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Cyber-Physical Systems – Modeling and Simulation ofHybrid Systems
Matthias Althoff
TU Munchen
05. June 2015
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 1 / 28
Overview
Overview
Hybrid Systems
Modeling as hybrid automata
Other modeling formalisms
Properties of hybrid systems
Numerical simulation of hybrid automata
Stability analysis of hybrid systems (next lecture)
Reachability analysis of hybrid systems (next lecture)
Hybrid systems are the most general class of systems considered in thiscourse. Timed automata can already be considered as hybrid systems witha simple continuous dynamics (ci = 1).
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 2 / 28
Modeling as Hybrid Automata
Why combine discrete and continuous dynamics?
In many cyber-physical systems, the continuous dynamics and the discretedynamics cannot be designed separately (see first lecture):
automated drivingsource: Carnegie Mellon University
automated farmingsource: Kesmac
human-robot collaborationsource: Rethink Robotics
surgical robotssource: daVinci
smart gridssource: Siemens
air traffic controlsource: NASA
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 3 / 28
Modeling as Hybrid Automata
Hybrid Automata
Hybrid automata describe the dynamics of systems that can be described by afinite set of discrete states zi and continuous state variables xi ∈ R. Starting froman initial state z(t0), initial continuous values xi (t0), a continuous input trajectoryuc(t), and a timed input sequence
u =((u(t0), t0), (u(t1), t1), (u(t2), t2), . . .
)
a finite state automaton creates a continuous output trajectory yc(t) and a timedoutput sequence
y =((y(t0), t0), (y(t1), t1), (y(t2), t2), . . .
),
where it is not required that the times ti and ti are synchronized.
u Hybridautomaton
y
uc(t) yc (t)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 4 / 28
Modeling as Hybrid Automata
Syntax of Hybrid Automata
Definition
A hybrid automaton HA is a tuple (ordered set):
HA = (Z,X ,U ,Y,Uc ,Yc ,T, inv, g, h, f, z0, x0),
where z0 is the discrete initial state, x0 is the continuous initial state
Z = {z1, . . . , zn} set of discrete statesX ⊆ Rn continuous state spaceU = {u1, . . . , up} set of input symbols (input alphabet)Y = {y1, . . . , yq} set of output symbols (output alphabet)Uc ⊆ Rm continuous input spaceYc ⊆ Ro continuous output spaceT ⊆ Z × U × Z × Y set of transitionsinv : Z → P(X ) invariant functiong : T → P(X ) guard functionh : T×X → X jump functionf : Z × X × Uc → Rn flow function
There exist many variations of definitions of hybrid automata in the literature. AMatthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 5 / 28
Modeling as Hybrid Automata
Semantics of Hybrid Automata
Our definition of a hybrid automaton has the following semantics:
The hybrid automaton starts at the discrete state z0 and thecontinuous state x0.
The continuous state evolves according to the flow function that isassigned to each location zi : x = f (zi , x , uc ).
As soon as the continuous state x is within a guard set g(z , u, z ′, y) ofa transition (z , u) → (z ′, y), the corresponding transition is activated.
As soon as the input event u of an activated transition occurs, thetransition is taken and the output event y is generated.
A transition is enforced if the continuous state would leave theinvariant inv(zi).
After a transition is taken, the jump function resets the continuousstate:
x ′ = h((z , u, z ′, y), x
)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 6 / 28
Modeling as Hybrid Automata
Animation of Hybrid Automata
initialcontinuous
set
trajectory
guards
invariant
x1
x2z1 z2
Continuous evolution
Start at z0 and x0
x(t) is the solution of x(t) = f (z(t), x(t), uc (t))
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 7 / 28
Modeling as Hybrid Automata
Animation of Hybrid Automata
initialcontinuous
set
trajectory
guards
jumpinvariant
x1
x2z1 z2
Activation of discrete transition
Transition (z , u, z ′, y) is activated when x(t) ∈ g(z , u, z ′, y) (z :before transition, z ′: after transition)
Transition is taken as soon as event u occurs
Transition is enforced when x(t) leaves inv(z)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 7 / 28
Modeling as Hybrid Automata
Animation of Hybrid Automata
initialcontinuous
set
trajectory
guards
jump
invariant
x1
x2z1 z2
Discrete transition and jump of continuous state
Location changes from z to z ′
Output event y is generated
Continuous state may jump: x ′ = h((z , u, z ′, y), x
)
(x ′: continuous state after jump)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 7 / 28
Modeling as Hybrid Automata
Animation of Hybrid Automata
initialcontinuous
set
trajectory
guards
jump
invariant
x1
x2z1 z2
... and so on ...
