1. Energy storage-aware prediction/control of mobile
systems with uncertain loads
Jonathan R. LeSage
Dissertation Defense
The University ofTexas atAustin
August 8th, 2013
Dr. Raul Longoria
Dr. MaruthiAkella
Dr. Joseph Beaman
Dr. Dongmei Chen
Dr. Dragan Djurdjanovic
Supervisor:
Committee Members:
2. Non-constant power loads imposed on battery
systems invalidate many existing methodologies
for run-time prediction/mission planning
Source: http://esrdc.caps.fsu.edu Source:Wikipedia commons
Existing limitations of literature particularly
significant for UGV/UAV field scenarios
Dynamic battery effects excited by transient
load jumps
Transient shutdown of battery protective
circuitry
Onboard battery as sole energy source
Battery rate-capacity effect [Linden 2010]
3. The nature of terramechanical interactions, driving style,
etc. increase complexity of power loads encountered in
the field
5. Summary of Contributions
An online method for transient load characterization via the self-supervised
Gaussian mixture and jump-Markov process
A method for battery remaining run-time prediction via the particle filter and
transient loading forecasts
A method for evaluating the probability of mission completion that accounts for
battery-mission dependency
An energy-aware predictive control scheme for battery transient shut-down
prevention
An experimental methodology for repeated ground vehicle discharge studies in
a stochastic terrain environment
A constructed stochastic terrain environment which physically simulates field
terrain
Theoretical Contributions:
Experimental Contributions:
6. Outline
Motivation
Mobile System Run-time Prediction
Limitations of prior art
Online transient load characterization
Particle filter-based run-time prediction
Power load design considerations
Battery-aware Mission Assessment
Probability of mission completion process
Online mission correlation estimation
ExperimentalValidation
Laboratory-based ground robot studies
Packbot Field Studies
Transient Shutdown Prevention Control
Conclusions/FutureWork
7. Prior art - Battery run-time prediction
Prediction intricacies with electrochemical batteries
Nonlinearities (discharge curve, temperature effects)
Dynamic system (rate-capacity effect, recovery effect)
8. Prior art remains particularly limited for small-
scale mobile systems with transient loads
Existing methods discount transient loading considerations
Model-based methods utilize load averaging/known future loads
Source: [Saha 2011,Saha 2012].
9. Transient load forecasts typically rely on a
priori load information
Transients via Markov process
Self-supervised online load clustering
Source: Lin 2004 Source: Clothe 2007
10. A Markov chain model can be developed online
by clustering loads as mixed Gaussian models
11. Example characterization of transient
battery load data
UGV Power Load: Cluster Selection: Transient Identification:
12. A particle filter can be used for long-term
prediction or prognostics
Advantages: nonlinear dynamics + non-Gaussian distributions
Disadvantages: computational complexity
Applications: crack propagation [Chaocaho2011], battery health [Goebel 2008],
semiconductor manufacturing tool degradation [Butler 2010], etc.
1 1: 1: 1( , ) ( | ) ( | )k p k p k k kP x z P x z P x x dx
∞
+ +
−∞
= ∫
Analytical Chapman-Kolmogorov:
( )1: 1 1
1
( | )
pN
i i
k k k k k
i
P x z w x f xδ− −
=
≈ −∑
Numerical Particle Filter:
13. The particle filter algorithm was modified to
include Markov chain load forecasts
14. A battery model was selected for online use that
balances accuracy vs. complexity
ModifiedThévenin equivalent circuit model [He 2011, LeSage 2013]
Power load-based state/output equations
0 0
1 2
( , , )
1 2
( , , )
D L D L
P D D
L D L
D
D V P V q P
R C C
q P V q P
q R q
V
q
=− + ϒ
=− + ϒ
[ ] 2 21 1
( ) 2 ( ) ( ) 4
2 2
B D D D I LV q V V V q q R P= Γ − + − Γ + Γ −
Power electronics
enforces constant load
power. [Rahn 2012]
L L BI P V=
( , , )L DP V qϒDiffusionVoltage:
Normalized Charge:
BatteryVoltage:
- nonlinear input
0Bq q q=
Notation:
- normalized charge
100%SOC q= × - state-of-charge
15. Under what loading conditions does the particle
filter outperform existing methodologies?
Monte Carlo simulation study of run-time prediction
fidelity versus power load transient magnitude
Related question: When is the additional computational
complexity of the modified particle filter justified?
