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Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach
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Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach Weiming Xianga,n, Jian Xiaob, Yangsheng Jianga a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China b School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China
Received 12 February 2014; received in revised form 17 July 2014; accepted 27 September 2014
Abstract In this paper, the traffic-responsive signalization problem for an oversaturated intersection is addressed. A static state feedback control strategy is presented to relieve the waiting vehicles in lanes. By modeling the intersection as a discrete-time switched system and with the aid of quadratic Lyapunov function method, the green time of each phase in signalization is determined by a set of linear state feedback controllers, and moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the boundary of waiting queue length is estimated and further minimized by the state feedback control. Then, linear matrix inequality (LMI) technique is employed to numerically tackle the design problem through solving a set of LMI optimization problems. Furthermore, the design results are extended to the case with unknown disturbance, and H1 controller is designed to achieve the disturbance attenuation performance. In the end, a simulation study in comparison with traditional Webster method is presented to illustrate the effectiveness of state feedback control in relieving the oversaturated situation for an intersection. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Traffic signal control is a long-lasting research problem in urban transportation network systems, e.g. in [1–3]. The effectiveness of a traffic signal system can reduce the incidence of delays, stops, fuel consumption, emission of pollutants, and accidents. Moreover, due to the n
Corresponding author. E-mail address: [email protected] (W. Xiang).
http://dx.doi.org/10.1016/j.jfranklin.2014.09.017 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Nomenclature Ai;gp number of vehicles joining the movement i during the green time gp ε boundary distinguishing oversaturated between unsaturated situation Di;gp number of vehicles departing the movement i during the green time gp gp(k) green time in phase state k gmin , gmax lower limit and upper limit of green time I identity matrix Kp state feedback control gain for phase p ωi ðkÞ unknown disturbance of movement i in phase state k ωðkÞ vector of disturbance in phase state k P > ðx > Þ transposition of matrix P (vector x) P g0ðP≽0Þ matrix P is positive definite (semi-positive definite) Pb0 all the elements pij are non-negative where pij is the element in (i, j) position of P qi input flow rate for movement i si saturation flow rate for the movement i up(k) control variables for phase p in phase state k xi(k) queue length of lane or movement i in phase state k xðkÞ vector of queue length in phase state k Jx J Frobenius norm of vector x y(k) output variable in phase state k
rapid growth of traffic congestion, an effective traffic signalization plays an important role of relieve the oversaturated situation such as related papers [4–11] and references cited therein. Most of the signal control strategies are based on fixed-time signal control [1,4–7], which are mainly based on historical data rather than real-time data. In a few recent papers, some on-line signalization methods have been proposed [2,8–13], which are more adaptive to the real-time traffic conditions. As for oversaturated condition, some significant results have been reported. The result in [14], which is probably the first result proposing the two-stage timing method, attempts to find an optimal switch-over point during the oversaturated period to interchange the timing of the approaches. Recently, a discrete delay type model is introduced in [8] to improve the results of [14]. In [9], an optimal traffic light switching scheme was presented. Generally, it resorts to a minimization problem over a set of an extended linear complementarity problem, which is not an easy task when switching cycles are large. And concerned with the recent notable result [10], a dissipative idea is applied into traffic signal design problem, a state feedback controller based on dissipativity-based control is derived. From the control point of view, the traffic-responsive control strategies are supposed to be able to achieve better control performance than fixed-time strategies. As to the real-time responsive control, the static state feedback scheme is the prevailing method for various controlled systems. In particular to linear dynamic systems in state space form, the static linear state feedback control strategy has been widely used in numerous applications, however, few results have been devoted to apply this simple but effective control rule to traffic control issues, e.g. in [15]. But the dynamics of vehicle movements in [15] is modeled as a single discrete-time state space model which is not completely controllable, thus the traffic-responsive control strategy is only applicable to controllable variables. In this paper, Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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we are going to introduce the static state feedback control strategy applicable to all variables in an intersection by means of modeling it as a switched system. Switched system can be efficiently used to model many practical systems which are inherently multi-model in the sense that several dynamical systems are required to describe their behaviors. For more details about switched systems, the reader is referred to some recent papers [16–24] and the references cited therein. Due to the essential nature of multi-mode contained in a signalized intersection, which results from the multiple phases decided by traffic lights, the intersection can be modeled as a switched system composed by several linear subsystems describing the multiple phases in signalization. Based on the switched system model, the key point is to set the green time appropriately to relieve the waiting vehicles within the oversaturated intersection. By considering the green time as the control input of the system, the signalization design is transformed to design a set of state feedback controllers ensuring the state of closed-loop system being able to decrease below a certain bound, which implies the oversaturated situation is relieved. Moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the maximal number of waiting vehicles during the relieving period needs to be considered and less of it would be better and required, thus the optimal state feedback controller guaranteeing the minimal bound of system state is required. The Lyapunov function approach is employed to fulfill above design task. By constructing a Lyapunov function in the quadratic form, sufficient conditions for the existence of state feedback controller are given. Then, to make the design procedure numerically tractable, the linear matrix inequality (LMI) technique is introduced to solve the optimization problem, and to obtain the optimal state feedback control scheme. Furthermore, as an extension of our results, the robust signalization subjected to unknown disturbance is considered. In presence of unknown disturbance, the influence caused by disturbance needs to be attenuated by the control strategy. In this paper, the disturbance attenuation is considered as H1 performance, and the H1 feedback controller is designed. At last, a simulation study is carried to illustrate our results. In comparison to the traditional Webster method, it shows the state feedback control strategy is an effective trafficresponsive approach to release the oversaturated situation. The remainder of this paper is organized as follows. The system modeling and control problem formulation for an oversaturated intersection are presented in Section 2. The main results on the state feedback controller design and LMI formulation are given in Section 3. As an extension to robust signalization, the H1 control strategy design is studied in Section 4. Simulation study is in Section 5, and conclusions are given in Section 6.
2. System description and problem formulation 2.1. System description and modeling The urban transportation system is composed by a network of intersections, and generally, a signalized intersection is operated by a traffic signal that decides the movements of vehicles to pass the intersections or have to stop to generate the queues. The movement may include vehicles going straight, turning left, turning right, or a combination of them. In general, a signalized intersection has n phases, where n Z 2, and each phase is allotted a certain amount of time (green time) during which a group of traffic lanes is allowed to proceed. Here, in order to show our control idea clearly, a typical single intersection with 8 lanes and the traffic signal has 4 phases is considered, which is illustrated in Fig. 1. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Fig. 1. An intersection with 8 lanes and 4 phases.