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 7 / 28
Modeling as Hybrid Automata
Hybrid Automaton of a Bouncing Ball
Given is a ball with dynamics s = −g , where s is the vertical position andg is the gravity constant. After impact with the ground at s = 0, thevelocity changes to v ′ = −αv (v = s) with α ∈ [0, 1].
s0
v0
g
Z = {z1}X = R+ × R (ball above ground)U = Y = {ǫ}Uc = Yc = {}T = {(z1, ǫ, z1, ǫ)}inv(z1) = {[x1, x2]T |x1 ∈ R+
0 , x2 ∈ R}g((z1, ǫ, z1, ǫ)
)= {[x1, x2]T |x1 = 0, x2 ∈ R−
0 }h((z1, ǫ, z1, ǫ), x
)=
[x1
−αx2
]
f(z1, x) =
[x2−g
]
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 8 / 28
Modeling as Hybrid Automata
Graphical Representation of the Bouncing Ball
A typical representation of hybrid automata is as follows:
Discrete states are represented by circles (or similar shapes).
Transitions are illustrated by arrows to which input and outputevents, guards, and jump functions are attached.
The continuous dynamics is written within the discrete states above adashed line.
The invariant is placed underneath the dashed line.
x1 = x2x2 = −g
x1 ≥ 0
x1 = 0 ∧ x2 ≤ 0
x2 := −αx2invariant
differentialequations
guard
jump function
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 9 / 28
Modeling as Hybrid Automata
Trajectory of the Bouncing Ball
The trajectories of the bouncing ball are plotted for s0 = 30 [m] (quitehigh, but then we can reasonably plot the result together with velocity),v0 = 0 [m/s], and α = 0.8 [-].
0 2 4 6 8 10 12 14 16 18 20−30
−20
−10
0
10
20
30
t
s(t),v(t)
s(t)
v(t)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 10 / 28
Other Modeling Formalisms
Hybrid Statecharts
When a hybrid automata has many locations, it is useful to group themusing statecharts.
Guards: Are modeled as conditions of discrete transitions.
Flow function: Is specified after the keyword throughout(MATLAB/Stateflow: during/du) within a state.
Jump function: Is specified after the keyword exit within a state orattached to a discrete transition.
Invariant: Most work does not specify invariants for statecharts.Instead, an urgent semantics is assumed, i.e. a transition is taken assoon as a state is in a guard. Why does one not require invariants inthis case?
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 11 / 28
Other Modeling Formalisms
Hybrid Statecharts: Electric Motor (I)
We model an electric motor with several operation modes. The torqueT = km i of our motor is proportional to the applied current i . We alsomodel friction as Tf = −kf ω and denote the disturbance torque by Td sothat the overall dynamics of the angular velocity ω for the rotationalinertia J is
Jω = T = km i − kf ω + Td .
To control the speed, we use a simple P-controller i = KP(ωd − ω) so thatwe obtain
ω =kmJKP(ωd − ω)− kf ω + Td .
The motor has the following modes:
The motor is switched off.
The motor is fully accelerating with current imax until the speed ωd isreached or when the speed drops below 0.8ωd .
The motor is controlled to keep the speed ωd .
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 12 / 28
Other Modeling Formalisms
Hybrid Statecharts: Electric Motor (II)
power off
power on
acceleration const speed
during:ω = f1(ω,Td )
during:ω = f2(ω,Td )
during:ω = f3(ω,Td )
on
off
ω ≥ ωd
ω < 0.8ωd
f1(ω,Td ) = −kf ω + Td ,
f2(ω,Td ) =kmJimax − kf ω + Td ,
f3(ω,Td ) =kmJKP(ωd − ω)− kf ω + Td .
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 13 / 28
Other Modeling Formalisms
Hybrid Statecharts: Bouncing Ball in MATLAB/Stateflow
In a similar way, one can model hybrid statecharts inMATLAB/Stateflow.
MATLAB also assume urgent semantics, i.e. a transition is taken assoon as a state is in a guard.
The bouncing ball example in MATLAB/Stateflow:
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 14 / 28
Other Modeling Formalisms
Interaction of Discrete and Continuous Components (I)
Many engineering tools realize hybrid systems by combining discreteand continuous components.No explicit modeling formalism: Discrete states, guards, etc. areimplicitly described by the interaction of continuous and discretecomponents.
Example: Bouncing ball modeled in MATLAB/Simulink
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 15 / 28
Other Modeling Formalisms
Interaction of Discrete and Continuous Components (II)
Advantages
Intuitive modeling.
Components can be easily exchanged, which might cause changing thediscrete transition structure of many locations in a hybrid automata.
Disadvantages
The modeling formalism is not suited for formal analysis.
The modeling formalism is not formally defined. What happens whenseveral discrete components switch at the same time?
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 16 / 28
Properties of Hybrid Automata
Deadlock and Livelock
Due to bad design, the undesired event of deadlock and livelock can occur.
Deadlock
A deadlock occurs when thecontinuous state leaves the invariantand is not in any guard set. x1
x2 guard
invariant
trajectory
deadlock
Livelock
A livelock occurs when the system switches infinitely often betweendiscrete states and no time passes in between discrete transitions.
