Invariance principles and passivity notions for switched DAE and hybrid DAE systems

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Invariance Principles and Passivity

Notions for Switched DAE and

Hybrid DAE Systems

Pablo ˜

Na˜

nez

Universidad de Los Andes

A thesis submitted for the degree of Doctor of Philosophy in Engineering (PhD)

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Members of the committee:

Nicanor QUIJANO PhD. (Universidad de Los Andes), Advisor

Ricardo G. SANFELICE PhD. (University of California, Santa Cruz), co-Advisor

Alain GAUTHIER PhD. (Universidad de Los Andes), jurado

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Abstract

Homogeneous and non-homogeneous linear switched differential algebraic equations (DAE) are studied. Particularly, a class of switched system where the value of a switching signal determines the current model of operation (among a finite number of them) and, for each mode, its dynamics are described by a DAE. Also, we consider a class of hybrid dynamical system with continuous dynamics that can be modeled as DAEs and with discrete dynamics given by difference inclusions. We denote such class of hybrid systems as hybrid DAE systems.

Motivated by the lack of invariance principles and input-output analysis (such as passivity) for switched DAE systems, a main goal of this dissertation is the sta-bility analysis of both switched DAE and hybrid DAE systems through invariance properties and passivity concepts.

We define the conditions on the data of the switched DAE and hybrid DAE systems that guarantee that both representations share the same set of solutions. The prop-erties of theω-limit set of a solution for hybrid DAE and switched DAE systems are characterized. Then, we extended the invariance principles for hybrid systems to hybrid DAE systems, and by using the aforementioned equivalence of solutions we develop such principles for switched DAE systems under arbitrary and dwell-time switching.

For non-homogeneous switched DAE with output maps, stability via passivity anal-ysis is studied, therefore, passivity notions for hybrid DAE systems are defined. Using notions of detectability from hybrid systems, we show the relationship be-tween passivity properties to zero-input stability of hybrid DAE and switched DAE systems by the usage of the novel invariance principle for hybrid DAE systems also developed in this dissertation.

Given a static output-feedback control law that satisfies some mild conditions and a (flow- or jump-) passive hybrid DAE with respect to a set of interest, we study the resulting feedback loop stability regarding the set of interest. We use the pas-sivity results for hybrid DAE systems to show that the control law renders such set asymptotically stable.

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Acknowledgements

Deseo agradecer a todos los que me apoyaron durante mi doctorado, en especial a mi esposa Catalina quien me acompa˜no en los buenos momentos e inspiro en mi el coraje necesario para afrontar los dif´ıciles. A mis padres, quienes siempre han cre´ıdo en mi y me han apoyado incondicionalmente. A mi hermana, su esposo y mis sobrinas, por sus buenos consejos y cari˜no. Sin ellos este logro no habr´ıa sido posible.

En especial, deseo expresar la mas profunda gratitud y respeto a mi mentor, Ricardo G. Sanfelice, quien mediante su ejemplo de integridad en la investigación me guio a través de la etapa mas importante de mi formación académica. Este trabajo no hubiera sido posible sin su asesor´ıa, brillantes aportes, y amistad. Es un honor trabajar a su lado y poder contarse entre los miembros de su laboratorio.

También, agradezco a los miembros del comité doctoral por la interesante discusión y sugerencias que ayudaron a mejorar este documento.

A mis compa˜neros del programa doctoral en la Universidad de los Andes, Ferney Maldonado, German Obando, Wilson Achicanoy, Andres Pantoja, y Javier Riascos, entre otros, muchas gracias por su amistad, por las interesantes discusiones, y sus innumerables comentarios.

I addition, I would like to thank the wonderful team at the University of Arizona during my stay there. To BharaniPrabha Malladi, Jun Chai, Nathalie Risso, Sean Phillips, Yuchun Li, Pedro Casau, Thomas Theunisse, and Jeffrey Koessler, thanks for all the support, interesting discussions, and friendship.

Es menester también agradecer a las personas e instituciones que me apoyaron fi-nancieramente durante el programa doctoral, entre las que se encuentran Colciencias, la Fundación Mariano Ospina Pérez, la Universidad de los Andes, y en especial al profesor Juan Saldarriaga.

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List of Figures

2.1 DC-DC boost converter circuit and Current-Voltage characteristic curve of the diode. . . 11 2.2 Level sets of functions V1 and V2, where R = 3, c = 0.2, L = 0.2, vcc(0) = 5,

v∗

C =u=e= 7,i∗L=u2/(Rvcc), λ= 0.3. . . 15

4.1 Two carts system. . . 37 4.2 C,D, and λ. . . 39 5.1 Representation of the invariant set M0_{, where the circle in blue (dashed)}

repre-sents _{x _∈ R2× {1,2} | ξ12 +ξ22 = r, σ = 1} and the points in red represents

{x ∈R2× {1,2} | ξ1 = 0, ξ22 =r, σ = 2}. The solid (blue) curve indicate flow.

The dashed (red) arc indicate jumps. . . 49 5.2 Consistency sets and trajectories for each mode. . . 50 5.3 Plot of uD in (5.18) for varyingδ. . . 51

5.4 Lyapunov function V and system’s trajectories. Color gradient represents Lya-punov function level sets for = 2.2_×10−6. . . 52 5.5 System’s trajectory forδ >δ¯and a level setV−1(¯r). For the system’s trajectory,

color gradient represents time. . . 55 5.6 System’s trajectories for δ ≈ ¯δ. Color gradient represents Lyapunov function

level sets. . . 56 5.7 System’s trajectories forδ = 0.9>¯δ. Color gradient represents time. . . 56 5.8 System’s trajectories forδ = 0.4<¯δ. Color gradient represents time. . . 57 5.9 Solutions of the switched DAE in Example 5.15 and Example 5.16 for different

dwell-time switching signals using the same initial condition ξ1(0) = ξ2(0) = 2

and σ(0) = 1. Color gradient represents time, the green triangle represents the initial condition, and the red _×symbol represents the final condition at t= 4. . . 62

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5.10 Two-carts system trajectories. Simulation parameters: t= [0,30], λ= 0.001, m1= m2=k1 =k2 = 1, rule= 2 (lite simulator), andx(0) = [0.3202, −0.4335, 0.3716,

−1.0915, 0, 1]>. . . 71 5.11 Results for the ˆHDAE system with the logic state σ. The green ∆ is the initial

condition, and the red ×is the state of the system at the end of the simulation.

t= [0,150], λ= 0.001, m1 =m2 =k1 =k2 = 1, rule = 2 (lite simulator), and x(0) = [0.3202, −0.4335, 0.3716, −1.0915, 0, 1]>. . . 71 6.1 Equivalent vector field ˜f(ξ, u) and level sets of functions γσ(ξ) for each σ ∈ Σ,

where R = 3, c = 0.2, L = 0.2, vcc(0) = 5, vC∗ = 0, u = ˜vC = e = 7, i∗L =

˜

v2

C/(Rvcc), λ = 0.3. The vector fields and the 0-level set are depicted by gray

arrows and a cyan line, respectively. The set value forvC andiLis given by a red

x. Blue and yellow lines mark the sections of the state space where the derivative of Ve along the trajectories of the equivalent vector field is negative and positive,

respectively. . . 100 6.2 Simulation of the solutions of the switched DAE in Example 6.11 for different

initial conditions. Color gradient represents time and the red×symbol represents the set point forvC. The level setsV1−1(0) andV2−1(0) that triggers the changes

inσ are shown. . . 101 6.3 Control Lyapunov Function approach using functions γ1(ξ) andγ2(ξ). . . 102

6.4 Set of interest and level sets of functions V1 and V2, where R = 3, c = 0.2, L= 0.2,vcc(0) = 5, v∗C =u=e= 7, iL∗ =u2/(Rvcc),λ= 0.3, andδ = 1/6. . . 103

6.5 Simulation of the states of the switched DAE in Example 6.11 for two initial conditions. . . 107

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1 Introduction

1.1 Background

In this dissertation, we consider a class of hybrid dynamical systems with continuous dynamics that can be modeled as differential-algebraic equations (DAEs) – also known as descriptor systems or singular systems – and with discrete dynamics given by difference inclusions. This type of systems, which we refer to as hybrid DAE systems, arise in several applications in engineering such as robot manipulators, vehicular traffic systems [1], power systems, biological systems, and mechanical systems [2]. The systems of interest in this paper include a logic variable which determines the current mode of the system (among finitely many of them) and that, during flows, the dynamics of the other state components evolve according to a linear DAE. Particularly, we also consider a class of switched systems with a state evolving according to a linear DAE in-between switching instants and, due to the algebraic constraints of the DAE in each mode, potentially exhibiting jumps in the state at switching instants. Switched DAE representations naturally appear when modeling electrical circuits, where algebraic constraints (e.g., due to Kirchhoff’s laws) are entangled with differential equations (e.g., governing the change of current and voltages in capacitors and inductors) as well as elements such as (ideal) switches or diodes [3].