16. For prediction evaluation, commonly used
accuracy/precision metrics were adopted
Relative Accuracy [Vachtsevanos 2006]: α-λ Performance [Saxena 2011]:
*( )sr t
*
*
( ) ( )1
1
( )
RT
p
t
s s
s tp s
r t r t
CRA
N r t=
−
= −
∑
Cumulative relative accuracy:
[ ]*( ) ( ) 1s st r tα λ α−∆ = +
[ ]( )sr t
α
α
π β
+
−
≥∫
1
( )
RT
p
t
s
s tp
CAL AL t
N =
= ∑
- ground truth (experimental) run-time
( )sr t - predicted run-time
Cumulative α-λ:
- α-λ window
( ) 1sAL t =
{ }( ) 0,1sAL t ∈
17. Increased load separation results in intensified
battery recovery and rate-capacity effects
1) Cluster separation ranges from 5W- 47W (4.63 to 43.5% of rated battery power)
2)Average power load enforced at 16.5W
18. Illustration of modified particle filter
predictions with transient load realizations
1) Single prediction at 40 min. of the particle filter
2) Cluster separation of 40W (37.02% of rated battery power)
19. A particle filter maintains prediction fidelity
across changing load separation
Power
Jumps
Current Jumps
(12V nominal)
Power Capacity
(max 108W)
10W 0.83A 9.26 %
20W 1.67A 18.52 %
30W 2.50A 27.78 %
40W 3.33A 37.04 %
50W 4.17A 46.30 %
Prediction
Method
Computation
Time
Percent Increase
over Peukert’s
Peukert 0.02 ± 0.004 ms -
SOC Reg. 2.47 ± 4.37 ms 122.5 %
EKF 0.19 ± 0.09 s 9.49×103 %
PF 4.55 ± 1.63 s 2.28×105 %
20. Outline
Motivation
Mobile System Run-time Prediction
Limitations of prior art
Online transient load characterization
Particle filter-based run-time prediction
Power load design considerations
Battery-aware Mission Assessment
Probability of mission completion process
Online mission correlation estimation
ExperimentalValidation
Laboratory-based ground robot studies
Packbot Field Studies
Transient Shutdown Prevention Control
Conclusions/FutureWork
21. Using online battery run-time predictions, can the
probability of completing a mission be estimated?
( ) ( , )
m
b m b m
t
P MT RT P t t dt dt
∞ ∞
−∞
> =∫ ∫
Reliability analysis:
Evaluating probability of mission
completion (PoMC):
1 ( )PoMC P MT RT=− >
Online characterization of the “mission process” enables for reliability
analysis of a mission (the probability of completing a mission)
Definition:The “mission process” denotes the statistical relationship
between battery run-time and required mission time
22. The “mission process” for unmanned ground vehicles
can be approximated as a bivariate normal
Single UGV Mission Realization:Monte Carlo analysis with jump transient loads
• Rolling resistances (f(1) = 0.01 and f(2) = 0.3)
• Throttle commands (u(1) = 10% and u(2) = 80%)
Mardia normality test
[Rencher 2002]
Bivariate normal
approximation holds.
23. A commonly used EWMA velocity estimate
was used for mission time prediction
1
ˆ ˆ( ) (1 ) kv v v vv kµ λ λ µ −
= + −
( )1 1
2
2 2
ˆ ˆ ˆ(1 ) ( )k kv v v v vv kσ λ σ λ µ− −
=− + −
EWMAVelocity Statistics: MissionTime Prediction:
ˆ
ˆ
m
v
MT
D
µ
µ
=
2 2
2 2
4
ˆ
ˆ ˆ ˆ ˆ4
ˆ4
v
v m vMT v
v
D
σ
σ σ µ σ
µ
= + +
Prediction of time required to complete an a priori specified mission given current operation
• Ignoring time required for tasks (only drive time considered here)
• Following the notation of [Sadrpour 2013]
EWMA = Exponentially-weighted moving average
25. Monte Carlo simulations demonstrate the
effectiveness of Bayesian correlation estimation
Correlated bivariate mission process with ρ = 0.7
Monte Carlo study with 500 mission realizations
Single Realization of
Correlation Estimation:
Mean Correlation
Estimation of 500 Realizations:
26. An online correlation estimate is essential
for mission reliability prediction
Probability of mission completion evaluation from UGV simulation study
Probability of mission completion as a mission decision metric
27. Outline
Motivation
Mobile System Run-time Prediction
Limitations of prior art
Online transient load characterization
Particle filter-based run-time prediction
Power load design considerations
Battery-aware Mission Assessment
Probability of mission completion process
Online mission correlation estimation
ExperimentalValidation
Laboratory-based ground robot studies
Packbot Field Studies
Transient Shutdown Prevention Control
Conclusions/FutureWork
28. A stochastic terrain test-stand was designed
and built to evaluate proposed methods
Design Objectives:
1) Repeatable discharge studies
2) Mimic field terrain demands
3) Automated ground vehicle
Test-stand Realization:
1) Varied/replaceable terrains
2) Turntable for terrain selection
3) Wall-following ground vehicle
29. A commercially available ground vehicle was
retrofitted for experimental discharge studies
National Instruments DaNIVehicle hardware:
Schematic of sensors:
• 3 A-h NiMH 12 nominal battery pack (1/2 charge)
• Differentially-driven/passive rear omniwheel
• 2x ultrasonic distance sensors for wall following
• Onboard 802.11g wireless router
Source: NI DaNI product service guide
31. Mission Process Discharge Results
N = 22 (25.33 hours of data)
Experiment #1 Data: UGV Discharge Studies:
Number of Studies: 22
Initial Battery Charge: 1500 mA-h
Run-times [min] 69.21 ± 4.34 min.
