A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

A New Solution Concept and Family of Relaxations for Hybrid Dynamical Systems

Tyler Westenbroek    S. Shankar Sastry T. Westenbroek and S. S. Sastry are with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. Email: \{westenbroekt, sastry\}@eecs.berkeley.edu.
Abstract

Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding their dynamical properties remain. In this paper, we develop a novel solution concept for a class of hybrid systems, which is a generalization of Filippov’s solution concept. In the mathematical theory, these hybrid Filippov solutions eliminate the notion of Zeno executions. Building on previous techniques for relaxing hybrid systems, we then introduce a family of smooth control systems that are used to approximate this solution concept. The trajectories of these relaxations vary differentiably with respect to initial conditions and inputs, may be numerically approximated using existing techniques, and are shown to converge to the hybrid Filippov solution in the limit. Finally, we outline how the results of this paper provide a foundation for future work to control hybrid systems using well-established techniques from Control Theory.

I Introduction

The hybrid dynamical systems framework has been used as an effective modeling technique for a wide range of engineering systems. However, the flexibility the framework provides does not come without its challenges. Despite considerable efforts to extract classic systems theoretic properties from hybrid systems [1], [2], [3], fundamental questions regarding even the existence and uniqueness of their executions remain, as the interplay between their discrete and continuous dynamics is not fully understood.

Zeno executions [4], executions which undergo an infinite number of discrete transitions in finite time, have proven particularly troublesome to analyze. Numerous frameworks have been proposed to regularize [5], relax [6], or otherwise transform Zeno hybrid systems [7], [8] into approximations or continuations which do not display Zeno phenomena. Yet, a single solution concept which directly describes executions past the Zeno point has remained elusive.

Meanwhile, significant progress has been made towards characterizing the topological structure of hybrid systems [1], [6], [9], [8]. In this paper, we consider a class of hybrid systems similar to those analyzed in [1], which may be endowed with the structure of a smooth manifold, or hybridfold. In particular, we find these systems appealing in light of the results from [9], which demonstrated hybrid models naturally reduce to this class of hybrid systems near periodic orbits, and broad efforts to control legged robots on low dimensional hybrid models [10].

As our first contribution, we generalize the solution concept of Filippov [11] to this class of hybrid systems. These hybrid Filippov solutions are defined using the solution to a single differential inclusion over a smooth topological manifold. These solutions are not defined using discrete transitions, thus the notion of Zeno executions cannot apply to this solution concept. We then relax the problem, introducing a family of smooth, stiff vector fields over the relaxed topology from [6], which can be used to approximate the hybrid Filippov solution in both continuous and discrete time. These relaxed vector fields are an extension of Teixeira’s method for regularizing classical Filippov systems [12], and intuitively these vector fields can be thought of as a generalization of the regularization in space from [5] and the smoothing techniques from [1] and [9]. Convergence guarantees for these relaxed hybrid trajectories to the hybrid Filppov solution are provided, suggesting hybrid dynamics are merely the limit of a family of stiff interactions. Numerous well-established control techniques [13], [14] immediately apply to our relaxations, opening new avenues to control hybrid dynamical systems using the results we establish here.

II Mathematical Notation

In this section we fix mathematical notation used throughout the paper. We assume a strong background in topology and differential geometry. If the reader is unfamiliar with any of the concepts used throughout the paper, they are refered to [15] or [14, Appendix C] for comprehensive introductions to these topics.

Given a set D, \partial D is the boundary of D and int(D) is the interior of D. For a topological space V, we let \mathcal{B}(V) denote all subsets of V. Given a metric space (X,d), we denote the ball of radius \delta centered at x\in X by B^{\delta}(x). The 2-norm is our metric of choice for finite-dimensional real spaces, unless otherwise noted. We use \overline{co}S to denote the convex closure of a set S, which is a subset of some vector space V. Given a collection of sets \mathopen{}\mathclose{{}\left\{D_{i}}\right\}_{i\in\mathcal{I}} the disjoint union of this collection is {\coprod_{i\in\mathcal{I}}D_{i}=\bigcup_{i\in\mathcal{I}}D_{i}\times\mathopen{% }\mathclose{{}\left\{i}\right\}}, which is endowed with the piecewise topology. For a topological space \mathcal{S} and a function f\colon A\to B, where A,B\subset\mathcal{S}, we define the following equivalence relation: {A\sim B=\mathopen{}\mathclose{{}\left\{(a,b)\in\mathcal{S}\times\mathcal{S}% \colon a\in f^{-1}(b)}\right\}}, and denote the set of equivalence classes of \mathcal{S} under \sim by \frac{\mathcal{S}}{\Lambda_{f}}. There is a natural quotient projection \pi\colon\mathcal{S}\to\frac{S}{\Lambda_{f}} taking each s\in S to its equivalence class [s]\in\frac{S}{\Lambda_{f}} and we endow \frac{\mathcal{S}}{\Lambda_{f}} with the finest topology that makes \pi continuous [15, Theorem A.27], the quotient toplogy. The reader is referred to [6] or [1] for details on how these concepts will used throughout this paper.

A topological space M is said to be a topological n-manifold if it is covered by an atlast of coordinate charts \mathopen{}\mathclose{{}\left\{U_{\alpha},\varphi_{\alpha}}\right\}_{\alpha\in% \mathcal{A}}, where each U_{\alpha}\subset\mathcal{M} is open and \varphi_{\alpha}\colon U_{\alpha}\to\mathbb{R}^{n} is a homeomorhpism onto its range. We say that M is a topological manifold with boundary if we instead remove the requirement that U_{\alpha} is open and we have \varphi_{\alpha}\colon U_{\alpha}\to H^{n}, for each \alpha\in\mathcal{A}, where H^{n}=\mathopen{}\mathclose{{}\left\{(x_{1},\dots,x_{n})\colon x_{n}\geq 0}\right\}. We say that a topological manifold is smooth if its coordninate charts are smooth. In particular, when we say the coordinate charts are smooth we mean that \varphi_{i}\circ\varphi_{j}^{-1} is a smooth diffeomorphism over \varphi_{j}(U_{i}\cap U_{j}), for i,j\in\mathcal{A} such that U_{i}\cap U_{j} is non-empty. Each point x\in M is endowed with a tangent space T_{x}M, which is an n-dimensional vector space, and we denote the tangent bundle of M as TM=\coprod_{x\in M}T_{x}M.

Inspired by [14, Definition 4.3.1], we define a control system to be a 3-tuple S=\mathopen{}\mathclose{{}\left(\mathcal{M},U,F}\right) where \mathcal{M} is a n-dimensional topological manifold, U\subset\mathbb{R}^{m} is a space of admissible inputs, and F\colon\mathcal{M}\times U\to T\mathcal{M} is a vector field defining the dynamics of the system, recalling that T\mathcal{M} is the tangent bundle of \mathcal{M}. We say that S is a smooth control system if \mathcal{M} is smooth topological manifold and F is a smooth map. Throughout the paper, we will consider input signals in the space of piecewise continuous controls, which we denote with PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right],U}\right). The trajectories of smooth control systems are unique, and vary differentiably with respect to their initial conditions and inputs (see e.g. [14, Chapter 4]). Given two smooth n-manifolds M,N and a mapping P\colon M\to N, then at each x\in M there is an associated linear map DP(x)\colon T_{x}M\to T_{P(x)}N, known as the pushforward. In coordinates, DP is simply the Jacobian of P. If P is a diffeomorphism, then a smooth vector field F\colon M\times U\to TM pushes forward to a unique smooth vector field DP\circ F\colon N\times U\to TN. Throughout the paper we use the term smooth to mean infinitely differentiable and it is understood that diffeomorphisms are smooth mappings.

III Filippov Solutions

We now briefly introduce Filippov’s solution concept [11] for differential equations with discontinuous right-hand sides. Let g\colon\mathbb{R}^{n}\to\mathbb{R} be a smooth regular map, and let {D_{+}=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\colon g(x)>0}\right\}}, D_{-}=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\colon g(x)<0}\right\}, and let \Sigma=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\colon g(x)=0}\right\} be a smooth (n-1)-manifold separating D_{+} and D_{-}. Define f\colon\mathbb{R}^{n}\times U\to T\mathbb{R}^{n} by

 f(x,u)=\begin{cases}f_{+}(x,u)&\text{ if }x\in D_{+}\\ f_{-}(x,u)&\text{ if }x\in D_{-},\end{cases} (1)

where f_{+} and f_{-} are smooth globally Lipschitz continuous functions from \mathbb{R}^{n}\times U\to T\mathbb{R}^{n}, and U is a space of admissible controls. Note that f is undefined and discontinuous along \Sigma. The Filippov Regularization of f is the set-valued map \mathcal{F}[f]\colon D\times U\to B(TR^{n}), where

 \mathcal{F}\mathopen{}\mathclose{{}\left[f}\right](x,u)=\overline{co}\bigcap_{% \delta>0}\bigcap_{\mu(S)=0}f(B^{\delta}(x)-S,u), (2)

and \bigcap_{\mu(S)=0} denotes the intersection over all sets of non-zero measure. We say that a Filippov solution for this system on the time interval \mathopen{}\mathclose{{}\left[0,T}\right], given data x_{0}\in D and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), is an absolutely continuous curve x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to D satisfying the differential inclusion with conditions x(0)=x_{0} and

 \dot{x}(t)\in\mathcal{F}\mathopen{}\mathclose{{}\left[f}\right](x(t),u(t))% \text{ a.e }t\in\mathopen{}\mathclose{{}\left[0,T}\right]. (3)

The following is a sufficient condition for the uniqueness of Filippov solutions from [11, Chapter 2.10,  Theorem 2].

