HybridSimulatorPresentation

Hybrid Equations (HyEQ) Toolbox
A Toolbox for Simulation of
Hybrid Dynamical Systems
in MATLAB/Simulink R
Ricardo G. Sanfelice, David A. Copp, and Pablo Ñañez
Hybrid Dynamics and Control Lab (HDC Lab)
Department of Aerospace and Mechanical Engineering
University of Arizona, USA
Ricardo Sanfelice, David Copp, Pablo Ñañez - Hybrid Equations (HyEQ) Toolbox - April 9, 2013

Hybrid Systems
Hybrid systems combine behaviors that are typical of
continuous-time dynamical systems with behaviors that are
typical of discrete-time dynamical systems.
In switched electrical circuits, voltages
and currents that change continuously
according to classical electrical
network laws also change
discontinuously due to switches
opening or closing.
Embedded systems and, more
generally, systems involving both
digital and analog components,
execute software code discretely while
interacting with their physical
environments continuously.
s1
s2
s4
s3
ell, r
il
c0
vc
ic
Hybrid
VDC
DC/AC
io so vg

Hybrid Systems
Velocities in a multibody dynamical
system change continuously according
to Newton’s second law, but undergo
instantaneous changes in velocity and
momentum due to collisions.
The concentrations in genetic
networks grow or decay continuously
according to binary variables that
change between 1 and 0 when the
concentrations reach certain
thresholds.

Hybrid Systems
Impulsive oscillators capture the
dynamics of certain networked
biological systems, such as ﬁreﬂies and
crickets, by means of timers with
event-driven resets.
Modern control algorithms often lead
to both kinds of behavior, due to
either digital components used in
implementation or logic and decision
making encoded in the control
algorithm.

Outline
1 Hybrid Systems
Challenges to Simulation
2 A Toolbox for Simulation of Hybrid Systems
Hybrid Equations (HyEQ) Toolbox Overview
Modeling Framework and Contributions
3 Components and Usage of the HyEQ Toolbox
Simulating Hybrid Systems
Simulating Interconnections
Internals
Conﬁguration, Initialization, Postprocessing
Simulation Procedure
4 Illustrative Example
5 Conclusion and References

Simulation of hybrid systems
Hybrid systems demand simulation algorithms capable of
Numerically integrate diﬀerential equations
Locate, detect, and manage events
Reset state variables discretely
Generate a time axis that jointly parameterizes continuous
and discrete behavior

Simulation of hybrid systems
Hybrid systems demand simulation algorithms capable of
Numerically integrate diﬀerential equations
Locate, detect, and manage events
Reset state variables discretely
Generate a time axis that jointly parameterizes continuous
and discrete behavior
Challenges to Simulation of Hybrid Systems:
Simultaneous simulation of continuous and discrete dynamics
Accurate event location and detection
Large number of events/jumps in ﬁnite time (e.g., Zeno)
Multiple events at the same time
Time sharing in simulation of interconnected hybrid systems

A hybrid simulator for MATLAB/Simulink called Hybrid Equations
(HyEQ) Toolbox with the following capabilities is introduced and
illustrated in examples:
Simulation of autonomous and non-autonomous hybrid
systems given by hybrid equations
Modular simulation of interconnections of hybrid systems
Simulation of systems with Zeno behavior and multiple jumps
at an instant
Generation of hybrid time axis indexing jumps
The HyEQ Toolbox is designed to simulate hybrid systems written
in a hybrid system framework amenable for control theory for
which a theory of robust stability has recently been developed.