Cumulative Distance [m] 655.67 ± 41.91 m
ShutdownVoltage [V] 8.24 ± 0.22V
Peak Current [A] 2.34 to 4.02A
32. Battery run-time prediction analysis for a single
discharge study illustrates particle filter efficacy
Prediction of run-time at 20 minutes into DaNI UGV discharge test #1
Prognostic horizon analysis of DaNI UGV discharge test #1
33. Statistically significant improvement using the
proposed particle filter for 770 predictions
Comparison of cumulative scores for discharge studies (all 22 studies)
Summary of Discharge Predictions:
1. Predictions - every min. from 15 to 50 min.
2. Characterized loads prior to each prediction
3. Data horizon – 10 minutes (identified via cross-validation study)
4. 770 total predictions for each scheme
34. Experimental data used for assessment of
mission reliability for different mission distances
35. Mean reliability prediction shown to converge to
actual PoMC for all UGV discharge studies
PoMC Prediction of Single Experiment:
(DaNI UGV Experiment #1)
Mean PoMC Predictions forAll Experiments:
(DaNI UGV Experiments #1-22)
Mission 550m 625m 650m 675m 725m
No correlation 6.50% 2.13% 9.58% 21.78% 12.06%
Bayesian update 0.82% 1.14$ 3.87% 8.44% 3.83
Mean PoMC Prediction Error [%]:
Actual dist. = 641 m
36. Incorporating online correlation estimates
improves infeasible mission classification
Mean Prediction Error versus Mission Distance
(DaNI UGV Experiments #1-22)
MissionAssessment as Binary Classification
(ROC Curve for Experiments #1-22)
Binary Mission Classification [Hastie 2001]:
Type I error: Prediction of mission success
when failed (false positive)
Type II error: Prediction of failure when mission
actually finished (false negative)
True
Pos.
False
Pos.
False
Neg.
True
Neg.
Bayesian 1806 314 124 2156
Ind. 1755 365 143 2137
Analysis for 4400 predictions (Threshold = 0.5)
37. Field study - PackBot
PackBot – 17.7 kg unmanned ground vehicle
14.4V Li-ion (UBI-2590) pack – rated 6.2A-h capacity
Line-of-sight control by operator
Generic desert terrain near 29 Palms, CA
39. Particle filter run-time prediction maintains
fidelity under field loading conditions
Summary of PackBot Predictions:
1. Predictions - every min. from 10 to 50 min.
2. Characterized loads prior to each prediction
3. Data horizon – 10 minutes
4. Number of particles = 1500
Single Battery Run-time Prediction:
(Mission time = 15 minutes)
Prognostic Horizon Run-time Prediction
(Run-time predictions every minute)
ComputationalTime Comparison
Prediction
Method
Characterization
Time
Prediction
Time
EKF 0.02 ± 0.004 ms 3.65 ± 0.23 s
PF 2.47 ± 4.37 ms 30.25 ± 10.41 s
% Increase +1225% +728.76%
40. Online mission reliability analysis provides 24.2
minutes warning of PackBot mission infeasibility
PackBot differential GPS measurements
(Periods of drive and rest)
PoMC Prediction forThree Missions
1) 725 m – success (37 m surplus)
2) 750 m – success (12 m surplus)
3) 775 m – failure (13 m deficit)
41. Outline
Motivation
Mobile System Run-time Prediction
Limitations of prior art
Online transient load characterization
Particle filter-based run-time prediction
Power load design considerations
Battery-aware Mission Assessment
Probability of mission completion process
Online mission correlation estimation
ExperimentalValidation
Laboratory-based ground robot studies
Packbot Field Studies
Transient Shutdown Prevention Control
Conclusions/FutureWork
42. Predictive control can be used to delay mobile
system failure by preventing transient shutdown
43. Drivetrain and battery dynamics required for
transient shutdown prevention control
Integrated battery/drivetrain dynamic model
No slip conditions, only rolling resistance
Longitudinal locomotion control via PWM duty cycle
[ ] 2
( )2
1
( ) ( ) ( )
sgn
v m I
m x m D m
w m m m m
w t
x x m i x
v w v w
K R R
i v i q V u t i u t
r L L L L
b K
v v i g f v
m r m r
=− − + Γ − −
=− + −
0 0 0
1 1 1
( ) ( )
1 1 1
( ) ( ) ( )
D D m d
p D D D
m d
D
V V i u t i t
R C C C
q q i u t i t
q R q q
=− + +
=− Γ + +
Battery Dynamics: Drivetrain/Vehicle Dynamics:
