h,

for all j ∈ J(x(t), y) :=⋂

h0>0

⋃0<h<h0

J(x(t) + hy). Due to closedness of Γj ,it holds that d(x(t), Γj) := infz∈Γj

|z − x(t)| > 0, when j 6∈ J(x(t)). Hence, forsufficiently small h, we have that x(t) + hy ∈ ⋃

j∈J(x(t)) Γj and thus J(x(t) +hy) ⊆ J(x(t)) for sufficiently small h and thus J(x(t), y)) ⊆ J(x(t)). Hence, wecan conclude that for x(t) with |x(t)| > χ(|u(t)|) that2

d

dtV (x(t)) 6 max

j∈J(x(t))limh→0

Vj(x(t) + hy)− Vj(x(t))h

=

= maxj∈J(x(t))

∇Vj(x(t))y(28)=

(28)= max

j∈J(x(t))

∑

i∈I(x(t),u(t))

αi∇Vj(x(t))fi(x(t), u(t)) 6

6 maxi∈I(x(t),u(t)),j∈J(x(t))

∇Vj(x(t))fi(x(t), u(t)) 6

6 −α(V (x(t))). (29)

Using the above theorem we can now prove that the existence of an ISSLyapunov function implies ISS of the system.

Theorem 3.4 If there exists an ISS Lyapunov function V of the form (20) forsystem (2) in the sense of Definition 3.2, then system (2) is ISS. Moreover, anexplicit expression for γ as in Definition 3.1 is

γ := ψ−11 ψ2 χ. (30)

Proof Consider initial condition x(0) = x0 and let u be a piecewise contin-uous bounded input function. Let x denote a corresponding extended Filip-pov solution (which might be non-unique) to (2). Define the set S := x ∈

2The right-hand side of the first inequality forms an upperbound on the upper Dini deriva-

tive of V at x in the direction of y, which is defined as V +(x, y) := lim suph↓0V (x+hy)−V (x)

hand equal to maxj∈J(x,y)∇Vj(x)y. Dini derivatives could also have been used. However,

as t 7→ V (x(t)) is differentiable almost everywhere, we use here the normal derivative of Vinstead of the Dini derivative.

14

Rn | V (x) 6 c with c := ψ2(χ(‖u‖)). Note that when x(t) 6∈ S, thenψ2(|x(t)|) > V (x(t)) > ψ2(χ(‖u‖)), which implies |x(t)| > χ(|u(t)|). Accordingto Theorem 3.3, inequality (26) holds for x(t) 6∈ S (provided x(t) and d

dtV (x(t))exist). We prove the following claim.

Claim: S is positively invariant, i.e. if there exists a t0 such that x(t0) ∈ S,then x(t) ∈ S for all t > t0.

Indeed, suppose this statement is not true. Due to closedness of S (continuityof V ) there is an ε > 0 and time t > t0 with V (x(t)) > c + ε. Let t∗ := inft >t0 | V (x(t)) > c + ε and t∗ := supt0 6 t 6 t∗ | V (x(t)) 6 c. Note thatt0 6 t∗ < t∗ and V (x(t∗)) = c by continuity of V and x. Since x(t) 6∈ S fort ∈ (t∗, t∗) (26) holds if both x(t) and dV (x(t))

dt exist. Since V is locally Lipschitzcontinuous and any solution to (11) is absolutely continuous, the compositefunction t 7→ V (x(t)) is absolutely continuous and consequently, t 7→ V (x(t)) isdifferentiable almost everywhere (a.e.) with respect to time t, and x(t) existsalso a.e. Therefore,

V (x(t∗))− V (x(t∗)) =∫ t∗

t∗

dV (x(τ))dt

dτ 6

6∫ t∗

t∗−α(V (x(τ)))dτ 6 0.

Hence, V (x(t∗)) 6 V (x(t∗)) = c, thereby contradicting that V (x(t∗)) > c + ε.This proves the claim.