HyEQ Toolbox: Modeling Framework
Hybrid systems are given by the hybrid equations
H : x ∈ Rn
, u ∈ Rm ˙x = f(x, u) (x, u) ∈ C
x+ = g(x, u) (x, u) ∈ D
or H = (C, f, D, g) for short.
C
D
Rn
˙x = f(x, u)
g(x, u)
×
Solutions are functions parameterized by hybrid time (t, j):
Flows parameterized by t ∈ R≥0 := [0, +∞)
Jumps parameterized by j ∈ N≥0 := {0, 1, 2, . . .}

A hybrid system is a dynamical system with continuous and
discrete dynamics.
Modeling Framework: A hybrid system H on a state space Rn
and input space Rm is defined by the following objects:
A set C ⊂ Rn × Rm called the flow set.
A function f : Rn → Rn called the flow map.
A set D ⊂ Rn × Rm called the jump set.
A function g : Rn → Rn called the jump map.
The flow map f defines the continuous dynamics on the flow set C.
The jump map g defines the discrete dynamics on the jump set D.
H can be explicitly denoted as H = (C, f, D, g).
Hybrid systems may have states that evolve both continuously
and jump, in which case the boundary of the flow and jump set
intersect.

Contributions to Hybrid Systems Theory
Autonomous Hybrid Systems
Notion of Solution
Lyapunov Theory and Invariance
Robustness to Small Perturbations
Nonautonomus Hybrid Systems
Notion of Solution
Control Lyapunov Functions
ISS and I/O Stability
Interconnections
Small gain Theorems
Hybrid Control Design
Tracking Control
Passivity-based Control
Minimum-norm Control
Backstepping
Numerical Simulation
Simulation theory
Simulation toolbox

HyEQ Toolbox: Examples
Examples of such systems include:
a bouncing ball on a moving platform,

a DC/AC inverter,
s1
s2
s4
s3
ell, r
il
c0
vc
ic
Hybrid
VDC
DC/AC
io so vg

a vehicle traveling along a track with boundaries,
−2
−1
1
2
0
the synchronization of multiple fireflies flashing.

Example: Impulsive biological oscillators
Figure 1 shows how two fireflies can be modeled as periodic
oscillators with timers. The timer in each firefly is denoted τ1 and
τ2, respectively. The stars indicate when resets of the timers occur.
The “timer threshold” defines when the timers are reset to zero,
which corresponds to the respective firefly flashing.
τ1, τ2
time
synchronizationreset
firefly1
firefly2threshold
Figure 1 : Fireflies with hybrid dynamics synchronizing

Example: Impulsive biological oscillators
A model for interacting fireflies can be described as a hybrid
system. The fireflies are modeled mathematically as impulsive
oscillators with timer variables exhibiting impulses due to flashes.
In this way, the oscillator in a firefly exhibits continuous and
discrete dynamics:
The flow map f enforces that the timer state counts time
during continuous evolution.
The jump map g induces jumps in the timer state as follows:
it resets the timer to zero when the firefly flashes.
it is incremented a small amount when a neighboring firefly
flashes.
The flow set C determines when the timer should count time
while the flow set D triggers the jumps of the timer.

Components and Usage of the HyEQ Toolbox

The Hybrid Equations (HyEQ) Toolbox includes the following:
HyEQsolver.m lite simulator script for systems w/o inputs.

Simulink library for systems w/inputs and interconnections.
HS with external
functions
HS with embedded
functions
HS with embedded
functions and inputs
Time
t
State
x
J umps
jHS
x
t
j
Time
t
State
x
J umps
jHS
x
t
j
u
Time
t
Terminator
State
x
J umps
j
HSu
x
t
j
x−(int)
u
u
HSu
x
t
j
x−(int)
u
HSu
x
t
j
x−(int)
u
1 2

A Simulink implementation in the HyEQ Toolbox is shown below.
This model simulates a hybrid system with input
x−(int)
4
j
3
t
2
x
1
u
jump set D
x
u
v
D
jump map g
x
u
xplus
g
flow set C
x
u
v
C
flow map f
x
u
xdot
f
Time
t
Terminator
State
x
Jumps
j
Integrator System
f
C
g
D
x
t
j
x−
HSu
x
t
j
x−(int)
u
u1
1
Double Click
to Initialize
Double Click to
Plot Solutions
Figure 2 : MATLAB/Simulink implementation of a hybrid system with
inputs. The blocks f and C implement the ﬂow equation while the
blocks g and D implement the jump equation.

