optimizing systems with conflicting objectives competing
TRANSCRIPT
Optimizing Systems with Conflicting ObjectivesCompeting for a Limited Resource
Radar waveform design, Unimodular QPUAV tracking optimization
Hans D. Mittelmann
School of Mathematical and Statistical SciencesArizona State University
AFOSR Optimization and Discrete Math Review
23 August 2019
COLRO problems Hans D. Mittelmann MATHEMATICS AND STATISTICS 1 / 36
collaborators
Shankarachary Ragi, South Dakota School of Mines & Technology
Daniel Bliss, Alex Chiriyath, Arizona State University
Edwin Chong, Colorado State University
Shawon Dey, Azam Md Ali, South Dakota School of Mines & Technology
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Outline
Waveform Design for Joint Radar-CommunicationsBackgroundWaveform Optimization MethodsNumerical Results
Unimodular Quadratic ProgramsBackgroundProblem DescriptionExisting MethodsOur MethodsNumerical Results
Ongoing ResearchCompeting Objective Optimization in Networked Swarm Systems
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Introduction
• Traditionally, wireless communications (0.3 - 3 GHz) and radar (3 - 30GHz) are spectrally separated
• Spectral congestion forcing co-existence
• Key performance factors: spectral shape of waveform, receiver design,signal decoupling strategies
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Radar: Preliminaries
Radar Transmitter
Radar Target
Transmitted signal
Waveform transmission → Scattered signal recovery → Matched filter response
• Signal delay→ Range detection• Doppler shift→ Speed detection
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Joint Radar-Comms System
Radar Target
Joint
Radar-Communications
Node
Communications
Transmitter
Radar Spectrum
Communications Spectrum
Key functions:• Sends radar pulses for target detection
• Receives a mixed signal - radar return and communications signal
• Decouples the signals
• Processes radar returns for ranging and speed
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Signal Decoupling
Successive-Interference Cancellation
TransmitRadar
Waveform
RadarChannel
CommsChannel
RemovePredicted
Return
DecodeComms
& Remove
ProcessRadarReturn
Comms InfoCommsSignal
Σ
Key step: Remove predicted radar return from the mixture
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Performance Indicators
• Communications performance: Shannon’s information rate bound
Rcom ≤ B log2
(1 +‖b‖2 Pcom
σ2int+n
)
• Radar performance: estimation rate bound [D. W. Bliss, 2014 IEEE RadarConference]
Rest ≤δ
2Tlog2
[1 +
σ2proc
σ2est
]
Spectral shape of the waveform directly influences the above per-formance indicators!
σ2int+n ∝ Brms σ2
est ∝1
Brms
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Spectral-Shape Affects Performance
−B/2 0 B/2
Sp
ectr
al
Mask
Weig
hti
ng
FrequencySNR (dB)
Radar optimal
Comms optimal
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Maximizing joint radar-communications performanceu controls the spectral-shape of the waveform!
maximizeu∈[0,1]N
[Rcom(u)]α [Rest(u)]1−α
subject to system constraints
Result: If u∗ is the optimal solution, then u∗ is pareto efficient[S. Ragi, A. Chiriyath, D. Bliss, H. Mittelmann, Optimization-Online pre-print]
Estimationrate
Commsrate
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Spectrum: α = 0 and α = 1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Frequency (Hz) 10 6
0
2
4
6
8
10
12
14
16
18
20A
mpl
itude
Frequency spectrum vs. Emphasis on radar( = 0)Emphasis on comms( = 1)
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Solver Performance
0 500 1000 1500Rest
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10R
com
10 6 Rest vs. R comm
KnitroFminconDE with 500 Iter
Increasing
= 1
= 0
[S. Ragi, A. Chiriyath, D. Bliss, H. Mittelmann, Optimization-Online pre-print]
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Solver Performance
10 -1 10 0 10 1 10 2 10 3
Execution time (sec, logscale)
0
0.2
0.4
0.6
0.8
1C
umul
ativ
e fr
eque
ncy
Empirical CDF
EvolFminconKnitro
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Outline
Waveform Design for Joint Radar-CommunicationsBackgroundWaveform Optimization MethodsNumerical Results
Unimodular Quadratic ProgramsBackgroundProblem DescriptionExisting MethodsOur MethodsNumerical Results
Ongoing ResearchCompeting Objective Optimization in Networked Swarm Systems
COLRO problems Hans D. Mittelmann MATHEMATICS AND STATISTICS 14 / 36
Monostatic Radar
• Transmits encoded pulse sequence
a(0)
a(1) a(2) a(3)
time
u(t)
• Objective: optimize c = (a(0), . . . ,a(N))T, where |a(i)| = 1 ∀i , tomaximize signal-to-noise ratio (SNR)
SNR = cHRc
where R = M−1 (ppH)∗, M = E[wwH]
• Code optimization leads to unimodular quadratic program (UQP)
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Unimodular Quadratic Program• Ω = x ∈ C, |x | = 1
1-1
-i
iΩ
• Unimodular quadratic program (UQP)
maximizes∈ΩN
sHRs
where R ∈ CN×N is a Hermitian matrix.