Now let t1 = inft > 0 | x(t) ∈ S 6 ∞ (note that t1 might be infinity).Then it follows from the above reasoning and (22) that ψ1(|x(t)|) 6 V (x(t)) 6c := ψ2(χ(‖u‖)) for all t > t1. Hence,

|x(t)| 6 γ(‖u‖) for all t > t1 (31)

with γ := ψ−11 ψ2 χ a K-function. For t < t1, x(t) 6∈ S and consequently, (26)

holds almost everywhere in [0, t1). This yields ddtV (x(t)) 6 −α(V (x)) a.e. in

[0, t1). Lemma 4.4. in [18] now gives that there exists a KL function β (onlydepending on α) such that V (x(t)) 6 β(V (x0), t)) for t 6 t1. Hence,

|x(t)| 6 β(|x0|, t) for all t 6 t1, (32)

where β(r, t) := ψ−11 (β(ψ2(r), t)) is a KL function as well. Combining (31) and

(32) yields (19) for this particular trajectory. Global existence of any trajectorycan also be proven via Theorem 2 page 78 [8] by using the bound (19) (thatshows that there cannot be “finite escape times.”) As β and γ do not rely onthe particular initial state nor on the input u, this proves ISS of the system.

Remark 3.5 The proof follows similar lines as the proof of [25, Lemma 2.14]with the necessary adaptations for the non-smoothness of V and the discontinu-ity of the dynamics using Theorem 3.3.

15

As a corollary, we can obtain a stability (GAS) result, if we consider theautonomous variant of the discontinuous system (2) given by

x(t) = f(x(t)) = fi(x(t)) when x(t) ∈ Ωi ⊆ Rn (33)

with a similar generalization in terms of a differential inclusion

x(t) ∈ Cf (x(t)). (34)

We assume that 0 is an equilibrium of (33) (or equivalently (34)), which meansthat fi(0) = 0 for all i ∈ I(0) and thus F (0) = 0. According to Theorem 2.5we have that Ff (x) = Ef (x) = Cf (x), which implies that classical Filippov andextended Filippov solutions coincide for this system.

Definition 3.6 The system (33) is said to be globally asymptotically stable(GAS), if there exists a function β of class KL such that for each x0 ∈ Rn,all Filippov solutions x of the system (2) with initial condition x(0) = x0 existon [0,∞) and satisfy:

|x(t)| 6 β(|x(0)|, t), ∀t > 0. (35)

As a corollary of Theorem 3.4 we obtain the following result.

Theorem 3.7 Consider the discontinuous dynamical system (33) and a Lips-chitz continuous function V of the form (20). Assume that

• there exist functions ψ1, ψ2 of class K∞ such that:

ψ1(|x|) 6 V (x) 6 ψ2(|x|), ∀x ∈ Rn

• there exists a continuous positive definite function α such that

∇Vj(x)fi(x) 6 −α(V (x)) when x ∈ Ωi ∩ Γj . (36)

Then the discontinuous dynamical system (33) is GAS.

4 Interconnection result

Consider the system Σ obtained by the interconnection of Σa in (8a) and Σb

in (8b) as given by (8) or (9). For this interconnected system Σ we wouldlike to derive input-to-state stability with state x = col(xa, xb) and input u =col(ua, ub) from input-to-state stability conditions on the subsystems Σa andΣb.

Theorem 4.1 Suppose that there exist ISS Lyapunov functions V a and V b ofthe form (20) for the systems (8a) and (8b), respectively, that satisfy:

16

• There exist functions ψa1 , ψa

2 , ψb1, ψ

b2 ∈ K∞ such that

ψa1 (|xa|) 6 V a(xa) 6 ψa

2 (|xa|) and

ψb1(|xb|) 6 V b(xb) 6 ψb

2(|xb|). (37)

• There exist functions αa positive definite and continuous, χa ∈ K∞ andγa ∈ K with

|xa| > max(χa(|xb|), γa(|ua|) implying∇V a

ja(xa)fa

ia(xa, xb, ua) 6 −αa(V a(xa)) (38)

for all ia ∈ Ia(xa, xb, ua) and all ja ∈ Ja(xa), where Ja(xa) denotes theindex set corresponding to the partitioning Γa

1 , . . . , ΓaMa of V a as in (21).

• There exist functions αb positive definite and continuous, χb ∈ K∞ andγb ∈ K with

|xb| > max(χb(|xa|), γb(|ub|) implying

∇V bjb

(xb)f bib

(xa, xb, ua) 6 −αb(V b(xb)) (39)

for all ib ∈ Ib(xa, xb, ub) and all jb ∈ Jb(xb), where Jb(xb) denotes theindex set corresponding to the partitioning Γb

1, . . . , ΓbMb of V b as in (21).