Lemma 1

Assume that for each x,u\in\Sigma\times U that either \nabla g^{T}(x)\cdot f_{-}(x,u)>0 or \nabla g^{T}(x)\cdot f_{+}(x,u)<0. Then the Filippov solutions for the discontinuous system (1) are unique.

IV Hybrid Dynamical Systems

In this section we introduce the class of hybrid dynamical systems considered in this paper. The following definition is inspired by [6].

Definition 1

A hybrid dynamical system is a seven-tuple

 \mathcal{H}=\mathopen{}\mathclose{{}\left(\mathcal{J},\Gamma,\mathcal{D},% \mathcal{U},\mathcal{F},\mathcal{G},\mathcal{R}}\right), (4)

where:

• \mathcal{J} is a finite set indexing the discrete states of \mathcal{H};

• \Gamma\subset\mathcal{J}\times\mathcal{J} is the set of edges, forming a graphical structure over \mathcal{J}, where edge e=(j,j^{\prime})\in\Gamma corresponds to a transition from j to j^{\prime};

• \mathcal{D}=\mathopen{}\mathclose{{}\left\{D_{j}}\right\}_{j\in\mathcal{J}} is the set of domains, where D_{j}\subset\mathbb{R}^{n} is a smooth n-dimensional manifold with boundary;

• \mathcal{U}=PC\mathopen{}\mathclose{{}\left(\mathbb{R},U}\right) is the space of piecewise continuous control signals to a set of allowable inputs U\subset\mathbb{R}^{m};

• \mathcal{F}=\mathopen{}\mathclose{{}\left\{f_{j}}\right\}_{j\in\mathcal{J}} is the set of vector fields, where each f_{j}\colon\mathbb{R}^{n}\times U\to T\mathbb{R}^{n} is smooth and globally Lipschitz continuous, and defines the continuous dynamics of the system on D_{j};

• \mathcal{G}=\mathopen{}\mathclose{{}\left\{G_{e}}\right\}_{e=(j,j^{\prime})\in\Gamma} is the set of guards, where each G_{(j,j^{\prime})}\subset\partial D_{j} is a smooth embedded (n-1)-manifold;

• \mathcal{R}=\mathopen{}\mathclose{{}\left\{R_{e}}\right\}_{e=(j,j^{\prime})\in\Gamma} is the set of reset maps where, for each e=(j,j^{\prime})\in\Gamma, R_{e}\colon\mathbb{R}^{n}\to\mathbb{R}^{n} is smooth and globally Lipschitz continuous and R_{e}(G_{e})\subset\partial D_{j^{\prime}}.

Taking after [9], let us define G=\coprod_{e\in\Gamma}G_{e} and D=\coprod_{j\in\mathcal{J}}D_{j}, where we note that \partial D=\coprod_{j\in\mathcal{J}}\partial D_{j}. Next, define the map R\colon G\to\partial D by R(x)=R_{e}(x) for each x\in G_{e}, and e\in\Gamma. We endow our hybrid systems with the quotient topology introduced in [1], but borrow our notation from [6]. We define the hybrid quotient space to be \mathcal{M}=\frac{\coprod_{j\in\mathcal{J}}D_{j}}{\Lambda_{R}}. The construction of the hybrid quotient space for a simple two-mode hybrid system is depicted in Figure 1. Intuitively, the hybrid quotient space is constructed by attaching G_{e} to R_{e}(G_{e}), so that \mathcal{M} is a connected topological space. When interpreted on this space, there are no loner any jumps in hybrid trajectories. In fact, trajectories of hybrid systems, to be defined formally in the sequel, are absolutely continuous on \mathcal{M} with respect to the following metric from [6]. Let d\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}_{+} be a metric, then define d_{\mathcal{M}}\colon\mathcal{M}\times\mathcal{M}\to\mathbb{R}_{+} for x,y\in\mathcal{M} by

 d_{\mathcal{M}}(x,y)=\inf_{k\in\mathbb{N}}\mathopen{}\mathclose{{}\left\{\sum_% {i=1}^{k}d(p_{i},q_{i})\colon x=p_{1},y=q_{k},q_{i}\sim p_{i+1}}\right\}. (5)

Though we refer the reader to [6] for further details, intuitively, for two points x,y\in\mathcal{M} the distance d_{\mathcal{M}}(x,y) is simply the shortest curve connecting the two points on \mathcal{M}. Throughout the rest of the paper, whenever we refer to trajectories on \mathcal{M} as absoloutely continuous, it is understood to be with respect to this metric. Next, we invoke an assumption inspired by those made in [1] and [9, Section 3] that ensures \mathcal{M} is a smooth manifold [9, Theorem   3].

Assumption 1

The map R is a diffeomorpshism onto its range, and G\coprod R(G)=\partial D. 111This Assumption has several implications. First, it implies that, for each e\in\Gamma, R_{e} is a diffeomorphism onto its range. Secondly, the Assumption implies that for each pair of distinct edges e,e^{\prime}\in\Gamma that if G_{e}\cap G_{e^{\prime}}\neq\emptyset, then e and e^{\prime} can be combined into a single edge \hat{e}, where G_{\hat{e}}=G_{e}\cup G_{e^{\prime}}, and \hat{R}_{\hat{e}}\colon G_{\hat{e}}\to\partial D is a diffeomorphism onto its range and R_{\hat{e}}(x)=R_{e}(x) if x\in G_{e} and R_{\hat{e}}(x)=R_{e^{\prime}}(x) if x\in G_{e^{\prime}}. Thus, as far as our notation in this paper is concerned, we will proceed as if guards do not overlap.

As demonstrated by our examples, hybrid models for mechanical systems undergoing elastic impacts satisfy this Assumption. It was also shown in [9] that hybrid systems collapse to (sub)-systems satisfying this hypothesis near periodic orbits. We make one final assumption about our hybrid systems, which is not made in [1] or [9]:

Assumption 2

For each e=(j,j^{\prime})\in\Gamma there exists unit vectors \hat{g}_{e},\hat{r}_{e}\in\mathbb{R}^{n} and scalars c_{e},d_{e} such that G_{e}\subset\hat{G}_{e} and R_{e}(G_{e})\subset\hat{R}_{e}, where

1. \hat{G}_{e}\colon=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\colon g_{e% }(x)\colon=\hat{g}_{e}^{T}x-c_{e}=0}\right\}; and,

2. \hat{R}_{e}\colon=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{n}\colon r_{e% }(x)\colon=\hat{r}_{e}^{T}x-d_{e}=0}\right\},

and g_{e}(x)\leq 0 for each x\in D_{j}, and r_{e}(x)\geq 0 for each x\in D_{j^{\prime}}.

In other words, each guard set and its image are subsets of (n-1)-dimensional planes. As was demonstrated in [16], it is often possible to transform a hybrid system with non-linear guard sets into an equivalent hybrid systems satisfying Assumption 2 by adding auxiliary continuous states to the hybrid system. Employing this additional Assumptions will enable us to develop a set of techniques which will greatly simplify our analysis of the executions of hybrid systems, as will become apparent in Section VI.

Notation: For the rest of the paper let \pi\colon D\to\mathcal{M} denote the quotient map induced by R. For each j\in J, define a_{j}\colon D_{j}\to D_{j^{\prime}}\times\mathopen{}\mathclose{{}\left\{j}\right\} by a_{j}(x)=x\times\mathopen{}\mathclose{{}\left\{j}\right\}, for each x\in D_{j}. Subsequently, let \pi_{j}\colon D_{j}\to\pi\circ a_{j}(D_{j}) be defined by {\pi_{j}=\pi\circ a_{j}}, a diffeomorphism which takes each domain from \mathbb{R}^{n} to the hybrid quotient space. For each j\in\mathcal{J} we define {\mathcal{N}_{j}=\mathopen{}\mathclose{{}\left\{e\in\Gamma\colon\exists j^{% \prime}\in\mathcal{J}\text{ s.t. }e=(j,j^{\prime})}\right\}}, to be the neighborhood of mode j as in [6].