• Many problems in radar and wireless communications lead to UQP• UQP is an NP-hard problem
[S. Zhang, et. al., “Complex quadratic optimization and semidefinite programming," SIAM J. Optimization,2006]
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Semi-Definite Relaxation (SDR)
• UQP can also be stated as (since sHRs = tr(sHRs) = tr(RssH))
maximizeS
tr(RS)
subject to S = ssH , s ∈ ΩN .
• If rank constraint is relaxed⇒ semidefinite program (SDP)
maximizeS
tr(RS)
subject to [S]k,k = 1, k = 1, . . . ,NS is positive semidefinite.
• SDP can be solved in polynomial time
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Phase-Matching with Dominant Eigenvector
• Pick d ∈ ΩN that “phase-matches” the dominant eigenvector of R
• Time complexity: O(N3)
• Example:
If (0.2eiπ/3, 0.6eiπ/4, 0.77eiπ/5)T is the dominant eigenvector, thend = (eiπ/3, eiπ/4, eiπ/5)T
[S. Ragi, E. K. P. Chong, H. D. Mittelmann, “Heuristic methods for designing unimodular code sequences withperformance guarantees,” ICASSP 2017.]
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Performance Bound
Result (S. Ragi, E. K. P. Chong, H. D. Mittelmann, ICASSP 2017)If VD = dHRd and Vopt is the optimal objective value, then
VDVopt
≥ λN + (N − 1)λ1
λNN
N λ1 λN Bound34 6.7 81.4 0.1196 8.7 64.6 0.1493 19.5 71.6 0.286 41.6 50.8 0.8574 40.2 99.2 0.4127 3 58.4 0.09
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Greedy Strategy
• Greedy solution g = (g(1), . . . ,g(N))T
g(k) = arg maxx∈Ω
[gk−1; x ]HRk [gk−1; x ],
k = 2, . . . ,N, g(1) = 1
gk = (g(1), . . . ,g(k))T
where Rk is the k × k principle sub-matrix of R.
Example: If R =
1 2 32 4 53 5 6
, then R2 =
[1 22 4
], and R3 = R
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Performance Bound for Greedy Strategy
If the objective function is string submodular, then
gHRg ≥ (1− 1/e) maxs∈ΩN
sHRs
where (1− 1/e) ≈ 0.63
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String Submodular FunctionsMonotonic functions with diminishing returns!
Input
Output
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Is UQP objective function string submodular?
f (Ak ) = AHk Rk Ak
• f is not string-submodular⇒ UQP for any R is not string-submodular
• But f (Ak ) = AHk Rk Ak is string submodular, where R is obtained from R
via manipulating diagonal entries of R
[S. Ragi, E. K. P. Chong, H. D. Mittelmann, “Heuristic methods for designing unimodular code sequences withperformance guarantees,” ICASSP 2017.]
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Bound for greedy method
Result (S. Ragi, E. K. P. Chong, H. D. Mittelmann, ICASSP 2017)If Tr(R) ≤ Tr(R), then
gHRg ≥(
1− 1e
)(maxs∈ΩN
sHRs),
where g is the solution from the greedy method.
