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

Radu Grosu Vienna University of Technology

Hybrid Systems Modeling, Analysis and Control

Lecture 4

A structure F = (F,+,⋅,0,1) such that:

(F,+,0) is a commutative monoid:

(ass) ∀x,y,z ∈F. (x + y) + z = x + (y + z) (com) ∀x,y ∈F. x + y = y + x (zer) ∀x ∈F. x + 0 = x

(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:

(disl) ∀x,y,z ∈F. x ⋅(y + z) = (x ⋅ y) + (x ⋅ z) (disr) ∀x,y,z ∈F. (x + y)⋅ z = (x ⋅ z) + (y ⋅ z) (ann) ∀x ∈F. 0 ⋅ x = x ⋅0 = 0

Semirings: Q, R,C and B, L

(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

A structure F = (F,≤) such that:

(ass) ∀x,y ∈F. x ≤ y ! ∃z ∈F. x + z = y

Canonical partial order in a semiring F:

− Additive inverse − Canonical partial order

Partial Order

Theorem: A semiring cannot have both

∀x,y. x ≤ y ≡ ∃(( − x) + y). x + ( − x) + y = y

Proof by contradiction:

Semirings F = (F,+,⋅,0,1)

Divergence of Mathematics

Fields F = (F,+,⋅,0,1,−, −1) Dioids F = (F,+,⋅,0,1,≤) Idempotent Dioids F = (F,+,⋅,0,1,≤) x + x = x Q,R,C and F

2

Continuous math B, L and P(S) Discrete math

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∈Aa)⋅b = +a∈A(a ⋅b) (ldis) ∀A ⊂ F,b ∈F. b⋅(+a∈Aa) = +a∈A(b⋅a)

Complete Dioid

Consequence:

∀A ⊂ F,B ⊂ F. (+a∈Aa)⋅(+b∈Bb) = +a∈A,b∈B(a ⋅b)

Top element: T = +a∈Fa

∀x ∈F. T + x = T , T ⋅0 = 0

Max-Plus complete dioid X = (R±∞, max, +,

− ∞, 0, ≤)

Examples of Complete Dioids

∀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

Consider the equation: x = xa + b

Least Fixpoint

Proof:

(fxp) (ba*)a + b = ba+ + b = b(a+ + 1) = ba*

Teorem: The least fixpoint of x = xa + b is ba*

(lfp) x = xa + b ⇒ b ≤ x (lfp) x = xaa + ba + b ⇒ ba ≤ x (lfp) ... ... ... ⇒ b ⋅(+n∈Nan) ≤ x

A structure V = (F ,T ,i) such that:

−F = (F,+,⋅,0,1) is a semiring (of scalars)

(dis1) ∀a ∈F, x,y ∈T. ai(x+ y) = aix +aiy (dis2) ∀a,b ∈F, x ∈T. (a + b)ix = aix +bix (cmp) ∀a,b ∈F, x ∈T. a ⋅(bix) = (a ⋅b)ix (ntr) ∀x ∈T. 1ix = x

Typical example: V = (B, Bn,∧)

[x1,x2] ∨ [y1,y2] = [x1 ∨ y1,x2 ∨ y2], a ∧ [x1,x2] = [a ∧ x1,a ∧ x2]

Left Semi-module: Bn, L

n and P(S)n

−T = (T,+,0) is commutative monoid (of vectors) −Scalar multiplication satisfies:

Consider the equation: x = xA + b

Least Fixpoint

Proof: Extension from semirings to semimodules Teorem: The least fixpoint of x = xA + x0 is x0A* Question: How do we compute A*

Time-Triggered Automata

x(n+1) = x(n)A, y(n) = x(n)C, x(0) = x0

Difference equations:

x = ( x(n)Dn)(δ)

n=0 ∞

∑

= x0D0(δ) + ( x(n +1)Dn+1)(δ)

n=0 ∞

∑

Power-series representation:

δ(n) = (n = 0)? 1 : 0

Impulse and delay:

Dm(x)(m + n) = x(n) x = x0D0(δ) + (x A D)(δ) = x0(A D)*(δ)

Time-Triggered Automata

Z{x(n)} = x(n) z−n

n=0 ∞

∑

= x0 + x(n +1) z−(n+1)

n=0 ∞

∑

Z-Transform:

X = x0 + X A z−1 = x0(A z−1)*

Block diagram: Automaton with explicit delay Inverse Z-Transform: x = x0(AD)*(δ)