V Relaxed Hybrid Topology

We construct our relaxations on the relaxed hybrid topology from [6], which is constructed by attaching an \varepsilon-thick strip to each of the guard sets. The novelty of our relaxations arise from the smooth vector fields we impart over this topology, to be defined in Section VII. Here, we simply introduce the relaxed topology from [6] for use later.

Concretely, for each e\in\Gamma we define the relaxed strip

 S_{e}^{\varepsilon}:=\mathopen{}\mathclose{{}\left\{p+\hat{g}_{e}q\in\mathbb{R% }^{n}\colon p\in G_{e}\text{ and }q\in\mathopen{}\mathclose{{}\left[0,% \varepsilon}\right]}\right\} (6)

and then for each j\in\mathcal{J} define the relaxed domain D_{j}^{\varepsilon}=D_{j}\cup_{e\in\mathcal{N}_{j}}S_{e}^{\varepsilon}. Next, for each e=(j,j^{\prime})\in\Gamma we then define the relaxed guard set

 G_{e}^{\varepsilon}:=\mathopen{}\mathclose{{}\left\{x\in S_{e}^{\varepsilon}:g% _{e}^{\varepsilon}(x)\colon=\hat{g}_{e}^{T}x-\mathopen{}\mathclose{{}\left(c_{% e}+\varepsilon}\right)=0}\right\}. (7)

We then define R_{e}^{\varepsilon}\colon\mathbb{R}^{n}\to\mathbb{R}^{n} by

 R_{e}^{\varepsilon}(x)=R_{e}\mathopen{}\mathclose{{}\left(x-\hat{g}_{e}% \varepsilon}\right), (8)

and we note that R_{e}^{\varepsilon}(G_{e}^{\varepsilon})=R_{e}(G_{e}).

Define D^{\varepsilon}=\coprod_{j\in\mathcal{J}}D_{j}^{\varepsilon} and let G^{\varepsilon}=\coprod_{e\in\Gamma}G_{e}^{\varepsilon}. Subsequently define R^{\varepsilon}\colon G^{\varepsilon}\to\partial D^{\varepsilon} by R^{\varepsilon}(x)=R_{e}^{\varepsilon}(x) if x\in G_{e}^{\varepsilon}, for each e\in\Gamma, and then define the relaxed hybrid quotient space to be \mathcal{M}^{\varepsilon}=\frac{D^{\varepsilon}}{\Lambda_{R^{\varepsilon}}}. The construction of the relaxed hybrid quotient space is depicted in Figure 2.

Proposition 1

Assume \mathcal{H} satisfies Assumption 1. Then, for each \varepsilon>0, \mathcal{M}^{\varepsilon} is a smooth topological manifold.

{proof}

Note that for each j\in\mathcal{J} the domain D_{j}^{\varepsilon} is a smooth topological manifold with boundary. Furthermore, if Assumption 1 holds for a given hybrid system, then it also holds for its \varepsilon-relaxations. Thus, \mathcal{M}^{\varepsilon} is a smooth topological manifold, by an appeal to [9, Theorem 3].

As shown in [6], a relaxed distance metric on \mathcal{M}^{\varepsilon}, which we denote d_{\mathcal{M}^{\varepsilon}}\colon\mathcal{M}^{\varepsilon}\times\mathcal{M}^% {\varepsilon}\to\mathbb{R}_{+}, may be defined analogously to how d_{\mathcal{M}} was defined in (5). We also borrow the following metric between curves (trajectories) on \mathcal{M}^{\varepsilon} from [6], which will be used to study the convergence of our relaxed trajectories. Let \gamma_{1},\gamma_{2}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to% \mathcal{M}^{\varepsilon}, then define

 \rho^{\varepsilon}\mathopen{}\mathclose{{}\left(\gamma_{1},\gamma_{2}}\right)=% \sup\mathopen{}\mathclose{{}\left\{d_{\mathcal{M}^{\varepsilon}}\mathopen{}% \mathclose{{}\left(\gamma_{1}(t),\gamma_{2}(t)}\right)\colon t\in\mathopen{}% \mathclose{{}\left[0,T}\right]}\right\}. (9)

Notation: Let \pi^{\varepsilon}\colon\coprod D_{j}^{\varepsilon}\to\mathcal{M}^{\varepsilon} denote the quotient projection induced by R^{\varepsilon}. For each j\in\mathcal{J}, define a_{j}^{\varepsilon}\colon D_{j}^{\varepsilon}\to D_{j}^{\varepsilon}\times% \mathopen{}\mathclose{{}\left\{j}\right\} by a_{j}^{\varepsilon}(x)=x\times\mathopen{}\mathclose{{}\left\{j}\right\}, then let \pi_{j}^{\varepsilon}\colon D_{j}^{\varepsilon}\to\pi^{\varepsilon}\circ a_{j}% ^{\varepsilon}(D_{j}^{\varepsilon}) be defined by \pi_{j}^{\varepsilon}=\pi^{\varepsilon}\circ a_{j}^{\varepsilon}.

VI Hybrid Filippov Solutions

In this section, we generalize Filippov’s solution concept to our class of hybrid systems. We begin with a number of definitions which will make our intuition for this solution concept clear. First, for each e=(j,j^{\prime})\in\Gamma we define

 \Sigma_{e}\colon=\pi_{j}\mathopen{}\mathclose{{}\left(G_{e}}\right)=\pi_{j^{% \prime}}\mathopen{}\mathclose{{}\left(R_{e}(G_{e})}\right), (10)

and then define

 D_{e}\colon=\pi_{j}(int(D_{j}))\bigcup\pi_{j^{\prime}}\mathopen{}\mathclose{{}% \left(int(D_{j^{\prime}})}\right)\bigcup\Sigma_{e}. (11)

Note that, as depicted on the right of Figure 4, \Sigma_{e} is an (n-1)-dimensional manifold, which forms a surface separating the two open sets \pi_{j}\mathopen{}\mathclose{{}\left(int(D_{j})}\right) and \pi_{j^{\prime}}\mathopen{}\mathclose{{}\left(int(D_{j^{\prime}})}\right). We will extend the Filippov regularization to the following vector field over \mathcal{M}, the hybrid vector field:

Definition 2

Let \mathcal{H} be a hybrid system. We define the hybrid vector field to be F\colon\mathcal{M}\times U\to T\mathcal{M} where

 F(\pi_{j}(x),u)=D\pi_{j}\circ f_{j}(x,u)\text{ if }x\in int(D_{j}). (12)

That is, we simply push forward each vector field f_{j} onto \mathcal{M} using \pi_{j}. Much like the piecewise smooth vector field (1), F is a piecewise smooth vector field, which is discontinuous and undefined along \Sigma_{e}, for each e\in\Gamma. Thus the tuple \mathopen{}\mathclose{{}\left(\mathcal{M},U,F}\right) is a non-smooth control system and we find it natural to extend the Filippov regularization to describe its trajectories.

In order to accomplish this, for each edge e\in\Gamma, we will construct a set \hat{D}_{e}\subset\mathbb{R}^{n} and a diffeomorphism \pi_{e}\colon\hat{D}_{e}\to D_{e}. We will then be able to represent the flow of F on \hat{D}_{e} using the pushed forward vector field f_{e}=D\pi_{e}^{-1}\circ F|_{D_{e}\times U}, and we can subsequently apply the classical Filippov regularization to this local representation of F. In order to define each of these objects, we require some intermediate definitions.

For each e=(j,j^{\prime})\in\Gamma, let p_{e}\colon\mathbb{R}^{n}\to\hat{R}_{e} be defined by p_{e}(x)=x-\hat{r}_{e}r_{e}(x), the Euclidian projection onto \hat{R}_{e}. Next, define the diffeomorphism P_{e}\colon\mathbb{R}^{n}\to\mathbb{R}^{n} by

