st 1 HYCON PhD School on Hybrid Systems www.ist-hycon.org Solution Concepts www.unisi.it and W ell-posedness of Hybrid Systems Maurice Heemels Embedded Systems Institute (NL) maurice.heemels@embeddedsystems.nl scimanyd suounitnoc enibmoc smetsys dirbyH lacipyt (snoitauqe ecnereffid ro laitnereffid) scimanyd etercsid dna stnalp lacisyhp fo fo lacipyt (snoitidnoc lacigol dna atamotua) fo senilpicsid gninibmoc yB .cigol lortnoc ,yroeht lortnoc dna smetsys dna ecneics retupmoc dilos a edivorp smetsys dirbyh no hcraeser ,sisylana eht rof sloot lanoitatupmoc dna yroeht fo ngised lortnoc dna ,noitacifirev ,noitalumis egral a ni desu era dna ,''smetsys deddebme`` ria ,smetsys evitomotua) snoitacilppa fo yteirav ssecorp ,smetsys lacigoloib ,tnemeganam ciffart .(srehto ynam dna ,seirtsudni HYSCOM IEEE CSS Technical Committee on Hybrid Systems 5 Siena, July 1 9-22, 2005 - Rectorate of the University of Siena
Solution Concepts and Well-posedness of Hybrid Systems Maurice Heemels Embedded Systems Institute Eindhoven, The Netherlands maurice.heemels@esi.nl HYCON Summer School on Hybrid Systems 1/62 1/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Key issues: • Solution concepts • Well-posedness: existence & uniqueness of solutions given an initial condition Outline lecture • Smooth systems: differential equations • Switched systems: Discontinuous differential equations: “classics” • Hybrid automata • Zenoness: importance of choice of solution concept • Some piecewise linear, linear relay and complementarity systems • Summary 2/62 2/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Solution concept Description format / syntax / model ↓ solutions / trajectories / executions/ semantics/ behavior ⇒ Well-posedness: given initial condition does there exists a solution and is it unique ? Let’s start simple ... 3/62 3/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Smooth differential equations Example ˙ x = f ( t, x ) x ( t 0 ) = x 0 . A solution trajectory is a function x : [ t 0 , t 1 ] �→ R n that is continuous, differentiable and satisfies x ( t 0 ) = x 0 and x ( t ) = f ( t, x ( t )) for all t ∈ ( t 0 , t 1 ) ˙ Well-posedness: given initial condition does there exists a solution and is it unique ? 4/62 4/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Well-posedness x = 2 √ x , x (0) = 0 . Solutions: x ( t ) = 0 and x ( t ) = t 2 . Example ˙ Local existence and uniqueness of solutions given an initial condition: Theorem 1 Let f ( t, x ) be piecewise continuous in t and satisfy the following Lipschitz condition: there exist an L > 0 and r > 0 such that � f ( t, x ) − f ( t, y ) � ≤ L � x − y � and all x and y in a neighborhood B := { x ∈ R n | � x − x 0 � < r } of x 0 and for all t ∈ [ t 0 , t 1 ] . ⇓ There is a δ > 0 s.t. a unique solution exists on [ t 0 , t 0 + δ ] starting in x 0 at t 0 . 5/62 5/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Global well-posedness x = x 2 + 1 , x (0) = 0 . Solution: x ( t ) = tan t . Local on [0 , π/ 2) . Example ˙ • Note that we have lim t ↑ π/ 2 x ( t ) = ∞ . Finite escape time! Theorem 2 (Global Lipschitz condition) Suppose f ( t, x ) is piecewise con- tinuous in t and satisfies � f ( t, x ) − f ( t, y ) � ≤ L � x − y � for all x , y in R n and for all t ∈ [ t 0 , t 1 ] . Then, a unique solution exists on [ t 0 , t 1 ] for any initial state x 0 at t 0 . x = − x 3 not glob. Lipsch., but unique global solutions. • Not necessary: ˙ • Also in hybrid systems, but even more awkward stuff (Zeno) 6/62 6/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Discontinuous differential equations: a class of switched systems C + x' = f (x) + φ (x)=0 C - � , if x ∈ C + := { x ∈ R n | φ ( x ) > 0 } f + ( x ) x' = f (x) x = ˙ - , if x ∈ C − := { x ∈ R n | φ ( x ) < 0 } f − ( x ) • x in interior of C − or C + : just follow! • f − ( x ) and f + ( x ) point in same direction: just follow! ∇ φ ( x ) �∇ φ ( x ) � then ( n ( x ) T f − ( x )) · ( n ( x ) T f + ( x )) > 0 n ( x ) = • n ( x ) T f + ( x ) > 0 ( f + ( x ) points towards C + ) and n ( x ) T f − ( x ) < 0 ( f − ( x ) points towards C − ): At least two trajectories 7/62 7/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Sliding modes f (x ) - 0 C + φ (x)=0 x 0 C - f (x ) + 0 n ( x ) T f + ( x ) < 0 ( f + ( x ) points