Optimization-Based Feedback Control for Pedestrian ... - IEEE Xplore

2 downloads 0 Views 317KB Size Report
Dec 5, 2011 - for pedestrian evacuation from an exit corridor. Exit corridors will have high usage demand during an evacuation, because all the pedestrians ...
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

1167

Optimization-Based Feedback Control for Pedestrian Evacuation From an Exit Corridor Apoorva Shende, Mahendra P. Singh, and Pushkin Kachroo

Abstract—The evacuation of pedestrians is the most important task when a building is subjected to a significant level of threat that compromises occupant safety. However, very few studies have dealt with the problem of controlling pedestrian evacuation in real time. Due to modern developments in sensor technology and computational facilities, it now seems possible to attempt a real-time controlled evacuation by instructing pedestrians to adjust their velocities according to an algorithm to effect an efficient evacuation. This paper deals with the development of such a control algorithm for an exit corridor where high congestion can be expected during evacuation. To accommodate the possible variation in the pedestrian density along the length, the corridor is divided into several sections. Using the conservation of pedestrian mass, ordinary differential equations that define the pedestrian flow in all sections are developed. For the system of state-space equations that define the flow in all the sections of the corridor, an optimization-based feedback control scheme is developed, which ensures the maximum input discharge subject to tracking the critical state and boundedness of the control variables. Simulation results are obtained, which indicate the superior performance of the controlled flow over the uncontrolled flow. The proposed flow control is also applicable to the regulation of vehicular traffic on a long section of a freeway in urban areas that receives input at several ramps along its length. Index Terms—Conservation of mass, feedback linearization, linear programming, pedestrian evacuation, traffic flow models.

I. I NTRODUCTION

T

HE EVACUATION of occupants is of primary concern when a building is engulfed in an emergency situation caused by fire, an earthquake, or a terrorist attack. In such situations, an organized evacuation of the occupants from the building is most desired. To accomplish such an efficient evacuation of pedestrians, we propose a feedback control algorithm for pedestrian evacuation from an exit corridor. Exit corridors will have high usage demand during an evacuation, because all the pedestrians in the building will have to use one exit

Manuscript received August 11, 2009; revised July 31, 2010 and February 20, 2011; accepted April 10, 2011. Date of publication May 27, 2011; date of current version December 5, 2011. This work was supported in part by the National Science Foundation (with Dr. S. C. Liu as the Program Director) under Grant CMS-0428196. Any opinion, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. The Associate Editor for this paper was Z. Li. A. Shende and M. P. Singh are with the Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, VA 24060 USA (e-mail: [email protected]; mpsingh@ vt.edu). P. Kachroo is with the Transportation Research Center and the Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, NV 89154 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TITS.2011.2146251

corridor or another to finally exit the building in a relatively short span of time. Pedestrians would enter such a corridor from its back end and adjacent rooms and exit from the front end into the free space. The broad objective of control here is to achieve a smooth flow of pedestrians without the development of any congestion that leads to stoppage. This paper must, of course, be complimented with the development of sensors (e.g., cameras) to sense the pedestrian congestion level and actuators (e.g., video displays, light arrays, and speakers) to convey the desired control actions to the pedestrians. In addition, a study of human responses to the signals from actuators is required for implementation in a real building. These issues, however, are not a part of this paper. This paper assumes that the pedestrian flow in an evacuation scenario is directional, i.e., it is only in the direction toward the exit. For such situations, it is possible to use the techniques of the control theory to improve the flow characteristics. The pedestrian flow will, indeed, be directional in an evacuation scenario, because all pedestrians share the common objective of leaving the building through the exit closest to them. This paper identifies the parameters that can be used to control the pedestrian motion during evacuation and develops a modelbased control algorithm to adjust these parameters to reach a desired outcome. In the study of pedestrian flow, both microscopic and macroscopic flow models have been considered. In microscopic flow models, the flow dynamics are based on the motion of individual pedestrians, which results from their purpose of performing the motion and their interactions with other pedestrians and surrounding influences. These models are best suited for including the behavioral aspects of pedestrians in flow modeling, thus generating detailed simulations. More details with regard to microscopic modeling can be found in [1]–[12]. A macroscopic model, on the other hand, gives a big-picture averaged view of the pedestrian flow situation. It is based on the assumption that the pedestrian flow can be treated as a continuum. This assumption directly results in the use of conservation equations in flow modeling. The detailed interactions of pedestrians are overlooked to attain the global flow scenario in terms of relatively fewer and simpler equations. Various macroscopic models that address different aspects of pedestrian flow can be found in [13]–[19]. In this paper, we adopt a macroscopic model to represent the pedestrian flow similar to the vehicular traffic flow model [20]. Although microscopic models may more accurately predict the pedestrian motion, they are typically rule based, and with a large number of people involved in an evacuation, they can be computationally intensive and, hence, unsuitable for real-time

