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SECTION I

MOTIVATION

Many questions still remain unanswered in the modelling and analysis of switched and hybrid systems with myriad interdependent and interlocking subsystems. These subsystems are entire systems in themselves, not only different operation modes from the whole system. In this scenario, the hybrid system has many different equilibria and some subsystems probably have no equilibrium point. Ignoring these details may lead to oversimplification. The real potential of hybrid automata lies in the capability to capture the dynamics of these kinds of systems: this is the motivation behind this work. More general than switched systems, hybrid automata explicitly consider the influence of the transition from one subsystem to another through guards, as well as impulses in the states represented by reset functions. We here define a framework to deal with multiple isolated equilibria in nonlinear hybrid automata and characterise some stability and dissipativity properties. the conditions proposed in this paper for stability and dissipativity can be automatically checked using recent formal verification techniques for hybrid systems [1].

Dissipativity in switched systems has been studied by means of common storage functions [2] and, with less restriction, multiple storage functions [3]. the expanded results of these are given in [4], [5], [6], and within the framework of differential inclusions [7]. there are also studies of feedback passivity of continuous and discrete-time switched systems [8], [9]. Dissipativity in hybrid automata has not attracted as much attention. Within hybrid systems, dissipativity has been successfully applied to study the asymptotic stability of compact sets in a general class of jump systems (see [10], [11] and references therein), the control of interconnected impulsive systems [12], or the control of impact mechanical systems [13]. The analysis of switched and hybrid systems with multiple equilibria is less common [14], [15], [16]. Our approach differs because we provide an alternative framework for hybrid automata, with reference to complex large-scale systems with different types of discontinuities, multiple isolated equilibria, and non-identical subsystem dynamic structures – which allows having different continuous state space for every subsystem. In this work, we do not consider Zeno equilibria as in [15].

In brief, the contribution of this paper is three-fold. First, we establish a framework within nonlinear hybrid automata to define different types of co-existing equilibrium points. Second, pre-existing stability conditions are adapted to illuminate the co-existence of different types of equilibria by combining common and multiple Lyapunov-like functions. Finally, we identify dissipative parts within a hybrid automaton and give the definition of group dissipativity for groups of locations of the hybrid automaton, and total dissipativity for the whole hybrid automaton. Dissipativity of the groups of discrete locations will not imply the dissipativity of the whole hybrid automaton. Additional cross-group-coupling conditions are established, and common and multiple storage-like functions are used.

SECTION II

PRELIMINARIES

Following [17], a hybrid automaton with inputs and outputs Formula TeX Source $$H=(Q, E, {\cal X}, {\cal U}, {\cal Y}, Dom, {\cal F}, Init, G, R, h)$$ is a model for a hybrid system with:

  • Discrete locations: Formula$Q=\{q_{1}, \, q_{2}, \, \ldots, \, q_{\rm N_{\rm q}}\}$.
  • Continuous state, input and output spaces: Formula${\cal X}\subseteq \BBR ^{\rm n}$, Formula${\cal U}\subseteq \BBR ^{\rm m}$ and Formula${\cal Y}\subseteq \BBR ^{\rm p}$.
  • Continuous inputs: for each Formula$q_{\rm i}\in Q$, there is one input space Formula${\cal U}_{\rm q_{\rm i}} \subseteq {\cal U}$, and Formula${\cal U}=\bigcup _{\rm q_{\rm i}\in Q} {{\cal U}_{\rm q_{\rm i}}}$.
  • Transitions: Formula$E \subseteq Q \times Q$, with Formula$E$ a finite set of edges.
  • Location domains: for each Formula$q_{\rm i}\in Q$, there is one continuous state space Formula${\cal X}_{\rm q_{\rm i}} \subseteq {\cal X}$, with Formula$\bigcup _{\rm q_{\rm i}\in Q} {{\cal X}_{\rm q_{\rm i}}}= {\cal X}$, and Formula${Dom}:Q \to 2^{{\cal X}_{\rm q_{\rm i}}}$. Formula${Dom}(q_{\rm i})\subseteq {\cal X}_{\rm q_{\rm i}}$.
  • Continuous dynamics: Formula${\cal F}=\{{\bf f}_{\rm q_{\rm i}}({\bf x},{\bf u}): q_{\rm i} \in Q\}$ is a collection of vector fields such that Formula$f_{\rm q_{\rm i}}: {{\cal X}_{\rm q_{\rm i}}}\times {{\cal U}_{\rm q_{\rm i}}} \to {\cal X}_{\rm q_{\rm i}}$. Each Formula${\bf f_{\rm q_{\rm i}}}({\bf x},\cdot)$ is Lipschitz continuous on Formula${{\cal X}_{\rm q_{\rm i}}}$ in order to ensure that in each Formula$q_{\rm i}$ the solution exists and is unique.
  • Set of initial states: Formula${Init}\subseteq \bigcup _{\rm q_{\rm i}\in Q} \, q_{\rm i}\times {{\cal X}_{\rm q_{\rm i}}}\subseteq Q\times {\cal X}$.
  • Guard maps: Formula$G:E \to 2^{\cal X}$.
  • Reset maps: Formula$R: E\times {\cal X}\times {\cal U}\to 2^{\cal X}$. For each Formula$e=(q_{\rm i},q_{\rm j})\in E$, Formula${\bf x} \in G(e)$ and Formula${\bf u} \in {{\cal U}_{\rm q_{\rm i}}}$, Formula$R(e,{\bf x},{\bf u}) \subset {{\cal X}_{\rm q_{\rm j}}}$.
  • Continuous outputs: Formula${\bf y}={\bf h}(q_{\rm i},{\bf x},{\bf u})$, Formula$h: Q \times {\cal X}_{\rm q_{\rm i}}\times {\cal U}_{\rm q_{\rm i}}\to {\cal Y}_{\rm q_{\rm i}}$. For each Formula$q_{\rm i}\in Q$, there is one output space Formula${\cal Y}_{\rm q_{\rm i}} \subseteq {\cal Y}$, and Formula${\cal Y}=\bigcup _{\rm q_{\rm i}\in Q} {{\cal Y}_{\rm q_{\rm i}}}$.

Consider the execution of Formula$H$, Formula$\phi =(\tau ,q,{\bf x})$, with hybrid time trajectory Formula$\tau =\{[t_{\rm i},t_{\rm i}^{\prime}]\}_{\rm i=0}^{\rm N}\in {\cal T}$, and Formula${\cal T}$ the set of all hybrid time trajectories [18]. We highlight that for all Formula$0\leq i< N$, Formula$t_{\rm i}\leq t_{\rm i}^{\prime}=t_{\rm i+1}$.

Definition 1

An input sequence of Formula$H$ is a collection Formula$\phi _{\rm u}=(\tau ,{\bf u})$ with hybrid time trajectory Formula$\tau =\{[t_{\rm i},t_{\rm i}^{\prime}]\}_{\rm i=0}^{\rm N}\in {\cal T}$, and the mapping Formula${\bf u}:\tau \rightarrow {\cal U}$, satisfying

  1. Initial condition. Formula${\bf u}(t_{0}) \in {\cal U}_{\rm q(t_{0})}$ with Formula$(q(t_{0}),{\bf x}(t_{0}))\in {Init}$ and Formula${\bf x}(t_{0}) \in {Dom}(q(t_{0}))$.
  2. Continuous evolution. For all Formula$i$: Formula$\forall t \in [t_{\rm i},t_{\rm i}^{\prime}]$, Formula$q(t)$ is constant and Formula$\forall t \in [t_{\rm i},t_{\rm i}^{\prime})$, Formula${\bf u}(t)\in {\cal U}_{\rm q(t)}$ is continuous.
  3. Discrete transitions. For all Formula$(q(t_{\rm i}^{\prime}),q(t_{\rm i+1}))\in E$, Formula$0\leq i \leq N-1$: Formula${\bf u}(t_{\rm i}^{\prime})\in {\cal U}_{\rm q(t_{\rm i}^{\prime})}$ and Formula$\exists {\bf u}(t_{\rm i+1}) \in {\cal U}_{\rm q(t_{\rm i+1})}$. Formula$\hfill \blacksquare$

Definition 2

An output sequence of Formula$H$ is a collection Formula$\phi _{\rm y}=(\tau ,{\bf y})$ with hybrid time trajectory Formula$\tau =\{[t_{\rm i},t_{\rm i}^{\prime}]\}_{\rm i=0}^{\rm N}\in {\cal T}$, and the mapping Formula${\bf y}:\tau \rightarrow {\cal Y}$, satisfying

