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Tom Henzinger University of California, Berkeley

The Symbolic Approach to Hybrid Systems. Tom Henzinger University of California, Berkeley. Discrete (transition) system. Discrete (transition) system. Continuous (dynamical) system. Q = R n.

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Tom Henzinger University of California, Berkeley

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  1. The Symbolic Approach to Hybrid Systems Tom Henzinger University of California, Berkeley

  2. Discrete (transition) system

  3. Discrete (transition) system Continuous (dynamical) system Q = Rn

  4. Discrete (transition) system Continuous (dynamical) system Q = Rn Hybrid system jumps flows

  5. Discrete (transition) system Continuous (dynamical) system Q = Rn Hybrid system jumps flows -nondeterministic -time abstract

  6. A Thermostat States x  R temperature h  { on, off } heat

  7. A Thermostat States x  R temperature h  { on, off } heat Flows f1Y h = on  x' = K(H-x) f2Y h = off  x' = -Kx invariants

  8. A Thermostat States x  R temperature h  { on, off } heat Flows f1Y h = on  x' = K(H-x) f2Y h = off  x' = -Kx Jumps j1Y h = on  h := off j2Y h = off  h := on guards

  9. f1 h = on j1 j2 h = off f2 x

  10. A Thermostat States x  R temperature h  { on, off } heat t  R timer Flows f1Y h = on  t  U  x' = K(H-x); t' := 1 f2Y h = off  t  U  x' = -Kx; t' := 1 Jumps j1Y h = on  t  L  h := off; t' := 0 j2Y h = off  t  L  h := on; t' := 0

  11. A Thermostat States x  R temperature h  { on, off } heat t  R timer Flows f1Y h = on  t  U  x' = K(H-x); t' = 1 f2Y h = off  t  U  x' = -Kx; t' = 1 Jumps j1Y h = on  t  L  h := off; t := 0 j2Y h = off  t  L  h := on; t := 0

  12. A Thermostat States x  R temperature h  { on, off } heat t  R timer Flows f1Y h = on  t  U  x' = K(H-x); t' = 1 f2Y h = off  t  U  x' = -Kx; t' = 1 Jumps j1Y h = on  t  L  h := off; t := 0 j2Y h = off  t  L  h := on; t := 0

  13. x f1 h = on t x h = off t L U

  14. x f1 h = on j1 t x h = off t L U

  15. x f1 h = on j1 t x h = off f2 t L U

  16. x f1 h = on j1 t x j2 h = off f2 t L U

  17. x f1 h = on j1 t x j2 h = off f2 t L U

  18. x f2 h = off t L U Step 1: Discretize

  19. Transition System Q set of states  set of actions post: Q    2Q successor function

  20. Transition System Q set of states  set of actions post: Q    2Q successor function Thermostat Q = R2 { on, off } = { f1, f2, j1, j2 } post ( x, t, on, j1) = { { (x, 0, off) } if t  L  if t < L

  21. Transition System Q set of states  set of actions post: Q    2Q successor function Thermostat Q = R2 { on, off } = { f1, f2, j1, j2 } post ( x, t, on, j1) = post ( x, t, on, f1) = { { (x, 0, off) } if t  L  if t < L infinite set if t < U { (x, 0, on) } if t = U  if t > U {

  22. x Step 2: Lift h = on t x R h = off t L U

  23. x Step 2: Lift h = on t x R h = off post(R,f2) t L U

  24. x Step 2: Lift post( post(R,f2), j2) h = on t x R h = off post(R,f2) t L U

  25. Lifted Transition System Q  post: 2Q    2Q post(R,) = UqQ post(q,)

  26. Lifted Transition System Q  post: 2Q    2Q post(R,) = UqQ post(q,) pre: 2Q    2Q pre(R,) = UqQ pre(q,)

  27. x h = on t x h = off R t L U

  28. x h = on t x pre(R,f2) h = off R t L U

  29. x pre( pre(R,f2), j2) h = on t x pre(R,f2) h = off R t L U

  30. x Step 3: Observe h = on t x 3 < x < 4 h = off 2 < x < 3 1 < x < 2 0 < x < 1 t L U

  31. Observed Transition System Q  pre, post: 2Q    2Q A = { a1, a2, a3, … } set of observations Q a3 a2 a1

  32. Observed Transition System Q  pre, post: 2Q    2Q A set of observations Thermostat A = { on, off } [ { x = c, c < x < c+1 | c  Z }

  33. Symbolic Transition System Q  pre, post A = { R1, R2, … } set of regions Ri  Q Region algebra: 1. A  2. pre, post:      computable 3. Å : 2   \ : 2   computable  : 2  { t, f }

  34. Symbolic Transition System 1. Local computation: Region Operations Compute pre, post, Å , \ , and  on regions in . 2. Global computation: Symbolic Semi-Algorithms Starting from the observations in A, compute new regions in  by applying the operations pre, post, Å , \ , and  .

