Modeling Hybrid Systems
Yerang Hur
CIS 640, October 10, 2002
Department of Computer and Information Science
University of Pennsylvania
Code generation demo: Jesung Kim
Some slides are edited from previous slides made
together with others in Hybrid Systems Group.
Outline
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Hybrid Systems
Simulink
State-machine-based Representation
CHARON Framework
CHARON Language
Examples
Tool Demo
Hybrid Systems
Continuous Behaviors
Discrete Events
Physical processes in control system
e.g. robots, aircraft, ...
Collisions
Decision logic
Discrete communication
(send/receive)
Represented with
Differential equations
Algebraic equations
Invariant constraints
Guarded transitions between modes
Hybrid Systems - two views
Control engineering-oriented view
reference
discrete
control command
generator
dynamical
system
output
Hybrid Systems - two views
(con’t)
Computer scientist-oriented view
- State machine embedding dynamics
dx/dt=f(x)
guard
reset x
dx/dt=g(x)
- Continuous state variables with discrete state variables
cont x
disc y
dx/dt=f(x,y)
Y=1
guard
reset x
update y
cont x
disc y
dx/dt=g(x,y)
Simulink
Simulink is block diagram-based notations for modeling
dynamical systems.
sources
system block
diagram
sinks
Elements of Simulink model
Simulink (cont’d)
Blocks for representing continuous components
k
x
y : algebraic equation
Gain Block
y=kx
a +
c : algebraic equation
Sum Block
b c=a-b
Derivative Block
x du/dt
y : differential equation
y=dx/dt
Integrator Block x
Y: Y = y0 +  x(t)dt

State-Space Block
u
.
X=Ax+Bu
Y=Cx+Du
y
Simulink (cont’d)
Hybrid system representation
reference
+
-
zero order
holder
Kp
discrete time
integrator
Ki
derivative
approximation
Kd
+
+
+
physical
plant
State-machine-based
Representation
x := 73.0
On
x>=82.0
dx/dt=-x+100
x > 80.0
x < 70.0
Thermostat
Off
68.0<=x
dx/dt=-x
Framework for Developing Hybrid
System
1. High-level modeling
• platform independent
• hierarchical and modular
– parallel composition of agents
– specify modes and constraints
– sequential composition of modes
2. Simulation, analysis, and automatic code
generation
3. Deploy the code on actual target platform
CHARON Framework
CHARON Code
(High level language)
CHARON to Java Translator
Drivers
Java
Libraries
Java Code
Control Code Generator
Simulator Generator
Human Interface
Analysis
CHARON Language
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Individual components described as agents
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Composition, Instantiation, and Hiding
Individual behaviors described as modes
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Encapsulation, Instantiation, and Scoping of variables
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Shared variables as well as message passing
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Support for discrete and continuous behavior
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Well-defined formal semantics
Example: TwoAgent
Behavioral Hierarchy
Architectural Hierarchy
Agent
Mode
ATop
choppy
a1 (10.0,0.0) top
TwoAgent
A
a2
(9.0,-1.0)
mode0
mode1
mode Atop(real iVal, real c)
{
read analog real vIn
;
write analog real vOut
;
mode mode0=choppy(2.0,-50.0,c);
mode mode1=choppy(2.0,1.0,c) ;
trans initTrans from default to mode0
when true do {vOut=iVal}
…
Example: TwoAgent
agent TwoAgent()
{
private analog real v1,v2;
agent a1 = A (10.0, 0.0)
[vIn, vOut := v2, v1] ;
agent a2 = A (9.0, -1.0)
[vIn, vOut := v1, v2] ; }
Agent
Mode
ATop
choppy
mode0
mode1
a1 (10.0,0.0) top
TwoAgent
A
a2
(9.0,-1.0)
agent A(real initValue, real c) {
read analog real vIn
;
write analog real vOut
;
mode top=Atop(initValue,c) ; }
Modes of Agent A1
variable
declaration
invariant:
flow constraint
transition
guard: g1
v1<8.0 &&
|v1-v2|>1.1
write analog real v1
write discrete int p1
do {p1:=1}
write analog real v1
write discrete int p1
inv invChoppy {v1>0.0}
diff fv1
{d(v1)==2.0*v1-50.0+c}
v1>12.0 &&
|v1-v2|>1.0
do {p1:=0}
inv invChoppy {v1>0.0}
diff fv1
{d(v1)==2.0*v1+1.0+c}
dynamics mode0
mode
name
mode1
ATop
Architectural Hierarchy
write analog position pos1, pos2
class position {double x; double y;}
Robots
pos1
pos2
Monitor
Variable Specifiers
Update: discrete/analog
I/O interface:
read/write/private
Architectural Hierarchy
Robots
r1Est1
Robots
pos1
r1Est2
pos2
Robot1
Monitor
pos1
r2Est1
r2Est2
Robot2
pos2
Behavioral Hierarchy
pos == target
private analog real timer
awTarget arrive atTarget
r2Est1
r2Est2
r1Est1
dPlan
iAway
Robot1
dTimer
.
