Computable analysis and control synthesis over complex dynamical systems via formal verification Alessandro Abate Department of Computer Science, University of Oxford Delft Center for Systems and Control, TU Delft September 25, 2013 Alessandro Abate 1 / 46
Outline Formal abstractions for verification of complex models 1 Formal verification of stochastic hybrid systems 2 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Formal verification of max-plus linear models 3 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Concluding remarks 4 Key references will appear here Alessandro Abate 2 / 46
Outline Formal abstractions for verification of complex models 1 Formal verification of stochastic hybrid systems 2 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Formal verification of max-plus linear models 3 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Concluding remarks 4 Alessandro Abate 2 / 46
Formal abstractions for verification of complex models concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models ✻ ǫ -quantitative abstraction concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models abstract simple ǫ -specification model ✻ ǫ -quantitative abstraction concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models automatic abstract verification ✲ simple ǫ -specification model control synthesis ✻ ǫ -quantitative abstraction concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ✲ simple ǫ -specification model control synthesis ✻ ǫ -quantitative abstraction concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ǫ -spec holds yes/no ✲ simple ǫ -specification policy µ → ǫ -spec model control synthesis ✻ ǫ -quantitative abstraction concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ǫ -spec holds yes/no ✲ simple ǫ -specification policy µ → ǫ -spec model control synthesis ✻ ǫ -quantitative refine back abstraction ❄ concrete property, complex specification, model cost or reward Alessandro Abate 3 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ǫ -spec holds yes/no ✲ simple ǫ -specification policy µ → ǫ -spec model control synthesis ✻ ǫ -quantitative refine back abstraction ❄ concrete property, spec holds yes/no complex specification, policy µ → spec model cost or reward (correct by design) Alessandro Abate 3 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ǫ -spec holds yes/no ✲ simple ǫ -specification policy µ → ǫ -spec model control synthesis ✻ ǫ -quantitative refine back abstraction ❄ ■ ❅ ❅ ❅ concrete property, spec holds yes/no ❅ complex specification, policy µ → spec if no, model cost or reward (correct by design) tune ǫ Alessandro Abate 3 / 46
Outline Formal abstractions for verification of complex models 1 Formal verification of stochastic hybrid systems 2 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Formal verification of max-plus linear models 3 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Concluding remarks 4 Alessandro Abate 4 / 46
Formal abstractions for verification of complex models model checking ❄ automatic abstract verification ǫ -spec holds yes/no ✲ ǫ -specification simple policy µ → ǫ -spec model control synthesis ✻ ǫ -quantitative refine back abstraction ❄ ■ ❅ ❅ ❅ concrete property, spec holds yes/no ❅ complex specification, policy µ → spec if no, model cost or reward (correct by design) tune ǫ Alessandro Abate 5 / 46
Formal abstractions for verification of dtSHS PRISM MRMC ❄ prob. model checking ǫ -spec holds dtMC relax’d/strenght’d PCTL ✲ policy max/min ǫ -spec dtMDP inflated LTL – ǫ -spec dynamic programming adaptive, ✻ approximate refine back sequential probabilistic abstractions bisimulations ❄ ❄ PCTL spec holds dtSHS LTL – spec policy max/min spec automata Alessandro Abate 6 / 46
