Transpn ges.--B Vol. 16B, No. 4, pp. 279-290, 1982 0191-2615/82]040279-12503.00/0 Printed in Great Britain. Pergamon Press Ltd.

A D E C E N T R A L I Z E D C O N T R O L S T R A T E G Y F O R F R E E W A Y R E G U L A T I O N t

N. B. GOLDSTEIN and K. S. P. KUMAR

Department of Electrical Engineering, 123 Church Street Southeast, University of Minnesota, Minneapolis, MN 55455, U.S.A.

(Received 16 January 1981)

Abstract--A multilevel decentralized control scheme, the cascading technique, with application to the regulation of traffic on an urban freeway is presented. Performance of the decentralized system is compared to the performance of a centralized and a fixed time control structure. It is shown that the decentralized structure performs better than the centralized structure when incidents (lane closures) occur on the freeway. The freeway is modeled in terms of the aggregate variables section density and section speed, and is considered as a system of interconnected subsystems.

INTRODUCTION

In order to provide a smooth flow of traffic on urban freeways, it is becoming common to use control techniques that regulate the on-ramp traffic and advise motorists of any diversions that seem desirable. Features that are basic to these control and advisory schemes are the availability of current traffic conditions, incident detection procedures and a computer to process the data. When computers are used, it is possible to use them in either a central mode or a decentralized mode. In the former, the traffic data is sent to the central computer where computations are carried out and the ramp metering strategies are sent back to the appropriate ramps. This system is reliable to the extent that the communication lines are and the control computer (hardware and software) is. Also, addition of new sections of the freeway under central computer control requires modifications in the software.

In the decentralized mode of control, the following set up is planned. The freeway is divided into sections (Fig. 1) with each section having an on- and off-ramp (perhaps only an on-ramp). The sections are modelled in terms of aggregate variables, density and speed. The ramp control strategy is developed for individual sections using a procedure called the cascading method. A higher level controller provides control signals in case one or more local controllers fail, thus ensuring reliable operation.

In designing the cofitrol structure, it is assumed that the state variables, section density and section speed, have been estimated by other procedures. This paper will not address the problem of state estimation. The paper is organized as follows: a brief review of the literature, the aggregate variable model, the cascading control method, simulation results and conclusions.

L I T E R A T U R E REVIEW

In the centralized approach to freeway traffic control, Kaya (1970) used travel time as a performance index (MOE) in a linear programming formulation. Sendula and Knapp (1972) used a distributed parameter model to study bottlenecks, Yagoda (1970) used a stochastic approach to control entrance ramps and Wang (1972) used dynamic programming to arrive at the ramp flow assignment. Although different optimization strategies were used in the above papers, a common feature characterizes them, i.e. the assumption that all relevant information about the system is available to the controller.

Among the pertinent references to the decentralized control approach to the freeway problem, the multilevel approach of Drew et al. (1969), the augmentation method of Isaksen and Payne (1973), the constrained parameter approach of Houpt (1974), Looze (1975) are notewor- thy. The augmentation method of Isaksen and Payne obtains suboptimum controls based on the optimum control laws for a set of overlapping subsystems derived from the original system dynamic description. This method, while attractive, does not guarantee reliability of operation

?This research was supported in part by the CONICIT and by NSF/INT 78-08778.

279

280 N. B. GOLDS'rEIN and K. S. P. KUMAR

Direct i0ta J / / S u b t y s t e m J ! / / / I • ~ I1--1 !D---s..=, of i i • iD ~rq

i ID • i [ " l I , / / / ' / / / - - J'"""m"i r [ Processing • Control J-I C ontrdler U Command and Control i Actuator

Fig. 1. Freeway control system.

when communication from the detectors to the computer is lost. The cascading technique of the present paper, while a variation of the augmentation technique, does provide a reliable control scheme. The methods of Houpt and Looze do not guarantee stability and convergence of the nonlinear optimization algorithms.

