Scalable Parametric Approximations of the Maximal Controlled - PowerPoint PPT Presentation

Scalable Parametric Approximations of the Maximal Controlled Invariant Set and Two Health Related Application Domains Ian M. Mitchell Department of Computer Science University of British Columbia

Formal Methods & Reachability • Formal Methods seek to bring the rigour of mathematical proof to the specification, development and verification of hardware and software systems • Safety is a commonly desired form of specification – The system will not do something bad • Reachability is a common approach to analyzing safety backward reach set target (uncontrollably unsafe) (unsafe) safe (under appropriate control) Dec 2014 Ian M. Mitchell (UBC Computer Science) 2

Evolving Continuous Reachable Sets • Modified Hamilton-Jacobi partial differential equation growing set final set Reachable set for the game of two identical vehicles Dec 2014 Ian M. Mitchell (UBC Computer Science) 3

The Toolbox of Level Set Methods • Collection of Matlab routines to approximate viscosity solutions of time-dependent Hamilton-Jacobi equations – First publicly released implementation of state-of-the-art level set algorithms – Only HJ package that works in arbitrary dimension – Vectorized code achieves reasonable speed – Direct access to Matlab debugging and visualization – Download package include 25+ examples and 120+ page user guide – Can compute reachable sets for nonlinear systems with free parameters / inputs in best case and/or worst case fashion • Hamilton-Jacobi framework and level set algorithms have limitations as a tool for formal verification – Cost of nonparametric, grid-based representation of sets scales exponentially with state space dimension Dec 2014 Ian M. Mitchell (UBC Computer Science) 4

Closed Loop Control of Anesthesia • Push to automate delivery of anesthesia in order to reduce cost and number of adverse events – Driven considerable improvement in sensing & modelling • Current systems are open-loop – Closed loop would be better able to handle patient variability and surgical stimuli • Key element for FDA / Health Canada: confidence that closed-loop system will maintain a suitable depth of hypnosis Dec 2014 Ian M. Mitchell (UBC Computer Science) 5

Smart Wheelchair for Older Adults • Cognitively (and mobility) impaired older adults in long term care (LTC) facilities – Heterogenous population – Constrained but navigable environment • Assistance with multiple objectives – Short term: Collision avoidance – Medium term: Wayfinding • Shared(?) control – Autonomous navigation (with supervisory control) can cause confusion or agitation in this population – What to call it? Collaborative control? Human-in-the-loop? • Low cost sensors • User trials with target population • Reproducible research Dec 2014 Ian M. Mitchell (UBC Computer Science) 6

Viability & Anesthesia Slides Dec 2014 Ian M. Mitchell (UBC Computer Science) 7

Scalable Techniques for Viability Kernels in LTI Systems with Application to Automated Control of Anesthesia Ian M. Mitchell Department of Computer Science University of British Columbia December 2014 Ian M. Mitchell — 1

Background: Dynamic System and Constraints  x = f ( x, u ) , ˙ x (0) = x 0 , t ∈ [0 , τ ] =: T   u ( t ) ∈ U (input constraint)  K ⊆ X (target set/state constraint)  • What is reachability analysis? ◮ [Tomlin, et al. 03; Aubin, et al. 11; Kurzhanski and Varaiya 00; Lygeros 04; Blanchini and Miani 08; ...] ◮ Maximal vs. minimal reachability [Mitchell 07] Ian M. Mitchell — 2

Background: Reachability Constructs • Maximal reach tube Reach ♯ T ( K , U ) := { x 0 ∈ X | ∃ u ( · ) , ∃ t, x ( t ) ∈ K} K Ian M. Mitchell — 3

Background: Reachability Constructs • Maximal reach set Reach ♯ t ( K , U ) := { x 0 ∈ X | ∃ u ( · ) , x ( t ) ∈ K} K Ian M. Mitchell — 4

Background: Maximal Reachability • Maximal reach tube vs. set [Lygeros 04; Mitchell 07] Reach ♯ � t ∈ T Reach ♯ T ( K , U ) = t ( K , U ) • Parametric methods to approximate ◮ e.g. [Frehse, et al. 11; Girard and Le Guernic 08; Girard, et al. 06; Han and Krogh 06; Kurzhanski and Varaiya 00; Kurzhanskiy and Varaiya 06] ◮ Scalable and computationally efficient (polynomial) Ian M. Mitchell — 5

Background: Minimal Reachability • Minimal reach tube Reach ♭ T ( K , U ) := { x 0 ∈ X | ∀ u ( · ) , ∃ t, x ( t ) ∈ K} K Ian M. Mitchell — 6

Background: Viability • Viability kernel (finite horizon) V iab T ( K , U ) := { x 0 ∈ X | ∃ u ( · ) , ∀ t, x ( t ) ∈ K} K • Infinite horizon viab kernel ≡ maximal controlled-invariant subset Ian M. Mitchell — 7

Introduction • Viability kernel vs. minimal reach tube [Cardaliaguet, et al. 99] ( V iab T ( K , U )) c = Reach ♭ T ( K c , U ) • The only constructs to prove existence of safety control laws [Mitchell 07; Lygeros 04] ◮ Applications: [Lygeros, et al. 98; Tomlin, et al. 03; Bayen, et al. 07; Gillula, et al. 10; Oishi, et al. 03; Aswani, et al. 11; Borrelli, et al. 10; Panagou, et al. 09; Del Vecchio, et al. 09; Ghaemi and Del Vecchio 11; ...] • Non-parametric methods to approximate ◮ [Mitchell, et al. 05; Cardaliaguet, et al. 99; Gao, et al. 06; Saint-Pierre 94] ◮ Computationally intensive (exponential) since grid-based Ian M. Mitchell — 8

Set-theoretic Methods • Efficient techniques (parametric) to compute maximal reach sets Reach ♯ t ( K , U ) := { x 0 ∈ X | ∃ u ( · ) , x ( t ) ∈ K} K • Approximate V iab T ( K , U ) via a nested sequence of sets reachable in small sub-time intervals of T Ian M. Mitchell — 10

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Start with an under-approximation K ↓ ( P ) of K ( P : interval partition of time; M : uniform bound on f ) K ↓ ( P ) := { x ∈ K | dist( x, K c ) ≥ M � P �} • Recursively compute K 0 ( P ) from: K | P | ( P ) = K ↓ ( P ) , K k − 1 ( P ) = K ↓ ( P ) ∩ Reach ♯ t k − t k − 1 ( K k ( P ) , U ) for k ∈ { 1 , . . . , | P |} . Ian M. Mitchell — 11

Set-theoretic Methods: Continuous-Time • Guaranteed under-approximation: K 0 ( P ) ⊆ V iab T ( K , U ) • Arbitrarily precise by choosing a sufficiently fine partition of time: ◦ � V iab T ( K , U ) ⊆ K 0 ( P ) ⊆ V iab T ( K , U ) P ∈ P ( T ) Ian M. Mitchell — 12

Set-theoretic Methods: Discrete-Time • Particular form of the continuous-time case • Recursively compute K 0 from: K n = K , K k − 1 = K ∩ Reach ♯ 1 ( K k , U ) for k ∈ { 1 , . . . , n } • Compute exactly: K 0 = V iab T ∩ Z + ( K , U ) • Closely related to discrete algorithms in e.g. [Saint-Pierre 94; Cardaliaguet, et al. 99; Blanchini and Miani 08] Ian M. Mitchell — 13