44. Monte Carlo simulations were used to compare
transient shutdown prevention of three controllers
Three control schemes for comparison
Direct vehicle control (DVC) – uk = uref
Energy-aware model predictive (5 sec. ahead) control - uk = δuk
*+uk
0
Command-governing PID - uk = uk
(c)- ε
Termination conditions – UGV failure [Carlson 2005]
1. Battery below 9.4V (1% less than specified shutdownVSD = 9.5V)
2. Velocity (10 second moving average) below 25% expected UGV velocity
3. Quadratic program infeasibility (MPC only)
[ ]min
1
( ( )) ( ) ( )
( )
k D I d
I m
u q k V V k R i k
R i k
≤ Γ − − +
45. Illustration of the transient shutdown
prevention capabilities of predictive control
46. Model predictive control extends UGV operation
time/distance beyond other methods
Controller Operating
Time [s]
Time Increase
over DVC [%]
Cumulative
Distance [m]
Distance Increase
over DVC [%]
DVC 139.11 ± 6.90 --- 161.63 ± 8.34 ---
PID 199.20 ± 6.06 + 43.2% 198.38 ± 5.53 + 22.74%
MPC 248.16 ± 11.80 + 78.39% 214.24 ± 5.70 + 32.55%
DVC = “Direct vehicle control”
47. Summary of Contributions
An online method for transient load characterization via the self-supervised
Gaussian mixture and jump-Markov process
A method for battery remaining run-time prediction via the particle filter and
transient loading forecasts
A method for evaluating the probability of mission completion that accounts for
battery-mission dependency
An energy-aware predictive control scheme for battery transient shut-down
prevention
An experimental methodology for repeated ground vehicle discharge studies in
a stochastic terrain environment
A constructed stochastic terrain environment which physically simulates field
terrain
Theoretical Contributions:
Experimental Contributions:
48. Promising Future Avenues of Research
Battery remaining run-time prediction
Reduce computational requirements by limiting numerical (particle filter)
requirements: Gaussian sum prediction [Terejanu 2011]
Incorporation of temperature and aging effects into battery model for more
accurate field prediction
Improve prediction given a priori knowledge of terrain characteristics via
Bayesian updating of loading [Sadrpour 2013]
Online mission reliability assessment
Extend mission assessment to include non-Gaussian bivariate distributions
through copula theory [Tang 2013]
Improve mission time prediction through Bayesian task time updating and
vehicle drive time prediction
Transient shutdown prevention control
Incorporate differential steering dynamics into vehicle model for planar
movement control
Implement rolling resistance parameter estimation algorithm to implement
MPC online
49. Publications
Journal Papers
J. LeSage and R. Longoria, "Characterization of load uncertainty in unstructured terrains and
applications to battery remaining-run time prediction," Journal of Field Robotics, 2013.
J. LeSage and R. Longoria,“Mission and energy storage correlation for online mission assessment for
unmanned ground vehicles operating in unstructured environments," Journal of Field Robotics, In
Preparation.
Government Reports
J. LeSage, et al., "Modeling and Synthesis Methods for Retrofit Design of SubmarineActuation
Systems. Energy Storage for ElectricActuators," Office of Naval Research,Arlington,VA, 2011.
Conference Papers
J. LeSage and R. Longoria, "Hybrid observer design for online battery state-of-charge estimation,"
presented at the American Control Conference,Washington, DC, 2013.
J. LeSage, et al., "Power system stability analysis of synthesized complex impedance loads on an
electric ship," presented at the Electric ShipTechnologies Symposium,Alexandria,VA, 2011.
J. LeSage, et al., "Two-Port Synthesis for Retrofit Design of Electric Ship Control SurfaceActuation
Systems," presented at the Dynamic Systems and Control Conference, Boston, MA, 2010.
R. Longoria and J. LeSage, "Modeling and Requirements Formulation for Submarine Control Surface
Actuation Systems," presented at the American Society of Naval Engineers Day,Arlington,VA., 2010.
