Download presentation
Presentation is loading. Please wait.
1
1 Verification and Synthesis of Hybrid Systems Thao Dang October 10, 2000
2
2 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
3
3 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
4
4 Hybrid systems Hybrid systems: systems which combine continuous-time dynamics and discrete-event dynamics Continuous processesDigital controllers, switches, gears.. (e.g., chemical reactions) Arisen virtually everywhere (due to the increasing use of computers)
5
5 Analysis of Hybrid Systems Formal verification: prove that the system satisfies a given property Controller synthesis: design controllers so that the controlled system satisfies a desired property We concentrate on invariance properties: all trajectories of the system stay in a subset of the state space Hybrid systems are difficult to analyze No existing general method
6
6 Illustrative Example: A Thermostat onoff Verification problem: prove that the temperature x [a,b] Characterize all behaviors Reachability Analysis
7
7 The Thermostat Example (cont’d) Two-phase behavior Non-deterministic behavior Set of initial states x t max min 00 0 How to characterize and represent “tubes” of trajectories of continuous dynamics in order to treat discrete transitions??
8
8 Algorithmic Analysis of Hybrid Systems Exact symbolic methods applicable for restricted classes of hybrid systems Our objective: verification method for general hybrid systems in any dimension
9
9 Algorithmic Verification of Hybrid Systems approximate reachability techniques represent reachable sets by orthogonal polyhedra What do we need?? a reachability technique which is applicable for arbitrary continuous systems can be extended to hybrid systems
10
10 Approximations by Orthogonal Polyhedra Non-convex orthogonal polyhedra (unions of hyperrectangles) Motivations canonical representation, efficient manipulation in any dimension easy extension to hybrid systems termination can be guaranteed Over-approximation Under-approximation
11
11 Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
12
12 Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
13
13 Reachability Analysis of Continuous Systems Problem Find an orthogonal polyhedron over-approximating the reachable set from F x(0) F, set of initial states
14
14 [0,r](F)[0,r](F) Successor Operator r(F)r(F) F Reachable set from F: (F) = [0, ) (F)
15
15 Abstract Algorithm for Calculating (F) P 0 := F ; repeat k = 0, 1, 2.. P k+1 := P k [0,r] (P k ) ; until P k+1 = P k Use orthogonal polyhedra to represent P k approximate [0,r] r : time step
16
16 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
17
17 Reachability of Linear Continuous Systems F is a convex polyhedron: F = conv{v 1,..,v m } r (F) = e Ar F F vivi r (v i )=e Ar v i F is the set of initial states r (F) = conv{ r (v 1 ),.., r (v m )}
18
18 Over-Approximating the Reachable Set [0,2r] (F) P 2 = G 1 G 2 X2X2 P2P2 [0,r] (F) G 1 P 1 =G 1 [r,2r] (F) G 2 X1X1 X2X2 G2G2 X0=FX0=F r(v2)r(v2) X 1 = r (X 0 ) v1v1 v2v2 r(v1)r(v1) X1X1 X1X1 X0X0 C 1 =conv{X 1,X 0 } C1C1 Cb 1 Extension to under-approximations
19
19 Example
20
20 Extension to Linear Systems with Uncertain Input Computation of r (F) [Varaiya 98] i (r) i F yi*(r)yi*(r) yiyi r(F)r(F) Bloating amount u1u1 u2u2 (Maximum Principle)
21
21 Example [Kurzhanski and Valyi 97] Advantage: time-efficiency
22
22 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
23
23 Principle of the Reachability Technique y F x ‘Face lifting’ technique, inspired by [Greenstreet 96] x(0) F, set of initial states Continuity of trajectories compute from the boundary of F The initial set F is a convex polyhedron The boundary of F: union of its faces
24
24 N(e) H(e) Over-Approximating [0,r] (F) Step 1: rough approximation N(F) F e f e : projection of f on the outward normal to face e : maximum of f e over the neighborhood N(e) of e H’(e) r e1e1 N(F) Step 2: more accurate approximation
25
25 Computation Procedure Decompose F into non-overlapping hyper-rectangles Apply the lifting operation to each hyper-rectangle (faces on the boundary of F) Make the union of the new hyper-rectangles F
26
26 Example: Airplane Safety [Lygeros et al. 98] P = [V min,V max ] [ min, max ]
27
27 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
28
