aae 666 final project

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


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



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)



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



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



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



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



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



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



Upper BoundEstimationExample, norm vs. for R_est = .95 R_actual 0

1H

0

1H

uHy

1

1H



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



“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



“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



“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



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



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



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



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



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



Time Simulation of IR Drop Compensation



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



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



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



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

aae 666 final project

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