 P_{e}(x)=R_{e}^{-1}\circ p_{e}(x)+\hat{g}_{e}r_{e}(x), (13)

and consider the domain P_{e}(D_{j^{\prime}}), which we depict in Figure 3, and is the result of smoothly attaching D_{j^{\prime}} to D_{j}, by passing D_{j^{\prime}} through P_{e}. To understand this action, we first note that, as depicted in Figure 3, each point x\in D_{j^{\prime}} may be decomposed as p_{e}(x)+\hat{r}_{e}r_{e}(x), where p_{e}(x)\in\hat{R}_{e} and \hat{r}_{e}r_{e}(x) is a vector of length r_{e}(x) units in the direction \hat{r}_{e}. The map P_{e} decomposes x into p_{e}(x)+\hat{r}_{e}r_{e}(x), and sends p_{e}(x) to R_{e}^{-1}\circ p_{e}(x)\in\hat{G}_{e}, while sending \hat{r}_{e}r_{e}(x) to \hat{g}_{e}r_{e}(x), which is a vector of length r_{e}(x) units in the direction \hat{g}_{e}. In other words, P_{e} attaches \hat{R}_{e} to \hat{G}_{e} via R_{e}^{-1}, and rotates the coordinate that is transverse to \hat{R}_{e} (the direction \hat{r}_{e}) to align with the coordinate that is transverse to \hat{G}_{e} (the direction \hat{g}_{e}). 222It is easy to verify that P_{e} is a diffeomorphism, since each of its terms are smooth and it has a closed-form inverse P_{e}^{-1}(x)=R_{e}\circ\mu_{e}(x)+\hat{r}_{e}g_{e}(x), where \mu_{e}(x)=x-\hat{g}_{e}g_{e}(x) is the Euclidian projection onto \hat{G}_{e}. Indeed, note that P_{e}^{-1}\circ P_{e}(x)=R_{e}\circ\mu_{e}(R_{e}^{-1}\circ p_{e}(x)+\hat{g}_{e% }r_{e}(x))+\hat{r}_{e}g_{e}(R_{e}^{-1}\circ p_{e}(x)+\hat{g}_{e}r_{e}(x)), but \mu_{e}(R_{e}^{-1}\circ p_{e}(x)+\hat{g}_{e}r_{e}(x))=\mu_{e}(R_{e}^{-1}\circ p% _{e}(x)) and g_{e}(R_{e}^{-1}\circ p_{e}(x)+\hat{g}_{e}r_{e}(x))=r_{e}(x), thus P_{e}^{-1}\circ P_{e}(x)=R_{e}\circ\mu_{e}(R_{e}^{-1}\circ p_{e}(x))+\hat{r}_{% e}r_{e}(x)=p_{e}(x)+\hat{r}_{e}r_{e}(x)=x, as desired. Finally, for each e=(j,j^{\prime})\in\Gamma, let

 \hat{D}_{e}:=int(D_{j})\cup int(P_{e}(D_{j^{\prime}}))\cup G_{e}. (14)
Proposition 2

For each e=(j,j^{\prime})\in\Gamma the the mapping \pi_{e}\colon\hat{D}_{e}\to D_{e}, where

 \pi_{e}(x)=\begin{cases}\pi_{j}(x)&\text{ if }x\in int(D_{j})\cup G_{e}\\ \pi_{j^{\prime}}\circ P_{e}^{-1}(x)&\text{ if }x\in int(P_{e}(D_{j^{\prime}}))% \end{cases} (15)

is a diffeomorphism.

{proof}

The argument largely parallels the proof of [15, Theorem 9.29], with a mild refactoring of notation. In particular, it is easily verified that \pi_{e} is smooth and full rank on int(D_{j}) and int(P_{e}(D_{j^{\prime}})), since \pi_{j}|_{int(D_{j})}, \pi_{j^{\prime}}|_{int(P_{e}(D_{j^{\prime}}))} and P_{e} are all diffeomorphisms. Continuity of \pi_{e} is also easy to establish, as if x\in\hat{G}_{e} then \pi_{j}(x)=\pi_{j^{\prime}}\circ R_{e}(x)=\pi_{j^{\prime}}\circ P_{e}^{-1}(x), which follows since P_{e}^{-1}|_{\hat{G}_{e}}=R_{e}|_{\hat{G}_{e}}. We are now left to determining the smoothness of \pi_{e} along \Sigma_{e}, which follows by an argument analogous to the one used to establish the smoothness of the map \tilde{\Phi} in [15, Theorem 9.29]. In particular, we note that if V is any open set in D_{e}, then (V,\pi_{e}^{-1}|V) is a smooth chart on \mathcal{M}.

Other authors [9], [1] have demonstrated that it is theoretically possible to smoothly attach one domain of a hybrid system to another. However, to the best of our knowledge, we provide the first explicit representation of the diffeomorphisms (15) required for this proccess, which we are able to construct largely by virtue of Assumption 2. When either G_{e} or R_{e}(G_{e}) is a general non-linear surface, it may not be possible to write down a closed-form expression for the projections needed to construct P_{e} or its inverse, and implicit techniques [1, Lemma 2.8], [6, Theorem 3] are required.

Resuming our construction, for each e=(j,j^{\prime})\in\Gamma define the piecewise smooth vector field f_{e}\colon\hat{D}_{e}\times U\to T\mathbb{R}^{n} by

 f_{e}(x,u)=D\pi_{e}^{-1}\circ F(\pi_{e}(x),u). (16)

By appropriately evaluating the arguments of D\pi_{e}^{-1} one can obtain the following explicit representation of f_{e}:

 f_{e}(x,u)=\begin{cases}f_{j}(x,u)&\text{ if }x\in int(D_{j})\\ DP_{e}\circ f_{j^{\prime}}(P_{e}^{-1}(x),u)&\text{ if }x\in int(P_{e}(D_{j^{% \prime}})).\end{cases} (17)

We now define the hybrid Filippov regularization, using the classical Filippov regularizations of the vector fields \mathopen{}\mathclose{{}\left\{f_{e}}\right\}_{e\in\Gamma} and the maps \mathopen{}\mathclose{{}\left\{\pi_{e}}\right\}_{e\in\Gamma}.

Definition 3

Let \mathcal{H} be a hybrid system. The hybrid Filippov regularization of F is the set-valued map {\widehat{\mathcal{F}}[F]\colon\mathcal{M}\times U\to\mathcal{B}\mathopen{}% \mathclose{{}\left(T\mathcal{M}}\right)} where

 \displaystyle\widehat{\mathcal{F}}\mathopen{}\mathclose{{}\left[F}\right](\pi_% {e}(x),u) \displaystyle=D\pi_{e}\mathopen{}\mathclose{{}\left(\mathcal{F}[f_{e}](x,u)}% \right)\text{ if }x\in\hat{D}_{e}. (18)

In other words, the hybrid Filippov regularization is constructed by taking the classical Filippov regularizations of the vector fields \mathopen{}\mathclose{{}\left\{f_{e}}\right\}_{e\in\Gamma}, and then pushing each element of the resulting set-valued maps forward to T\mathcal{M}.

Definition 4

Let x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right). We say that an absolutely continuous curve x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} is a hybrid Filippov solution for this data if x(0)=x_{0} and

 \dot{x}(t)\in\widehat{\mathcal{F}}\mathopen{}\mathclose{{}\left[F}\right](x(t)% ,u(t)),\text{ a.e. }t\in\mathopen{}\mathclose{{}\left[0,T}\right]. (19)
Example 1

(Bouncing Ball) Consider the following simplified model of a ball that is bouncing vertically and loses a fraction of its energy during each bounce, which we borrow from [1]. The ball has two identical modes, \mathcal{J}_{bb}=\mathopen{}\mathclose{{}\left\{1,2}\right\}. 333As noted in [1], the use of two identical modes ensures the hybrid quotient space for the bouncing ball is a smooth manifold. For j\in\mathopen{}\mathclose{{}\left\{1,2}\right\} the continuous dynamics are given by

 D_{j}=\mathopen{}\mathclose{{}\left\{(x_{1},x_{2})\colon x_{1}\geq 0}\right\}% \text{ and }f_{j}(x_{1},x_{2})=\mathopen{}\mathclose{{}\left(x_{2},-g}\right)^% {T},

where g is the gravitational constant. Each mode has a single edge leaving it to the other mode:

 G_{(j,1-j)}=\mathopen{}\mathclose{{}\left\{(x_{1},x_{2})\colon x_{1}=0,~{}x_{2% }\leq 0}\right\}\text{ and }
 R_{(j,1-j)}(x_{1},x_{2})=(x_{1},-cx_{2})^{T},

where c\in(0,1] is the coefficient of restitution.

The hybrid quotient space for the bouncing ball, \mathcal{M}_{bb}, as well as a hybrid Filippov solution for this system are depicted on the left in Figure 5, for c<1. The trajectory flows between the two modes an infinite number of times by some finite time t_{\infty} (see [5] for details), before coming to rest at \pi_{1}((0,0))=\pi_{2}((0,0)) for all t\geq t_{\infty}. In other words, previous notions of hybrid executions (e.g. [5], [2], [1], [6], [3]) for the bouncing ball are Zeno, as they require an infinite number of reset map evaluations to define. On the other hand, the hybrid Filippov solution defines such trajectories using the solution of a single differential inclusion. 444This solution concept should not be conflated with solutions of the ”Hybrid Inclusion” [3], which defines solutions using a (possibly infinite) sequence of flows and jumps. Of course, in practice constructing such trajectories poses numerous challenges, as even classical Filippov solutions are non-trivial to simulate [17], motivating the construction of our relaxations in the following section. We conclude this section by presenting conditions under which the Hybrid Filippov solution is unique.

Assumption 3

Let \mathcal{H} be a hybrid system. For each e=(j,j^{\prime}) and (x,u)\in G_{e}\times U either \hat{g}_{e}^{T}\cdot f_{j}(x,u)>0 or \hat{r}_{e}^{T}\cdot f_{j^{\prime}}(R_{e}(x),u)<0.