The discrete time state space system model for the intersection in Fig. 1 is going to be proposed to characterize the evolution of queue lengths between the phase switching instant k and k þ 1. Taking the Phase 1 for example, the queue length xi(k) evolves according to xi ðk þ 1Þ ¼ xi ðkÞ þ Ai;g1 Di;g1 . For movements 1 and 5 which have same characteristics, it is obtained that Ai;g1 ¼ qi g1 ðkÞ, i¼ 1,5 and Di;g1 ¼ si g1 ðkÞ, i¼ 1,5 , thus we have the following equation in Phase 1: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg1 ðkÞ;
i ¼ 1; 5
ð1Þ
And for other movements i ¼ 2; 3; 4; 6; 7; 8, since the movements are stopped to generate queues, which implies Ai;g1 ¼ qi g1 ðkÞ, i ¼ 2; 3; 4; 6; 7; 8 and Di;g1 ¼ 0, i ¼ 2; 3; 4; 6; 7; 8, it is therefore obtained as xi ðk þ 1Þ ¼ xi ðkÞ þ qi g1 ðkÞ;
i ¼ 2; 3; 4; 6; 7; 8
ð2Þ
Following the similar guideline in Phase 1, the rest of evolution equations for Phase 2–4 can be derived as follows: Phase2: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg2 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g2 ðkÞ;
i ¼ 2; 6
i ¼ 1; 3; 4; 5; 7; 8
ð3Þ ð4Þ
Phase3: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg3 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g3 ðkÞ;
i ¼ 3; 7
i ¼ 1; 2; 4; 5; 6; 8
ð5Þ ð6Þ
Phase4: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg4 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g4 ðkÞ;
i ¼ 4; 8
i ¼ 1; 2; 3; 5; 6; 7
ð7Þ ð8Þ
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Among the 4 subsystems concerned with 4 phases, there is a switching signal σðkÞ determining which subsystem is activated at each instant k. Define the phase indication function as 1 σðkÞ ¼ p ηp ðkÞ ¼ ð9Þ 0 otherwise Then, augmenting the dynamics in Phases 1–4, the above equations can be restated compactly in state space form as follows: 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ þ Bp up ðkÞ
ð10Þ
p¼1
where up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 and 3 x1 ðkÞ 7 6 6 x2 ðkÞ 7 7 6 6 x3 ðkÞ 7 7 6 7 6 6 x4 ðkÞ 7 7 xðkÞ ¼ 6 6 x5 ðkÞ 7; 7 6 7 6 6 x6 ðkÞ 7 7 6 6 x7 ðkÞ 7 5 4 x8 ðkÞ 2
3 q 1 s1 6 q 7 7 6 2 7 6 6 q3 7 7 6 7 6 6 q4 7 7 B1 ¼ 6 6 q 5 s5 7 ; 7 6 7 6 6 q6 7 7 6 6 q 7 7 5 4 q8 2
2
q1
3
6 q s 7 6 2 27 7 6 6 q3 7 7 6 6 q 7 7 6 4 7 6 7; q B2 ¼ 6 5 7 6 7 6 6 q6 s6 7 7 6 6 q7 7 7 6 7 6 4 q8 5
2
q1 q2
3
7 6 7 6 7 6 6 q3 s3 7 7 6 7 6 6 q4 7 7 B3 ¼ 6 6 q5 7; 7 6 7 6 6 q6 7 7 6 6 q s7 7 5 4 7 q8
2
q1 q2
3
7 6 7 6 7 6 6 q3 7 7 6 7 6 6 q4 s4 7 7 B4 ¼ 6 6 q5 7 7 6 7 6 6 q6 7 7 6 6 q 7 7 5 4 q8 s8
Obviously, system (10) is a typical discrete-time switched system, and since the phases work in turns as Phase 1-Phase 2-Phase 3-Phase 4-Phase 1-⋯ in the intersection system model, the switching signal σðkÞ is described as 8 1 σðk 1Þ ¼ 4 > > > > < 2 σðk 1Þ ¼ 1 ð11Þ σðkÞ ¼ 3 σðk 1Þ ¼ 2 > > > > : 4 σðk 1Þ ¼ 3 The switched system model (10) represents an easy way to interpret and reflect the real operation of a crossroads. An obvious advantage of this model is that it can be easily extended to the case of n phases and m lanes, that is to say, a general modeling method is provided here. With n phases and m lanes, the state space system is presented to be consisted of n subsystems with m variables in vector xðkÞ. Nonetheless, there are 8 state variables in model (10) due to the 8 lanes in the intersection. If more lanes are involved in the intersection, more dimension of system state will be added into the system model. Because of the difficulties in analyzing and design for system model with large dimension, the reduced-order system model is necessary to be developed based on model (10). Since the control objective for oversaturated intersection is to relieve all the waiting vehicles in the intersection, the summation of waiting vehicles in the lanes can be used as the new state variables. For example, the summation of waiting vehicles in horizontal direction in Fig. 1 can be defined as the new variable x~ 1 ðkÞ ¼ x1 ðkÞ þ x2 ðkÞ þ x5 ðkÞ þ x6 ðkÞ, and the summation of waiting vehicles in vertical direction in Fig. 1 as x~ 2 ðkÞ ¼ x3 ðkÞ þ x4 ðkÞ þ x7 ðkÞ þ x8 ðkÞ is defined as another new variable. Then, the reduced-order system whose dimension is reduced from 8 to 2 can be expressed as follows: Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Phase1:
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql ∑ sl g1 ðkÞ
ð12Þ
l ¼ 1;5
! x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
ql g1 ðkÞ
Phase2:
ð13Þ !