Simulating Hybrid Systems in the HyEQ Toolbox
In the HyEQ Toolbox, the hybrid equations above are implemented
in Simulink using Embedded MATLAB Functions. The data of the
hybrid system is coded as follows:
The ﬂow set C is captured by the embedded function C
The ﬂow map f is captured by the embedded function f
The jump set D is captured by the embedded function D
The jump map g is captured by the embedded function g
Each of the blocks in Figure 2 implement the embedded functions
above.

Simulating Interconnections in the HyEQ Toolbox
The simulator implementation in HyEQ with inputs uses
embedded MATLAB functions within the main Simulink file
HyEQsimulator.mdl.
This allows the user to modify the data of each hybrid system
inside the model without having to edit several files.
With such an implementation, multiple hybrid systems can be
incorporated in a single Simulink file.
Then, interconnections of N hybrid systems is possible. Each
systems can be identical, as in the case of multiple fireflies
synchronizing flashing, or they can be different, such as a
bouncing ball impacting on a moving floor, which can be
interpreted as an interconnection of two systems.

In the model shown below, the input of the second system is
deﬁned as the output of the ﬁrst system, and viceversa. Other
types of interconnections are also possible.
This model simulates interconnected hybrid systems.
time1
t2
time
t1
state1
x2
state
x1
jumps1
j2
jumps
j1
Hybrid System 2
x
t
j
x−(int)
u
Hybrid System 1
x
t
j
x−(int)
u
Double Click
to Initialize
Double Click to
Plot Solutions
Figure 3 : Implementation of interconnected hybrid systems, H1 and H2.

Figure 4 : Bouncing ball on a
moving floor
0 2 4 6 8 10 12 14 16 18
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18
−1
−0.5
0
0.5
1
heightvelocity
flows [t]
flows [t]
Figure 5 : Simulation trajectories: height
and velocity

plotflows.m, plotjumps.m, plotHybridArc.m files for
plotting solutions,
initialization.m, postprocessing.m files for
configuring the simulation and solutions,
Files for several examples of individual and interconnected
hybrid systems.

Internal Structure of Simulink Implementation

Internals of the HyEQ Toolbox simulator
Four basic blocks are used to deﬁne
the dynamics of the hybrid system H:
Flow map f
Flow set C
Jump map g
Jump set D
x
u
v
D
jump map g
x
u
xplus
g
x
u
v
C
flow map f
x
u
xdot
f

Integrator System
x−
4
j
3
t
2
x
1
Update logic
g(x−,u)
j−
t−
update law
Stop logic
t
j
C
D
stop
Stop
Simulation
STOP
Save
In1
In2
Jump logic
C
D
r
jump
1
s
xo
ICx0
[0; 0; x0(:)]
ICx
[x0]
Constant
save
CT dynamics
f(x,u) dot
D
4
g
3
C
2
f
1

CT Dynamics
dot
1
Jumps
0
Flows
1
f(x,u)
1
This block deﬁnes the continuous dynamics by assembling the time
derivative of the state [t j x ] .
States t and j are considered states of the system because
they need to be updated throughout the simulation in order to
keep track of the time and number of jumps.
The dynamics of these states is given by
˙t = 1, ˙j = 0, ˙x = f(x, u) .

Jump Logic
rule=1: out = D
rlule=2: out = D and ~C
rule=3: out = (D and ~C) or (D and C and r>0.5)
other : out = 0
jump
1
jump
Multiport
Switch
1
2
3
*
AND
OR
NOT
AND
0
rule
Compare
To Constant
>= 0.5
r
3
D
2
C
1
The inputs are the outputs of blocks C and D indicating
whether the state is in those sets or not, and a random signal
with uniform distribution in [0, 1].
Output of 1 indicates a jump of H and triggers a reset of the
integrator.

