hyper A Useful Toolbox for Hybrid Systems Shankar Sastry, Jonathan Sprinkle, Mike Eklund, Ian Mitchell UC Berkeley November 2004 Shankar Sastry, UC Berkeley

What Are Hybrid Systems? Dynamical systems with interacting continuous and discrete dynamics November 2004 Shankar Sastry, UC Berkeley

Why are Hybrid Systems Hard? All implicit error in continuous analysis, coupled with switching criteria, leads to possible catastrophic error X1 Numerical Solution Real Solution t = t* X2 November 2004 Shankar Sastry, UC Berkeley Slide by Rafael Garcia

Why are Hybrid Systems Hard? Zeno behavior Event transitions happen so rapidly that the simulator grinds to a halt A Toolbox should detect this November 2004 Shankar Sastry, UC Berkeley Slide by Edward Lee

Shankar Sastry, UC Berkeley What are the pieces? Modeling & Simulation Control: classify discrete phenomena, existence and uniqueness of execution, Zeno [Branicky, Brockett, van der Schaft, Astrom] Computer Science: composition and abstraction operations [Alur-Henzinger, Lynch, Sifakis, Sztipanovits,Varaiya] Analysis & Verification Control: stability, Lyapunov techniques [Antsaklis, Branicky, Michel], LMI techniques [Johansson-Rantzer], optimal control [Branicky, Sussmann, Caines] Computer Science: Algorithmic [Alur-Henzinger, Sifakis, Pappas-Lafferrier-Sastry] or deductive methods [Lynch, Manna, Pnuelli] Controller Synthesis Control: optimal control [Branicky-Mitter, Bensoussan-Menaldi], hierarchical control [Caines, Pappas-Sastry], supervisory control [Lemmon-Antsaklis], model predictive techniques [Morari Bemporad], safety specifications [Lygeros-Tomlin-Sastry] Computer Science: algorithmic synthesis [Maler, Pnueli, Asarin, Wong-Toi] Observability and Diagnosability Control: observers[Bemporad, Koutsoukos, Vidal] Computer Science[ Biswas, Karsai, Zhao] November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Tool Integration No one tool exists for all of these Several academic toolmakers created HSIF Hybrid Systems Interchange Format Failed to mature for a few reasons Tool-specific, and Tool-driven, not capability driven Not enough programming power behind it No thought to coverage of “corner cases” (Lee, Zheng) November 2004 Shankar Sastry, UC Berkeley

Tool Integration Example correct output: What is needed is a framework that utilizes interchange format Must support what we know about hybrid systems semantics, encourage tools integration Implicit tool semantics makes fully meaningful translation impossible, or impractical The proper specification of the semantics of an interchange format would ease this difficulty Leverage HSIF as a learning tool for the semantic specification of the hyper core RK 2 - 3 variable - step solver and breakpoint solver determine sample times: Note two values at Note two values at the same time: the same time: Incorrect output: November 2004 Shankar Sastry, UC Berkeley Slide by Edward Lee

Shankar Sastry, UC Berkeley Tool Integration What about industry-standard tools? Many big industries are bound to Matlab, but clamoring for verification/synthesis capabilities Suggests a real need for getting this interchange piece right and showing examples for how to do the interchange with existing tools or toolboxes November 2004 Shankar Sastry, UC Berkeley

Example Integration: HyVisual Ptolemy II’s HyVisual http://ptolemy.eecs.berkeley.edu/hyvisual/ November 2004 Shankar Sastry, UC Berkeley Slide by Edward Lee

Shankar Sastry, UC Berkeley HyVisual Clear contribution well-defined simulation and semantic behavior Assets Platform-independent model editing Visualization tools Reasoned meaning of hierarchical containment Well-formulated for the business of numerical solutions and execution of complex, nonlinear, deterministic hybrid systems November 2004 Shankar Sastry, UC Berkeley

HyVisual: Integration Benefits Contains a rich set of examples of hybrid systems Useful for other tool integrations Platform independent, open source Lacks synthesis/runtime tools, so could then integrate with other tools with these capabilities November 2004 Shankar Sastry, UC Berkeley

Example Integration: A Toolbox of LSM Ian Mitchell’s Toolbox of Level Set Methods http://www.cs.ubc.ca/~mitchell/ToolboxLS/ Numerical algorithms for dynamic implicit surfaces and Hamilton-Jacobi partial differential equations November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Level Set Methods Clear contribution Leverage level sets to perform reachability calculations for hybrid systems Assets Computationally sound for low dimensions Works in Matlab (no toolboxes required) November 2004 Shankar Sastry, UC Berkeley

Reachable sets: how they are calculated Time dependent H-J equation: First order hyperbolic PDE Solution can form kinks (discontinuous derivatives) For the backwards reachable set, find the “viscosity” solution [Crandall, Evans, Lions, …] Level set methods Convergent numerical algorithms to compute the viscosity solution [Osher, Sethian, …] Non-oscillatory, high accuracy spatial derivative approximation Stable, consistent numerical Hamiltonian Variation diminishing, high order, explicit time integration November 2004 Shankar Sastry, UC Berkeley

Level Set Methods: Integration Benefits Give additional capability to tools such as HyVisual Add a hybrid systems (rather than numerical) front-end to the LSM Toolbox Provide controller synthesis capability, as well as safety calculations November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Hyper Framework A new toolbox/toolsuite with the following characteristics High performance simulation High robustness factor High level modeling (with refinement) High number of interacting tools Provide a formal interchange between tools Low-level fundamental model specifications (a “core”) Requires a set of “implementable” functions to call Add a base package with interfaces for interoperability, and a lightweight editor Include industrial-strength solvers through transformations - Should be highly robust, with respect to many different KINDS of systems (hybrid, discrete, stochastically driven, etc.) - November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Hyper Framework Interoperability Interfaces November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Hyper Framework Extensible to other tools Existing examples for integration through HyVisual/T-LSM A more focused, useful, core interchange format When integrated, allows persistence of legacy models in industry (Matlab/Simulink), now with advantage(s) of synthesis/verification Newer/faster tools can be tested against “known true” Check for same behavior Can be used for regression testing November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley Conclusions Not trying to build Rome in 1 day Focused application to certain capabilities at first Concentration on not solving all the world’s problems at once Restricting initial release to useful and mature tools Immediately providing two diverse toolbox integrations as examples to others November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley November 2004 Shankar Sastry, UC Berkeley

Shankar Sastry, UC Berkeley BASE Package Simulators Matlab, HyVisual Simulators Matlab, HyVisual Simulators Matlab, HyVisual CORE Extraction model database Verification Engine HyTech, LSM, CheckMate Verification Engine HyTech, LSM, CheckMate Verification Engine HyTech, LSM, CheckMate Manipulation Visualization Editor Transformations XSL, GReAT Transformations XSL, GReAT Transformations XSL, GReAT November 2004 Shankar Sastry, UC Berkeley