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql ∑ sl g2 ðkÞ
ð14Þ
l ¼ 2;6
! x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
Phase3:
ql g2 ðkÞ
ð15Þ
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql g3 ðkÞ
ð16Þ !
x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
Phase4:
ql ∑ sl g3 ðkÞ
ð17Þ
l ¼ 3;7
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql g4 ðkÞ
ð18Þ !
x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
ql ∑ sl g4 ðkÞ
ð19Þ
l ¼ 4;8
The above reduced-order system can be rewritten into the following compact form: 4
~ þ B~ p up ðkÞ ~ þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ xðk
ð20Þ
p¼1
~ ¼ ½x~ 1 ðkÞ x~ 2 ðkÞ > and, where ηp ðkÞ, up ðkÞ, p ¼ 1; 2; 3; 4 are same as in Eqs. (9) and (10), xðkÞ 2 3 2 3 ∑ ql ∑ s l ∑ ql ∑ s l l ¼ 2;6 l ¼ 1;2;5;6 l ¼ 1;5 l ¼ 1;2;5;6 6 7 6 7 B~ 1 ¼ 4 5; B~ 2 ¼ 4 5 ∑ ql ∑ ql 2 6 B~ 3 ¼ 4
l ¼ 3;4;7;8
∑
ql
l ¼ 1;2;5;6
∑
l ¼ 3;4;7;8
ql ∑ s l l ¼ 3;7
3
2
7 5;
6 B~ 4 ¼ 4
l ¼ 3;4;7;8
∑
ql
l ¼ 1;2;5;6
∑
l ¼ 3;4;7;8
ql ∑ s l
3 7 5
l ¼ 4;8
The reduced-order system (20) differs somewhat from the full-order system (10), for instance, the exact information of queue length in each lane is lost in Eq. (20), which is replaced by the summation of queue lengths of several lanes. But, since the oversaturated situation considered in Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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this paper is concerned with the number of all vehicles waiting in the intersection (see the problem formulation later), the reduced-order system is suitable to be used in the oversaturated intersection problems as well. The reduced-order system can simplify the problem significantly in numerical aspect, which is an obvious advantage. 2.2. Control problem formulation The first control purpose of oversaturated intersection is to relieve the intersection out of the oversaturated situation quickly, i.e. reduce the number of the waiting vehicles within the intersection quickly. Now, the fundamental question arises naturally. How to distinguish between the oversaturated situation and unsaturated situation for a signal intersection? In this paper, the oversaturated situation refers norm of vector of waiting vehicles xðkÞ in the to the Frobenius 1=2 intersection, i.e. JxðkÞ J ¼ ∑8l ¼ 1 x2l ðkÞ . Given a positive scalar ε40 denoting the boundary between oversaturated and unsaturated situation, the oversaturated situation and unsaturated situation are distinguished by the following rule:
If JxðkÞJ rε, the intersection is unsaturated; If JxðkÞJ 4ε, the intersection is oversaturated.
where ε can be chosen by empirical observation for an intersection. Thus, the control problem to relieve the intersection out of the oversaturated situation can be formulated as follows: Problem 1. Design the green time of each phase gp(k), p ¼ 1; 2; 3; 4, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 of system (10), which guarantees that there exists a time K such that JxðkÞJ r ε, 8k Z K. Sometimes, the time K which is called relieving time is preferred to be estimated, so in the rest of this paper, not only the existence of relieving time K is considered to solve Problem 1, but also the estimation of relieving time K is involved in our solution. In addition, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the maximal number of waiting vehicles during the relieving period needs to be considered and less of it would be better and required. Given the different significance of different lanes considered, the weighted matrix R g 0 is introduced to notify the significance of lanes. Hence, the value of weighted number of waiting vehicles x > ðkÞRxðkÞ during relieving period ½0; K is of interest. In particular, if R ¼ I, it is concerned with JxðkÞJ 2 . An improved problem on the basis of Problem 1 is presented, in which the risk of overflow of waiting vehicles is also taken into account. Problem 2. Design the green time of each phase gp(k), p ¼ 1; 2; 3; 4, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 of system (10) such that J xðKÞ J r ε, 8k Z K, and the x > ðkÞRxðkÞ r ρ2 , 8k A ½0; K, where ρ40 is the prescribed bound avoiding risk of overflow. In some practice, the prescribed bound ρ has to be minimized by the obtained optimal controller. Thus, an optimization problem for ρ is required, which is also an important issue considered in our control strategy design. The above problems are the main concerns in our paper. At last, to achieve the control objective in problems, the state feedback control scheme adaptive to the number of waiting vehicles is employed. Thus, the following necessary assumption should be satisfied. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Assumption 1. The queue length of all movements (number of vehicles) can be measured in real-time, i.e. the value of vector xðkÞ can be obtained instantaneously. The value of queue length xi(k) can be measured in real-time, which can be measured in realtime by a traffic detector installed at the corresponding lanes or be obtained when video detection systems are utilized, otherwise, the local occupancy measurements oi, collected in real time by traditional detector loops, can be transformed into (approximate) numbers of vehicles via suitable nonlinear functions xi ¼ f i ðoi Þ such as the approach proposed in [3,25]. In this section, the system model of an oversaturated intersection is constructed, and the concerned problem is formulated. In next sections, the solutions for Problems 1 and 2 will be presented. 3. State feedback signalization strategy based on quadratic Lyapunov function 3.1. Static state feedback signalization In this section, the real time signalization for oversaturated intersection is considered to solve Problems 1 and 2 with online state feedback control policy. Under Assumption 1, the linear static state feedback scheme is used to set the green time as gp ðkÞ ¼ up ðkÞ ¼ Kp xðkÞ;
p ¼ 1; 2; 3; 4
ð21Þ
where Kp are feedback gains for each phase, which needs to be determined in our signalization strategy design. Substituting feedback control input (21) into system model (10), the closed-loop system becomes 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½ðI þ Bp Kp ÞxðkÞ
ð22Þ
p¼1
For the closed-loop system (22), a natural idea is to study the evolution of JxðkÞ J directly. According to the statements of Problem 1, the strategy only needs to ensure J xðKÞJ r ε, 8 k Z K, one choice is to make JxðkÞJ strictly decrease, however, the monotonic property of JxðkÞJ is not necessarily required, and moreover it is not easy to find such control strategy letting J xðkÞJ always decrease. Thus, we resort to the well known Lyapunov approach, whose basic idea is to construct a positive scalar function to study the convergence of system state during its evolution time, without requiring the actual solution of state trajectories. Herein, the Lyapunov function candidate can be chosen in the following quadratic form: 4