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Saha, B., E. Koshimoto, et al. (2011). Battery health management system for electric UAVs. Aerospace Conference, 2011 IEEE.
Saxena, A., J. Celaya, et al. (2008). Metrics for evaluating performance of prognostic techniques. Prognostics and Health Management, 2008. PHM 2008. International
Conference on.
Teleke, S., M. E. Baran, et al. (2010). "Optimal Control of Battery Energy Storage for Wind Farm Dispatching." 25(3): 787--794.
Zhang, F., G. Liu, et al. (2008). A battery State of Charge estimation method with extended Kalman filter. Advanced Intelligent Mechatronics, 2008. AIM 2008. IEEE/ASME
International Conference on.
Zhang, S. and K. Chatha (2009). "Near optimal battery-aware energy management." Proceedings of the 14th ACM/IEEE: 249--254.
55. The particle filter algorithm was
modified to include Markov chain loads
( ) ( ) ( ) ( ) ( )
1 1 1
1
( | ) ,
N
b b i i i
k p p k k
i
P V V w g x p+ + +
=
≈ ∑
( ) ( ) ( )
1 ,i i i
k k kx f x p+ =
Load forecast: State/load particles: Predicted PDF at k+1:
( ) ( )
1
i i
k kp p− →
- particles( )i
kx
[.]f - battery model
Load process
realization
One-step ahead prediction:
[.]g - output equation
- constant particle weights( )i
pw
56. Energy-aware MPC Implementation
LabVIEW implementation of energy-aware MPC through mathscript
Online state-of-charge estimation through EKF
Direct measurement of encoder rates and motor currents
MPC quadratic program solved through NI-coded QP solver
57. Energy-aware MPC Implementation
Constant rolling resistance load (f1 ≈ 0.0085) on “smooth cardboard”
Enforce VB > 9.5 constraint on battery voltage
Compare green MPC (with battery SOC estimation) with blue direct drive
59. Power load implicit battery model
Assuming a constant power load
Discharge curve equates with decreased terminal voltage over time
Resultantly, increase in current draw with same load
Load current forecasting introduces bias in prediction
Elimination of bias
Constrained battery model for PF prediction
Forecast power loads via the GMJM process
60. Battery Model Dynamic Equations
Unconstrained differential equations
Adding the input power constraint
Solve implicit algebraic loop
0 0
1
( ) 1
D P D D
D
D
L
V R C C
q q
V
I
q R q
= +
−
−
− Γ
( )B D I LV q V R I=Γ − −
State Dynamics Output equation
L
L
B
P
I
V
=
[ ]
[ ]
2
2 2
( )
( ) 0
1 1
( ) 2 ( ) ( ) 4
2 2
L
B D I
B
B D B I L
B D D D I L
P
V q V R
V
V V q V R P
V q V V V q q R P
=Γ − −
+ − Γ + =
= Γ − + − Γ + Γ −
0 0
1
( ) 1
D P D D L
D
D
B
V R C C P
q q R
V
q q V
= +
−
−Γ −
61. Power load constrained dynamics
Nonlinear state equations (no longer affine with input)
Nonlinear output equation
Necessary for shutdown conditions
0 0
1 2
( , , )
1 2
( , , )
D L D L
P D D
L D L
D
D V P V q P
R C C
q P V q P
q R q
V
q
=− + ϒ
=− + ϒ
1
2 2
( , , ) ( ) 2 ( ) ( ) 4L D D D D I LP V q q V V V q q R P
−
ϒ =Γ − + − Γ + Γ −
[ ] 2 21 1
( ) 2 ( ) ( ) 4
2 2
B D D D I LV q V V V q q R P= Γ − + − Γ + Γ −
62. Model Limitations
Implicit power relationship into the dynamics
Rate limiting effect on power loads (max of 456W for these parameters)
State estimation complicated with power input model (I continue to use the
current input model)
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
-10
-5
0
5
10
15
20
X: 455.7
Y: 6.75
Load Power [W]
SSTerminalVoltage[V]
Positive
Negative
63. Nonlinear Dynamics for Estimation
0 0
1
( ) 1
D P D D
D
D
L
V R C C
q q
V
I
q R q
+
−
−
− Γ
ˆˆ( )k B D I Le V q V R I= − Γ + +
State Dynamics
Output equation
Parameter Opt.Value
RP 0.00038 Ω
CD 1520.5 F
RI 0.562 Ω
RD 2590 Ω
q0 6.39 kC
64. Discretization of Dynamics
Linearization of nonlinear dynamics
Discretization for computational efficiency
(1) 0 0ˆ( )
1 0
0 ( )
P D
Dx t
R Cf
A
q q Rx
− ∂
= = −Γ∂
0
(1) 0
( )
( )
q
q
q
∂Γ
Γ =
∂
ˆ 0( )
1
1
D
x t
Cg
B
qx
∂
= = −∂
(1) 0 0( )
0
0
D P
D
T C R
A T
D q T q R
e
A e
e
∆
∆
−Γ ∆
= =
( )
( )(1) 0 0( )0
(1) 0
1
1
( )
D P
D
T C R
P
T
A
D q T q RD
R e
B e d B R
e
q
σ
σ
−∆
∆
−Γ ∆
− −
= = − Γ
∫
65. Experimental Battery Discharge
System
Application of hybrid observer to nonlinear battery system
Periodic discharge of ten cell 3A-h NiMH battery pack
Square discharge profile between 0-1 amp at 0.25 Hz
66. Optimization of battery parameters
Minimization of cost function (via parameters)
Dynamic system constraints
Constrained nonlinear prog.