  1. Initial condition. Formula${\bf y}(t_{0}) \in {\cal Y}_{\rm q(t_{0})}$ with Formula${\bf y}(t_{0})={\bf h}(q(t_{0}),{\bf x}(t_{0}),{\bf u}(t_{0}))$, and Formula$(q(t_{0}),{\bf x}(t_{0}))\in {Init}$, Formula${\bf x}(t_{0})\in {Dom}(q(t_{0}))$, Formula${\bf u}(t_{0})\in {\cal U}_{\rm q(t_{0})}$.
  2. Continuous evolution. For all Formula$i$: Formula$\forall t \in [t_{\rm i},t_{\rm i}^{\prime}]$, Formula$q(t)$ is constant, and Formula$\forall t \in [t_{\rm i},t_{\rm i}^{\prime})$ we have that Formula${\bf y}(t)={\bf h}(q(t),{\bf x}(t),{\bf u}(t))$, Formula${\bf h}$ is smooth, Formula${\bf y}(t)\in {\cal Y}_{\rm q(t)}$, Formula${\bf x}(t)\in {Dom}(q(t))$, and Formula${\bf u}(t)\in {\cal U}_{\rm q(t)}$.
  3. Discrete transitions. For all Formula$e=(q(t_{\rm i}^{\prime}),q(t_{\rm i+1}))\in E$, Formula$0\leq i \leq N-1$: Formula${\bf y}(t_{\rm i}^{\prime})={\bf h}(q(t_{\rm i}^{\prime}),{\bf x}(t_{\rm i}^{\prime}),{\bf u}(t_{\rm i}^{\prime}))$ with Formula${\bf y}(t_{\rm i}^{\prime})\in {\cal Y}_{\rm q(t_{\rm i}^{\prime})}$, Formula${\bf x}(t_{\rm i}^{\prime})=G(e)$ and Formula${\bf u}(t_{\rm i}^{\prime})\in {\cal U}_{\rm q(t_{\rm i}^{\prime})}$, and Formula$\exists {\bf y}(t_{\rm i+1}) \in {\cal Y}_{\rm q(t_{\rm i+1})}$ obtained by using Formula${\bf x}(t_{\rm i+1}) \in R(e,{\bf x}(t_{\rm i}^{\prime}),{\bf u}(t_{\rm i}^{\prime}))$. Formula$\hfill \blacksquare$

An execution Formula$\phi$, an input sequence Formula$\phi _{\rm u}$ or an output sequence Formula$\phi _{\rm y}$ is finite if Formula$\tau$ is a finite sequence ending with a closed interval, that is Formula$N< \infty$, Formula$I_{\rm N}=[t_{\rm N},t_{\rm N}^{\prime}]$ with Formula$t_{\rm N}^{\prime}< \infty$, and is infinite if Formula$\tau$ is (i) a finite sequence ending with an infinite interval (Formula$N< \infty, I_{\rm N}=[t_{\rm N},t_{\rm N}^{\prime}), t_{\rm N}^{\prime}=\infty$) or (ii) an infinite sequence (Formula$N=\infty$). The set of executions with initial condition Formula$(q(t_{0}),{\bf x}(t_{0}))$ is Formula${\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$. It is Formula${\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}^{\rm F}$ for finite executions and Formula${\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}^{\infty }$ for infinite executions.

For any Formula$q_{\rm i}\in Q$, we consider Formula$T \vert q_{\rm i}=\{t_{\rm {qi}_{1}},t_{\rm {qi}_{2}},\ldots, t_{\rm {qi}_{\rm k}},\ldots, t_{\rm {qi}_{\rm N_{\rm q_{\rm i}}}};~q(t_{\rm {qi}_{\rm k}})=q_{\rm i},k\in \BBN \}$, as the sequence of times when the location Formula$q_{\rm i}$ becomes ACTIVE, and Formula$T^{\prime} \vert q_{\rm i}=\{t_{\rm {qi}_{1}}^{\prime},t_{\rm {qi}_{2}}^{\prime},\ldots, t_{\rm {qi}_{\rm k}}^{\prime},\ldots,~t_{\rm {qi}_{\rm M_{\rm q_{\rm i}}}}^{\prime}; \, q(t_{\rm {qi}_{\rm k}})=q_{\rm i}, k\in \BBN \}$, as the sequence of times when the location Formula$q_{\rm i}$ becomes INACTIVE, with Formula$N_{\rm q_{\rm i}}$ and Formula$M_{\rm q_{\rm i}}$ the number of entrances to and exits from Formula$q_{\rm i}$, respectively. For instance, if Formula$t \in [t_{\rm {qi}_{\rm k}},t_{\rm {qi}_{\rm k}}^{\prime}]\in \tau$, Formula$q_{\rm i}$ is active, for the Formula$k$th time. We also use Formula$T^{\prime} \vert _{\rm q_{\rm i}}^{\rm q_{\rm j}}$ to denote the sequence of times when Formula${\rm q_{\rm i}}$ becomes inactive to change to another location Formula${\rm q_{\rm j}}$. We define Formula${\cal I}(T \vert q_{\rm i})$ as the set of time intervals during which location Formula$q_{\rm i}$ is active: that is, Formula${\cal I}(T \vert q_{\rm i})=\bigcup _{\rm k=1}^{\rm N_{\rm q_{\rm i}}} [t_{\rm {qi}_{\rm k}},t_{\rm {qi}_{\rm k}}^{\prime}]$.

Consider the following systems: Formula TeX Source $$\eqalignno{{\mathdot {\bf x}}(t) =&\, {\bf f}({\bf x}(t),{\bf u}(t)), ~{\bf y}(t)={\bf h}({\bf x}(t),{\bf u}(t)), &{\hbox{(1)}}\cr{\bf x}(k+1) =&\, {\bf F}({\bf x}(k),{\bf u}(k)), ~{\bf y}(k)={\bf H}({\bf x}(k),{\bf u}(k)), &{\hbox{(2)}}}$$ with Formula${\bf x} \in \BBR ^{\rm n}$, Formula${\bf u} \in \BBR ^{\rm m}$, Formula${\bf y} \in \BBR ^{\rm p}$ and Formula${\bf f},{\bf h},{\bf F},{\bf H}$ are smooth mappings and maps. the system (1) is dissipative w.r.t. the supply rate function Formula$s({\bf y},{\bf u})$, with Formula$\int _{0}^{\rm t}{\vert (s({\bf y}(\sigma),{\bf u}(\sigma))\vert \, d\sigma }< \infty, \forall t\geq 0$, if there exists a positive definite storage function Formula$V:\BBR ^{\rm n}\to \BBR$, such that for any Formula$t_{0}$ and any Formula$t_{\rm f}>t_{0}$, the following relation is satisfied for all Formula${\bf x}(t_{0})$ [19]: Formula TeX Source $$V({\bf x}(t_{\rm f}))-V({\bf x}(t_{0}))\leq \int ^{\rm t_{\rm f}}_{\rm t_{0}} s({\bf y}(\sigma),{\bf u}(\sigma)) \, d\sigma, ~\forall ({\bf x},{\bf u}).\eqno{\hbox{(3)}}$$ For Formula$V \in {\cal C}^{1}$, inequality (3) is equivalent to [19], Formula TeX Source $${{\partial V({\bf x})}\over {\partial {\bf x}}} {\bf f}({\bf x},{\bf u})\leq s(h({\bf x},{\bf u}),{\bf u}), \, \forall {\bf x} \in \BBR ^{\rm n}, \forall {\bf u} \in \BBR^{\rm m}.\eqno{\hbox{(4)}}$$ System (2) is dissipative w.r.t. the supply rate function Formula$s$ if there exists a positive definite storage function Formula$V$, such that Formula$\forall x(0)$, Formula$\forall k\in \{0,1,2,\ldots \}$ [20]: Formula TeX Source $$V({\bf x}(k+1))-V({\bf x}(k))\leq s({\bf y}(k),{\bf u}(k)), ~\forall ({\bf x}(k),{\bf u}(k)).\eqno{\hbox{(5)}}$$

SECTION III

MOTIVATING EXAMPLE

To illustrate the results in this paper, we consider a simplified model of the torsional behaviour of a conventional vertical oilwell drillstring that has multiple equilibria and is given in [17]. The system may exhibit self-excited stick-slip oscillations depending on the values of the control input to the system, Formula$u$, and the weight on the bit, Formula$W_{\rm ob}$, which is a varying parameter. the drillstring with Formula$u$ a constant can be modelled as a 5-location hybrid automaton [17].

As also shown in [17], the oscillations in the system can be eliminated using a switching controller that drives the angular velocity of the top-rotary system to a desired value Formula$x_{3\rm r}>0$. The switching control mechanism is driven by the changing sign of a function Formula$s^{\rm r}({\bf x},t)$, which is an integral function of the angular velocities. Based on this model, the closed-loop system, given in [17], can be represented by the 15-location hybrid automaton of Fig. 1, with: Formula TeX Source $$\eqalignno{q_{1}=&\, \{slip^{+}_{\rm b},slip^{+}_{\rm r}\},\, q_{2}=\{slip^{+}_{\rm b},slip^{-}_{\rm r}\}, \cr q_{3}=&\, \{slip^{-}_{\rm b},slip^{+}_{\rm r}\},\cr q_{4}=&\, \{slip^{-}_{\rm b},slip^{-}_{\rm r}\},\, q_{5}=\{slip^{+}_{\rm b},stick_{\rm r}\},\cr q_{6}=&\, \{stick_{\rm b},stick_{\rm r}\}, \cr q_{7}=&\, \{slip^{-}_{\rm b},stick_{\rm r}\},\, q_{8}=\{stick_{\rm b},slip^{+}_{\rm r}\},\cr q_{9}=&\, \{stick_{\rm b},slip^{-}_{\rm r}\},\cr q_{10}=&\, \{tr^{+},slip^{+}_{\rm r}\},\, q_{11}=\{tr^{+},stick_{\rm r}\},\, q_{12}\!=\! \{tr^{+},slip^{-}_{\rm r}\},\cr q_{13}=&\, \{tr^{-},slip^{+}_{\rm r}\},\, q_{14}=\{tr^{-},stick_{\rm r}\},\, q_{15}\!=\! \{tr^{-},slip^{-}_{\rm r}\}.}$$Formula$stick_{\rm r}$ stands for Formula$S^{\rm r}_{0}\equiv \{\vert s^{\rm r}\vert \leq \delta \}$, Formula$stick_{\rm b}$ for Formula$G_{0}^{\delta }\equiv \{\vert x_{3} \vert \leq \delta ,\,\vert u_{\rm eq}({\bf x})\vert\leq T_{\rm s_{\rm b}}\}$, Formula$slip^{+}_{\rm r}$ for Formula$S^{\rm r}_{+} \equiv \{s^{\rm r}>\delta \}$, Formula$slip^{+}_{\rm b}$ for Formula$G_{+} \{x_{3} > \delta \}$, Formula$slip^{-}_{\rm r}$ for Formula$S^{\rm r}_{-}\equiv \{s^{\rm r}< -\delta \}$, and Formula$slip^{-}_{\rm b}$ for Formula$\{x_{3} < -\delta \}$; Formula$tr^{+}$ denotes Formula$G_{+}^{\delta } \equiv \{\vert x_{3}\vert \leq \delta, \, u_{\rm eq}({\bf x}) >T_{\rm s_{\rm b}}\}$, and Formula$tr^{-}$ denotes Formula$G_{-}^{\delta } \equiv \{\vert x_{3}\vert \leq \delta, \, u_{\rm eq}({\bf x}) < -T_{\rm s_{\rm b}}\}$.