Defineχa := ψa

2 χa [ψb1]−1, χb := ψb

2 χb [ψa1 ]−1

and assume that the coupling condition

χa χb(r) < r (40)

holds for all r > 0. Then the interconnected system (9) is input-to-state stablewith state x = col(xa, xb) and input u = col(ua, ub).

Proof Due to the coupling condition it holds that χb(r) < [χa]−1(r) forr > 0. According to Lemma A.1 in [13] there exists a K∞-function σ, which iscontinuously differentiable and satisfies χb(r) < σ(r) < [χa]−1(r) for all r > 0and σ′(r) > 0 for all r > 0 (thus the derivative σ′ is positive definite andcontinuous).

Define the Lipschitz continuous function V similar as in [13] with x =col(xa, xb)

V (x) = maxσ(V a(xa)), V b(xb). (41)

This function will be proven to be an ISS Lyapunov function for (8) in the senseof Definition 3.2. The function V is in the form (20) with a partitioning inducedby the partitioning Γa

1 , . . . , ΓaMa of V a, the partitioning Γb

1, . . . , ΓbMb of V b

and the additional split given by σ(V a(xa)) > V b(xb) or σ(V a(xa)) 6 V b(xb).Hence, we can define regions in the combined state space Rn as

Γja,jb,p := x = col(xa, xb) | xa ∈ Γaja

, xb ∈ Γbjb

and

(−1)p[σ(V a(xa))− V b(xb)] > 0 (42)

17

with ja = 1, . . . , Ma, jb = 1, . . . , Mb and p = 0, 1.Hence, V (x) = V(ja,jb,0)(x) := σ(V a

ja(xa)), when x ∈ Γja,jb,0 and V (x) =

V(ja,jb,1)(x) := V bjb

(xb), when x ∈ Γja,jb,1. Note that there is a slight abuse ofnotation as we characterize (21) using “indices” consisting of triples (ja, jb, p).The function V is of the form (20) as the closed sets Γja,jb,p, ja = 1, . . . , Ma,jb = 1, . . . , Mb and p = 0, 1 form a partitioning of the state space Rn.

We will now show that V is an ISS Lyapunov function in the sense of Defi-nition 3.2 for the system (9)

Note thatmax(σ(ψa

1 (|xa|)), ψb1(|xb|))6V (x)6 max(σ(ψa

2 (|x|)), ψb2(|x|)). (43)

As either |x|2 = |xa|2 + |xb|2 6 2|xa|2 or |x|2 6 2|xb|2, we obtain that

max(σ(ψa1 (|xa|)), ψb

1(|xb|)) > 12[σ(ψa

1 (|xa|)) + ψb1(|xb|)] >

> 12

min[σ(ψa1 (

12

√2|x|)), ψb

1(12

√2|x|)].

Since the minimum of two K∞-functions is also a K∞-function, we obtain thatV is lower bounded by K∞-functions. Since the right-hand side of (43) providesan upper bound, this proves the first condition of Definition 3.2.

Next we show that the second hypothesis of Definition 3.2 is satisfied for theK-function χ given by

χ(s) =√

2max(γa(s), γb(s), [χa]−1 γa(s), [χb]−1 γb(s)) (44)

To verify (24), suppose col(xa, xb, ua, ub) ∈ Ωia,ib, x ∈ Γja,jb,p and |x| >

χ(|u|). We consider two cases being to p = 0 and p = 1.Case 1 (p = 0) As p = 0, it holds that V b(xb) 6 σ(V a(xa)) and thus V (x) =σ(V a(xa)). Using the properties of σ and the definition of χa yields

ψb1(|xb|) 6 V b(xb) 6 σ(V a(xa)) < [χa]−1(V a(xa)) 6

6 [χa]−1 ψa2 (|xa|) 6 ψb

1 [χa]−1(|xa|).This implies that |xa| > χa(|xb|). As above, we have that either 2|xa|2 > |x|2

or 2|xb|2 > |x|2. Hence, we can distinguish two subcases:

• In the first case this means that

|xa| > 12

√2|x| > 1

2

√2χ(|u|)

(44)

> γa(|x|) > γa(|xa|).

• In case 2|xb|2 > |x|2, we have from |xa| > χa(|xb|)|xa| > χa(|xb|) > χa(

1

2

√2|x|) > χa(

1

2

√2χ(|u|))

(44)

> γa(|u|).

