ITS World Congress :: Vienna, Oct 2012
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Transcript of ITS World Congress :: Vienna, Oct 2012
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Iterative adaptive compensation of modeling uncertainties in emission control of freeway
traffic
József K. Tar, Imre J. Rudas,László Nádai, Teréz A. Várkonyi
Óbuda University, H-1034 Budapest, Bécsi út 96/B, Hungary
19th ITS World Congress, 22-26 October 2012, Vienna, Austria
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Motivations
It is expedient to find less complicated design methodology thatdoes not need „artistic skills” by the designer;• contains little number of arbitrary parameters and more easy to
be „automated” by standardized procedures;• Doesn’t need exact analytical system model (Freeway Traffic).
• Lyapunov’s 2nd Method is a very sophisticated and complicatedmodel-based technique for designing globally (sometimes asymptotically) stable controllers.
• Its use is dubious if the analytical form of the available system model is ambiguous besides the parameter uncertainties.
• It needs designers well skilled in Math. Finding an appropriate Lyapunov function is an art.
• It works with a great number of non-optimally set control parameters. Parameter optimization may happen via ample computations (e.g. by GAs or Evolutionary Computation)
Aims
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Macroscopic Dynamic Model of Freeway Traffic
• Deals with average traffic data as vehicle density, mean velocity; No information is contained for individual vehicles.
• The analytical model is based on the flow of compressible fluid and discretization of the spatial variable.
• Conservation of the vehicles is guaranteed by the continuity equation.
• The model’s form is dubious (backward, forward or central differences may be used for discretization).
• The model’s parameters may depend on various circumstances, they are only of approximate nature.
• The resulting model necessarily provides highly nonlinear coupled differential equations for the variables of the individual segments;
• The segments are embedded in an environment that determines the ingress flow rates and they must „swallow” their inputs.
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0 1 2 3 4 5
0
v0
q0:= 0v0
1
v1
0
v0
2
v2
3
v3
4
v4
5
v5
r2 additional input
L LL L L L
Discretized Dynamic Model of Freeway Traffic
output0=input1
output1=input2
output2=input3
output3=input4
output4=input5
ii outputinputLdtd
i
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The Continuity Equation prescribes:
Dynamic equations for one-sided discretization: (v4 is assumed to be constant):
Papageorgiou’s model
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Dynamic equations for central discretization: (v4, v5 are assumed to be constant, 5 is directlydetermined by the last equation):
Stationary solution: obtained for constant environmental data and r2 control signal. If it is stable, instead dynamic control the idea of Quasi-Stationary Process in Classical Thermodynamics can be applied for obtaining a simple adaptive control: after a small jump in r2 the new state stabilizes itself. Adaptive iterative control is possible!
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Finding the Stationary Solutions:
• By the use of MS EXCEL, Visual Basic, and SOLVER it is veryeasy to prescribe zero value for one of the derivatives while theother zero derivatives can be prescribed at constraints.
• By using Lagrange Multipliers and Reduced Gradient it is easyto find the solutions.
• It was found that simple 3rd order polynomial approximation inq0:=0v0 and r2 the stationary solutions can be well described.
• So we have only a few coefficients in the polynomial approximation that can be copied into a SCILAB/SCICOS simulator program as common text for further simulations and developing of the iterative adaptive controller.
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ddr rfrer
dre Calculatedexcitation
Desiredresponse
Rough systemmodel
Realizedresponse
Actual System’s Response
Unknown function with known input and measurable
output values.
The Adaptive Control Approach Developed at Óbuda University
rd rr
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Precise Realizatio
n
Introduction of „Robust Transformations” to create localdeformations
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Good fixed point: if f(r*)=rd then G(r*;rd)=r*False fixed point: G(-K;rd)=-K
KrrfABKrrrG dd tanh1;
d
drrfAB
rrfArfBAKrG
tanh1cosh
'2
A possibility is the utilization of the strongly saturated natureof sigmoid functions with (0)=0 for SISO systems
1' rfBAKrrG
The derivative easily can be made small enough in the fixed point to obtain convergent iteration:
For this purpose the manipulation of three adaptive control parameters (A,B,K) is needed. The design of the control parameters can be done in a few simple steps via simulation:
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Convergence Issues: Contractive Mappings in Banach Spaces
Seeking the Fixed Point of the function g(x)via iteration in the case of a contractive mapping:
The fixed point u is the limit of the iteration:
abKdttgagbg,dttgagbg,Kxgb
a
b
a
1
0
...
01
21111
nn
nnnnnnnn
xxK
xgxgKxxKxgxgxx
01
1
nnn
nnnnnn
uxxuK
uxxguguxxuguxxuguug
Cauchy Sequence in a Complete Metric Space! It is convergent to some
value u!
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Design of the control parameters
:
• Design a common non-adaptive controller for the available approximate stationary dynamic model and record the responses;• Let• Give a little negative contribution to 1 by setting a small A!
1,100max
BrK
5.0rfKA
The main factors determining the emission of CO2 Controlling the overall emission rate of exhaust fumes at two segments of a road.
Drag force for 1 carPower cons. for 1 car
Power cons. for L cars in thesegment for an average,unknown drag coeff.
