control of vehicular platoons: limitations and tradeoffs
TRANSCRIPT
Control of vehicular platoons:limitations and tradeoffs
Mihailo Jovanovicwww.umn.edu/∼mihailo
ELECTRICAL AND COMPUTER ENGINEERING
UNIVERSITY OF MINNESOTA
CDC-ECC’05 Workshop – December 11, 2005
M. JOVANOVIC, UMN 1
Control of vehicular platoons
• ACTIVE RESEARCH AREA FOR ≈ 40 YEARS
(Levine & Athans, Melzer & Kuo, Chu, Ioannou, Varaiya, Hedrick, Swaroop, etc.)
• SPATIO-TEMPORAL SYSTEMSsignals depend on time & discrete spatial variable n
arrays of micro-cantileversarrays of micro-mirrorsunmanned aerial vehicle formationssatellite constellations
• INTERACTIONS CAUSE COMPLEX BEHAVIOR‘string instability’ in vehicular platoons
• SPECIAL STRUCTUREevery unit has sensors and actuators
M. JOVANOVIC, UMN 2
Controller architectures: platoonsCENTRALIZED:
G0 G1 G2
K
6
?
6
?
6
?
BEST PERFORMANCEEXCESSIVE COMMUNICATION
FULLY DECENTRALIZED:
G0 G1 G2
K0 K1 K2
6
?
6
?
6
?
NOT SAFE!
LOCALIZED:
G0 G1 G2
-K0
-K1
-K2
-����
6
?
6
?
6
?
MANY POSSIBLE ARCHITECTURES
M. JOVANOVIC, UMN 3
IssuesCENTRALIZED:
G0 G1 G2
K
6
?
6
?
6
?
PERFORMANCE VS. SIZE
FUNDAMENTAL LIMITATIONS/TRADEOFFS
LOCALIZED:
G0 G1 G2
-K0
-K1
-K2
-����
6
?
6
?
6
?
IS IT ENOUGH TO LOOK ONLY
AT NEAREST NEIGHBORS?
M. JOVANOVIC, UMN 4
Outline
¶ ILL-POSEDNESS OF SEVERAL WIDELY CITED RESULTS
? small scale GAP−−→ large scale ≈ infinite
FORMULATION OF WELL-POSED CONTROL PROBLEMS
? spatially invariant theory
· PEAKING IN VEHICULAR PLATOONS
large platoonsuniform convergence rates
⇒ large transient peaks
CONTROL IN THE PRESENCE OF SATURATION
? explicit constraints on feedback gains to avoid magnitude and rate saturation
¸ CONCLUDING REMARKS
M. JOVANOVIC, UMN 5
Optimal control of vehicular platoons
• FINITE PLATOONS
Levine & Athans (LA), IEEE TAC’66Melzer & Kuo (MK1), IEEE TAC’71
• INFINITE PLATOONS
Melzer & Kuo (MK2), Automatica’71
M. JOVANOVIC, UMN 6
Control objective
DYNAMICS OF n-TH VEHICLE: xn = un
CONTROL OBJECTIVE:desired cruising velocity vd := const.inter-vehicular distance L := const.
+ COUPLING ONLY THROUGH FEEDBACK CONTROLS
ABSOLUTE DESIRED TRAJECTORYxnd(t) := vdt − nL
M. JOVANOVIC, UMN 7
Control of finite platoons
absolute position error: ψn(t) := xn(t) − vdt + nL
absolute velocity error: φn(t) := xn(t) − vd
}n ∈ {1, . . . ,M}
⇓
MK1:
[ΨΦ
]=[
0 I
0 0
] [ΨΦ
]+[
0I
]U
relative position error: ζn(t) := xn(t) − xn−1(t) + L
= ψn(t) − ψn−1(t)
}n ∈ {2, . . . ,M}
⇓
LA:
[ZΦ
]=[
0 A12
0 0
] [ZΦ
]+[
0I
]U
M. JOVANOVIC, UMN 8
Optimal control of finite platoons
LA:
[ZΦ
]=
[0 A12
0 0
] [ZΦ
]+[
0I
]U
J :=12
∫ ∞
0
(ZT (t)Z(t) + ΦT (t)Φ(t) + UT (t)U(t)
)dt
maxRe(λ{Acl}) :
0 10 20 30 40 50 60 70 80 90
−1
−0.8
−0.6
−0.4
−0.2
0
M
max
Re
( λ {
Acl
} )
M. JOVANOVIC, UMN 9
MK1:
[ΨΦ
]=[
0 I
0 0
] [ΨΦ
]+[
0I
]U
J :=12
∫ ∞
0
(M+1∑n=1
(ψn(t) − ψn−1(t))2 +M∑
n=1
(φ2n(t) + u2
n(t))
)dt
maxRe(λ{Acl}) :
0 50 100 150
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
M
max
Re
( λ {
Acl
} )