VðxðkÞÞ ¼ ∑ ηp ðkÞx > ðkÞPp xðkÞ
ð23Þ
p¼1
where Pp g 0, 8 p ¼ 1; 2; 3; 4. Letting ΔVðxðkÞÞ ¼ Vðxðk þ 1ÞÞ VðxðkÞÞ, and q ¼ sðpÞ maps the phase p to the successive phase q, the following derivation can be obtained along with the state trajectory: ΔVðxðkÞÞ ¼ x > ðk þ 1ÞPq xðk þ 1Þ x > ðkÞPp xðkÞ ¼ x > ðkÞðI þ Bp Kp Þ > Pq ðI þ Bp Kp ÞxðkÞ Pp xðkÞ ¼ x > ðkÞ½ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ Pp xðkÞ
ð24Þ
where the phase q ¼ sðpÞ. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Following the basic idea of Lyapunov method, it requires the value of Lyapunov function VðxðkÞÞ strictly decreasing, thus, we have to let ΔVðxðkÞÞo0, moreover, in order to estimate the decay of VðxðkÞÞ, it is further required as ΔVðxðkÞÞo ηVðxðkÞÞ
ð25Þ
where 0oηo1. With the aid of (24), the following condition is sufficient to guarantee Eq. (25) satisfying ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ ð1 ηÞPp !0
ð26Þ
and q ¼ sðpÞ. By what indicates in Eq. (25), and iterating Eq. (25), it straightforwardly arrives Vðxðk þ 1ÞÞoð1 ηÞVðxðkÞÞ ) VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ
ð27Þ
It is easy to be seen that there always exist scalars 0obmin obmax so that bmin I⪯Pp ⪯bmax I, 8p ¼ 1; 2; 3; 4. So, we have V ðxðkÞÞoð1 ηÞk V ðxð0ÞÞ ) bmin J xðkÞJ obmax ð1 ηÞk Jxð0ÞJ bmax ð1 ηÞk ) JxðkÞJ o Jxð0ÞJ bmin
ð28Þ
Given the ε40, the time of relieving oversaturated situation K should satisfy JxðkÞJ oε, 8k Z K, which can be ensured by the following condition: bmax ð1 ηÞK Jxð0Þ Joε bmin
ð29Þ
And the relieving time K can be estimated by ð1 ηÞK o
εbmin bmax J xð0ÞJ
εbmin bmax Jxð0ÞJ lnðbmin =bmax Þ þ lnðε=J xð0ÞJÞ ) K4 lnð1 ηÞ
) K lnð1 ηÞo ln
ð30Þ
Summarizing above discussion, the following statements can be concluded. Proposition 1. Consider the signalized oversaturated intersection (10), if there exist positive matrices Pp g0, 8 p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ ð1 ηÞPp !0;
8 p ¼ 1; 2; 3; 4
ð31Þ
where the phase q ¼ sðpÞ is the successive phase after phase p. Then, the oversaturated situation can be relieved by green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4. If the initial number of waiting vehicles are xð0Þ and the boundary between oversaturated and unsaturated situation is given as ε40, the relieving time can be estimated by K4
lnðbmin =bmax Þ þ lnðε=J xð0Þ JÞ lnð1 ηÞ
ð32Þ
where bmin and bmax satisfy bmin I⪯Pp ⪯bmax I, 8p ¼ 1; 2; 3; 4. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Sometimes, the time K which is called relieving time is preferred to be estimated, and in this case the decay rate is often required to be prescribed before designing the control strategy. In Proposition 1, it is seen that η stands for the decay rate. If η ¼ 0, the existence of relieving tine K can be ensured, but the estimation of it stays unsolved. Thus, the decay rate 0oηo1 is usually prescribed in advance to obtain a quantitative estimation of relieving time K according to Eq. (32). In addition, though a value of η close to 1 would make a fast decay and quick relief, it may not find feasible solutions for Eq. (31). Hence, regarding to the decay rate 0oηo1, it should be carefully selected. In Eq. (32), the effect of boundary parameter ε and initial length of waiting queue Jxð0ÞJ can be observed. The first point is that the larger value of εleads to less time to relieve the oversaturated situation, this means a relaxed definition of oversaturated situation can be easier to be achieved, which meets the actual situation. The second point is that the larger Jxð0ÞJ which indicates the more waiting vehicles involved in the oversaturated intersection requires more time to release the waiting queue, this is consistent with the actual observation. The above Proposition 1 provides a state feedback control scheme to solve Problem 1, which can relieve the oversaturated situation of an intersection, and the relieving time can be estimated according to Eq. (32). Then, we attend towards Problem 2 to avoid the risk of overflow of waiting vehicles. Again considering the quadratic Lyapunov function (23) and assuming Eq. (26) holds, it therefore yields VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ
ð33Þ
Before solving Problem 2, some explicit facts are recalled. For a symmetric positive definite matrix R ARnn , it is easy to verify that R can be factorized according to R ¼ ðR1=2 Þ > R1=2 , where R1=2 A Rnn is a symmetric positive definite matrix. And for any positive definite matrix R A Rnn , there always exists R 1 A Rnn which is positive definite. Then, let Wp ¼ R 1=2 PR 1=2 , if the weighted matrix R≽0 is given for Problem 2, it is obtained that VðxðkÞÞ ¼ x > ðkÞR1=2 Wp R1=2 xðkÞZ inf λmin ðWp Þ x > ðkÞRxðkÞ ð34Þ p ¼ 1;2;3;4
Vðxð0ÞÞ ¼ x > ðk0 ÞR1=2 Wp R1=2 xðk0 Þ r
sup
p ¼ 1;2;3;4
λmax ðWp Þ x > ðk 0 ÞRxðk0 Þ
ð35Þ
where λmin ðWi Þ, λmax ðWi Þ stand for the smallest and the largest eigenvalue of matrix, respectively. Moreover, since Wp ¼ R 1=2 Pp R 1=2 g0, it always exists 0ocmin r cmax such that cmin I r Wp r cmax I;
8 p ¼ 1; 2; 3; 4
ð36Þ are constrained by cmin r inf p ¼ 1;2;3;4 λmin ðWp Þ and cmax Z supp ¼ 1;2;3;
where cmin , cmax 4 λmax ðWp Þ . Then, due to 0oηo1, combining Eqs. (34)–(36), it arrives that VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ ) inf λmin ðWp Þ x > ðkÞRxðkÞr p ¼ 1;2;3;4
sup
p ¼ 1;2;3;4
λmax ðWp Þ x > ðk0 ÞRxðk 0 Þ
) cmin x > ðkÞRxðkÞr cmax x > ðk0 ÞRxðk 0 Þ Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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) x > ðkÞRxðkÞr ðcmax =cmin Þx > ðk 0 ÞRxðk0 Þ
11
ð37Þ
Thus, the condition cmax x > ðk0 ÞRxðk 0 Þ cmin ρ2 o0, where ρ40, can ensure x > ðkÞRxðkÞr ρ2 , 8k A ½0; K be satisfied. In conclusion, the solution of Problem 2 can be presented as what follows. Proposition 2. Consider the signalized oversaturated intersection (10), if there exist positive matrices Pp g0, 8p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that Eq. (31) is satisfied, and moreover, given R g0, if the following condition is satisfied: cmax x > ðk0 ÞRxðk0 Þ cmin ρ2 o0
ð38Þ
where ρ40, cmin Ir R 1=2 Pp R 1=2 r cmax I. Then, the green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees x > ðkÞRxðkÞ rρ2 , 8 k A ½0; K, where K is the relieving time estimated by Eq. (32).
