Interconnections of hybrid systems

Interconnections of hybrid systems
Michael Kosmykov
3. February 2012, Fachhochschule Erfurt

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Industrial Mathematics
Outline
1 Motivation
2 Hybrid system
3 Extension of hybrid time domain
4 Conclusion and outlook
2 / 40Motivation Hybrid system Hybrid domain Summary

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Logistics network as hybrid system from PhD thesis:
3 production sites (D, F, E) for
Liquidring-Vaccum (LRVP), Industrial (IND)
and Side-channel (SC) pumps
5 distribution centers (D, NL, B, F, E)
33 ﬁrst and second-tier suppliers for the
production of pumps
90 suppliers for components that are needed
for the assembly of pump sets
More than 1000 customers

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Logistics network as hybrid system from PhD thesis:
Instability leads to:
High inventory costs
Large number of unsatisﬁed orders
Loss of customers

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Analysis steps:
1 Mathematical modelling
2 Model reduction, if the model size is large
3 Stability analysis

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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

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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

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

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

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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

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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

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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.

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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.

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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.

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Stability notions
Global stability (GS)
Hybrid system is called globally stable (GS), if there exists σ ∈ K∞
such that any solution x satisﬁes
|x(t, k)| ≤ σ(|x0|), ∀(t, k) ∈ dom x.
Global attractivity (GA)
Hybrid system is called globally attractive (GA), if for each > 0
and r > 0 there exists T > 0 such that, for any solution x,
|x(0, 0)| ≤ r, (t, k) ∈ dom, x, and t + k ≥ T imply |x(t, k)| ≤ .
Global asymptotical stability (GAS)
Hybrid system is called globally asymptotically stable (GAS), if it is
both GS and GA.

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Example: Bouncing ball
˙x=
x2
−γ
=:f (x), x∈C,
x+=
x1
−λx2
=:g(x), x∈D.
x1 - height
x2 - velocity
C := {(x, u) ∈ R2 × U : x1 ≥ 0},
D := {(x, u) ∈ R2 × U : x1 =
0, x2 ≤ 0},
γ - gravitation force,
λ ∈ (0, 1) - restitution coeﬃcient

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Example: Bouncing ball
Figure: Trajectory of the
bouncing ball.
Figure: Time domain of the
bouncing ball.

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Zeno solutions
Definition (Ames et. al. 2006)
The solution (x, u) possesses:
chattering Zeno behaviour, if there exists a finite K ≥ 0 such
that (t, k), (t, k + 1) ∈ dom x for all k ≥ K;
genuinely Zeno behaviour, if there exists a finite T ≥ 0 such
that for all (s, k), (t, k + 1) ∈ dom x, s < t < T.
Bouncing ball possesses the genuinely Zeno behaviour with
T = tmax = 2h
γ + 2λ
1−λ
2h
γ .
Furthermore, it is GAS (Sanfelice and Teel 2008).

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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 .

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Example of interconnection of hybrid systems
two bouncing balls with states x1 ∈ R2 and x2 ∈ R2
respectively.
they are interconnected by an elastic elastic coeﬃcient k ≥ 0
horizontal distance between the balls is neglected
the balls do not hit each other.
The interaction force due to the elastic spring is given by
±k(x1
1 − x2
1 ).

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Their dynamics is given by:
First ball
˙x1
=
x1
2
−γ − k(x1
1 − x2
1 )
, (x1
, x2
) ∈ C1
={(x1
, x2
) ∈ R4
: x1
1 ≥ 0},
x1+
=
x1
1
−λx1
2
, (x1
, x2
) ∈ D1
={(x1
, x2
) ∈ R4
: x1
1 = 0, x1
2 ≤ 0},
Second ball
˙x2
=
x2
2
−γ + k(x1
1 − x2
1 )
, (x1
, x2
) ∈ C2
={(x1
, x2
) ∈ R4
: x2
1 ≥ 0},
x2+
=
x2
1
−λx2
2
, (x1
, x2
) ∈ D2
={(x1
, x2
) ∈ R4
: x2
1 = 0, x2
2 ≤ 0},

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Dynamics of interconnection
˙z = f (z), z ∈ C := C1 ∩ C2,
z+ = g(z), z ∈ D := D1 ∪ D2 ∈ R4,
where z := (x1T
, x2T
)T , f (z) := (f 1T , f 2T )T ,
g(z) := (g1T , g2T )T ,
gi
(z) = gi
(x1
, x2
) :=
gi (x1, x2), if (x1, x2) ∈ Di ,
xi , otherwise .