M. JOVANOVIC, UMN 10
• LEVINE & ATHANS:maxRe(λ{Acl}) :
0 10 20 30 40 50 60 70 80 90
−1
−0.8
−0.6
−0.4
−0.2
0
Mm
ax R
e (
λ {
Acl
} )
• MELZER & KUO:maxRe(λ{Acl}) :
0 50 100 150
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
M
max
Re
( λ
{ A
cl }
)
M. JOVANOVIC, UMN 11
Optimal control of infinite platoons+ GOOD APPROXIMATION OF LARGE-BUT-FINITE PLATOONS
MAIN IDEA: EXPLOIT SPATIAL INVARIANCE
LA:
[ζn
φn
]=
[0 1 − T−1
0 0
] [ζnφn
]+[
01
]un, n ∈ Z
ySPATIAL Zθ-TRANSFORM[ζθ
φθ
]=[
0 1 − e−jθ
0 0
] [ζθφθ
]+[
01
]uθ, 0 ≤ θ < 2π
+ NOT STABILIZABLE AT θ = 0
M. JOVANOVIC, UMN 12
MK2:
[ψn
φn
]=
[0 10 0
] [ψn
φn
]+[
01
]un, n ∈ Z
J :=12
∫ ∞
0
∑n∈Z
((ψn(t) − ψn−1(t))2 + φ2
n(t) + u2n(t)
)dt
ySPATIAL Zθ-TRANSFORM
Aθ =[
0 10 0
], Qθ =
[2(1 − cos θ) 0
0 1
], 0 ≤ θ < 2π
+ PAIR (Qθ, Aθ) NOT DETECTABLE AT θ = 0
+ A FIX: PENALIZE ABSOLUTE POSITION ERRORS IN J
J :=12
∫ ∞
0
∑n∈Z
(qψ2
n(t) + (ψn(t) − ψn−1(t))2 + φ2n(t) + u2
n(t))
dt
M. JOVANOVIC, UMN 13
[ψn
φn
]=
[0 10 0
] [ψn
φn
]+[
01
]un, n ∈ Z
J :=12
∫ ∞
0
∑n∈Z
(qψ2
n(t) + (ψn(t) − ψn−1(t))2 + φ2n(t) + u2
n(t))
dt
CLOSED-LOOP SPECTRUM:
q = 0: q = 1:
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -1
-0.5
0
0.5
1
Re ( λ { A cl
})
Im (
λ { A
cl }
)
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -1
-0.5
0
0.5
1
Re ( λ { A cl
})
Im (
λ {
A cl
})
M. JOVANOVIC, UMN 14
M = 20, q = 0: M = 50, q = 0:
0 10 20 30 40 500
0.5
1
1.5
2
t
ψ n(t)
0 10 20 30 40 500
0.5
1
1.5
2
t
ψ n(t)
M = 20, q = 1: M = 50, q = 1:
0 2 4 6 8 10 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
ψ n(t)
0 2 4 6 8 10 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
ψ n(t)
M. JOVANOVIC, UMN 15
‘Problematic’ initial conditions for LA & MK• LA & MK FOR INFINITE PLATOONS:
non-zero mean initial conditions cannot be driven to zero!
MK:
∑n∈Z
ψn(0) 6= 0 ⇒ limt→∞
∑n∈Z
ψn(t) 6= 0
+ MANY MODES HAVE VERY SLOW RATES OF CONVERGENCE
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -1
-0.5
0
0.5
1
Re ( λ { A cl
})
Im (
λ { A
cl }
)
M. JOVANOVIC, UMN 16
Remarks
• CONTROL OF VEHICULAR PLATOONS
Jovanovic & Bamieh, IEEE TAC’05:
? analytically showed ill-posedness of several widely cited results
? formulated well-posed control problems
SMALL SCALECAREFUL−−−−−→ LARGE SCALE
SMALL SCALE 6= LARGE SCALE ≈ INFINITE DIMENSIONAL
large-but-finite systems ⇒ large scale computations
infinite dimensional abstractionswith spatial invariance ⇒ almost analytical results
M. JOVANOVIC, UMN 17
Outline
¶ ILL-POSEDNESS OF SEVERAL WIDELY CITED RESULTS
? small scale GAP−−→ large scale ≈ infinite
FORMULATION OF WELL-POSED CONTROL PROBLEMS
? spatially invariant theory
· PEAKING IN VEHICULAR PLATOONS
large platoonsuniform convergence rates
⇒ large transient peaks
CONTROL IN THE PRESENCE OF SATURATION
? explicit constraints on feedback gains to avoid magnitude and rate saturation
¸ CONCLUDING REMARKS
M. JOVANOVIC, UMN 18
LQR design
absolute position error: ψn(t) := xn(t) − vdt + nL