Stochastic hybrid (discrete/continuous) systems PRISM MRMC ❄ prob. model checking ǫ -spec holds dtMC relax’d/strenght’d PCTL ✲ policy max/min ǫ -spec dtMDP inflated LTL – ǫ -spec dynamic programming adaptive, ✻ approximate refine back sequential probabilistic abstractions bisimulations ❄ ❄ PCTL spec holds dtSHS LTL – spec policy max/min spec automata Alessandro Abate 7 / 46
Stochastic hybrid (discrete/continuous) systems discrete-time models finite-space Markov chain uncountable-space Markov process ( Z , T ) ( S , T s ) S = R 2 Z = ( z 1 , z 2 , z 3 ) p 11 p 12 p 13 T s ( x | s ) = e − 1 2 ( x − m ( s )) T Σ − 1 ( s )( x − m ( s )) T = · · · · · · p 21 √ 2 π | Σ( s ) | 1 / 2 · · · · · · · · · � P ( z 1 , { z 2 , z 3 } ) = p 12 + p 13 P ( s , A ) = A T s ( dx | s ) , A ∈ B ( S ) Alessandro Abate 7 / 46
Stochastic hybrid (discrete/continuous) systems discrete-time models finite-space Markov chain uncountable-space Markov process ( Z , T ) ( S , T s ) S = R 2 Z = ( z 1 , z 2 , z 3 ) p 11 p 12 p 13 T s ( x | s ) = e − 1 2 ( x − m ( s )) T Σ − 1 ( s )( x − m ( s )) T = · · · · · · p 21 √ 2 π | Σ( s ) | 1 / 2 · · · · · · · · · � P ( z 1 , { z 2 , z 3 } ) = p 12 + p 13 P ( s , A ) = A T s ( dx | s ) , A ∈ B ( S ) ⇒ discrete-time, stochastic hybrid systems Alessandro Abate 7 / 46
Stochastic hybrid (discrete/continuous) systems Definition A discrete-time stochastic hybrid system is a pair ( S , T s ) , where S = ∪ q ∈ Q ( { q } × R n ( q ) ) , Q a discrete set of modes, n : Q → N T s : S × S → [ 0 , 1 ] specifies the dynamics of process at point s = ( q , x ) : if q ′ = q (no transition) T x ( dx ′ | ( q , x )) T q ( q | ( q , x )) , � T s ( ds ′ | s ) = if q ′ � = q (transition) T r ( dx ′ | ( q , x ) , q ′ ) T q ( q ′ | ( q , x )) , initial state π : S → [ 0 , 1 ] [AA et al - Automatica 08] Alessandro Abate 7 / 46
Stochastic hybrid (discrete/continuous) systems Definition A discrete-time stochastic hybrid system is a pair ( S , T s ) , where S = ∪ q ∈ Q ( { q } × R n ( q ) ) , Q a discrete set of modes, n : Q → N T s : S × S → [ 0 , 1 ] specifies the dynamics of process at point s = ( q , x ) : if q ′ = q (no transition) � T x ( dx ′ | ( q , x )) T q ( q | ( q , x )) , T s ( ds ′ | s ) = if q ′ � = q (transition) T r ( dx ′ | ( q , x ) , q ′ ) T q ( q ′ | ( q , x )) , initial state π : S → [ 0 , 1 ] can be control dependent ( u ∈ U ): if q ′ = q (no transition) � T x ( dx ′ | ( q , x ) , u ) T q ( q | ( q , x ) , u ) , T s ( ds ′ | s , u ) = if q ′ � = q (transition) T r ( dx ′ | ( q , x ) , u , q ′ ) T q ( q ′ | ( q , x ) , u ) , policy µ : “string” of controls equivalent dynamical representation: s k + 1 = f ( s k , ξ k , u k ) related to other models, e.g. LMP [AA et al - Automatica 08] Alessandro Abate 7 / 46
Stochastic hybrid systems in risk analysis � Z n + 1 = g ( Z n , θ n ) Z n ∈ R , ← capital θ n + 1 = h ( Z n , θ n , ξ n ) θ n ∈ { Θ 1 , . . . , Θ N } , ← interest where ξ n i.i.d. random variables; g , h measurable; ( Z 0 , θ 0 ) given [I. Tkachev, AA - CDC 11 ] Alessandro Abate 8 / 46
Stochastic hybrid systems in risk analysis � Z n + 1 = g ( Z n , θ n ) Z n ∈ R , ← capital θ n + 1 = h ( Z n , θ n , ξ n ) θ n ∈ { Θ 1 , . . . , Θ N } , ← interest where ξ n i.i.d. random variables; g , h measurable; ( Z 0 , θ 0 ) given Capital 15 y � 16 x � 12 10 5 t 50 100 150 200 objective: what is the probability that, starting from initial capital Z 0 = x , high capitalization y is reached, while company’s bankruptcy is avoided [I. Tkachev, AA - CDC 11 ] Alessandro Abate 8 / 46
Outline Formal abstractions for verification of complex models 1 Formal verification of stochastic hybrid systems 2 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Formal verification of max-plus linear models 3 Analysis and control synthesis problems Computable analysis and control synthesis via formal abstractions Concluding remarks 4 Alessandro Abate 9 / 46
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