Remark 1

For a given edge e=(j,j^{\prime})\in\Gamma, by carefully inspecting P_{e}, one observes that if Assumption 3 holds then the hypothesis of Lemma 1 is saticefied for f_{e}. Intuitively, this follows from the fact that P_{e} rotates vectors in the direction \hat{r}_{e} to align with the vector \hat{g}_{e}.

Theorem 1

Let \mathcal{H} be a hybrid system satisfying Assumption 3. Then the hybrid Filippov solutions of \mathcal{H} are unique.

{proof}

Let x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} be a hybrid Filippov solution for some data x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right). By our construction of \mathopen{}\mathclose{{}\left\{\pi_{e}}\right\}_{e\in\Gamma} and \mathopen{}\mathclose{{}\left\{f_{e}}\right\}_{e\in\Gamma}, there is a diffeomorphic correspondence between each segment of the curve x and a segment of a classical Filippov solution for one of the regularizations \mathopen{}\mathclose{{}\left\{\mathcal{F}[f_{e}]}\right\}_{e\in\Gamma}, as depicted in Figure 4. By Remark 1 we conclude that each of these segments is unique.

Note that the conditions for uniqueness in Theorem 1 are sufficient but not necessary. For example, though we do not prove it formally, it is possible to show that the boucing ball admits a unique, infinite hybrid Filippov solution, even though careful inspection reveals it does not satisfy the hypothesis of Theorem 1.

VII Relaxed Hybrid Vector Fields

In this section we introduce the relaxed hybrid vector fields which we use to approximate the hybrid Filippov solution. This family of vector fields will be constructed by extending Teixeira’s method [12], which approximates classical Filippov solutions using a family of smooth, stiff vector fields. The result of this relaxation will be a vector field F^{\varepsilon} such that the tuple \mathopen{}\mathclose{{}\left(\mathcal{M}^{\varepsilon},U,F^{\varepsilon}}\right) is a smooth control system. We begin by defining relaxed analogues to some of the objects defined in the previous section. First, for each edge e=(j,j^{\prime})\in\Gamma, we define

 \Sigma_{e}^{\varepsilon}=\pi_{j}^{\varepsilon}(S_{e}^{\varepsilon}), (20)

and then define

 D_{e}^{\varepsilon}\colon=\pi_{j}^{\varepsilon}(D_{j})\bigcup\pi_{j^{\prime}}^% {\varepsilon}(D_{j^{\prime}})\bigcup\Sigma_{e}^{\varepsilon}. (21)

As depicted in Figure 6, \Sigma_{(j,j^{\prime})}^{\varepsilon} forms a strip separating \pi_{j}(D_{j}) and \pi_{j^{\prime}}(D_{j^{\prime}}). The main idea behind our relaxation technique is to smoothly transition between the dynamics of mode j and the dynamics of mode j^{\prime} along \Sigma_{(j,j^{\prime})}. For each edge e=(j,j^{\prime}) we define the diffeomorphism P_{e}^{\varepsilon}\colon\mathbb{R}^{n}\to\mathbb{R}^{n} by

 P_{e}^{\varepsilon}(x)=\mathopen{}\mathclose{{}\left(R_{e}^{\varepsilon}}% \right)^{-1}\circ p_{e}(x)+\hat{g}_{e}r_{e}(x), (22)

and consider the domain P_{e}^{\varepsilon}(D_{j^{\prime}}), the result of attaching D_{j^{\prime}} to D_{j}^{\varepsilon} via P_{e}^{\varepsilon}, which is depicted on the left of Figure 6. 555It is again easy to check that P_{e}^{\varepsilon} is a diffeomorphism, whose inverse is \mathopen{}\mathclose{{}\left(P_{e}^{\varepsilon}}\right)^{-1}(x)=R_{e}^{% \varepsilon}\circ\mu_{e}^{\varepsilon}(x)+\hat{r}_{e}g_{e}^{\varepsilon}(x), where \mu_{e}^{\varepsilon}(x)=x-\hat{g}_{e}g_{e}^{\varepsilon}(x). We then define

 \hat{D}_{e}^{\varepsilon}:=int(D_{j})\cup int(P_{e}^{\varepsilon}(D_{j^{\prime% }}))\cup S_{e}^{\varepsilon}. (23)

The proof of the following result is analogous to that of Proposition 2.

Proposition 3

For each e=(j,j^{\prime})\in\Gamma the mapping \pi_{e}^{\varepsilon}\colon\hat{D}_{e}^{\varepsilon}\to D_{e}^{\varepsilon} where

 \pi_{e}^{\varepsilon}(x)=\begin{cases}\pi_{j}^{\varepsilon}(x)&\text{ if }x\in int% (D_{j}^{\varepsilon})\cup S_{e}^{\varepsilon}\\ \pi_{j^{\prime}}^{\varepsilon}\circ\mathopen{}\mathclose{{}\left(P_{e}^{% \varepsilon}}\right)^{-1}(x)&\text{ if }x\in int(P_{e}^{\varepsilon}(D_{j^{% \prime}})).\end{cases} (24)

is a diffeomorphism.

Next, we will construct a set of smooth vector fields \mathopen{}\mathclose{{}\left\{f_{e}^{\varepsilon}}\right\}_{e\in\Gamma} where f_{e}^{\varepsilon}\colon\hat{D}_{e}^{\varepsilon}\times U\to T\mathbb{R}^{n}, that will be used with the maps \mathopen{}\mathclose{{}\left\{\pi_{e}^{\varepsilon}}\right\}_{e\in\Gamma} to define F^{\varepsilon}. We use the following set of functions to smoothly transition between the dynamics of neighboring modes along the relaxed strips:

Definition 5

[18] We say that \varphi\colon\mathbb{R}\to\mathopen{}\mathclose{{}\left[0,1}\right] is a transition function if it is smooth and

1. \varphi(a)=0 if a\leq 0;

2. \varphi(a)=1 if a\geq 1; and,

3. \varphi is monotonically increasing on \mathopen{}\mathclose{{}\left(0,1}\right).

For the rest of the paper, we assume a single transition function has been chosen. Then, for each e=(j,j^{\prime})\in\Gamma and \varepsilon>0 let us define \varphi_{e}^{\varepsilon}\colon\mathbb{R}^{n}\to\mathopen{}\mathclose{{}\left[% 0,1}\right] by

 \varphi_{e}^{\varepsilon}(x)=\varphi\mathopen{}\mathclose{{}\left(\frac{g_{e}(% x)}{\varepsilon}}\right), (25)

and then define f_{e}^{\varepsilon}\colon\hat{D}_{e}^{\varepsilon}\times U\to T\mathbb{R}^{n} by

 \displaystyle f_{e}^{\varepsilon}(x,u)=\mathopen{}\mathclose{{}\left(1-\varphi% _{e}^{\varepsilon}(x)}\right)f_{j}(x,u)\\ \displaystyle+\varphi_{e}^{\varepsilon}(x)DP_{e}\circ f_{j^{\prime}}\mathopen{% }\mathclose{{}\left((P_{e}^{\varepsilon})^{-1}(x),u}\right). (26)

For edge e=(j,j^{\prime}), note that when g_{e}(x)\leq 0 (and x\in D_{j}) we have f_{e}^{\varepsilon}(x,u)=f_{j}(x,u). When g_{e}(x)\geq\varepsilon (and x\in P_{e}(D_{j^{\prime}})) we have that f_{e}^{\varepsilon}(x,u)=DP_{e}^{\varepsilon}\circ f_{j}(\mathopen{}\mathclose% {{}\left(P_{e}^{\varepsilon}}\right)^{-1}(x),u). And finally when 0\leq g_{e}(x)\leq\varepsilon (and x\in S_{e}^{\varepsilon}) f_{e}^{\varepsilon} produces a convex combination of these two vector fields. The following result follows from the main construction in [12].

Lemma 2

For each e\in\Gamma the vector field f_{e}^{\varepsilon} is smooth.

We are now ready to define the relaxed vector field we impart over \mathcal{M}^{\varepsilon}, for each \varepsilon>0.

Definition 6

Let \mathcal{H} be a hybrid system. We define the relaxed hybrid vector field F^{\varepsilon}\colon\mathcal{M}^{\varepsilon}\times U\to T\mathcal{M}^{\varepsilon} by

 F^{\varepsilon}(\pi_{e}^{\varepsilon}(x),u)=D\pi_{e}^{\varepsilon}\circ f_{e}^% {\varepsilon}(x,u)\text{ if }x\in\hat{D}_{e}^{\varepsilon}. (27)

In other words, F^{\varepsilon} is constructed by pushing forward the smooth vector fields \mathopen{}\mathclose{{}\left\{f_{e}^{\varepsilon}}\right\}_{e\in\Gamma} to \mathcal{M}^{\varepsilon}. Note that F^{\varepsilon}(\pi_{j}^{\varepsilon}(x),u)=D\pi_{j}^{\varepsilon}\circ f_{j}(% x,u) if x\in D_{j}.

