Hybrid dynamics in large-scale logistics networks

Hybrid dynamics in large-scale logistics
networks
Mykhaylo Kosmykov
Promotionskolloquium
19. September 2011, Universit¨at Bremen

Centre for
Industrial Mathematics
Outline
1 Motivation
2 Modelling of logistics networks
3 Stability conditions for interconnected hybrid systems
4 Structure-preserving model reduction
5 Conclusion and outlook
2 / 33Motivation Model Stability Reduction Summary

Centre for
Logistics networks
Production facilities Distribution center Retailers Customers
Instability leads to:
High inventory costs
Large number of
unsatisﬁed orders
Loss of customers
Analysis steps:
1 Mathematical modelling
2 Model reduction, if the
model size is large
3 Stability analysis

Centre for
Modelling approaches
Discrete system:
Decentralized supply chain
Re-entrant/queueing
system
”Bucket brigade”
Continuous system:
Ordinary differential equations
- Damped oscillator model
Multilevel network
Partial differential equations
Hybrid model:
Hybrid system
Switched system
Stochastic system:
Stochastic system
Queueing/fluid network

Centre for
Hybrid system
xi - state of logistics location Σi (stock level)
u - external input (customer orders, raw material)
State xi changes continuously during production.
Σi
t0
xi (0)
xi (t)
xi

Centre for
Hybrid system
When a truck picks up ﬁnished material, state xi ”jumps”.
Σi
t0
xi (0)
xi (t)
t1
xi

Centre for
Hybrid system
After the jump state xi changes again continuously.
Σi
t0
xi (0)
xi (t)
t1
xi

Centre for
Hybrid system
Hybrid dynamics of location Σi :
˙xi = fi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Ci production
x+
i = gi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Di transportation
Σi
t0
xi (0)
xi (t)
t1 t2 t3
xi

Centre for
Hybrid system
The whole network Σ is given by interconnection of individual
locations:
Hybrid dynamics of the whole logistics network Σ:
˙x = ?, (x1, . . . , xn, u) ∈ C =? overall production
x+ = ?, (x1, . . . , xn, u) ∈ D =? overall transportation

Centre for
Hybrid system (Teel’s framework)
Σi :
˙xi = fi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Ci
x+
i = gi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Di
xi ∈χi ⊂RNi ,fi :Ci →RNi ,gi :Di →χi ,Ci , Di ⊂χ1×. . .×χn×U1×. . .×Un.

Centre for
Σi :
˙xi = fi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Ci
x+
i = gi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Di
Basic regularity conditions for ∃ of solutions (Goebel & Teel 2006):
fi , gi are continuous;
Ci , Di are closed.

Centre for
Σi :
˙xi = fi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Ci
x+
i = gi (x1, . . . , xn, u), (x1, . . . , xn, u)∈Di
Basic regularity conditions for ∃ of solutions (Goebel & Teel 2006):
fi , gi are continuous;
Ci , Di are closed.
Solution xi (t, k) is deﬁned on hybrid time domain:
dom xi := ∪[tk, tk+1]×{k}
t is time and k is number of the last jump.

Centre for
Interconnection of hybrid systems
Σ:
˙x = f (x, u), (x, u)∈C
x+ = g(x, u), (x, u)∈D
χ:=χ1×. . .×χn, x:=(xT
1 , . . ., xT
n )T ∈χ⊂RN, N:= Ni , C:= ∩ Ci ,
D:= ∪ Di , f :=(f T
1 , . . ., f T
n )T , g:=(gT
1 , . . ., gT
n )T with
gi (x, u) :=
gi (x, ui ), if (x, u) ∈ Di ,
xi , otherwise .

Centre for
Comparison functions
γ is p.d.f γ ∈ K γ ∈ K∞
β ∈ KLL

Centre for
Input-to-state stability (ISS) in maximization formulation
Deﬁnition (Input-to-state stability, Cai & Teel 2005)
Σ is input-to-state stable (ISS), if ∃β ∈ KLL, γ ∈ K∞ ∪ {0}, such
that for all initial values x0 all solution pairs (x, u) ∈ Su(x0) satisfy
|x(t, k)| ≤ max{β(|x0
|, t, k), γ( u (t,k))}, ∀(t, k) ∈ dom x.
Function γ is called ISS nonlinear gain.
β(|x0
|, t, k)
γ( u (t,k))
t0
|x0
|
x