Figure 1
Fig. 1. 15-location hybrid automaton of the closed-loop drillstring.

Note that in the specification of the domains, to avoid numerical problems with zero detection in the simulation, we define a neighbourhood around zero with a small Formula$\delta > 0$.

The letters on the edges represent the 12 guards of Formula$H$: Formula${\bf a} \Leftrightarrow G_{0}^{\delta } \cap S^{\rm r}_{0}$, Formula${\bf b} \Leftrightarrow G_{-}^{\delta } \cap S^{\rm r}_{0}$, Formula${\bf c} \Leftrightarrow G_{+}^{\delta } \cap S^{\rm r}_{0}$, Formula${\bf d} \Leftrightarrow G_{0}^{\delta } \cap S^{\rm r}_{+}$, Formula${\bf e} \Leftrightarrow G_{-}^{\delta } \cap S^{\rm r}_{+}$, Formula${\bf f} \Leftrightarrow G_{+}^{\delta } \cap S^{\rm r}_{+}$, Formula${\bf g} \Leftrightarrow G_{0}^{\delta } \cap S^{\rm r}_{-}$, Formula${\bf h} \Leftrightarrow G_{-}^{\delta } \cap S^{\rm r}_{-}$, Formula${\bf i} \Leftrightarrow G_{+}^{\delta } \cap S^{\rm r}_{-}$, Formula${\bf j} \Leftrightarrow \{{\bf x}\in \BBR ^{3}:\, \vert x_{3}\vert > \delta \} \cap S^{\rm r}_{0}$, Formula${\bf k} \Leftrightarrow \{{\bf x}\in \BBR ^{3}:\, \vert x_{3}\vert > \delta \} \cap S^{\rm r}_{-}$, Formula${\bf l} \Leftrightarrow \{{\bf x}\in \BBR ^{3}:\, \vert x_{3}\vert > \delta \} \cap S^{\rm r}_{+}$.

SECTION IV

CLASSIFICATION OF EQUILIBRIA

Inspired by [21], we propose several types of equilibria, and split the discrete locations of Formula$H$ into groups, depending on these equilibria (see Fig. 2).

Figure 2
Fig. 2. Example of the division of the state space Formula${\cal X}$ of a hybrid automaton with 5 discrete locations, 3 groups of locations and 3 group equilibria. Formula${\cal X}_{\rm g_{1}}={\cal X}_{\rm q_{1}}$, Formula${\cal X}_{\rm g_{2}}={\cal X}_{\rm q_{2}}$, Formula${\cal X}_{\rm g_{3}}={\cal X}_{\rm q_{3}}\bigcup {\cal X}_{\rm q_{4}} \bigcup {\cal X}_{\rm q_{5}}$. The unique equilibrium point within group Formula$g_{3}$ belongs to Formula$q_{3}$ and Formula$q_{4}$. Moreover, Formula${\cal X}_{\rm q_{3}} \bigcap {\cal X}_{\rm q_{4}}\ne \emptyset$.

Definition 3

Formula$\overline {\bf x}_{\rm q_{\rm i}} \in \BBR ^{\rm n}$ is a non-virtual equilibrium of a discrete location Formula$q_{\rm i} \in Q$ if: (i) Formula$\exists \overline {\bf u}_{\rm q_{\rm i}} \in {\cal U}_{\rm q_{\rm i}}$ such that Formula${\bf f}_{\rm q_{\rm i}}(\overline {\bf x}_{\rm q_{\rm i}},\overline {\bf u}_{\rm q_{\rm i}})=0$ and Formula$\overline {\bf x}_{\rm q_{\rm i}} \in {\rm cl}({\cal X}_{\rm q_{\rm i}})$; (ii) Formula$\forall e \in (q_{\rm i}\times Q)\cap E$ with Formula$\overline {\bf x}_{\rm q_{\rm i}} \in G(e)$, Formula$R(e,\overline {\bf x}_{\rm q_{\rm i}},\overline {\bf u}_{\rm q_{\rm i}})=\{\overline {\bf x}_{\rm q_{\rm i}}\}$. Formula$\overline {\bf x}_{\rm q_{\rm i}}$ is isolated if it has a neighbourhood in Formula${\cal X}_{\rm q_{\rm i}}$ which contains no other equilibria. The equilibrium output for Formula$q_{\rm i}$ is Formula$\overline {\bf y}_{\rm q_{\rm i}}={\bf h}(q_{\rm i},\overline {\bf x}_{\rm q_{\rm i}},\overline {\bf u}_{\rm q_{\rm i}})$. Formula$\hfill \blacksquare$

Definition 4

Formula$\overline {\bf x}_{\rm q_{\rm i}} \in \BBR ^{\rm n}$ is a virtual equilibrium of location Formula$q_{\rm i} \in Q$ if Formula$\exists \overline {\bf u}_{\rm q_{\rm i}} \in {\cal U}_{\rm q_{\rm i}}$ such that Formula${\bf f}_{\rm q_{\rm i}}(\overline {\bf x}_{\rm q_{\rm i}},\overline {\bf u}_{\rm q_{\rm i}})=0$ and Formula$\overline {\bf x}_{\rm q_{\rm i}} \not \in {\rm cl}({\cal X}_{\rm q_{\rm i}})$, but Formula$\overline {\bf x}_{\rm q_{\rm i}} \in {\rm cl}({\cal X}_{\rm q_{\rm j}})$ for some Formula$q_{\rm j} \in Q$, Formula$q_{\rm j} \ne q_{\rm i}$. Formula$\hfill \blacksquare$

Definition 5

Let Formula$N_{\rm q}$ be the number of discrete locations of the hybrid automaton Formula$H$. Consider a partition Formula$P \subset Q$, with Formula$P=\{g_{1},g_{2},\ldots, g_{\rm N_{\rm g}}\}$ and Formula$N_{\rm g}\leq N_{\rm q}$, such that Formula$\bigcup _{\rm i=1}^{\rm N_{\rm g}} \, g_{\rm i}=Q$ and Formula$\bigcap _{\rm i=1}^{\rm N_{\rm g}} \, g_{\rm i}=\emptyset$. Let Formula$N_{\rm g_{\rm i}}$ be the number of locations within each group Formula$g_{\rm i}$, with Formula$1\leq N_{\rm g_{\rm i}} \leq N_{\rm q}$ for all Formula$i$. We associate with each group Formula$g_{\rm i}$ a subset of the state space Formula${\cal X}_{\rm g_{\rm i}}$ such that Formula$\bigcup _{\rm q_{\rm j} \in g_{\rm i} } \, {\cal X}_{\rm q_{\rm j}}={\cal X}_{\rm g_{\rm i}}$, Formula$\bigcup _{\rm i=1}^{\rm N_{\rm g}} \, {\cal X}_{\rm g_{\rm i}}={\cal X}$ and Formula$\bigcap _{\rm i=1}^{\rm N_{\rm g}} \, {\cal X}_{\rm g_{\rm i}}=\emptyset$. Then, Formula$\overline {\bf x}_{\rm g_{\rm i}} \in {\cal X}_{\rm g_{\rm i}}$ is a group equilibrium for Formula$H$ if:

  1. There exists at least one Formula$q_{\rm i} \in g_{\rm i}$ for which Formula$\overline {\bf x}_{\rm g_{\rm i}}$ is an isolated non-virtual equilibrium for Formula$q_{\rm i}$;
  2. Formula$\overline {\bf x}_{\rm g_{\rm i}}$ is the unique non-virtual equilibrium point for the discrete locations of the group Formula$g_{\rm i}$;
  3. Formula$\overline {\bf x}_{\rm g_{\rm i}}$ is not a non-virtual equilibrium for any discrete location outside group Formula$g_{\rm i}$;
  4. for Formula$e \in (g_{\rm i}\times g_{\rm i})\cap E$ with Formula$\overline {\bf x}_{\rm g_{\rm i}} \in G(e)$, Formula$R(e,\overline {\bf x}_{\rm g_{\rm i}},\cdot)=\{\overline {\bf x}_{\rm g_{\rm i}}\}$;
  5. for all Formula$e \in (g_{\rm i}\times (Q\setminus g_{\rm i}))\cap E$, Formula$\overline {\bf x}_{\rm g_{\rm i}} \notin G(e)$. Formula$\hfill \blacksquare$

Remark 1

Condition (i) of Definition 5 allows a shared non-virtual equilibrium for several discrete locations Formula$q_{\rm i} \in g_{\rm i}$. This also allows locations with no equilibrium within the same group. Note that Formula$\bigcap _{\rm q_{\rm j} \in g_{\rm i}} \, {\cal X}_{\rm q_{\rm j}}$ can be a non-empty set, allowing the situation shown in Fig. 2.