1524-9050/$26.00 © 2011 IEEE

1168

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

application. A control algorithm based on such models will require communicating instructions to individual pedestrians, making it difficult for practical implementation. For the purpose of this paper, a traffic-flow-based macroscopic model is more suitable, because it can represent the directed nature of pedestrian flow in an exit corridor. A comprehensive compilation of various pedestrian-flow-modeling techniques based on trafficflow-modeling concepts can be found in [21]. Our focus in this paper is to develop a feedback control strategy for pedestrian evacuation. Toward that end, we formulate a system of statespace equations using a macroscopic model. The novelty of our approach lies in using an optimization procedure in the feedback control loop to attain state tracking and certain other desirable outcomes by adjusting the flow rate and flow velocity control variables. Thus, our approach is significantly different from traffic signal timing optimization techniques in the traffic literature, e.g., [22] and references therein, where the control variables of signal timing are used to optimize the traffic by inducing stop-and-go waves along a long stretch of road. In Section II, we present the traffic-flow-based model of pedestrian movement in an exit corridor. For this flow model, a likely uncontrolled-flow scenario is described in Section III for its use in the pedestrian-flow simulation. For such a likely uncontrolled scenario, the simulation results indicate the development of congestion very soon, thus providing the motivation for implementing control strategies to avoid such undesirable occurrences. In Section IV, an optimization-based control algorithm is presented. To ensure the existence of a solution to the optimization problem proposed in Section IV, an optimal gain-scaling technique is proposed in Section V. This approach is followed by Section VI, which presents the simulation results for the controlled- and the uncontrolled-flow scenarios. This paper is summarized and concluded in Section VII. II. P EDESTRIAN -F LOW M ODEL IN A C ORRIDOR As aforementioned, we model the flow of pedestrians at the macroscopic level. The flow modeling in an exit corridor differs from the flow modeling in a normal corridor, because we do not have to account for the congestion beyond the exit end. To account for the possible variation of the pedestrian density along the length of the corridor, we divide the corridor into n sections and assume that pedestrian density in a section is uniform along its length. Fig. 1 shows a corridor that is divided into three sections that can easily be generalized to n sections. Let the uniform pedestrian density associated with section i be ρi . These pedestrian densities constitute the state of the system described by the n-dimensional state vector that consists of n section densities. We assume that the section flow velocity vi and section density ρi in a section i satisfy the following functional relationship:   ρi vi = f (1) vf i ρm where ρm is the jam density, i.e., the density at which the section flow velocity vi = 0, and vfi is the free-flow velocity in section i. Free-flow velocity vfi is the velocity at section density ρi = 0. We require that f (.) is a decreasing function

Fig. 1.

Flow in a corridor divided into sections.

of its argument and that f (0) = 1 and f (1) = 0, implying that zero density corresponds to the free-flow velocity and jam density corresponds to zero velocity, respectively. Equation (1) represents the fundamental diagram of pedestrian flow. A discussion on various fundamental diagrams are available in the pedestrian-flow literature (see [23]–[27]). These different fundamental diagrams will result in a different functional forms for f (.). In addition to the pedestrian density, the urgency in performing the motion affects the pedestrian velocity. We assume that this urgency can be conveyed through devices such as an array of flashing light bulbs or video displays or speakers and will directly impact the free-flow velocity parameter. Thus, we will use the section free-flow velocities vfi for 1 ≤ i ≤ n as a set of control input parameters in this paper. Using (1), we get the following relation for the output discharge qi from section i:   ρi qi = ρi vi = ρi f (2) vf i . ρm Let ρcr denote the density that corresponds to the maximum discharge for a given function f (.). Thus, based on (2), we get   ρcr = ρm arg max xf (x) (3) x

and this maximum value of the discharge is equal to qmax = ρcr f (ρcr /ρm )vfi .. To develop the state-space equations that govern the evolution of the section densities ρi for 1 ≤ i ≤ n, we apply the principle of the conservation of mass to each section i. Here, the term conservation of mass is meant to signify the “conservation of pedestrians,” and we use the term pedestrian mass to signify the number of pedestrians. Let the output discharge from section i be denoted by qi and the input discharge into section i from section i − 1 be denoted by qi−1 . For 1 ≤ i ≤ n, qi is given by (2). The input discharge into the first section q0 represents the input discharge into the corridor from the rest of the corridors that adjoin the exit corridor at the back end, and thus, it is not specified by (2). In fact, this discharge can be used as a control variable. In addition to these input and output discharges, we denote the total discharge from the rooms attached to section i by q r(i) . Note that the back input discharge q0 is absent if there are no pedestrians behind the corridor. Similarly, the section room input discharge q r(i) is absent if no pedestrians are present in the rooms adjacent to the ith section. We introduce a parameter δ0 that corresponds to

SHENDE et al.: OPTIMIZATION-BASED FEEDBACK CONTROL FOR PEDESTRIAN EVACUATION

the back input discharge q0 and parameters δi that correspond to the room discharges q r(i) to indicate their presence or absence. δ0 assumes the values   0 if back input discharge is absent δ0 = (4) 1 if back input discharge is present. Similarly, δi assumes the values   0 if the ith room discharge is absent δi = 1 if the ith room discharge is present

(5)

for 1 ≤ i ≤ n. The parameters δ0 and δi will be given for a control or a simulation problem. At the beginning of the evacuation process, δ0 = 1 and δi = 1, implying that there are people in the adjacent rooms and at the back end. Let Li denote the length of section i. By a simple application of the conservation of pedestrian mass for sections 1 ≤ i ≤ n, we obtain the following equations that describe the rate of change of the section density in terms of input and output discharges and section length Li :   δ0 q0 + δ1 q r(1) − ρ1 f ρρm1 vf1 ρ˙ 1 = (6) L1     vfi−1 − ρi f ρρmi vfi δi q r(i) + ρi−1 f ρρi−1 m ρ˙ i = Li 2 ≤ i ≤ n.

(7)

From the simulation and control development point of view, we are more interested in knowing the level of density in a section relative to the maximum density (jam density) that can occur rather than the actual density at every instant of time. We will thus express the flow equations in terms of the normalized values of the section densities and other variables. For this approach, we normalize the flow equations (6) and (7) by appropriately dividing by the maximum density ρm . After minor algebraic operations, we obtain the following normalized state-space equations:   vf 1 b1 (8) ρ˜˙ 1 = δ0 q˜0 + δ1 q˜r(1) − ρ˜1 f (ρ˜1 )˜   ρi−1 )˜ vfi−1 − ρ˜i f (˜ ρi )˜ vf i bi ρ˜˙ i = δi q˜r(i) + ρ˜i−1 f (˜ 2 ≤ i ≤ n. (9) In (8) and (9), ρ˜i = ρi /ρm denotes the normalized section density, q˜0 = q0 /ρm L denotes the normalized back input discharge, q˜r(i) = q r(i) /ρm L for 1 ≤ i ≤ n denotes the normalized room discharge associated with section i, and v˜fi = vfi /L for 1 ≤ i ≤ n denotes the normalized free-flow velocity associated with section i. The normalization constants bi = L/Li for 1 ≤ i ≤ n are also introduced in the normalization. Hereafter, we will refer to (8) and (9) as the state-space equations. The normalized section densities ρ˜i for 1 ≤ i ≤ n are the state variables. The normalized back input discharge q˜0 , the normalized room discharges q˜r(i) for 1 ≤ i ≤ n, and the normalized free-flow velocities v˜fi for 1 ≤ i ≤ n are the control variables for our problem. To ensure the implementability

1169

of the controls, we impose the following bounds on the control variables: 0 ≤ v˜fi ≤ v˜fm , 0 ≤ q˜0 ≤ q˜m q˜m , 0 ≤ q˜r(i) ≤ n

1≤i≤n 1 ≤ i ≤ n.