Example: A continuous statejumps in between guard setsthat cause transitions inbetween each other.
x1
x2
inv(z1) inv(z2)
g((z1, ǫ, z2, ǫ)
)g((z2, ǫ, z1, ǫ)
)
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 17 / 28
Properties of Hybrid Automata
Nondeterminism
Since finite state automata are a special case of hybrid automata, it is obviousthat hybrid automata can be nondeterministic. The sources of nondeterminismare manifold:
Guard regions can overlap so that several goal locations are possible.
The jump function can be nondeterministic.
When the guard set is full-dimensional (see figure below), the switching timeis nondeterministic.
The differential equations have uncertain continuous inputs.
One can define hybrid automata with nondeterministic initial states.
x1
x2
g((z1, ǫ, z2, ǫ)
)g((z1, ǫ, z2, ǫ)
)
inv(z1)inv(z1)
deterministic non-deterministic
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 18 / 28
Properties of Hybrid Automata
Zeno Behavior (I)
Zeno behavior occurs when the duration δi between the i th and the(i + 1)th transition decreases and
∑∞i=0 δi is finite. With other words, an
infinite number of transitions occurs in finite time.
Example: Bouncing Ball
Let us introduce the velocity vi and the time ti at the i th transition:
vi+1 = αvi ti+1 = ti +2α
gvi
vi = αiv0 ti = t0 +2v0g
(α− αi+1
1− α
)
vzeno = limi→∞
vi = 0 tzeno = limi→∞
ti = t0 +2v0g
(α
1− α
)
(we use∑n−1
k=0 axk = a 1−xn
1−x (x 6= 1) )
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 19 / 28
Properties of Hybrid Automata
Zeno Behavior (II)
For the values v0 = 1 [m/s] (here: velocity at initial contact), α = 0.8 wehave
tzeno =2
g· 4 ≈ 0.82 [s]
The ball dynamics cannot proceed beyond 0.82 [s]. In reality, the elasticityof the ball causes the ball not take off after a certain time.
Zeno of Elea
Greek philosopher who is famous for his paradoxes, such as the one ofAchilles and the tortoise: A tortoise wants to race against Achilles and hegives it a head start. After both start running, the tortoise has alreadymoved to s1 when Achilles arrives at its initial position s0. Then Achillesruns to s1 when the tortoise is at s2. Zeno claims that by thisargumentation, Achilles can never overtake the tortoise.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 20 / 28
Properties of Hybrid Automata
Finite Escape Time
One speaks of finite escape time, when ‖x‖ → ∞ in finite time. Finiteescape time is also possible for purely continuous systems when they arenonlinear:
x = 1 + x2(t), x0 = 0
The solution of the differential equation is
x(t) = tan(t),
which has an ”explosion time” at t = π2 .
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 21 / 28
Numerical Simulation of Hybrid Automata
Numerical Simulation of Hybrid Automata
As for nonlinear continuous systems, for most hybrid systems modelingreal world problems, there exists no analytical solution.
Steps in hybrid system simulation
1 Simulation of the continuous dynamics within the current location(see lecture ”Modeling and Simulation of Continuous Systems”) aslong as the state is in the invariant;
2 Detection whether the current state is within a guard set and whetherit is activated by the required input event;
3 Update of the discrete state once the transition is taken and generatethe output event;
4 Update of the continuous state according to the jump function;
5 Continue with step 1.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 22 / 28
Numerical Simulation of Hybrid Automata
Guard detection
Step 1 (continuous evolution) has been previously discussed and ise.g. performed via Runge-Kutta methods.
Step 3 and 4 (discrete and continuous update) are trivial.
We need to focus on step 2 (guard detection).
For simplicity we only consider deterministic guards, i.e. guards that canonly be hit at one point in time. Those guards are usually only activatedby the state: g
((z , ǫ, z ′, ǫ)
). Why?
Reminder:
x1
x2
g((z1, ǫ, z2, ǫ)
)g((z1, ǫ, z2, ǫ)
)
inv(z1)inv(z1)
deterministic non-deterministic
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 23 / 28
Numerical Simulation of Hybrid Automata
Modeling of Guards
We model the guard by a level set function l(x), which allows arbitraryshapes:
g((z , ǫ, z ′, ǫ)
)= {x |l(x) = 0}.
