Sliding mode for user equilibrium dynamic traffic routing control

Faculty Publications (ECE) Electrical & Computer Engineering 1-1-1997 Sliding mode for user equilibrium dynamic traffic routing control P. Kachroo University of Nevada Las Vegas, Department of Electrical & Computer Engineering, pushkin@unlv.edu K. Ozbay Repository Citation Kachroo, P. and Ozbay, K., "Sliding mode for user equilibrium dynamic traffic routing control" (1997). Faculty Publications (ECE). Paper 80. http://digitalcommons.library.unlv.edu/ece_fac_articles/80 This Conference Proceeding is brought to you for free and open access by the Electrical & Computer Engineering at University Libraries. It has been accepted for inclusion in Faculty Publications (ECE) by an authorized administrator of University Libraries. For more information, please contact marianne.buehler@unlv.edu. zyxwvutsr zyxwvutsr SLIDING MODE FOR USER EQUILIBRIUM DYNAMIC TRAFFIC ROUTING CONTROL Pushkin Kachroo Bradley Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-01 11 Kaan Ozbay Dept. of Civil & Environmental Engineering Rutgers, the State University of New Jersey Piscataway, NJ 08855-0909 Keywords Traffic Assignment, Feedback Control ABSTRACT This paper presents a solution to the user equilibrium Dynamic Traffic Routing (DTR) problem for a point diversion case using feedback control methodology. The sliding mode control technique which is a robust control methodology applicable to nonlinear systems in canonical form is employed to solve the user equilibrium DTR problem. The canonical form for this problem is obtained by using feedback linearization technique, and the uncertainties of the system are countered by using sliding mode principle. Simulation results show promising results. 1. INTRODUCTION Real-time control of point diversion is an important topic as part of an overall incident management process. The advent of Intelligent Transportation Systems (ITS) has made clear the need for real time control for diversion. Researchers have used expert systems [ I , 21, mathematical programming and more recently feedback control techniques [4, 51 for solving this real-time diversion problem. The method of “rolling horizon” which uses finite horizon nonlinear optimization technique at different sampling times to achieve feedback configuration has also been tried as one of the solution approaches [3, 6, 71. Recently some feedback control methodologies have been designed to address the Dynamic Traffic Assignment and DTR problem. [lo-131. The feedback linearization technique works on exact cancellation [ 121. This paper enhances that work by using sliding mode control, which takes care of a class of uncertainties in the system. 2. SYSTEM DYNAMICS Many researchers have studied and designed optimal open loop controllers utilizing space and time discretized models of traffic flow [8, 9, 141. In order to utilize the various linear and 7 nonlinear [ 15 - 161 control techniques available for lumped parameter systems, the distributed parameter model of the original hydrodynamic model of traffic flow is space discretized [14]. zyx zyxwvut Consider the following traffic model.. d 1 -p,=-[q,(t)-q,+,(t)+r,(t)-s,(t)l, dt 6, zyx i = LL..n (1) v = vc (1 --) P Pmax Here, rl (t)ands,(t) terms indicate the on-ramp and off-ramp flows, p(t)is the density of the traffic as a function of x, and time t, and q(t) is the flow at given x, and t, vf is the free flow speed, and p,, is the jam density. Equation (1) and the output equations (4) give the mathematical model for a highway, which can be represented in a standard nonlinear state space form for control design purposes. y,=g,(p,,p 2,...,pn), j = 1,2,...,p, (4) The standard state space form is d -x(t) dt = f[x(t),u(t)], zyx zy FEEDBACK CONTROL FOR THE TRAFFIC In the discretized traffic flow model, the freeway is divided into sections with aggregate traffic densities. Sensors are used to measure variables such as densities, traffic flow and traffic average speeds in these sections, which can be used