For instance, for the 15-location hybrid automaton shown in Fig. 1, we have:

  • Virtual equilibrium for Formula$q_{1}$ and Formula$q_{3}$ (for any value of Formula$x_{3\rm r}$, Formula$\eta$ and Formula$\lambda$), and for Formula$q_{8},q_{10},q_{13}$ (only if Formula$x_{3\rm r}= {{\eta }/{\lambda}}$): Formula TeX Source $${\bf \overline v}_{1}\!=\! \left(x_{3\rm r}-\! {{\eta }\over {\lambda}},~{{c_{\rm b}\left(x_{3\rm r}- {\eta \over \lambda} \right)}\over {k_{\rm t}}}+ {{T_{\rm f_{\rm b}}\left(x_{3\rm r}- {\eta \over \lambda }\right)}\over{k_{\rm t}}},~x_{3\rm r}-\! {{\eta }\over {\lambda}}\right)^{\!\!{\rm T}}\!.$$
  • Virtual equilibrium for Formula$q_{2}$: Formula TeX Source $${\bf \overline v}_{2}\!=\! \left(\! x_{3\rm r}- {{\eta }\over {\lambda}},~{{c_{\rm b}\left(x_{3\rm r}+ {\eta \over \lambda }\right)}\over {k_{\rm t}}}+ {{T_{\rm f_{\rm b}}\left(x_{3\rm r}+ {\eta \over \lambda }\right)}\over{k_{\rm t}}},~x_{3\rm r}+ {{\eta }\over {\lambda}}\right)^{\!\!{\rm T}}\!.$$
  • Group equilibrium within location Formula$q_{5}$: Formula${\bf \overline x}_{\rm g_{1}}=(x_{3\rm r},~{{(c_{\rm b}x_{3\rm r} +T_{\rm f_{\rm b}}^{+}(x_{3\rm r},W_{\rm ob}))}/ {k_{\rm t}}}, ~x_{3\rm r})^{\rm T}$.

Locations Formula$q_{6},q_{7},q_{11},q_{14},q_{4},q_{9},q_{12}$ and Formula$q_{15}$ have no equilibrium point. All the discrete locations of the 15-location hybrid automaton are grouped together in Formula$g_{1}$.

SECTION V

TOTAL STABILITY OF GROUP EQUILIBRIA IN HYBRID AUTOMATA

The stability conditions presented in this section are adapted from [14], [22], [23], [24] for nonlinear hybrid automata. Whilst in these works, a different Lyapunov function is considered for each subsystem, we have a different common Lyapunov function for each group of locations. We define a ball of radius Formula$r>0$ around a point Formula$p \in \BBR ^{\rm n}$ as Formula$B(r,{\bf p})=\{{\bf x} \in \BBR ^{\rm n}: \, \Vert {\bf x}-{\bf p}\Vert < r\}$, with Formula$\Vert \cdot \Vert$ the Euclidean 2-norm.

Definition 6

Given Formula$\overline {\bf x}_{\rm g_{\rm j}}$ a group equilibrium of Formula$H$. Formula$\overline {\bf x}_{\rm g_{\rm j}}\in {\cal X}_{\rm q_{\rm i}}$, for some Formula$q_{\rm i}\in g_{\rm j}\subseteq Q$, is: (i) stable iff for all Formula$\epsilon >0$ There exists Formula$\delta (\epsilon)>0$ such that Formula$\forall \phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$, Formula TeX Source $${\bf x}(t_{0}) \in B(\delta ,\overline {\bf x}_{\rm g_{\rm j}})\cap {\cal X}_{\rm q(t_{0})} \Rightarrow{\bf x}(t) \in B(\epsilon ,\overline {\bf x}_{\rm g_{\rm j}}),~\forall \,t \in \tau.$$(ii) attractive iff there exists Formula$\delta _{1}>0$ such that Formula$\forall \phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}^{\infty }, ~t_{\infty} =\sum _{\rm i} (t_{\rm i}^{\prime}-t_{\rm i})$, Formula TeX Source $${\bf x}(t_{0}) \in B(\delta _{1},\overline {\bf x}_{\rm g_{\rm j}})\cap {\cal X}_{\rm q(t_{0})} \Rightarrow\lim _{\rm t \to t_{\infty} } {\bf x}(t)=\overline {\bf x}_{\rm g_{\rm j}}.$$(iii) asymptotically stable if it is stable and attractive. Formula$\hfill \blacksquare$

Stability is defined for any executions, whether finite or infinite, but attractivity is defined for infinite executions only since it is a property of convergence to a certain value.

Definition 7

Consider any group of locations Formula$g_{\rm j}$ within Formula$H$, and its associated group equilibrium Formula$\overline {\bf x}_{\rm g_{\rm j}} \in {\cal X}_{\rm g_{\rm j}}$, with Formula${\cal X}_{\rm g_{\rm j}}=\bigcup _{\rm q_{\rm i} \in g_{\rm j} } \, {\cal X}_{\rm q_{\rm i}}$. A function Formula$V_{\rm g_{\rm j}}:{\cal X}_{\rm g_{\rm j}} \to \BBR$ such that: (i) Formula$V_{\rm g_{\rm j}}$ is continuously differentiable within every Formula$q_{\rm i}\in g_{\rm j}$; (ii) Formula$V_{\rm g_{\rm j}}({\bf x})>0$, Formula$\forall {\bf x}\in {\cal X}_{\rm g_{\rm j}}\setminus \{\overline {\bf x}_{\rm g_{\rm j}}\}$; (iii) Formula$V_{\rm g_{\rm j}}({\bf x})=0 \Leftrightarrow {\bf x}={\bf \overline x_{\rm g_{\rm j}}}$, is referred to as group candidate Lyapunov function for the group Formula$g_{\rm j}$ of Formula$H$. Formula$\hfill \blacksquare$

Assumption 1

Formula$H$ switches from one location to another a finite number of times Formula$S_{\rm H}$ on any finite time interval. For any finite time Formula$T$, with Formula$t_{0} < T\leq t_{\rm N}$, and Formula$T \in I_{\rm i}$ for some time interval Formula$I_{\rm i}\in \tau$, there exists Formula$K_{\rm T} \in \BBZ ^{+}$, such that during the time interval Formula$[t_{0},T]$, Formula$S_{\rm H} \leq K_{\rm T}$.

Let define Formula$Q_{\rm i}:=\{q_{\rm i}\in g_{\rm j}:\, \overline {\bf x}_{\rm g_{\rm j}}$ is a group equilibrium and non-virtual equilibrium of Formula$q_{\rm i}\}$, Formula$T \vert Q_{\rm i}=\left \{t_{\rm {Qi}_{1}},t_{\rm {Qi}_{2}},\right.~\left.\ldots, t_{\rm {Qi}_{\rm k}},\,\ldots \right \}$ as the sequence of times when any location within Formula$Q_{\rm i}$ becomes active and Formula$\left \vert T \vert Q_{\rm i} \right \vert =N_{\rm in,Qi}$. Consider Formula$\Omega _{\rm g_{\rm j}}$ as the set within Formula${\cal X}_{\rm g_{\rm j}}$ where Formula$V_{\rm g_{\rm j}}$ is a Lyapunov function: Formula TeX Source $$\eqalignno{\Omega _{\rm q_{\rm i}}=&\, \biggl \{{\bf x} \in {\cal X}_{\rm q_{\rm i}}: \, q_{\rm i}\in Q_{\rm i},\, {{\partial V_{\rm g_{\rm j}}({\bf x})}\over {\partial {\bf x}}} f_{\rm q_{\rm i}} ({\bf x},\cdot) \leq 0 \biggr \},\cr \Omega _{\rm g_{\rm j}}=&\, \{{\bf\overline x}_{\rm g_{\rm j}}\} \bigcup _{\rm q_{\rm i} \in Q_{\rm i}} \Omega _{\rm q_{\rm i}}.&{\hbox{(6)}}}$$

Now, we state a result on total stability of a group equilibrium against all co-existing equilibria in Formula$H$ for a particular case of hybrid automata, in which executions start at a location whose domain does not contain the domain of attraction of other group equilibrium different from Formula$\overline {\bf x}_{\rm g_{\rm j}}$.