Centre for
Input-to-state stability (ISS) in summation formulation
Deﬁnition
Σ is input-to-state stable (ISS), if ∃β ∈ KLL, γ ∈ K∞ ∪ {0}, such
that for all initial values x0 all solution pairs (x, u) ∈ Su(x0) satisfy
|x(t, k)| ≤ β(|x0
|, t, k) + γ( u (t,k))}, ∀(t, k) ∈ dom x.
Equivalence by max{β, γ}≤β+γ≤ max{2β, 2γ}.
β(|x0
|, t, k)
γ( u (t,k))
t0
|x0
|
x

Centre for
ISS for subsystems
Σi is ISS, if ∃βi ∈ KLL, γij , γi ∈ K∞ ∪ {0}:
|xi (t, k)|≤βi (|x0
i |, t, k)+
j,j=i
γij ( xj (t,k))+γi ( u (t,k)), i∈IΣ,
|xi (t, k)|≤ max{βi (|x0
i |, t, k), max
j,j=i
γij ( xj (t,k)),γi ( u (t,k))}, i∈Imax.
IΣ, Imax ⊂ {1, . . . , n}, IΣ ∪ Imax = {1, . . . , n}, IΣ ∩ Imax = ∅.

Centre for
ISS for subsystems
Σi is ISS, if ∃βi ∈ KLL, γij , γi ∈ K∞ ∪ {0}:
|xi (t, k)|≤βi (|x0
i |, t, k)+
j,j=i
γij ( xj (t,k))+γi ( u (t,k)), i∈IΣ,
|xi (t, k)|≤ max{βi (|x0
i |, t, k), max
j,j=i
γij ( xj (t,k)),γi ( u (t,k))}, i∈Imax.
IΣ, Imax ⊂ {1, . . . , n}, IΣ ∪ Imax = {1, . . . , n}, IΣ ∩ Imax = ∅.
Let Σ1, . . . , Σn be ISS with IΣ = {1, . . . , k}, Imax = {k + 1, . . . , n}.
Interconnection of ISS subsystems is in general not ISS!
⇒ we need conditions that guarantee ISS of the interconnection.

Centre for
Small gain condition
Consider Γ=(γij )n×n with γii ≡ 0, i = 1, . . . , n. Operator
Γ : Rn
+ → Rn
+ is deﬁned by:
Γ(s) :=










γ12(s2) + · · · + γ1n(sn)
...
γk1(s1) + · · · + γkn(sn)
max{γk+1,1(s1), . . . , γk+1,n(sn)}
...
max{γn1(s1), . . . , γn,n−1(sn−1)}










, s ∈ Rn
+.
Given an α ∈ K∞ consider the operator D : Rn
+ → Rn
+ deﬁned by
D(s) :=



s1 + α(s1)
...
sn + α(sn)


 , s ∈ Rn
+.

Centre for
Known results:
Hybrid systems, n = 2
Liberzon, Nesic 2006
IΣ={1, 2}: If ∃α∈K∞ s.t. Γ◦D(s)≥s, ∀s∈Rn
+{0} ⇒ Σ is ISS.
a ≥ b, a, b ∈ Rn ⇒ ∃ i ∈ {1, . . . , n} : ai < bi .
Continuous and discrete systems, n ≥ 2
Dashkovskiy, R¨uﬀer, Wirth 2007
1 Imax={1, . . . , n}: If Γ(s) ≥ s, ∀s ∈ Rn
+{0} ⇒ Σ is ISS.
2 IΣ={1, . . . , n}: If ∃α∈K∞ s.t. Γ◦D(s)≥s, ∀s∈Rn
+{0} ⇒ Σ is ISS.
For the case 0 < k < n (mixed case) we can use
maxi=1,...,n{xi } ≤ n
i=1 xi ≤ n maxi=1,...,n{xi }
The gains are conservative!

Centre for
Main theorem
D : Rn
+ → Rn
+, D(s):=










s1 + α(s1)
...
sk + α(sk)
sk+1
...
sn










, s∈Rn
+,
IΣ={1, . . . , k}, Imax={k + 1, . . . , n}.
Small gain theorem (in terms of trajectories)
Assume Di = D, i = 1, . . . , n and {f (x, u) : u ∈ U ∩ B} is convex
for each x ∈ χ, > 0. If all subsystems Σi are ISS and ∃α ∈ K∞
s.t. Γ satisﬁes Γ ◦ D(s) ≥ s, ∀s ∈ Rn
+{0}, then Σ is ISS.
If Di = D, then the assertion does not hold in general!