Update Logic
update law
1
1
t−
3
j−
2
g(x−,u)
1
The update logic uses the state port information of the integrator.
This port reports the value of the state of the integrator, [t j x ] ,
at the exact instant that the reset condition becomes true.
The ﬂow time t is kept constant at jumps and j is incremented by
one. More precisely:
t+
= t−
, j+
= j−
+ 1, x+
= g(x−
, u)
where [t− j− x− ] is the state that triggers the jump.

Stop Logic
stop
1
Logical
Operator1
OR
Logical
Operator
NOR
Jump Horizon, J
>= J
Flow Horizon, T
>= T
D
4
C
3
j
2
t
1
This block stops the simulation under any of the following events:
1. flow time ≥ maximum flow time specified by T.
2. jump time ≥ maximum number of jumps specified by J.
3. state of the hybrid system x is neither in C nor in D.
Under any of these events, the output of the logic operator
connected to the Stop block becomes one, stopping the
simulation.

Configuration
Before a simulation is started, the following parameters should be
defined
Determine the integrator scheme, zero-cross detection
settings, precision, and other tolerances. Using the default
settings does not always give the most efficient or most
accurate simulations. One way to edit these settings is to
open the Simulink Model, select
Simulation>Configuration Parameters>Solver
An easy way to adjust some settings is to include a line of
code in the initialization.m file like that given below.
% configuration of solver
RelTol = 1e-8; MaxStep = .001;
In this code, RelTol = 1e-8 and MaxStep = .001 define
variables for the relative tolerance and maximum step size of
the ODE solver, respectively.

Initialization
Double-click block labeled Double Click to Initialize to initialize
variables. This calls initialization.m which defines the initial
conditions by defining the initial values of
the state components
any necessary parameters
the maximum flow time specified by T
the maximum number of jumps specified by J, and
tolerances used when simulating.

Postprocessing and Plotting solutions
Solutions are plotted by double-clicking on Double Click to Plot
Solutions block which calls the script postprocessing.m. The
script postprocessing.m may be changed to include the desired
postprocessing and solution plots.
The following functions are available to generate plots:
plotflows(t,j,x):
plots (in blue) the projection of the trajectory x onto the flow time
axis t.
The value of the trajectory for trivial flow intervals (zero
length) is marked with ∗ (in blue).
Dashed lines (in red) connect the value of the trajectory
before and after the jump.

plotjumps(t,j,x):
plots (in red) the projection of the trajectory x onto the jump time
axis j.
The initial and final value of the trajectory on each flow
interval is denoted by ∗ (in red).
The continuous evolution of the trajectory on each flow
interval is depicted with a dashed line (in blue).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0
0.2
0.4
0.6
0.8
1
x1
x1
flows [t]
jumps [j]

plotHybridArc(t,j,x):
plots (in blue) the trajectory x on hybrid time domains.
The ﬂow intervals indexed by the corresponding j are depicted
in the t − j plane (in red).
0
2
4
6
8
10
12
14
16
18
200 1 2 3 4
0
0.5
1
x1
t
j

Using the HyEQ Simulink Toolbox

The following procedure is used to simulate an example:
Step 1)
The main Simulink ﬁle HyEQsimulator.mdl is opened in
MATLAB/Simulink.
Step 2)
The Embedded MATLAB function blocks f, C, g, D are edited. In
each embedded function block, global variables must be added as
input variables and deﬁned as parameters by selecting Tools>Edit
Data/Ports>Add>Data. Name the variables the same as they are
called in the function block, and set the scope to Parameter.

Step 3)
The initialization script initialization.m is edited. The flow
time and jump horizons, T and J are defined as well as the initial
conditions for the state vector x0, a rule for jumps, and the
tolerances for simulation.
Step 4)
The integrator system is double-clicked and the flow time T, jump
horizon J, and initial condition x0 are defined on the window that
appears.

