Hybrid Systems Modeling, Analysis and Control Radu Grosu Vienna - PowerPoint PPT Presentation

Hybrid Systems Modeling, Analysis and Control Radu Grosu Vienna University of Technology Lecture 4

Semirings: Q , R , C and B , L A structure F = (F, + , ⋅ ,0,1) such that: (F, + ,0) is a commutative monoid: ∀ x,y,z ∈ F. (x + y) + z = x + (y + z) (ass) (com) ∀ x,y ∈ F. x + y = y + x ∀ x ∈ F. x + 0 = x (zer) (F \ {0}, ⋅ ,1) is a monoid: (ass) ∀ x,y,z ∈ F. (x ⋅ y) ⋅ z = x ⋅ (y ⋅ z) (zer) ∀ x ∈ F. x ⋅ 1 = x = 1 ⋅ x Compatibility of addition and multiplication: (dis l ) ∀ x,y,z ∈ F. x ⋅ (y + z) = (x ⋅ y) + (x ⋅ z) (dis r ) ∀ x,y,z ∈ F. (x + y) ⋅ z = (x ⋅ z) + (y ⋅ z) (ann) ∀ x ∈ F. 0 ⋅ x = x ⋅ 0 = 0

Partial Order A structure F = (F, ≤ ) such that: (ref) ∀ x ∈ F. x ≤ x (ant) ∀ x,y ∈ F. (x ≤ y) ∧ (y ≤ x) ⇒ x = y (tra) ∀ x,y,z ∈ F. (x ≤ y) ∧ (y ≤ z) = x ≤ z Canonical partial order in a semiring F : (ass) ∀ x,y ∈ F. x ≤ y ! ∃ z ∈ F. x + z = y Theorem: A semiring cannot have both − Additive inverse − Canonical partial order Proof by contradiction: ∀ x,y. x ≤ y ≡ ∃ (( − x) + y). x + ( − x) + y = y

Divergence of Mathematics Semirings F = (F, + , ⋅ ,0,1) Fields Dioids F = (F, + , ⋅ ,0,1, − , − 1 ) F = (F, + , ⋅ ,0,1, ≤ ) Continuous math Idempotent Dioids F = (F, + , ⋅ ,0,1, ≤ ) Q , R , C and F 2 x + x = x Discrete math B , L and P (S)

Complete Dioid A dioid F = (F, + , ⋅ ,0,1, ≤ ) such that: (F, ≤ ) is complete as an ordered set F satisfies the following infinite distributivity: (rdis) ∀ A ⊂ F,b ∈ F. ( + a ∈ A a) ⋅ b = + a ∈ A (a ⋅ b) (ldis) ∀ A ⊂ F,b ∈ F. b ⋅ ( + a ∈ A a) = + a ∈ A (b ⋅ a) Consequence: ∀ A ⊂ F,B ⊂ F. ( + a ∈ A a) ⋅ ( + b ∈ B b) = + a ∈ A,b ∈ B (a ⋅ b) Top element: T = + a ∈ F a ∀ x ∈ F. T + x = T , T ⋅ 0 = 0

Examples of Complete Dioids Max-Plus complete dioid X = ( R ± ∞ , max , + , − ∞ , 0 , ≤ ) ∀ x ∈ F . max( ∞ ,x) = ∞ , ∞ + ( −∞ ) ! −∞ Min-Plus complete dioid N = ( R ± ∞ , min , + , ∞ , 0 , ≤ ) ∀ x ∈ F . min ( −∞ , x ) = −∞ , −∞ + ( ∞ ) ! ∞ Languages complete dioid L = ( P ( Σ * ), + , ⋅ , 0 , 1 , ⊆ ) ∀ L ∈ P ( Σ * ). L + Σ * = Σ * , Σ * ⋅ 0 = 0

Least Fixpoint Consider the equation: x = xa + b Teorem: The least fixpoint of x = xa + b is ba * Proof: (fxp) (ba * )a + b = ba + + b = b ( a + + 1 ) = ba * (lfp) x = xa + b ⇒ b ≤ x ⇒ ba ≤ x (lfp) x = xaa + ba + b ... ... (lfp) ... ⇒ b ⋅ ( + n ∈ N a n ) ≤ x

Left Semi-module: B n , L n and P (S) n A structure V = ( F , T , i ) such that: − F = (F, + , ⋅ , 0 , 1 ) is a semiring (of scalars) − T = (T,+,0) is commutative monoid (of vectors) − Scalar multiplication satisfies: (dis 1 ) ∀ a ∈ F, x,y ∈ T. a i (x+ y) = a i x + a i y (dis 2 ) ∀ a , b ∈ F, x ∈ T. ( a + b ) i x = a i x + b i x (cmp) ∀ a , b ∈ F, x ∈ T. a ⋅ ( b i x) = ( a ⋅ b ) i x ∀ x ∈ T. = x (ntr) 1 i x Typical example: V = ( B , B n , ∧ ) [x 1 ,x 2 ] ∨ [y 1 ,y 2 ] = [x 1 ∨ y 1 ,x 2 ∨ y 2 ] , a ∧ [x 1 ,x 2 ] = [a ∧ x 1 ,a ∧ x 2 ]

Least Fixpoint Consider the equation: x = xA + b Teorem: The least fixpoint of x = xA + x 0 is x 0 A * Proof: Extension from semirings to semimodules Question: How do we compute A *