Examples:
hyperplane: l(x) = nT x − d , where n ∈ Rn is the normal vector andd ∈ R is the distance from the origin to the hyperplane.
x1
x2d
nT
hyperplane
hypersphere: l(x) = ‖x − c‖2 − r , where c ∈ Rn is the center andr ∈ R is the radius of the hypersphere.
x1
x2 c
r hypersphere
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 24 / 28
Numerical Simulation of Hybrid Automata
Guard Detection without Hitting Time Detection
A guard has been crossed when the level set function l(x) changes its sign:
x1
x2
guard: l(x) = 0trajectory
in z1
z1
z2
trajectory in z2without hitting time detection
x(tn)x(tn+1)
l(x) < 0
l(x) > 0
trajectory with exacthitting time detection
A simple method is to perform a discrete transition as soon as a signchange of l(x) is detected, without determining the exact switching time.This is computationally cheap, but creates larger errors.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 25 / 28
Numerical Simulation of Hybrid Automata
Guard Detection with Hitting Time Detection
More accurate results are obtained when the solver iteratively searches forthe exact hitting time until the value of l(x) is in a ǫ-region: ‖l(x)‖2 ≤ ǫ.
x1
x2
guard: l(x) = 0trajectory
in z1
z1
z2
x(tn)x(tn+1)
l(x) < 0
l(x) > 0
trajectory with exacthitting time detection
1
2
3
4
An iterative method for hitting time detection is presented in the exercise.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 26 / 28
Numerical Simulation of Hybrid Automata
Further Reading
A. van der Schaft and H. Schumacher: An Introduction to HybridDynamical Systems, Springer, 2000.
R. Alur, C. Coucoubetis, N. Halbwachs, T.A. Henzinger, P.H. Ho, X.Nicolin, A. Olivero, J. Sifakis, S. Yovine: The Algorithmic Analysis ofHybrid Systems, Theoretical Computer Science, 1995, 138, pages3-34.
Y. Kesten and A. Pnueli: Timed and Hybrid Statecharts and theirtextual representation, Formal Techniques in Real-Time andFault-Tolerant Systems, LNCS 571, 1991, pages 591-620.
M. Otter, H. Elmqvist, and Sven Erik Mattsson: Hybrid Modeling inModelica based on the Synchronous Data Flow Principle, Proc. ofComputer Aided Control System Design, 1999.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 27 / 28
Numerical Simulation of Hybrid Automata
Conclusions
In many cyber-physical systems, the continuous dynamics and thediscrete dynamics cannot be designed separately.
Hybrid automata are an extension of finite state automata bycontinuous dynamics.
There exists a large number of alternative modeling formalisms:Hybrid statecharts, hybrid Petri nets, hybrid bond graphs, etc.
Hybrid systems can exhibit a variety of phenomena:DeadlockLivelockNondeterminismZeno behaviorFinite escape time
The main difficulty in extending numerical solvers for continuoussystems is guard detection.
Matthias Althoff Modeling and Simulation of Hybrid Systems 05. June 2015 28 / 28
Cyber-Physical Systems – Analysis of Hybrid Systems
Matthias Althoff
TU Munchen
12. June 2015
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 1 / 39
Overview
Overview
Hybrid Systems
Stability analysis of hybrid systems:
Common Lyapunov functionMultiple Lyapunov function
Reachability analysis of hybrid systems
Applications
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 2 / 39
Stability Analysis of Hybrid Systems
Motivating Example (I)
Warning
Even if a hybrid system is Lyapunov stable in all locations, the hybridsystem is not necessarily stable!
Example: Hybrid automaton with two locations Z = {z1, z2}, twocontinuous state variables X = {x1, x2}, and no inputs and outputs:
T = {(z1, ǫ, z2, ǫ), (z2, ǫ, z1, ǫ)}inv(z1) = {[x1, x2]T |x1x2 ≤ 0, x1 ∈ R, x2 ∈ R}inv(z2) = {[x1, x2]T |x1x2 ≥ 0, x1 ∈ R, x2 ∈ R}g((z1, ǫ, z2, ǫ)
)= g
((z2, ǫ, z1, ǫ)
)= {[x1, x2]T |x1x2 = 0, x1 ∈ R, x2 ∈ R}
h((z1, ǫ, z2, ǫ), x
)= h
((z2, ǫ, z1, ǫ), x
)= x
f(z1, x) =
[−1 4−1 −1
]x
f(z2, x) =
[−1 1−4 −1
]x
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 3 / 39
Stability Analysis of Hybrid Systems
Motivating Example (II)
The phase portraits of each subsystem are as follows:location z1:
−1 0 1
−3
−2
−1
0
1
2
3
x1
x 2
location z2:
−3 −2 −1 0 1 2 3−1.5
−1
−0.5
0
0.5
1
1.5
x1
x 2
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 4 / 39
Stability Analysis of Hybrid Systems
Motivating Example (III)
It can be seen from the phase portrait, that the system is unstable(left figure).
When exchanging the flow functions, the system is stabilized (rightfigure).
Trajectory of original switchingsequence:
−90 −80 −70 −60 −50 −40 −30 −20 −10 0 10−40
−20
0
20
40
60
80
100
x1
x 2
Trajectory of modified switchingsequence:
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x1
x 2
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 5 / 39
Common Lyapunov Function
Common Lyapunov Function
We first address the problem that we have a system of which themodes can be arbitrarily switched.
Arbitrary switching can be modeled by defining ∀i : g(Ti ) = Rn,where Ti is the i th transition. External events are used to perform theswitching.