In modeling the freeway by means of aggregate variables following Payne (1971), it becomes necessary to estimate the section density and section speed from the detector measurements. Gazis and Knapp (1971), using an estimate of vehicle travel time estimate vehicle densities on a section of roadway from speed and flow measurements at the entrance and exit points of the section. Mikhalkin et al. (1972) developed an unbiased, minimum mean square error estimate of speed. The approach that closely fits in with the centralized philosophy is the extended Kalman Filtering approach of Nahi and Trivedi (1973). Their algorithm is recursive in nature and the experimental results provide good deal of confidence in their algorithm. To use the cascading control technique of the present paper, one will want to use the estimates of the aggregate variables as provided by the Nahi-Trivedi algorithm or a variation thereof.

In the general area of decentralized control, there exists many contributions. Among the books, the ones authored by Siljak (1978), Singh (1977), Findeisen et al. (1980) provide excellent general theoretical and practical treatment. These books contain a number of references to papers in the literature and we will not include these in this section.

AGGREGATE V A R I A B L E MODEL

In this model, the freeway under control is divided into many sections. The equations of motion in each section are written in terms of the aggregate variables, section density and section speed. With respect to a given section, see Fig. 2, the aggregate variables are defined as follows.

Volume, qi", is the number of vehicles passing through location j between times t,_~ and t,. Section density, p~", is the number of vehicles per lane mile in section j at time t. Section mean speed, up, is the average of instantaneous speeds of all vehicles located within

section j during the interval t, - t, ~.

n*l I / / DIRECTION OF I r °~ - ' / "~ TRAVEL .

x n - ~ / n

Pj-= n

uj-i

n. I X ; nJ~lJ/ ~ ro: n n*l n n qj-i / J-I qj = pj.., u.

j - i

Fig. 2. Freeway section and associated variables.

A decentralized control strategy for freeway regulation 281

ro~ut,j is the number of vehicles leaving section ./through the ramp during the time interval t . - t,_~. Similarly, ri~,j is the number of vehicles entering section j through the ramp during t . - t . - v

T is the reaction time that it takes a group of drivers to accelerate or decelerate in order to attain the desired speed u~(p).

In terms of these variables, Payne (1971) derived the dynamic equations of motion which are given below

n+l / n+l n+l n+l p"n +~ = pj" + A t tj-~,qj - tjqi, +~ + A t rin,j - rom,./ (1)

tj[Xi+ 1 -- Xj) l j(Xi_ 1 -- Xi)

u7 +~ = u~" - Atu~" up - uT_~ A t , , A t 1 pj+~ - pj 1/2(xi+2_ x i _ O - - - ~ [ u j - Ue(pj") ) - U T p~" 1/2(Xj+2- xj)lj (2)

where u,(p) is the equilibrium speed-density relationship, I~ is the number of lanes in section j and At = t . - t~-l. The factor l I T represents the temporal delay in response of a group of vehicles to changing traffic conditions• The term involving the parameter u represents anti- cipation of the changing density ahead. The relation between flow, density and speed under uniform flow is given by

q~+~ = p~-l u~'_,. (3)

Assuming nominal conditions about which to regulate the traffic, Payne linearized the above equations around the nominal conditions and obtained the linearized equations of motion as

= A x ( t ) + B y ( t ) (4)

where A is a 2n × 2n matrix for a freeway partitioned into n sections, B is a 2n × m matrix for a freeway with m ramp controls, x is the state vector with components section density and section speed• For example, for a freeway divided into 4 sections and having 2 controlled on-ramps, the linearized equations assume the formt

Pl ul

P2 U2

P3 U3

P~ //4

all a12 bzl b22 b23 a3! a32 a33

b4~ b43 a53

Pl Ul

a34 0 P2 b44 b45 u2 a54 a55 a56 P3 b64 b65 b66 b67 u3

a75 ~/76 a77 a78 P4

b86 b87 ba8 u4

+ (5)

It is observed that the system matrix assumes a diagonal band structure with two charac- teristics. First, the odd numbered rows constitute the density equations and the even numbered rows the velocity equations. Second, the off-diagonal terms such as b23, a31, b45, etc. constitute the interconnection terms. These terms represent the effects one section has on the adjacent ones or on a section further down stream. The control vector has either zero or unit elements representing the absence or presence of controls.