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Drawback
Taking x1
1 (0) = x1
2 (0) = 0 and x2
1 (0) = h > 0, x2
2 (0) = v ∈ R the
hybrid arc
x1
1 (t, j) = x1
2 (t, j) = 0, x2
1 (t, j) = h, x2
2 (t, j) = v
is a solution with hybrid time domain {(0, j)}∞
j=0.
Thus tmax = 0 and the system jumps inﬁnitely many times from a
non-zero state to the same state ⇒ the solution possesses
chattering Zeno solution ⇒ No physical meaning! and furthermore
is no more GAS.

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Our approach
We take into account which of the system can jump by:
IC (x, u):={i : (x, u) ∈ Ci }, ID(x, u):={i : (x, u) ∈ Di }.
Then dynamics is given by
˙xi = fi (x, u), i ∈ IC (x, u),
x+
i = gi (x, u), i ∈ ID(x, u).
Hybrid time domain considers the jumps of the subsystems
separately:
domk1,...,kn := ∪[tk, tk+1] × {k1, . . . , kn} ⊂ R+ × Nn
+,
where k = k1 + · · · + kn and ki ∈ N+ calculates the jumps of
the ith subsystem.
ti
max

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Solution
(i) (x(0, 0), u(0, 0)) ∈ Ci ∪ Di , ∀i
(ii) for i ∈ IC (x1
1 (t, k), . . . , xn(t, k), u(t, k)),
˙xi (t, k)=fi (x1
1 (min{t, t1
max}, k), . . ., xn(min{t, tn
max}, k), u(t, k))
(iii) for all (t, k) ∈ domk x with (t, k + p) ∈ domk x p ≥ 1
for i ∈ ID(x1
1 (t, k), . . . , xn(t, k), u(t, k),
x+
i (min{t, tk
max}, k+p) =
gi (x1
1 (min{t, t1
max}, k), . . . , xn(min{t, tn
max}, k), u(t, k))
k = (k1, . . . , kn) ∈ Nn
+

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Application to an example
The sets IC (x) and ID(x) in the example with an interconnection
of bouncing balls are given by
IC (x) = {1, 2}, ID(x) = ∅, if x1
1 > 0, x2
1 > 0,
IC (x) = {1, 2}, ID(x) = {1}, if x1
1 = 0, x2
1 > 0,
IC (x) = {1, 2}, ID(x) = {2}, if x1
1 > 0, x2
1 = 0,
IC (x) = {1, 2}, ID(x) = {1, 2}, if x1
1 = 0, x2
1 = 0.
Then the arc
x1
1 (t, j) = x1
2 (t, j) = 0, x2
1 (t, j) = h, x2
2 (t, j) = v
is not a solution, because it corresponds to IC = {1, 2}, ID = {1},
i.e., the second subsystem is not allowed to jump.

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GAS
Adaptation of Matrosov’s theorem
Let a hybrid system be GS. Then, it is GAS if ∃m∈N, and for each
0 < δ < ∆,
a number µ > 0,
continuous wc,j : (∪i
¯Ci ) ∩ ΩIC (x)(δ, ∆) → R,
wd,j : (∪i
¯Di ) ∩ ΩID (x)(δ, ∆) → R, j ∈ {1, . . . , m},
Vj : RN 0 → R, j ∈ {1, . . . , m} are C1 on an open set
containing (∪i
¯Ci ) ∩ ΩIC (x)(δ, ∆),

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GAS
Adaptation of Matrosov’s theorem (continued)
such that, for each j ∈ {1, . . . , m}
i) |Vj (x)| ≤ µ ∀x ∈ (∪i
¯Ci )∪(∪i Di ) ∪ (∪i gi (Di ))∩Ω(δ, ∆)
ii) Vj (x)IC
, (f T
1 , . . ., f T
n )T
IC
≤wc,j (x), ∀x∈(∪i Ci )∩ΩIC (x)(δ, ∆)
iii)Vj ((gT
1 , . . . , gT
n )T
(˜x))−Vj (˜x)≤wd,j (x), ∀x∈(∪i Di )∩ΩID (x)(δ, ∆)
and with wc,0, wd,0 : RN → {0} and wc,m+1, wd,m+1 : RN → {1}
such that for each l ∈ {0, . . . , m},
1) if x ∈ (∪i
¯Ci ) ∩ Ω(δ, ∆) and wc,j (x) = 0 for all j ∈ {0, . . . , l}
then wc,l+1(x) ≤ 0,
2) if x ∈ (∪i
¯Di ) ∩ Ω(δ, ∆)ID (x) and wd,j (x) = 0 for all
j ∈ {0, . . . , l} then wd,l+1(x) ≤ 0.