Common Lyapunov function is sufficient (proof omitted)
If the continuous systems of all locations share a common Lyapunovfunction, the hybrid dynamics is stable.
Common Lyapunov function is necessary (proof omitted)
If a hybrid system is stable for arbitrary sequences of locations, alllocations share a common Lyapunov function.
A common Lyapunov function is necessary and sufficient.Matthias Althoff Analysis of Hybrid Systems 12. June 2015 6 / 39
Common Lyapunov Function
Lyapunov Function for Linear Systems
Given is a linear systemx(t) = Ax(t). (1)
Lyapunov function for LTI systems
The Lyapunov function V (x) = xTPx , P > 0 proves that an LTI system isstable if
PA+ ATP < 0
Proof: Using (AB)T = BTAT , we have that
V (x) = xTPx + xTPx = xTPAx + xTATPx = xT (PA + ATP)x
so that PA+ ATP < 0 when the system is stable.
Lyapunov function is necessary (no proof)
One can show that if (1) is stable → there has to exist a P such thatPA+ ATP < 0.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 7 / 39
Common Lyapunov Function
Common Lyapunov Function: Switched Linear Systems
For switched linear systems
x(t) = A(i)x(t)
where i refers to the i th location, it is natural to use the quadraticLyapunov function
V (x) = xTPx , P > 0
so that PA(i) + (A(i))TP < 0 when the i th location has a stable dynamics.This problem can be written as linear matrix inequalities for whichpowerful solvers exist:
P > 0
∀i : A(i)P + PA(i) < 0
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 8 / 39
Common Lyapunov Function
Common Lyapunov Function: Infeasibility Test
For switched linear systems there exists an infeasibility test for quadraticLyapunov functions:
Infeasibility Test (no proof)
If there exist M positive definite matrices R (i) > 0 (M: number oflocations) such that
M∑
i=1
R (i)(A(i))T + A(i)R (i) > 0
then there is no P > 0 such that
∀i ∈ {1, . . . ,M} : (A(i))TP + PA(i) < 0
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 9 / 39
Common Lyapunov Function
Example for the Infeasibility Test
Does stability of a switched linear system imply existence of a commonquadratic Lyapunov function?
No, the system
A(1) =
[−1 −11 −1
], A(2) =
[−1 −100.1 −1
]
is stable for arbitrary switching, but does not have a common quadraticLyapunov function since
R (1) =
[0.2996 0.70480.7048 2.4704
], R (2) =
[0.2123 −0.5532−0.5532 1.9719
]
satisfy the infeasibility condition.
However, there is a common piecewise quadratic Lyapunov function.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 10 / 39
Multiple Lyapunov Function
Multiple Lyapunov Function
It is often easier to use a different Lyapunov function V (zi , x) for eachlocation zi .
Lyapunov’s stability theorem for hybrid systems
The origin is a stable equilibrium of a hybrid automata if for all zi ∈ Z andx ∈ D
1 V (zi , 0) = 0, ∀x ∈ D \ {0} : V (zi , x) > 0
2 V (zi , x) ≤ 0, ∀x ∈ D
3 For all discrete transition times ti we have that for ti > tj andz(ti) = z(tj) that V (z(ti), x(ti )) < V (z(tj), x(tj )).
One of the difficulties is that one has to know the discrete sequences inadvance and that they strongly depend on the initial state.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 11 / 39
Multiple Lyapunov Function
Possible Evolution of Lyapunov Function Values
t
V (z , t)
t0 t1 t2 t3 t4
V (z1, t)
V (z2, t)
active inactive
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 12 / 39
Reachability Analysis of Hybrid Systems
Reachability Analysis
possibletrajectory
exactreachable set
jump
steady state
initial set
x1
x2
Informal Definition
A reachable set is the set of states that can be reached by a dynamicalsystem in finite or infinite time for a
set of initial states,
uncertain inputs,
and uncertain parameters.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 13 / 39
Reachability Analysis of Hybrid Systems
Verification Task
overapproximativereachable set exact
reachable setinvariant set
unsafe set
initial set
x1
x2
Verification Task
Check if a set of unsafe states is never reached.
Exact reachable set only for special classes computable→ overapproximation computed for consecutive time intervals.
Overapproximation might lead to spurious counterexamples.
Simulation cannot prove correctness.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 14 / 39
Reachability Analysis of Hybrid Systems Linear Systems
Linear Systems: Overview of Reachable Set Computation
x(t) = Ax(t) + u(t), A ∈ Rn×n, x(t) ∈ Rn, x(0) ∈ R(0), u(t) ∈ uc ⊕ U
1 Compute reachable set H(r) = eArR(0)⊕∫ r
t=0eA(r−t)dt uc at time r neglecting
the uncertain input (C ⊕ D := {c + d |c ∈ C, d ∈ D}).2 Obtain convex hull of initial set R(0) and H(r).