Complete details of the various elements in the A and B matrices are contained in Goldstein (1980).

CASCADING TECHNIQUE

Since the problem is one of regulating the section densities and speeds around a nominal value, the ramp control strategy is chosen so as to minimize a quadratic performance index. This index is quite useful in arriving at linear feedback control strategies and the results are

tA detailed description of alj & bij is given in the Appendix.

282 N.B. GOLDSTEIN and K. S. P. KUMAR

easily implementable. As the simulation examples demonstrates, this index is quite satisfactory for purposes of regulation. The same index has been used by Isaksen and Payne (1973), Houpt (1974) and Looze (1975). For the linearized system dynamics given in equation (4), we associate a performance index

io J(xo, u(. )) = x ' ( t )D'Dx( t ) + v ' ( t )Rv(t) ldt (6)

where R is a positive definite matrix and D'D is a positive semi-definite matrix. Under the assumption of complete controllability of the pair [A, B] and complete observability of the pair [A, D], it is well known, Kalman (1960), that the optimal feedback control v*(t) that minimizes the performance index is

v*(t) = - R-~B 'Kx( t ) (7)

where K is the unique positive definite solution of the algebraic matrix Riccati equation

0 = A ' K + K A ' - K B R - ~ B ' K + D'D. (8)

While this solution is good from a theoretical point of view, it has several shortcomings for our problem. The computational burden is too much for a centralized solution and if some communication is lost (i.e. some state variables are not fed back), the total system may break down. In order to avoid these problems and to make decisions in a decentralized framework, the cascading technique is introduced. The central idea of this technique is to enlarge the original system in a systematic way. For a square matrix of size n, rows 2 to n - 1 are duplicated by 2 and two partitioned zero columns are introduced. For example, consider a fifth order matrix A and the associated matrix B given by Iaa2013a4as EO0!I

a21 a22 a23 a24 a2s | 1 0 A = a31 a32 a33 a34 a35 / B = 0 1 (9)

a41 a4~ a43 a44 a45| 0 0 as~ asz a53 a54 a55 __1 0 0

now duplicate rows 2,3 and 3,4 to get the enlarged matrix A ~

Ae=

all a12 a13 a14 a15 a21 a22 a23 024 a25

a31 a32 a33 a34 a35 a21 a22 a23 a24 a25 a31 a32 a33 a34 a35 a41 a42 a43 a,14 a45 a31 a32 a33 a34 a35 a41 a42 a43 an a45 asi a52 a53 a54 a55

(lo)

Introducing the partitioned zero columns now yields the cascaded matrix A c

a . a~z a~3 0 0 a~4 0 0 aa5

a2~ az2 azs 0 0 a24 0 0 a25 a31 a32 a33 0 0 a34 0 0 a35 azl 0 0 a2z a23 az4 0 0 az5

a31 0 0 a32 a33 a34 0 0 a35 a41 0 0 a42 a43 a44 0 0 a45 a3] 0 0 a32 0 0 a33 a34 a35 a4~ 0 0 a42 0 0 a43 a44 a45 as~ 0 0 a52 0 0 a53 a54 a55

(11)

A decentralized control strategy for freeway regulation

and the cascaded B matrix is

BC =

Eo,o ooo oo!1 0 0 0 0 1 0 O 0 ,

0 0 0 0 0 0 O 0

(I) (II) (III)

283

(12)

Then, subsystems (I), (II) and (III) are represented by

Yci = (Aia)xi + (Bij)vi (13)

For subsystem (I), i = 1,2,3 j = 1,2,3 For subsystem (II), i = 2,3,4 j = 2,3,4 For subsystem (III), i = 3,4,5 j = 3,4,5

A graphic representation of the cascading technique is shown in Fig. 3. The cascading technique offers the possibility of controlling a section when the controller in

that section breaks down. For example, in Fig. 3, if controller II is not functioning the control for section 3 can be obtained either from controller I or controller III, whereas in the augmentation technique of Isaksen and Payne (1973) if controller I breaks down there is no way of recuperating a control for sections 2 and 3.