by 3. zyxwvut 0-7803=4269-0/97/S10.00 0 1998 IEEE Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply. zyxwvut the feedback controller to give appropriate commands to actuators like VMS, HAR, etc. 3.1 User Equilibrium Formulation of the DTR Problem We present here a DTR formulation for the two alternate routes problem. The two routes are divided into n I , and n2 sections respectively. For simplicity, we are considering static velocity relationship, and ignoring the effect of downstream flow. Hence, the model used is i=l i=l 4. FEEDBACK LINEARIZATION Feedback linearizaton is an appropriate technique for developing feedback controllers for nonlinear systems similar to the DTR model described above. The feedback linearization technique is applicable to an input affine square multiple input multiple output (MIMO), system. The details on exact nonlinear decoupling technique (feedback linearization) can be found in [19-221, and is briefly summarized here for the DTR application. Let us consider the following square MIMO system: zyxwvutsr zy zyxwvutsr zyxwvutsrqponml zyxwvutsrq zyxwv The flow U(t) is measured as a function of time, and the splitting rate p(t) is the control input. The output measurement could be the full state vector, i.e., vector of flows of all the sections, or a subset of that. The control problem can be stated as: find P o ( t ) , the optimal P(t), which minimizes I’ X(t) = f ( x ) + C g i ( x ) u i i=l y , = h j ( x ) j = 1,2,...,p This can be written in a compact form as (15) X ( t ) = f(x) + g(x)u c: 0 i=l x(.,.) m+l where is the travel time function and t, is the final time. The controller should provide m ni+o i=l in+] and some transient behavior characteristics like some specified settling time, percent overshoot, etc. For the generalized case with n alternate route problem we have: Find pb, i=1,2,...,n, which minimize Problem: (k=1,2,...,n, p=1,2 ,...,n, and the summations are taken over total number of combinations of n and p, and not permutations so that (k,p)=(1,2) is considered the same as (k,p)=(2,1), and hence only one of these two will be in the summation) ), or which guarantee L t , , s -+ 0, ( I 1) with some transient behavior characteristics like some specified settling time, percent overshoot, etc. for the system where, x Y = h(x) E R”,f(x):R” + R”, g(x) : R” + R”, U E RP, and y E RP. The vector fields of f(x) and g(x) are analytic functions. c, It is assumed that for the system each output Y; h as a defined relative degree y,. The concept of relative degree implies that if the output is differentiated with respect to time yj times, then the control input appears in the equation. This can be succinctly represented using Lie derivatives. Definition of a Lie derivative is given below, after which the definition of relative degree in terms of Lie derivatives is stated. Definitioiz (Lie Derivative): Lie derivative of a smooth scalar function h : R” + R with respect to a smooth vector field f : RI’ -+ R” is given by ah L,h = - f . ax Here, L,h denotes the Lie derivative of order zero. Higher order Lie derivatives are given by L’,h = L,(L‘;’h). Definition (Relative Degree): The output y, of the system has a relative degree y j if, 3 an integer, s.t. 71 Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply. zy zyxwv zyxwvutsrqponmlk zyxwv in the system, the control law (21) can not be directly utilized, but it will have to be modified. Let a single input nonlinear system be defined as (23) x(n) = f(x,t) + b(x,t)u(t) Here, x(t) = [x(t) x(t) ... x("-I)lT is the state vector, U is the control input and x is the output state. The superscript n on x(t) signifies the order of differentiation. A time varying surface S(t) is defined by equating variable s(t) to zero, where -y;iy;? L:?h,(x) - yO ;, -Lg,LT-'hl(x) ... Lg,,Ly-'hl(x)LgpL:?-'h,(x) ... LgnL:?-'h,(x) -L:h,(x)- + L?h,,(x) (16) U Lg,L7-'hl(x) ... LgpL7-'hl(x) UL Here, y is a design constant and ?