3 Enlarge reachable set to account for (1) uncertain inputs, (2) curvature oftrajectories.
4 Continue with further time intervals [kr , (k + 1)r ], k ∈ N.
Known algorithm, similar to work of A. Girard at HSCC’05.
R(0)
H(r)convexhull of
R(0), H(r) R([0, r ])
➀ ➁ ➂
enlargement
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 15 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Nonlinear Reachability Analysis: Overall Algorithm
initial set R(0), input set U , time step k = 1
linearize system
compute reachable set Rlin without linearization error
obtain set of linearization errors L based onRlin and L (L: set of admissible linearization errors)
L ⊆ L ? enlarge L
compute reachable set Rerr due to L
R = Rlin ⊕ Rerr
k:=
k+
1
yes
no
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 16 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
R(0)
linearize system
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
Rlin([0, r ])
compute reachable set Rlin
without linearization error
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
Rlin([0, r ])⊕Rerr ([0, r ])
Rerr : reachable set due to L
obtain set of linearizationerrors L based on
Rlin([0, r ]) ⊕Rerr ([0, r ])
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
R([0, r ]) =
Rlin([0, r ])⊕Rerr ([0, r ])
L ⊆ L ?
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
R([r , 2r ])
reachable set ofnext time interval
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Overall Algorithm: Animation
R([0, tf ])
reachable set ofthe complete time horizon tf
possibletrajectories
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 17 / 39
Reachability Analysis of Hybrid Systems Nonlinear Systems
Scalability of the Linearization Approach
x1
xn−1
xn
u
... (more tanks)
Water tank system.
1 2 3 4
2
3
4
5
6
x1
x6
initial set
possibletrajectories
Projected reachable set(n = 6).
Complexity with respect to the number of continuous state variables n: O(n3).
Dimension n 5 10 20 50 100
CPU-time [sec] 1.19 1.73 3.11 11.59 35.78
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 18 / 39
Reachability Analysis of Hybrid Systems Hybrid Systems
Reachability Analysis of Hybrid Systems
Hybrid systems additionally require intersections of guard sets:
x1
x2
R(0)
Rg
R([tk , tk+1])
guard
(a) Classical approach.x1
x2
R(0)
Rg
R([tk , tk+1])
R(tη)
guard
(b) New approach.
tη: last point in time before intersecting the hyperplane.
Rg : Overapproximation of the guard set intersection.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 19 / 39
Reachability Analysis of Hybrid Systems Hybrid Systems
Scalability of the Mapping-Based Approach
Jm J1 J2 JθJl
ks k1 k2 kθ
Θm
Θ1 Θ2 Θθ
Θl
gear
enginedynamics
uTm
Θs
2α
Powertrain with backlash.
−0.1 0 0.1 0.2
0
20
40
60
80
Θs −Θ1
Θref
guard set
R(0)
sampletraj.
Projected reachable set(n = 101).
Complexity with respect to the number of continuous state variables n: O(n5).
Dimension n 11 21 31 41 51 101
CPU time [sec] 8.122 14.31 23.72 31.83 53.74 1550
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 20 / 39
Reachability Analysis of Hybrid Systems Hybrid Systems
Comparison With SpaceEx
SpaceEx: state of the art tool for reachability analysis of hybrid systems.
Uses geometric guard intersection.
Example sensitive to overapproximation → comparison for initial set with5% of initial size and n = 7.
−0.05 0 0.05 0.1−20
0
20
40
60
80
Θs −Θ1
Tm
SpaceEx
mappingapproach
guard
R0.05(0)
−0.05 0 0.05 0.1
0
20
40
60
80
Θs −Θ1
Θref
SpaceEx
mappingapproachguard
R0.05(0)
Computational times: 10023 s (new approach: 0.133 s).
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 21 / 39
Applications
Ensuring Safety for Complete Vehicle Control
➀ occupancy prediction ➁ trajectory planning
➂ collision checking➃ trajectory tracking
controller
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 22 / 39
Applications
Consideration of Uncertainty
obstacle
reference trajectory
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 23 / 39
Applications
Consideration of Uncertainty
obstacle
reference trajectoryvehicle occupation
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 23 / 39
Applications
Consideration of Uncertainty
obstacle
reference trajectoryreachable set of the center
Robust Safety Problem
Is the planned maneuver of the autonomous vehicle still safe under
uncertain initial states,uncertain measurements,and disturbances?
Objective: Guarantee safety when bounds on uncertainties are known.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 23 / 39
Applications
Consideration of Uncertainty
reachable set of the center vehicle occupation
possible collision
Robust Safety Problem
Is the planned maneuver of the autonomous vehicle still safe under
uncertain initial states,uncertain measurements,and disturbances?