Step 5)
The simulation stop time and simulation tolerances in the Simulink
model are set to the values deﬁned in initialization.m by
selecting Simulation>Configuration Parameters and inputing
the values T, RelTol, and MaxStep.
Step 6)
The postprocessing script postprocessing.m is to be edited.
Flows and jumps may be plotted by calling the functions
plotflows.m and plotjumps.m, respectively. The hybrid arc may
be plotted by calling the function plotHybridArc.m.

Step 7)
The block labeled Double Click to Initialize is double-clicked to
initialize the variables.
Step 8)
The play button is clicked in the Simulink model to run the
simulation.
Step 9)
The block labeled Double Click to Plot Solutions is
double-clicked to plot the desired solutions.

Example: Back to Synchronization of Two Fireflies
The synchronization of the fireflies can be modeled as an
interconnection of two hybrid systems where each firefly can be
modeled as a hybrid system given by
fi(τi, ui) := 1
Ci := {(τi, ui) : 0 ≤ τi ≤ 1} ∩ {(τi, ui) : 0 ≤ ui ≤ 1}
gi(τi, ui) :=
(1 + ε)τi if (1 + ε)τi ≤ 1
0 if (1 + ε)τi ≥ 1
Di := {(τi, ui) : τi ≥ 1} ∪ {(τi, ui) : ui ≥ 1}
A state value of τi = 1 corresponds to a flash, and after each
flash, the firefly automatically resets its timer to zero.
When one of the fireflies flashes, the other tries to synchronize
its flash by jumping ahead in its periodic cycle. This behavior
is captured by the coefficient ε.

Example: initialization.m
%initialization for fireflies example
% initial conditions
tau1_0 = .5;
tau2_0 = 0;
%biological coefficient
e = 0.3;
% simulation horizon
T = 15; J = 15;
% rule for jumps
rule = 1; % priority for jumps
%solver tolerances
RelTol = 1e-8;
MaxStep = .005;

Example: Flow Map
fi(τi, ui) := 1
function taudot = f(tau, u)
% flow map
taudot = 1;

Example: Flow Set
Ci := {(τi, ui) : 0 ≤ τi ≤ 1} ∩ {(τi, ui) : 0 ≤ ui ≤ 1}
function v = C(tau, u)
% flow set
if ((tau >= 0) && (tau <= 1)) || ((u >= 0) && (u <= 1))
v = 1; % report flow
else
v = 0; % do not report flow
end

Example: Jump Map
gi(τi, ui) :=
(1 + ε)τi if (1 + ε)τi ≤ 1
0 if (1 + ε)τi ≥ 1
function out = g(tau, u, e)
% jump map
if (1+e)*tau <= 1
tauplus = (1+e)*tau;
else (1+e)*tau >= 1
tauplus = 0;
end
out = tauplus;

Example: Jump Set
Di := {(τi, ui) : τi ≥ 1} ∪ {(τi, ui) : ui ≥ 1}
function v = D(tau, u)
% jump set
if (tau >= 1) || (u >= 1) % jump condition
v = 1; % report jump
else
v = 0; % do not report jump
end

0 2 4 6 8 10 12
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0.5
1
1.5
τ1
τ1
flows [t]
jumps [j]
Figure 6 : Solution of firefly system
H1
0 2 4 6 8 10 12
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0.5
1
1.5
τ2
τ2
flows [t]
jumps [j]
Figure 7 : Solution of firefly system
H2

0 2 4 6 8 10 12
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0.5
1
1.5
τ1,τ2τ1,τ2
flows [t]
jumps [j]
Figure 8 : Solution of firefly systemfor interconnection of H1 and H2
A solution to the interconnection of hybrid systems H1 and H2
with T = 15, J = 15, rule = 1, ε = 0.3 is depicted in Figure 8.
The fireflies initially flash out of phase with one another and then
synchronize to flash with same phase.

