Bond Graph for Modelling, Analysis, Control Design, Fault Diagnosis
Geneviève Dauphin-Tanguy Christophe Sueur Laboratoire d’Automatique et d’Informatique Industrielle de Lille Bond Graph Research Group Laboratoire d’Automatique et d’Informatique Industrielle de Lille Ecole Centrale de Lille
• 6 academics (2 Profs, 2 associate profs, 2 assistant profs) • 10 PhD students
Application areas : power systems (electrical machines, photovoltaic systems, fuel cells), thermofluid process, car industry Studies performed in collaboration with Peugeot -Citroën
• Mechatronic design of an automatic gear box • Clutch management and drive comfort • Mechatronic design of an active hydraulic suspension • Thermal comfort regulation in a car interior • Modelling and simulation of a fuel cell system • Analysis of structural properties of bond graph models • Robustness of control laws for systems with parametric uncertainties • …..
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 3 Why a bond graph approach ?
• Multidisciplinary systems à need for a communication language between people from different physical domains • Need for models with physical insight (virtual testing facility) • Unified modelling methodology for knowledge storage in model libraries • Integrated (« mechatronic ») design of controlled systems
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 4 Mechatronic design
Objectives
Identification Simplification Analysis Modelling Control Actuators Validation Sensors Simulation
Maintaining Faulty mode
Diagnosis
Fault Detection and Isolation
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 5 1 - Mechatronic design of an automatic gear box
computer
engine torque automatic vehicle converter gear box
Differential, Transmission shaft Wheels, Vehicle mass, Complete driveline air resistance, road friction
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 6 1 - Mechatronic design of an automatic gear box Problem statement
• Design control laws for the driving of an automatic gear box by a computer with the following objectives: – Complete satisfaction of the customer corresponding to a variation of the engine torque as continue as possible (no jerk in acceleration during a shift) – Respect of technological constraints (actuator response duration, minimization of the energy dissipated in the clutchs)
When to shift ? How to shift ?
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 7 1 - Mechatronic design of an automatic gear box Automatic gear box : physical scheme
Arrangement of 2 epicyclic gear trains which allows 3 ratios plus one reverse w Planet
wPlanet wring carrier w -w Different ratios : one element blocked R PC = m or 2 elements maintained at the same speed wP -wPC
Clutches between 2 rotating elements Brakes between one element and the housing block
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 8 1 - Mechatronic design of an automatic gear box Bond graph model of the automatic gear box
brake 1
:housing :housing
Free wheel :PC2 brake 2 clutch 2 :P1 :m2
:Jinp :1-m2
:input :1-m1
:m1 :load
:R1 :PC1_R2 :friction Clutch1
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 9 1 - Mechatronic design of an automatic gear box Bond graph model of a clutch or a brake
:friction_coeff :pressure torque pressure
slip
Coulomb friction depending on the pressure applied on the clutch disks, defining the « limited torque »
If clutch torque < limited torque à clutch closed, all the torque transmitted If clutch torque = limited torque à clutch opened, slipping velocity
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 10 1 - Mechatronic design of an automatic gear box Decision block
Contains the schift schedule (diagram throttle position vs vehicle speed) which permits to know « when to shift »
Different programs: economical, sport, snow
When a shift is decided, the different pressures in the clutches are controlled
pressure « How to shift » : - action on only 2 clutches or 2 brakes at the same time, - control of the pressure to have a smooth shift time and no jerk in acceleration
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 11 1 - Mechatronic design of an automatic gear box Bond graph model of the vehicle
left_shaft left_wheel vehicle : AGB :Jshaft_l :Cshaft_l :Jwheel_l :Cwheel_l output torque :Mveh
:rwh_1
:diff :slope
:Idiff :rwh_2 :air_friction
:Jshaft_r :Cshaft_r :Jwheel_r :Cwheel_r right_wheel differential right_shaft
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 12 2- Clutch management and drive comfort
• Ojectives: – Reduce the well-known fore and aft oscillation of a vehicle occuring when a sudden torque variation takes place in the transmission (throttle step sollicitation) – Satisfy comfort and driving pleasure
• Means: – Define an hydraulic-electronic-mechanical actuator transforming the numerical output into pressure on the plates of the clutch – Design control laws for this electrohydraulic servovalve
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 13 3 - Thermal comfort regulation in a car interior
Usual climate control
• Try to reach and maintain the passenger compartment temperature to a specified target temperature.