Definition 8

Consider a hybrid automaton Formula$H$. Assume there is a group equilibrium of Formula$H$, Formula$\overline {\bf x}_{\rm g_{\rm j}}$, associated with the group Formula$g_{\rm j}$, with Formula$\overline {\bf x}_{\rm g_{\rm j}}$ a non-virtual equilibrium of Formula$q_{\rm i}$, for at least one Formula$q_{\rm i}\in g_{\rm j}$. Then, Formula$H$ is an Init-constrained hybrid automaton if Formula TeX Source $$\bigl (q(t_{0}),{\bf x}(t_{0})\bigr)\in \bigl ((Q_{\rm i}\times \Omega _{\rm g_{\rm j}})\bigcap {Init} \bigr),~{\bf u}(t_{0}) \in \bigcup _{\rm q_{\rm i} \in Q_{\rm i}} \, {\cal U}_{\rm q_{\rm i}}$$ for all executions Formula$\phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$, and all input sequences Formula$\phi _{\rm u}=(\tau ,{\bf u})$. Formula$\hfill \blacksquare$

Theorem 1

(Total stability of a group equilibrium of Formula$H$) Consider an Init-constrained hybrid automaton Formula$H$. Let Formula$N_{\rm g}$ be the number of groups of locations in Formula$H$. Consider Formula${\cal I}(T \vert q_{\rm i})$ as the set of time intervals during which location Formula$q_{\rm i}$ is active. Let define Formula$T \vert g_{\rm j}=\break\{t_{\rm {gj}_{1}},t_{\rm {gj}_{2}},\,\ldots, t_{\rm {gj}_{\rm k}},\,\ldots\}$ as the sequence of times when any location of group Formula$g_{\rm j}$ becomes active, and Formula$T \vert g_{\rm s}=\{t_{\rm {gs}_{1}},t_{\rm {gs}_{2}},\,\ldots, t_{\rm {gs}_{\rm k}},\,\ldots\}$ as the sequence of times when any location Formula$q_{\rm s}$, that does not belong to group Formula$g_{\rm j}$ becomes active, with Formula$q_{\rm s}\in g_{\rm s}, \, g_{\rm s} \subset Q\setminus g_{\rm j}$. Let Assumption 1 hold. Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is totally stable if there exist Formula$N_{\rm g}$ group candidate Lyapunov functions Formula$\{V_{\rm g_{1}},\ldots, V_{\rm g_{\rm N_{\rm g}}}\}$ such that Formula$\forall \phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$ and Formula$\forall \phi _{\rm u}=(\tau ,{\bf u})$, the following conditions hold:

  1. Condition related to locations within Formula$g_{\rm j}$ for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a non-virtual equilibrium. Formula$\forall t \in \bigcup _{\rm q_{\rm i}\in Q_{\rm i}}{\cal I}(T \vert q_{\rm i})$, Formula$\forall q_{\rm i}\in Q_{\rm i}$, and for Formula${\bf x}(t)\in {Dom}(q_{\rm i})$, Formula${\bf u}(t)\in {\cal U}_{\rm q_{\rm i}}$: Formula TeX Source $${{\partial V_{\rm g_{\rm j}}({\bf x}(t))}\over {\partial {\bf x}}} f_{\rm q_{\rm i}} ({\bf x}(t),{\bf u}(t)) \leq 0.\eqno{\hbox{(7)}}$$
  2. Condition related to the entrances to any Formula$q_{\rm i}$ for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a non-virtual equilibrium. Formula$\forall k \in \{1,\ldots, N_{\rm in,Qi}-1\}$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}(t_{\rm {Qi}_{\rm k+1}}))-\! V_{\rm g_{\rm j}}({\bf x}(t_{\rm {Qi}_{\rm k}})) \leq0, ~{\rm with~} t_{\rm {Qi}_{\rm k}}, t_{\rm {Qi}_{\rm k+1}}\! \in \! T \vert Q_{\rm i}.\eqno{\hbox{(8)}}$$
  3. Conditions related to locations of Formula$H$ for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is not a non-virtual equilibrium. For every Formula$q_{\rm r}\in Q\setminus Q_{\rm i}$:

    1. There exists Formula$\lambda >0$ such that for every time Formula$q_{\rm r}$ is active: Formula TeX Source $$V_{\rm g_{\rm r}}({\bf x}(t)) \leq \lambda V_{\rm g_{\rm j}}({\bf x}(t^{\ast }_{\rm {Qi}_{\rm k}})), ~{\rm for~} t\in [t_{\rm {qr}_{\rm k}},t_{\rm {qr}_{\rm k}}^{\prime}],\eqno{\hbox{(9)}}$$ with Formula$q_{\rm r}\in g_{\rm r}$, and Formula$t^{\ast }_{\rm {Qi}_{\rm k}}=\max _{\rm k} \{t_{\rm {Qi}_{\rm k}} \in T \vert Q_{\rm i}: \, t_{\rm {Qi}_{\rm k}} \leq t_{\rm {qr}_{\rm k}}\}$ the last switch-on time before entering Formula$q_{\rm r}$ of any location Formula$q_{\rm i} \in Q_{\rm i}$, with Formula$t_{0}\leq t^{\ast }_{\rm {Qi}_{\rm k}}\leq t_{\rm {qr}_{\rm k}}$. Note that if Formula$q_{\rm r}\in g_{\rm j}\setminus Q_{\rm i}$, we substitute Formula$V_{\rm g_{\rm r}}$ by Formula$V_{\rm g_{\rm j}}$ in (9);
    2. Formula$\forall t \in {\cal I}(T \vert q_{\rm r}), {\bf x}(t)$ does not exhibit finite escape times, i.e., Formula$\nexists t, \Vert {\bf x}(t) \Vert \to \infty$ as Formula$t\to t_{\rm e} < \infty$.
  4. Cross-group-coupling conditions when entering Formula$g_{\rm j}$ from any other group.

    1. For every group Formula$g_{\rm s} \subset Q\setminus g_{\rm j}$, if Formula$\exists t_{\rm {gs}_{\rm k}}$, with Formula$t_{\rm {gs}_{\rm k}} \in T \vert g_{\rm s}$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}(t_{\rm {gj}_{\rm k}})) \leq V_{\rm g_{\rm s}}({\bf x}(t_{\rm {gs}_{\rm k}}^{\ast })), \eqno{\hbox{(10)}}$$Formula$\forall t_{\rm {gj}_{\rm k}} \in T \vert g_{\rm j}$ for which any location of Formula$g_{\rm j}$ becomes active coming from any location of group Formula$g_{\rm s}$, with Formula$t^{\ast }_{\rm {gs}_{\rm k}}=\max _{\rm k} \{t_{\rm {gs}_{\rm k}} \in T \vert g_{\rm s}: \, t_{\rm {gs}_{\rm k}} \leq t_{\rm {gj}_{\rm k}}\}$ the last time when a location within Formula$g_{\rm s}$ became active before entering any location of Formula$g_{\rm j}$.
    2. Condition on resets. For every Formula$(q_{\rm s},q_{\rm j})\in E$, with Formula$q_{\rm s}\in g_{\rm s}\subset Q \setminus g_{\rm j}$ and Formula$q_{\rm j}\in g_{\rm j}$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}^{+}) -V_{\rm g_{\rm s}}({\bf x}(t^{\prime}_{\rm {qs}_{\rm k}})) \leq0,\eqno{\hbox{(11)}}$$ for all Formula$t^{\prime}_{\rm {qs}_{\rm k}} \in T^{\prime} \vert _{\rm q_{\rm s}}^{\rm q_{\rm j}}$ such that Formula${\bf x}(t^{\prime}_{\rm {qs}_{\rm k}})\in G(q_{\rm s},q_{\rm j})$, Formula${\bf x}^{+}=R(q_{\rm s},q_{\rm j},{\bf x}(t^{\prime}_{\rm {qs}_{\rm k}}),{\bf u}(t^{\prime}_{\rm {qs}_{\rm k}}))$, Formula${\bf u}(t^{\prime}_{\rm {qs}_{\rm k}}) \in {\cal U}_{\rm q_{\rm s}}$, with Formula$t^{\prime}_{\rm {qs}_{\rm k}}$ the time when Formula$q_{\rm s}$ becomes inactive to change to any Formula$q_{\rm j} \in g_{\rm j}$. Formula$\hfill \blacksquare$

The total stability conditions can be strengthened to total asymptotic stability as stated next.

Theorem 2

(Total asymptotic stability of a group equilibrium of Formula$H$) In addition to conditions of Theorem 1, if (7) is a strict inequality and one of the following conditions is satisfied for all Formula$\phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}^{\infty }$ and their associated infinite input sequences: (i) condition (8) is substituted by the fact that for all Formula$q_{\rm i}\in Q_{\rm i}$ the sequence Formula$\{V_{\rm g_{\rm j}}({\bf x}(t_{\rm {Qi}_{\rm k}}))\}$ converges to zero as Formula$k \rightarrow \infty$; or (ii) for some Formula$q_{\rm i}\in Q_{\rm i}$, the set Formula$T \vert q_{\rm i}$ is finite and Formula$q(t)=q_{\rm i}$ for all Formula$t \in [t_{\rm {qi}_{\rm N_{\rm q_{\rm i}}}},\infty)$, with Formula$t_{\rm {qi}_{\rm N_{\rm q_{\rm i}}}} \in T \vert q_{\rm i}$ the last switch-on time for Formula$q_{\rm i}$, then Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a totally asymptotically stable equilibrium of Formula$H$ in the sense of Lyapunov. Formula$\hfill \blacksquare$

Remark 2

The case of having the same non-virtual equilibrium point for all the locations of Formula$H$ is a special case of our grouping of locations.

