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AAE 666 Final Project. Disturbance Gain Analysis of Electric Drive System on Wheelchair Chuck Sullivan 4/30/2005. Disturbance Gain Estimation for Electric Wheel Chair Drive. Background Electric Drive propulsion for wheel chair, medical application by Delphi Corporation (Kokomo IN, Flint MI) - PowerPoint PPT Presentation
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04/22/23 CJS AAE 666 Final Project 1
AAE 666 Final Project
Disturbance Gain Analysis of Electric Drive System on Wheelchair
Chuck Sullivan4/30/2005
04/22/23 CJS AAE 666 Final Project 2
Disturbance Gain Estimation for Electric Wheel Chair Drive
Background
Electric Drive propulsion for wheel chair, medical application by Delphi Corporation (Kokomo IN, Flint MI)
Chair consists of :Wheel chair chassis24V battery, electric drive motor on each rear wheel (2)Hand controller (joystick)Puff and Sip controllerCentral controller (motor controller, accessory controls)
Electric propulsionDC motors and drivesControl of dc motors
04/22/23 CJS AAE 666 Final Project 3
Disturbance Gain Estimation for Electric Wheel Chair Drive
Motivation
2 main electric drive modes for motorsSpeed and current control with speed sensorsIR drop compensation for operation without speed sensors
Drivability in both modes is both a “quality of feel” and safety issue
IR drop compensation is target of stability and drivability (no speed sensor)
Stability and steady state gain due to driver commandDisturbance rejection capability (obstacles , incline/decline surfaces)
04/22/23 CJS AAE 666 Final Project 4
Disturbance Gain Estimation for Electric Wheel Chair Drive
Objective
Analyze disturbance rejection capability of wheel chair under IR Drop Compensation control
Application and demonstration of both linear and non-linear analysis tools presented in class
Analysis techniques: Disturbance gain from estimation from Lyapunov equation Disturbance gain from through simulation of state space representation LMI characterization of gain Comparison of these techniques to detailed model simulation
1H
L1H
04/22/23 CJS AAE 666 Final Project 5
Disturbance Gain Estimation for Electric Wheel Chair Drive
System Representation
DC machine system relationships (open loop)
Using , after simplification:
LVw
LKi
Lr
ti
JTi
JKw
JB
tw
ar
va
aa
La
Tr
mr
dddd
aTe iKT
04/22/23 CJS AAE 666 Final Project 6
Disturbance Gain Estimation for Electric Wheel Chair Drive
System Representation
With IR drop compensation
Using into previous equations:
a
rr
*v
L
a
r
av
vm
a
r
iw
01wy
wL
KV
VT
00
0J1
iw
L)1(r
LK
JK
JB
tit
w
r
kdddd
aa*
va riwKVr
Tv KK
04/22/23 CJS AAE 666 Final Project 7
Disturbance Gain Estimation for Electric Wheel Chair Drive
System Parametersradius=.178 mGR=15.46Jmot=.00106 kg*m^2J=GR^2*Jmot kg*m^2Bmot=.46/1000 Nm/rpmBtot=GR^2*Bmot = 1.05 Nm/(rad/sec)Mrider=100Mchair=40Mass=Mrider+Mchair = 140 kgJgear=0Jm=Mass*radius^2 = 4.426 kgm^2Jtotal=J+Jgear+Jm/2 = 2.466 kgm^2Kv=.059 Nm/A (V/(rad/sec))R=.24 ohmfactor=.8Rest=factor*RL=.0002
04/22/23 CJS AAE 666 Final Project 8
Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper BoundEstimation For convolution system where impulse response “H” is L1,
system is Lp stable with (AAE 666 Notes, Corless, p225)
y = w (angular speed of wheel) u = TL (load torque) = disturbance H = transfer function y/u (output-to-noise)
For standard state space representation (A,B,C,D), an upper bound for (Corless, notes):
Where is any scalar for which is A.S., and
(Lyapunov Equation)
uHy
1
C'CS2
1)H( 21
0 IA
0BB'S2A'SAS
1H
1H
04/22/23 CJS AAE 666 Final Project 9
Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper BoundEstimationR_est is estimated armature resistance in control algorithm
With R_actual fixed and known, assess disturbance gain due to inaccuracy of R_est during IR Drop Compensation control operation
Strategy for analysis: Choose various % error values for R_est For each value of R_est, apply , check ( ) Solve Lyapunov equation for = disturbance gain Minimization problem: choose such that Lyapunov equation is
feasible and is minimized.
