MoBIES meeting Deerfield Beach ETC Challenge Problem ETC Model Requirements Simulation results Parametric verification Results Towards a Checkmate model

MoBIES meeting Deerfield Beach

ETC Challenge Problem

ETC Model

Requirements

Simulation results

Parametric verification

Results

Towards a Checkmate model

OEP vs Checkmate model


ETC Hardware

Components

• D.C. motor

• Return spring

• Throttle body & Plate

• Potentiometer (TPS)


ETC Hardware


Simulink/Stateflow

Driver Electric Sys

Mech. Sys

Back EMF

Sensors

Controller

Misc. Inputs

Current FeedbackTop Level Simulation


Hardware Model: Plant• Input: throttle torque: ea• 2nd Order nonlinear System• Coulomb friction adds non-linearity

Coulomb Friction Return Spring

Voltage Input

Viscous Damping

• Output : throttle angle , back EMF Kt


Hardware Parameters• Parameters estimated from system step response and electrical measurements of motor

Hardware Model: Actuator




• input: Back emf , pwm • switches between motor on (pwm=1) and off (pwm=0)

on: 2

1( )

( )

1

( ) ( ) ( )

bat a batbat c

bat a bat bat bata a bat

bat c bat c bat c

v v R e VC R R

R R R R Re v e V

L R R L R R L R R L

1

a

c aa a

ev

LR R

e v eL L L

off:

• output: motor current ea



Hardware Parameters

• Parameters of the simulink model for the acuator

Ra 1.7 Ohm resistance of motor windings

Rc 1.5 Ohm resistance of RC filter

Rbat 0.5 Ohm internal resistance of battery

L 1.5e-3 Henry motor winding inductance

C 1.5e-3 Farad capacitance of RC filter


Pulse Width Modulation

Time

Delay introduced by PWM

• outputs 1 if dc>mc at begin of PWM cycle

• outputs 0 if dc mc

Input: motor current dc (=ea), desired motor current mc

dc>mcdc mc

pwm cycle

pwm=1pwm=0

(hypothetical input)


Hardware Model:Sensors (I)


A sliding mode controller tries to reach the desired throttle angle

The Lyapunov function and sliding surface

Human Control Mode


Reminder


A sliding mode controller which tries to reach the desired throttle angle.

Human Control Mode


Outline

• How to get formal requirements?

• How to get a model suitable for verification?

• How does the verification model compare to the OEP model?• First results

• More results


Performance Requirements

1. Rise time smaller than 100ms

2. Fall time smaller than 60ms

3. Settling time ( ±5%) smaller than 40ms

4. Steady state error smaller than 2%

5. Angle always in [0,90º]

Problem: Transforming these requirements into formal specifications.

Solution: (part of) Discussion with phase 2 participant c.q. UCB

1 to 4 only in human/cruise control

mode


Performance RequirementsRise Time, defined as the time required to rise from 10% of fully open to 90% for the throttle plate angle response to a step change in pedal position of the steady state value. The rise time for step changes from closed to is 100ms.

Settle Time is defined as the minimum time after which the throttle plate angle remains within +5% of steady-state value. ETC shall guarantee that the settle time is less than 40ms after the throttle plate angle reaches 90% of the steady-state value

input , internal clock x

A<10x’=0

Bx<=100

x’=1

Cx<=40x’=1

D95105

x’=0

violatesettletime

x:=0

>=10

x>=100<90

>=90x:=0

x>=40<95

x>=40>=95

<95 v >105

violate risetime>=90

x:=0 G not(violate rise time)

G not(violate settle time)


The ETC model simplified

The aim is to prove properties that deal withthe angle when the sliding-mode controller is used

OEP model can be simplified• Contains control logic for switching modes• Models internal communication• Contains place holders• Contains implementation details with limited effect

on



Omitting the PWM

How does this effect the behavior?

Reducing chattering

Removing delays (about 2 ms)

Replaced a 5th order filter by a 2nd order filter

Replacing numeric derivatives by exact ones



output of pwm/actuatoroutput of gain and saturation block

Omitting actuator and PWM


The ETC model simplifiedReducing chattering in sliding mode

slidingsurface

Behaves close to surface approximately as a given

equivalent controller

Introducing a boundary layer with =0.05. Within this layer we

apply the equivalent controller

slidingsurface



1

-1

1

-1

-

Within the boundary layer with =0.05 we apply the

function s/

Reducing chattering in sliding mode

OEP model uses a sign-function to represent

the modes

ss


The ETC model simplified• Reducing chattering in sliding mode• Removing communication delays (about 2ms)

OEP modelwithout chattering, delay and pwm

alp

ha

om

eg

a


A 5th order filter is part of the controller

If we reduce it to a 2nd order filter we get slightly different behavior

as before but with 2nd order filterOEP model

The ETC model simplifiedalp

ha

om

eg

a


OEP vs Checkmate model

• checkmate model separates discrete part from continuous part• switching in behavior triggered by hitting thresholds• sliding-mode controller and coulomb friction modeled by modes • continuous behavior and controller modeled by the same switching continuous function


Checkmate model

switched continuous system

discrete input


Checkmate modelSaturation of output current


Checkmate modelSliding mode switching and coulomb friction


Checkmate modelSliding mode switching and coulomb friction


Requirements


Requirements

switching conditions

timer

angle


Requirements

• Some Requirements can be proven by simulation (e.g. Rise time)


Requirements

• Some Requirements can be proven by simulation (e.g. Rise time)• Other can be proven not to hold, by simulation

angle

overshoot

filtered input

input


angle

input

filtered input

Requirements

• Some Requirements can be proven by simulation (e.g. Rise time)• Other can be proven not to hold, by simulation

steady state tracking error


What can verification add, if simulation gives the answer, already?

Verification allows to deal with uncertain initial conditions on the state.

Parametric verification allows to deal with uncertain parameters

For example: Does the rise time requirement hold if spring constant or coulomb friction range over an

interval?

Verification


Parametric Verification

1. Propagate vertices for each vertex of the parameter range




2. Determine enclosing polyhedron




2. Determine enclosing polyhedron

3. Enlarge polyhedron by optimization over the initial set, the time interval and the parameter range


Parametric VerificationFirst experiments (multi-rate automata)


Parametric VerificationFirst ETC results with Checkmate validation tool

error trace

angle below 95%

Documents

MoBIES meeting Deerfield Beach ETC Challenge Problem ETC Model Requirements Simulation results Parametric verification Results Towards a Checkmate model