SECTION VI

DISSIPATIVE GROUPS WITHIN A HYBRID AUTOMATON AND TOTAL DISSIPATIVITY

We introduce the notion of group dissipativity for each group of locations of Formula$H$ and total dissipativity for the whole hybrid automaton. Two key differences from previous works are: 1) multiple isolated equilibria are present in the system, and some locations might have no equilibrium, 2) due to the nature of hybrid automata, jumps between locations at switching times are considered. We use multiple storage functions, different for each group, whilst a group of locations will share a common storage function.

To study the dissipativity in hybrid automata, we can exploit the dissipativity of groups of locations to state the dissipativity of the whole hybrid automaton. This is done by establishing appropriate input and output relationships between the groups of locations.

Definition 9

Let Formula$T \vert g_{\rm j}$ be the sequence of times when any location of group Formula$g_{\rm j}$ becomes active and Formula$N_{\rm in,gj}$ the number of these entrances to any location Formula$q_{\rm j} \in g_{\rm j}$. Under Assumption 1, a group of locations Formula$g_{\rm j}$ of Formula$H$ is group dissipative w.r.t. the supply functions Formula$s_{\rm q_{\rm j}}({\bf y},{\bf u})$ defined for each Formula$q_{\rm j}\in g_{\rm j}$, if there exists a group storage-like function Formula$V_{\rm g_{\rm j}}({\bf x})$ satisfying the conditions of a group candidate Lyapunov function, such that for all executions Formula$\phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$, and all input and output sequences Formula$\phi _{\rm u}=(\tau ,{\bf u})$, Formula$\phi _{\rm y}=(\tau ,{\bf y})$:

  1. Condition on discrete locations. Formula$\forall q_{\rm j} \in g_{\rm j}$, and Formula$\forall t \in \tau$ for which Formula$q(t)=q_{\rm j}$: Formula TeX Source $${{\partial V_{\rm g_{\rm j}}({\bf x}(t))}\over {\partial {\bf x}}} f_{\rm q_{\rm j}} ({\bf x}(t),{\bf u}(t)) \leq s_{\rm q_{\rm j}} ({\bf y}(t),{\bf u}(t)), \eqno{\hbox{(12)}}$$ with Formula${\bf x}(t)\in {Dom}(q_{\rm j})$, Formula${\bf u}(t)\in {\cal U}_{\rm q_{\rm j}}$, Formula${\bf y}(t) \in {\cal Y}_{\rm q_{\rm j}}$.
  2. Condition related to the entrances to locations within Formula$g_{\rm j}$. Formula$\forall k \in \{1,\ldots, N_{\rm in,gj}-1\}$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}(t_{\rm {gj}_{\rm k+1}}))-V_{\rm g_{\rm j}}({\bf x}(t_{\rm {gj}_{\rm k}})) \leq s ({\bf y}(t_{\rm {gj}_{\rm k}}),{\bf u}(t_{\rm {gj}_{\rm k}})),\eqno{\hbox{(13)}}$$ with Formula$t_{\rm {gj}_{\rm k}}, t_{\rm {gj}_{\rm k+1}} \in T \vert g_{\rm j}$, and Formula$s=s_{\rm q_{\rm j}} ({\bf y}(t_{\rm {gj}_{\rm k}}),{\bf u}(t_{\rm {gj}_{\rm k}}))$ if Formula$q_{\rm j} \in g_{\rm j}$ became active at Formula$t_{\rm {gj}_{\rm k}}$. Formula$\hfill \blacksquare$

Condition (12) is equivalent to the dissipation inequality (4), and must be verified for all time intervals that every discrete location in Formula$g_{\rm j}$ is active. Furthermore, condition (13) generalises for dissipative systems the passivity conditions given in [3]. This is an extra condition which guarantees that the switching sequence only adds a bounded amount of energy into the system. Since sequences of values of the group storage functions are considered in discrete time, it is more appropriate to use the dissipation inequality for discrete-time systems (5). The time gap between consecutive entrances to any Formula$q_{\rm j}$ in Formula$g_{\rm j}$ includes the time when Formula$q_{\rm j}$ is active and inactive. Thus, (13) considers the energy stored by the location while inactive, and is bounded by the supplied energy calculated at the most recent entrance to Formula$q_{\rm j}$.

Inspired by the results of [4] and [5], condition (12) can be relaxed as follows.

Definition 10

Let Formula$N_{\rm q_{\rm j}}$ and Formula$M_{\rm q_{\rm j}}$ be the number of entrances to and exits from Formula$q_{\rm j}$, respectively; and consider the set-up of Definition 9. A group of locations Formula$g_{\rm j}$ of Formula$H$ is weakly group dissipative w.r.t. the supply functions Formula$s_{\rm q_{\rm j}}({\bf y},{\bf u})$ of all Formula$q_{\rm j}\in g_{\rm j}$, if:

  1. Condition on discrete locations. Formula$\forall q_{\rm j} \in g_{\rm j}$, and Formula$\forall t \in \tau$ for which Formula$q(t)=q_{\rm j}$, instead of (12):

    1. If Formula$N_{\rm q_{\rm j}}=M_{\rm q_{\rm j}}$: Formula TeX Source $$\displaylines{\sum\limits _{\rm \forall q_{\rm j}\in g_{\rm j}} \sum\limits _{\rm k=1}^{\rm N_{\rm q_{\rm j}}} \biggl[V_{\rm g_{\rm j}}({\bf x}(t_{\rm {qj}_{\rm k}}^{\prime}))-V_{\rm g_{\rm j}}({\bf x}(t_{\rm {qj}_{\rm k}}))\hfill\cr\hfill-\int _{\rm t_{\rm {qj}_{\rm k}}}^{\rm t_{\rm {qj}_{\rm k}}^{\prime}} s_{\rm q_{\rm j}} ({\bf y}(\tau),{\bf u}(\tau)) \, d\tau\biggr ]\leq 0.\quad{\hbox{(14)}}}$$
    2. If Formula$N_{\rm q_{\rm j}}>M_{\rm q_{\rm j}}$, then the execution has entered Formula$q_{\rm j}$ and remains there until terminal time Formula$t_{\rm N}^{\prime}$, with Formula$t_{\rm N}^{\prime} \geq t_{\rm {qj}_{\rm N_{\rm q_{\rm j}}}}$, Formula$t_{\rm {qj}_{\rm N_{\rm q_{\rm j}}}} \in T \vert q_{\rm j}$ is the last switch-on instant time of Formula$q_{\rm j}$, and: Formula TeX Source $$\displaylines{\sum\limits _{\rm \forall q_{\rm j} \in g_{\rm j}} \sum\limits _{\rm k=1}^{\rm N_{\rm q_{\rm j}}} \biggl[V_{\rm g_{\rm j}}({\bf x}(t_{\rm {qj}_{\rm k}}^{\prime}))-V_{\rm g_{\rm j}}({\bf x}(t_{\rm {qj}_{\rm k}}))\hfill\cr\hfill-\int _{\rm t_{\rm {qj}_{\rm k}}}^{\rm t_{\rm {qj}_{\rm k}}^{\prime}}s_{\rm q_{\rm j}} ({\bf y}(\tau),{\bf u}(\tau)) \, d\tau \biggr ] +\sum\limits _{\rm \forall q_{\rm j} \in g_{\rm j}}{{\cal B}_{\rm q_{\rm j}}}\leq 0,\quad{\hbox{(15)}}}$$ with Formula TeX Source $$\displaylines{{\cal B}_{\rm q_{\rm j}}=V_{\rm g_{\rm j}}({\bf x}(t_{\rm N}^{\prime}))-V_{\rm g_{\rm j}}({\bf x}(t_{\rm {qj}_{\rm N_{\rm q_{\rm j}}}}))- \hfill \cr\hfill -\int _{\rm t_{\rm {qj}_{\rm N_{\rm q_{\rm j}}}}}^{\rm {\rm t}_{\rm N}^{\prime}} s_{\rm q_{\rm j}} ({\bf y}(\tau),{\bf u}(\tau)) \, d\tau.}$$
  2. Condition related to the entrances to any Formula$q_{\rm j}\in g_{\rm j}$. Condition (13) is satisfied. Formula$\hfill \blacksquare$

With conditions (14) and (15), during the time intervals any Formula$q_{\rm j}$ is active, the balance of stored and supplied energy for Formula$g_{\rm j}$ is allowed to grow for all Formula$q_{\rm j} \in g_{\rm j}$, and the dissipativity of each group is obtained as the total balance of stored and supplied energy when each Formula$q_{\rm j} \in g_{\rm j}$ is active.

To expand group dissipativity to the whole hybrid automaton, we define total dissipativity.