( )
2
( ) ( ) ( )B m
t
J V t V tθ= −∑
( ) ( )
( ) ( )B
x f x g x u
V h x j x u
= +
= +
0 200 400 600 800 1000
1.5
2
2.5
3
3.5
4
4.5
Optimal Parameter Simulation Comparison with data
Time [sec]
TerminalVoltage[V]
Data
Optimization
Parameter Opt.Value
RP 0.00038 Ω
CD 1520.5 F
RI 0.562 Ω
q0 6.39 kC
67. Line regulating converter
DC-DC power converter - Boost converter
Step-up converter, battery voltage -> bus line
Switched-mode power supply
Assumed fast dynamics relative to loading (fast averaging)
1
1
out inV V
D
=
−
1
1
in outI I
D
=
−
( )
1
out
L
in
L
out
out
di
L q t V v
dt
dv
C i v
dt R
= −
= −
Converter dynamics
70. Extended Kalman Filter Estimation
Prediction Step (a priori estimate)
Update (a posteriori estimate after measurement)
| 1 1| 1 1
| 1 | 1 1
ˆ ˆk k D k k D k
T
k k D k k D k
x A x B u
P A P A Q
− − − −
− − −
= +
= +
| 1
ˆ( )k k k ke y h x −= −
[ ]
| 1| 1
| | 1
ˆ ˆk k k k k k
k k k D k k
x x K e
P I K B P
− −
−
= +
= −
1
| 1 | 1
T T
k k k D D k k D kK P B B P B R
−
− − = +
73. Nonlinear Observability
Lie Derivatives – rate of change
of function h as we “flow” along
manifold f
Successive Lie derivatives of
both f(x) and g(x), the state and
input dynamics respectively
0
1
2 1
( )
( )
( ) ( )
f
f
f f
L h h
h
L h f
x
h
L h f f L h f
x x x
=
∂
=
∂
∂ ∂ ∂
= = ∂ ∂ ∂
0 (0)
1
1 (1) (0)
02
2 2
1 (1) (0) (2) (0)2 2 2 2
0
( )
1 1
( )
( )
1 1
f
X f
D P D
f
D P D
L h
G L h x
C R q R
L h
x
C R q R
Γ
= = − Γ Γ
− + Γ Γ + Γ Γ
(0) 2
0
(1) 2 2
2
2 2 0
(2) 2 2 2
( )
( )
( )
( )
( )