Simulation of N interconnected fireflies
The same procedure can be
used to simulate any number of
fireflies. The main system can
be copied N times, and then
those systems can be
interconnected where the input
to every system is the
maximum state value of all of
the outputs. This
interconnection is shown for 10
fireflies in Figure 9.
This model simulates the synchronization of ten fireflies.
x6
x5
x4
x3
x2
j1
j10
j9
t1
j8
j7
j6
j5
j4
j3
j2
t10
t9
t8
x1
t7
t6
t5
t4
t3
t2
x10
x9
x8
x7
Max
u y
fcn
Hybrid System 9
x
t
j
x−(int)
u
Hybrid System 8
x
t
j
x−(int)
u
Hybrid System 7
x
t
j
x−(int)
u
Hybrid System 6
x
t
j
x−(int)
u
Hybrid System 5
x
t
j
x−(int)
u
Hybrid System 4
x
t
j
x−(int)
u
Hybrid System 3
x
t
j
x−(int)
u
Hybrid System 2
x
t
j
x−(int)
u
Hybrid System 10
x
t
j
x−(int)
u
Hybrid System 1
x
t
j
x−(int)
u
Double Click
to Initialize
Double Click to
Plot Solutions
Figure 9 : Model for 10 Fireflies

Simulation of 10 interconnected fireflies
0 5 10 15
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839404142434445
0
0.5
1
1.5
ττ
flows [t]
jumps [j]
Figure 10 : Solution for 10 fireflies with T = 15, J = 45, rule = 1,
ε = 0.1
It can be seen that all 10 fireflies start with different initial
conditions and eventually all synchronize.

Conclusion
HyEQ Toolbox installation ﬁles can be found at MATLAB
Central’s ﬁle exchange website by accessing
http://www.mathworks.com/matlabcentral
and searching for HyEQ Toolbox or through the “Software” tab at
the author’s website
http://www.u.arizona.edu/~sricardo/.

Conclusion
We would like to thank Giampiero Campa for his thoughtful
feedback and advice as well as Torstein Ingebrigtsen Bo for his
comments and ﬁrst version of the HyEQsolver script.
MATLAB/Simulink are registered trademarks of The MathWorks, Inc.

References
[1] David A. Copp and Ricardo G. Sanfelice, Hybrid Equations
(HyEQ) Toolbox v1.0 - A Toolbox for Simulating Hybrid Systems
in MATLAB/Simulink. Hybrid Dynamics and Control Laboratory,
University of Arizona.
[2] http://control.ee.ethz.ch/~ifaatic/ex/example1.m.
Institut f¨ur Automatik - Automatic Control Laboratory, ETH
Zurich, 2011.
[3] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid dynamical
systems. IEEE Control Systems Magazine, 28-93, 2009.
[4] R. G. Sanfelice and A. R. Teel, Dynamical Properties of Hybrid
Systems Simulators. Automatica, 46, No. 2, 239–248, 2010.
[5] Sanfelice, R. G., Interconnections of Hybrid Systems: Some
Challenges and Recent Results Journal of Nonlinear Systems and
Applications, 111–121, 2011.

Extra Slides

Conclusion
We would like to thank NSF and AFOSR for partially funding
this project, Giampiero Campa from The Mathworks for helpful
suggestions, and Torstein Ingebrigtsen Bo for his comments and
ﬁrst version of the HyEQsolver script.
MATLAB/Simulink are registered trademarks of The MathWorks, Inc.