Definition 11

Let Formula$T \vert g_{\rm s}=\{t_{\rm {gs}_{1}},t_{\rm {gs}_{2}},\,\ldots, t_{\rm {gs}_{\rm k}},\,\ldots\}$ be the sequence of times when any location of group Formula$g_{\rm s}$ becomes active. Under Assumption 1, the hybrid automaton Formula$H$ is totally dissipative w.r.t. a set of supply functions Formula$\{s_{\rm q_{1}}({\bf y},{\bf u}),\ldots ,~s_{\rm q_{\rm N_{\rm q}}}({\bf y},{\bf u})\}$, if there exists a set of group storage-like functions Formula$\{V_{\rm g_{1}}({\bf x}),\ldots, V_{\rm g_{\rm N_{\rm g}}}({\bf x})\}$ satisfying the conditions of a group candidate Lyapunov function, such that for all Formula$\phi =(\tau ,q,{\bf x})\in {\cal E}_{\rm (q(t_{0}),{\bf x}(t_{0}))}$, and all Formula$\phi _{\rm u}=(\tau ,{\bf u})$, Formula$\phi _{\rm y}=(\tau ,{\bf y})$:

  1. Condition on groups of locations. All Formula$g_{\rm j}$ are group dissipative, with Formula$1\leq j\leq N_{\rm g}$.
  2. Cross-group coupling when changing from one group Formula$g_{\rm s}$ to another Formula$g_{\rm j}$. Formula$\forall g_{\rm j}$ and Formula$\forall g_{\rm s} \subset Q\setminus g_{\rm j}$, if Formula$\exists t_{\rm {gs}_{\rm k}}$, with Formula$t_{\rm {gs}_{\rm k}} \in T \vert g_{\rm s}$, such that:

    1. Formula$\forall t_{\rm {gj}_{\rm k}} \in T \vert g_{\rm j}$ for which any location of Formula$g_{\rm j}$ becomes active coming from any location of group Formula$g_{\rm s}$, with Formula$t^{\ast }_{\rm {gs}_{\rm k}}=\max _{\rm k} \{t_{\rm {gs}_{\rm k}} \in T \vert g_{\rm s}: \, t_{\rm {gs}_{\rm k}} \leq t_{\rm {gj}_{\rm k}}\}$ the last time when a location Formula$q_{\rm s}$ within any group Formula$g_{\rm s}$ was active before entering any location of Formula$g_{\rm j}$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}(t_{\rm {gj}_{\rm k}})) -V_{\rm g_{\rm s}}({\bf x}(t_{\rm {gs}_{\rm k}}^{\ast }))\leq s_{\rm q_{\rm s}} \bigl ({\bf y}(t^{\ast }_{\rm {gs}_{\rm k}}),{\bf u}(t^{\ast }_{\rm {gs}_{\rm k}})\bigr).\eqno{\hbox{(16)}}$$
    2. Condition on resets. For every Formula$q_{\rm j}\in g_{\rm j}$ and every Formula$q_{\rm s}\in g_{\rm s}$ such that Formula$(q_{\rm s},q_{\rm j})\in E$: Formula TeX Source $$V_{\rm g_{\rm j}}({\bf x}^{+}) -V_{\rm g_{\rm s}}({\bf x}(t^{\prime}_{\rm {qs}_{\rm k}})) \leq s_{\rm q_{\rm s}} \bigl ({\bf y}(t^{\prime}_{\rm {qs}_{\rm k}}),{\bf u}(t^{\prime}_{\rm {qs}_{\rm k}})\bigr),\eqno{\hbox{(17)}}$$ for all Formula$t^{\prime}_{\rm {qs}_{\rm k}} \in T^{\prime} \vert _{\rm q_{\rm s}}^{\rm q_{\rm j}}$ such that Formula${\bf x}(t^{\prime}_{\rm {qs}_{\rm k}})\in G(q_{\rm s},q_{\rm j})$, Formula${\bf x}^{+}=R(q_{\rm s},q_{\rm j},{\bf x}(t^{\prime}_{\rm {qs}_{\rm k}}),{\bf u}(t^{\prime}_{\rm {qs}_{\rm k}}))$, Formula${\bf y}(t^{\prime}_{\rm {qs}_{\rm k}}) \in {\cal Y}_{\rm q_{\rm s}}, {\bf u}(t^{\prime}_{\rm {qs}_{\rm k}}) \in{\cal U}_{\rm q_{\rm s}}$, with Formula$t^{\prime}_{\rm {qs}_{\rm k}}$ the time when Formula$q_{\rm s}$ becomes inactive to change to any Formula$q_{\rm j} \in g_{\rm j}$. Formula$\hfill \blacksquare$

Note that conditions (16) and (17) are required to take into account the impact of the stored and supplied energy at one group in the past, on the stored energy in the most recently active group of locations. These conditions are only checked when we change group.

Definition 12

The hybrid automaton Formula$H$ is weakly totally dissipative under all the assumptions considered in Definition 11 if one of the following conditions holds:

  1. At least one group location is weakly group dissipative, and the others are group dissipative.
  2. All the group locations are group dissipative, and instead of (16) and/or (17) we have:

    • Formula$\forall g_{\rm j}, \forall g_{\rm s} \ne g_{\rm j}$, with Formula$t^{\ast }_{\rm {gs}_{\rm k}}$ as in Definition 11, the following holds instead of (16): Formula TeX Source $$\displaylines{\sum\limits _{\scriptstyle\rm {\forall t_{\rm {gj}_{\rm k}}: g_{\rm j}~ {\rm becomes}}\atop\scriptstyle {\rm active~after~} g_{\rm s}} \left [V_{\rm g_{\rm j}}({\bf x}(t_{\rm {gj}_{\rm k}})) -V_{\rm g_{\rm s}}({\bf x}(t_{\rm {gs}_{\rm k}}^{\ast }))\right.\hfill\cr\hfill\left.-s_{\rm q_{\rm s}} \bigl({\bf y}(t^{\ast }_{\rm {gs}_{\rm k}}),{\bf u}(t^{\ast }_{\rm {gs}_{\rm k}})\bigr)\right] \leq 0;\quad{\hbox{(18)}}}$$
    • Formula$\forall q_{\rm j} \in g_{\rm j}, \forall g_{\rm j}, \forall g_{\rm s} \ne g_{\rm j}$, Formula$\forall q_{\rm s}\in g_{\rm s}$ such that Formula$(q_{\rm s},q_{\rm j})\in E$, the following holds instead of (17): Formula TeX Source $$\displaylines{\sum\limits _{\rm (\forall q_{\rm s}: \, (q_{\rm s},q_{\rm j})\in E) } \sum\limits_{\rm k=1}^{\rm M_{\rm q_{\rm s}}^{\rm q_{\rm j}}} \left [V_{\rm g_{\rm j}}({\bf x}^{+}) - V_{\rm g_{\rm j}}({\bf x}(t^{\prime}_{\rm {qs}_{\rm k}}))\right.\hfill\cr\hfill\left.-s_{\rm q_{\rm s}} \bigl ({\bf y}(t^{\prime}_{\rm {qs}_{\rm k}}),{\bf u}(t^{\prime}_{\rm {qs}_{\rm k}})\bigr)\right ] \leq 0,\quad{\hbox{(19)}}}$$Formula$\forall t^{\prime}_{\rm {qs}_{\rm k}} \in T^{\prime} \vert _{\rm q_{\rm s}}^{\rm q_{\rm j}}$ such that Formula${\bf x}(t^{\prime}_{\rm {qs}_{\rm k}})\in G(q_{\rm s},q_{\rm j})$, Formula${\bf x}^{+}=R(q_{\rm s},q_{\rm j},{\bf x}(t^{\prime}_{\rm {qs}_{\rm k}}),{\bf u}(t^{\prime}_{\rm {qs}_{\rm k}}))$, Formula${\bf y} \in {\cal Y}_{\rm q_{\rm s}}, {\bf u} \in {\cal U}_{\rm q_{\rm s}}$, with Formula$M_{\rm q_{\rm s}}^{\rm q_{\rm j}}$ the number of exits from Formula$q_{\rm s}$ to Formula$q_{\rm j}$, and Formula$t^{\prime}_{\rm {qs}_{\rm k}}$ the time when Formula$q_{\rm s}$ changes to Formula$q_{\rm j} \in g_{\rm j}$. Formula$\hfill \blacksquare$

Definition 13

A group of locations Formula$g_{\rm j}$ of Formula$H$ is group passive if it is group dissipative w.r.t. the supply functions Formula$s_{\rm q_{\rm i}}({\bf y},{\bf u})={\bf y}^{\rm T} {\bf u}$, Formula$\forall q_{\rm i}\in g_{\rm j}$. The hybrid automaton Formula$H$ is totally passive if it is totally dissipative with Formula$s_{\rm q_{\rm i}}({\bf y},{\bf u})={\bf y}^{\rm T} {\bf u}$, Formula$\forall q_{\rm i}\in Q$. Formula$\hfill \blacksquare$

From the classical theory of dissipative systems [20], it is well known that dissipative systems exhibit some stability properties for some specific inputs, outputs and supply functions. Similarly, from our dissipativity definitions, we can conclude some of the stability properties given in Section V for particular classes of hybrid automata. For example, if a hybrid automaton Formula$H$ which is Init-constrained to some group Formula$g_{\rm j}$ is totally dissipative with respect to supply functions Formula$\{s_{\rm q_{1}}({\bf y},{\bf u}),\ldots, s_{\rm q_{\rm N_{\rm q}}}({\bf y},{\bf u})\}$ which are zero for zero inputs (i.e., Formula$\forall q_{\rm i}$, Formula$s_{\rm q_{\rm i}}({\bf h}(q_{\rm i},{\bf x},0),0)=0, \forall {\bf x}\in {\cal X}_{\rm q_{\rm i}}$) then for some Formula$q_{\rm i}\in g_{\rm j}$, the equilibrium point of the zero-input dynamics Formula${\mathdot {\bf x}}=f_{\rm q_{\rm i}}({\bf x},0)$ coincides with the group equilibrium point Formula$\overline {\bf x}_{\rm g_{\rm j}}$, and Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is totally stable.