The controls obtained by the cascading technique are also more sensitive to local variations than the ones obtained by either augmentation or by a centralized controller. This is due to the fact that the cascaded controls take into account only what happens in the closest neighboring sections, while in the other techniques, augmentation and centralized, the conditions of sections further upstream or downstream is also considered.

Due to the fact that the size of the matrices in the cascading technique is smaller, in general, than in the centralized or augmentation techniques, the time required to solve for a control is less and therefore it may respond better to the dynamic changes that occur in the system.

The cascading technique is also easier to implement since it requires only one controller for every three sections which can be copied throughout the system, while in the augmentation technique different controllers might have to be designed.

It has already been demonstrated, Siljak (1978), that with regard to reliability, the decen- tralized control schemes are superior to their centralized counterparts. In the cascading technique redundancy is introduced into the control, that is, more than one controller is used per section.

Although for the purposes of this paper, only one control command was issued from each controller, more controls were available in case of controller failure. To clarify the previous statement, we will refer to the section following where the cascaded system for the freeway traffic control is solved. The matrix A for the complete systems is 10 × 10 and the control matrix B is 10×3, and controls are provided for sections 2--4. When the augmentation for the cascading system is completed, sections 1-3 comprise the first subsystem, with controls in

r - - - ~ COOROI NATOR~- - - .-.:

xI u2 x:t I x5 5

L 51 Fig. 3. The cascading technique block diagram.

284 N.B. GOLDSTEIN and K. S. P. KUMAR

sections 2 and 3, sections 2-4 comprise the second subsystem with controls in sections 2-4 and sections 3-5 comprise subsystem 3 with controls in sections 3 and 4. As can be seen, sections 2 and 4 have two different controller sources that can supply their controls and section 3 has three different sources of control. This is the place where the coordinator shown in Fig. 2 plays an important role. when a controller failure is detected, the coordinator issues a command so that an alternate controller provides the control for that section.

Ths structure just shown is very similar in form to the generic case of multiple control systems introduced in Siljak (1979) and shown in Fig. 4. When a controller failure occurs, the other controller is capable of acting on the plant without any interruption of the function of the overall system.

Siljak (1979) has shown that stability of the decentralized (cascaded) matrix A e of eqn (11) implies stability of the original matrix A. For the purposes of this paper the cascaded matrix of the freeway system was examined and all eigenvalues were found to lie in the left half of the complex plane.

SIMULATION RESULTS

In this section, a simulation example will be worked out in which the cascading technique is used for ramp metering and compared to controls using fixed time and centralized techniques.

The data for the simulation is the one used by Isaksen (1972). The data corresponds to the northbound San Diego Freeway from the Harbor Freeway to Crenshaw Boulevard. The freeway section is 2.06 miles in length and has three on-ramps and two off-ramps. The section is divided into five subsections, sections 1, 3, 4 and 5 have four lanes and section 2 has five lanes.

The dynamic equation requires knowledge of the parameters T and u. The time constant T is taken to be 30 sec. The sensitivity coetticient u is taken as 7.5 mi: per hour. This implies that a group of drivers accelerates witha reaction time of 30 sec in order to attain the desired speed ue(p), i.e. an acceleration of approx. 0.2 miles per hour per second.

The upstream value entering the first section is taken as 1900 vehicles per hour per lane. A time increment of 5 sec was used in all calculations and simulations.