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Application of Matrosov’s theorem to an interconnection
of bouncing balls
System is GS (Sanfelice and Teel 2008).
Deﬁne z := (x1, x2)T and
V1(z) := V1(x1, x2) := 1
2(x1
2
2
+ x2
2
2
) + γx1
1 + γx2
1 + k
2 (x2
1 − x1
1 )2
V2(z) := V2(x1, x2) = γx1
2 + γx2
2 .
Consider the following four cases:
both components of the state ﬂow continuously:
˙V1(z) = x1
2 (−kx1
1 + kx2
1 − γ) + x2
2 (kx1
1 − kx2
1 − γ) + γx1
2 +
γx2
2 + k(x2
1 − x1
1 )(x2
2 − x1
2 ) = 0,
˙V2(z) = −2γ2

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Application of Matrosov’s theorem to an interconnection
of bouncing balls
both components of the state jump:
V1(z+) − V1(z) = −1
2(1 − λ2)x1
2
2
− 1
2(1 − λ2)x2
2
2
,
V2(z+) − V2(z) = −(1 + λ)γ(x1
2 + x2
2 )
the first component of the state jumps and the second flows
continuously:
V1(z+) − V1(z) = −1
2(1 − λ2)x1
2
2
,
V2(z+) − V2(z) = −(1 + λ)γx1
2
the first component of the state flows continuously and the
second jumps:
V1(z+) − V1(z) = −1
2(1 − λ2)x2
2
2
,
V2(z+) − V2(z) = −(1 + λ)γx2
2
Then the interconnected system is GAS.

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Trajectories
Figure: Trajectories two bouncing
balls connected by a spring.
Figure: Time domain of two
interconnected balls.

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Existence of solutions
Basic regularity conditions:
1 χi is open, U is closed, and Ci , Di ⊂ χ1 × · · · × χn × U are
relatively closed in χ1 × · · · × χn × U;
2 fi , gi are continuous.
Theorem (Existence of solutions)
Assume the basic regularity conditions hold.
If one of the following conditions holds:
(i) (x0, u0) ∈ Di for all i ∈ {1, . . . , n};
(ii) (x0, u0) ∈ Ci and for some neighborhood P of (x0, u0), for all
(x , u0) ∈ P ∩ Ci , TCi
(x , u0) ∩ fi (x , u0) = ∅, for all i ∈ {1, . . . , n} ;
(iii) 1 ≤ |IC (x0, u0)| < n, 1 ≤ |ID(x0, u0)| < n and for some
neighborhood P of (x0, u0), for all i ∈ ICi
(x0, u0),
(x , u0) ∈ P ∩ Ci , TCi
(x , u0) ∩ fi (x , u0) = ∅,

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Theorem (Existence of solutions, continued)
then there exists a solution pair (x, u) for hybrid system with
(t, ¯k) ∈ domx for some t > 0 or ¯k ≡ (0, . . . , 0)T ∈ Nn
+.
Furthermore, if for all i ∈ {1, . . . , n}, gi (Di ) ∈ Ci ∪ Di , then there
exists a solution with t > 0, ¯k ∈ Nn
+ such that
(x(t, ¯k), u(t, ¯k)) ∈ Ci ∪ Di .
Proof
We consider all the cases separately and construct solutions for
them explicitly.

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Another problem (semi-genuinely Zeno solution)
Example:
two balls with masses m1 and m2 such that m2 > m1.
the ﬁrst ball was launched at an angle 0 < θ < π
2 to the
horizontal line and then bounces with initial velocity v1
towards the second ball
the second balls rolls with constant velocity v2 towards the
ﬁrst ball.

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Dynamics in C
For
(x1, y1, v1, x2, y2, v2, q)∈C={(x1, y1, v1, x2, y2, v2, q)∈R7 : y1 ≥ 0}:



˙x1 =
v1 cos θ, if q = 0,
−v1, if q = 1
˙y1 =
v1 sin θ − gt, if q = 0,
0, if q = 1
˙v1 = 0



˙x2 =
−v2, if q = 0,
v2, if q = 1
˙y2 = 0
˙v2 = 0
˙q = 0
q - logical variable that determines, whether they have already
collided.