à The regulator acts on the mixing flap to increase or decrease the blown air temperature. Usually, a proportional strategy is used to control the mixing flap
T T a,target e b,target Mixing Tb + Prop. CAB - Flap
Ta
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 14 3 - Thermal comfort regulation in a car interior Usual climate control
! Usually the air temperature in the compartment does not reach the target temperature.
25 25 Tcons 20 24
15 23
10 22 (°C) (°C)
Thab 5 Thab 21
0 20 Tcons
-5 19
Text = -10°C Text = 25°C -10 18 0 500 1000 1500 0 500 1000 1500 temps(s) temps(s)
heating cooling)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 15 3 - Thermal comfort regulation in a car interior Comfort strategy
• comfort : much more than only thermal comfort. Our five senses, our cerebral state, our thermal state have an influence on our comfort estimating. • thermal sensations rather than thermal comfort - a very subjective notion. • in PSA Peugeot-Citroën, quantitative scale to evaluate a thermal sensation : an integer between 1 (very cold) and 9 (very hot) sensation.
Objectives: define a regulation strategy for a climate controller for car interior, taking into account the car passenger's thermal sensations.
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 16 3 - Thermal comfort regulation in a car interior Block representation of the model
climatic driving conditions conditions
ACTUATORS
recycling flap air temperature physiological thermal mixing flap thermal walls temperature model feelings blower tension car air speed compressor on/off interior spreading air flaps model clothes activity
interior materials geometry properties
e REGULATOR SENSORS
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 17 3 - Thermal comfort regulation in a car interior Physiological model
à human body divided into seven parts called segments: the head, the trunk, the left arm, the right arm, the hands, the legs and the feet.
à head and hands segments are bare.
center bare segment center dressed segment muscle muscle fat skin fat air skin clothes
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 18 3 - Thermal comfort regulation in a car interior Physiological model
respiration
BLOOD
convection convection MEDIUM AIR AMBIENT AIR evaporation convection convection Sun OR radiation conduction conduction conduction CENTER MUSCLE FAT SKIN walls radiation Activity convection shivers
METABOLISM
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 19 3 - Thermal comfort regulation in a car interior Physiological model
the
To blood
R
Bc
1
C:C(m)
C: Cblood Td (c) 0 Td (m) 1 to the fat For each T(m) Sf segment Bc(c) 0 Bc(s) Qb (m) (m) :Qa R Bc Sf (f) Tblood (m) Sf :Qsh (m) Bc
Blood layer Muscle layer
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 20 3 - Thermal comfort regulation in a car interior Physiological mathematical model
• x state vector (order = 57) : temperature, water mass, sudation production and water partial pressure of the layers. • u input vector (size = 14) : air temperature and air speed of the ambient air close to the 7 segments. • d disturbance vector (size = 35) : ambient air humidity, sun and wall radiation on clothes and skin layers. • y output vector (size = 7) : thermal feelings of the 7 segments.
ìx& = f (x,u,d) í îy = g(x,u,d)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 21 3 - Thermal comfort regulation in a car interior Linearized physiological mathematical model
• A nominal functioning point defined as a comfortable situation for the human (air temperature=299K, air speed=0.5m/s and humidity=50%). • Two linear models containing saturations, because heat transfers are different whether the body is warm or cold.
ìx& = (bA + bA )x + B ×u + E × d + F ï c w íy = C × x + D × u ï îm < K × x < M
– b = 1 if the body is cold, and 0 if not. – F vector that results from constant thresholds due to saturations. – K matrix selecting the x components concerning the saturations.
