ITS World Congress :: Vienna, Oct 2012

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]


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


Adaptive tracking of fcompr [vehicle/km] vs. time [h] (=0)


[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)


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.

ITS World Congress :: Vienna, Oct 2012

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