Dimensionless variables were defined to ease numerical computations. Let the scale factors S~, S~ and S,, be defined as

Xd =Sxx, (14)

qd =Sqqa (15)

Ud =S, ua (16)

where the subscripts d and a refer to dimensionless and actual values respectively. The scale

I I S U B S Y S T E M I I

'1 I I I C O N T R O L L E R I i I

I

,,, I PLANT I I k _ _

CONTROLLER 21 SUBSYSTEM 2

Fig. 4. Multiple control system.

- - I I I I I I I I ! I

- 1

_ _ J

A decentralized control strategy for freeway regulation

factors for density Sp and for time S, are defined by

285

Sp = SJS. (17)

S, = SxlS.. (18)

The following values were chosen for the scale factors: Sx = 2 1/mile S, = 0.02 hr/mile Sq = 1/1800 hr lane vehicle Sp = 1/36 mile lane/vehicle St = 100 1/hr.

A simulation routine was used to determine the nominal conditions of density and speed. The routine was run using the constant on- and off-ramp rates given in Table 2. The initial densities and speeds in the five sections of the freeway were given the value of one in dimensionless variables and the program was allowed to run until a steady state condition was obtained. The nominal speeds and densities are given in Table 2. In Table 3 the priority parameters a and/3 are shown.

The nominal speed is very similar for the five sections (approx. 52.75 miles/hr), while the nominal density varies due to different number of lanes in the sections and on- and off-ramp rates. The lowest density is encountered in section 2 with 29.51 vehicles/mile/lane and the highest density in section 3 with 36.0 vehicles/mile/lane. Speeds between 40 and 55 miles/hr were considered satisfactory.

Table 1.

Section Section Number of Number Length (ft) Lanes

1 1585 4

2 1140 5

3 2720 4

4 2720 4

5 2640 4

Table 2. Nominal freeway conditions

Section Volume Density Speed On-ramp Off-ramp Number veh/hr/lane veh/mi/lane mi/hr veh/hr veh/hr

1 1901 36.01 52.78 0 0

2 1557 29.51 52.77 180 0

3 1916 36.33 52.V5 180 300

4 1756 33.30 52.73 180 820

5 1756 33.33 52.72 0 0

TR(B) Vol. 16, No. 4 ~

Table 3. Nominal ramp coefficients

Section Number a 8

1 0 0

2 .023 0

3 .023 .039

4 .025 .107

5 0 0

286 N.B. GOLDSTEIN and K. S. P. KUMAR

The numerical value of the A and B matrices are shown below:

The A matrix is

- 1.7585 - 1.6667 0 0 0.5710 - 1.2000 -0.6967 0 1.9554 1.8538 -2.3879 - 1.8543

0 2.0447 0.6060 -3.2441 0 0 1 .2305 0.9559 0 0 0 1.4428 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

-0.6014 0 0 0 - 1.0009 -0.9567 0 0

0.3250 -2.6427 -0.3516 0 0.9424 0.9012 - 1.0284 -0.9021

0 1 .0394 0.3600 -2.2386 0 0 1 .0547 0.9249 0 0 0 1.0543

0 0- 0 0 0 0 0 0 0 0 0 0 0 0

-0.3891 0 - 1.0542 -0.9258

0.2870 - 2.2540

The B matrix is

0 0 0 0 0 0

1.000 0 0 0 0 0 0 1.000 0 0 0 1.000 0 0 0 0 0 0 0 0 0

In Fig. 5 the density change in section 3, where an accident occurs at t = 5 rain (one lane is closed), is shown.

In the fixed time control solution, the density increased to a maximum of 71.5 vehi- cles/mile/hr, while in the cascaded solution it reached only to 63.6 vehicles/mile/hr, an improvement of 12.4%.