0 IA 0)Re(
S 1H
01
H
1H
04/22/23 CJS AAE 666 Final Project 10
Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper BoundEstimationExample, norm vs. for R_est = .95 R_actual 0
1H
0
1H
uHy
1
1H
04/22/23 CJS AAE 666 Final Project 11
Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper BoundEstimation
Minimization results:
R_est H1_norm
0.99 0.0487
0.96 0.017
0.95 0.0166
0.9 0.0293
0.8 0.0583
0.6 0.1159
0.4 0.1727
0.2 0.2288
0 0.2843
uHy
1
1H
04/22/23 CJS AAE 666 Final Project 12
Disturbance Gain Estimation for Electric Wheel Chair Drive
“True” y = w (angular speed of wheel) u = TL (load torque) = disturbance as input H = transfer function y/u (output-to-noise)
An alternate approach to find an upper bound for(From time simulation):
uHy
1
notes) course 224, (page H(t)BCeCx(t)y(t)
x(0)ex(t)
CxyBx(0)
Axtx
At
At
dd
1tHη)(lim
0η(0)
ytη
dd
1H
1H
04/22/23 CJS AAE 666 Final Project 13
Disturbance Gain Estimation for Electric Wheel Chair Drive
“True” Comparison to Lyapunov equation technique forStrategy for analysis:
Choose various % error values for R_est For each value of R_est, simulate alternate state space in Simulink Steady state output value = 1
H
1H
1H
04/22/23 CJS AAE 666 Final Project 14
Disturbance Gain Estimation for Electric Wheel Chair Drive
“True” Time simulation vs. Lyapunov minimization:
uHy
1
R_estH1_normLyapunov
H1_normSimulation
0.99 0.0487 0.0439
0.96 0.017 0.01619
0.95 0.0166 0.01629
0.9 0.0293 0.0292
0.8 0.0583 0.0583
0.6 0.1159 0.1159
0.4 0.1727 0.1727
0.2 0.2288 0.2288
0 0.2843 0.2843
1H
04/22/23 CJS AAE 666 Final Project 15
Disturbance Gain Estimation for Electric Wheel Chair Drive
LMI Characterization of L gain (chapter 18, AAE 666 Notes, Corless) y = w (angular speed of wheel) u = TL (load torque) = disturbance as input Non-linear model (motor armature resistance Vs temperature)
Background: suppose symmetric P 0, scalars 0, 1,2 0 such that
(18.7, 18.8 AAE 666 Notes, Corless)
Then, (18.9, 18.10 AAE 666 Notes,
Corless)
0DD'I-CD'
DC'CC'PA-
0I2α-PB'
PBP2PA'PA
2
1
uγβy0t
)μ(μγ
uγe)Px(xy(t)
21
21
αt21
00
04/22/23 CJS AAE 666 Final Project 16
Disturbance Gain Estimation for Electric Wheel Chair Drive
LMI Characterization of L gain (chapter 18 Notes)Approach
u = TL (load torque) = disturbance as input Take x0 = 0, disturbance gain normalized around steady state
equilibrium point For this application, D=0 LMI applied to non-linear system (temperature effects modeled)
Now, effect of temperature on motor armature resistance:
For 25T 125:
1A
ΔAbA0A2ΔAaA0A1
bΨ(x)aΨ(x)ΔAA0A(x)
25))(T.00385(1RR 25
2A
04/22/23 CJS AAE 666 Final Project 17
Disturbance Gain Estimation for Electric Wheel Chair Drive
LMI Characterization of L gain (chapter 18 Notes)
(continued….) Finally
Then,0
0I-00CC'PA-
0I2α-PB'
PBP2P'APA
0I2α-PB'
PBP2P'APA
2
1
22
1
11
uγβy0t
)μ(μγ
uγe)Px(xy(t)
21
11
αt21
00
0D
04/22/23 CJS AAE 666 Final Project 18
Disturbance Gain Estimation for Electric Wheel Chair Drive
LMI Characterization of L gain (chapter 18 Notes)Using LMI Toolbox
Fix R_est, determine A1,A2 (i.e. Actual R varies with temp.) Adjust to minimize
Results:
uγy 0x0
04/22/23 CJS AAE 666 Final Project 19
Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensationy = w (angular speed of wheel)u = TL (load torque) = disturbance as inputEstablish steady state operation, then apply load, quantify change
in speed
04/22/23 CJS AAE 666 Final Project 20
Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation
04/22/23 CJS AAE 666 Final Project 21
Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation Results:
uHy
1
Load Torque (Disturbance Response)
0.00000.05000.10000.15000.20000.25000.30000.35000.4000
0.0000 0.5000 1.0000 1.5000
R_estimated (%)
Gai
n
H1_normLyapunovH1_normSimulation
H1 norm time simulation
R_estH1_normLyapunov
H1_normSimulation
H1 norm time simulation
0.9900 0.0487 0.0439 #DIV/0!0.9600 0.0170 0.0162 0.0019440.9500 0.0166 0.0163 0.0148610.9000 0.0293 0.0292 0.0297220.8000 0.0583 0.0583 0.0590560.6000 0.1159 0.1159 0.1748060.4000 0.1727 0.1727 0.2237220.2000 0.2288 0.2288 0.2879890.0000 0.2843 0.2843 0.346589
1H
04/22/23 CJS AAE 666 Final Project 22
Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation L gain Results (Vs LMI)
R_estimate fixed at .95 R_actual @ 25C R_actual simulated at 25C, 125C R_estimate = .66 R_actual @ 125C
uγy
R_est
Gamma: LMI Temp Effect on R_act
Gamma: Time simulation Temp Effect on R_act
0.9500 .126 0.075944
04/22/23 CJS AAE 666 Final Project 23
Disturbance Gain Estimation for Electric Wheel Chair Drive
SummaryDemonstrating a few different analysis techniques from class, the
disturbance gain was characterized for IR Drop Compensation control on an electrically driven wheel chair.
Disturbance was treated as input, and disturbance gain Vs R_estimation inaccuracy was analyzed using:
estimation using Lyapunov equation and minimization:
estimation from state space simulation
LMI characterization of gain of non-linear system System time simulation
L
C'CS2
1H1 0BB'S2A'SAS
1H
1H
04/22/23 CJS AAE 666 Final Project 24
Disturbance Gain Estimation for Electric Wheel Chair Drive
Summary (cont.)
System exhibits good disturbance rejection, even for very inaccurate estimation of R_armature
Methods showed similar trends and values, as disturbance gain was minimized for more accurate R_estimate values (near 95% of actual armature resistance)
Assuming system model is complete and accurate estimation methods (Vs simulation) proved viable but with some measurable deviation (future investigation?)
Various techniques to estimate disturbance gain demonstrated decent correlation