(10) (11) (12)

In (10), v˜fm = vfm /L, where vfm is the maximum possible free-flow velocity. In (11) and (12), q˜m = qm /ρm L, where qm = ρcr f (ρcr /ρm )vfm is the maximum possible discharge allowable for a given f (.) in (1) that corresponds to ρ = ρcr and vf = vfm . In (12), we have set the upper bound that corresponds to the section room discharge to q˜m /n, because the corridor is divided into n sections, and we assume that the total maximum room discharge from all the n sections is qm . This total maximum discharge that corresponds to the entire corridor when equally divided into n sections is assumed to be the upper bound for the individual section room discharges and, when expressed in the normalized form, is equal to q˜m /n. The model developed in this section is used to develop a control algorithm for the case when there is at least one input discharge present, i.e., at least one of δi = 1 for 0 ≤ i ≤ n. The feedback control algorithm for the case when no input discharge is present, i.e., all δi = 0 for 0 ≤ i ≤ n, is presented in [21, Ch. 7]. III. U NCONTROLLED -F LOW S CENARIO In an evacuation situation in an uncontrolled scenario, it is reasonable to assume that the pedestrians would try to move at the maximum possible speed that corresponds to the section density. Such an action implies that the free-flow velocities in all sections initially start at their maximum values, i.e., v˜fi = v˜fm .

(13)

We also expect the pedestrians to enter the corridor from the back end and from adjoining rooms at the maximum rate, whenever they are present. This situation can be represented by assigning the maximum possible value to the normalized backend input discharge and the room discharges as follows: q˜0 = q˜m q˜m , q˜r(i) = n

(14) 1 ≤ i ≤ n.

(15)

Thus, for the uncontrolled-flow scenario, we use (13)–(15) to determine the time evolution from (8) and (9). However, if any of the section i reaches jam density, then we assume that the flow in that section completely stops. We notionally set the free-flow velocity in that section to 0, indicating the stoppage of flow. A direct consequence of a section i getting jammed will be that the section room discharge associated with it is zero, i.e., q˜r(i) = 0. This case would happen, because the pedestrians from adjoining rooms will not enter a section that is jammed. Another important consequence of a section i ≥ 2 getting jammed is that the pedestrians in the preceding section i − 1 will not move, because their forward motion is completely blocked. This case would happen even if the density in section i − 1, ρ˜i−1 ≤ 1 or the section i − 1 is not jammed. Thus, to

1170

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

account for this stoppage of motion in the section i − 1, we set the free-flow velocity in that section v˜fi−1 = 0. In case in which the first section (i = 1) gets jammed, we set the back-end input discharge into the corridor q˜0 = 0. The numerical results that will later be presented for the uncontrolled scenario indicate that, within a few seconds of the start of evacuation, various locations in the corridor get jammed. This case indicates a need for adjusting the control variables, which were set at their maximum values to represent the uncontrolled scenario. A systematic procedure for computing such control variable values is described in the following section. IV. C ONTROL A LGORITHM Because our ultimate aim is to evacuate all the pedestrians from the building, it might seem like a good control objective to track all the section densities to 0. However, a control algorithm that accomplishes this condition would result in preventing the pedestrians from the back end and the adjacent rooms from entering into the corridor. Hence, a better control objective is to track the congestion in the corridor at some optimal level that ensures high pedestrian discharge. The maximum discharge for a given free-flow velocity value is attained for the critical density value of ρcr , which corresponds to the assumed function f (.) in (1). Thus, one of the control objectives here is to attain the exponential convergence of the normalized section densities ρ˜i in every section i to the critical normalized density value of ρ˜cr = ρcr /ρm that corresponds to the assumed model. Approaching this critical density ensures that, for a given free-flow velocity, the corresponding discharge exponentially approaches the critical discharge. Along with this exponential convergence to the critical density, we also want the control algorithm to achieve maximum input discharge into the corridor and ensure that the explicitly specified control variable bounds are met. To achieve the exponential convergence, we need the time derivatives of the section densities ρ˜i to satisfy the following condition: ρi − ρ˜cr ), ρ˜˙ i = −ki (˜

1≤i≤n

= −ki (˜ ρi − ρ˜cr ),

n 

2 ≤ i ≤ n.

(18)

That is, we have manipulated the control inputs such that the right-hand sides of (8) and (9) that were nonlinear in the state and control variables become equal to the right-hand sides of (16) that are affine in the state variables. Let us assume that we have p inflow discharges into the corridor. Because we have at most n section room input discharges and at most one back input discharge, we have p ≤ n + 1. In addition, because we assume that at least one inflow discharge exists, we have 1 ≤ p ≤ n + 1. Each of the p discharges can be used as a control variable, and δi = 1 for i, which corresponds to these input discharges. δi = 0 for all other i. In addition to these input discharge control parameters, we have n freeflow velocity control parameters vfi . Thus, (17) and (18) are n independent equations with n + p unknown control variables. Therefore, this expression is an overdeterminate system of equations in control variables. To solve for the control variables, we formulate an optimization problem, the solution of which satisfies (17) and (18) and the control bounds (10)–(12) and, at the same time, optimizes a certain function of these control variables. Because we would like to maximize the total inflow discharge into the corridor to speed up the evacuation, we solve an optimization problem that maximizes this quantity at every time instant. The total inflow discharge into the corridor comprises the back-end input discharge q˜0 and the room discharges q˜r(i) into different corridor sections, depending on their availability. Thus, our payoff function h can be defined as follows: h = δ0 q˜0 +