The stability analysis of switched DAE systems has its foundations on the stability analysis of descriptor systems (also referred to as differential-algebraic systems, singular systems or semi-state systems), e.g., a survey of results on linear singular systems is given in [4]. Also, a historical review of linear singular systems is presented in [5, 6]. The consistency of initial conditions together with Lyapunov theory is used to study the stability properties of the origin for linear singular systems in [7]. Several authors have analyzed switched DAE systems from many perspectives, with most of the research being focused in establishing asymptotic stability

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of the origin. In [3], Lyapunov’s direct method is used to analyze the asymptotic stability of the origin for switched DAE systems. Sufficient conditions for exponential stability of switched singular system with stable subsystems are presented in [8]. It is important to note that a typical assumption that is enforced in such works is that solutions are given by piecewise (right or left) continuous functions, so as to preclude the presence of impulses in the solutions at the times when switches occur. As stated in [9] in the context of switched systems, to deal with such impulses in the solutions, one approach is to consider distributional solutions or weak solutions; see, e.g., [3, 10]. However, unless explicitly assumed, neither concept of solution leads to a set of solutions with the so-called sequential compactness property, which is key in the development of invariance-like and passivity results [11] (see [12]).

1.2 Contributions

In this document, we consider dynamical systems with multiple modes of operation and state jumps. Within each mode, the dynamics are given by DAEs. State jumps can occur when: in a fixed mode, when transitioning between modes, or when they are be induced by an inconsistent initial condition. We refer to this class of hybrid systems as hybrid DAEs. Motivated by the lack of results regarding the invariance properties of switched DAE systems, being perhaps the main reason the difficulty in guaranteeing a sequential compactness property of the solutions to such systems [11], we build from the concept of solution to hybrid systems in [13] (referred here as hybrid inclusions) and propose a model for switched DAE systems using the framework of

hybrid DAEs introduced here. For the proposed hybrid DAE model, we establish that when its data satisfy certain mild conditions, the system has a set of solutions with structural properties enabling the development of invariance results. Also, we propose a model for switched DAE systems under certain classes of switching signals and we establish conditions on its data for an invariance principle to hold.

Surprisingly, there is a lack of results regarding the passivity properties of switched DAE systems (or input/output analysis in general), being the main reason the absence of invariance principles for such systems. To fill that gap, we build passivity concepts for hybrid DAE systems based on the newly invariance principles developed in this dissertation and the passivity analysis for hybrid systems in [14, 15, 16, 17, 18], particularly from the ideas related to passivity-based control for hybrid systems in [19]. Thanks to the heritage of the concept of solution from hybrid systems, we link our novel passivity properties for hybrid DAE systems to stability. Namely, given a static output-feedback control law, we establish conditions to guarantee asymptotic stability of a set of interest for the closed loop via passivity analysis.

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More precisely, the contributions of this dissertation include the following:

1. Building from results for switched DAE systems [3, 8], we propose a dynamical model that allows for jumps when initial conditions are not consistent with the algebraic condi-tions. In fact, our model uses concepts from the literature of switched DAE systems to keep the special structure of the algebraic restrictions in DAE systems, in particular, the so-called consistency jumps driven by the consistency projectors and inconsistent initial conditions [3, 4, 10]. Using ideas for hybrid systems [13], the proposed model captures the jumps triggered by state conditions. Our model uses the concept of solution, the invariance notions, and passivity concepts for the class of hybrid systems in [13].

2. For hybrid DAEs with linear dynamics during flows, we determine the properties of ω -limit sets of bounded and complete solutions, and establish an invariance principle. The invariance principle resembles the classical one for continuous-time systems.

3. Building from the results for hybrid DAE systems in item 2 (see also [12]) and for switched systems in [20], we propose invariance principles for switched DAE systems under arbitrary and dwell-time switching.

4. Considering switched DAE and hybrid DAE systems with inputs and output maps, based on passivity analysis in [19] and [16], we define passivity concepts for such systems. Namely, two passivity notions are described flow- and jump-passivity, where the energy dissipation takes place in the continuous and discrete behaviour, respectively.

5. Applying the recently introduced invariance principle in item 2, we link the strict and output versions of the aforementioned passivity properties to asymptotic stability via static output-feedback.

It is important to note that tools for the study of invariance properties of hybrid systems in [13] are used in this document, but the invariance principles in [13] cannot be applied directly to the class of systems of interest, in particular switched DAE systems as in (2.1). This is mainly because the systems we consider do not fulfil the required conditions in [13]. Another important point to highlight is that, unlike [3], the concept of solution used in this paper does not take into account impulses. As we point out in Section 5.2 and illustrate in Example 4.2 and Example 4.3, this notion of solution is required so that the ω-limit set of bounded and complete solutions for switched DAE systems have the needed structural properties, particularly, weak invariance. Examples 4.2 and 4.3 point out that impulses in the solutions do not necessarily lead toω-limit sets with such key property.

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To the best of our knowledge, invariance properties and passivity analysis for DAE systems with discontinuous coefficient matrices and with jumps in its states, such as the hybrid DAE and switched DAE models considered here, are not available in the literature.

The remainder of this document is organized as follows. In Chapter 2, motivational ex-amples for both invariance and passivity concepts are presented, also the required modelling background is presented in Section 2.3. In Chapter 4, a series of application examples are presented, where some of those examples are revisited in the forthcoming Chapters. In Chap-ter 5, the main invariance results are presented; namely, a description of hybrid DAE systems and the invariance principle for such systems, and, in Section 5.1, an invariance principle for switched DAE systems under arbitrary, and an invariance principle for switched DAE systems under dwell-time switching. In Section 5.2, we present examples where the definitions and the invariance principles are exercised. The technical prerequisites needed in the proofs of the main results are summarised in Section 5.3.1, while the proofs of the main passivity results are given in Section 5.3.2. In Chapter 6, a description of hybrid DAE systems with inputs and outputs maps is presented, which is followed in Section 6.2 by the introduction of the passivity and stability definitions for such systems. In Section 6.3, we present the results that link passivity with zero-input stability for hybrid DAE systems, and in Section 6.4, we link the properties of flow and jump passivity to stability of a closed loop given by a static output-feedback control law. Also in Section 6.5, we revisit the motivational Example 2.2 where the definitions and the results are exercised. Conclusions and future directions are discussed in Chapter 7.

1.3 Outline of the contributions and the related publications.

The following list of publications is related with each of the contributions listed in the previous section

1. and 2. Invariance principle and model description of the so-called hybrid DAEsystems.

Related publication:

• P. Nanez and R. G. Sanfelice. An Invariance Principle for Differential-Algebraic Equations with Jumps. InProceedings of the 2014 American Control Conference, pages 1426–1431. IEEE, 2014.

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Related publications:

• P. Nanez, R. G. Sanfelice, and N. Quijano. Invariance Principles for Switched Differential-Algebraic Systems Under Arbitrary and Dwell-time

Switching. InProceedings of the 2015 American Control Conference. IEEE, 2015.

• P. Nanez, R. G. Sanfelice, and N. Quijano. On an Invariance Principle for Differential-Algebraic Equations with Jumps and its Application to

Switched Differential-Algebraic Equations. Mathematics of Control, Signals, and Systems, 2017.

4. and 5. Passivity for switched DAE systems using concepts of passivity for hybrid DAE systems.

Related publication:

• P. Nanez, N. Quijano, and R. G. Sanfelice. Passivity for Hybrid DAE Systems and its Applications to Switched DAE systems. Submitted IEEE Conference on Decision and Control, 2017.