Event-Triggered Automata

x(wσ) = x(w)A(σ), y(w) = x(w)C, x(ε) = x0

Difference equations:

N = {a}*, V = {a,b}* = V*

Instead of linear time, one has multiform time:

a a … a a b a b a b

x = ( x(w) Dw)(δ)

w∈V

∑

= x0ε(δ) + ( x(wσ) Dwσ)(δ)

σ∈V,w∈V

∑

Power-series representation:

Event-Triggered Automata

x(wσ) = x(w)A(σ), y(w) = x(w)C, x(ε) = x0

Difference equations:

δ(w) = (w = ε)? 1 : 0

Impulse and delay:

w(x)(wu) ! Dw(x)(wu) ! x(u) = x0Dε(δ) + ( x(w)A(σ)Dwσ)(δ)

σ∈V,w∈V

∑

= x0Dε(δ) + ( x(w)DwA(σ)Dσ)(δ)

σ∈V,w∈V

∑

x = x0ε(δ) + (x A )(δ) = x0(A )*(δ)

Z{x(w)} = x(w) w

w∈V

∑

= x0 + x(wσ) wσ

σ∈V,w∈V

∑

Z-Transform:

X = x0 + X A = x0A *

Block diagram: Finite automaton. Inverse Z-Transform: x = x0A*(δ)

Event-Triggered Automata

= x0 + x(w)A(σ) wσ

σ∈V,w∈V

∑

= x0 + ( x(w)w)A

w∈V

∑

Z{x(w)} = x(w) w

w∈V

∑

= x0 + x(wσ) wσ

σ∈V,w∈V

∑

Z-Transform:

X = x0 + X A = x0A *

Block diagram: Finite automaton. Inverse Z-Transform: x = x0A*(δ)

Event-Triggered Automata

= x0 + x(w)A(σ) wσ

σ∈V,w∈V

∑

= x0 + ( x(w)w)A

w∈V

∑

DT-Signal as Formal Power Series

Linear time: A = {a}, A = A*, A ≅ N

x : N → K K is a semiring

x(a) x(ε) x(a2)

…

N ε 1 a 2 a2 A x = x(n)Dn(δ)

n∈N

∑

= x(w)D|w|(δ)

w∈A

∑

= x(w)w

w∈A

∑

Formal-power-series representation:

D(x)(aw) ! a(x)(aw) ! x(w) δ(w) ! ε(w) ! (w = ε)? 1 : 0

DT-Signal as Formal Power Series

(superposition) (x + y)(w) = x(w) + y(w) (convolution) (x ∗y)(w) = x(u)y(v) = x(k)y(n − k)

k=0 n=|w|

∑

uv=w

∑

Discrete-time-signals semiring:

f(x)(k) = x(n)Dn(y)(k) =

n=0 k

∑

x(n)y(k − n)

n=0 k

∑

Convolution

f(x) = x(n)f(Dn(δ)

n∈ N

∑

) = x(n)Dn(f(δ))

n∈ N

∑

) = x(n)Dn(y)

n∈ N

∑

Output of linear time-invariant systems:

Impulse response Time invariance Linearity

MT-Signal as Formal Power Series

Branching time: A = {a,b}, A = A*, A ≅ T (a tree)

x(ab)

x : A → K K is a semiring

x(ε) x(a) x(aa) x(b) x(bb) x(ba) a a b b ε b

… …

a x = x(w)Dw(δ)

w∈A

∑

= x(w)w

w∈A

∑

Formal power-series representation:

Da(x)(aw) ! a(x)(aw) ! x(w) Db(x)(bw) ! b(x)(bw) ! x(w) δ(w) ! ε(w) ! (w = ε)? 1 : 0

BT-Signal as Formal Power Series

(superposition) (x + y)(w) = x(w) + y(w) (convolution) (x ∗y)(w) = x(u)y(v)

uv=w

∑

Branching-time-signals semiring:

f(x)(w) = x(u)Du(y)(w) =

uv=w

∑

x(u)y(v)

uv=w

∑

Convolution

NFA A: L(A) is a BT-B-signal (K = B in the FP series)!

f(x) = x(w)f(Dw(δ)

w∈A

∑

) = x(w)Dw(f(δ))

w∈A

∑

) = x(n)Dw(y)

w∈A

∑

Output of linear time-invariant systems:

Impulse response Time invariance Linearity