(t) = x(t) - xd(t) is the error in the output state where xd(t) is the desired output state. The switching condition (25) I -(s(t)2) d 5 -qls(t)l, T) > 0 2 dt makes the surface S(t) an invariant set. All trajectories outside S(t) point towards the surface, and trajectories on the surface remain there. It takes finite time to reach the surface S(t) from outside. Moreover the definition (24) implies that once the surface is reached, the convergence to zero error is exponential. Chattering is caused by non-ideal switching around the switching surface. Delay in digital implementation causes s(t) to pass to the other side of the surface, which in turn produces chattering. zyxwvutsrq A(x) = [Lph,(x) Lyr'h2(x) ... L:h,(x)] (19) LgILy hI (x) ... Lgl,Ly h, (x) Lgl,Ly-'h1(x) ... Lg,,Ly-'h,(x) B(x) = zyxwvuts zyx LgILY,'-lh,(x) ... LgpL:-IhI (x) If the ecoupliw matrix B(x) is invei we can use &the-feedback control law (21) to obtain the decoupled dynamics (22). (21) U = (B(x))-'[-A(x) + V] yy = v (22) where .I v=[v, v* ... v,] The vector can be chosen to render the decoupled system (1 7) stable with desired transient behavior. 5. Y L l U l N t i MWUK CWN'I'KWL The point diversion problem, as will be illustrated in the following sections, has a special structure. The special structure is such that the matrix B(x) is diagonal, and the relative degree of each output in (16) is one. This special structure will be utilized to design sliding mode control law for the diversion law. We can also analyze and design control when there are uncertainties in the system, which is bound to be happen in practice. When there are uncertainties 7 Consider a second order system (26) X(t) = f(x,t) + u(t) where f(x,t) is generally nonlinear and/or time z varying and is estimated as €(x,t), u(t) is the control input, and x(t) is the output, desired to follow trajectory xd(t). The estimation error on f(x,t) is assumed to be bounded by some known function F=F(x,t), so that (27) I?(x,t) - f(x,t)l 5 F(x,t). We define a sliding variable according to (4) UL The next two theorems give controls that guarantee the satisfaction of the switching condition (25). Theorem I : For a single input second order nonlinear lumped parameter system, affine in control, given by (26), where x E R 2 , U E R , x E R , and f : R2 x R' R , choosing control law as: (29) u(t) =G(t) - k(x,t) sgn(s(t)) with k(x,t) = F(x,t) + q, and A G(t) = -f + x, - * Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply. zyxwvu zyxwvut zyxwvutsrq zyxwvutsrq zyxwvuts zyxwvuts zyxw zyxwvu satisfies the invariant condition of Equation (25). Results for a second order system with uncertain control gain are given by the following theorem. Theorem 2 : For a single input second order nonlinear lumped parameter system, affine in control, given by (30) x(t) = f(x,t) + b(x,t)u(t) where 0 I bmjn(X,t) I b(x,t) 5 bmax(x,t) where X E R ' , U E R , X E R , b : R 2 X R + + R , and f : R2 x R' 3 R , control law (31) u(t) = g(x,t)-'[;(t) - k(x,t) sgn(s(t))] where k(x,t) s a(x,t)(F(x,t) + q) .t (a(x,t) - I)IG(t)l (32) * (33) b ( x , t ) = ibmin(x,t)bmax(x,t> and (34) I' a(x,t) = bmax(x,t)/bmin(x,t> ensures the invariant condition of Equation ( 5 ) . introduces the input split factor into the dynamic equation. The variable y is equal to the difference in the travel time on the two sections. k3 (39) y = - k, (k2 - PJ - (k, This equation can be differentiated with respect to time to give the travel time difference dynamics. By substituting (35) and (36) i n (40) we obtain k, v,, PL(l- 4) - p" + 6,0, I; v t 2p2(I-&)+ pu - + 6,0, J (41) )'= 6. SAMPLE PROBLEM (TWO ALTERNATE ROUTES WITH ONE SECTION) In order to illustrate the ideas discussed above, we have designed a feedback control law for two alternate routes problem with single section each. The space discretized flow equations used for the two alternate routes are: lr 1 (35) - q::r where U, and 0, are the terms representing uncertainties of the system equations. We have considered a simple first order travel time function, which is obtained by dividing the length of a section by average velocity of vehicles on