towards C − ) and n ( x ) T f − ( x ) > 0 ( f − ( x ) points towards C + ). No classical solution • Relaxation: spatial (hysteresis) ∆ , time delay τ , smoothing ε • Chattering / infinitely fast switching (limit case ∆ ↓ 0 , ε ↓ 0 , and τ ↓ 0 ) Filippov’s convex definition : convex combination of both dynamics x = λf + ( x ) + (1 − λ ) f − ( x ) with 0 ≤ λ ≤ 1 ˙ such that x moves (“slides”) along φ ( x ) = 0 . “Third mode ... ” 8/62 8/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Differential inclusions f + ( x ) , if φ ( x ) > 0 λf + ( x ) + (1 − λ ) f − ( x ) , if φ ( x ) = 0 , 0 ≤ λ ≤ 1 x = ˙ f − ( x ) , if φ ( x ) < 0 , x ∈ F ( x ) with set-valued Differential inclusion ˙ { f + ( x ) } , φ ( x ) > 0 F ( x ) = { λf + ( x ) + (1 − λ ) f − ( x ) | λ ∈ [0 , 1] } , φ ( x ) = 0 { f − ( x ) } , φ ( x ) < 0 Definition 3 A function x : [ a, b ] �→ R n is a solution of ˙ x ∈ F ( x ) , if x is x ( t ) ∈ F ( x ( t )) for almost all t ∈ [ a, b ] . absolutely continuous and satisfies ˙ 9/62 9/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
A well-posedness result C + x' = f (x) + φ (x)=0 C - x' = f (x) - • f − and f + are continuously differentiable ( C 1 ) • φ is C 2 • the discontinuity vector h ( x ) := f + ( x ) − f − ( x ) is C 1 If for each point x with φ ( x ) = 0 at least one of the two inequalities n ( x ) T f + ( x ) < 0 or n ( x ) T f − ( x ) > 0 (for different points a different in- equality may hold), then the Filippov solutions exist and are unique. 10/62 10/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Alternative: Utkin’s equivalent control definition � g + ( x ) , ξ ( x ) > 0 x = f ( x, u ) with u = ˙ g − ( x ) , ξ ( x ) < 0 • Sliding mode: f + ( x ) := f ( x, g + ( x )) and f − ( x ) := f ( x, g − ( x )) point outside C + and C − , resp. { g + ( x ) } , if ξ ( x ) > 0 u equiv ∈ U ( x ) := { λg + ( x ) + (1 − λ ) g − ( x ) | λ ∈ [0 , 1] } , if ξ ( x ) = 0 { g − ( x ) } , if ξ ( x ) < 0 Differential inclusion x ∈ F ( x ) := f ( x, U ( x )) = { f ( x, u ) | u ∈ U ( x ) } ˙ “Idealization” determines Filippov/ Utkin / different solution concept! 11/62 11/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Example x 1 = − x 1 + x 2 − u ˙ x 2 = 2 x 2 ( u 2 − u − 1) ˙ � 1 , if x 1 > 0 u = − 1 , if x 1 < 0 . Two “original” dynamics: • C + : x 1 > 0 : • C − : x 1 < 0 : x = f + ( x ) ˙ x = f − ( x ) ˙ x 1 = − x 1 + x 2 − 1 x 1 = − x 1 + x 2 + 1 ˙ ˙ x 2 = − 2 x 2 ˙ x 2 = 2 x 2 ˙ 12/62 12/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Vector fields 13/62 13/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Vector fields: zoom 14/62 14/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Sliding modes? Two “original” dynamics: • C + : x 1 > 0 : x = f + ( x ) ˙ • C − : x 1 < 0 : x = f − ( x ) ˙ x 1 = − x 1 + x 2 − 1 ˙ x 1 = − x 1 + x 2 + 1 ˙ x 2 = − 2 x 2 ˙ x 2 = 2 x 2 ˙ • n ( x ) T f + ( x ) = x 2 − 1 < 0 − → x 2 < 1 • n ( x ) T f − ( x ) = x 2 + 1 > 0 − → x 2 > − 1 • Sliding possible in x 1 = 0 and x 2 ∈ [ − 1 , 1] . 15/62 15/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Filippov’s solution concept Two “original” dynamics: • C + : x 1 > 0 : x = f + ( x ) ˙ • C − : x 1 < 0 : x = f − ( x ) ˙ x 1 = − x 1 + x 2 − 1 ˙ x 1 = − x 1 + x 2 + 1 ˙ x 2 = − 2 x 2 ˙ x 2 = 2 x 2 ˙ • Filippov: Take convex combination of dynamics such that state slides on x 1 = 0 : Hence, x 1 = ˙ x 1 = 0 . • λ ( x 2 − 1) + (1 − λ )( x 2 + 1) = 0 implies λ = 1 2 ( x 2 + 1) x 2 = λ ( − 2 x 2 ) + (1 − λ )(2 x 2 ) = − 2 x 2 • Hence, ˙ 2 • 0 is unstable equilibrium. 16/62 16/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Vector fields: Filippov’s case 17/62 17/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Utkin’s solution concept x 1 = − x 1 + x 2 − u ˙ x 2 = 2 x 2 ( u 2 − u − 1) ˙ � 1 , if x 1 > 0 u = − 1 , if x 1 < 0 . • The equivalent control u equiv is such that state slides along x 1 = 0 . Hence, x 1 = ˙ x 1 = 0 and thus u equiv = x 2 and x 2 = 2 x 2 ( x 2 ˙ 2 − x 2 − 1) • Equilibria: -0.618 (unstable) and 0 (stable) 18/62 18/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Vector fields 19/62 19/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
Solution trajectories 20/62 20/62 ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
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