Figure 6 shows the density response in the section immediately downstream from the section where the accident occurred. Using the fixed time control strategy, a maximum density of 49.8 vehicles/mile/lane is reached, while using a cascaded control solution the maximum is 44.5 vehicles/mile/lane. It takes 4 min for the fixed time control strategy to return to nominal conditions, it takes only 2.5 min for the cascaded solution to return to nominal.

7 2

t . 6 4 _J

W

~ 4 8 - r

~ 4 0 i¢)

z 3 2 9 ( . I w 2 4

_z 16 > - I,.- ~ 8 Z w

" 0

- 8

\

/ - - I

CASCADING

. . . . . FIXED TIME

1 I I I I I t I

5 I0 15 20 25 30 3.5 40

TIME AXIS (MINUTES)

I

45 50

Fig. 5. Density in section 3.

A decentralized control strategy for freeway regulation

G Z

.J

~ 6 4

s s

J 48 - r tAJ > 4 0 q J- ~

3 2 ,

v.- ~'4 Or)

1 6 ,~ CASCADING

>. 8 . . . . FIXED TIME I - -

0 J i Z w 5 I 0 £3

I I I I I I I

15 20 ;)5 30 35 40 45 50 - 8

*lIME AXIS (MINUTES)

Fig. 6. Density in section 4.

287

6 4

56

4 8

4 0

~ 3z I..- U ~ 24 Z - - 16

I,- 8 , U

0 . J w 0

- 8

._F'

CASCADING

. . . . RXED T,ME

I I I I I I I I I

5 I0 15 20 25 30 35 4 0 45 50

TIME AXIS (MINUTES)

Fig. 7. Input rate to section 4.

180

i6o ..r

i4o hl

J

iJ 120 la.i >

I00

8 0 Z 0

~ 6 0 U

~o 4 0

~ 20

~ 0 n z - - 2 0

CASCADING . . . . CENTRALIZ ED

I I I l l I I I I

5 I0 15 20 25 30 35 4 0 45 50

TIME AXIS (MINUTES)

Fig. 8. Velocity in section 4.

288 N.B. GOLDSTEIN and K. S. P. KUMAR

Figure 7 shows the input rate to section 4, the section immediately downstream from the section where the accident occurred. This time the comparison is made between the centralized solution and the cascaded solution. It can be seen that the cascaded solution gives a much better response to the situation on the freeway than the centralized response. The reason being that in the cascaded type of solution the system allows more vehicles to enter through the ramp in section 4 after noticing less cars coming in from section 3, 165 vehicles/hr as compared with 90 vehicles/hr in the centralized solution. Clearly the cascading type of control is more sensitive to local variations than the centralized one. This is a big advantage in this case since it is desired to keep a maximum number of vehicles coming into the freeway at all times while keeping the density and speed on the freeway close to the nominal conditions. In Fig. 8 the velocity profile for section 4 is shown. More simulation results are contained in Goldstein (1980).

CONCLUSIONS

In this paper, the freeway system was first modeled from a macroscopic point of view. The model was appropriate for simulation of small traffic perturbations about the nominal conditions and also for incidents involving lane closures. The results obtained with the model appear to reproduce the characteristics of traffic flow around nominal conditions as well as the flow when an incident occurs. Correlation between different sections is consistent and the relationship between a congested section and its neighboring sections corresponds to what is observed in a real traffic situation. The model can be improved by gathering further information and data on different traffic situations on freeways and observing the behavior of the model under similar conditions. By comparing the response of the model with the observed situation on the freeway differences may be corrected by changes in the variables of the model like v and T or more terms may be added to the dynamic equations.

The interconnection terms have also to be related to the cascading technique used in this research. It is seen that in the equations that describe the dynamics of the freeway, a relationship exists between section j and section j - 1 with the exception of the speed equation where a relationship also exists with section j + 1 through the density term. It may be important to include interconnection terms that would include a relation to the speed and density in section j - 1 in both equations so as to be consistent with the cascading technique in which every three consecutive sections are considered.