Centre for
Auxiliary results
µ(w, v) :=










w1 + v1
...
wk + vk
max{wk+1, vk+1}
...
max{wn, vn}










, w ∈ Rn
+, v ∈ Rn
+.
Lemma
Let Γ satisfy Γ ◦ D(s) ≥ s, ∀s ∈ Rn
+{0}. Then ∃φ ∈ K∞ s.t. for
all w, v ∈ Rn
+,
w ≤ µ(Γ(w), v)
implies |w| ≤ φ(|v|).

Centre for
Auxiliary results
Small gain theorem for pre-GS (global pre-stability)
Assume all Σi are pre-GS. If ∃α ∈ K∞ s.t. Γ satisfies Γ ◦ D(s) ≥ s,
∀s ∈ Rn
+{0}, then Σ is pre-GS.
Small gain theorem for AG (asymptotic gain property)
Assume Di =D, i = 1, . . . , n and all Σi have the AG property and
solutions of Σ exist, are bounded and some of them are complete.
If ∃α ∈ K∞ s.t. Γ satisfies Γ ◦ D(s) ≥ s, ∀s ∈ Rn
+{0}, then Σ
satisfies the AG property.
pre-GS ⇒ 0-input pre-stability
Proposition (Cai & Teel 2009)
Assume that {f (x, u) : u∈U ∩ B} is convex ∀ε>0 and for any
x∈χ. Then ISS ⇔ AG and 0-input pre-stability.

Centre for
Proof of the theorem
Small gain theorem for trajectories
Assume Di = D, i = 1, . . . , n and {f (x, u) : u ∈ U ∩ B} is convex
for each x ∈ χ, > 0. If all subsystems Σi are ISS and ∃α ∈ K∞
s.t. Γ satisﬁes Γ ◦ D(s) ≥ s, ∀s ∈ Rn
+{0}, then Σ is ISS.
Proof.
ISS of subsystems Σi ⇒ pre-GS and AG of subsystems Σi .
pre-GS and AG of Σi and Γ ◦ D(s) ≥ s ⇒ pre-GS and AG of
Σ.
pre-GS of Σ ⇒ 0-input pre-stability of Σ
0-input pre-stability and AG of Σ ⇒ ISS of Σ.

Centre for
ISS-Lyapunov function
ISS-Lyapunov function
Locally Lipschitz continuous (LipLoc) function V : χ→R+ is an
ISS-Lyapunov function for Σ, if
1) ∃ψ1, ψ2 ∈ K∞:
ψ1(|x|) ≤ V (x) ≤ ψ2(|x|) for any x ∈ χ;
2) ∃γ ∈ K, and cont., p.d.f. α, λ with λ(s) < s for all s > 0:
V (x)≥γ(|u|)⇒ ∀ζ∈∂V (x): ζ, f (x, u) ≤ − α(V (x)), (x, u)∈C,
V (x)≥γ(|u|)⇒ V (g(x, u)) ≤ λ(V (x)), (x, u)∈D.
γ is called ISS-Lyapunov gain.
Proposition, Cai & Teel 2009
∃ ISS-Lyapunov function ⇒ ISS
ISS ⇒ ∃ ISS-Lyapunov function!

Centre for
ISS-Lyapunov function for subsystems
1) ψi1(|xi |) ≤ Vi (xi ) ≤ ψi2(|xi |) for any xi ∈ χi ;
2) for (x, u) ∈ Ci : Vi (xi )≥
j
γij (Vj (xj ))+γi (|u|)⇒
∀ζi ∈∂Vi (xi ): ζi , fi (x, ui ) ≤−αi (Vi (xi )), i ∈ IΣ,
Vi (xi ) ≥ max{max
j
{γij (Vj (xj ))}, γi (|u|)} ⇒
∀ζi ∈ ∂Vi (xi ) : ζi , fi (x, ui ) ≤ −αi (Vi (xi )), i ∈ Imax;
3) for (x, u) ∈ Di : Vi (xi ) ≥
j
γij (Vj (xj )) + γi (|u|) ⇒
Vi (gi (x, ui )) ≤ λi (Vi (xi )), i ∈ IΣ,
Vi (xi ) ≥ max{max
j
{γij (Vj (xj ))}, γi (|u|)} ⇒
Vi (gi (x, ui )) ≤ λi (Vi (xi )), i ∈ Imax.