(16)

where ki are the control gain parameters. The larger the value of ki is, the faster the normalized densities will approach the critical values of ρ˜cr . The time derivatives of the normalized section densities ρ˜i are given by (8) and (9). However, we can adjust the control variables q˜0 (input discharge at the back end), v˜fi (section free-

Maximize h = δ0 q˜0 +

flow velocities), and q˜r(i) (room discharge in sections) such that the right-hand sides of (8) and (9) are equal to the right-hand sides of (16) for all 1 ≤ i ≤ n. That is, the control inputs variables need to satisfy the following equations at every time instant:   δ0 q˜0 + δ1 q˜r(1) − ρ˜1 f (˜ ρ1 )˜ vf1 b1 = −k1 (˜ ρ1 − ρ˜cr ) (17)   δi q˜r(i) + ρ˜i−1 f (˜ ρi−1 )˜ vfi−1 − ρ˜i f (˜ ρi )˜ vf i bi

n 

δi q˜r(i) .

(19)

i=1

We note that (17) and (18) are linear in the control variables q˜0 , v˜fi , and q˜r(i) . In addition, the payoff function in (19) is a linear function of the control variables. Thus, the optimization problem consists of a linear payoff subject to linear equality and inequality constraints. This linear-programming problem can be stated as (20), shown at the bottom of the page.

δi q˜r(i)

i=1

subject to

vf1 b1 = −k1 (˜ ρ − ρ˜cr ) δ0 q˜0 + δ1 q˜r(1) − ρ˜1 f (ρ˜1 )˜

1 r(i) δi q˜ + ρ˜i−1 f (˜ ρi−1 )˜ vfi−1 − ρ˜i f (˜ ρi )˜ vf i b i = −ki (˜ ρi − ρ˜cr ) 2≤i≤n 1≤i≤n 0 ≤ v˜fi ≤ v˜fm 0 ≤ q˜0 ≤ q˜m 1≤i≤n 0 ≤ q˜r(i) ≤ q˜nm

(17) (20) (18) (10) (11) (12)

SHENDE et al.: OPTIMIZATION-BASED FEEDBACK CONTROL FOR PEDESTRIAN EVACUATION

The solution of this linear-programming problem at every time instant gives us the control variable values. We note that, for any optimization problem, there is a risk of running into constraints that are infeasible. This case can happen if the control gain parameter values are very high. In such cases, we need to scale the gains ki such that the constraints become feasible. A systematic procedure to do this approach is outlined in the next section. For the moment, we assume that the constraints of the linearprogramming problem of (20) are feasible, and thus, there exists a solution to this problem. The following theorem states an important property about this solution. Theorem 1: The solution of the linear-programming problem [see (20)] q˜0 , q˜r(i) , and v˜fi for i = 1, . . . , n also maximizes the normalized output discharge from the corridor at every time instant given by vf n o˜d = ρ˜n f (ρ˜n )˜

(21)

subject to the same set of constraints [see (10)–(12), (17), and (18)] as in the linear-programming problem (20). Proof: To prove this result, we show that, if the variables q˜0 , q˜r(i) , and v˜fi for i = 1, . . . , n of the linear-programming problem [see (20)] satisfy its constraints [see (10)–(12), (17), and (18)], then the objective h and o˜d differ by a constant in terms of these variables. This case will imply that, for the same set of constraints, the same solution will maximize both h and o˜d . To show this case, we divide all the equality constraints (17) and (18) by bi for each 1 ≤ i ≤ n and add them up. In doing this approach, all the terms, except for δ0 q˜0 , ρn )˜ vfn , cancel out, resulting in the following δi q˜r(i) , and ρ˜n f (˜ expression: n n   ki − (˜ ρi − ρ˜cr ) = δ0 q˜0 + δi q˜r(i) − ρ˜n f (˜ ρn )˜ vf n . b i i=1 i=1

(22)

Based on (19) and (21), we can rewrite (22) as −

ki (˜ ρi − ρ˜cr ) = h − o˜d . bi

(23)

The left-hand side of (23) is fully determined by the normalized state variables ρ˜i , normalization parameters bi , gains ki for i = 1, . . . , n, and the normalized critical density ρ˜cr , which are given parameters for the linear-programming problem in (20). Thus, it is a constant in terms of the optimization variables q˜0 , q˜r(i) , and v˜fi for i = 1, . . . , n. The right-hand side, on the other hand, is a difference h and o˜d . Thus, we have shown that h and o˜d differ by a constant in terms of the variables of the optimization problem. This case implies that the solution of the linear-programming problem that maximizes h under its constraints [see (10)–(12), (17), and (18)] also maximizes o˜d , subject to the same set of constraints.  Remark: The aforementioned theorem implies the dual optimality of our evacuation control algorithm, because it shows that, while tracking the critical density that corresponds to