Time-Triggered Automata Difference equations: x (n+1) = x (n) A , y (n) = x (n) C , x (0) = x 0 Impulse and delay: δ (n) = (n = 0)? 1 : 0 D m (x)(m + n) = x(n) Power-series representation: ∞ ∞ ∑ ∑ x = ( x (n)D n )( δ ) = x 0 D 0 ( δ ) + ( x (n + 1)D n + 1 )( δ ) n = 0 n = 0 x = x 0 D 0 ( δ ) + ( x A D)( δ ) = x 0 ( A D) * ( δ )

Time-Triggered Automata Z-Transform: ∞ ∞ ∑ ∑ Z{ x (n)} = x (n) z − n = x 0 + x (n + 1) z − (n + 1) n = 0 n = 0 X = x 0 + X A z − 1 = x 0 ( A z − 1 ) * Inverse Z-Transform: x = x 0 ( A D) * ( δ ) Block diagram: Automaton with explicit delay

Event-Triggered Automata Difference equations: x (w σ ) = x (w) A ( σ ), y (w) = x (w) C , x ( ε ) = x 0 Instead of linear time, one has multiform time: N = {a} * , V = {a,b} * = V * a a a … a a b b a b

Event-Triggered Automata Difference equations: x (w σ ) = x (w) A ( σ ), y (w) = x (w) C , x ( ε ) = x 0 Impulse and delay: δ (w) = (w = ε )? 1 : 0 w( x )(wu) ! D w ( x )(wu) ! x (u) Power-series representation: ∑ ∑ x = ( x (w) D w )( δ ) = x 0 ε ( δ ) + ( x (w σ ) D w σ )( δ ) w ∈ V σ∈ V,w ∈ V ∑ ∑ = x 0 D ε ( δ ) + ( x (w) A ( σ )D w σ )( δ ) = x 0 D ε ( δ ) + ( x (w)D w A ( σ )D σ )( δ ) σ∈ V,w ∈ V σ∈ V,w ∈ V x = x 0 ε ( δ ) + ( x A )( δ ) = x 0 ( A ) * ( δ )

Event-Triggered Automata Z-Transform: ∑ ∑ Z{ x (w)} = = x 0 + x (w σ ) w σ x (w) w w ∈ V σ∈ V,w ∈ V ∑ ∑ = x 0 + x (w) A ( σ ) w σ = x 0 + ( x (w)w) A σ∈ V,w ∈ V w ∈ V X = x 0 + X A = x 0 A * Inverse Z-Transform: x = x 0 A * ( δ ) Block diagram: Finite automaton.

Event-Triggered Automata Z-Transform: ∑ ∑ Z{ x (w)} = = x 0 + x (w σ ) w σ x (w) w w ∈ V σ∈ V,w ∈ V ∑ ∑ = x 0 + x (w) A ( σ ) w σ = x 0 + ( x (w)w) A σ∈ V,w ∈ V w ∈ V X = x 0 + X A = x 0 A * Inverse Z-Transform: x = x 0 A * ( δ ) Block diagram: Finite automaton.

DT-Signal as Formal Power Series Linear time: A = {a}, A = A * , A ≅ N x(a) x : N → K x(a 2 ) x( ε ) K is a semiring … 0 1 N 2 D( x )(aw) ! a ( x )(aw) ! x (w) ε A a a 2 δ (w) ! ε (w) ! (w = ε )? 1 : 0 Formal-power-series representation: ∑ ∑ ∑ x = x (n)D n ( δ ) = x (w)D |w| ( δ ) = x (w) w n ∈ N w ∈ A w ∈ A

DT-Signal as Formal Power Series Discrete-time-signals semiring: (superposition) ( x + y )(w) = x (w) + y (w) n = |w| ∑ ∑ (convolution) ( x ∗ y )(w) = x (u) y (v) = x (k) y (n − k) uv = w k = 0 Output of linear time-invariant systems: ∑ ∑ ∑ f( x ) = x (n)f(D n ( δ ) ) = x (n)D n (f( δ )) ) = x (n)D n ( y ) n ∈ n ∈ n ∈ N N N Linearity Time invariance Impulse response k k ∑ ∑ f( x )(k) = x (n)D n ( y )(k) = x (n) y (k − n) Convolution n = 0 n = 0

MT-Signal as Formal Power Series Branching time: A = {a,b}, A = A * , A ≅ T (a tree) x(a) x(aa) x : A → K x( ε ) a x(ab) K is a semiring a x(b) b … ε D a (x)(aw) ! a (x)(aw) ! x(w) x(bb) b D b (x)(bw) ! b (x)(bw) ! x(w) x(ba) … b δ (w) ! ε (w) ! (w = ε )? 1 : 0 a Formal power-series representation: ∑ ∑ x = x (w)D w ( δ ) = x (w) w w ∈ A w ∈ A

BT-Signal as Formal Power Series Branching-time-signals semiring: (superposition) ( x + y )(w) = x (w) + y (w) ∑ (convolution) ( x ∗ y )(w) = x (u) y (v) uv = w Output of linear time-invariant systems: ∑ ∑ ∑ f( x ) = x (w)f(D w ( δ ) ) = x (w)D w (f( δ )) ) = x (n)D w ( y ) w ∈ A w ∈ A w ∈ A Linearity Time invariance Impulse response ∑ ∑ f( x )(w) = x (u)D u ( y )(w) = x (u) y (v) Convolution uv = w uv = w NFA A: L (A) is a BT- B -signal ( K = B in the FP series)!