Other Contributions

The following publications had a significant influence on some of the thesis contributions, but they are not covered in this document.

• R. G. Sanfelice, D. A. Copp, and P. Nanez. A Toolbox for Simulation of Hybrid Systems in Matlab/Simulink: Hybrid Equations (HyEQ) Toolbox. In

Proceedings of Hybrid Systems: Computation and Control Conference, pages 101–106, 2013.

• P. Nanez, N. Risso, and R. G. Sanfelice. A Symbolic Simulator for Hybrid Equations. In Proceedings of the 2014 Summer Simulation Multiconference, Summer-Sim ’14, pages 18:1–18:8, San Diego, CA, USA, 2014. Society for Computer Summer-Simulation International.

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2 Motivation and Preliminaries

Notation

The notation used throughout the document is as follows. Given a setS _⊂Rn, the closure of S is the intersection of all closed sets containingS, denoted byS. Given a setS_⊂Rn×Rm, we

denote Γ0(S) :={x ∈Rn : (x,0)∈S} and Γ(S) :={x ∈Rn : ∃u∈Rm s.t. (x, u) ∈S}. We

defineR≥0 := [0,∞) andN:={0,1, . . .}. Given vectorsν ∈Rn, ω ∈Rm, [ν>ω>]>is equivalent

to (ν, ω), where (_·)>_{denotes the transpose operation. Given a function}_f _:

Rm →Rn, its domain

of definition is denoted by domf, i.e., domf :=_{x_∈Rm | f(x) is defined}. The range of f is

denoted by rgef, i.e., rgef :=_{f(x) _|x_∈domf_}. The right limit of the function f is defined as f+₍_x_{) := lim}

ν→0+f(x+ν) if it exists. The notation f−1(r) stands for the r−level set of f

on domf, i.e., f−1₍_r_{) :=}_{_z_∈_dom_f _|_f₍_z_{) =}_r_}_{. The n-th derivative of a function} _f _:

R→Rn

(if exists) is denoted by f(n)₍_·_{). The}_r_{-sublevel set of the function} _V _{: dom}_V _→

R, namely, the

set of points _{x_∈domV _|V(x)_≤r_}, is denoted byLV(r). Given two functions f :Rm →Rn

and h : Rm → Rn, hf, hi denotes the inner product. We denote the distance from a vector y_∈Rn to a closed set A⊂Rn by|y|A, which is given by|y|A := infx∈_A|x−y|. Given a matrix

P _∈Rn×n, the determinant of P is denoted by detP. The column space of the matrixP, i.e.,

the set of all possible linear combinations of the column vectors ofP, is denoted byim(P). The span of a set of vectorsQis defined as the set of all finite linear combinations of elements ofQ, e.g., span(Q) = nPk

i=1αivi |k∈N\ {0}, vi ∈Q, αi ∈R o

. Given two matrices P, Q _∈ Rn×n,

the set of finite eigenvalues _{λi}i∈{1,2,...,n1}, where n1 ≤ n, is denoted by λ(P, Q) := {s | s ∈ C, s finite,|sP −Q|= 0}. Givenn∈N, the matrix 0n∈Rn×n denotes the zeron×n matrix,

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2.1 Motivation invariance principle for switched DAE systems

Next, as a motivation, we present an example where the analysis of the asymptotic stability of the origin is not suitable and a notion of invariance is required to analyze the solutions of a fairly simple switched DAE system.

Example 2.1 (Motivational example, invariance principle) Consider a switched DAE system with two modes of operation determined by σ_{∈ {}1,2_}and dynamics

Eσξ˙=Aσξ (2.1)

whereξ _∈R2 is the state of the system and E1=

"

1 0 0 1

#

, A1= "

0 1

−1 0

#

, E2 = "

0 0 0 1

#

, A2 = "

1 0

0 ₋1

#

(2.2)

Let the switching signal σ : R≥0 → {1,2} be a piecewise-constant right-continuous function.

Consider the function

V(ξ, σ) = (Eσξ)>Eσξ ∀(ξ, σ)∈R2× {1,2} (2.3)

and note that, when σ remains constant, the change of the function V is given as follows:

• Ifσ = 1, then

˙

z }| {

V(ξ, σ) = (E1ξ˙)>E1ξ+ (E1ξ)>E1ξ˙= 0 ∀ξ ∈R2

• Ifσ = 2, then ˙

z }| {

V(ξ, σ) = (E2ξ˙)>E2ξ+ (E2ξ)>E2ξ˙=−2ξ22 ∀ξ ∈R2 (2.4)

Since (2.1) for σ= 2 reduces to ˙

ξ2 =−ξ2, ξ1 = 0

we have that (2.4) implies exponential stability of the origin during that mode. On the other hand, ifσ jumps at timets, the stateξ is mapped to a point inR2 given byξ(t+s) = Πσ(t+s)ξ(ts),

where the subsequent algebraic restrictions are fulfilled. (These maps are given by the so-called consistency projectors; se also [3, Definition 3.7].) Using the definitions of (Eσ, Aσ) above, for

changes from σ = 2 to 1 (i.e., σ+ = 1), the consistency projector is given by Π1, while, for

changes fromσ = 1 to 2 (i.e.,σ+= 2), the projector is given by Π2. These projectors are given

by

Π1= "

1 0 0 1

#

, Π2= "

0 0 0 1

#

. (2.5)

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• Ifσ = 1, thenσ+ _{= 2 and, for each}_ξ_∈ R2,

V(Π2ξ,2)−V(ξ,1) =ξ>

Π>₂E₂>E2Π2−E1>E1

ξ

=₋ξ₁2

• Ifσ = 2, thenσ+ _{= 1 and, for each}_ξ_∈ R2,

V(Π1ξ,1)−V(ξ,2) =ξ>

Π>₁E₁>E1Π1−E2>E2

ξ

=ξ₁2 = 0,

where we used the fact that ξ1= 0 if σ = 2.

Denoting byV+_{the value of}_V _{after the jump at (}_{ξ, σ}_{), the change of}_V _{during flows and jumps}

is given by

˙

z }| { V(ξ, σ) =

(

0

−2ξ₂2

ifσ = 1 ifσ = 2

V(ξ, σ)+₋V(ξ, σ) =

(

−ξ₁2

0

ifσ= 1 ifσ= 2

Note thatV is not strictly decreasing during flows or jumps. Depending on the law triggering the change of σ, solutions can either approach the origin or stay away from it for all time. In fact, for any initial condition away from the origin, if σ eventually remains at 1, then the solution would remain at a level set ofV for all future time.

Due to the nonstrict decrease of V during flows, asymptotic stability1 _{of the origin of (2.1)}

cannot be established using the tools in [3, 5, 7, 8, 21] for particular classes of switching signals. The main reason is the lack of a tool to characterize the ω-limit set of bounded and complete solutions to (2.1). One could be tempted to recast this system as a hybrid inclusion as defined in [13, 19] and apply the invariance principle in [13]. The resulting hybrid inclusion is given by

Hs     

˙

ξ =fs(ξ, σ) (ξ, σ)∈Cs ξ+ =gs(ξ, σ) (ξ, σ)∈Ds

(2.6)

1_{Stability can be inferred from the properties of}_V_{, but attractivity requires the application of an invariance}

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where

Cs:={(ξ, σ)∈R2×Σ|σ= 1} ∪ {(ξ, σ)∈R2×Σ|σ = 2, ξ1= 0} (2.7a)

fs(ξ, σ) :=           

Πdiff 1 A1 =

" ξ2

−ξ1 #

ξ if σ = 1

Πdiff 2 A2 =

"

0

−ξ2 #

ξ if σ = 2

(2.7b)

Ds:={(ξ, σ)∈R2× {1,2} | σ = 2, ξ1 6= 0} (2.7c)

gs(ξ, σ) := Πσξ. (2.7d)

Notice that this hybrid inclusion has the same solutions as the original switched DAE system. The next step would be to define an autonomous system generating the switching signal and apply the invariance principle in [13]; however, since the setDsis open, the assumptions in [13]

would not hold. In fact, as we will show in detail in Example 5.2, the hybrid inclusion model associated to (2.1) is not nominally well-posed, and therefore the invariance principles in [13] are not applicable.