However, Problem 2 has not been fully solved by above Proposition 2 regardless of consideration to minimize ρ. Thus, further investigation on obtaining optimal state feedback strategy subjected to ρ is required. Moreover, both Propositions 1 and 2 only provide sufficient conditions on the existence of state feedback controller to set proper green time relieving the waiting vehicles, however, the computation on the feedback gains are generally in non-convex form and known NP-hard, e.g. positive matrix Pq is coupled with the controller matrix Kp , it yields a typical bilinear matrix inequality (BMI) problems, which is rather unlikely to find a polynomial time algorithm for solving such general BMI problems. In order to obtain numerically tractable conditions, the conditions in Propositions 1 and 2 will be turned into convex forms mainly expressed in linear matrix inequality (LMI) problems in next subsection. 3.2. LMI formulation Firstly, we present the following technical lemma that will be essential for the proofs in our following results. Lemma 1 (Boyd et al. [26]). The linear matrix inequality " # S11 S12 S¼ T ⪯0 S12 S22 > > where S11 ¼ S11 and S22 ¼ S22 are equivalent to 1 > S22 ⪯0; S11 S12 S22 S12 ⪯0;
Lemma 1 is the well-known Schur complement lemma. Based on Lemma 1, the condition (31) in Proposition 1 can be transformed into " # ð1 ηÞPp ðI þ Bp Kp Þ > !0; 8p ¼ 1; 2; 3; 4 I þ Bp Kp Pq 1
ð39Þ
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Letting Qp ¼ Pp 1 , p ¼ 1; 2; 3; 4 and performing a congruence transformation to Eq. (39) via diagfQp ; Ig, we have " # ð1 ηÞQp Qp þ Xp> Bp> !0; 8p ¼ 1; 2; 3; 4 ð40Þ Qp þ Bp Xp Qq where Xp ¼ Kp Qp . The parameter 0oηo1 indicates the decay rate of waiting vehicles, which is assumed to be prescribed in the following discussion. Therefore, the state feedback control can be obtained by solving following LMI feasibility problem, if the 0oηo1 is fixed. Proposition 3. Consider the signalized oversaturated intersection (10), if there exist positive matrices Qp g 0, 8 p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that " # ð1 ηÞQp Qp þ Xp> Bp> !0; 8p ¼ 1; 2; 3; 4 ð41Þ Qp þ Bp Xp Qq where the phase q ¼ sðpÞ is the successive phase after phase p. Then, the oversaturated situation can be relieved by green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4, where Kp ¼ Xp Qp 1 . If the initial vector of waiting vehicles are xð0Þ and the boundary between oversaturated and unsaturated situation is given as ε40, the relieving time can be estimated by K4
lnðbmin =bmax Þ þ lnðε= Jxð0ÞJ Þ lnð1 ηÞ
ð42Þ
where bmin and bmax satisfy bmin I⪯Qp 1 ⪯bmax I, 8 p ¼ 1; 2; 3; 4. Then, Proposition 2 is considered. At first, it is assumed that there exist Qp g 0, 8 p ¼ 1; 2; 3; 4 such that Eq. (41) is established. Since in Eq. (38), the parameter ρ is coupled with cmin , we let cmin ¼ 1 and cmax ¼ μ, then Eq. (38) becomes ρ2 4μx > ðk0 ÞRxðk 0 Þ
ð43Þ
The main purpose is to formulate LMI-based conditions and these conditions are convenient to minimize parameter ρ, which is equivalent to minimize μ according to Eq. (43). To fulfill this aim, by using Lemma 1 again, it has " # Qp Qp ⪯0 ) Qp þ Qp RQp r 0 ) R r Qp 1 ) Ir R 1=2 Qp 1 R 1=2 Qp R1 "
μR
I
I
Qp
# ⪯0 ) μR þ Qp 1 r 0 ) R 1=2 Qp 1 R 1=2 r μI
Thus, the following optimization problem with a fixed 0oηo1 can be formulated for 8 p ¼ 1; 2; 3; 4: minμ
"
ð1 ηÞQp s:t: Qp þ Bp Xp
# Qp þ Xp> Bp> !0 Qq
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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" "
Qp
Qp
#
⪯0 R1 # I ⪯0 Qp
Qp μR I
13
ð44Þ
where the phase q ¼ sðpÞ is the successive phase after phase p. On the basis of optimization problem in the LMI form, the following result can be derived to get the optimal green time setting with respect to the minimized ρ. Proposition 4. Consider the signalized oversaturated intersection (10), the optimal feedback controller gains Kp ¼ Xp Qp 1 can be obtained by solving optimization problem (44) with μmin . Then, the green time of each phase p asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees x > ðkÞRxðkÞr ρ2min , 8 k A ½0; K, where ρmin ¼ μmin x > ðk 0 ÞRxðk0 Þ and K is the relieving time estimated by Eq. (42). Propositions 3 and 4 are the LMI formation of Propositions 1 and 2 with a prescribed decay rate η. Though the parameter η is fixed in above discussion, it is possible to set a variation and run optimization (44) with respect to all the values in ð0; 1Þ with discretized step Δη. All the LMI conditions in this subsection can be efficiently solved by existing software tools such as LMI toolbox in MATLAB. 4. Robust signalization subjected to unknown disturbances In the previous sections, the unknown disturbances are not involved in system model (10). However, since the disturbances inevitably exist in practice such as the uncertainties or disturbances in the input traffic flow to the intersection, the previous results should be extended to robust signalization subject to disturbances. At first, the system model with disturbances is presented. By adding the unknown disturbance term ωðkÞ into system (10), the system with unknown input disturbances can be expressed by 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ þ Bp up ðkÞ þ ωðkÞ
ð45Þ
p¼1
where ωðkÞ ¼ ½ω1 ðkÞ ω2 ðkÞ ω3 ðkÞ ω4 ðkÞ ω5 ðkÞ ω6 ðkÞ ω7 ðkÞ ω8 ðkÞ > is the vector of unknown input disturbances and xðkÞ, Bq , up(k), ηp ðkÞ are defined same as in Eq. (10). Furthermore, the reduced-order system can be presented as 4
~ þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ ~ þ B~ p up ðkÞ þ ωðkÞ ~ xðk
ð46Þ
p¼1