• Time complexity of greedy method: O(N)
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Performance Comparison
0 20 40 600
0.2
0.4
0.6
0.8
1
F(x
)
Obj Val N=10GreedyDom. Eig.Row-SwapGreedySDRPowerMethod
0 50 100 150
Objective value
0
0.2
0.4
0.6
0.8
1Obj Val N=20
0 100 200 3000
0.2
0.4
0.6
0.8
1Obj Val N=30
10 -5 10 00
0.2
0.4
0.6
0.8
1
F(x
)
Exec Time N=10
10 -4 10 -2 10 0 10 2
Execution time (sec, logscale)
0
0.2
0.4
0.6
0.8
1Exec Time N=20
10 -5 10 00
0.2
0.4
0.6
0.8
1Exec Time N=30
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Outline
Waveform Design for Joint Radar-CommunicationsBackgroundWaveform Optimization MethodsNumerical Results
Unimodular Quadratic ProgramsBackgroundProblem DescriptionExisting MethodsOur MethodsNumerical Results
Ongoing ResearchCompeting Objective Optimization in Networked Swarm Systems
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COLRO Problems
• COLRO: competing objective limited resource optimization
• COLRO problems appear naturally in many applications includingdecision making in autonomous systems
• We explore novel methods to solve COLRO problems in real-time
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UAV swarm control• Goal: control the motion of a networked swarm of UAVs while tracking a
target• Minimizing the energy costs and maximizing the tracking performance
are conflicting objectives
Target Target’s trajectory
Swarm centroid
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COLRO formulation
• Goal: optimize the motion of UAVs to maximize target trackingperformance while minimizing the network energy costs
• Decision variables: swarm centroid Ck and Gk
minGk ,Ck ,k=0,..,H−1
H−1∑k=0
E[wftrack (Gk ,Ck , χk )+
(1− w)fenergy (Gk ,Ck , χk )]
(1)
• Objective function is hard to evaluate exactly!
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COLRO cost functions
minGk ,Ck ,k=0,..,H−1
H−1∑k=0
E[wftrack (Gk ,Ck , χk )+
(1− w)fenergy (Gk ,Ck , χk )]
(2)
• ftrack measuresI benefits of data fusion between a pair of UAVs - GkI benefits of having the swarm staying close to the target - Ck
• fenergy measuresI benefits of using the communications network sparingly - Gk
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Solution Approach
• Nominal belief-state optimization - an approximate dynamic programmingapproachI Replace future noise variables with “nominal” valuesI Replace the expectation with “nominal” trajectory of the posterior distribution
into the future
• Apply receding horizon control approach
minGk ,Ck ,k=0,..,H−1
H−1∑k=0
[wftrack (Gk ,Ck , ψk )+
(1− w)fenergy (Gk ,Ck , ψk )]
(3)
where ftrack and fenergy are deterministic approximations.
• Mixed integer nonlinear program - solution is obtained via a commercialsolver Knitro
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Converting G∗k and C∗k to UAV kinematic controls
Desired centroid
Repulsive force to avoid collisions
Attractive force to form desired network
Attractive force toward centroid
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3 UAVs and 1 target (H = 6)
-1000 -500 0 500 1000 1500 2000 2500 3000
0
500
1000
1500
2000
Target trajectory
UAV trajectory
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Performance against centralized approach (3 UAVs)
0 50 100 150
Time (sec)
0
5
10
15
20
25
30
35
40
45
Tar
get l
ocat
ion
erro
r (m
)
Target tracking performance vs. centralized approach
centralized fusionapproachsensor 1sensor 2sensor 3
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Future Work
• Incorporate “belief consensus” into the COLRO frameworkI Running consensus algorithms leads to increased network energy costs, but
improves cooperativeness of the agentsI Belief consensus can be time consuming - we will develop fast heuristic
approaches
• Dealing with heterogeneous data from sensors on-board the agents, e.g.,imagery, video, and audio. We need new data fusion techniques, e.g.,fusion in feature space
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Thank you for your attention!
For papers see http://plato.asu.edu/papers.html no.s 142-152
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