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Dynamics in D = D1 ∪ D2
At the bounce in
(x1, y1, v1, x2, y2, v2, q)∈D1={(x1, y1, v1, x2, y2, v2, q)∈R7 : y1 =
0, v1 ≥ 0, q = 0}: 


x+
1 = x1
y+
1 = y1
v+
1 = µv1



x+
2 = x1
y+
2 = y1
v+
2 = v2
q+
= q

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Dynamics in D = D1 ∪ D2
At the collision in (x1, y1, v1, x2, y2, v2, q) ∈ D2 =
{(x1, y1, v1, x2, y2, v2, q) ∈ R7 : x1 = x2, v1 ≥ 0, q = 0}:



x+
1 = x1
y+
1 = y1
v+
1 = v1 = 2m2
m1+m2
v2



x+
2 = x1
y+
2 = y1
v+
2 = v2 = m2−m1
m1+m2
v2
q+
= 1

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Trajectories
Figure: Trajectories before the stop of the ﬁrst ball.
The ﬁrst ball ”stops” earlier due to the loss of energy, i.e. at
t1
max < t2
max

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Trajectories
Figure: Trajectories before the collision.
After some time the second ball reaches the ﬁrst, i.e. at
(x1, y1, v1, x2, y2, v2, q) ∈ D2 and they collide.

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Trajectories
Figure: Trajectories after the collision.
As m2 > m1, the ﬁrst ball begins to move again in opposite
direction to the second ball.
Thus hybrid time domain of the ﬁrst ball is extended beyond t1
max!

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Possible solution
Instead of considering min{t, ti
max}, we consider intervals
[ti,ki
min, ti,ki
max], ki = 0, 1, 2, . . .
where system i is ”active”, and intervals
[ti,ki
max, ti,ki +1
min ],
where system i is ”passive” (in physical sense).
θi (t) :=
t, ti,ki
min < t < ti,ki
max
ti,ki
max, ti,ki
max < t < ti,ki +1
min
identiﬁes, whether at time t the system i is ”active” or ”passive”.

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Deﬁnition of solution
Hybrid arc x and hybrid input u are a solution pair, if
(i) (x(0, 0), u(0, 0)) ∈ Ci ∪ Di , ∀i
(ii) for i ∈ IC (x1
1 (t, k), . . . , xn(t, k), u(t, k)),
˙xi (t, k) = fi (x1
1 (θ1(t), k), . . . , xn(θn(t), k), u(t, k));
(iii) for all (t, k) ∈ domk x with (t, k + p) ∈ domk x where p ≥ 1
for i ∈ ID(x1
1 (t, k), . . . , xn(t, k), u(t, k),
x+
i (θi (t), k + p) = gi (x1
1 (θ1(t), k), . . . , xn(θn(t), k), u(t, k)).
k := (k1, . . . , kn) ∈ Nn
+

Centre for
Application to an example
1 The second ball is always ”active” ⇒ k2 = 0, t2,k2
min = 0,
t2,k2
max = ∞.
2 The first ball first bounces from t1,0
min = 0 till t1,0
max.
3 Then it lies on the floor.
4 At t1,1
min the two balls collide
5 And the first ball flows till t1,1
max = ∞ ⇒ Thus k1 = 1.

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Theorem (Existence of solutions, continued)
then there exists a solution pair (x, u) for hybrid system with
(t, ¯k) ∈ domx for some t > 0 or ¯k ≡ (0, . . . , 0)T ∈ Nn
+.
Furthermore, if for all i ∈ {1, . . . , n}, gi (Di ) ∈ Ci ∪ Di , then there
exists a solution with t > 0, ¯k ∈ Nn
+ such that
(x(t, ¯k), u(t, ¯k)) ∈ Ci ∪ Di .
Proof
For each interval [ti,ki
min, ti,ki +1
min ] we consider all the cases separately
and construct solutions for them explicitly. Then we concatenate
the solutions on the intervals.

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Conclusion
Hybrid systems allow to describe complex dynamics like
behaviour of logistics networks
Hybrid systems may possess solutions with no physical
meaning (Zeno behaviour) ⇒ hybrid systems have to be used
very carefully
If subsystems in interconnections of hybrid systems possess
Zeno behaviour, one can extend the notion of time domain to
exclude such solutions

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Thank you for your attention!

Interconnections of hybrid systems

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