1171

the maximum discharge, it maximizes both the total input and the output discharges of the corridor. Section densities that asymptotically approach the critical density is a highly desirable aspect of the control, because critical density implies the maximum flow discharge for a given free-flow velocity. Although it is true that we cannot enable the absolute maximum ρcr )˜ vfm , we should, in general, output discharge given by ρ˜cr f (˜ expect a very high level of discharge due to Theorem 1 and the tracking of critical densities in every section. Because we can affect the section densities only through the control input parameters q˜0 , q˜r(i) , and v˜fi for i = 1, . . . , n, we cannot expect the end-section normalized density ρ˜n to suddenly jump to critical density ρ˜cr at one time instant. Thus, the control methodology that results in an asymptotic approach of the congestion to the critical density is well suited for this problem. In addition, we ensure the moderation of the evacuee movement effort through bound constraints on the control variables given by (10)–(12). Thus, our technique and results in Section VI, which are indicative of highly improved flow conditions for the control algorithm, further reinforce the finding in [28] that moderate effort to evacuate results in far better evacuation flow patterns than a hasty panic-based movement. V. G AIN S CALING FOR I NFEASIBLE C ONSTRAINTS In this section, we describe a strategy for adjusting the gains in case the constraints (equality and bound) in the linearprogramming problem in (20) are infeasible. The idea again exploits the linearity of the equality constraints in (17) and (18) in the control variables v˜fi , q˜0 , and q˜r(i) . This linearity q0 )0 , and (˜ q r(i) )0 satisfy (17) and (18) implies that, if (˜ vfi )0 , (˜ 0 0 vfi ) /ν, (˜ q0 )0 /ν, and (˜ q r(i) )0 /ν satisfy with gains kρi , then (˜ 0 (17) and (18) with gains kρi /ν. Thus, in case the constraints (equality and bound) in the linear-programming problem (20) are infeasible because of an improper selection of gains kρ0i , then we find a solution (˜ q0 )0 ≥ 0, (˜ vfi )0 ≥ 0, and q˜r(i) ≥ 0, satisfying only (17) and (18) with these values of the gains kρ0i . Then, we find a least scaling factor ν > 0 such that (˜ vfi )0 /ν, q r(i) )0 /ν not only satisfy (17) and (18) with (˜ q0 )0 /ν, and (˜ gains kρ0i /ν but the bounds (10)–(12) as well. However, the goal is to find the initial solution such that the cumulative violation of the original bounds is a minimum. We now describe the procedure of finding the optimal solution that violates the original bounds such that the cumulative violations of the original bounds are minimum. For this condition, we formulate a linear-programming problem as follows. First, we relax the upper bound on all the bound constraints of (10)–(12) by shifting the upper bound all the way to ∞, resulting in the following set of new relaxed constraints: 0 ≤ v˜fi ≤ ∞,

1≤i≤n

0 ≤ q˜0 ≤ ∞ 0 ≤ q˜r(i) ≤ ∞

(24) (25)

1 ≤ i ≤ n.

(26)

Next, we proceed to quantify the violations of the original bounds that were relaxed. To do this approach, we introduce

1172

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

a slack and a surplus variable that corresponds to each bound that has been relaxed. Because the upper bound on all the control variables is relaxed, we need to introduce a slack and a surplus variable that corresponds to each variable. The slack variables denote the amount by which the original bound has been satisfied, and the surplus variable denotes the amount by which the original bound has been violated. Each of the slack and the surplus variables satisfy the nonnegativity constraint. Let the slack variables that correspond to v˜fi , q˜0 , and q˜r(i) be denoted by (vfi )sl , (q0 )sl , and (q r(i) )sl , respectively. Let the surplus variables that correspond to v˜fi , q˜0 , and q˜r(i) be denoted by (vfi )su , (q0 )su , and (q r(i) )su , respectively. The nonnegativity of the slack and the surplus variables results in the following bound constraints on these variables: 0 ≤ (vfi )sl ≤ ∞,

1≤i≤n

0 ≤ (vfi )su ≤ ∞,

1≤i≤n

(27) (28)

0 ≤ (q0 )sl ≤ ∞

(29)

0 ≤ (q0 )su ≤ ∞

(30)

0 ≤ (q r(i) )sl ≤ ∞,

1≤i≤n

(31)

0 ≤ (q r(i) )su ≤ ∞,

1 ≤ i ≤ n.

(32)

Because the slack and the surplus variables, respectively, represent the amount by which a relaxed constraint has been satisfied or violated, we have the following equality relations between the optimization variables, the original upper bound, and the slack and the surplus variables: v˜fi = v˜fm + (vfi )su − (vfi )sl ,

1≤i≤n

q˜0 = q˜m + (q0 )su − (q0 )sl q˜r(i)

  q˜m  r(i)  + q = − q r(i) , su sl n

n   i=1

1 ≤ i ≤ n. (35)

su

Minimize n



 (vfi )su + δi q r(i) su g = δ0 (q0 )su + i=1

subject to

vf1 b1 = −k1 (˜ ρ1 − ρ˜cr ) δ0 q˜0 +δ1 q˜r(1) − ρ˜1 f (ρ˜1 )˜

r(i) δi q˜ + ρ˜i−1 f (˜ ρi−1 )˜ vfi−1 − ρ˜i f (˜ ρi )˜ vf i b i = −ki (˜ ρi − ρ˜cr ) 2≤i≤n 1≤i≤n v˜fi = v˜fm + (vfi )su − (vfi )sl q˜0 = q˜m + (q0 )su − (q0 )sl

q˜r(i) = q˜nm + q r(i) su − q r(i) sl 1 ≤ i ≤ n 1≤i≤n 0 ≤ v˜fi ≤ ∞, 0 ≤ q˜0 ≤ ∞ 0 ≤ q˜r(i) ≤ ∞ 1≤i≤n 1≤i≤n 0 ≤ (vfi )sl ≤ ∞ 0 ≤ (vfi )su ≤ ∞ 1≤i≤n 0 ≤ (q0 )sl ≤ ∞ 0 ≤ (q0 )su ≤ ∞ 1≤i≤n 0 ≤ (q r(i) )sl ≤ ∞ 1 ≤ i ≤ n. 0 ≤ (q r(i) )su ≤ ∞

(17) (18) (33) (34) (35) (24) (25) (26) (27) (28) (29) (30) (31) (32) (37)

(34)

   (vfi )su + δi q r(i) .

problem that we propose to solve to obtain the solution is thus defined as

(33)

To find the solution to (17) and (18) that violates the original bounds in a minimum cumulative sense, we formulate a linearprogramming problem using the aforementioned slack and surplus variables. Because the individual surplus variables define the amount by which a relaxed constraint has been violated, the sum of all the surplus variables denotes the cumulative constraint violations. Thus, the cost to be minimized in the new linear-programming problem is g = δ0 (q0 )su +

Fig. 2. Five sections of the corridor indicated in different symbols. These symbols will be used to plot various variables that correspond to the sections.