x
q
x x
q
q
x x
q
Γ =Γ
∂Γ
Γ = ∂Γ ∂ =
∂
∂ Γ
Γ =∂ Γ ∂ =
∂
74. Nonlinear Observability (cont.)
Take Lie derivatives of affine input dynamics
In event of unobservability due to the dynamics, control effort
could force observability
0
(1) 0
01
2
(1) (2) (0)2 2
0
0
(0)
1
( ( ))
( ( ))
1 1
g
U g f D
D
g f
D P D
L
G L L h C q
q C
L L h
C R q R
= = − Γ +
+ Γ + Γ Γ
(0) 0
0
(1)
2
0
(2) 2
( )
( )
( )
q
q
q
q
q
Γ =Γ
∂Γ
Γ =
∂
∂ Γ
Γ =
∂
75. Observability Conditions
Gradient of Lie Derivatives must be full rank for observability
Weakly observable for all x, other than when q = 0
(1)
2
(1) (0) (2) (0)
0
3
(1) (2) (1) (0) (3) (0)2 2 2 2
0
1
1 1
1 1
4
X
D P D
D P D
G
C R q R
C R q R
− Γ
∇= − Γ Γ + Γ Γ
− Γ + Γ Γ Γ + Γ Γ
(2)
0
(3) (0) (2) (1)2
0
0 0
1
0
1
0 3
U
D
G
q
q R
∇= − Γ
Γ Γ + Γ Γ
76. Bivariate Mission Process
( ) ( )11 1
( , ) exp
2 2
T
b mP t t
π
−
= − − −
t μ Σ t μ
Σ
[ ]T
b mt t=t
[ ]T
RT MTµ µ=μ
2
2
ˆ
ˆ
RT k RT MT
k RT MT MT
σ ρ σ σ
ρ σ σ σ
=
Σ
( ) ( , )
m
b m b m
t
P MT RT P t t dt dt
∞ ∞
−∞
> =∫ ∫
Multivariate normal mathematical notation:
Mission reliability analysis:
Evaluating probability of mission completion (PoMC)
1 ( )PoMC P MT RT=− >
78. Determining MT and RT Correlation
Online
Stochastic process for MT and RT remains complex
Updating mission parameters/objectives online
Vehicle terramechanical interactions complex for varied terrains
Operating at different vehicle regions of efficiencies
A priori unknown transient load profiles
Energy storage transient shutdown characteristics and dynamic load effects
Using successive predictions, we “sample” the stochastic process
One potential estimate of correlation (ρ) – Pearson coefficient
95% confidence interval bounds on Pearson estimate (using Fisher r to z)
( )( )
( ) ( )
2 2
i ii
i ii i
x x y y
r
x x y y
− −
=
− −
∑
∑ ∑
95%1 1
ln
2 1 3
r
z
r n
σ
±
+
= ± − −
2
2
1
1
z
z
e
r
e
σ
±
±
±
−
=
+
79. Bayesian Correlation Estimation
Pearson requires ample samples (or predictions) for
accuracy
With 100 samples, 0.2 to -0.2 correlation not significantly difference
from 0 correlation
Bayesian inference to determine ρ online with no a priori
information (other than 0 < ρ < 1)
Assume means/standard deviations known from
predictions
Performing a Bayesian inference:
1: 1: 1: 1:
1: 1:
1
( | , ) ( , | ) ( )
( , )
i i i i
i i
P x y P x y P
P x y
ρ ρ ρ=
Uninformative (objective) prior
Marginal likelihood (typically ignored if using particles)
Sampling distribution (likelihood)
1: 1: 1: 1:( | , ) ( , | ) ( )i i i iP x y P x y Pρ ρ ρ∝
80. Bayesian Correlation Estimation
Sampling distribution (likelihood of each observation)
For all observations
Simplifying sampling distribution
( )
2
1: 1: 2
( , | ) 2 1 exp
2 1
n
ii
i i x y
Q
P x y ρ πσ σ ρ
ρ
−
= − −
−
∑
( ) ( ) ( )( )
22
1: 1: 1:1:
2 2
2
2
i y i x i yi x
ii i i i
x y x y
i x y xyi
y x yx
Q
Q Q Q Q
µ µ µµ
ρ
σ σ σ σ
ρ
− − −−
= + −
= + −
∑ ∑ ∑ ∑
∑
( )
2
1: 1: 2
2
( , | ) 2 1 exp
2 1
n
x y xy
i i x y
Q Q Q
P x y
ρ
ρ πσ σ ρ
ρ
− + − = − −
−
81. Bayesian Correlation Estimation
What about an unbiased prior distribution?