Centre for
Small gain theorem for ISS-Lyapunov functions
Case n = 2 with Imax = {1, 2}:
Teel, Nesic, 2008
∃ ISS-Lyapunov functions for Σi and Γ(s) ≥ s, ∀s ∈ Rn
+{0} ⇒ ∃
ISS-Lyapunov function for Σ.
Case n ≥ 2, mixed case:
Small gain theorem (in terms of Lyapunov functions)
Assume Di = D, i = 1, . . . , n and ∃ ISS-Lyapunov functions Vi for
Σi . If ∃α ∈ K∞ s.t. Γ ◦ D(s) ≥ s for all s = 0, s ≥ 0 is satisﬁed,
then ∃ ISS-Lyapunov function V for Σ:
V (x) = max
i=1,...,n
σ−1
i (Vi (xi )),
where σ satisﬁes:
Γ(σ(r), φ(r)) < σ(r), ∀ r > 0, φ ∈ K∞.

Centre for
Small gain theorem for ISS-Lyapunov functions
Proof.
Take V (x) := maxi {σ−1
i (Vi (xi ))},
γ(|u|) = max
j
{φ−1(γj (|u|))}, ψ1(|x|) := mini σ−1
i (ψi1(L1|x|)),
ψ2(|x|) := maxi σ−1
i (ψi2(L2|x|)), where ∃ of σ and φ is
guaranteed by Γ◦D(s) ≥ s (DRW 2010).
Assume V (x) ≥ γ(|u|) and consider separately:
1 (x, u) ∈ C
2 (x, u) ∈ D
In both cases we can show that such V satisﬁes conditions on
an ISS-Lyapunov for Σ using Γ ◦ D(s) ≥ s, properties of Vi
and σ.

Centre for
Example



˙x1 = −x1 + max
x2
2 +2x2
6x2+8
,
x2
3 +x3
8x3+4
+ u1
˙x2 = −x2 + max
2x2
1 +x1
8x1+5
,
x2
3 +5x3
3x3+11
+ u2
˙x3 = −x3 + max
x2
1 +x1
5x1+3
,
x2
2 +3x2
4x2+7
+ u3
, (x, u)∈C= xi ∈R+, x1≥
1
2
max{x2, x3}



x+
1 = 1
4
x1+1
8
max{x2, x3}
x+
2 = 1
4
x2
x+
3 = 1
4
x3
, (x, u)∈D= xi ∈R+, x1≤
1
2
max{x2, x3}

Centre for
Example



˙x1 = −x1 + max
x2
2 +2x2
6x2+8
,
x2
3 +x3
8x3+4
+ u1
˙x2 = −x2 + max
2x2
1 +x1
8x1+5
,
x2
3 +5x3
3x3+11
+ u2
˙x3 = −x3 + max
x2
1 +x1
5x1+3
,
x2
2 +3x2
4x2+7
+ u3
, (x, u)∈C= xi ∈R+, x1≥
1
2
max{x2, x3}



x+
1 = 1
4
x1+1
8
max{x2, x3}
x+
2 = 1
4
x2
x+
3 = 1
4
x3
, (x, u)∈D= xi ∈R+, x1≤
1
2
max{x2, x3}
Take Vi (xi ) = |xi | as a candidate for an ISS-Lyapunov function for Σi . Consider
Imax = {1, 2, 3} and take γ12(r) := max{
r(r + 2)
(3r + 4)(1 − 1)
,
1
2(3 − 4 1)
r},
γ13(r) := max{
r(r + 1)
(4r + 2)(1 − 1)
,
1
2(3 − 4 1)
r}, γ1(|u|) :=
2
(1 − 1)
|u|.
From V1(x1) ≥ max{γ12(|x2|), γ13(|x3|), γ1(|u|)} follows that
for (x, u)∈C ˙V1(x1) ≤ 1|x1|
for (x, u)∈D V1(g1(x, ui )) ≤ 1
2
|x1|.

Centre for
Example
Consider Γ =


0 γ12 γ13
γ21 0 γ23
γ31 γ32 0

. As all γij < id and Imax = {1, 2, 3}, then
Γ(r) < r and σ := (id, id, id)T . Thus Γ ◦ D(r) = Γ(r) ≥ r.
By the small gain theorem V (x) = max{V1(x1), V2(x2), V3(x3)} = max{|x1|, |x2|, |x3|}
is an ISS-Lyapunov function for Σ and thus Σ is ISS.
x(0) = (9; 3; 1)T , input u(t) = (0.5(1 + sin t); |2cos(t + 2)|; cos2 t)T :
Figure: Trajectory of the whole system.

