Data-driven and Model-based Verification via Bayesian Identification and Reachability Analysis

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Experiment Setup

S

u(t)
u(t)u(t)
\tilde{y}(t)
y~(t)\tilde{y}(t)
y(t)
y(t)y(t)
e(t)
e(t)e(t)
S \models \psi
S⊨ψS \models \psi

???

\text{Parameter set } \Theta
Parameter set Θ\text{Parameter set } \Theta
\text{Parametrised model } M(\theta) \mid \theta \in \Theta
Parametrised model M(θ)∣θ∈Θ\text{Parametrised model } M(\theta) \mid \theta \in \Theta
\text{Family of parametrised models } \mathcal{G} = \{ M(\theta) \mid \theta \in \Theta \}
Family of parametrised models G={M(θ)∣θ∈Θ}\text{Family of parametrised models } \mathcal{G} = \{ M(\theta) \mid \theta \in \Theta \}
\text{Satisfaction function } f_{\psi}(\theta) = \textbf{P}(M(\theta)) \models \psi)
Satisfaction function fψ(θ)=P(M(θ))⊨ψ)\text{Satisfaction function } f_{\psi}(\theta) = \textbf{P}(M(\theta)) \models \psi)
\text{mapping to } \{0,1\} \text{ or } [0,1]
mapping to {0,1} or [0,1]\text{mapping to } \{0,1\} \text{ or } [0,1]
\text{Sample size } N_s
Sample size Ns\text{Sample size } N_s
\text{Sample } Z^{N_{s}} = \{ u(t), \tilde{y}(t) \}_{t=1}^{t=N_s}
Sample ZNs={u(t),y~(t)}t=1t=Ns\text{Sample } Z^{N_{s}} = \{ u(t), \tilde{y}(t) \}_{t=1}^{t=N_s}
\textbf{P} (S \models \psi | Z^{N_s}) = \int_{\Theta} f_{\psi}(\theta) p(\theta | Z^{N_s}) d\theta
P(S⊨ψ∣ZNs)=∫Θfψ(θ)p(θ∣ZNs)dθ\textbf{P} (S \models \psi | Z^{N_s}) = \int_{\Theta} f_{\psi}(\theta) p(\theta | Z^{N_s}) d\theta
\text{Bayesian confidence: }
Bayesian confidence: \text{Bayesian confidence: }
p(\theta | Z^{N_s}) = \frac{p(Z^{N_s}|\theta)p(\theta)}{\int_{\Theta}p(Z^{N_s}|\theta)p(\theta)d\theta}
p(θ∣ZNs)=p(ZNs∣θ)p(θ)∫Θp(ZNs∣θ)p(θ)dθp(\theta | Z^{N_s}) = \frac{p(Z^{N_s}|\theta)p(\theta)}{\int_{\Theta}p(Z^{N_s}|\theta)p(\theta)d\theta}

??

??

LTL over continuous signals

Atomic propositions: linear constraints in output space

Alphabet: Sets of atomic propositions (convex polytopes)

Models (LTI Systems)

x(t+1) = A \cdot x(t) + B \cdot u(t)
x(t+1)=A⋅x(t)+B⋅u(t)x(t+1) = A \cdot x(t) + B \cdot u(t)

Difference equations + output equations

y(t) = C(\theta) \cdot x(t) + D(\theta) \cdot u(t)
y(t)=C(θ)⋅x(t)+D(θ)⋅u(t)y(t) = C(\theta) \cdot x(t) + D(\theta) \cdot u(t)

Only C and D are parametrised!

Non-linearly parametrised models can be approximated using orthonormal basis functions

How do we compute this?

How do we compute that?! 

Several maths later...

Bottom line: It is computable and the dimensionality depends on model and property

Bottom line: It is computable, assuming we know R, which can be approximated.