28 Hybrid Systems Hybrid automata continuous dynamics: linear with uncertain input, non-linear staying and switching conditions: convex polyhedra reset functions : affine of the form R qq’ (x) = D qq’ x + J qq’ q0q0 q1q1 switching condition reset function discrete state staying condition continuous dynamics
29
29 Reachability of Hybrid Automata The state (q, x) of the system can change in two ways: continuous evolution: q remains constant, and x changes continuously according to the diff. eq. at q discrete evolution (by making a transition): q changes, and x changes according to the reset function. Reachability analysis continuous-successors discrete-successors approximations by orthogonal polyhedra
30
30 Over-approximating Continuous-Successors Use the reachability algorithms for continuous systems Take into account the staying conditions HqHq F [0,r] (F) P
31
31 F g F G qq’ Over-approximating Discrete-Successors R qq’ (b) H q’ F qq’ (q, F) = (q’, R qq’ (F G qq’ ) H q’ ) b G qq’ FgFg
32
32 q0q0 q1q1 q0q0 Example q0q0 q1q1
33
33 Plan 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
34
34 Switching Controller Synthesis: Introduction q1q1 q2q2 q3q3 f1f1 f2f2 f3f3 qx Mode selection Plant Discrete Switching Controller q3q3 q1q1 q2q2
35
35 The Safety Synthesis Problem Given a hybrid automaton A and a set F How to restrict the guards and the staying conditions of A so that all trajectories of the resulting automaton A * stay in F Solution: Compute the maximal invariant set (set of ‘winning’ states)
36
36 Operator Given F={(q, F q ) | q Q}, (F) consists of states from which all trajectories stay indefinitely in F without switching OR stay in F for some time and then make a transition to another discrete state and still in F G qq’ F q’ FqFq x1x1 x2x2 x3x3
37
37 Calculation of the Maximal Invariant Set P 0 := F ; repeat k = 1, 2,.. P k+1 := P k (P k ) ; until P k+1 = P k P * = P k ; P * : maximal invariant set A * : H * =H P *, G * =G P *
38
38 Effective Approximate Synthesis Algorithm Use our reachability techniques for hybrid automata to approximate (F) Under-approximations Effective approximate synthesis algorithm for hybrid systems with linear continuous dynamics To approximate the maximal invariant set:
39
39 F0F0 F1F1 G 10 G 01 G 10 F 0 F1F1 F0F0 G 01 F 1 G 01 =[-0.2,-0.01] [-0.2,-0.01] G 10 =[0.01,0.32] [-0.01,0.1]
40
40 Plan 1- Approach to Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation
41
41 The tool d/dt Three types of automatic analysis for hybrid systems with linear differential inclusions Reachability Analysis: compute an over-approximation of the reachable set from a given initial set Safety Verification: check whether the system reaches a set of bad states Safety Controller Synthesis: synthesize a switching controller so that the controlled system always remains inside a given set
42
42 Implementation OpenGL LEDA Interface Verification Algorithms Controller Synthesis Algorithms Numerical Integration CVODE Geometric Algorithms Qhull, Polka, Cubes Orthogonal Approximations d/dt
43
43 The tool d/dt
44
44 Conclusions Generality of Systems Complexity of continuous and discrete dynamics High dimensional systems Variety of Problems Safety Verification and Synthesis Applications collision avoidance ( 4 continuous variables, 1 discrete state ) double pendulum ( 3 continuous variables, 7 discrete states ) freezing system ( 6 continuous variables, 9 discrete states )
45
45 Perspectives More efficient analysis techniques - Combining with analytic/qualitative methods - Adapting existing techniques for discrete/timed systems More classes of problems - more properties to verify, more synthesis criteria - controller synthesis for more general systems, e.g linear diff. games Tool - more interactive analysis, simulation features - experimentation: real-life problems
46
46 Related Work Reachability Analysis Polygonal Projections [Greenstreet and Mitchell 99] Ellipsoidal Techniques [Kurzhanski and Varaiya 00] Approximations via Parallelotopes [Kostoukova 99] Verification CheckMate [Chutinan and Krogh 99] HyperTech [Henzinger et al. 00] VeriShift [Botchkarev and Tripakis 00] Symbolic Method [Lafferriere, Pappas, and Yovine 99] Synthesis Synthesis for timed automata [Asarin, Maler, Pnueli, and Sifakis 98] Hamilton Jacobi Partial Diff. Eq. [Lygeros, Tomlin, and Sastry 98] Computer Algebra [Shakernia, Pappas, and Sastry 00]
47
47 Fin Merci
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.