Hence, this yields |xa| > max(χa(|xb|), γa(|u|)) and thus (38) for subsystem (8a)can be used to arrive at

∇V(ja,jb,0)(xa, xb)f(ia,ib)(x, u) == σ′(V a

ja(xa))∇V a

ja(xa)fa

ia(xa, xb, ua) 6

6 −σ′(V aja

(xa))αa(V aja

(xa)) = −αa(V (x)), (45)

18

if we set αa(r) := σ′(σ−1(r))αa(σ−1(r)), which is a positive definite and con-tinuous function.Case 2 (p = 1) Since V b(xb) > σ(V a(xa)), we have that V (x) = V b(xb). Thenusing the properties of σ we obtain

ψb2 χb(|xa|) 6 χb ψa

1 (|xa|) 6 χb(V a(xa)) 6 σ(V a(xa)) 66 V b(xb) 6 ψb

2(|xb|),

which implies that |xb| > χb(|xa|). Similarly as in Case 1, we can derive thatwe have that |xa| > max(χb(|xa|), γb(|u|)). Hence, (39) holds, which implies

∇V(ja,jb,1)(xa, xb)f(ia,ib)(x, u) = ∇V bjb

(xb)fbib

(xa, xb, ub) 66 −αb(V b(xb)) = −αb(V (x))). (46)

Hence, V is an ISS Lyapunov function for system (8) for α(s) = min[αa(s), αb(s)]which is a positive definite and continuous function. Hence, ISS follows fromTheorem 3.4.

Remark 4.2 A benefit of using non-smooth Lyapunov functions is that it sim-plifies the proof of the corresponding theorem for locally Lipschitz systems (seee.g. [13]). A major part of the proof in [13] aims at constructing a smooth ISSLyapunov from the non-smooth V in (41). Using the framework of non-smoothISS Lyapunov functions, as presented here, this step becomes obsolete (althoughone might possibly prefer a smooth ISS Lyapunov function over a non-smoothone in some situations).

5 Computational aspects for PWL systems

The algebraic conditions (24) for ISS of system (2) can be transformed into amore “dissipation” type of characterization for ISS:

col(x, u) ∈ Ωi and x ∈ Γj ⇒∇Vj(x)fi(x, u) 6 −α(V (x)) + δ(χ(|u|)l − |x|l) (47)

for all i = 1, . . . , N and j = 1, . . . , M in which δ > 0 is a constant and l is somepositive integer. One can interpret the term δ(χ(|u|) − |x|) in the right-handside also as an S-procedure relaxation [15]. In particular, for piecewise linearsystems these conditions can be transformed into a convenient computational interms of linear matrix inequalities when piecewise quadratic (PWQ) Lyapunovfunctions are used. Indeed, consider the following PWL system [23, 10], whichfor ease of exposition is chosen to switch only on the basis of the state3:

x(t) = Aix(t) + Biu(t), when Exi x(t) > 0, (48)

3The case of switching also on the basis of the input can be treated in a similar manneras outlined below with the only difference that the selection of the regions Γj for the ISSLyapunov function is less obvious.

19

for matrices Exi , Ai and Bi of appropriate dimensions for i ∈ 1, . . . , N. In

view of (2), this means that Ωi = Ωxi ×Rm with Ωx

i = x | Exi x > 0, which are

convex cones, fi(x, u) = Aix + Biu and Ω1 . . . , ΩN is a partitioning of Rn.We take as a candidate ISS Lyapunov function the piecewise quadratic func-

tionV (x) = xT Pjx when x ∈ Ωx

j , (49)

where we selected the regions of V in accordance with the PWL system (48),i.e. Γj = Ωx

j , j = 1, . . . , N .We assume that there exist full column rank matrices Fi, i = 1, . . . , N

such that Fix = Fjx for all x ∈ Ωxi ∩ Ωx

j . The matrices Fi, i = 1, . . . , Nare used to obtain a parameterized PWQ function that is continuous (see proofof Theorem 5.1 and [15] for more details). Moreover, we assume that there existmatrices Hij that satisfy Ωx

i ∩ Ωxj ⊆ x | Hijx = 0 for any pair (i, j) ∈ S with

S = (i, j) ∈ 1, . . . , N2 | i 6= j and Ωxi ∩ Ωx

j 6= 0.