absolute velocity error: φn(t) := xn(t) − vd
⇓[ψn
φn
]=[
0 10 0
] [ψn
φn
]+[
01
]un
n ∈ {0, . . . ,M − 1}
PERFORMANCE INDEX:
J :=12
∫ ∞
0
(M−1∑n=1
(ψn(t) − ψn−1(t))2 +M−1∑n=0
(ψ2n(t) + φ2
n(t) + u2n(t))
)dt
M. JOVANOVIC, UMN 19
Example
+ PEAKING: CERTAIN INITIAL CONDITIONS CAN LEAD TO LARGE CONTROLS
xn(0) = −n(L + S) ⇒ ψn(0) = −nSxn(0) = vd ⇒ φn(0) = 0
M. JOVANOVIC, UMN 20
POSITION: VELOCITY:
0 2 4 6 8 10
-25
-20
-15
-10
-5
0
t
ψ n (t)
0 2 4 6 8 10
0
2
4
6
8
10
t
φ n (t
)
CONTROL: supt un(t):
0 2 4 6 8 10−5
0
5
10
15
20
25
t
u n(t)
0 10 20 30 40 500
5
10
15
20
25
n
sup t u
n(t)
M. JOVANOVIC, UMN 21
LQR design for a platoon on a circle
[ψn
φn
]=
[0 10 0
] [ψn
φn
]+[
01
]un
=: Anξn + Bnun
SPATIALLY INVARIANT PERFORMANCE INDEX:
J :=12
∫ ∞
0
M−1∑n=0
M−1∑m=0
(ξ∗n(t)Qn−mξm(t) + u∗n(t)Rn−mum(t)) dt
MAIN IDEA: EXPLOIT SPATIAL INVARIANCE
M. JOVANOVIC, UMN 22
LARGE SCALE PROBLEM:[ψn
φn
]=[
0 10 0
] [ψn
φn
]+[
01
]un =: Anξn + Bnun
J :=12
∫ ∞
0
M−1∑n=0
M−1∑m=0
(ξ∗n(t)Qn−mξm(t) + u∗n(t)Rn−mum(t)) dt
ySPATIAL DFT
BLOCK DIAGONAL PROBLEM:[ ˙ψk
˙φk
]=[
0 10 0
][ψk
φk
]+[
01
]uk =: Akξk + Bkuk
J =√M
2
∫ ∞
0
M−1∑k=0
(ξ∗k(t)Qkξk(t) + u∗k(t)Rkuk(t)
)dt
Rk > 0, Qk :=[q11k q∗21k
q21k q22k
]≥ 0, k ∈ {0, . . . ,M − 1}
M. JOVANOVIC, UMN 23
[ψn
φn
]=
[0 10 0
] [ψn
φn
]+[
01
]un
=: Anξn + Bnun
{ψn(0) = −nS, 0 < S < L; xn(0) ≡ vd}ymagnitude saturation unless
umax ≥ S2
6(M − 1)(2M − 1) inf
k
q11k
Rk
IMPORTANT:
DETECTABILITY ⇔ q11k > 0
M. JOVANOVIC, UMN 24
Trajectory generation
¨xn = un
rn(t) := xn(t) − vdt + nL
}⇒ rn = un
un = − p2nrn − 2pnrn ⇒
rn(t) = (cn + dnt)e−pnt
rn(t) = (dn − cnpn − dnpnt)e−pnt
un(t) = (cnp2n − 2dnpn + dnp
2nt)e
−pnt
+ DERIVED EXPLICIT CONSTRAINTS ON pn TO GUARANTEE:
|rn(t)| ≤ rmax
|rn(t)| ≤ vmax
|un(t)| ≤ umax
M. JOVANOVIC, UMN 25
FUNDAMENTAL TRADEOFFS:large pn’s ⇒
{fast stabilizationsmall position overshoots
small pn’s ⇒
slow stabilizationsmall velocity overshootssmall control effort
CONTROL AROUND GENERATED TRAJECTORIES:
ηn(t) := xn(t) − vdt + nL − rn(t)
⇓
ηn = un − un =: un
⇓
un = un + un
↓ ↓from trajectory generation from e.g. LQR design
PERFECT KNOWLEDGE OF INITIAL CONDITIONS: un = un
M. JOVANOVIC, UMN 26
POSITION: VELOCITY:
0 10 20 30 40 50 60
-25
-20
-15
-10
-5
0
t
ψ n (t)
0 10 20 30 40 50 60 -0.5
0
0.5
1
1.5
2
t
φ n (t
)
CONTROL:
0 10 20 30 40 50 60−1
0
1
2
3
4
5
t
u n(t)
M. JOVANOVIC, UMN 27
RemarksJovanovic, Fowler, Bamieh, & D’Andrea, Automatica’05 (submitted):
? trajectory planing to avoid magnitude and rate peaking
large platoonsuniform convergence rates
⇒ large transient peaks
fast convergence rates ⇒ small position overshoots
slow convergence rates ⇒small velocity overshoots
small control efforts
• ONGOING/FUTURE RESEARCH
? design of distributed controllers with favorable architectures
? control under communication constraints
? limitations & tradeoffs in the control of 2D & 3D formationsflocking & swarming
M. JOVANOVIC, UMN 28
Additional info
Mihailo Jovanovicwww.umn.edu/∼mihailo