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 22 3 - Thermal comfort regulation in a car interior Physiological model : simplifications
comparison of linear and non-linear comparison of order 57 and order 9 linear models 34.6 4.8 order 9 model 34.4
4.75 34.2 non linear model tru 34 nk 4.7 ski 33.8 linear model order 57 model n te 33.6 mp trunk sensation air temperature=22°C 4.65 er atu33.4 air humidity=50% 33.2 4.6 air speed=0.1m/s 33
32.8 4.55 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 7000 8000 time (s ) time(s)
Trunk skin temperature Trunk sensations
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 23 3 - Thermal comfort regulation in a car interior Proposed comfort strategy
Control strategy based on the thermal feelings :
• Determine the air temperature close to the head driver to ensure him a comfortable thermal feeling (level 5 in PSA scale) • Take into account the wall temperatures and the air flows in the compartment. • Compute the best air temperature target for the driver to be comfortable by using the inverted human model
In case of chilly driver, a target sensation superior to 5 can be asked for
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 24 3 - Thermal comfort regulation in a car interior Proposed comfort strategy
Ta
- Thermal Ta,target Tb,target T Inverted e Mixing b Thermal feelings Regulator Cab Human human + Flap feelings target
Wall temp. model
Air temp. model
Tair,model
Twall
Predictive control (GPC)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 25 3 - Thermal comfort regulation in a car interior Comfort strategy - Air temperature model
linear convex formulation:
a 1-a Ta = ( + )(laTout + (1- la )Tb 1+t1s 1+t 2 s
• Ta air temperature,
• Tout outside air temperature,
• Tb blown air temperature, • V convex parameter depending essentially of the blown air flow and the vehicle speed. There are two dynamic modes : a fast mode caused by the mass transfer and a slow mode caused by the thermal transfers with the outside.
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 26 3 - Thermal comfort regulation in a car interior Comfort strategy - Wall temperature model
first order model:
1 Tw = ( )(lwTout + (1- lw )Ta ) 1+t w s
• Tw the surface temperature, • Ta the air compartment temperature close to the surface.
Experiments in a wind tunnel to identify the parameters of each air temperature model and wall temperature model : several wind-tunnel air flows (for the air flow due to the vehicle speed), several outside temperatures, several blown air temperatures and flows
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 27 3 - Thermal comfort regulation in a car interior Comfort strategy
HEATING COOLING 6 9
8 4 7
6 2
thermal feeling thermal feeling 5
0 4 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 time(s) time(s) 30 45
40 20 35 10 Ta(°C) Ta(°C) 30
0 25
-10 20 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 time(s) time(s)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 28 4 – Modelling of a fuel cell system
engine
H 2 reforming Compressor U Pmax ? i
H 2 fuel Humidifier humid (natural gaz, Fuel recovering methanol) recycling air air Cell Compressor
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 29 4 – Modelling of a fuel cell Fuel cell : principle
electrolyte anode cathode O2-
H 2 O2
2- - 1 - 2- H2 +O ® H 2O + 2e 2 O2 + 2e ® O
- e load
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 30 4 – Modelling of a fuel cell Fuel cell : tubular design
mesh 1 mesh 2 a1
½ anode interconnect
fuel,i a2 fuel,o interconnect Anode canal
Anode H2Ov gaz,diffa diffusion zone producted
anode active layer cathode Reaction zone Q& electrolyte air anode electrolyte
interconnect post combustion Cathode active layer
air,diffc Cathode diffusion layer
air,e Cathode canal air,s Fuel flow (hydrogen) a 2 ½ cathode i nterconnect
a1 environment Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 31 4 – Modelling of a fuel cell Variables
Pressure: P Temperature: Fluid T Mass or molar flow rate: m& or n& Enthalpy flow & H = m&.cp.T
Chemical Molar flow rate: n& reaction Chemical potential : m