> ~ where ωðkÞ ¼ ∑l ¼ 1;2;5;6 ωl ðkÞ ∑l ¼ 3;4;7;8 ωl ðkÞ . In the presence of disturbances, besides the relief of oversaturated situation of interest, the disturbance attenuation performance of control strategy also draws great attention by us. In this paper, the disturbance attenuation performance is considered. With respect to the output as a convex combination of waiting vehicles in each lanes as yðkÞ ¼ CxðkÞ
ð47Þ
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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where C A R1n , the definition of H1 disturbance attenuation performance describing the relationship between disturbance and output is given as follows:
If under zero initial condition, i.e. xð0Þ ¼ 0, the H1 disturbance attenuation problem is defined by the following inequality: K
K
k¼0
k¼0
∑ JyðkÞ J 2 r γ 2 ∑ J ωðkÞJ 2 ;
8 K40
ð48Þ
where γ is the H1 performance level. Obviously, the control strategy is required to reduce performance level γ as small as possible to achieve better disturbance attenuation performance. Combining the ρ in Problem 2 which is also required to be minimized, the control objective here is formulated as to minimize the value of convex combination of γ2 and ρ2. The objective function to be minimized can be described as Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2
ð49Þ
where 0oθo1. Problem 3. Design the green time of each phase, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 such that (1) For ωðkÞ ¼ 0: There exists a time K such that JxðKÞJ r ε, 8k Z K, where the estimated K is the relieving time; x > ðkÞRxðkÞr ρ2 , 8k A ½0; K, where ρ40 is the prescribed bound avoiding risk of overflow; For ωðkÞa 0: (2) Minimize objective function Hðρ; γÞ in (49).
On the basis of quadratic Lyapunov function approach, the following derivation can be obtained. Considering J ¼ JyðkÞ J γ JωðkÞJ þ ΔVðxðkÞÞ, we have J ¼ JyðkÞJ 2 γ 2 JωðkÞ J 2 þ ΔVðxðkÞÞ ¼ ½x > ðkÞC > CxðkÞ γ 2 ωðkÞ > ðkÞωðkÞ þ ΔVðxðkÞÞ " # h i xðkÞ > > ¼ x ðkÞ ω ðkÞ Ξ p;q ωðkÞ where Ξ p;q ¼
"
ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ Pp þ C > C
ðI þ Bp Kp Þ > Pq
Pq ½ðI þ Bp Kp Þ
Pq γ 2 I
# :
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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If we have Ξ p;q !0, it arrives J yðkÞJ 2 γ 2 J ωðkÞ J 2 þ ΔVðxðkÞÞo0, which leads to K
∑
k¼0
J yðkÞJ 2 γ 2 J ωðkÞJ 2 þ VðxðK þ 1ÞÞ Vðxð0ÞÞo0
Due to the fact that VðxðK þ 1ÞÞ40 and Vðxð0ÞÞ ¼ 0, it has K
K
k¼0
k¼0
∑ J yðkÞJ 2 o ∑ γ 2 JωðkÞ J 2 ;
8 K40
Therefore, the H1 disturbance attenuation performance can be ensured by Ξ p;q ! 0, q ¼ sðpÞ, 8p ¼ 1; 2; 3; 4. Then, in order to apply Ξ p;q ! 0, q ¼ sðpÞ, 8 p ¼ 1; 2; 3; 4 into controller design, Lemma 1 is used to equivalently transform Ξ p;q !0, q ¼ sðpÞ, 8 p ¼ 1; 2; 3; 4 into what follows. 2 3 0 ðI þ Bp Kp Þ > C > Pp 6 0 γ2I I 0 7 6 7 ð50Þ 6 7 !0; q ¼ sðpÞ; 8 p ¼ 1; 2; 3; 4 1 4 I þ Bp Kp I Pq 0 5 C
0
0
I
By letting Qp ¼ Pp 1 , p ¼ 1; 2; 3; 4 and performing a congruence transformation to Eq. (39) via diagfQp ; I; I; Ig, we have 2 3 Qp 0 Qp þ Xp> Bp> Qp C > 6 7 6 0 γ2 I I 0 7 6 7 ! 0; q ¼ sðpÞ; 8p ¼ 1; 2; 3; 4 ð51Þ 6 Q þ Bp Xp I Qq 0 7 4 p 5 CQp 0 0 I So, Eq. (51) can guarantee the H1 disturbance attenuation performance with prescribed level γ. Moreover, to estimate relieving time K such that JxðKÞJ r ε, 8 k Z K, the decay rate 0oηo1 is added into Eq. (51), and the following condition is obtained: 3 2 ð1 ηÞQp 0 Qp þ Xp> Bp> Qp C > 7 6 6 0 γ2I I 0 7 7 ! 0; q ¼ sðpÞ; 8p ¼ 1; 2; 3; 4 ð52Þ 6 6 Q þ Bp Xp I Qq 0 7 5 4 p CQp 0 0 I It is obvious that Eq. (51) is a particular case of Eq. (52) with η ¼ 0, and Eq. (51) can be always established if Eq. (52) holds. And when ωðkÞ ¼ 0, it implies that there exists a time K such that JxðKÞJ r ε, 8 k Z K, where the relieving time K is estimated by Eq. (42). Then, by the same steps in deriving Proposition 2, if the following conditions are satisfied: " # Qp Qp ⪯0; 8p ¼ 1; 2; 3; 4 ð53Þ Qp R1 "
μR I
# I ⪯0; Qp
8 p ¼ 1; 2; 3; 4
ð54Þ
We can conclude that x > ðkÞRxðkÞr ρ2 , 8 k A ½0; K, where ρ2 Z μx > ðk0 ÞRxðk 0 Þ. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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By summarizing above procedures, the convex optimization problem to minimize the objective function Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2 with a fixed decay rate 0oηo1 can be formulated as follows: minθρ2 þ ð1 θÞγ 2 2 0 Qp þ Xp> Bp> ð1 ηÞQp 6 6 0 γ2 I I s:t:6 6 Q þ Bp Xp I Qq 4 p CQp 0 0 " # Qp Qp ⪯0 Qp R1 " # μR I ⪯0 I Qp μx > ðk0 ÞRxðk 0 Þ ρ2 r 0
Qp C > 0 0 I
3 7 7 7!0 7 5
ð55Þ
As to the improved optimization (55) concerned with both H1 disturbance attenuation performance γ and state boundary ρ, the LMI-based solution to Problem 3 is proposed based on Eq. (55). Proposition 5. Consider the signalized oversaturated intersection (45) with disturbances ωðkÞ, the optimal feedback controller gains Kp ¼ Xp Qp 1 can be obtained by solving optimization problem (55) with H min . Then, the green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees Hðρ; γÞ r H min , 8 k A ½0; K, where Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2 and K is the relieving time estimated by Eq. (42). In this section, an extension from system without disturbances to the one with unknown disturbances is presented. By Proposition 5, it is seen that the green time setting via state feedback strategy not only provides the oversaturated situation relieving performance as in the previous section, the disturbance attenuation in the sense of H1 performance is also considered to minimize the affect caused by the disturbances