Some suggest uniform distribution [-1,1] (Fosdick 2012)
Jeffreys prior proven to be unbiased (based on Fisher information) (Wang 2012)
Combining terms: ( )
3
2 2( ) 1x yP ρ σ σ ρ
−
∝ −
1: 1: 1: 1:( | , ) ( , | ) ( )i i i iP x y P x y Pρ ρ ρ∝
( )
( )
3
2 22
1: 1: 2
2
( | , ) 1 2 1 exp
2 1
n
x y xy
i i x y x y
Q Q Q
P x y
ρ
ρ σ σ ρ πσ σ ρ
ρ
−− + − ∝ − − −
−
[ ]
( )
( )
1
31 22
1: 1: 2
2
( | , ) 2 1 exp
2 1
nnn x y xy
i i x y
Q Q Q
P x y
ρ
ρ π σ σ ρ
ρ
− +− −−
+ − ∝ − − −
( )
( )
1
3
22
1: 1: 2
2
( | , ) 1 exp
2 1
n
x y xy
i i
Q Q Q
P x y
ρ
ρ ρ
ρ
− + + − ∝ − −
−
82. Bayesian Correlation Estimation
Correlation inference via uniform particle sampling
( )
( )
1
3
22
1: 1: 2
2
( | , ) 1 exp
2 1
n
x y xy
i i
Q Q Q
P x y
ρ
ρ ρ
ρ
− + + − ∝ − −
−
5 10 15 20 25
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Number of Samples [#]
CorrelationCoefficient
-1
-0.5
0
0.5
1
5
10
15
20
25
0
2
4
6
8
x 10
-3
Correlation EstimateNumber of Sample [#]
PDF
Actual
Bayesian
84. A stochastic terrain test-stand was designed
and built to evaluate proposed methods
Stochastic terrain test-stand design
Repeatable discharge studies on stochastic terrain
Charging station provides for uniform initial charge (1.5 A-h)
Three terrains with distinct terramechanical properties
Loose Gravel – deformable terrain, induces slip
Rough bumps – increased rolling resistance/slow navigation
Incline tile – elevated grade resistance
Turntable assembly selects terrain
Terrain selected via Markov process
Transition matrix selected to mimic field terrain
1k k ijX X T−= ( )k kx X ω=
Compute terrain
probability:
Weighted random
number realization:
85. Run-time Prediction Fidelity for 22
discharge studies
Comparison of cumulative scores for discharge studies (all 22 studies)
Prediction
Method
Computation
Time
Percent Increase
over Peukert’s
Peukert 0.02 ± 0.004 ms -
SOC Reg. 2.47 ± 4.37 ms 122.5 %
EKF 0.19 ± 0.09 s 9.49×103 %
PF 4.55 ± 1.63 s 2.28×105 %
ComputationalTime ComparisonSummary of Discharge Predictions:
1. Predictions - every min. from 15 to 50 min.
2. Characterized loads prior to each prediction
3. Data horizon – 10 minutes
4. 770 total predictions for each scheme
Cumulative α-λ coefficients [Saxena 2008]:
• α = 0.1 - α-λ window modifier
• β = 0.3 - PDF area required in window
86. Mission process
Onboard available energy depends on mission operation
Rate-capacity effect decreases apparent energy for high current operation
Recovery effect depends on transient power demands
Mission time and battery run-time correlated processes
Source: NCGIA Website
87. A Bayesian method for Online Correlation
Estimation was implemented
Perform Bayesian inference to update estimate of correlation [Fosdick 2012]
Unbiased (uninformative) prior distribution → Jeffery’s prior [Wang 2012]
( ) ( )1: 1:
ˆ ˆ| | ( )k kP P Pρ ρ ρ∝μ μ1: 1:
1:
1
ˆ ˆ( | ) ( | ) ( )
ˆ( )
k k
k
P P P
P
ρ ρ ρ=μ μ
μ
( )
( )
1 ( ) ( ) ( )3
22 1: 1: 1:
1: 2
2
ˆ( | ) 1 exp
2 1
rt mt ck
k k k
k
Q Q Q
P
ρ
ρ ρ
ρ
− + + − ∝ − −
−
μ
1:
ˆ ˆargmax ( | )k kP
ρ
ρ ρ= μ
Correlation Estimate:
88. Model predictive control with constraints
1. Generate trajectories for
linearization
2. Linearize plant about x0 trajectory
3. Solve the quadratic program to find
the optimal control trajectory
4. Apply optimal control at time, k
90. Mobile Energy Systems
Global market demand increase/emerging applications
Consumer electronics (up ~100% by 2020) [Kim 2012]
Grid energy storage systems [Masaud 2010]
Electric vehicles (27% of global battery demand by 2020) [Kim 2012]
Mobile robotics, UAVs, ROVs, etc. (31% US military aircraft) [Gertler 2012]
Li-ion demand (5.4 GWh in 2011 to 24.2 GWh in 2015) [Jaffe 2011]
How much longer can I operate?
What can be done (within mission parameters)?
Remove?
91. Uncertain Environments
Limited source applications in uncertain environments
Mobile robotics – Pathfinder power range (2.5 - 24.9W) [Liu 2001]
Electric vehicles – urban/highway driving and idling [Lin 2004]
Emergency backup power such as electric ship vital systems
Existing remaining run time (RRT) prediction techniques
Constant current loads [Doeffel 2006, Kerasiotis 2010]
A priori known load profile [Rakhmatov 2003, Chen 2006, Saha 2011]
Source: http://esrdc.caps.fsu.eduSource:Wikipedia commons
Remove?
![Non-constant power loads imposed on battery
systems invalidate many existing methodologies
for run-time prediction/mission planning
Source: http://esrdc.caps.fsu.edu Source:Wikipedia commons
Existing limitations of literature particularly
significant for UGV/UAV field scenarios
Dynamic battery effects excited by transient
load jumps
Transient shutdown of battery protective
circuitry
Onboard battery as sole energy source
Battery rate-capacity effect [Linden 2010]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)