Current: i Electrical Voltage: U
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 32 FC Word BG load
U i
n& Electro-chemical reaction O 2 ,cons
n&H ,cons 2 n& & H 2 Ov,prod Qohm ,e m& air ,e MSf H MSf m& H air,e m& cathode anode gaz,e + MSf + T anode H& Cathode MSf T T air,e & electrolyte & air ,e Qohm ,c Qohm ,a canal H& canal gaz,e + T gf + interconnect Tgf Se interconnect & & Qohm ,elc Qohm ,ela Se Twf T environt wf environt
T Pext Text Pext ext MSe MSe MSe MSe
m& air ,s T air ,s m& gaz ,s T gaz ,s
Pext Text Pext T ext
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 33 n& n& H 2 ,diffa H 2,diffa m& gaz,e E n& n& H 2 Ov,diffa le H 2 Ov,diffa anode c diffusion n& & n& H 2 Ov,prod.diffa H Anode H 2Ov,prod.diffa gaz,e tr zone Anode o & active canal Qohm,ela & layer n&Ar,diffa nAr,diffa l Tgf y te & Anode Qdiffa & & Qdiffa Qpba intercon Twf & & Hgaz ,diffa H gaz,diffa nect
Q& n& m& T ohm,a H 2,cons n& gaz,s P T gaz,s H 2 Ov,prod ext ext
n&O ,cons Electrochemical 2 réaction FC Anode U i Word BG
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 34 C cnlaj & H gaz,cnlaj xi ,cnlaj , M i * P xi ,ej , M i H 2, cnlaj * P n&H , cnlaj n&H Ov ,cnlaj H 2Ov,cnlaj 2 2 Fuel supply 0 n&H sj * n&H ej 2 , P n& 2 , m& Ar ,cnlaj Ar , cnlaj gaz ,ej Pcnlaj 0 n& Ar ,sj m& gaz, sj n& Ar ,ej m& 0 gaz, ej n&H Ov ej n& 2 , H 2Ov ,sj Tcnlaj n& = m& . x . M H2 ,ej gaz ,ej H2 ,ej H2
Tcnlaj & 0 & H gaz,sj H gaz,ej
n&H Ov proddiffaj 2 , . & Qdiffaj & H gaz, diffaj n& H 2Ov,diffaj n&H diffaj 2 , & Qdpbaj n& Ar ,difafj
Anode canal anode jth anode BG J-th diffusion zone interconnect
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 35 E l e c CC caaj m t Ar , caaj Tcaaj x ou x & & i,e i,cnla ( j -1) r nAr,caaj & H nH2 Ov,caaj caaj
m n&H ,caaj * o H 2Ov, caaj 2 mH ,caaj 2 PAr,cnlaj CC cnlaj * mH 2 ,caaj PH2 ,cnlaj c 1 * T 0 n& 1 PH , cnlaj & cnlaj n& H2 ,diffaj 2 nAr,cnlaj H 2, consj n& n& H2 ,cnlaj H& m H2 ,diffaj gaz, cnlaj h H 2Ov, caaj 1 m& & 0 n& * 0 & gaz,e nH2Ov, prodj H2 Ov,diffaj 1 P * nH ,ej e H2 Ov, cnlaj 2 ou n& PH 2Ov,cnlaj n& m H2 Ov,diffaj H2 Ov,cnlaj H 2Ov, caaj P * m& m & 1 1 H 2Ov,cnlaj gaz, ( j -1) nH2 Ov,prod.diffaj n& H2 Ov,prod.diffaj 0 n& i H2Ov,ej * 0 n&Ar,ej PAr , cnlaj n& & & H2Ov,sj c nAr,diffaj nAr,sj m Ar ,caaj n& 1 R H 2,sj h 0 & 1 & & nAr,diffaj Rdiffaj & H gaz,e ou H gaz( j -1) Qdiffaj 0 a & Tcaaj Q dpbaj 0 1 1 T l Q& Q& cnlaj ohm,aj diffaj & & H gaz,diffaj H gaz,diffaj 1 1 Tcaaj r & H gaz, s e x M i,cnlaj , i ou & a H gazj & Q Pcnlaj m& ou m& c ohm,elaj gaz,s gazj 1 R t R pdcj thermj m& gaz,s ou m& gazj i P P T T o ext ou cnla ( j +1) ext ou cnla ( j +1) n environment m& T electrolyte gaz,s if last mesh or gaz,s if last mesh mesh (j+1)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 36 ì ì O2 ,cac H2 ,caa cathode TF 1 TF 1/uO 1/u n& n& 2 H2 H 2 ,cons active O2,cons layer - ÄG & Qohm,c v TF u RS H 2Ov Rc TF neF E i anode h active U pol R ì load pol H 2Ov,caa 1 layer i i h i n& ohm hdiff H 2Ov,prod electrolyte RS & Rel RS R Qohm,el diff Rc
Tcaa & Qohm,a
Electrochemical reaction
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 37 C cnlaj & H gaz,cnlaj xi ,cnlaj , M i * P xi ,ej , M i H 2, cnlaj * P n&H , cnlaj n& H 2Ov,cnlaj 2 H 2 Ov ,cnlaj 0 n&H sj * n&H ej 2 , P & 2 , Ar ,cnlaj nAr , cnlaj Pcnlaj 0 n& Ar ,sj m& gaz, sj n& Ar ,ej m& 0 gaz, ej n&H Ov ej n& 2 , H 2Ov ,sj
Tcnlaj
Tcnlaj & 0 & H gaz,sj H gaz,ej
n&H Ov proddiffaj 2 , . & Qdiffaj & H gaz, diffaj n& H 2Ov,diffaj n&H diffaj 2 , & Qdpbaj n& Ar ,difafj
interconnect anode anode j Diffusion zone j BG to Simulink
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 38 4 – Modelling of a fuel cell Results
à Complete dynamic model of the fuel cell system (no similar result in the literature, only static models)
à Simulation results validated by comparison with experimental data
à Work with PSA is running for: à Control designing : how to maximize the power delivered by the fuel cell system à Fault diagnosis
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 39 5 - Structural properties of bond graph models
z B A x G (m, J) theta LSS bs1 RSS bs 2
mw1 mw2
Cw1 bw1 Cw2 bw2