Theorem 5.1 If one can find symmetric matrices T , Wi, Ui, i = 1, . . . , Nand Yij for (i, j) ∈ S with Ui and Wi having nonnegative entries such thatPi := F>i TFi, i = 1, . . . , N satisfy

1. for all i = 1, . . . , N−A>i Pi − PiAi − µPi − I − Ex>i UiE

xi −PiBi

−B>i Pi εI

> 0

2. for all (i, j) ∈ S−A>i Pj − PjAi − µPj − I −H>ijYijHij −PjBi

−B>i Pj εI

> 0

3. Pi − [Exi ]>WiE

xi > 0, i = 1, . . . , N

Then the system (48) is input-to-state stable. Moreover, Definition 3.2 is sat-isfied for V as in (49) with χ(|u|) =

√ε|u|, α(V (x)) = µV (x), ψ1(|x|) = c1|x|2

and ψ2(|x|) = c2|x|2, where

c1 := minj=1,...,M

min|x|=1,Ex

j x>0x>Pjx > 0 and

c2 := maxj=1,...,M

max|x|=1,Ex

j x>0x>Pjx > 0. (50)

Proof Due to the form Pi = F>i TFi the function V as in (49) is continuous.The third statement shows that V is upper and lower bounded by the K∞-functions ψ1 and ψ2. The first hypothesis implies (47) with χ(|u|) =

√ε|u|,

α(V (x)) = µV (x) and l = 2 for i = j, where we used an additional S-procedurerelaxation related to x ∈ Ωx

i . The second hypothesis implies (47) with χ(|u|) =√ε|u|, α(V (x)) = µV (x) and l = 2 when i 6= j. In this case the S-procedure

relaxation (Finsler’s lemma) is applied for x ∈ Ωxi ∩ Γj = Ωx

i ∩ Ωxj , which is

contained in x | Hijx = 0.Note that the matrices Yij have no restrictions on their entries (as opposed

to Wj and Uj). Moreover, the above conditions are linear matrix inequalitiesonce µ is fixed.

20

Remark 5.2 Theorem 5.1 generalizes the result in [15] (if one considers sta-bility instead of ISS) in the sense that it includes sliding behaviour due to theadoption of Filippov’s solution concept. Indeed, statement 2 is used to accommo-date for possible sliding mode solutions (see also discussion after Definition 3.2).This condition was not used in [15] as they considered only ordinary piecewisecontinuously differentiable solutions without sliding modes.

Remark 5.3 In case one searches a quadratic Lyapunov function V (x) = x>Px(in terms of Theorem 5.1, Pi = P , i = 1, . . . , N), then one does not have toimpose hypothesis 2 in Theorem 5.1 as these are already satisfied via hypothesis1.

Remark 5.4 An alternative to the parameterization of Pi as F>i TFi to warrantcontinuity of V as in (49), is to use in addition to the kernel representationsΩx

i ∩ Ωxj ⊆ x | Hijx = 0 for any pair (i, j) ∈ S, also image representations

Ωxi ∩ Ωx

j ⊆ imZij := Zijz | z ∈ Rnij for matrices Zij ∈ Rn×nij . Continuityof V can now also be guaranteed by the condition Z>ij [Pi − Pj ]Zij = 0 for all(i, j) ∈ S.

6 Example

This section will illustrate the use of Theorem 4.1 and the computational ma-chinery in Section 5 for the interconnection of two PWL systems inspired bythe ‘flower system’ of [15]. Consider a piecewise linear system of the form (48)with system matrices

A1 = A3 =

−0.1 1−5 −0.1

; A2 = A4 =

−0.1 5−1 −0.1

;

B1 = B3 =

0.0667 0

0 0.0667

; B2 = B4 =

0 0.050 0

and the partitioning given by

Ex1 =

−1 1−1 −1

; Ex

2 =

−1 11 1

; Ex

3 = −Ex1 ; Ex

4 = −Ex2 .

The regions are depicted in Figure 4. The switching planes are given byH12 = H21 = H34 = H43 = [1 1] and H23 = H32 = H41 = H14 = [−1 1] in ker-nel representation and can be put in image representation by using the matricesZ12 = Z21 = Z34 = Z43 = [−1 1]T and Z23 = Z32 = Z41 = Z14 = [1 1]T . Wesearch for an ISS Lyapunov function of the form V (x) = x>Pjx, when Ex

j x > 0with P1 = P3 and P2 = P4 and where we adopt the idea in Remark 5.4 toguarantee continuity of V . By fixing µ = 0.01 we solve the LMIs given in The-orem 5.1. Due to the symmetry in the system and in the PWQ ISS Lyapunovfunction we have to consider 6 LMIs (2 of each type in Theorem 5.1) togetherwith the elementwise conditions on the corresponding matrices Ui and Wi. Solv-ing this via the SeDuMi solver using the yalmip interface and minimizing ε yieldsε = 0.364 and