  35. Region Algebra If -Q is the valuations for a set X:Vals of typed variables, -the effect of transitions can be expressed using Ops on Vals, -the first-order theory FO(Vals,Ops) admits quantifier elimination, then the quantifier-free fragment ZO(Vals,Ops) is a region algebra. This is because each pre and post operation is a quantifier elimination: pre(R(X)) = (X) (Trans(X,X)  R(X))

  36. Example: Polyhedral Hybrid Automata Q = Bm Rn Invariants and guards: boolean and linear constraints, e.g. a  (3x1 + x2 7) Flows: rectangular differential inclusions, e.g. x'1 [1,2] Jumps: boolean and linear constraints, e.g. x2 := 2x1 + x2 +1

  37. Example: Polyhedral Hybrid Automata Q = Bm Rn Invariants and guards: boolean and linear constraints, e.g. a  (3x1 + x2 7) Flows: rectangular differential inclusions, e.g. x'1 [1,2] Jumps: boolean and linear constraints, e.g. x2 := 2x1 + x2 +1 A = set of boolean valuations and integral polyhedra in Rn

  38. Example: Polyhedral Hybrid Automata Q = Bm Rn Invariants and guards: boolean and linear constraints, e.g. a  (3x1 + x2 7) Flows: rectangular differential inclusions, e.g. x'1 [1,2] Jumps: boolean and linear constraints, e.g. x2 := 2x1 + x2 +1 A = set of boolean valuations and integral polyhedra in Rn  = set of boolean valuations and rational polyhedra in Rn x= … ZO(Q,,+)

  39. Example: Polyhedral Hybrid Automata FO(Q,,+) admits quantifier elimination, hence ZO(Q,,+) is a region algebra. Jump j: Y x1 x2  x2 := 2x1-1 pre( 1 x1  x2  2, j ) = ( x1, x2) (x1 x2  x1 = x1  x2 = 2x1-1  1 x1 x2  2)

  40. Example: Polyhedral Hybrid Automata FO(Q,,+) admits quantifier elimination, hence ZO(Q,,+) is a region algebra. Jump j: Y x1 x2  x2 := 2x1-1 pre( 1 x1  x2  2, j ) = ( x1, x2) (x1 x2  x1 = x1  x2 = 2x1-1  1 x1 x2  2) = x1 x2  1  x1  2x1-1  2 = x1 x2  1  x1  3/2

  41. x2 R x1

  42. x2 pre(R,j) R x1

  43. Example: Polyhedral Hybrid Automata Flow f: Y x1 x2  x'1 [1,2]; x'2 = 1 pre( 1 x1  x2  2, f ) = ( 1 k1 2) (   0) (1 x1+k1  x2+  2 ( 0    ) (x1+k1  x2+))

  44. Example: Polyhedral Hybrid Automata Flow f: Y x1 x2  x'1 [1,2]; x'2 = 1 pre( 1 x1  x2  2, f ) = ( 1 k1 2) (   0) (1 x1+k1  x2+  2 ( 0    ) (x1+k1  x2+)) = (   0) (  d1 2) (1 x1+d1  x2+  2 x1 x2  x1+d1  x2+) convex guards

  45. Example: Polyhedral Hybrid Automata Flow f: Y x1 x2  x'1 [1,2]; x'2 = 1 pre( 1 x1  x2  2, f ) = ( 1 k1 2) (   0) (1 x1+k1  x2+  2 ( 0    ) (x1+k1  x2+)) = (   0) (  d1 2) (1 x1+d1  x2+  2 x1 x2  x1+d1  x2+) = (   0) (x1 x2  1 x1+2  1 x2+ 2) =x1 x2  x2  2  2x2  x1+3 convex guards

  46. x2 R pre(R,f) x1 f

  47. y x

  48. far near x = 1000 x’[-50,-40] x’[-50,-30] app! app x  1000 x  0 exit! x = 100  x : [2000,) x = 0 exit past x’[30,50] x  100 train x: initialized rectangular variable

  49. up open y = 90 y’ = 9 y’ = 0 y  90 raise raise? lower? raise? lower down closed y = 0 y’ = -9 y’ = 0 y  0 gate lower? y: uninitialized singular variable

  50. app exit t := 0 t := 0 t’ = 1 app? exit? t’ = 1 idle t   t   lower! raise! controller lower raise t: clock variable : design parameter

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