Simulations and bisimulations for analysis of stability with respect to inputs of hybrid systems

Abstract

Simulation and bisimulation relations define pre-orders on processes which serve as the basis for approximation based verification techniques, and have been extended towards the design of continuous and hybrid systems with complex logic specifications. We study pre-orders between hybrid systems which preserve stability properties with respect to input. We show that these properties are not bisimulation invariant, and hence propose stronger notions which strengthen simulation and bisimulation relations with uniform continuity constraints. We show that uniform continuity is necessary on the relations corresponding to both the state-space and the input-space, and continuity itself does not suffice. Finally, we demonstrate the satisfiability of our definitions by casting the well-known Lyapunov function based techniques for stability analysis as constructing a simple one-dimensional system which is stable and uniformly continuously simulates the original system.

Appendices

Appendix A: Proof of the super-position Theorem 1

Proof

δ ISS ⇒ (C1) − (C3): It is straightforwardto check that δ ISS implies conditions (C1) − (C3).In fact, choosing u ₁ = u ₂ = u and δ such that β(δ, 0) < 𝜖 in Eq. 3implies

$$|{{\mathbf{x}}}(\zeta_{1}, \mathbf{u})(t) -{{\mathbf{x}}}(\zeta_{2}, \mathbf{u})(t)| \le \beta(|\zeta_{1} - \zeta_{2}|, t)< \beta(\delta, 0)<\epsilon, $$

provided that |ζ ₁ − ζ ₂| < δ. This shows (C1)is true. Moreover,since

$$|{{\mathbf{x}}}(\zeta_{1}, \mathbf{u})(t) -{{\mathbf{x}}}(\zeta_{2}, \mathbf{u})(t)| \le \beta(|\zeta_{1} - \zeta_{2}|, t)\rightarrow 0 $$

as t →∞, for anygiven 𝜖 and ζ ₁,ζ ₂, we can choose T independent of u suchthat

$$|{{\mathbf{x}}}(\zeta_{1}, \mathbf{u})(t) -{{\mathbf{x}}}(\zeta_{2}, \mathbf{u})(t)|<\epsilon $$

for all t > T. This shows (C2)is true. Finally,choosing ζ ₁ = ζ ₂ = ζ and δ such that γ(δ) < 𝜖 in Eq. 3implies

$$|{{\mathbf{x}}}(\zeta, \mathbf{u})(t) -{{\mathbf{x}}}(\zeta, \mathbf{u})(t)| \le \gamma(|\mathbf{u}_{1} - \mathbf{u}_{2}|_{\infty})< \gamma(\delta)<\epsilon, $$

providedthat | |u ₁ −u ₂| | _∞ < δ. Thisshows (C3)is true.

(C1) − (C3) ⇒ δ ISS: The prooffor the opposite implication essentially follows from the proof of Lemma 4.5 in Khalil (1996). Therefore, thedetailed argument is omitted and the following is an outline of the proof. First, by (C3), we can prove thereexists a ${\mathcal {K}_{\infty }}$function γ such that

$$ |{{\mathbf{x}}}(\zeta, \mathbf{u}_{1})(t) - {{\mathbf{x}}}(\zeta, \mathbf{u}_{2})(t)| \leq \gamma({|\!|\mathbf{u}_{1} - \mathbf{u}_{2}|\!|_{\infty}}),\quad t\geq 0, $$

(14)

holds for all initial states ζ ₁,ζ ₂.Second, conditions $(C1)$and (C2)imply thatthere exist a ${\mathcal {K}\mathcal {L}}$function β such that

$$ |{{\mathbf{x}}}(\zeta_{1}, \mathbf{u})(t) - {{\mathbf{x}}}(\zeta_{2}, \mathbf{u})(t)| \leq \beta(|\zeta_{1} - \zeta_{2}|, t),\quad t\geq 0, $$

(15)

holds for all input trajectory u. Now given any pair of initial states ζ ₁, ζ ₂and any pair ofinput trajectories u ₁, u ₂,it follows from Eqs. 14and 15that

$$\begin{array}{@{}rcl@{}} |{{\mathbf{x}}}(\zeta_{1}, \mathbf{u}_{1})(t) - {{\mathbf{x}}}(\zeta_{2}, \mathbf{u}_{2})(t)| &\le& |{{\mathbf{x}}}(\zeta_{1}, \mathbf{u}_{1})(t) - {{\mathbf{x}}}(\zeta_{2}, \mathbf{u}_{1})(t) + {{\mathbf{x}}}(\zeta_{2}, \mathbf{u}_{1})(t)- {{\mathbf{x}}}(\zeta_{2}, \mathbf{u}_{2})(t)|\\ &\le& \beta(|\zeta_{1} - \zeta_{2}|, t) + \gamma({|\!|\mathbf{u}_{1} - \mathbf{u}_{2}|\!|_{\infty}}),\quad t\geq 0. \end{array} $$