Emission Factor:
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The control task with contradictionMain health issues: • Pollution of hazardous materials and that causing greenhouse effects: mainly influenced by the emission factor Ef;• Damages caused by accidents, collisions: mainly depend on the velocity and vehicle density: for higher speeds lower vehicle densities are desirable; in our case the control of the density seems to be realistic;Contradiction:Our sytem is „underactuated”: we have a single control signal r2, and we wish to simultaneously control Ef and .Contradiction resolution: Find a compromise between the simultaeously prescribed Ef and values by controlling the compound „compromise factor” ]1,0[,1: fscompr EKf
Scaling factor bringing KsEf to the same order of magnitude as
Significance factor
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Simulations for segment 3 (SCILAB/SCICOS)
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Non-adaptive tracking of 3 [vehicle/km] vs. time [h] (=0)
[vehicle/km]
2.0
10
15
1.51.00.50.0
20
25
5
rho3 nominal and rho3 versus time [h]
[vehicle/km]
0.0
5
4
3
2.01.51.0
2
1
00.5
Tracking error versus time [h]
2.01.51.00.50.0
500
450
400
350
300
[vehicle/h]
q0 versus time [h]
[km/h]
2.0
106108
1.51.00.50.0
110112114116118120
104
v1, v2, v3 versus time [h]
[vehicle/km]
0.0
181614
2.01.51.0
121086420 0.5
rho1, rho2, rho3, rho4 versus time [h]
2.01.51.00.50.0
16001400120010008006004002000
[vehicle/h]
r2 versus time [h]
Nominal Simulated
1 2 3 4
This chart reveals the effects of the modeling errors
Sampling time: 20 s, K= 104, B=1, A=0.25×104, Ks=106
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[vehicle/km]
2.0
10
15
1.51.00.50.0
20
5
rho3 nominal and rho3 versus time [h]
[vehicle/km]
0.0
3
2
1
2.01.51.0
0
-1
-20.5
Tracking error versus time [h]
2.01.51.00.50.0
500
450
400
350
300
[vehicle/h]
q0 versus time [h]
Adaptive tracking of 3 [vehicle/km] vs. time [h] (=0)
Nominal Simulated
1 2 3 4
This chart reveals the effects of adaptivity
[km/h]
2.0
105
110
1.51.00.50.0
115
120
100
v1, v2, v3 versus time [h]
[vehicle/km]
0.0
20
15
10
2.01.51.0
5
0.5
rho1, rho2, rho3, rho4 versus time [h]
2.01.51.00.50.0
18001600140012001000800600400200
[vehicle/h]
r2 versus time [h]
Sampling time: 20 s, K= 104, B=1, A=0.25×104, Ks=106
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2.01.5
1.2e+007
1.3e+007
1.00.50.0
1.4e+007
1.5e+007
1.6e+007
1.1e+007
Ef nominal and simulated [vehicle x km^2/h^3]
2.0
1e+0060e+000-1e+006-2e+006
1.51.00.5
-3e+006
-4e+0060.0
Tracking error [vehicle x km^2/h^3] versus time [h]
2.01.51.00.50.0
500
450
400
350
300
[vehicle/h]
q0 versus time [h]
Tracking of Ef [vehicle×km2/h3] vs. time [h] (=1)
1.61.4
1.6e+0071.5e+007
1.21.00.80.60.4
1.4e+0071.3e+0071.2e+0071.1e+0071.0e+0079.0e+006
0.20.0
Tracking of E.F. vs. time
1.60.60.40.20.0 1.41.21.0
4e+0063e+0062e+0061e+0060e+000-1e+006-2e+006-3e+006
0.8
Tracking error vs. time
1.60.60.40.20.0 1.41.21.0
450
350
250
150
0.8
q0 vs. time
Nominal
Simulated NominalSimulated
Non-adaptiveAdaptive
Sampling time: 20 s, K= 104, B=1, A=0.25×104, Ks=106
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Adaptive tracking of fcompr [vehicle/km] vs. time [h] (=0)
Sampling time: 20 s, K= 104, B=1, A=0.25×104, Ks=106
[veh
icle
/km]
0.0
16
15
14
2.01.51.0
13
12
11
10
90.5
fcompr desired, simulated, required versus time [h]
Desired
Simulated
Required: adaptively deformed
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[veh
icle
/km]
0.0
18
16
14
2.01.51.0
12
10
8
0.5
fcompr desired, simulated, required versus time [h]
Adaptive tracking of fcompr [vehicle/km] vs. time [h] (=0.4)
Sampling time: 20 s, K= 104, B=1, A=0.25×104, Ks=106
DesiredSimulated
Required: adaptively deformed
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Thank you for
your attention!!!
Conclusions• Commonly available and cheap software/hardware sets seem
to be satisfactory for the design of a Robust Fixed Point Transformation based iterative adaptive controller for freewaytraffic using the stability of the stationary states.
• Simple 3rd order polynomial approximation in the main ingress and control rate seems to be satisfactory to well describe the stationary solutions.
• The real difficulties in finding appropriate compromises in multi objective optimization stem from the strongly nonlinear nature of the phenomenon under consideration.
• Considerations for bigger lumps (more segments) may be of interest.