Theorem 2

Let \mathcal{H} be a hybrid system. Then the tuple \mathopen{}\mathclose{{}\left(\mathcal{M}^{\varepsilon},U,F^{\varepsilon}}\right) is a smooth control system.

{proof}

The result follows from a straightforward application of Proposition 3 and Lemma 2.

The construction of the relaxed vector field F^{\varepsilon} can be thought of as a generalization of the regularization in space introduced in [5]. This vector field also approximates the smoothing discussed in [1, Section 5] and [9, Section 3], but we emphasize that we provide a constructive, explicit means to accomplish this smoothing, unlike either of these works. We also consider hybrid systems with continuous control inputs, a necessary development for the Control and Robotics communities.

We define relaxed hybrid trajectories as flows of F^{\varepsilon}:

Definition 7

Given x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), we say that the absolutely continuous curve x^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M}^{\varepsilon} is a relaxed hybrid trajectory for this data if x(0)=x_{0} and

 \dot{x}^{\varepsilon}(t)=F^{\varepsilon}(x^{\varepsilon}(t),u(t)),~{}\forall t% \in\mathopen{}\mathclose{{}\left[0,T}\right]. (28)

Relaxed hybrid trajectories are unique and vary differentiably with respect to their initial conditions and inputs, due to \mathopen{}\mathclose{{}\left(\mathcal{M}^{\varepsilon},U,F^{\varepsilon}}\right) being a smooth control system, as noted in Section II. As depicted in Figure 6, each segment of a relaxed hybrid trajectory may be explicitly constructed using an integral flow of f_{e}^{\varepsilon} and the map \pi_{e}^{\varepsilon}, for some e\in\Gamma. The algorithmic technique for integrating vector fields on the relaxed hybrid quotient space presented in [6] can be used to explicitly construct a full relaxed hybrid trajectory. We close this section by studying the convergence of relaxed hybrid trajectories to the hybrid Filippov solution. We leave the proofs of the following results to the Appendix, as they rely on supportive lemmas contained therein.

Theorem 3

Let Assumption 3 holds for \mathcal{H}. Let x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), and let x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} be the corresponding hybrid Filippov solution, guaranteed to be unique by Theorem 1. Let x^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M}^{\varepsilon} be the relaxed hybrid trajectory corresponding to this data. Then \exists C>0 such that \rho^{\varepsilon}\mathopen{}\mathclose{{}\left(x,x^{\varepsilon}}\right)\leq C\varepsilon for each \varepsilon small enough.

In other words, when Assumption 3 is satisfied, and the hybrid Filippov solution is unique, relaxed hybrid trajectories converge to the hybrid Filippov solution at a rate that is linear in \varepsilon. Next, we demonstrate that our relaxations always converge to a unique, well-defined limit even when the corresponding hybrid Filippov solution may be non-unique.

Theorem 4

For x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), let x^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M}^{\varepsilon} be the resulting relaxed hybrid trajectory, for each \varepsilon>0. Then there exists an absolutely continuous x^{0}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} such that \lim_{\varepsilon\to 0}\rho^{\varepsilon}\mathopen{}\mathclose{{}\left(x^{% \varepsilon},x^{0}}\right)=0.

Due to the uniqueness of this limit, we have found it convenient to think of hybrid systems as the limit of our relaxations, analogously to [8]. In Figure 5 we depict the relaxed hybrid quotient space of the bouncing ball, \mathcal{M}^{\varepsilon}_{bb}, and a relaxed hybrid trajectory evolving on this space. The trajectory spirals towards an equilibrium point on the relaxed strips separating the two domains. Although we do not prove it formally, it can be shown that the relaxed trajectories for the bouncing ball converge to the corresponding hybrid Filippov solution as one takes \varepsilon\to 0, even though Theorem 3 does not apply to this system. A similar result holds for our second example presented in the following section.

VIII Numerical Implementation

In this section we demonstrate how to implement the of numerical approximations for relaxed hybrid systems in [6] for the smooth vector fields we introduced in the previous section. Before defining these integrators, we first require some additional notation.

For each j\in\mathcal{J} we then define

 \hat{D}_{j}^{\varepsilon}=D_{j}^{\varepsilon}\bigcup_{e=(j,j^{\prime})\in% \mathcal{N}_{j}}P_{e}^{\varepsilon}(D_{j^{\prime}}), (29)

then define \hat{\pi}_{j}^{\varepsilon}\colon\hat{D}_{j}\to\cup_{e\in\mathcal{N}_{j}}D_{e} by

 \hat{\pi}_{j}^{\varepsilon}(x)=\begin{cases}\pi_{j}^{\varepsilon}(x)&\text{if % }x\in D_{j}^{\varepsilon}\\ \pi_{j^{\prime}}^{\varepsilon}\circ\mathopen{}\mathclose{{}\left(P_{e}^{% \varepsilon}}\right)^{-1}(x)&\text{if }x\in P_{e}(D_{j^{\prime}}),\end{cases} (30)

and finally define \hat{f}_{j}^{\varepsilon}\colon\hat{D}_{j}^{\varepsilon}\times U\to T\mathbb{R% }^{n} by

 \hat{f}_{j}^{\varepsilon}(x,u)=f_{e}^{\varepsilon}(x,u)\text{ if }x\in D_{e}^{% \varepsilon},~{}\forall e\in\mathcal{N}_{j}. (31)

These constructions will prove use full since we may not a priori which of the edges in \mathcal{N}_{j} a trajectory might leave mode j through. We will extend the following class of integrators to construct numerical approximations for relaxed hybrid trajectories.

Definition 8

[6] Given a hybrid system \mathcal{H}, we say \mathcal{A}\colon\mathbb{R}^{n}\times U\times\mathcal{J}\times\mathbb{R}\to% \mathbb{R}^{n} is a numerical integrator of order \omega, if for each j\in\mathcal{J} and h=T/N (where N\in\mathbb{N}), and each x_{0}\in D_{j} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right) we have

 \sup\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\lVert x(kh)-z^% {\varepsilon,h}(kh)}\right\rVert\colon k\in\mathopen{}\mathclose{{}\left\{0,1,% \dots,N}\right\}}\right)=O(h^{\omega}),

where x(0)=x_{0} and {\frac{\mathrm{d}}{\mathrm{d}t}}x=f_{j}^{\varepsilon}(x,u), and z^{\varepsilon,h}(0) and
z^{\varepsilon,h}((k+1)h)=\mathcal{A}(z(kh),u(kh),j,h).

As was noted in [6], this definition of a numerical integrator is compatible with a large class of discretization schemes, including Euler and the Runge-Kutta family.

Definition 9

Given a relaxed hybrid system \mathcal{H}, initial condition \pi_{j}^{\varepsilon}(x_{0})\in\pi_{j}^{\varepsilon}(D_{j}), input u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), step size h=\frac{T}{N} (where N\in\mathbb{N}), we construct the discrete approximation z^{\varepsilon,h}\colon\mathopen{}\mathclose{{}\left[0,t}\right]\to\mathcal{M}% ^{\varepsilon} according to the following algorithm.

1. Let z^{\varepsilon,h}(0)=x_{0}, t=0, k=0 and j\in\mathcal{J}.

2. If k=N, terminate the execution. Otherwise, let \gamma^{k+1}=\mathcal{A}\mathopen{}\mathclose{{}\left(z^{\varepsilon,k}(kh),u(% kh),j,h}\right).

3. For each t\in[kh,(k+1)h) set
z^{\varepsilon,h}(t)=\pi_{j}^{\varepsilon}\mathopen{}\mathclose{{}\left(\frac{% (k+1)h-t}{h}\gamma^{k+1}+\frac{t-kh}{h}z^{\varepsilon,h}(kh)}\right).

4. If \gamma^{k+1}\notin\hat{D}_{j}^{\varepsilon}, then let \bar{t}=\inf\mathopen{}\mathclose{{}\left\{t\colon z^{\varepsilon,h}(t)\in\hat% {D}_{j}^{\varepsilon}}\right\} and return z^{\varepsilon,h}|_{\mathopen{}\mathclose{{}\left[0,\bar{t}}\right]}. Terminate the execution.

5. If \exists e=(j,j^{\prime})\in\gamma such that \gamma^{k+1}\in D_{j^{\prime},e}^{\varepsilon}, set
z^{\varepsilon,h}((k+1)h)=\mathopen{}\mathclose{{}\left(P_{e}^{\varepsilon}}% \right)^{-1}(\gamma^{k+1}), set k = k+1, and set j=j^{\prime}. Go to step 2.

6. Otherwise, set z^{\varepsilon,h}((k+1)h) and k = k+1. Go to step 2.

We also recover the convergence results from [6] for completeness.