This fact motivates the development of invariance principles for systems of the form (2.1) with jumps on ξ and σ generated by a state-space model.

2.2 Motivation passivity for switched DAE systems and hybrid

DAE systems

Example 2.2 (Motivational example, passivity properties) We consider the DC-DC boost converter shown in Figure 2.1(a) and model it as a switched DAE as in (2.14). More interestingly, using concepts of passivity, a given set-point for the voltage, and an appropriate selection of inputs and outputs, we will show that a passivity-based control law renders a set of interest asymptotically stable. There is a fair amount of literature related to the control of DC-DC boost converters from many perspectives, particularly, in [22] the authors present a complete comparison between several types of hybrid control techniques. The authors in [23] follow an energy-based hybrid control approach to design controllers for impulsive dynamical systems, and in [24] the authors propose a Control Lyapunov Function (CLF) approach for a DC-DC boost converter.

The converter is composed by an inductorL, a capacitorc, a resistorR, a voltage sourcevcc,

a switchS, and a dioded. The voltage across the inductor, diode, and capacitor are denoted as

vL,vD, and vC, respectively. The current through the inductor, switch, and diode are denoted

asiL,iS, andiD, respectively. In this example, we consider the model of the diode depicted in

Figure 2.1(b) where theλis the forward bias voltage of the diode. Now, consider the switching signal σ : [0,_∞) _→ Σ, where each element in Σ represents a mode of operation of the boost

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vcc

L vL

+/₋ +/₋

iL iS

S

vD

d

iD

+

−

vC C R

(a) Boost converter circuit.

λ vD

iD

(b)vDVs. iD curve.

Figure 2.1: DC-DC boost converter circuit and Current-Voltage characteristic curve of the diode.

converter. More precisely, given the nature of the switch and the diode, the four different modes can described by differential-algebraic equations as follows:

Mode 1 (σ = 1) Mode 2 (σ = 2)

d

dtvcc= 0

c_dtdvC= iD− _R1vC L_dtdiL= vL

0 = iL−iS−iD

0 = iS

0 = vcc−vL−vD−vC d

dtvD= 0

d

dtvcc= 0

c_dtdvC = iD− _R1vC L_dtdiL= vL

0 = iL−iS−iD

0 = iD

0 = vcc−vL

0 = vD+vC

Mode 3 (σ = 3) Mode 4 (σ = 4)

d

dtvcc= 0

c_dtdvC= iD− _R1vC L_dtdiL= vL

0 = iL−iS−iD

0 = iS

0 = vcc−vL−vD−vC

0 = iL

d

dtvcc= 0

c_dtdvC = iD− _R1vC L_dtdiL= vL

0 = iL−iS−iD

0 = vD+vC

0 = vcc−vL d

dtvD= 0

(2.8)

The state conditions where each one of the modes is valid are as follows:

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the current through the diode is positive. This requires that the voltagevD is larger than

the threshold voltage (or forward voltage λ >0). In this mode, the diode is modeled as a short circuit.

• Mode 2: (Switch is closed and diode is blocking) In this mode the switch is closed and the voltage in the capacitor is larger or equal than the forward bias voltage (orvC ≥ −λ).

In this case, the voltage across the diode is always smaller than λ. Hence the diode is modeled as an open circuit. Notice that the circuit can be modeled as two separated loops. In the right, we have a RC circuit where the capacitor is discharging through resistor, then the voltage in the capacitor reaches 0 at no finite time (while vD ≤λ). In the left,

we have that the voltage at the inductor (vL) is equal to de source vcc and the current

through the inductor grows unboundly. This mode is valid when the switch is closed and

vD ≤λ.

• Mode 3: (Switch is open and diode is blocking) When the switch is open, the voltage in the diode could become smaller than λ, in which case the current through the diode become zero. In this mode, the diode is modeled as an open circuit. This mode is valid only if the switch is open,vD ≤λand iL= 0. GivenvD =vcc−vC, that impliesvC ≥vcc−λ.

• Mode 4: (Switch is closed and diode is conducting) In this mode the switch is closed and the voltage in the capacitor smaller than the forward bias voltage of the diode (or

vC ≤ −λ). In this case, the voltage in the diode is equal to λ. Then, the diode can

be modeled as a short circuit. Given the charge in the capacitor and that the diode is grounding the capacitor, no current can circulate through the diode, hence the capacitor neither charges or discharges. This mode is valid only if the switch is closed andvD =λ

(orvC ≤ −λ).

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differential-algebraic equation system described above given by

ˆ

E1= ˆE4=          

1 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

         

, Eˆ2 = ˆE3 =          

1 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

         

, Aˆ1 =          

0 0 0 0 0 0 0

0 0 0 −1

R 0 0 1

0 1 0 0 0 0 0

0 0 0 0 1₋1 ₋1

0 0 0 0 0 1 0

1 ₋1 ₋1 ₋1 0 0 0

0 0 0 0 0 0 0

          , ˆ

A2 =          

0 0 0 0 0 0 0 0 0 0 −1

R 0 0 1

0 1 0 0 0 0 0 0 0 0 0 1 ₋1 ₋1 0 0 0 0 0 0 1 1 −1 0 0 0 0 0 0 0 1 1 0 0 0

         

, Aˆ3=          

0 0 0 0 0 0 0

0 0 0 −1

R 0 0 1

0 1 0 0 0 0 0

0 0 0 0 1 ₋1 ₋1

0 0 0 0 0 1 0

1−1 −1 −1 0 0 0

0 0 0 0 1 0 0

         

, Aˆ4=          

0 0 0 0 0 0 0 0 0 0 −1

R 0 0 1

0 1 0 0 0 0 0 0 0 0 0 1₋1 ₋1 0 0 1 1 0 0 0 1−1 0 0 0 0 0 0 0 0 0 0 0 0

          (2.9)

We model this system as a switched DAE in (2.14a). The DC-DC boost converter is designed (and controlled) to deliver a desired DC voltage at its output, which is typically the voltage of the capacitor (vC). Therefore, its input is the set-point for the voltage, denoted as u ∈ R.

Consider the input u being generated by an exosystem, where u is relevant only at switching instants. To proceed with a CLF approach, we add an extra state ewhich is reset to the value of the input at switching instants. The dynamics of the state e in-between switching instants and at switching instants are given by ˙e= 0 ande+₌_u_{+ ˜}_v

C,

˙

e= 0

e+=u+ ˜vC,

respectively, where ˜vC ∈ R≥0 is a constant such that ˜vC ≥ vcc+λ. From the data in (2.9)

and an extended state of the system given byξ = [vcc, vL, vD, vC, iL, iS, iD, e]>, the data of the

switched DAE system in (2.14a) is given by Σ ={1,2,3,4},

Eσ = "

ˆ

Eσ 0

0 1

#

, Aσ = "

ˆ

Aσ 0

0 0

#

, Bσ = h

ˆ

Bσ 0 i>

(2.10)

Where ˆEσ, ˆAσ, and ˆBσ are given by the differential-algebraic equations describing each mode.

We will restrict the state space of this system to the set{ξ ∈R8 |vC >0, iL≥0}.