(36)

With the introduction of the slack and surplus variables, (33)–(35) now need to be satisfied in addition to the original equalities (17) and (18). In addition, the bounds that correspond to the slack and surplus variables given by (27)–(32) need to be satisfied in addition to the relaxed bounds on the original variables given by (24)–(26). The complete linear-programming

The solution to this problem given by (37) will violate the original bounds in a minimum cumulative sense. To obtain the scaling factor (ν)0 for the infeasible gains kρ0i , we scale the components of the solution that correspond to v˜fi , q˜0 , and q˜r(i) by a factor ν > 0 such that it satisfies the original bounds given by (10)–(12). The minimum of all possible ν > 0 will give us the optimal scaling factor (ν)0 . The feasible gain that will make the original bound feasible is then equal to kρ0i /(ν)0 . VI. N UMERICAL S IMULATION In this section, we present the numerical simulation results obtained for the uncontrolled and controlled scenarios for the case where the corridor receives input from the back end and from all the connecting rooms. Thus, we have δi = 1 for all 0 ≤ i ≤ n. We have used MATLAB to run the simulations. The linear-programming problems were solved using the MATLAB optimization toolbox to obtain the control inputs. As a test case, we consider a corridor that is divided into five equal-length sections, as displayed in Fig. 2. The symbols of the sections indicated in the figure are used to plot the results that correspond to the sections. For example, the numerical results that pertain to section 1 will be represented by a cross, and

SHENDE et al.: OPTIMIZATION-BASED FEEDBACK CONTROL FOR PEDESTRIAN EVACUATION

1173

Fig. 3. Normalized corridor section densities ρ˜i for all sections in the controlled- and uncontrolled-flow scenarios. (a) Uncontrolled-flow case. (b) Controlledflow case.

the numerical results for section 3 will be represented by a triangle. We choose the Greenshields model for traffic flow [29] to define the dependence of the average section flow velocity on the average link density. The Greenshields model defines a linearly decreasing relationship between the average velocity and average density. This relationship for the fundamental diagram has been proposed for pedestrian-flow modeling in the Society of Fire Protection Engineers (SFPE) Handbook of Fire Protection Engineering, which is published by the National Fire Protection Association (NFPA) [26]. It can be stated for section i as follows:   ρi . (38) v i = vf i 1 − ρm Thus, the function f (.) in (1) takes the following form: f (x) = 1 − x.

(39)

The critical density for the Greenshields model is given by ρcr = ρm /2, and the corresponding critical discharge is given by qm = ρm vfm /4. In [26], vfm = 1.4 m/s and ρm = 3.8 pedestrians/m2 are the values for the free-flow velocity and the jam density parameters, respectively. In [26], the jam density is given in terms of (in pedestrians/m2 ). However, because we assume that uniform pedestrian density across the width and the density that we model is per unit length of the corridor, the jam density that corresponds to our model is given by ρm = 3.8 ∗ W pedestrians/m, where W is the width of the corridor (in meters). We assume the physical parameters to be the length of the corridor L = 50 m, the maximum freeflow velocity vfm = 1.4 m/s with a corresponding normalized value of v˜fm = (vfm /L) = 0.028, the normalized maximum vfm /4) = 7 × 10−3 , and the initial pedestrian discharge q˜m = (˜ densities in different sections as ρ˜01 = 0.6933, ρ˜02 = 0.5850, ρ˜03 = 0.2670, ρ˜04 = 0.8000, and ρ˜05 = 0.0290. The initial densities were randomly selected, and the maximum value was scaled to be 0.8. All room discharges are bound by the max-

imum value q˜m /n. For the controlled simulation case, we assume the same gain value for all the sections as ki = k = (1/250) = 4 × 10−3 . However, we present the results in terms of normalized quantities, because they can later be generalized for different jam densities and widths. First, we present the results for the evolution of the normalized densities in different sections in Fig. 3(a) and (b) for the uncontrolled- and controlled-flow scenarios, respectively. The results for the uncontrolled-flow scenario in Fig. 3(a) clearly show that all sections, except for section 5, reach the jam density. Section 4 reaches jam density in about 10 s from the start of the evacuation, section 1 in about 15 s, section 3 in about 30 s, and section 2 in about 140 s. Note that once section 4 has been jammed, the flow behind this section becomes completely obstructed. Because every section receives input from the connected rooms, the density in a section behind the jammed section would eventually reach the jam density. Although section densities in sections 1, 3, and 4 continually increase until they reach the jam density, the density in section 2 initially decreases and then suddenly starts to increase toward the jam density once section 3 has been jammed. This case can be attributed to the fact that until section 3 was unjammed, pedestrians in section 2 can move into section 3, resulting in the decrease in the density of section 2. However, once section 3 has been jammed, the pedestrians in section 2 cannot move into section 3, resulting in an increase in the density in section 2 due to the room input discharge in section 2. It is shown that the density in section 5 stabilizes at a constant value, because pedestrians can uninterruptedly exit from this section. This constant density value is sustained by the room input discharge into this section. The blockages in different sections that were observed in the uncontrolled-flow scenario, however, never occur in the controlled-flow scenario [see Fig. 3(b)]; all section densities exponentially approach the normalized critical density of 0.5, thus ensuring a smooth flow. Fig. 4(a) and (b) shows the evolution of the normalized freeflow velocities in different sections for the uncontrolled- and

1174

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

Fig. 4. Normalized corridor section free-flow velocity v˜fi for all sections in the controlled-flow and uncontrolled-flow scenarios. (a) Uncontrolled-flow case. (b) Controlled-flow case.