Theorem 5

Let \mathcal{H} be a hybrid system. For a given initial condition x_{0}\in D_{j} and input u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), let x^{\varepsilon} the corresponding relaxed trajectory, and let z^{\varepsilon,h} be its numerical approximation. Then, \exists C>0 such that \rho\mathopen{}\mathclose{{}\left(x^{\varepsilon},z^{\varepsilon,h}}\right)% \leq Ch^{\omega} for each h small enough.

{proof}

The proof is analogous to that of Theorem 3, except that instead of continually appealing to Lemma 3, we note that we incur an numerical error that of order h^{\omega} each time we integrate one of the vector fields \mathopen{}\mathclose{{}\left\{\hat{f}_{j}^{\varepsilon}}\right\}_{j\in% \mathcal{J}}, by the convergence of A on each of the domains \mathopen{}\mathclose{{}\left\{\hat{D}_{j}}\right\}_{j\in\mathcal{J}}.

In other words, numerical approximations of relaxed hybrid systems retain the convergence rate of the integrator \mathcal{A}. Furthermore, when Assumption 3 is satisfied, our discrete approximations converge to the hybrid Filippov solution.

Corollary 1

Let \mathcal{H} be a hybrid system that satisfies Assumption 3. For data x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), let x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} be the unique hybrid Filippov solution for this data. For each \varepsilon>0 and h>0 let z^{\varepsilon,h}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M}% ^{\varepsilon} be the numerical approximation of a relaxed execution corresponding to this data. Then \lim_{\varepsilon\to 0}\lim_{h\to 0}\rho^{\varepsilon}\mathopen{}\mathclose{{}% \left(x,z^{\varepsilon,h}}\right)=0.

{proof}

By a straightforward Application of the triangle equality and Theorems 3 and 5 we have that

 \rho^{\varepsilon}\mathopen{}\mathclose{{}\left(x,z^{\varepsilon,h}}\right)% \leq C\mathopen{}\mathclose{{}\left(\varepsilon+h^{\omega}}\right) (32)

for some C>0. The desired result follows by taking the appropriate limits.

Note that the proof of Corollary 1 also demonstrates the rate of convergence. In cases where the hybrid Filippov solution may be non-unique, our discrete approximations still converge to a well-defined limit.

Corollary 2

Let \mathcal{H} be a hybrid system. For data x_{0}\in\mathcal{M} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right), let x\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M} let x^{0}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\mathcal{M} be the limiting trajectory in Theorem 4. For each \varepsilon>0 and h>0 let z^{\varepsilon,h}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\mathcal{M}% ^{\varepsilon} be the numerical approximation of a relaxed execution corresponding to this data. Then \lim_{\varepsilon\to 0}\lim_{h\to 0}\rho^{\varepsilon}\mathopen{}\mathclose{{}% \left(x^{0},z^{\varepsilon,h}}\right)=0.

{proof}

The result follows from Theorems 4 and 5 and taking the appropriate limits.

It should be noted that since the vector fields \mathopen{}\mathclose{{}\left\{f_{j}^{\varepsilon}}\right\}_{j\in\mathcal{J}} are stiff, thus in practice in practice we expect the constant C in Theorem 5 to be large. However, there are two practical ways to overcome this issue. The first strategy is to simply use a high-order integrator to, such as a member of the Runge-Kutta family, to offset integrating vector fields with large Lipschitz constants. The other strategy is to reduce the step-size of the integrator when the discrete approximation is in a relaxed strip. Fortunately, these two effects compound.

Example 2

(Double Pendulum With a Mechanical Stop)Again, the double pendulum has two identical modes, \mathcal{J}_{dp}=\mathopen{}\mathclose{{}\left\{1,2}\right\}. For j\in\mathopen{}\mathclose{{}\left\{1,2}\right\} the continuous dynamics are given by

 D_{j}=\mathopen{}\mathclose{{}\left\{x\in\mathbb{R}^{x}\colon x_{1}\geq 0}% \right\}\text{ and }f_{j}(x)=f_{L}(x)^{T},

where the state is ordered x=\mathopen{}\mathclose{{}\left(\theta_{1},\dot{\theta}_{1},\theta_{2},\dot{% \theta}_{2}}\right), and an explicit representation of the Lagrangian dynamics prescribed by f_{L} may be found in [7]. Each mode has a single edge leaving it to the other mode:

 G_{(j,1-j)}=\mathopen{}\mathclose{{}\left\{x\in R^{4}\colon x_{3}=0,~{}x_{4}% \leq 0}\right\}\text{ and }
 R_{(j,1-j)}(x)=(x_{1},\dot{\theta}_{1}+Mc\dot{\theta}_{2},x_{3},-cx_{4})^{T},

where c\in(0,1] is the coefficient of restitution. We formally define the corresponding hybrid system in [19]. It was shown in [7] that when c\in(0,1) classical constructions of hybrid execution for this system yield Zeno trajectories near points where \theta_{2}=\dot{\theta}_{2}=0 and \theta_{1}\leq 0. Physically, such trajectories correspond to the second arm being locked in place against the stop for a time by an imaginary force. The results of a numerical simulation of the system is depicted on the right of Figure 7. Time steps where the simulation is in a relaxed strip are colored black. The second arm is repeatedly locked into place (during intervals when the blackened time-steps accumulate) until the imaginary force dissipates, at which point the second arm swings freely again. In other words, our relaxation procedure automatically recovers the completion of Lagrangian Hybrid Systems [7], but does so using a smooth dynamical system.

IX Conclusion and Future Work

In this paper we developed a novel solution concept for hybrid systems, which is a generalization of Filippov’s method, and allows us to described each of the trajectories of the system using a single differential inclusion. We then demonstrated that these trajectories can be approximated using the flows of a parameterized family of smooth control systems, and discussed how these two solution concepts compare in the limit. Controller synthesis for smooth control systems is an established discipline [13], [14] providing a new angle to control hybrid systems using our relaxations. In particular, we are currently investigating algorithmic techniques for explicitly computing variations on relaxed trajectories, developing gradient-based techniques (see e.g. [20]) to solve optimal control problems over our relaxations, and constructing feedback controllers to stabilize mechanical systems undergoing impacts.

X Acknowledgments

The results of this paper arose from a long-standing collaboration with Humberto Gonzalez.

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This Appendix contains proofs for Theorems 3 and 4. Each proof relies on a supportive lemma, which we develop before each of the main results.

-A Proof of Theorem 3

Lemma 3

Let e=(j,j^{\prime})\in\Gamma and assume that the hypothesis of Lemma 1 holds for f_{e}. Consider two initial conditions x_{0},x_{0}^{\varepsilon}\in\hat{D}_{e}^{\varepsilon} such that \mathopen{}\mathclose{{}\left\lVert x_{0}-x_{0}^{\varepsilon}}\right\rVert\leq% \bar{C}\varepsilon for some \bar{C}>0, and consider a single input u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right). Let \gamma\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\hat{D}_{e} be defined by {\frac{\mathrm{d}}{\mathrm{d}t}}\gamma(t)\in\mathcal{F}[f_{e}](\gamma(t),u(t)) and \gamma(0)=x_{0}, and let \gamma^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\hat{D}_% {e}^{\varepsilon} be defined by {\frac{\mathrm{d}}{\mathrm{d}t}}\gamma^{\varepsilon}(t)=f_{e}^{\varepsilon}(% \gamma^{\varepsilon}(t),u(t)) and \gamma^{\varepsilon}(0)=x_{0}^{\varepsilon}. Then for each \varepsilon small enough \mathopen{}\mathclose{{}\left\lVert\gamma-\gamma^{\varepsilon}}\right\rVert_{% \infty}\leq C\varepsilon, for some C>0.