To show that a passivity property holds for this system, consider the functionsVe :R2→R≥0

and Ve :R8 →R≥0 defined as

Ve(e,v˜C) =

1

2(e−˜vC)

2₊1

2

e2 vccR −

˜

v2 C vccR

2

(2.11)

e

V(ξ) = 1

2(vC −e)

2

+1 2

iL−

e2 vccR

2

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Notice that for each (ξ, σ) such that the algebraic constraints onξ for the mode of operation

σ are meet, we can always pickσ such that h∇Ve(ξ),f˜(ξ, u)i=:γ_σ(ξ)≤0, where the equivalent

vector field ˜f is given in Lemma 2.10 by (2.18) or (2.19). More precisely, we have

• ifσ= 1,

h∇Ve(ξ),f˜(ξ, u)i= (v_C−e)

iL c −

vC cR

+

iL−

e2 vccR

vcc−λ L −

vC L

=:γ1(ξ)

(2.13a)

• ifσ= 2,

−_cRvC+

iL−

e2 vccR

v_cc

L

=:γ2(ξ) (2.13b)

• ifσ= 3,

−_cRvC=:γ3(ξ) (2.13c)

• ifσ= 4,

h∇Ve(ξ),f˜(ξ, u)i=

iL− e2 vccR

v_cc

L

=:γ4(ξ) (2.13d)

In addition, notice that by picking the output of the system as y = p

−γσ(ξ) and the input u = −ky, where 0 < k ≤ 1, we have that h∇Ve(ξ),f˜(ξ, u)i ≤ uy, which describes a passivity

property of the system during the continuous regime. At switching instants, by making use of the consistency projectors in Definition 2.8, as we show in Example 6.11, we have thatv_C+=vC

and i+_L =iL for eachσ, σ+∈Σ. By choosingu=−ky andy= ˜vC−eit can be shown that the

following holds: Ve([v+_cc, v+_L, v_D+, v+_C, i+_L, i+_S, i+_D,−ky+ ˜v_C])≤Ve(ξ), which givesVe(ξ+)−Ve(ξ)≤0

and corresponds to a lossless property at jumps. This motivates the development of passivity-based control techniques for such kind of switching DAE systems.

The question that remains open is whether the passivity-like property shown above can be exploited to design a static output-feedback control law that renders asymptotically stable a given set of interest1_{. In the upcoming sections, we propose a static output-feedback control}

law for the switched DAE system in the previous example. Using concepts of passivity, a given set-point for the voltage ˜vC, and an appropriate selection of inputs and outputs, we will show

that such control law renders a set of interest asymptotic stable. To this end, we develop a general set of notions and results for linking the passivity properties presented here with the asymptotic stability of a set of interest.

1_{Notice that by considering}_u_{= ˜}_v

C+v∗Cwith ˜vC≥vcc+λandvC∗ such thatu≥vcc+λholds, wherev∗C is

a new input and ˜vC a constant, and considering the aforementioned consistency projectors, the stability of the

system whenv∗

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-2 -1 0 1 2 3 4 5 6 7 8 9 -2

0 2 4 6 8 10 12 14 16

V₁−1 (15)

V₁−1 (0)

V₁−1 (₋₁₅₎

vC

V

−

1 2

(1₅ )

V

−

1 2

(0₎

V

−

1 2

(₋ 1₅

)

iL

Figure 2.2: Level sets of functions V1 and V2, where R = 3, c = 0.2, L = 0.2, vcc(0) = 5,

vC∗ =u=e= 7,i∗L=u2/(Rvcc),λ= 0.3.

To check the previously defined vector field ˜f(ξ, u), for the collection {(Eσ, Aσ, Bσ)}σ∈Σ,

the consistency, differential, and impulse projectors in Definition 2.8 are given by

Π1=            

1 0 0 0 0 0 0 0 1 0 ₋1 ₋1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

           

, Π2=

           

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0−1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

            ,

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Π3 =            

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ₋1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

           

, Π4 =

           

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ₋1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1

R 0 1 0 0 0

0 0 −1

R 0 0 0 0 0

0 0 0 0 0 0 0 1

            , Πdiff 1 =            

1 0 0 0 0 0 0 0

1₋1 c 0 −

1 c −

1

c 0 −1 0

0 0 0 0 0 0 1 0

0 1 c 0

1 c

1

c 0 0 0

0 0 _L1 0 0 _L1 0 0

0 0 0 0 0 0 0 0

0 0 1

L 0 0 1 L 0 0

0 0 0 0 0 0 0 1

           

, Πdiff 2 =            

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ₋1

c 0 0 1 c 0 0 0

0 1 c 0 0 −

1 c 0 0 0

0 0 _L1 0 0 _L1 0 0 0 0 1

L 0 0 1 L 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

            , Πdiff 3 =            

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ₋1

c 0 − 1 c − 1 c 0 1 c 0 0 1 c 0 1 c 1 c 0 −

1 c 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

           

, Πdiff 4 =            

1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −1 0 0 0 1

L 0 0 1 L 0 0

0 0 1 L 0 0

1 L

1 R 0

0 0 0 0 0 0 −1 R 0

0 0 0 0 0 0 0 1

            , Πimp 1 =            

0 0 0 0 0 0 0 0 0 0 0 0 0 ₋1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0−1 −1 0 0 0 0 0 0 0 0 0 0 0

           

, Πimp 2 =            

0 0 0 0 0 0 0 0 0 0 0 0 0 ₋1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0₋1 ₋1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

            , Πimp 3 =            

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 −1 −1 0 1 0 0 0 0 0 0 0 0 0

           

, Πimp 4 =            

0 0 0 0 0 0 0 0

0 0 0 0 0 ₋1 0 0

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0

0 ₋1 0 ₋1 ₋1

R 0−c 0

0 1 0 0 _R1 0 c 0

0 0 0 0 0 0 0 0

           

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2.3 Preliminaries

2.3.1 Modeling switched differential-algebraic systems

In this document, we consider the class of linear switched DAE systems given by

Eσξ˙=Aσξ+Bσu (2.14a)

y=hσ(ξ, u), (2.14b)

whereξ ∈Rn is the state, u:R→Rp is the input, y∈Rq is the output, hσ :Rn×Rp →Rq is

the output map, σ:R≥0 →Σ is the switching signal, Σ is a finite discrete set,Eσ, Aσ ∈Rn×n,

and Bσ ∈ Rn×p. Solutions to (2.14a) are typically given by (right or left) continuous

func-tions (see [10] and references therein). Definition 2.13 below introduces the notion of solution to (2.14a) employed here.

Definition 2.3 (DAE regularity [4, Definition 1-2.1])The collection_{(Eσ, Aσ)}σ∈Σ is regular

if, for eachσ ∈Σ, the matrix pencilsEσ−Aσ ∈R˜n×nis regular, wheres∈C. The matrix pencil sEσ−Aσis calledregularifn= ˜nand there exists a constants∈Csuch thatdet(sEσ−Aσ)6= 0,

or, equivalently, det(sEσ −Aσ) is not the zero polynomial. The matrix pair (Ep, Ap) and the

corresponding DAE is called regular whenever(Eσ, Aσ) is regular.

To define a switched DAE system as in [3], we recall first some concepts regarding the linear subspaces where solutions to (2.14a) belong. Due to the algebraic constraints in (2.14a), the solutions to (2.14a) evolve within a linear subspace (or manifold) called theconsistency space. If the initial value is not consistent, that is,ξ(0) does not belong to the consistency space, then, jumps may appear in the solution, similar to the jumps induced by switching between modes.

Definition 2.4 (Consistency set1₎ _Given_q_∈_Σ_{, the consistency set for} _(2.14a) _when_σ ₌_q _is

given by

Oq :={ξ0 ∈Rn | ∃continuously differentiable functions ξ : [0, τ)→Rn, u: [0, τ)→Rp

s.t. Eqξ˙(t) =Aqξ(t) +Bqu(t) ∀t∈(0, τ), ξ(0) =ξ0, τ >0}

For a linear switched DAE system as in (2.14a), for eachσ ∈Σ, the consistency set is given by a linear subspace. The consistency set can be characterized using the so called consistency space Cσ, the impulse projector Πimpσ , and the function u, for each σ ∈Σ as it is shown below.

The definition of the so called differential, consistency, and impulse projectors are presented first. The consistency spaces, differential, consistency, and impulse projectors can be computed

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using the quasi-Weierstrass form and the Wong Sequences, which are introduced in [25]. The Wong sequences are used to compute the consistency space, which is calculated from the basis of linear subspaces (see [3, 26]). Notice that, for eachσ_∈Σ, the consistency spaceCσ is computed

directly from the system data of (2.14a), which is given by the matrix pair (Eσ, Aσ). Next, as

a reference for the reader, we recall the Wong sequences and the quasi-Weierstrass form.

Definition 2.5 (Wong sequences1₎_{Consider a regular matrix pair}₍_E

σ, Aσ)whereσ∈Σ. The

associated Wong sequences are defined by

ν0:=Rn, νi+1 :=Aσ−1(Eσνi), νσ∗ := \ i∈_N

νi, ∀i∈N ω0:={0}, ωi+1:=Eσ−1(Aσωi), ω∗σ :=

[ i∈N

ωi, ∀i∈N

which are nested subspaces and get stationary after exactlyςσ steps, namely,νi+1 ⊆νi, ωi+1⊇ ωi for each i ∈ N and νi+1 = νi =νσ∗, ωi+1 = ωi = ωσ∗ for all i ≥ ςσ, where the integer ςσ is

called the index of theσ-th DAE.