Fig. 5. Normalized corridor back-end input discharge q˜0 in the controlled-flow and uncontrolled-flow scenarios. (a) Uncontrolled-flow case. (b) Controlledflow case.

controlled-flow scenarios, respectively. Based on the results for the uncontrolled-flow case shown in Fig. 4(a), we observe that as sections 1–4 are jammed, the normalized free-flow velocities in these sections fall to 0 from their initial maximum values at which they start. Section 5 is never jammed, and its free-flow velocity constantly remains at its maximum value. The line for section 3 is shown to overlap with the line for section 4, because the jamming of section 4 results in both the section free-flow velocities dropping to 0. For the controlled case, on the other hand, the free-flow velocities in all sections gradually change to ensure a smooth uninterrupted flow. The free-flow velocity in section 5 stays all through at the maximum possible value. This case is a desirable consequence of control, because it ensures the highest possible exit discharge that corresponds to the given exit density. Fig. 5(a) and (b) shows the time evolution of the back input discharge in the uncontrolled- and controlled-flow scenarios, respectively. The results for the uncontrolled-flow scenario

shown in Fig. 5(a) clearly indicate that the back input discharge falls to 0 as soon as section 1 is blocked. On the other hand, in Fig. 5(b), the back input discharge gradually increases to a high value of about half the maximum possible value (critically discharge) without any flow interruption. Thus, the corridor remains available to accept input from the back end all the time in the controlled-flow scenario. In Fig. 6(a) and (b), we show the evolution of the normalized room discharges in different corridor sections for the uncontrolled- and controlled-flow cases. The results for the uncontrolled-flow scenario given in Fig. 6(a) show that room inputs to sections 1–4 cease as soon as these sections are jammed. Section 5, which is next to the exit, can accept pedestrians from connected rooms at the maximum possible rate. In the controlled-flow scenario, on the other hand, the corridor sections can accept the input from rooms throughout the simulation. The control algorithms are clearly effective in controlling the discharge values to remain with the established bounds.

SHENDE et al.: OPTIMIZATION-BASED FEEDBACK CONTROL FOR PEDESTRIAN EVACUATION

1175

Fig. 6. Normalized corridor section room input discharge q˜r(i) in the controlled-flow and uncontrolled-flow scenarios. (a) Uncontrolled-flow case. (b) Controlled-flow case.

Fig. 7. Normalized exit discharge and total input discharge in the controlledflow and uncontrolled-flow cases.

Finally, in Fig. 7, we show the plots for the exit discharge and the total input discharge obtained for the uncontrolledand controlled-flow scenarios. The dark lines show the results for the uncontrolled-flow case, and the faint lines show the results for the controlled-flow case. The critical discharge is indicated by the dotted line. In the uncontrolled-flow scenario, we observe that the total input discharge (from the back end and rooms) starts at its maximum possible value, but then, it drops off in four steps to a value that is equal to the maximum room discharge, which occurs in section 5. The first drop coincides with the blocking of section 4 and is equal to the maximum room discharge in this section. The second drop, which is relatively larger than the other drops, occurs when section 1 is jammed, which blocks the back input discharge and the room discharge in section 1. Consequently, the magnitude of this drop is equal to the sum of the room discharge in section 1 plus the back-end input. The third drop occurs when section 3 is blocked, and the last drop occurs when section 2 is blocked. Both the drops are equal to the maximum possible room discharge in the respective sections. The final value of the

total input discharge for the uncontrolled-flow case is equal to the maximum possible room discharge from section 5, which remains open throughout. The figure also shows the exit discharge, which is noted to remain at a low value throughout, with its final value being equal to the final exit discharge, which is the room discharge from section 5. The results for the controlledflow case are, however, quite different from the results of the uncontrolled case. The total input discharge to the system and the exit discharge from the system are shown to smoothly vary and gradually increase to the maximum value in the last section. We also note that the input and exit discharges are initially very close to each other, and finally, they asymptotically approach the same maximum value. Note that the average time taken to solve the optimization problem for computing the control algorithm for 45 runs was tavg = 0.0161 s, which is quite negligible for the pedestrian evacuation system and points toward the implementability of the proposed control algorithm. VII. C ONCLUSION It is critical to control the pedestrian flow in the exit corridors during an evacuation, because there is a likelihood of high congestion due to high demand. Thus, in this paper, we have considered the problem of pedestrian flow in an evacuation situation in an exit corridor. In the proposed model, the corridor is divided into sections to consider the variability in the pedestrian density in different sections. Dividing the corridor into sections also enables us to assign sectionwise control variables. The pedestrian density evolution in the corridor sections is governed by ordinary differential equations obtained by applying the sectionwise conservation of pedestrian mass. For the proposed model, we define the uncontrolled-flow scenario, wherein we assume the fastest possible speed at a given density in the corridor sections. In addition, the flow discharges from the rooms and the back end are assumed to be at their maximum value. However, when a section is jammed, the velocity and the discharges that correspond to that section

1176

IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 12, NO. 4, DECEMBER 2011

drop to 0. Next, we develop the control algorithm to avoid jamming and maximize the pedestrian flow in the corridor. An optimization-based control algorithm is defined to attain tracking, boundedness, and optimality objectives. To ensure the feasibility of the constraints in the optimization procedure of this algorithm, an optimal gain-scaling procedure is presented. The simulation results for the evolution of different flow variables are obtained for the uncontrolled- and controlledflow scenarios. Note that the time taken to compute the controls is negligibly small, which makes it possible to implement the controls in a real intelligent evacuation system. Although we can observe jamming-related flow hindrances in the uncontrolled-flow scenario, no such interruptions are observed in the controlled-flow simulations. Rather, in the controlled-flow scenario, we observe a smooth evolution of density to the critical density, indicating the clear advantages of using a control algorithm for pedestrian-flow regulation.

