This document discusses the development of an iterative adaptive control system for managing freeway traffic emissions, utilizing simplified mathematical methodologies and macroscopic dynamic models. It highlights the challenges of modeling uncertainties and the need for adaptive control strategies amidst non-linear dynamics. The findings suggest that basic computational tools can effectively implement robust linear transformations to achieve satisfactory traffic management outcomes.

1. 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

2. Motivations
• Lyapunov’s 2nd Method is a very sophisticated and complicated
model-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
It is expedient to find less complicated design methodology that
does 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).

3. 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.

4. Discretized Dynamic Model of
Freeway Traffic
output1=input2 output3=input4
output0=input1 output2=input3 output4=input5
ρ0 ρ1 ρ2 ρ3 ρ4 ρ5
v0 v1 v2 v3 v4 v5
0 1 2 3 4 5
L L L L L L
ρ0
d
v0 ( Lρi ) = input i − outputi
dt
q0:= ρ0v0 r additional input
2

5. The Continuity Equation prescribes:
Dynamic equations for one-sided discretization:
(v4 is assumed to be constant):
Papageorgiou’s
model

6. Dynamic equations for central discretization:
(v4, v5 are assumed to be constant, ρ 5 is directly
determined 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!

7. Finding the Stationary Solutions:
• By the use of MS EXCEL, Visual Basic, and SOLVER it is very
easy to prescribe zero value for one of the derivatives while the
other zero derivatives can be prescribed at constraints.
• By using Lagrange Multipliers and Reduced Gradient it is easy
to find the solutions.
• It was found that simple 3rd order polynomial approximation in
q0:=ρ0v0 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.

8. The Adaptive Control Approach Developed
at Óbuda University
Calculated
excitation e =ϕ r ( ) d Desired
response
Actual System’s Rough system
Response model
r = ψ ( e) = ψ ϕ r
r
( ( ) ) = f (r )
d d
Realized
response Unknown function with
known input and measurable
≠r
d r output values.
r

9. Introduction of „Robust Transformations” to create local
deformations
Precise
Precise
Realizatio
Realizatio
nn

10. A possibility is the utilization of the strongly saturated nature
of sigmoid functions with σ(0)=0 for SISO systems
G ( r; r d
) = ( r + K )[1 + B tanh ( A[ f ( r ) − r ]) ] − K d
BAf ′( r )
G' = ( r + K ) + [1 + B tanh ( A[ f ( r ) − r d ] ) ]
cosh 2 ( A[ f ( r ) − r d ] )
False fixed point: G(-K;rd)=-K Good fixed point: if f(r*)=rd then G(r*;rd)=r*
The derivative easily can be made small enough in the fixed point
to obtain convergent iteration:
G ' ( r∗ ) = ( r∗ + K ) BAf ′( r∗ ) + 1
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:

11. 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:
b b
g ′( x ) ≤ K < 1, g ( b ) − g ( a ) = ∫ g ′( t ) dt , g ( b ) − g ( a ) ≤ ∫ g ′( t ) dt ≤ K b − a
a a
Cauchy Sequence in a Complete
Metric Space! It is convergent to some
value u!
The fixed point u is the limit of the iteration:
g ( u ) − u ≤ g ( u ) − xn + xn − u ≤ g ( u ) − xn + xn − u = g ( u ) − g ( xn−1 ) + xn − u ≤
≤ K u − xn−1 + xn − u n→ 0
→∞

12. Design of the control parameters
• Design a common non-adaptive controller for the available
approximate stationary dynamic model and record the
responses;
• Let K ≈ 100 × r max , B = ±1
• Give a little negative contribution to 1 by setting a small A!
∂f
KA ⋅ ≈ 0.5
: ∂r
The main factors determining the emission of CO 2
Controlling the overall emission rate of exhaust fumes at two
segments of a road.
Drag force for 1 car Emission Factor:
Power cons. for 1 car
Power cons. for Lρ cars in the
segment for an average,
unknown drag coeff.

13. The control task with contradiction
Main 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 ρ
seems to be realistic;
Contradiction:
Our sytem is „underactuated”: we have a single control signal r2,
and we wish to simultaneously control Ef and ρ.
Contradiction resolution:
Find a compromise between the simultaeously prescribed Ef and
ρ values by controlling the compound „compromise factor”
Significance
factor
Scaling factor bringing KsEf to the same order of
magnitude as ρ

14. Simulations for segment 3 (SCILAB/SCICOS)

15. Non-adaptive tracking of ρ3 [vehicle/km] vs. time [h] (ξ=0)
v1, v2, v3 versus time [h]
Nominal Simulated 120
118
[km/h]
116
114
112
110
108
106
104
1234
0.0 0.5 1.0 1.5 2.0
rho1, rho2, rho3, rho4 versus time [h]
18
[vehicle/km]
16
14
12
10
8
6
4
2
0
0.0 0.5 1.0 1.5 2.0
r2 versus time [h]
1600
[vehicle/h]
1400
1200
1000
800
600
400
200
0
0.0 0.5 1.0 1.5 2.0
Sampling time: 20 s, K= −104, B=1, A=0.25×10−4, Ks=10−6
This chart reveals the effects of
the modeling errors

16. Adaptive tracking of ρ3 [vehicle/km] vs. time [h] (ξ=0)
Nominal Simulated
1234
Sampling time: 20 s, K= −104, B=1, A=0.25×10−4, Ks=10−6
This chart reveals the effects of
adaptivity

17. Tracking of Ef [vehicle×km2/h3] vs. time [h] (ξ=1)
Nominal Tracking of E.F. vs. time
1.6e+007
1.5e+007
1.4e+007 Simulated Nominal
1.3e+007
1.2e+007
1.1e+007 Simulated
1.0e+007
9.0e+006
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Tracking error vs. time
4e+006
3e+006
2e+006
1e+006 Non-adaptive
0e+000
-1e+006
Adaptive
-2e+006
-3e+006
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
q0 vs. time
450
350
250
150
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Sampling time: 20 s, K= −104, B=1, A=0.25×10−4, Ks=10−6

18. Adaptive tracking of fcompr [vehicle/km] vs. time [h] (ξ=0)
Simulated
Required: adaptively deformed
Desired
Sampling time: 20 s, K= −104, B=1, A=0.25×10−4, Ks=10−6

19. Adaptive tracking of fcompr [vehicle/km] vs. time [h] (ξ=0.4)
f o p d s r d s m l t d r q i e v r u t m [ ]
c m r e i e , i u a e , e u r d e s s i e h
18
Desired
16 Simulated
v h c e k ]
[ e i l / m
14
12
10
8
Required: adaptively deformed
0 0
. 0 5
. 1 0
. 1 5
. 2 0
.
Sampling time: 20 s, K= −104, B=1, A=0.25×10−4, Ks=10−6

20. 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 freeway
traffic 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.
Thank you for
your attention!!!