For any full rank matrix [Vσ, Wσ]∈Rn×n, whereVσ ∈Rn×nσ1, W_σ ∈ _Rn×nσ2, nσ

1 +nσ2 = n,

with im(Vσ) =νσ∗ and im(Wσ) =ωσ∗, the matricesTσ := [Vσ, Wσ] and Sσ := [EσVσ, AσWσ]−1

are invertible and put the matrix pair (Eσ, Aσ) into the quasi-Weierstrass form given by

Theo-rem 2.6.

Theorem 2.6 (The quasi-Weierstrass form2)Suppose(Eσ, Aσ)is regular, whereσ ∈Σ. Then,

there exist matricesSσ and Tσ that transform (Eσ, Aσ) into the quasi-Weierstrass form

(SσEσTσ, SσAσTσ) = "

Inσ 1 0

0 Nσ #

, "

Jσ 0

0 Inσ 2

#!

for some Jσ ∈Rnσ1×nσ1 and a nilpotent matrixN_σ ∈_Rnσ2×nσ2, where (N_σ)nσ2 = 0, nσ

1 +nσ2 =n, ςσ ∈N is the smallest number such that (Nσ)ςσ = 0. The identity matrices Inσ

1, Inσ2 and zero

matrices have the proper dimensions.

Given the homogeneous linear DAE system case in (2.14a) (with a fixed σ and u ≡ 0), an equivalent ODE system and its algebraic restrictions can be obtained using the well-known quasi-Weierstrass transformation for regular matrix pencils. (see [27] and references therein). Note that this transformation does not change the solutions of the homogeneous system, but

1_{This definition is adapted from [7, Lemma 2.1], [3, Theorem 2.3], [27, Theorem 2.4], and [28, Proposition}

1]. For an explicit representation ofνiand ωi, see [10, Lemma 4.2.1] and [27, Lemma 2.2]. Also, for a Matlab

script to compute these subspaces, see [29].

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instead, exposes important features of it. For each σ _∈ Σ, the consistency space for (2.14a) (with u_≡0) is given by

Cσ :=im(νσ∗)

Even though ν_σ∗ is the collection of infinitely many elements, as shown in [27], we have that

ν∗

σ := T

i∈_Nνiis a subspace ofRn. Additionally, for the homogeneous system (2.14a),νσ∗ is given

by a linear subspace of Rn (see, e.g., [3, Remark 2.2]). Then, the basis Cσ is given by a finite

set of column vectors. In the following sections, we need to describe the intersection between the consistency space and some sets in Rn. To do so, we recast the consistency space as a set

inRn as in Proposition 2.12.

Remark 2.7 Explicit forms of matrices Sσ and Tσ are given after Definition 2.5. With this

transformation, for eachσ_∈Σ, system(2.14a)is equivalent to

Eσξ˙=Aσξ+Bσu EσTσ

"

˙

v

˙

w #

=AσTσ "

v w #

+Bσu

SσEσTσ "

˙

v

˙

w #

=SσAσTσ "

v w #

+SσBσu "

I 0 0 Nσ

# "

˙

v

˙

w #

=

" Jσ 0

0 I # "

v w #

+SσBσu,

where

" v w

#

= T_σ−1ξ, and Sσ and Tσ are invertible matrices. Then, the matrix pencil of the

equivalent system is given by (SσEσTσ, SσAσTσ) [27]. Based on this equivalence, it is possible

to find a matrix pencil that, for example, decouples the differential of the algebraic variables of the system (e.g., the Kronecker normal form and the Weierstrass normal form).

Next, the consistency, differential, and impulse projectors are defined.

Definition 2.8 (Consistency, differential, and impulse projectors [30, Definition 6.4.1])Given the regular collection _{(Eσ, Aσ)}σ∈Σ as in Definition 2.3, for eachσ ∈Σ, let the matrix pencil

(SσEσTσ, SσAσTσ) andnσ1 and nσ2 be given by Definition 2.5 and Theorem 2.6. Then, for each σ_∈Σ, we define the following:

• Consistency projector: Πσ :=Tσ "

Inσ 1 0

0 0nσ 2 #

T−1 σ

• Differential projector: Πdiff

σ :=Tσ "

Inσ 1 0

0 0nσ 2 #

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• Impulse projector: Πimp

σ :=Tσ "

0nσ 1 0

0 Inσ 2 #

Sσ

Remark 2.9 Notice that for each q _∈Σ,Cq=im(Πq).

In order to describe the consistency set (or manifold) where solutions to (2.14a) belong, consider the non-homogeneous DAE system (for someq _∈Σ)

Eqξ˙=Aqξ+Bqu, (2.15)

Next in Lemma 2.10, the solution to (2.15) in a given nonempty open intervalI is provided.

Lemma 2.10 (Explicit solution formula for switched DAE systems [30, Theorem 6.4.4]) Let the open interval I := _{t : t _∈ R≥0, t < t < t} has nonempty interior, with 0 < t < t,

and the regular matrix pair (Eq, Aq). Also, let the projectors Πq, Πdiffq ,and Πimpq as defined in

Definition 2.8. The smooth functions ξ, u:I →Rn satisfy

Eqξ˙(t) =Aqξ(t) +Bqu(t)

if and only if they satisfy,

ξ(t) =eΠdiffq Aq(t−t)_Π

qc+ Z t

t

eΠdiffq Aq(t−τ)_Πdiff

q Bqu(τ)dτ− ς X k=0

Πimp

q Eq k

Πimp

q Bqu(k)(t), (2.16)

forc_∈Rn and an initial conditionξ(t) =ξ0 such that

ξ0+ ς X k=0

Πimp

q Eq k

Πimp

q Bqu(k)(t)∈Oq (2.17)

Furthermore, if t7→u(t)≡0 then(2.16)can be replaced by

˙

ξ(t) := ˜f(ξ(t)) = Πdiff

q Aqξ(t) (2.18)

Additionally, if Πimp

q Bq ≡0 then(2.16)can be replaced by

˙

ξ(t) := ˜f(ξ(t), u(t)) = Πdiff

q Aqξ(t) + Πdiffq Bqu(t) (2.19) Proof.(see proof in Appendix A)

Before introducing Proposition 2.12, where we compute the consistency set Oq for each q_∈Σ, consider the following assumption

Assumption 2.11 For eachσ_∈Σ,Πimp

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Next, let us consider three cases regarding the input uand the matrix Πimp q Bq. Proposition 2.12 (Consistency set1) Givenq_∈Σand

(D1) (Homogeneous case) considering the homogeneous system in(2.15), that ist7→u(t)≡0, theconsistencyset for the system (2.15) is given by

Oq :={ξ |ξ∈span(Cq)}

(D2) (No impulse projection) that the data of (2.15)satisfy Assumption 2.11, namelyΠimp

q Bq≡

0, theconsistencyset for the system (2.15) is given by

Oq :={ξ |ξ∈span(Cq)}

(D3) (Impulse projection) the non-homogeneous case withΠimp

q Bq 6= 0, theconsistencyset for

the system (2.15), for a given inputu(t), is given by

Oq:= (

ξ(t) ξ(t) +

ς X k=0

Πimp

q Eq k

Πimp

q Bqu(k)(t)∈Cq )

Proof. According to Lemma 2.10 (and its proof), for an initial conditioncand q∈Σ we have,

ξ(t) = Πqc− ς X k=0

Πimp q Eq

k

Πimp

q Bqu(k)(t)

Also, consider the matricesTqandSq, which are given by the Wong sequences (Lemma 2.5),

and that put (SqEqTq, SqAqTq) into the quasi-Weierstrass form (see Theorem 2.6). The linear

transformation Tq is given by

ξ(t) =Tq

v(t)

w(t)

, (2.20)

According to the proof of Lemma 2.10 in Appendix A, we know thatξ, u:I →Rnsatisfy (2.15)

if and only if v : I → Rn

q

1, w : I → _Rnq2, and u : I → _Rn satisfies (A.4) and (A.5), with

v0 w0

=T_q−1c.