[21] P. Kachroo, S. J. Al-Nasur, S. A. Wadoo, and A. Shende, Pedestrian Dynamics: Feedback Control of Crowd Evacuation. New York: SpringerVerlag, 2008. [22] E. Tomer, L. Safonov, N. Madar, and S. Havlin, “Optimization of congested traffic by controlling stop-and-go waves,” Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 65, no. 6, p. 065101, Jun. 2002. [23] A. Schadschneider, W. Klingsch, H. Klupfel, T. Kretz, C. Rogsch, and A. Seyfried, “Evacuation dynamics: Empirical results, modeling and applications,” in Encyclopedia of Complexity and System Science. New York: Springer-Verlag, 2009, pp. 3142–3176. [24] V. Predtechenskii and A. Milinskii, Planning for Foot Traffic. New Delhi, India: Amerind, 1969. [25] D. Helbing, A. Johansson, and H. Al-Abideen, “The dynamics of crowd disasters: An empirical study,” Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 75, no. 4, p. 046109, 2007. [26] H. Nelson and F. Mowrer, “Emergency movement,” in SFPE Handbook of Fire Protection Engineering., 3rd ed. Quincy, MA: Nat. Fire Protection Assoc., 2002. [27] S. J. Older, “Movement of pedestrians on footways in shopping streets,” Traffic Eng. Control, vol. 10, pp. 160–163, 1968. [28] D. Helbing, I. Farkas, and T. Viscek, “Simulating dynamic features of escape panic,” Nature, vol. 407, no. 6803, pp. 487–490, Sep. 2000. [29] B. D. Greenshields, “A study in highway capacity,” in Proc. Highway Res. Board, 1935, vol. 14, pp. 458–468.

R EFERENCES [1] S. P. Hoogendoorn, M. Hauser, and N. Rodrigues, “Applying microscopic pedestrian flow simulation to railway station design evaluation in Lisbon, Portugal,” Transp. Res. Rec.: J. Transp. Res. Board, vol. 1878, pp. 83–94, 2004. [2] V. Blue and J. Adler, “Emergent fundamental pedestrian flows from cellular automata microsimulation,” Transp. Res. Rec., vol. 1644, pp. 29–36, 1998. [3] P. G. Gips and B. Marksjo, “A microsimulation model for pedestrian flows,” Math. Comput. Simul., vol. 27, no. 2/3, pp. 95–105, Apr. 1985. [4] D. Helbing, “A mathematical model for the behavior of pedestrians,” Behav. Sci., vol. 36, no. 4, pp. 298–310, Oct. 1991. [5] D. Helbing and P. Molnar, “Social force model for pedestrian dynamics,” Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 51, no. 5, pp. 4282–4286, May 1995. [6] D. Helbing and T. Vicsek, “Optimal self-organization,” New J. Phys., vol. 1, no. 1, pp. 13.1–13.17, 1999. [7] G. G. Lovas, “Modeling and simulation of pedestrian traffic flow,” Transp. Res., vol. 28B, no. 6, pp. 429–443, Dec. 1994. [8] S. Okazaki, “A study of pedestrian movement in architectural space— Part 1: Pedestrian movement by the application on magnetic models,” Trans. A.I.J., no. 283, pp. 111–119, 1979. [9] S. Okazaki and S. Matsushita, “A study of simulation model for pedestrian movement with evacuation and queuing,” in Proc. Int. Conf. Eng. Crowd Safety, 1993, vol. 12, pp. 271–280. [10] P. Thompson and E. Marchant, “A computer model of the evacuation of large-building populations,” Fire Safety J., vol. 24, no. 2, pp. 131–148, 1995. [11] P. Thompson and E. Marchant, “Testing and application of the computer model ‘Simulex,”’ Fire Safety J., vol. 24, no. 2, pp. 149–166, 1995. [12] J. Watts, “Computer models for evacuation analysis,” Fire Safety J., vol. 12, no. 3, pp. 237–245, Dec. 1987. [13] J. J. Fruin, “Designing for pedestrians: A level of service concept,” Highway Res. Rec., no. 355, pp. 1–15, 1971. [14] J. J. Fruin, Pedestrian Planning and Design. New York: Metropolitan Assoc. Urban Designers Environ. Planners, 1971. [15] Transp. Res. Board, Special Report 204 TRB, Highway Capacity Manual, Washington, DC, 1985. [16] R. L. Hughes, “A continuum theory for the flow of pedestrians,” Trans. Res. Part B, vol. 36, no. 6, pp. 507–535, Jul. 2002. [17] R. L. Hughes, “The flow of human crowds,” Annu. Rev. Fluid Mech., vol. 35, no. 1, pp. 169–182, 2003. [18] E. M. Cepolina, “Phased evacuation: An optimization model which takes into account the capacity drop phenomenon in pedestrian flows,” Fire Safety J., vol. 44, no. 4, pp. 532–544, May 2009. [19] A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht, and W. Klingsch, “New insights into pedestrian flow through bottlenecks,” Transp. Sci., vol. 43, no. 3, pp. 395–406, Aug. 2009. [20] A. C. May, Traffic Flow Fundamentals. Englewood Cliffs, NJ: PrenticeHall, 1990.

Apoorva Shende received the B.Tech. and M.Tech. degrees in civil engineering from the Indian Institute of Technology, Bombay, India, in 2003 and the Ph.D. degree in engineering mechanics from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 2008 for his work on optimization-based control for pedestrian evacuation. He is currently a Postdoctoral Associate with the Department of Electrical and Computer Engineering, Virginia Tech. His research interests include autonomous vehicles, optimization-based control, path planning, and stochastic control.

Mahendra P. Singh received the B.Eng. degree in civil engineering and the M.Eng. degree in structural engineering from the University of Roorkee [now Indian Institute of Technology (IIT) Roorkee], Roorkee, India, in 1962 and 1966, respectively, and the Ph.D. degree in civil engineering from the University of Illinois, Urbana-Champaign, in 1972. He is currently the Preston Wade Professor of Engineering with the Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg. His research interests include structural and pedestrian control, structural health monitoring, and structural dynamics.

Pushkin Kachroo received the Ph.D. degree from the University of California, Berkeley, in 1993, conducting research in vehicle control, and the Ph.D. degree in traffic control and evacuation from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, in 2007. He is currently the Director of the Transportation Research Center, University of Nevada, Las Vegas, where he is also a Professor with the Department of Electrical and Computer Engineering. He is the author of 10 books on traffic and vehicle control and a coauthor of more than 100 publications, including books, research papers, and edited volumes. He has graduated more than 30 graduate students and has been the Principal Investigator (PI) or a Co-PI on projects worth more than $3 million. Dr. Kachroo received the Most Outstanding New Professor Award from Virginia Tech and several teaching awards and certificates.