The decentralized control generated by the cascading technique was presented and tested in the freeway system model with satisfactory results. It performed very similar to the centralized solution with regards to keeping speeds and densities near their nominal values, and it had a significantly better performance than the centralized solution in generating a control for the on-ramp in a section after an incident occurred. This better response is due to the fact that the cascading technique is more sensitive to local variations than the larger centralized solution.

Although the simulation results for the freeway system presented in this paper are encouraging, it is not possible to conclude that the system developed here is a better one than those already existing. This conclusion can only come when "on the road" tests are performed However, based on the results obtained in this paper, the cascading technique together with the aggregate variable model appear to be more effective than fixed time controls or the aug- mentation technique.

There is another sense in which the cascading technique has an advantage over other techniques. The cascading technique requires communications between neighboring sections only, which makes communication lines shorter and thus more economical. Secondly, since all the controllers needed in the cascading technique are similar, only one design has to be made for the microprocessor hardware and then repeated throughout the system.

It has been shown that the cascading technique is an important option to be considered in freeway traffic systems and that it is a reliable method.

REFERENCES Drew D. R., Brewer K. A., Buhr J. H. and Whitson R. H. (1%9) Multilevel approach to the design of a freeway control

system. HRB No. 279, 70-86. Findeisen W. et aL (1980) Control and Coordination in Heirarchical Systems. Wiley, New York.

A decentralized control strategy for freeway regulation 289

Gazis D. C. and Knapp C. H. (1971) On line estimation of traffic densities from time series of flow and speed data. Transpn Sci. 5, 283-301.

Goldstein N. B. (1980) Decentralized control strategies for linear systems with application to freeway traffic regulation. Ph.D. Thesis, University of Minnesota.

Houpt P. K. (1974) Decentralized stochastic control of finite state systems with application to vehicular traffic flow. Ph.D. Thesis, MIT.

Isaksen L. (1972) Suboptimal control of large scale systems with application to freeway traffic regulation. Ph.D. Thesis, University of Southern California.

Isaksen L. and Payne H. J. (1973) Sub-optimal control of linear systems by augmentation with application to freeway traffic regulation. IEEE Trans. Automatic Control AC-18, 210-219.

Kalman R. E. (1960) Contributions to the theory of optimal control. BoL Soc. Mat. Mex. 5, 102-119. Kaya A. (1970) Optimization of traffic flow for an urban freeway transportation system. Ph.D. Thesis, University of

Minnesota. Looze D. P. (1975) Decentralized control of a freeway traffic corridor. M.S.Thesis, MIT. Mikhalkin B., Payne H. J. and Isaksen L. (1972) Estimation of speed from presence detectors. HRB, Rec. 388, 73-82. Nahi N. E. and Trivedi A. N. (1973) Recursive estimation of traffic variables: section density and average speed. Transpn Sci. 7,

269-286. Payne H. J. (1971) Models of freeway traffic and control. Simulation Council Proc. 1, 51-61. Sendula H. M. and Knapp C. H. (1972) Optimal control of hyperbolic distributed parameter systems with applications to

traffic bottlenecks. IEEE SMC Conf., Washington, D.C. Singh M. G. (1977) Dynamical Hierarchical Control. North-Holland, Amsterdam. Siljak D. D. (1978) Large-Scale Dynamic Systems: Stability and Structure. North-Holland, Amsterdam. Siljak D. D. (1979) On reliability of control. Proc. 17th IEEE Conf. on Decision and Control San Diego, California, pp.

687-694. Wang C. F. (1972) On a ramp flow assignment problem. Transpn Sci. 6, 114-130. Yagoda H. N. (1970) The dynamic control of automotive traffic at a freeway entrance ramp. Automatica 6, 385-393.

A P P E N D I X Nominal conditions t

The nominal condition is the condition the freeway would attain for constant on- off-ramp rates in the absence of disturbances.

Assume that under the nominal steady state condition, on- and off-ramp flow rates are dependent on existing flow conditions in the main stream.