This completes the proof. □

Appendix B: Proof of Theorem 4

Proof

Let us assume $\mathcal {H}_{2}$is hISS with respect to ${\mathcal {T}}_{2}$.We need to show that $\mathcal {H}_{1}$is hISS with respect to ${\mathcal {T}}_{1}$.We will show that $\mathcal {H}_{1}$satisfies conditions (D1) − (D3).Proof of satisfaction of Condition (D1) Let us fix an 𝜖 ₁ > 0.We need to find a δ ₁ > 0such that Condition (D1)holds in $\mathcal {H}_{1}$and ${\mathcal {T}}_{1}$.Let 𝜖 ₂be the uniformity constant of $R^{-1}_{1}$corresponding to 𝜖 ₁.Let δ ₂be the constant satisfying Condition (D1)for $\mathcal {H}_{2}$corresponding to 𝜖 ₂.Set δ ₁to be the uniformity constant of R ₁corresponding to δ ₂.

Let us fix an input signal σ ^u ₁and states $\eta \in {\textit {First}({{{\textit {Exec}}(\mathcal {H}_{1})}|_{{\sigma ^{u}}_{1}}})}$and $\hat {\eta }_{1} \in {\textit {First}({{{\mathcal {T}}_{1}}|_{{\sigma ^{u}}_{1}}})}$such that ${d^{s}}(\hat {\eta }_{1}, \eta _{1}) < \delta _{1}$.Let σ ^s ₁be such that $({\sigma ^{u}}_{1}, {\sigma ^{s}}_{1}) \in {\textit {Exec}}(\mathcal {H}_{1})$and First(σ ^s ₁) = η ₁.We need to find ${\hat {\sigma }^{s}}_{1}$such that $({\sigma ^{u}}_{1}, \hat {\sigma }_{1}^{s}) \in {\mathcal {T}}$,${\textit {First}({\hat {\sigma }^{s}}_{1})} = \hat {\eta }_{1}$and ${d^{s}}({\sigma ^{s}}_{1}, \hat {\sigma }_{1}^{s}) < \epsilon _{1}$.

First, we establish that R ₂(σ ^u ₁)is a singleton set. This follows from (A3).Let R ₂(σ ^u ₁) = {σ ^u ₂}.Hence, from (A1),we have that there exists $\hat {\eta }_{2} \in {\textit {First}({{{\mathcal {T}}_{2}}|_{{\sigma ^{u}}_{2}}})}$such that $R_{1}(\hat {\eta }_{1}, \hat {\eta }_{2})$.From uniform continuity, we have $R_{1}(B_{\delta _{1}}(\eta _{1})) \subseteq B_{\delta _{1}}(R_{1}(\eta _{1}))$.Hence, $\hat {\eta }_{2} \in B_{\delta _{2}}(\eta _{2})$for some η ₂ ∈ R ₁(η ₁).From the definition of input simulation and uniqueness of the choice of σ ^u ₂,we have that there exists σ ^s ₂such that First(σ ^s ₂) = η ₂,$({\sigma ^{u}}_{2}, {\sigma ^{s}}_{2}) \in {\textit {Exec}}(\mathcal {H})$and R ₁(σ ^s ₁,σ ^s ₂).Then, from the hISS of $\mathcal {H}_{2}$with respect to ${\mathcal {T}}_{2}$,we obtain that there exists $({\sigma ^{u}}, \hat {\sigma }_{2}^{s}) \in {\mathcal {T}}_{2}$such that ${\textit {First}({\hat {\sigma }^{s}}_{2})} = \hat {\eta }_{2}$and ${d^{s}}(\hat {\sigma }_{2}^{s}, {\sigma ^{s}}_{2}) < \epsilon _{2}$.From (A2),there exists $\hat {\sigma }_{1}^{s}$such that ${\textit {First}(\hat {\sigma }_{1}^{s})} = \hat {\eta }_{1}$,$({\sigma ^{u}}_{1}, \hat {\sigma }_{1}^{s}) \in {\mathcal {T}}_{1}$and $R_{1}({\hat {\sigma }^{s}}_{1}, {\hat {\sigma }^{s}}_{2})$.