{proof}

First, we show that the result holds for each \hat{u}\in PCD\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}% \right],U}\right) (where PCD denotes the class of piecewise-continuously differentiable functions), and then extend the result to our desired class of inputs. We transform each of the vector fields into a corresponding autonomous vector field, so that we can inductively call the result from [18, Lemma 2]. Consider the autonomous vector field \bar{f}_{e}(\gamma,z)=(f_{e}(\gamma,\hat{u}(z)),1)^{T}, and the solution to the differential equation {\frac{\mathrm{d}}{\mathrm{d}t}}(\gamma,z)\in\mathcal{F}[\bar{f}_{e}](\gamma,z) where (\gamma(0),z(0))=(x_{0},0). Note that z(t)=t,\forall t\in\mathopen{}\mathclose{{}\left[0,T}\right], and thus {\frac{\mathrm{d}}{\mathrm{d}t}}\gamma(t)\in\mathcal{F}[f_{e}](\gamma(t),u(t)), \forall t\in\mathopen{}\mathclose{{}\left[0,T}\right], as desired. Let \bar{f}_{e}^{\varepsilon} be the \varepsilon-relaxation of \bar{f}_{e}, namely, \bar{f}_{e}^{\varepsilon}(\gamma^{\varepsilon},z^{\varepsilon})=(f_{e}^{% \varepsilon}(\gamma^{\varepsilon},\bar{u}(z^{\varepsilon})),1)^{T}, and note that if we let {\frac{\mathrm{d}}{\mathrm{d}t}}(\gamma^{\varepsilon},z^{\varepsilon})=\bar{f}% _{e}^{\varepsilon}(\gamma^{\varepsilon},z^{\varepsilon}) with initial condition (\gamma^{\varepsilon}(0),z^{\varepsilon}(0))=(x_{0}^{\varepsilon},0), then the solution of \gamma^{\varepsilon} is as desired. Next, note that \hat{u} must be non differentiable on a finite number of points 0=t_{1}<t_{2}<...<t_{p}=T, p\in\mathbb{N}. Thus, on each interval (t_{i},t_{i+1}),\forall i=1,2,\dots,p-1, \bar{f}_{e} is continuously differentiable in z. Thus, restricting both trajectories to the time interval \mathopen{}\mathclose{{}\left[t_{1},t_{2}}\right], we have \mathopen{}\mathclose{{}\left\lVert(\gamma,z)|_{\mathopen{}\mathclose{{}\left[% t_{1},t_{2}}\right]}-(\gamma^{\varepsilon},z^{\varepsilon})|_{\mathopen{}% \mathclose{{}\left[t_{1},t_{2}}\right]}}\right\rVert_{\infty}\leq C_{1}\varepsilon for each \varepsilon small enough and some C_{1}>0, by an argument similar to [18, Lemma 2]. Then by a straightforward inductive argument we obtain \mathopen{}\mathclose{{}\left\lVert(\gamma,z)-(\gamma^{\varepsilon},z^{% \varepsilon})}\right\rVert_{\infty}\leq C_{2}\varepsilon, for some C_{2}>0, and thus \mathopen{}\mathclose{{}\left\lVert\gamma-\gamma^{\varepsilon}}\right\rVert_{% \infty}\leq C\varepsilon, for some C>0. The result for our desired u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right) follows from noting that PCD\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right],U}\right) is dense in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right],U}\right) under the L^{2} norm, meaning we may select \hat{u} such that \mathopen{}\mathclose{{}\left\lVert u-\hat{u}}\right\rVert_{2}<\delta for arbitrarily small \delta>0, and then note that \gamma^{\varepsilon} depends continuously on its input [20, Lemma 5.6.7].

(Proof of Theorem 3): We will prove the result using the representations we constructed in Sections VI and VII, namely the vector fields \mathopen{}\mathclose{{}\left\{f_{e}}\right\}_{e\in\Gamma} and \mathopen{}\mathclose{{}\left\{f_{e}^{\varepsilon}}\right\}_{e\in\Gamma}, and by inductive appeals to Lemma 3. Assume that for some t^{\prime}\in\mathopen{}\mathclose{{}\left[0,T}\right] that x(t^{\prime}),x^{\varepsilon}(t^{\prime})\in D_{e}^{\varepsilon}, and that d_{\mathcal{M}^{\varepsilon}}(x(t),x^{\varepsilon}(t))\leq C_{1}\varepsilon, for some C_{1}>0. Here, as in [6], we view x as a piecewise continuous curve on \mathcal{M}^{\varepsilon}. Let \gamma\colon\mathopen{}\mathclose{{}\left[t^{\prime},t^{\prime\prime}}\right]% \to\hat{D}_{e} (for some t^{\prime}>0) be the maximal Filippov solution of \mathcal{F}[f_{e}](\gamma(t),u(t)) with initial condition \gamma(t^{\prime})=\pi_{e}^{-1}(x(t^{\prime})), and let \gamma^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[t^{\prime},t_{% \varepsilon}^{\prime\prime}}\right]\to\hat{D}_{e}^{\varepsilon} (for some t_{\varepsilon}^{\prime\prime}>0) be the maximal integral curve of f_{e}^{\varepsilon} with initial condition \gamma^{\varepsilon}(t^{\prime})=(\pi_{e}^{\varepsilon})^{-1}(x^{\varepsilon}(% t^{\prime})), and note that \mathopen{}\mathclose{{}\left\lVert\gamma(t^{\prime})-\gamma^{\varepsilon}(t^{% \prime})}\right\rVert\leq C_{2}\varepsilon, for some C_{2}>0. Let \bar{t}=\min\mathopen{}\mathclose{{}\left\{t^{\prime\prime},t_{\varepsilon}^{% \prime\prime}}\right\}. Then by Lemma 3, \mathopen{}\mathclose{{}\left\lVert\gamma|_{\mathopen{}\mathclose{{}\left[t^{% \prime},\bar{t}}\right]}-\gamma^{\varepsilon}|_{\mathopen{}\mathclose{{}\left[% t^{\prime},\bar{t}}\right]}}\right\rVert_{\infty}\leq C_{3}\varepsilon for some C_{3}>0. This implies that d_{\mathcal{M}^{\varepsilon}}(x(t),x^{\varepsilon}(t))\leq C_{4}\varepsilon, for some C_{4}>0, and each t\in\mathopen{}\mathclose{{}\left[t^{\prime},\bar{t}}\right], since we have that x|_{\mathopen{}\mathclose{{}\left[t^{\prime},\bar{t}}\right]}=\pi_{e}\circ% \gamma|_{\mathopen{}\mathclose{{}\left[t^{\prime},\bar{t}}\right]} and x^{\varepsilon}|_{\mathopen{}\mathclose{{}\left[t^{\prime},\bar{t}}\right]}=% \pi_{e}^{\varepsilon}\circ\gamma^{\varepsilon}|_{\mathopen{}\mathclose{{}\left% [t^{\prime},\bar{t}}\right]}, and the result of [6, Theorem 21] guarantees we incur an additional error of order \varepsilon by including x on \mathcal{M}. Note that if \varepsilon is small enough then we can find e^{\prime}\in\Gamma such that x(\bar{t}),x^{\varepsilon}(\bar{t})\in D_{e^{\prime}}^{\varepsilon}. The results follows by noting that x(0)=x^{\varepsilon}(0), an inductive application of the preceding argument.

-B Proof of Theorem 4

Lemma 4

Let e\in\Gamma, x_{0}\in\hat{D}_{e} and u\in PC\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[0,T}\right]% ,U}\right). For each \varepsilon>0 let \gamma^{\varepsilon}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\hat{D}_% {e}^{\varepsilon} be the solution to {\frac{\mathrm{d}}{\mathrm{d}t}}\gamma^{\varepsilon}(t)=f_{e}^{\varepsilon}(% \gamma^{\varepsilon}(t),u(t)) with initial condition x_{0}^{\varepsilon}, and let the map \varepsilon\to x_{0}^{\varepsilon} be Lipschitz continuous and such that {x^{0}_{0}=x_{0}}. Then there exists \gamma^{0}\colon\mathopen{}\mathclose{{}\left[0,T}\right]\to\hat{D}_{e} such that \gamma^{0}(0)=x_{0} and \lim_{\varepsilon\to 0}\mathopen{}\mathclose{{}\left\lVert\gamma^{\varepsilon}% -\gamma^{0}}\right\rVert_{\infty}=0.

{proof}

Note that f_{e}^{\varepsilon} is continuously differentiable in \varepsilon for each \varepsilon>0, since it is constructed using a finite number of compositions and multiplications of functions which are each continuously differentiable in \varepsilon. Thus, {\frac{\partial f_{e}^{\varepsilon}}{\partial\varepsilon}} must be Lipschitz continuous for each \varepsilon\in\mathopen{}\mathclose{{}\left[\text{\@text@baccent{$\epsilon$}},% \bar{\varepsilon}}\right], where \bar{\varepsilon}>\text{\@text@baccent{$\epsilon$}}>0, since continuous functions are Lipschitz on compact domains. By Lemma 5.6.8 of [20], \gamma^{\varepsilon}(t) is a Lipschitz continuous function of \varepsilon, \forall t\in\mathopen{}\mathclose{{}\left[0,T}\right] and \varepsilon\in(\text{\@text@baccent{$\epsilon$}},\bar{\varepsilon}). Thus we have \lim_{\varepsilon\to\text{\@text@baccent{$\epsilon$}}}\mathopen{}\mathclose{{}% \left\lVert\gamma-\gamma^{\text{\@text@baccent{$\epsilon$}}}}\right\rVert_{% \infty}=0 for some \gamma^{\text{\@text@baccent{$\epsilon$}}}\colon\mathopen{}\mathclose{{}\left[% 0,T}\right]\to\hat{D}_{e}^{\text{\@text@baccent{$\epsilon$}}}. The desired result follows by noting that $̱\varepsilon$ is arbitrary. (Proof of Theorem 4): The proof is entirely analogous to that of Theorem 3, except Lemma 4 is called inductively in place of Lemma 3.

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