Vr1 Vr 2 Bicycle heave and pitch physical model
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 40 5 - Structural properties of bond graph models
I : m Se : Fmt
1 C : LSS C : RSS 1 0 0 1 TF TF 1 0 1 0 1
R : bs1 1 I : mw1 I : J I : mw2 1 R : bs 2
C : Cw1 C : Cw2
0 0 1 1 Bicycle heave
R : bw1 and pitch R : bw2 Sf : Vr1 Sf : Vr 2 BG model
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 41 5 - Structural properties of bond graph models Passive model
• State equation
x& = Ax + Ed
æ pI ö æ inertia impulses ö x = ç ÷ = ç ÷ è qC ø èspring displacementsø
éd1 ù éV ù é ù ê road1 ú d = ê ú d = êF ú d = ê ú 1 ë masstransfer û 2 êV ú ëd2 û ëê road2 ûú
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 42 5 - Structural properties of bond graph models Passive model
• State equation x& = Ax + Ed
• order n of a model : number of I and C elements in integral causality when a preferred integral causality is assigned to the bond graph model
• BG-rank q of the state space matrix A : number of I and C elements in derivative causality when a preferred derivative causality is assigned to the bond graph model.
• number of structurally null modes of A-matrix : number of I and C elements which have to stay in integral causality when a preferred derivative causality is assigned to the bond graph model
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 43 5 - Structural properties of bond graph models Passive model
• State equation x& = Ax + Ed
• minimum number of actuators for the model to be controllable : – If BG-rank A = n, the model is controllable with a single actuator – If BG-rank A = n-k, for the model to be controllable, k well-located actuators are needed
• minimum number of sensors for the model to be observable : – idem
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 44 5 - Structural properties of bond graph models Design of the measurement and control architecture for the active system
• definition of the control objectives – what variables to be controlled? – for what performances (dynamical or frequential criteria)? – with what strategy (pole placement, disturbance rejection, …)?
• what type of control law? – state feedback? • Is the state measurable? – output feedback?
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 45 5 - Structural properties of bond graph models Design of the measurement and control architecture for the active system
Choice here
State feedback for pole placement and rejection of the disturbance corresponding to the mass transfer due to driver actions (braking or accelerating) on the 2 velocity variables (heave and pitch)
! the 2 variables (absolute velocities) to be controlled are not measurable à an observer is needed
! We want to perform input/output decoupling à 2 control inputs are needed
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 46 5 - Structural properties of bond graph models Design of the measurement and control architecture for the active system
é y ù éVm ù à 2 outputs to be controlled : not measurable y = ê 1ú = ê ú (Df*) ê y ú êw ú § ë 2û ëê J ûú é z ù éV ù à measurement vector z = ê 1 ú = ê rel1 ú (Df) êz ú ê ú ë 2 û ëêVrel2 ûú
é u ù é F ù à 2 control inputs u = ê 1ú = ê act1ú (MSe) êu ú êF ú § ë 2û ëê act2ûú à disturbance vector
é ù measurable to be rejected d = êF ú (Se) 1 ë masstransfer û
éV ù non- measurable d = ê road1 ú (Sf) 2 êV ú ëê road2 ûú
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 47 5 - Structural properties of bond graph models Design of the measurement and control architecture for the active system
Df* : Vm I : m Se : Fmt
1 C : LSS C : RSS Df : Vs1 1 0 0 1 Df : Vs 2 TF TF 1 0 1 0 1
Df* :w J MSe : Fa2 MSe: Fa1 R : bs1 1 I : mw1 I : J I : mw2 1 R : bs 2
C : Cw1 C : Cw2
1 0 0 1
R : bw1 R : bw2 Sf : Vr1 Sf : Vr 2
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 48 Conclusion
* bond graph : language quite « strange » which needs a learning time
* Could appear difficult to implement, but what is difficult is PHYSICS
* more and more introduced in the industrial world in France (better than in the academic world!)
Rosario - 22/11/02 LAIL - Ecole Centrale de Lille 49