SECTION VII

DISSIPATIVITY PROPERTIES IN THE EXAMPLE

We will check if the 15-location hybrid automaton is totally passive w.r.t. Formula$s_{\rm q_{\rm i}}=y_{\rm q_{\rm i}} u$, with Formula$y_{\rm q_{\rm i}}=x_{1}-x_{3}$ for Formula$i\in \{1,2,3,4,5,7\}$, and Formula$y_{\rm q_{\rm i}}=x_{1}$ for all other locations. Formula$x_{1}$ and Formula$x_{3}$ are the angular velocities of the top-rotary system and the bit, respectively, and Formula$x_{2}$ is the difference between the two angular displacements. For all the locations, Formula$u=W_{\rm ob} R_{\rm b} \left [\mu _{\rm c_{\rm b}}+(\mu _{\rm s_{\rm b}}-\mu _{\rm c_{\rm b}})\exp^{- {({\gamma _{\rm b}}/ {v_{\rm f}})} x_{3}}\right ]$, with Formula$R_{\rm b} > 0$ the bit radius, Formula$\mu _{\rm s_{\rm b}}, \mu _{\rm c_{\rm b}} \in (0,1)$ the static and Coulomb friction coefficients associated with the bit, Formula$0< \gamma _{\rm b}< 1$ and Formula$v_{\rm f} >0$. Note that there is only one group of locations, and we choose the following group storage-like function: Formula$V_{\rm g_{1}}\!=\! {({1}/ {2})}\left [(x_{1}\!-\! \overline x_{\rm g_{1,1}})^{2}\!+\! (x_{2}\!-\! \overline x_{\rm g_{1,2}})^{2}+\right.~\left. + (x_{3}\!-\! \overline x_{\rm g_{1,3}})^{2}\right ]$, with Formula${\bf \overline x}_{\rm g_{1}}$ as given at the end of Section IV. The parameters used are: Formula$J_{\rm r} =2122~{\rm kg}~{\rm m}^{2}$, Formula$J_{\rm b}=471.9698~{\rm kg}~{\rm m}^{2}$, Formula$R_{\rm b}= 0.155~{\rm m}$, Formula$k_{\rm t} =861.5336~{\rm N}~{\rm m}/{\rm rad}$, Formula$c_{\rm t}=172.3067~{\rm N}~ {\rm m \,s}/{\rm rad}$, Formula$c_{\rm r} = 425~{\rm N \,m \,s}/{\rm rad}$, Formula$c_{\rm b} = 50~{\rm N \,m \,s}/{\rm rad}$, Formula$\mu _{\rm c_{\rm b}} = 0.5$, Formula$\mu _{\rm s_{\rm b}}=0.8$, Formula$\delta = 10^{-6}$, Formula$\gamma _{\rm b}=0.9$, Formula$v_{\rm f}=1$. We will show how Formula$x_{3\rm r}$, Formula$\lambda$ and Formula$W_{\rm ob}$ affect the passivity of Formula$H$.

Fig. 3 shows the case where Formula$H$ is not totally passive but only weakly totally passive. In Fig. 4, we show the case in which the trajectories of Formula$H$ converge to the group equilibrium point in Formula$q_{5}$, although it is non-totally passive. For the non-passive locations, condition (12) does not hold; and for Formula$q_{2}$ condition (14) of weak passivity also fails. Finally, for the stick-slip situation shown in Fig. 5, Formula$H$ is not totally passive because conditions (12) and (16) do not hold for Formula$q_{5}$.

Figure 3
Fig. 3. 15-location hybrid automaton is weakly totally passive for: Formula$\lambda =0.9$, Formula$W_{\rm ob}=20~ {\rm kN}$ and Formula$x_{3\rm r}=12~ {\rm rad}/{\rm s}$.
Figure 4
Fig. 4. 15-location hybrid automaton is: 1) weakly totally passive with Formula$\lambda =0.9$, Formula$W_{\rm ob}=20~ {\rm kN}$ and Formula$x_{3\rm r}=12~ {\rm rad}/ {\rm s}$, in grey thick lines; 2) non-totally passive, but with trajectories converging to Formula${\overline x}_{\rm g_{1}}$ with Formula$\lambda =0.9$, Formula$W_{\rm ob}=65~{\rm kN}$ and Formula$x_{3\rm r}=12~{\rm rad}/{\rm s}$; 3) non-totally passive with stick-slip oscillations with Formula$\lambda =0.9$, Formula$W_{\rm ob}=20~ {\rm kN}$ and Formula$x_{3\rm r}=1~ {\rm rad}/{\rm s}$.
Figure 5
Fig. 5. Stick-slip situation for the 15-location hybrid automaton: Formula$H$ is not totally passive for small Formula$x_{3\rm r}$'s.
SECTION VIII

CONCLUSIONS

We propose a new classification of equilibria in hybrid automata and based on this, a partition of the continuous state space is given. Some stability properties of co-existing isolated equilibria for a type of hybrid automata are given, leading to what is called total stability. Finally, group and total dissipativity properties of hybrid automata are proposed. The example illustrates how the use of hybrid automata can be useful in the analysis of complex hybrid systems.

APPENDIX

Proof of Theorem 1

If the conditions for total stability for Formula$\overline {\bf x}_{\rm g_{\rm j}}$ hold, the stability of Formula$\overline {\bf x}_{\rm g_{\rm j}}$, as given in Definition 6, is guaranteed. We divide the sketch of the proof into four cases.

Case 1

The executions only visit one location for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a non-virtual equilibrium. From condition (7), the proof corresponds to the well-known proof of stability for smooth systems.

Case 2

The executions travel along locations (all or some locations) within Formula$g_{\rm j}$ for wich Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a non-virtual equilibrium. With conditions (7) and (8), the proof follows the same arguments as Branicky's proof of Theorem 2.3 in [22] considering the common candidate Lyapunov function Formula$V_{\rm g_{\rm j}}({\bf x})$ for all the locations within Formula$g_{\rm j}$. In addition, (7) and (8) ensure that after a reset in every change of location, Formula$V_{\rm g_{\rm j}}$ is decreased/maintained – just as in [24]. Then, if Formula${\bf x}$ starts in Formula$B(\delta ,\overline {\bf x}_{\rm g_{\rm j}})$ just before the reset, then Formula${\bf x}^{+}$ starts in Formula$B(\delta ,\overline {\bf x}_{\rm g_{\rm j}})$, and hence, Formula${\bf x}^{+}$ stays in Formula$B(\epsilon ,\overline {\bf x}_{\rm g_{\rm j}})$ at the time of the reset, with Formula$\delta \in (0,\epsilon)$. In brief, Formula$V_{\rm g_{\rm j}}$ decreases or is maintained as time progresses.

Case 3

The executions switch between locations, within the same group, that do not contain Formula$\overline {\bf x}_{\rm g_{\rm j}}$ and the locations for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is a non-virtual equilibrium. Since Formula$\bigl (q(t_{0}),{\bf x}(t_{0})\bigr)\in \bigl ((Q_{\rm i}\times \Omega _{\rm g_{\rm j}})\bigcap {Init}\bigr)$, we will always start at Formula$\Omega _{\rm g_{\rm j}}$, a discrete location whose domain satisfies condition (7). Bearing in mind conditions of Case 2 and conditions (iii).a and (iii).b of our Theorem 1, the proof follows the same arguments as given in Theorem 1 of [23] for the case of having a common Lyapunov function and switchings with resets.

Case 4

The executions travel along locations from different groups with different group equilibria. In addition to conditions of the three cases above, the cross-group-coupling conditions (10) and (11), one for each different group Formula$g_{\rm s}$ visited, are considered. Notice that in this case, condition (9) is applied to any location in any group of Formula$H$ for which Formula$\overline {\bf x}_{\rm g_{\rm j}}$ is not a non-virtual equilibrium. Following similar arguments as those given in the proof of Theorem 1 and Corollary 1 of [14], we can prove that Formula${\bf x}$ does not move away from the union of the closed level sets for all the group candidate Lyapunov functions of Formula$H$. In addition, from conditions of Theorem 1, it is ensured that Formula$\exists t^{\ast }>0$, such that Formula$\forall t\geq t^{\ast }$, Formula${\bf x}(t)$ remains close to Formula$\overline {\bf x}_{\rm g}$, if Formula${\bf x}(t)$ starts close to Formula$\overline {\bf x}_{\rm g}$.

Proof of Theorem 2

It follows similar steps to the proof of Theorem 1.

Footnotes

This work was supported by the EPSRC-funded project DYVERSE: A New Kind of Control for Hybrid Systems under Grant EP/I001689/1 and the RCUK under Grant EP/E50048/1. Recommended by Associate Editor L. Zaccarian.

E. M. Navarro-López is with the School of Computer Science, The University of Manchester, Manchester M13 9PL, U.K. (e-mail: eva.navarro@cs.man.ac.uk).

D. S. Laila is with the Faculty of Engineering and the Environment, The University of Southampton, Highfield, Southampton SO17 1BJ, U.K. (e-mail: d.laila@soton.ac.uk).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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Eva M. Navarro-López

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Dina. S. Laila

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