it. According to that, the travel time can be calculated as where, d, and d, are section lengths, vfl and vf2 are the free flow speeds of each section, and pml and pm? are the maximum (jam) densities of each section. Since we need to equate the travel times in the two freeways, we take the new transformed state variable y as the difference in travel times. Differentiating the equation representing y in terms of the state variables I+ U Pm2 $( w 4 -PA> This equation can be rewritten in the following form. y = F + Gp +f where f=9 - P2) where kIU, k302 (45) (k.2 - PJ2 - (k4 - P 2 I 2 Hence a feedback linearization control law can be designed to cancel the non-linearities and provide the desired error dynamics. The sliding mode control law used is where $ is a known function which gives the error estimate bound on f. If there were explicit uncertainties on G, then we would use control law similar to (31). 7. SOLUTION FOR THE GENERALIZED DTR PROBLEM FOR MULTIPLE ROUTES WITH MULTIPLE SECTIONS In this section, we give a generalize solution for the n alternate route DTR problem described in section 3.1. The space discretized flow equations used for the n alternate routes and n sections are given by (12) and (13). Number of sections for each alternate route I is denoted by ni . We are considering full state observation, which is used for estimating (sensing) the travel 73 Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply. times on the various alternate routes. dynamics can be written as The (47) We have considered a simple first order travel time function, which is obtained by dividing the length of a section by average velocity of vehicles on it. According to that, we amroximate travel time for a route as zy the boundary layer method to replace a continuous controller inside the boundary layer [23], which essentially produces a filter for the high frequency content of the error signal. 8 ti*- 0.1 0.2 8.3 0.0 9.9 9.8 8.7 zyxwvutsr zyxwvu zyxwv zyxwvut 0.6 4 t"ij The system can be written in the standard nonlinear input affine form X(t) = f(x, t) ig(x, t)u(t) y(t) = h(x,t ) (50) where (5 1) x = [Prl...P1,,,... P,,l ... P,,,,1' >U([)= [Pl...P,,-Il CONCLUSIONS In this paper, we have addressed the real-time traffic control problem for point diversion. A feedback model is developed for control purposes, and sliding mode technique is used to design this feedback controller. First, the simplest case with two alternate routes consisted of a single section each, is studied and a feedback controller using sliding mode technique is developed. Second, the case with two alternate routes with two discrete sections is analyzed and a feedback controller using sliding mode technique is also developed. Finally, the general case with multiple alternate routes divided into multiple sections is analyzed and a general solution is proposed. To illustrate the feasibility of above models, a simulation run is performed for a network topology of two alternate routes. The feedback controller developed for this test network performed well. zyxwvutsrqponm zyxwv zyx The output vector is denoted by y, and is givcn by : Y = [ Y I ~2 Y, * * * ~ n - i l (52) where, (53) Y 1 = Xi+, (t) - X I (t) This equation can be differentiated with respect to time to give the travel time difference dynamics. y, kl+l,lPi+II k,,lPII (54) Pl+Il P '=I kl12(l -2)' =z I=! -2 k 1+1,2(1 --)' P m i + ~1 Pmlj where k,,, denotes a constant p=l or 2 that belongs to section j of route i, similar to the constant k described for (39). This system is input-output square and can be solved since each output equation is decoupled and can be solved by the sliding mode control law. 8. SIMULATION The test network consists of two alternate routes The demand function is a constant flow of 325 vehicles per 15 minutes and it is then increased to 1000 vehicles per fifteen minutes. We have assumed full compliance of users for this sample case. The results bhown in the figure below illustrate that the travel times become equal after the successful