Now, let us consider the consistency set for the system (2.15) in each case next

(D1) (Homogeneous case2) Considering the homogeneous system in (2.15), that ist_7→u(t)_≡0, the solution (2.16) to (2.15) is reduced to

ξ(t) =eΠdiffq Aq(t−t)_Π

qc 1_{See the definition of a consistency set in Definition 2.4.}

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For an initial condition c we have ξ(t) = Πqc, thus ξ0 = Πqc. With c ∈ Oq being

a consistent initial condition, i.e., Πσc = c = ξ0. Furthermore, according to

Defini-tions 2.4, 2.5, and [3, Definition 2.1], given a consistent initial condition, the solution to

Eqξ˙(t) =Aq(ξ(t)) belongs to the consistency space Cq for the open interval I.

Notice that given t_7→ u(t) _≡0, Equation (A.5) is reduced to w(t) = 0. Thus, given the linear transformationTq in (2.20), we have

T_q−1ξ(t) =

v(t)

w(t)

=

v(t)

0

I 0 0 0

T_q−1ξ(t) =

v(t)

0

Tq

I 0 0 0

T_q−1ξ(t) =Tq

v(t) 0

Πqξ(t) =ξ(t)

Therefore, following Remark 2.9, in the open intervalI the solution ξ:I →Rn to (2.15)

is such that

ξ(t)_∈im(Πq) =Cq t∈I

Notice that from Definition 2.5 and Remark 2.9, for the homogeneous case, the consistency set for the homogeneous system (2.15) is defined solely by the consistency spaceCq, namely

Oq:={ξ |ξ∈span(Cq)}, (2.21)

where Cq := im(νq∗). Given that Cq is a basis with finitely many column vectors, the

operatorspan overCq leads to a closed set Oq.

(D2) (No impulse projection) Lets assume that the data of (2.15) satisfy Assumption 2.11, namely Πimp

q Bq ≡0, in such case the solution (2.16) to (2.15) is reduced to ξ(t) =eΠdiffq Aq(t−t)_Π

qc+ Z t

t

eΠdiffq Aq(t−τ)_Πdiff

q Bqu(τ)dτ

Notice that given Πimp

q Bq≡0 and the invertible matrixTq, we have Tq

"

0_nq 1 0

0 I_nq 2 #

SqBq = 0 "

0_nq 1 0

0 I_nq 2 #

SqBq = 0 h

0 I_nq 2 i

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Thus, Equation (A.5) is reduced tow(t) = 0. Therefore, similar to the homogeneous case, given an initial conditionc_{∈ {}ν _|ν _∈span(Cq)}, the solution toEqξ˙(t) =Aqξ(t) +Bqu(t)

belongs to the consistency space Cq=im(Πq) for the open interval I.

Similarly as in the homogeneous case, the consistency set Oq is given by (2.21) and is

defined solely by the consistency spaceCq.

(D3) (Impulse projection) For the non-homogeneous case with Πimp

q Bq 6= 0, we have that the

solution to (2.15) is given by (2.16). Considering the linear transformation in (2.20), notice thatvandware given by Equations (A.4) and (A.5), respectively, and particularly

w is not reduced to zero equation. Following the linear transformationTq, we have

T_q−1ξ(t) =

v(t)

w(t)

ξ(t) =Tq

v(t) 0

+Tq

0

w(t)

ξ(t)₋Tq

0

w(t)

=Tq

v(t) 0

Following the steps of the proof of Lemma 2.10 in Appendix A and the consistency space of the homogeneous systems Cq, we have

ξ(t) +

ς X k=0

Πimp q Eq

k

Πimp

q Bqu(k)(t) =Tq

v(t) 0

∈Cq

Finally, given anς₋differentiable input t_7→u(t)_∈Rp, the consistency set is given by

Oq:= (

ξ ξ+

ς X k=0

Πimp q Eq

k

Πimp

q Bqu(k)(t)∈Cq )

, (2.22)

In this document, we discuss homogeneous and non-homogeneous switched DAE systems without impulse projection (cases (D1) and (D2)). In [31] and [12], the authors have discussed homogeneous switched DAE systems (case (D1)). In Section 5.1, we present invariance prin-ciples for hybrid DAE systems and homogeneous switched DAE systems. In Section 6.2, we describe the passivity properties of non-homogeneous (without impulse projection) switched DAE systems using passivity concepts for hybrid DAE systems, also we link such passivity properties to static output-feedback stability. The case (D3) is not considered in this docu-ment.

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Definition 2.13 (Solution to a non-homogeneous switched DAE system)A solution pair(φ, u)

to the switched DAE system(2.14a)that satisfies Assumption 2.11, whereφ= (φξ, σ), consists

of a piecewise constant function t_7→σ(t)_∈Σ, a piecewise continuously differentiable function

t_7→φξ(t)∈Oσ(t), andt7→u(t)∈Rp, all right continuous, such that Eσ(t)φ˙ξ(t) =Aσ(t)φξ(t) + Bσ(t)u(t)for almost allt∈domφξ, withdomφ= domφξ= domσ = domu. Let the switching

instants be denoted as ti, which satisfy 0 = t0 < t1 < t2 < . . . < ti < ti+1. The function σ:R≥0 →Σis constant over the intervals [ti, ti+1)for each i∈N. Thus, at switching instants,

the jumps in the state of the system between consistency sets are given by the consistency projectors in Definition 2.8, namely,

φ+_ξ(ti) = Πσ(ti)φ

−

ξ(ti)

where the superscript + denotes value of the state after an instantaneous change and φ−_ξ(ti)

denotes the limit from the left of the value of the state, namely φ−_ξ(ti) := lim_t→t−_i φξ(t).

Fi-nally, the derivative of the solution of the DAE(Eσ(t), Aσ(t)) is computed in the interior of the

aforementioned intervals; therefore, this notion of solution does not consider impulses.

A solution pair (φ, u) to the switched DAE system in (2.14) iscompleteif its domain domφξ

is equal toR≥0andprecompactif the solution itself is complete and bounded, where by bounded

we mean that there exists a bounded setK such that rgeφξ ⊂K.

2.3.2 Modeling hybrid systems as hybrid inclusions

The hybrid system modeling framework employed here is the one in [19] and [13], where a hybrid system is given by a hybrid inclusion H of the form

H

            

x _∈ F(x, uc) (x, uc)∈C x+ _∈ G(x, ud) (x, ud)∈D yc =hc(x)

yd =hd(x)

with state x ∈ Rq, input u = [uc>, ud>]> ∈ Rmc+md in which uc ∈ Rmc and ud ∈ Rmd are

respectively the inputs for flows and jumps. The data of H is given by a set C _⊂ Rq×Rmc,

called the flow set; a set-valued mapping F : Rq ×Rmc ⇒ Rq, called the flow map; a set D⊂Rq×Rmd, called the jump set; and a set-valued mapping G:Rq×Rmd ⇒Rq, called the jump map. The flow mapF defines the continuous dynamics on the flow setC, while the jump map Gdefines the discrete dynamics on the jump set D. At times, the dataof H is explicitly denoted as H = (C, F, D, G, hc, hd). Finally, we let y = [yc>, yd>]> ∈ Rmc+md be the output

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Sometimes it is useful to describe hybrid inclusions without inputs and output maps, in such cases we consider the hybrid inclusion given by

H: x∈Rp   

x ∈ C x˙ ∈ F(x)

x _∈D x+ _∈ _G₍_x₎

where the data of H is explicitly denoted as H = (C, F, D, G). The data of H is given by

C_⊂Rp,F :Rp ⇒Rp,D⊂Rp, and G:Rp ⇒Rp.

To guarantee a sequential compactness property of the solutions to hybrid systems, some mild conditions on the data of the hybrid inclusion are required. These conditions are listed in Assumption 2.16 below. It requires Definitions 2.14 and 2.15 which are presented first.

Definition 2.14 (Outer semicontinuity [32, Definition 5.4])A set-valued mappingM :Rp ⇒ Rq is outer semicontinuous (osc) at x ∈ Rp if for every sequence of pointsxi convergent to x

and any convergent sequence of points yi ∈ M(xi), one has y ∈ M(x), where limi→∞yi = y.