Let off-ramp rate coefficients/3, j = 1 ..... n be defined by

ronffj = ~3in q~- I ( A I . I )

Equation (AI.1) indicates that a fraction/3r of the cars upstream of an off-ramp will exit./3i is zero when no off-ramp exists.

The on-ramps can be treated in a similar manner. For nominal steady conditions define the priority parameters aj, j = l ..... n by

r~,, = C qi~ (AI.2)

Linearized model Traffic responsive metering has the objective of maintaining the nominal conditions in the presence of small

disturbances. An extension of the aggregate variable model is presented. Let the nominal steady state variables be written as pok~, Uoki and qokj. Let the actual conditions on the freeway be given by' p ~ / v ~ j and qaki. Let the differences between the actual and

nominal conditions be written as p1 ~, ui k, and qji. ' • ' The variables pi k, ui k and qjk are perturbed quantities where

p ko, i k k = Pod + Pi (A2.1)

t _ Uoki + Ua~-- , U~ k (A2.2)

q~j = qkoj + qj'. (A2.3)

To model ramp rates in the presence of disturbances using traffic responsive metering, let

rn+l _ n+l n+l off/ -- /31 qa, i -1 (A2.4)

where B7 l is defined by eqn (AI.2) for the nominal steady state condition. Similarly write

r ~ ' = a~-' q~,~ l - vi" (A2.5)

where a~ '-~ is defined by eqn (A1.2) for the nominal steady state condition and v/' is the control being exerted. In order to utilize the above results for the freeway control problem it is necessary to obtain a linearized model for

freeway traffic with a continuous time dependence.

tSections A.1 and A.2 follow the treatment given by Isaksen (1972).

290 N.B. GOLDSTE1N and K. S. P. KUMAR

To obtain such a model the actual conditions (A2.1)-(A2.3) are substituted in eqns (1)--(3) where the time variables are again taken as continuous.

qj(t) = poj_l( t )us_l( t) + pj_~(t)Uo,i_~(t) (A2.6)

du j [ , , . IJi - Ui-I 2._ ,,. UOd -- Uod-I I r ~ I "] d t = - k "°a l/2(xj+ , - xj_,) " " l/2(xi+ , - xj_,) - - T [ u s - Oor( t) [.j=o~,oPsJ

{ 1 ,i+,-pJ _ poj+,-poj &o~ - T \~. j 1/2(x~+2 - xj) I/2(xi+2 - xj) p~., ~'J] (A2.7)

1) j n

d t (xi+ _ x i ) l i ( . ,_ ,q j - , :~ ,+, + t.. . . (A2.8)

xi+~ - xj

Substituting eqn (A2.6) into eqn (A2.8) and combining terms in eqn (A2.8) and eqn (A2.7) yields

doi(t) _ v." - aj.i_zpj_~(t) + aja_~u~_~(t) + a~aps(t) + ai.i+l uj(t) + x i + / - x i (A2.9) d t

dut(t) = bsj_2uj_~(t) + bia_lpj(t) + bjaui(t) - bij+l(t) (A2.10) d t '

where

/,_, t~.-i Uod-1 aid-2 = ~ ( l - ( A 2 . 1 1 ) " " xj+~-xj

= ~.~ (1 - a .~ p°~-~ dj, j - I (A2.12) "S" xs+~- x~

aj.j = (a i - 1) x~+U° a xj (A2.13)

aj,i+ I = (o ' i - - 1) Po,i (A2.14) x j+~- xj

bj.,-2 = l/2(x,+U~ x,_,) (A2.15)

I 4 . ' , .o. , (A2.16) ' - T Opj _pi=ooj 7"Po-j(xs+z-x~) Poj

1 1 bj.j I/2(X~+l--Xj_l) (u°J-~ - 2 u ° J ) - ~ (A2.17)

v 1 1 bsj+1 T l /2 (x j+z_ xi) po. / (A2.18)