Sample hybrid systems simulated in the HyEQ Toolbox:
s1
s2
s4
s3
ell, r
il
c0
vc
ic
Hybrid
VDC
DC/AC
io so vg
−2
−1
1
2
0

Examples of dynamical systems that may be simulated in Simulink
without using an implementation like the HyEQ Toolbox are shown
below.
Systems with continuous states only, e.g.,
˙x1 = x2, ˙x2 = u,
which can be simulated by
ScopeIntegrator1
1
s
Integrator
1
s
u
1

Systems with discrete states only, e.g.,
q(t + 1) = q + 1,
which can be simulated by
Unit Delay
z
1
Scope
Constant
1

Certain systems with combined continuous and discrete
states, e.g.,
q(t + 1) = q + 1, ˙x = q
Unit Delay
z
1
Scope
Constant
Integrator
1
s
1

Certain systems with combined continuous and discrete
states, e.g.,
q(t + 1) = q + 1, ˙x = q
Unit Delay
z
1
Scope
Constant
Integrator
1
s
1
However, if some of the system states evolve both continuously and
discretely (that is they evolve continuously but might also jump,
according to known hybrid state equations), then it might be easier
to simulate the system using the HyEQ toolbox.

HyEQ Toolbox: Several methods for simulation
As previously noted, there are multiple methods available in this
toolbox for simulating diﬀerent hybrid systems. They include:
1. Simulink implementation with embedded functions:
for simulation of individual or interconnected hybrid systems
with or without inputs.
2. Simulink implementation with external functions:
for simulation of individual hybrid systems with or without
inputs without needing a C-compiler installed with MATLAB.
3. Lite simulator
for simulation of individual hybrid systems without inputs
straight from the command line.

HyEQ Toolbox: No inputs
An implementation for HyEQ without inputs is as follows:
This model simulates a hybrid system.
j
3
t
2
x
1
jump set D
Interpreted
MATLAB Fcn
jump map g
Interpreted
MATLAB Fcn
flow set C
Interpreted
MATLAB Fcn
flow map f
Interpreted
MATLAB Fcn
Time
t
State
x
Jumps
j
Integrator System
f
C
g
D
x
t
j
x−
HS
x
t
j
Double Click
to Initialize
Double Click to
Plot Solutions
Figure 11 : MATLAB/Simulink implementation of a hybrid system

A Simulator for Hybrid Systems: No inputs
The data is implemented in using .m files. The data of the hybrid
system is coded as follows:
The flow set C is captured by the function C.m
The flow map f is captured by the function f.m
The jump set D is captured by the function D.m
The jump map g is captured by the function g.m
Each of the blocks in Figure 11 implement the .m functions above.

A Simulator for Hybrid Systems: No inputs
The data is implemented in using .m files. The data of the hybrid
system is coded as follows:
The flow set C is captured by the function C.m
The flow map f is captured by the function f.m
The jump set D is captured by the function D.m
The jump map g is captured by the function g.m
Each of the blocks in Figure 11 implement the .m functions above.
A stand alone script called HyEQsolver.m is also available. It
evaluates these functions and finds a solution to the hybrid system
with the given dynamics.

Example: Bouncing Ball
For the simulation of the bouncing ball system with a constant
input and regular data given by
f(x, u) :=
x2
−γ
, C := (x, u) ∈ R2
× R : x1 ≥ u
g(x, u) :=
u
−λx2
, D := (x, u) ∈ R2
× R : x1 ≤ u , x2 ≤ 0
γ > 0 is the gravity constant, u is the input constant, and
λ ∈ [0, 1) is the restitution coeﬃcient. An input was chosen to be
u(t, j) = 0.2 for all (t, j). The constants for the bouncing ball
system are chosen to be γ = 9.81 and λ = 0.8.
Note that for this example, the jump set may be implemented
using Simulink logic blocks with Zero Cross Detection. See the
ﬁgure below.