It remains to show that ${d^{s}}({\sigma ^{s}}_{1},\hat {\sigma }_{1}^{s}) < \epsilon _{1}$.Note that ${d^{s}}({\sigma ^{s}}_{2}, \hat {\sigma }_{2}^{s}) < \epsilon _{2}$and $R_{1}^{-1}(\hat {\sigma }_{2}^{s}) = \{\hat {\sigma }_{1}^{s}\}$(a singleton, from (A4)).To show that ${d^{s}}({\sigma ^{s}}_{1},{\hat {\sigma }^{s}}_{1}) < \epsilon _{1}$,we need to show that ${d^{s}}({\mathit {gr}}({\sigma ^{s}}_{1}),{\mathit {gr}}(\hat {\sigma }_{1}^{s})) < \epsilon _{1}$.Consider (t ₁,i ₁,x ₁) ∈g r(σ ^s ₁),we need to find $(t^{\prime }_{1}, i^{\prime }_{1}, x^{\prime }_{1}) \in {\mathit {gr}}(\hat {\sigma }_{1}^{s})$such that d ^s((t ₁,i ₁,x ₁), (t1′,i1′,x1′)) < 𝜖 ₁.Since R ₁(σ ^s ₁,σ ^s ₂),there exists (t ₂,i ₂,x ₂) ∈g r(σ ^s ₂),such that R ₁((t ₁,i ₁,x ₁), (t ₂,i ₁,x ₂)).Note, in fact, that $t_{1} = t_{2}$and i ₁ = i ₂,and R ₁(x ₁,x ₂).Since, ${d^{s}}({\sigma ^{s}}_{2}, \hat {\sigma }_{2}^{s}) < \epsilon _{2}$,there exists $(t^{\prime }_{2}, i^{\prime }_{2}, x^{\prime }_{2}) \in {\mathit {gr}}({\hat {\sigma }^{s}}_{2})$such that d ^s((t ₂,i ₂,x ₂), (t2′,i2′,x2′)) < 𝜖 ₂.Then, from the continuity of $R_{1}^{-1}$, (t ₂,i ₂,x ₂)is within 𝜖 ₁of some element in $R_{1}^{-1}(t^{\prime }_{2}, i^{\prime }_{2}, x^{\prime }_{2})$.But, since, the latter is a singleton set (from (A4)),namely, (t1′,i1′,x1′),we obtain that d ^s((t ₁,i ₁,x ₁), (t2′,i2′,x2′)) < 𝜖 ₁.Note that $t^{\prime }_{1} = t^{\prime }_{2}$,and $i^{\prime }_{1} = i^{\prime }_{2}$.The other part where given $(t^{\prime }_{1}, i^{\prime }_{1}, x^{\prime }_{1}) \in {\mathit {gr}}(\hat {\sigma }_{1}^{s})$,we need to find (t ₁,i ₁,x ₁) ∈g r(σ ^s ₁)such that d ^s((t ₁,i ₁,x ₁), (t1′,i1′,x1′)) < 𝜖 ₁is similar.Proof of satisfaction of Condition (D2) Let us fix an $\delta _{1}, \epsilon _{1} > 0$and T ₁ ≥ 0,such that Condition (D2)holds in $\mathcal {H}_{1}$and ${\mathcal {T}}_{1}$.Let 𝜖 ₂be the uniformity constant of $R^{-1}_{1}$corresponding to 𝜖 ₁.Let δ ₂be the constant satisfying Condition (D1)for $\mathcal {H}_{2}$corresponding to 𝜖 ₂.Set δ ₁to be the uniformity constant of R ₁corresponding to δ ₂.Let T ₂ = T ₁.We will show that Condition (D2)holds in $\mathcal {H}_{2}$and ${\mathcal {T}}_{2}$for $\delta _{1}, \epsilon _{2}$and T ₂.

The proof is similar to that of Condition (D1).Here we need to show that ${d^{s}}({{{\sigma ^{s}}_{1}}|_{T_{1}}},{\hat {\sigma }_{1}^{s}|_{T_{1}}}) < \epsilon _{1}$instead of ${d^{s}}({\sigma ^{s}}_{1},\hat {\sigma }_{1}^{s}) < \epsilon _{1}$.Note that in the previous proof the choice of the times whenwe move from one system to the other are the same. That is,$t_{1} = t_{2}$and t1′ = t2′.We mainly need to check if the triples being chosen at different stepsbelong to the time restricted signals rather than the complete signalsas required in the previous proof. Hence, if we consider the triple$(t_{1}, i_{1}, x_{1}) \in {\mathit {gr}}({{{\sigma ^{s}}_{1}}|_{T_{1}}})$,the corresponding triple (t ₂,i ₂,x ₂)will belong to ${\mathit {gr}}({{{\sigma ^{s}}_{1}}|_{T_{1}}})$,since, T ₁ = T ₂and the time stamp t ₂ ≥ T ₂.Here, we can choose the triple $(t^{\prime }_{2}, i^{\prime }_{2}, x^{\prime }_{2})$such that its time stamp $t^{\prime }_{2} \geq T_{2}$,because we have ${d^{s}}({{{\sigma ^{s}}_{2}}|_{T_{2}}},{\hat {\sigma }_{2}^{s}|_{T_{2}}}) < \epsilon _{2}$instead of ${d^{s}}({\sigma ^{s}}_{2},\hat {\sigma }_{2}^{s}) < \epsilon _{2}$.Finally, the triple (t1′,i1′,x1′)will have time stamp $t^{\prime }_{1} = t^{\prime }_{2} \geq T_{2} = T_{1}$.Hence, $(t^{\prime }_{1}, i^{\prime }_{1}, x^{\prime }_{1}) \in {\mathit {gr}}({\hat {\sigma }_{1}^{s}|_{T_{1}}})$.Proof of satisfaction of Condition (D3) Let us fix an 𝜖 ₁ > 0.We need to find a δ ₁ > 0such that Condition (D3)holds in $\mathcal {H}_{1}$and ${\mathcal {T}}_{1}$.Let 𝜖 ₂be the uniformity constant of $R^{-1}_{1}$corresponding to 𝜖 ₁.Let δ ₂be the constant satisfying Condition (D3)for $\mathcal {H}_{2}$corresponding to 𝜖 ₂.Set δ ₁to be the uniformity constant of R ₂corresponding to δ ₂.

