1. Disturbance Gain Analysis of Electric Drive System on Wheelchair
AAE 666 Final Project
Disturbance Gain Analysis of Electric Drive System on Wheelchair
Chuck Sullivan
4/30/2005
11/28/2018
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 chassis
24V battery, electric drive motor on each rear wheel (2)
Hand controller (joystick)
Puff and Sip controller
Central controller (motor controller, accessory controls)
Electric propulsion
DC motors and drives
Control of dc motors
11/28/2018
CJS AAE 666 Final Project

3. Disturbance Gain Estimation for Electric Wheel Chair Drive
Motivation
2 main electric drive modes for motors
Speed and current control with speed sensors
IR 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 command
Disturbance rejection capability (obstacles , incline/decline surfaces)
11/28/2018
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
11/28/2018
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:
11/28/2018
CJS AAE 666 Final Project

6. Disturbance Gain Estimation for Electric Wheel Chair Drive
System Representation
With IR drop compensation
Using into previous equations:
11/28/2018
CJS AAE 666 Final Project

7. Disturbance Gain Estimation for Electric Wheel Chair Drive
System Parameters
radius=.178 m
GR=15.46
Jmot= kg*m^2
J=GR^2*Jmot kg*m^2
Bmot=.46/1000 Nm/rpm
Btot=GR^2*Bmot = 1.05 Nm/(rad/sec)
Mrider=100
Mchair=40
Mass=Mrider+Mchair = 140 kg
Jgear=0
Jm=Mass*radius^2 = kgm^2
Jtotal=J+Jgear+Jm/2 = kgm^2
Kv=.059 Nm/A (V/(rad/sec))
R=.24 ohm
factor=.8
Rest=factor*R
L=.0002
11/28/2018
CJS AAE 666 Final Project

8. Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper Bound Estimation
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)
11/28/2018
CJS AAE 666 Final Project

9. Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper Bound Estimation
R_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.
11/28/2018
CJS AAE 666 Final Project

10. Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper Bound Estimation
Example, norm vs for R_est = .95 R_actual
11/28/2018
CJS AAE 666 Final Project

11. Disturbance Gain Estimation for Electric Wheel Chair Drive
Upper Bound Estimation
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
0.2843
11/28/2018
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):
11/28/2018
CJS AAE 666 Final Project

13. Disturbance Gain Estimation for Electric Wheel Chair Drive
“True”
Comparison to Lyapunov equation technique for
Strategy 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 =
11/28/2018
CJS AAE 666 Final Project

14. Disturbance Gain Estimation for Electric Wheel Chair Drive
“True”
Time simulation vs. Lyapunov minimization:
R_est
H1_norm Lyapunov
H1_norm Simulation
0.99
0.0487
0.0439
0.96
0.017
0.95
0.0166
0.9
0.0293
0.0292
0.8
0.0583
0.6
0.1159
0.4
0.1727
0.2
0.2288
0.2843
11/28/2018
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, AAE 666 Notes, Corless)
11/28/2018
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 25T 125:
11/28/2018
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,
11/28/2018
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:
11/28/2018
CJS AAE 666 Final Project

19. Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation
y = w (angular speed of wheel)
u = TL (load torque) = disturbance as input
Establish steady state operation, then apply load, quantify change in speed
11/28/2018
CJS AAE 666 Final Project

20. Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation
11/28/2018
CJS AAE 666 Final Project

21. Disturbance Gain Estimation for Electric Wheel Chair Drive
Time Simulation of IR Drop Compensation
Results:
11/28/2018
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 C
R_actual simulated at 25C, 125C
R_estimate = C
R_est
Gamma: LMI Temp Effect on R_act
Gamma: Time simulation Temp Effect on R_act
0.9500
 .126
11/28/2018
CJS AAE 666 Final Project

23. Disturbance Gain Estimation for Electric Wheel Chair Drive
Summary
Demonstrating 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
11/28/2018
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
11/28/2018
CJS AAE 666 Final Project