Transcript
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
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
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.
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
The Continuity Equation prescribes:
Dynamic equations for one-sided discretization: (v4 is assumed to be constant):
Papageorgiou’s model
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!
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.
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
Precise Realizatio
n
Introduction of „Robust Transformations” to create localdeformations
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:
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!
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:
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
Simulations for segment 3 (SCILAB/SCICOS)
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
[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
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
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
[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
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.
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