This model simulates a hybrid system with input
x (int)
4
j
3
t
2
x
1
u
jump map g
x
u
xplus
g
flow set C
x
u
v
C
flow map f
x
u
xdot
f
Time
t
Terminator
State
x
<=
<=
AND
Jumps
j
Integrator System
f
C
g
D
x
t
j
x
HSu
x
t
j
x (int)
u
u1
1
Double Click
to Initialize
Double Click to
Plot Solutions
Figure 12 : Implementation with Simulink logic blocks for jump set

A solution to the bouncing ball system from x(0, 0) = [1, 0] and
with T = 10, J = 20, rule = 1, is depicted in Figure 13 (height)
and Figure 14 (velocity). Both the projection onto t and j are
shown. Figure 15 depicts the corresponding hybrid arc.
These simulations reﬂect the expected behavior of the bouncing
ball model. Note the only diﬀerence between this example and the
example of a bouncing ball without a constant input is that, in this
example, the ball bounces on a platform at a height of the chosen
input value 0.2 rather than the ground at a value of 0.

0 0.5 1 1.5 2 2.5 3 3.5 4
0.2
0.4
0.6
0.8
1
flows [t]
x
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.2
0.4
0.6
0.8
1
jumps [j]
x
1
Figure 13 : Solution of Bouncing
Ball Example: height
0 0.5 1 1.5 2 2.5 3 3.5 4
−4
−2
0
2
4
flows [t]
x2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
−4
−2
0
2
4
jumps [j]
x2
Figure 14 : Solution of Bouncing
Ball Example: velocity

0
5
10
15
20
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
x1
t
j
Figure 15 : Hybrid arc corresponding to a solution of Bouncing Ball
Example: height

Outline
(1) Installation of the simulator (step by step)
(2) C compiler?
(3) A couple of examples from scratch (Boost converter (from
Thomas Theunisse))
(3.1) File organization
(3.2) Variables
(3.3) Debuging
(4) The use of the plot commands

[(1)] Installation of the simulator (step by step)

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS

[(3)] Examples
Example: Boost converter
Boost converter, (S = OFF, iD > 0, q = 1) (MODE 1)
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
Let’s write down the equations of the system...

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 1
C iL

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 1
C iL
(2) ˙iL = − 1
LVc + 1E
L

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 1
C iL
(2) ˙iL = − 1
LVc + 1E
L
(3) ˙E = 0
(4) ˙q = 0

[(3)] Examples
Boost converter, (S = ON, iD = 0, q = 2) (MODE 2)
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 0
C iL

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 0
C iL
(2) ˙iL = − 0
LVc + 1E
L

[(3)] Examples
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 0
C iL
(2) ˙iL = − 0
LVc + 1E
L
(3) ˙E = 0
(4) ˙q = 0

[(3)] Examples
Boost converter, (S = OFF, iD = 0, q = 1) (MODE 3)
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + 0
C iL
(2) ˙iL = − 0
LVc + 0E
L
(3) ˙E = 0
(4) ˙q = 0

[(3)] Examples
Boost converter,
E
S
L
C
R
+ -
VL
VC
iL
+ -
VD
D
+
−
iDiS
(1) ˙Vc = − 1
RC Vc + ρ(q)
C iL
(2) ˙iL = −ρ(q)
L Vc + λ(q)E
L
(3) ˙E = 0
(4) ˙q = 0

Flow Map
F(Vc, iL, E, q) =





− 1
RC Vc + ρ(q)
C iL
−ρ(q)
L Vc + λ(q)E
L
0
0





where
ρ(q) =
0 if q ∈ {2, 3}
1 if q = 1
λ(q) =
0 if q = 3
1 if q =∈ {1, 2}

Flow Set
C :={x ∈ 4
| q = 1, u = 0}
∪ {x ∈ 4
| q = 2, u = 1}
∪ {x ∈ 4
| q = 3, u = 0, iL = 0}

Jump Set and Jump Map
mode  mode 
mode 
S=
S=
S=0,
iL
=0,
vc
>E
vc
<E
iL
=0,
(S=0)
(S=0)
(S=)
S=0,
iL
=0
iL
>0
Figure 16 : Switching behavior of the Boost circuit.

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