The mappingM is outer semicontinuous if it is outer semicontinuous at each x∈Rp. Given a

set S_⊂Rp,M :Rp ⇒Rq isouter semicontinuous relative to S if the set-valued mapping from Rp toRq defined byM(x) forx∈S and∅ forx6∈S is outer semicontinuous at each x∈S.

Definition 2.15 (Local boundedness [32, Definition 5.14]) A set-valued mappingM :Rp ⇒ Rq is locally bounded atx ∈Rp if there exists a neighborhood Ux of x such that M(Ux) ⊂Rq

is bounded. The mappingM islocally bounded if it is locally bounded at eachx∈Rp. Given

a set S _⊂Rp, the mappingM is locally bounded relative to S if the set-valued mapping from Rp toRq defined byM(x) forx∈S and∅ forx6∈S is locally bounded at each x∈S.

Assumption 2.16 (Hybrid basic conditions [13, Assumption 6.5]) The hybrid inclusionH= (C, F, D, G) satisfies the hybrid basic conditions if

(A1) C and Dare closed subsets of Rp;

(A2) F : Rp ⇒ Rp is outer semicontinuous, locally bounded relative to C, C ⊂ domF, and F(x) is convex for every x_∈C;

(A3) G:Rp⇒Rp is outer semicontinuous, locally bounded relative toD, and D⊂domG.

A hybrid inclusion that meets the conditions in Assumption 2.16 is said to be nominally well-posed, see [13, Definition 6.2].

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3 Introduction to hybrid DAE systems

In this section, we introduce a class of hybrid systems that model DAE systems with state-triggered jumps1. We refer to these systems as hybrid DAE systems and denote them as HDAE. In this section, we formulate an invariance principle for hybrid DAE systems.

The state vector of an HDAE is given by

x:= (ξ, χ, σ)_∈Rn×Rm×Σ,

where ξ is the state component associated with the DAE system, χ is the state component driven by a differential/difference inclusion, and σ is the state component associated with the switching signal, where Σ is a finite discrete set as defined in Section 2.3.1. The hybrid DAE system is given by the hybrid inclusion

HDAE                 

Eσ 0 0

0 Im 0

0 0 1

 

 

˙

ξ

˙

χ

˙

σ   ∈

 

Aσξ ρ(x) 0



=:F(x) x∈C 

 ξ+ χ+ σ+ 

 ∈

[ ˜ σ∈ϕ(x)

 

g(x,σ˜)

γ(x) ˜

σ 

=:G(x) x∈D

(3.1a)

where

C:= [

σ∈Σ

(Cσ∩(Oσ×Rm× {σ})) (3.1b)

D:= [

σ∈Σ

(Dσ ∩(Oσ×Rm× {σ}))∪((Rn\Oσ)×Rm× {σ}) (3.1c) g(x,σ˜) :=gD(x,σ˜)∪gO(x,σ˜) ∀x∈D,σ˜∈ϕ(x) (3.1d)

1_{A simplified version of this model was introduced in our preliminary work in [12] and [33]. The model in [12]}

considers single-valued functionsϕσ. In [33] and [34], we generalize the hybrid DAE model to allow for set-valued

mapsϕσ to make it suitable for the study of switched DAE systems. Note that the models in [12] and [33] do

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and

gD((ξ, χ, σ),σ˜) =gD(x,σ˜) :=

∅

Πσ˜gσ(ξ, χ)

ifx∈(Rn\Oσ)×Rm× {σ}

ifx_∈Dσ∩(Oσ×Rm× {σ}), (3.1e) gO((ξ, χ, σ),σ˜) =gO(x,σ˜) :=

Πσ˜ξ

∅

ifx_∈(Rn\Oσ)×Rm× {σ}

ifx∈Dσ∩(Oσ×Rm× {σ}), (3.1f) γ((ξ, χ, σ)) =γ(x) :=

χ gγ(x)

ifx_∈(Rn\Oσ)×Rm× {σ}

ifx_∈Dσ∩(Oσ×Rm× {σ}), (3.1g)

The elements of data that represent HDAE on the state space Rn ×Rm ×Σ are given by

(Eσ, Cσ, Aσ, ρ, Dσ, gσ, gγ, ϕ) and have the following properties:

• The sets Oσ are the consistency sets and the matrices Πσ are the consistency

projec-tors, which are generated using Eσ and Aσ for each σ ∈ Σ, as in Proposition 2.12 and

Definition 2.8, respectively.

• GivenAσ for eachσ ∈Σ, and a set valued mapping ρ:Rn+m+1 ⇒Rm, the flow mapF is

a set valued mapping F :Rn+m+1⇒Rn+m+1 that defines the continuous evolution of x.

• At jumps, the map g defines the changes of ξ while γ defines the changes of χ. The set-valued map ϕdetermines the changes ofσ. The jump mapGis a set-valued mapping

G:Rn+m+1 ⇒Rn+m+1 that defines the changes ofx at jumps.

• For each σ ∈ Σ, the sets Cσ and Dσ are subsets of Rn+m+1 that, together with the

consistency sets Oσ, define where the evolution of the system according to F and G are

possible, respectively. Namely, the sets C and D model where the state of the system can change according to the differential inclusion or the difference inclusion in (3.1a), respectively.

• The conditions on the state related to the algebraic equations in the DAE are given by the consistency set Oσ, while conditions not related with such algebraic equations are

modeled by Cσ and Dσ.

• The set Cσ ∩(Oσ ×Rm × {σ}) is the collection of points in Rn+m+1 where the system

is allowed to flow (since Oσ is the set of points where flow is “consistent”). The set Dσ∩(Oσ×Rm×{σ}) is from where jumps ofξandχaccording togσ andgγ, respectively,

are allowed.

For convenience, the dimensionm of the state component χis allowed to be zero, in which case the state of the system becomesx= (ξ, σ), and the state variable χas well as the mapsρ,

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is motivated (and becomes very useful) in modeling switched DAEs under dwell time switching in Section 5.1.2.

As in the framework in [13], we define solutions to hybrid DAEs using hybrid time do-mains. Therefore, during flows, solutions are parametrized byt∈R≥0, while at jumps they are

parametrized by j∈N.

Definition 3.1 (Hybrid time domain [13, Definition 2.3])A subsetT_⊂R≥0×Nis acompact

hybrid time domainif

T=

I−1 [ j=0

([tj, tj+1], j)

for some finite sequence of times 0 =t0 ≤t1 ≤t2 ≤...≤tI. It is a hybrid time domain if for each (τ, α)_∈T,T_∩([0, τ]_{× {}0,1, . . . , α_}) is a compact hybrid time domain.

To define the notion of solution for hybrid DAE systems, first we recall the definition of hybrid arc from [13].

Definition 3.2 (Hybrid arc [13, Definition 2.4]) A functionφ :T_→ Rn+m+1 is a hybrid arc

ifTis a hybrid time domain and if for each j∈N, the functiont7→φ(t, j) is locally absolutely

continuous on the intervalIj _:=_{_t_{: (}_{t, j}₎_∈_T_}_.

A hybrid arc is a solution to a hybrid DAE system if it satisfies the system dynamics. This notion is made precise next.

Definition 3.3 (Solution) A hybrid arc φ = (φξ, φχ, φσ) is a solution to HDAE if φ(0,0) ∈ C_∪D and

(S1) (Flow condition) for each j∈Nsuch that Ij :={t: (t, j)∈T}has nonempty interior 

 

φξ(t, j) φχ(t, j) φσ(t, j)  

∈C for allt∈intI

j_, _t_7→   

φξ(t, j) φχ(t, j) φσ(t, j)  

 satisfies 

 

E_φ_σ_(t,j) 0 0

0 Im 0

0 0 1

  

  

˙

φξ(t, j)

˙

φχ(t, j)

˙

φσ(t, j)   =

  

A_φ_σ_(t,j)(φξ(t, j)) ρ((φξ(t, j), φχ(t, j), φσ(t, j)))

0

  

for almost all t∈Ij

(S2) (Jump condition) for each(t, j)∈domφsuch that (t, j+ 1)∈domφ,

Invariance principles and passivity notions for switched DAE and hybrid DAE systems