to the traffic flow. 5. Simulation study The basic mechanism of our approach is that the green time of each phase is capable of adjusting adaptively to the length of waiting queue, namely the state feedback control scheme. Compared with some methods with fixed time setting, our approach is supposed to have better control performance. In this section, a comparison through simulation will be presented to illustrate the advantages of approach proposed in this paper. By the signalization strategy proposed in the previous sections, the oversaturated situation will be relieved by the state feedback control scheme, and the risk of overflow can be avoided as well. However, in practical point of view, it is worth noting that, in order to fit physically reasonable signalization, the controller must respect the boundary conditions on green time gp(k). Generally, two boundary values gmin and gmax have to be selected so that gmin r gp ðkÞ r gmax , p ¼ 1; 2; 3; 4. The values of gmin and gmax must be well selected. For instance, too short effective green lights are impractical and too long effective green lights are unacceptable to the stopped drivers of the other approach. In addition, the approach proposed in this paper mainly focuses on the Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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oversaturated situation, if the intersection works in the unsaturated situation, it remains choosing some traditional signalization methods such as Webster method to determine green time gnp , p ¼ 1; 2; 3; 4. Without loss of generality, it is assumed that gmin r gnp r gmax , p ¼ 1; 2; 3; 4. In conclusion, with respect to the boundary condition on the control light and signalization for unsaturated situation, the operation of the controller leads to the following cases: 8 n if JxðKÞJ r ε g > > > p > < Kp xðkÞ if JxðkÞJ 4ε and gmin ogp ðkÞogmax ð56Þ gp ðkÞ ¼ gmax ifgp ðkÞ Z gmax > > > > : gmin ifgp ðkÞ r gmin The crucial point in designing controller (56) is to determine state feedback gain Kp , i ¼ 1; 2; 3; 4, which can be obtained by solving LMI feasibility problem in Proposition 3, 4 or 5. Here, one additional constraint must be considered in practical green time setting, that is the obtained green time gp(k), 8 p ¼ 1; 2; 3; 4 has to be positive. Since the length of waiting queue is always non-negative which implies the variables defined by state vector xðkÞ are always confined to the positive orthant, i. e. xðkÞ c 0. Therefore, the positivity of green time gp(k), 8 p ¼ 1; 2; 3; 4 is equivalent to Kp c 0, 8p ¼ 1; 2; 3; 4. By the design results in previous sections where the controller gains are determined by Kp ¼ Xp Qp 1 , herein we further choose positive matrices Qp ¼ diagfqp1 ; qp2 ; …; qpn g and confine matrices Xp c 0, 8p ¼ 1; 2; 3; 4 in our LMI. Thus, it can be seen that Qp 1 ¼ diagf1=qp1 ; 1=qp2 ; ⋯; 1=qpn g and Xp Qp 1 c 0, which directly leads to the positivity of green time gp(k), 8 p ¼ 1; 2; 3; 4. At first, the relief of an oversaturated intersection is considered. It is assumed that the intersection is depicted by Fig. 1 which involves 8 lanes and 4 phases and the simulation will be carried out using the following data:
Input flow rate: q1 ¼ q3 ¼ q5 ¼ q7 ¼ 0:35 veh=s, q2 ¼ q4 ¼ q6 ¼ q8 ¼ 0:3 veh=s. Saturation flow rate : s1 ¼ s3 ¼ s5 ¼ s7 ¼ 1:5 veh=s, s2 ¼ s4 ¼ s6 ¼ s8 ¼ 1:3 veh=s. Minimal and maximal green time: gmin ¼ 15 s, gmax ¼ 120 s.
At first, the green time of each phase by Webster method is obtained as gn1 ¼ gn3 ¼ 17:5069, g1 ¼ gn3 ¼ 17:3145. Then, the state feedback gains Kp , p ¼ 1; 2; 3; 4 are determined by optimization (44). In order to find feasible solution, we consider the reduced-order system form (20). With the prescribed decay rate η ¼ 0.9 and R ¼ I, the following feedback gains are obtained: n
K1 ¼ ½0:3203 0:1080;
K2 ¼ ½0:0610 0:0740;
K3 ¼ ½0:1080 0:3203;
K4 ¼ ½0:0740 0:0610
~ with the x~ > ðkÞRxðkÞr ρ2min , where ρmin ¼ 549:2136. Assuming the initial waiting vehicles > x ð0Þ ¼ ½70 50 60 50 60 50 80 70 and the boundary between oversaturated and unsaturated situation ε ¼ 100, the relieving process by our approach are shown in Fig. 2, in which it can be observed that the waiting vehicles can be effectively relieved by our approach. ~ To show the advantages of our feedback control strategy, the evolution of J xðkÞJ by Webster method and our approach are shown in Fig. 3. Comparing the simulation results in Fig. 3, the oversaturated situation can be relieved more quickly by our approach. The relieving time is given in Table 1. The main reason is that the green Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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time set by feedback scheme can adjust dynamically with respect to the length of waiting queue, which is more adaptive to the number of waiting vehicles and relieve them quickly. Then, the unknown disturbance input is considered. In this example, the output vector is selected as C ¼ ½1 1 and the concerned output is therefore standing for the total waiting vehicles ~ ¼ ∑8l ¼ 1 xl . The parameter θ in objective function is chosen as θ ¼ 0.5. To as yðkÞ ¼ CxðkÞ achieve the optimal control performance involving H1 disturbance attenuation performance γ and state boundary ρ, optimization (55) is executed to get the following state feedback gains: K1 ¼ ½0:5869 0:0309;
K2 ¼ ½0:7278 0:0150;
K3 ¼ ½0:0309 0:5869;
K4 ¼ ½0:0150 0:7278
450 x1+x2+x5+x6
400
x +x +x +x 3
4
7
8
350 300
x
250 200 150 100 50 0
0
20
40
60
80
100
k Fig. 2. Evolution of waiting vehicles relieved by our control method.