Centre for
Particular classes of hybrid systems and ISS
Similar small gain results can be proved for subclasses of hybrid
systems:
Impulsive systems:
˙x(t) = f (x(t), u(t)), t = tk,
x+(t) = g(x(t), u(t)), t = tk.
Systems with stability with respect to the part of the state:
˙xs = f s(xs, u)
˙xt = f t(xs, xt, u)
, (x, u) ∈ C
xs+
= gs(xs, u)
xt+
= gt(xs, xt, u)
, (x, u) ∈ D
Comparison systems:
s(l + 1) = µ(Γ(s(l)), u(l))

Centre for
Reduction of gain matrix
Suﬃcient condition for ISS of interconnected system
ISS (ISS-Lyapunov function) of Σi + Small gain condition
Γ ◦ D(s) ≥ s ⇒ ISS of Σ
Ho to verify this condition if the size of the network is large?
Idea
Reduce the size of matrix Γ:
Γ=



0 γ12 . . . γ1,n−1 γ1n
...
...
...
...
...
γn1 γn2 . . . γn,n−1 0


 →Γ=



0 γ12 . . . γ1,k−1 γ1k
...
...
...
...
...
γk1 γk2 . . . γk,k−1 0


 , k < n.
How to obtain Γ?
SGC for Γ ⇒ SGC for Γ?

Centre for
Typical interconnections in the network (motifs)
almost disconnected subgraph
v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6
v7v7v7 v8v8v8 v9v9v9 v10v10v10
v11v11v11 v12v12v12 v13v13v13 v14v14v14 v15v15v15
v16v16v16 v17v17v17
v18v18v18 v19v19v19
sequential connection
parallel connection
v20v20v20 v21v21v21 v22v22v22 v23v23v23

Centre for
Aggregation of certain motifs
Parallel connections:
vvv
v1v1v1 vkvkvk
vvv
... ⇒
JJJ vvv
γv ,J
Theorem
Consider gain matrix Γ and Imax = I. Let Γ be such that
γv ,J := max{γv ,v1
◦ γv1,v , . . . , γv ,vl
◦ γvl ,v , γv ,v },
γJ,v := γv,v , γJ,j := γv,j , γj,J := γj,J , j ∈ V (VJ ∪ {v, v }).
Then Γ(s) ≥ s ⇒ Γ(s) ≥ s.

Centre for
Sequential connections:
vvv v1v1v1 ......... vkvkvk vvv ⇒
JJJ
vvv
γv ,J
Theorem
Consider gain matrix Γ and Imax = I. Let Γ be such that
γv ,J := max{γv ,vl
◦ · · · ◦ γv2,v1
◦ γv1,v , γv ,v },
γJ,v := γv,v , γJ,j := γv,j , γj,J := γj,J , j ∈ V (VJ ∪ {v, v }).

Centre for
Almost disconnected subgraphs:
v1v1v1
v2v2v2
v3v3v3
v∗v∗
v∗
⇒ v∗v∗
v∗
JJJ
Theorem
γJ,v∗ := max
(k1,...,kr )∈{v1,...,vl ,v∗}r ,k1=kr
{γk1,k2
◦ γk2,k3
◦ · · · ◦ γkr−1,kr
},
γv∗,J = id, γij := γij , i, j = J.

Centre for
Application of aggregation rules
almost disconnected subgraph
parallel connection
sequential connection
v1v1v1 v2v2v2 v3v3v3 v6v6v6
v7v7v7 v8v8v8 vvv
v11v11v11 v14v14v14
v16v16v16 v17v17v17
v18v18v18 v19v19v19
v20v20v20 v21v21v21 v22v22v22 v23v23v23

Centre for
Conclusion and Outlook
Conclusion:
1 Hybrid systems combine two types of dynamics in the model
of a logistics networks
2 ISS of interconnected hybrid systems can be established using
small gain condition Γ ◦ D(s) ≥ s, where matrix Γ describes
interconnection structure and D is diagonal matrix
3 Size of Γ can be reduced by aggregating certain motifs that
preserves the structure of the network
Outlook:
1 Hybrid model of logistics networks with diﬀerent policies
2 Properties of interconnection of hybrid systems
3 Controllers based small gain conditions
4 Extension of reduction rules and their analysis

Centre for
Thank you for your attention!

Hybrid dynamics in large-scale logistics networks

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à Hybrid dynamics in large-scale logistics networks

Similaire à Hybrid dynamics in large-scale logistics networks (20)

Dernier

Dernier (20)

Hybrid dynamics in large-scale logistics networks