P1 =

16.80 0.220.22 3.36

; P2 =

3.36 0.220.22 16.80

21

−6 −5 −4 −3 −2 −1 0 1 2 3−2

0

2

4

6

8

10

12

14

16

x1

x2

Flower system with constant input u1(t) = u2(t) = −150

sliding modes

region 2

region 1region 3

Figure 3: First part of solution trajectory of flower system

By solving the optimization problems given in (50) we find c1 = 9.86 andc2 = 16.80 and thus

ψ1(x) = 9.86|x|2 6 V (x) 6 16.80|x|2 = ψ2(x).

From this we can compute the ISS gain γ via the expression (30) that yields forGISS = 0.788

γ(‖u‖) = ψ−11 ψ2 χ(‖u‖) = ‖u‖

√c2

c1ε = GISS‖u‖.

To illustrate the ISS property, we simulated the solution trajectory for initialcondition x0 = col(3, 9) and input function u1(t) = u2(t) = −150 for 0 6 t 6 100and u1(t) = u2(t) = 0 for 100 < t 6 200. We plotted the solution trajectoryin the phase plane in two parts: Figure 3 shows the first 100 time units andFigure 4 the next 100 time units. Note that there are two sliding phases inthe trajectory (Figure 3). In Figure 5 we show the norm of the state for thistrajectory as a function of time. We observe that there is a transient phase afterwhich the state remains bounded in the first 100 time units of the simulation.Once the control input u is set to zero, we observe that the state trajectoryconverges to the origin as is guaranteed by the ISS property.

We interconnect two flower systems with the state vector of one systemforming the input for the other, which leads to an autonomous interconnection(without external signals). Based on Theorem 4.1 (applied for autonomous in-terconnections) we obtain GAS of the interconnection, as the coupling condition(40) given by (since ψa

1 = ψb1, ψa

2 = ψb2 and χa = χb)

ψ2 χ [ψ1]−1 ψ2 χ [ψ1]−1(r) =c22

c21

ε2r = G4ISSr < r

is satisfied. This is demonstrated by a simulation of the interconnected systemin Figure 6.

22

−8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

12

1415

x1

x2

Flower system for piecewise constant input and time larger than 100

Region 2

Region 1

Region 4

Region 3

Figure 4: Second part of solution trajectory of flower system

0 50 100 150 2000

2

4

6

8

10

12

14

16

18

time

norm

of t

he s

tate

Flower system with piecewise constant input

Figure 5: Norm of the state for the solution trajectory

0 50 100 150 200−15

−10

−5

0

5

10

15

time

Sta

te v

aria

bles

A simulation of the interconnected system

Figure 6: Solution trajectory of the interconnected system

23

7 Conclusions

The well known ISS framework was extended in the current paper to continuous-time discontinuous dynamical systems and non-smooth ISS Lyapunov functions.The main motivation for the use of non-smooth ISS Lyapunov function is thecommon application of ‘multiple Lyapunov functions’ in the stability theoryfor hybrid systems. We introduced a new solution concept that was shownto be suitable for the interconnection of ‘open’ hybrid systems. This solutionconcept extends the famous Filippov’s convex definition, which turned out tohave undesirable properties for interconnections. After the formal introduc-tion of non-smooth ISS Lyapunov function, we proved that this type of ISSLyapunov functions can be used to guarantee ISS for discontinuous dynamicalsystems adopting extended Filippov solutions. Moreover, we proved that theinterconnection of two discontinuous dynamical systems, which both admit anon-smooth ISS Lyapunov function, is ISS with respect to the remaining exter-nal signals under a small gain condition. This enables the application of the ISSmachinery together with the available computational means based on ‘multipleLyapunov functions’ for hybrid systems. In particular for piecewise linear sys-tems, we presented conditions based on linear matrix inequalities to verify ISSfor PWL systems and their interconnections. A generalization of the ‘flower’system in [15] illustrated the effectiveness of the approach.