Let us fix input signals ${\sigma ^{u}}_{1}, \hat {\sigma }_{1}^{u}$,state $\eta _{1} \in {{{\mathcal {T}}_{1}}|_{{\hat {\sigma }^{u}}}}$.Let $d({\sigma ^{u}}_{1}, \hat {\sigma }_{1}^{u}) < \delta _{1}$and $({\sigma ^{u}}_{1}, {\sigma ^{s}}_{1}) \in {\textit {Exec}}(\mathcal {H})$with First(σ ^s ₁) = η ₁.We need to find $\hat {\sigma }_{1}^{s}$such that $(\hat {\sigma }_{1}^{u}, \hat {\sigma }_{1}^{s}) \in {\mathcal {T}}_{1}$,${\textit {First}(\hat {\sigma }_{1}^{s})} = \eta _{1}$and ${d^{s}}({\sigma ^{s}}_{1}, \hat {\sigma }_{1}^{s}) < \epsilon _{1}$.

Since $\eta _{1} \in {{{\mathcal {T}}_{1}}|_{{\hat {\sigma }^{u}}_{1}}}$,we obtain from (A1)that there exists $\hat {\sigma }_{2}^{u}$and $\eta _{2} \in {{{\mathcal {T}}_{2}}|_{{\hat {\sigma }^{u}}_{2}}}$such that $R_{1}(\eta _{1}, \eta _{2})$and $R_{2}(\hat {\sigma }_{1}^{u}, \hat {\sigma }_{2}^{u})$.Again, since, R ₁(η ₁,η ₂),from the definition of simulation, there exists $({\sigma ^{u}}_{2}, {\sigma ^{s}}_{2}) \in {\textit {Exec}}(\mathcal {H}_{2})$such that First(σ ^s ₂) = η ₂, R ₁(σ ^s ₁,σ ^s ₂)and R ₂(σ ^u ₁,σ ^u ₂).From the choice of δ ₁,we have that ${\sigma ^{u}}_{2} \in B_{\delta _{1}}(R_{2}(\hat {\sigma }_{1}^{u}))$.However, since, $R_{2}(\hat {\sigma }_{1}^{u})$is unique, namely, $\hat {\sigma }_{2}^{u}$,we obtain that ${d^{u}}(\hat {\sigma }_{2}^{u}, {\sigma ^{u}}_{2}) < \delta _{2}$.Therefore, from the hISS of $\mathcal {H}_{2}$with respect to ${\mathcal {T}}_{2}$,there exists $\hat {\sigma }_{2}^{s}$such that $(\hat {\sigma }_{2}^{u}, \hat {\sigma }_{2}^{s}) \in {\mathcal {T}}_{2}$,${\textit {First}(\hat {\sigma }_{2}^{s})} = \eta _{2}$and ${d^{s}}({\sigma ^{s}}_{2}, \hat {\sigma }_{2}^{s}) < \epsilon _{2}$.Finally, from (A2),there exists ${\hat {\sigma }^{s}}_{1}$such that ${\textit {First}(\hat {\sigma }_{1}^{s})} = \eta _{1}$,$R_{1}(\hat {\sigma }_{1}^{s}, \hat {\sigma }_{2}^{s})$and $(\hat {\sigma }_{1}^{u}, \hat {\sigma }_{1}^{s}) \in {\mathcal {T}}_{1}$.

It remains to show that ${d^{s}}({\sigma ^{s}}_{1}, {\hat {\sigma }^{s}}_{1}) < \epsilon$.The argument is the same as that in the proof of part D1. □