Objective: Guarantee safety when bounds on uncertainties are known.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 23 / 39
Applications
Online Verification Of Automated Driving
lane change
maneuver B
lane change
maneuver A
Test site Test vehicle
−20 0 20 40 60 80 100 120−5
0
5
reference trajectory
other vehicle
ego vehicle ego vehicle (braking part)
initial occupancy
final occupancyobstacle
x-position [m]
y-position[m
]
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 24 / 39
Applications
Test Drive Results
[sxsy
] Ψ
β
δ
x
y
v
sx , sy [m] x- and y-positionΨ [rad] orientationβ [rad] slip angle at center of massδ [rad] front wheel anglev [m/s] velocity
2.5 3−0.5
0
0.5
Ψ [rad]
Ψ[rad
/s]
lc B lc A−0.2 0 0.2
2.4
2.6
2.8
3
δ [rad]
Ψ[rad
]
−0.2 0 0.2−0.5
0
0.5
δ [rad]
Ψ[rad
/s]
lc A
lc B
computation time: ≈ 1.8 times faster than maneuver time (Intel i7, 1.6GHz)
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 25 / 39
Applications
Use Cases for Power Systems
Transient stability analysis(specific fault)
Transient stability analysis(region of attraction)
Stability prediction underuncertain power demandand production
final set
pre-faultset
post-faultset
x1
x2
pre-faultset
post-faultset
x1
x2
reachableset
allowedvoltage/phase
limits
time t
voltage/phase
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 26 / 39
Applications
Abstraction of the Dynamic Model
Original dynamic model (semi-explicit, index-1 DAEs)
x = f (x(t), y(t), u(t))
0 = g(x(t), y(t), u(t)),
[xT (0), yT (0)]T ∈ R(0), u(t) ∈ U ,x ∈ Rnd , y ∈ Rna : differential & algebraic states, u ∈ Rm: inputs,R(0): set of initial states, U : set of uncertain inputs
Abstraction by a linear differential inclusion
For t ∈ τk = [tk , tk+1] (k : time step):
˙x ∈ A(k)x ⊕ U(k),
x ∈ Rnd new differential states, U : new set of uncertain inputs
The algebraic states are extracted from the differential states (see later).Matthias Althoff Analysis of Hybrid Systems 12. June 2015 27 / 39
Applications
IEEE 14-Bus Benchmark System
GG
G
G
G
1
2
3
76
4
12
13
14
1110
9
58
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 28 / 39
Applications
Dynamic and Algebraic Equations
The algebraic equations are obtained from the standard equations of abus-network.
Generator/Synchronous Condenser Dynamics
The dynamics for each generator and synchronous condenser are described by thefollowing set of equations:
δi = ωi − ω1
ωi = −Di
Mi(ωi − ω1) +
1
MiTm,i −
1
MiPg ,i
Tm,i = − 1
TSV ,iRD,iωs(ωi − ωs)−
1
TSV ,iTm,i +
1
TSV ,iPc,i ,
Mi [MJ/Hz2] is the rotational inertia, Di [s/rad] the damping coefficient, TSV ,i [s]is the time constant of the governor, and 1
RD,i[-] is the proportional gain of the
governor.
Overall, the system has 14 dynamic state variables and 28 algebraic statevariables.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 29 / 39
Applications
Reachable Set of the Dynamic Variables
Black lines: random simulations; gray area: reachable set; white box: initial set
−0.6 −0.4 −0.2374.5
375
375.5
376
376.5
377
377.5
378
δ2
ω2
−0.8 −0.6 −0.4
375.5
376
376.5
377
377.5
δ3
ω3
−0.8 −0.6 −0.4
376
376.5
377
δ4
ω4
−0.8 −0.6 −0.4375.8
376
376.2
376.4
376.6
376.8
377
δ5
ω5
374 376 378 380
2.02
2.025
2.03
2.035
2.04
2.045
ω1
Tm,1
375 376 377 378
0.425
0.43
0.435
0.44
ω2
Tm,2
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 30 / 39
Applications
Reachable Set of the Algebraic Variables
Black lines: random simulations; dark gray area: pre- and post-fault reachable set; lightgray area: fault-on reachable set
1.05 1.1 1.15
−0.7
−0.6
−0.5
−0.4
−0.3
E1
Θ1
1.1 1.12 1.14 1.16 1.18−0.7
−0.6
−0.5
−0.4
−0.3
E2
Θ2
1.02 1.04 1.06 1.08−0.8
−0.7
−0.6
−0.5
−0.4
E3
Θ3
1.08 1.09 1.1
−0.8
−0.7
−0.6
−0.5
−0.4
E4
Θ4
1.12 1.122 1.124
−0.8
−0.7
−0.6
−0.5
−0.4
E5
Θ5
1.015 1.02 1.025
−0.7
−0.6
−0.5
−0.4
V7
Θ7
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 31 / 39
Applications
Verification of a Phase-Locked Loop (PLL)
Digital phase-locked loop with charge pumps:
Ci
Cp1
CP
Rp2
Rp3
frequencydivider1/N
Cp3
vi
vp1 vpip
ii
Φref
Φv
Φoutphase
frequencydetector(PFD)
RefUP
VCO
DN