implementation of the controller developed in this paper. One drawback which is evident from the figure is the high frequency chattering encountered using the sliding mode control. To eliminate this problem, we can use REFERENCES Gupta, A., Maslanka, V., and Spring, G., ' Development of a Prototype KBES in the Management of nonrecurrent Congestion on the Massachusetts Turnpike', 71st Annual TRB, Jan. 1992. Ketselidou, Z., 'Potential use of Kzowledge-Based Expert System for Freeway Incident Management,' ITS, University of California, Irvine, 1993. Peeta S., and Mahmassani H. ' Multiple user classes real-time traffic assignment for on-line operations: a rolling horizon solution framework'. Transportation Research Part C. Volume 3C, Issue 2, pp. 83-98, April 1995. Papageorgiou, M., and Messmer, A., 'Dynamic Network Traffic Assignment and Route Guidance via 7 Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply. zy zyxwvutsr zyxwvutsr zyxw Feedback Regulation’, Transportation Research Board, Washington, D.C., Jan. 1991. Papageorgiou, M., ‘Dynamic Modeling, Assignment and Route Guidance Traffic Networks’, Transportation Research-B, Vol. 24B, No. 6, 471-495, 1990. Messmer, A , and Papageorgiou, M., ‘Route Diversion Control in Motorway Networks via Nonlinear Optimization’, IEEE Trans. on Control Systems Tech., Vol. 3, No. 1, March 1995. Messmer, A , and Papageorgiou, M., ‘Optimal Freeway Network Control via Route Recommendation’, Vehicle Navigation & Information Systems Conference Proc., 1994. Friesz, T. L., Luque, J., Tobin, R. L., and Wie, B. W., “ Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem”, Operations Research, 37, 893-901, 1989. Tan, B., Boyce, D. E., and LeBlanc, L. J., “ A New Class of Instantaneous Dynamics User-Optimal Traffic Assignment Models”, Operations Research, 41, 192-202, 1993. Pushkin Kachroo, Kaan Ozbay, Sungkwon Kang, and John A. Burns, “System Dynamics and Feedback Control Formulations for Real Time Dynamic Traffic Routing with an Application Example” conditionally accepted for Intelligent Transportation Systems - Traffic Sensing and Management, Journal on mathematical and Computer Modelling (MCM). Pushkin Kachroo and Kaan Ozbay, “Real Time Dynamic Traffic Routing Based on Fuzzy Feedback Control Methodology”, Transportation Research Record 1556, 1996. Pushkin Kachroo and Kaan Ozbay, I’ Solution to the User Equilibrium Dynamic Traffic Routing Problem using Feedback Linearization under review for Transportation Research: Part B. Pushkin Kachroo, and Kaan Ozbay “Feedback Control Solutions to Network Level User-Equilibrium RealTime Dynamic Traffic Assignment Problems”, IEEE Southeastcon ‘97, Blacksburg, VA, April 12-14, 1997. Papageorgiou, Applications of Automatic Control Concepts to Traffic Flow Modeling and Control, SpringerVerlag, 1983. Kuo, B. C., Autoinntic Control Systems, Prentice Hall, Inc., 1987. Mosca, E., Optimal Predictive and Adaptive Control, Prentice Hall, 1995. Berger, C.R., and Shaw, L., Discrete event simulation of freeway traffic. Simulation council Proc., Series 7, 1977, 85-93. Payne, H. J., Models of freeway traffic and control. Simulation Council Proc., 1971, 51-61. 191 Isidori, A., Nonlinear Control Systems, Springer-Verlag, 1989. 201 Slotine, J. J. E., and Li, Weiping, Applied Nonlinear Control, Prentice Hall, 1991. 211 Lighthill, M. J., and Whitham, G. B., ‘On kinematic waves 11. A theory of traffic flow on long crowded roads’, Proc. of the Royal Society of London, Series A 229, 1955, 317-345. Godbole, D. N., and Sastry, S. S., “Approximate Decoupling and Asymptotic Tracking for MIMO Systems”, IEEE Transactions on Automatic Control, No. 3, Vol. 40, pp. 44 1-450, March 1995. Pushkin Kachroo, and Masayoshi Tomizuka, “Chattering Reduction and Error Convergence in the Sliding Mode Control of a Class of Nonlinear Systems”, IEEE Transactions on Automatic Control, vol. 41, no. 7, July 1996. zyxwvu zyxwvutsrq ’I, 75 Authorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 19:59:08 UTC from IEEE Xplore. Restrictions apply.