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A Appendix: Proofs of Theorems 2.4 and 2.5

In the appendix we will use index sets IF (x, u) and IC(x, u) similarly as theindex set I(x, u) in (14) for the set (13):

IF (x, u) := i | ∀ε>0∀M⊆Rn,µ(M)=0[(Bε(x) \M)× u] ∩ Ωi 6= ∅ (51)

IE(x, u) := i | ∀ε>0∀M⊆Rn+m,µ(M)=0[Bε(x, u) \ M] ∩ Ωi 6= ∅ (52)

such that

Ff (x, u) = cofi(x, u) | i ∈ IF (x, u) and (53)Ef (x, u) = cofi(x, u) | i ∈ IE(x, u). (54)

Proof of Theorem 2.4(1) Let col(x, u) ∈ Rn+m be arbitrary. To prove the inclusion Ef (x, u) ⊆Cf (x, u), it suffices to show that IE(x, u) ⊆ I(x, u). Suppose i ∈ IE(x, u), whichmeans that for all ε > 0 there exists col(xε, uε) with col(xε, uε) ∈ Bε(x, u) ∩ Ωi

(by taking M = ∅). Since col(xε, uε) → col(x, u) when ε → 0 and Ωi is closed,it follows that col(x, u) ∈ Ωi and thus i ∈ I(x, u). The proof of the inclusionFf (x, u) ⊆ Cf (x, u) is similar.(2) To prove (2), we only have to show Cf (x, u) ⊆ Ef (x, u) as the rest followsfrom (1). To prove this, let i ∈ I(x, u) and ε > 0. Since cl(int(Ωi)) = Ωi,we can always find a point col(xε, uε) ∈ int(Ωi) ∩ Bε/2(x, u). Since col(xε, uε)lies in the interior of Ωi there exists 0 < δ < ε/2 such that Bδ(xε, uε) ⊆ Ωi.This implies that Bδ(xε, uε) ⊆ Bε(x, u)∩Ωi and thus for all M with µ(M) = 0it holds that [Bε(x, u) \ M] ∩ Ωi 6= ∅. Hence, i ∈ IE(x, u) and consequently

26

Cf (x, u) ⊆ Ef (x, u).(3) In view of (2), we only have to prove I(x, u) ⊆ IF (x, u). Let i ∈ I(x, u)and thus col(x, u) ∈ Ωi. Let ε > 0. Since cl(int(Ωi)) = Ωi, we can al-ways find a point col(xε, uε) ∈ int(Ωi) ∩ Bε/2(x, u). Since Ωi = Ωx

i × Rm,this means that xε lies in the interior of Ωx

i (considered as a subset of Rn).Hence, there exists a 0 < δ < ε/2 such that Bδ(xε) ⊆ Ωx

i . This implies thatBδ(xε)× u ⊆ [Bε(x)× u] ∩ Ωi and thus for all M ⊂ Rn with µ(M) = 0 itholds that [(Bε(x) \M)× u] ∩ Ωi 6= ∅. Hence, i ∈ IF (x, u).

Proof of Theorem 2.5.(1) The index set I(x, u) as in (14) for the interconnected system (9) can bewritten as I(x, u) = (ia, ib) | ia ∈ Ia, ib ∈ Ib, where we used the abbreviationsIa for Ia(xa, xb, ua) and Ib for Ib(xa, xb, ub). This is easily seen from the regions(10) in the partitioning of the interconnected system. Note that there is a slightabuse of notation in I(x, u) as we use pairs of indices (ia, ib) instead of a singlecounter. From this we have for (9) that

Cf (x, u) = co(

fia(xa, xb, ua)

fib(xa, xb, ub)

)| ia ∈ Ia and ib ∈ Ib

= cofia(xa, xb, ua) | ia ∈ Ia × cofib(xa, xb, ub) | ib ∈ Ib

= Cfa(xa, xb, ua)× Cfb(xa, xb, ub).

(2) If (8a) and (8b) have non-degenerate regions, then, by statement (2) of The-orem 2.4, we have that Efa(xa, xb, ua) = Cfa(xa, xb, ua) and Efb(xa, xb, ub) =Cfb(xa, xb, ub). Moreover, from Theorem 2.4, (1) we infer that the inclusionEf (x, u) ⊆ Cf (x, u). Statement (1) of Theorem 2.5 completes the proof of thispart.(3) If the interconnected system is autonomous, Theorem 2.4 (3) implies thatFf (x) = Cf (x). The result is then immediate from statement (1) of Theorem 2.5and statement (2) of Theorem 2.4.

27