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 32 / 39
Applications
Hybrid Automaton Description of the PLL
both offUP = 0,DW = 0
up activeUP = 1,DW = 0
dw activeUP = 0,DW = 1
both activeUP = 1,DW = 1
guard: Φref == 2πreset: Φv := Φv − 2π
Φref := 0
guard: Φv == 2πreset: Φref := Φref − 2π
Φv := 0
guard: Φv == 0reset: t := 0
guard: Φref == 0reset: t := 0
guard: t == td
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 33 / 39
Applications
Hybrid Automaton Description of the PLL
both offUP = 0,DW = 0
up activeUP = 1,DW = 0
both activeUP = 1,DW = 1
guard: Φref == 2πreset: Φv := Φv − 2π
Φref := 0
guard: Φv == 0reset: t := 0
guard: t == td
IUPi
ton td
Φref
Φv
ii
t
t
t2π
2π
dw active
0
0
0
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 33 / 39
Applications
Continuous Dynamics of the PLL
x = Ax + Bu + c ,
A =
0 0 0 0 0
0 − 1Cp1
(1
Rp2+ 1
Rp3
)1
Cp1Rp30 0
0 1Cp3Rp3
− 1Cp3Rp3
0 0Ki
N 0Kp
N 0 00 0 0 0 0
,B =
1Ci
0
0 1Cp1
0 00 00 0
, c =
000
2πN f0
2πfref
Input values vary depending on the signals leaving the phase-frequency detector:
u =
[IUPi , IUP
p ], if UP = 1, DW = 0
[IDWi , IDW
p ], if UP = 0, DW = 1
[IUPi + IDW
i , IUPp + IDW
p ], if UP = 1, DW = 1
[0, 0], if UP = 0, DW = 0
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 34 / 39
Applications
Specification
Transient BehaviorGiven any initial state and any valid set of parameters, verify that thelocked condition (|Φref −Φv | < ∆Φlock) is reached in less than kcycles.
Invariant BehaviorGiven a set of states in the locked condition, show that the lockedcondition is an invariant.
0 500 1000 1500
−0.3
−0.2
−0.1
0
cycle number
ph
ase
di"
ere
nce
∆
Φ
transientpart
invariantpart
allowed∆Φ
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 35 / 39
Applications
Reachable Sets of the Phase-Locked Loop (first 200 cycles)
0.35 0.4 0.45 0.5 0.55 0.6 0.65−4
−2
0
2
4
6
8
10
vi in [V]
v p1 in
[V]
0.35 0.4 0.45 0.5 0.55 0.6 0.65−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
vi in [V]
(Φv −
Φre
f)/2Π
in [r
ad]
−4 −2 0 2 4 6 8 10−4
−2
0
2
4
6
8
10
vp1
in [V]
v p in [V
]
−4 −2 0 2 4 6 8 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
vp in [V]
(Φv −
Φre
f)/2Π
in [r
ad]
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 36 / 39
Applications
Computation Times
reachability analysis: avg. MATLAB simulation:∆Φ(0) total [s] 1 cycle [s] total [s] 1 cycle [s]
[−1,−0.8]π 55.0461 0.0270 48.3297 0.0237[−0.8,−0.6]π 54.4418 0.0275 47.9096 0.0242[−0.6,−0.4]π 53.4820 0.0280 46.2673 0.0242[−0.4,−0.2]π 47.8208 0.0264 44.4596 0.0245[−0.2, 0]π 42.9191 0.0260 38.5102 0.0233
Show videos...
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 37 / 39
Applications
Further Reading
Stability:
M. Johansson: Piecewise Linear Control Systems – A ComputationalApproach, Springer Lecture Notes in Control and InformationSciences no. 284, 2002.
R.A. DeCarlo, M.S. Branicky, S. Pettersson, and B. Lennartsson:Perspectives and Results on the Stability and Stabilizability of HybridSystems, Proceedings of the IEEE, Vol 88, No. 7, 2000.
Reachability:
M. Althoff: Reachability Analysis and its Application to the SafetyAssessment of Autonomous Cars, Technische Universitat Munchen,2010.
E. Asarin, T. Dang, G. Frehse, A. Girard, C. Le Guernic, O. Maler:Recent Progress in Continuous and Hybrid Reachability Analysis,Proc. of the IEEE Conference on Computer Aided Control SystemsDesign, 2006, pages 1582-1587.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 38 / 39
Conclusions
Conclusions
Stability:
Switching between stable subsystems can destabilize a system.
A common Lyapunov function is necessary and sufficient for arbitrarilyswitched systems.
When the system is not arbitrarily switched, one often requiresmultiple Lyapunov functions to prove stability.
Reachability:
For most hybrid systems it is theoretically impossible to exactlycompute the reachable set.
Overapproximations of reachable sets can prove the correctness ofhybrid systems. This is not possible with simulation techniques.
Matthias Althoff Analysis of Hybrid Systems 12. June 2015 39 / 39
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