pos.x == v * cos(phi)
.
pos.y == v * sin(phi)
r1Est2
.
dStop
iAt
timer/updateFreq == 0
pos
timer == 1
moving
dSteer
aOmega
iFreq
omega == k * (theta – phi)
sense
move
sensing
dStop
iConst
arrive
Example: the Simplex Architecturebased Inverted Pendulum
us
Safety
Controller
Baseline
Controller
Experimental
Controller
ub
Decision
Module
Physical
System
x
ue
x0
Equilibrium
state
u
Experimental
Controller
Safety Controller
Baseline Controller
The Inverted Pendulum
g
l
q
M
uf
x
m
Control Task and CHARON
Description
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Control objective: to move the cart from one position to a desired
target position xt maintaining the pendulum at the upright position.
Planner: generates a set point xs every T seconds with rate c until
the generated set point reaches the target position.
Lower level: stabilizes the system about [xs, 0, 0, 0 ]
Controllers
Physical System
Decision Module
Simulation Results
5
ctrlCmd
4
angle
VaBC
3
VaSC
2
position
1
VaEC
cVelocity
0
-2
-1
0
2
4
6
8
aVelocity
time
Initial Position : 0.0
Initial Cart Velocity : 0.0 m/s
Initial Angle : 5.0 deg
Initial Angular Velocity : 0.0 deg/s
Sampling Frequency 50Hz
Maximum Control Value : 3.78V < 4.96V
Maximum Position Tracking Error : 0.17m < 0.2m
Maximum Cart Velocity : 0.32m/s < 1.0m/s
Maximum Angle : 5.0 deg< 10.0 deg
CHARON Simulator Generation
Differential Equations
Process
Assertions
Algebraic Equations
Agents
Invariants
CHARON
Agent and Mode
generator
Guards
Modes
Simulator main
generator
Simulator
main
model
Actions
User-defined
external classes
Numerical integrator
Trace
generator
Trace
Plotter
Wrap-up:
Two Worlds Affect Each Other
- Simulink is augemented with a hierarchical state machine.
We call it Simulink/Stateflow.
- Controllers/Decision Module/Physical system case study
in CHARON: E.g. the Simplex-Architecture-based
inverted pendulum controller[RTSSWIP 2000] and
autonomous distributed multi-robots.
- Among other design tools of interest are Ptolemy II
with continuous time domain, SHIFT/Teja, Modelica,
and RT-UML.
Wrap-up: Features of CHARON
- Discrete event/discrete state representation: discrete int/bool
- Discrete event/continuous state representation: discrete real
- Continuous time/continuous state representation: analog real
- Behavior description: mode with hierarchy
- Architecture description: agent with hierarchy
-Structured modern programming language
: variable scoping, type-checking rules, formal semantics
Wrap-up: Features of CHARON
(cont’d)
- Non-determinism
: non-deterministic execution of agents when more than one agent
observe enabled transitions at once, currently interleaving semantics
: non-deterministic choice of transitions when more than one
transitions are enabled at once, currently randomly selected or
user-guided
: non-deterministic non-urgent jump when a guard is enabled while
the invariant is not violated, currently probabilistically taken and
80/20 rule is embedded
: non-deterministic differential and algebraic constraints
with inequalities, currently only front-end supports this
- Software engineering tool
: reusable agent/mode definition,
: simulator generator, verification tool box, code generator
Wrap-up: Interface Qualifiers
of Agent Variables
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Input from environment:
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Output to environment:
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input (v0.1) --> read (v0.37)
output (v0.1) --> write (v0.37)
--> write/readwrite (v0.4)
--> write exclusive/write shared (v0.7)
Local
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private (v0.1) --> hiding operation (v0.12)
--> private (v0.6)
Wrap-up: Other Type Qualifiers
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Update Qualifiers:
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analog for continuous update or reset
discrete for discrete update
Channel Qualifiers:
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buffered non-blocking communication
channel
combined with buffer size, message type,
buffer management policy
Example: write channel [2] of real cold chan1;
Tool Demo
For manuals and more examples,
visit http://www.cis.upenn.edu/mobies/experiments.php3.