600
Feedback method Webster method
500 Estimated boundary ρmin
||x||
400
300
200 124
100 25 0
0
50
100
150
200
k ~ J with two methods and estimated boundary. Fig. 3. Evolution of vehicles J xðkÞ Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Given the same initial waiting vehicles and randomly generated disturbance in Fig. 4, and the outputs by both feedback control and Webster method are shown in Fig. 5. By comparing the curves in Fig. 5, the feedback control has better disturbance attenuation performance, since the Table 1 Relieving time of two methods. Method
Feedback method
Webster method
Relieving time K
25
124
40 w (k) 1
w (k)
30
2
20
w(k)
10 0 −10 −20 −30 −40
0
50
100
150
200
k
Fig. 4. Randomly generated disturbance.
600 Feedback method Webster method 500
y
400
300
200
100
0
0
50
100
150
k
Fig. 5. Output responses subjected to unknown disturbance. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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output response under feedback scheme is much less sensitive to the unknown disturbance method during the relieving period. 6. Conclusions This paper presents a static state feedback control for a signal oversaturated intersection. By employing Lyapunov function method, sufficient conditions for the existence of state feedback controller relieving the oversaturated situation are obtained, and moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the boundary of waiting queue length is estimated and further minimized by state feedback control. As an extension to the case with unknown disturbance, robust signalization via state feedback control is presented. All the design procedure can be executed through a set of LMI optimization problems. Through simulation study, the effectiveness and advantages of state feedback control are shown by comparison with traditional Webster method. In this paper, the state feedback control for a single intersection is investigated, but the network of intersections has not been considered here. It is worthwhile to extend our results to signalization for a network of several intersections.
References [1] M. Kaltenbach, H. Koivo, Modelling and control of urban traffic flow, in: Proceedings of the Joint Automatic Control Conference, University of Texas, Houston, 1974, pp. 147–154. [2] Y. Tangsheng, On Line Traffic Signalization Using Robust Feedback Control (Ph.D. dissertation), Faculty of the Virginia Polytechnic Institute and State University, Virginia, 1997. [3] C. Diakaki, M. Papageorgiou, K. Aboudolas, A multivariable regulator approach to traffic-responsive network-wide signal control, Control Eng. Pract. 10 (2002) 183–195. [4] D. Gazis, Optimal control of a system of oversaturated intersections, Oper. Res. 12 (1964) 815–831. [5] D. Gazis, R. Potts, The oversaturated intersection, in: Proceedings of the Second International Symposium on the Theory of Road Traffic Flow Organization for Economic Cooperation and Development, Paris, 1965, pp. 221–237. [6] D. Green, Control of oversaturated intersections, Oper. Res. Q. 18 (2) (1968) 161–173. [7] G. Dans, D. Gazis, Optimal control of oversaturated store-and-forward transportation networks, Transport. Sci. 1 (1976) 1–19. [8] T. Chang, J. Lin, Optimal signal timing for an oversaturated intersection, Transport. Res. B (2000) 471–491. [9] B. Schutter, B. Moor, Optimal traffic light for a single intersection, Eur. J. Control 4 (3) (1998) 260–276. [10] F. Motawej, R. Bouyekhf, A. El Moudni, A dissipativity-based approach to traffic signal control for an oversaturated intersection, J. Frankl. Inst. 348 (2011) 703–717. [11] F. Boillot, J.M. Blosseville, J.B. Lesort, V. Motyka, M. Papageorgiou, S. Sellam, Optimal signal control of urban traffic networks, in: Proceedings of the Sixth IEE International Conference on Road Traffic Monitoring and Control, London, 1992, pp. 75–79. [12] F. Dion, S. Yagar, Real-time control of signalized networks different approaches for different needs, in: Proceedings of the Eighth IEE International Conference on Road Traffic Monitoring and Control, Savoy Place, London, 1996, pp. 56–60. [13] P.B. Hunt, D.L. Robertson, R.D. Bretherton, M.C. Royle, The SCOOT on-line traffic signal optimisation technique, Traffic Eng. Control 23 (1982) 190–199. [14] P. Michalopoulos, G. Stephanopolos, Optimal control of oversaturated intersections theoretical and practical consideration, Traffic Eng. Control (1978) 216–221. [15] H. Zhou, B. Rachid, M. Adbellah, Modelling and H 1 control of urban transportation network, in: 13th IFAC Symposium on Control in Transportation Systems, The International Federation of Automatic Control, September 12–14, 2012. Sofia, Bulgaria, pp. 72–77. [16] R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilization of hybrid systems, Proc. IEEE 88 (2000) 1069–1082. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
W. Xiang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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[17] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control 54 (2009) 308–322. [18] W. Xiang, J. Xiao, M.N. Iqbal, Asymptotic stability, l2 gain boundness analysis and control synthesis for switched systems: a switching frequency approach, Int. J. Adapt. Control Signal Process. 26 (4) (2012) 350–373. [19] W. Xiang, J. Xiao, Stability analysis and control synthesis of switched impulsive systems, Int. J. Robust Nonlinear Control 22 (2012) 1440–1459. [20] W. Xiang, J. Xiao, Stabilization of switched continuous-time system with all modes unstable via dwell time switching, Automatica 50 (2014) 940–945. [21] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (1998) 475–482. [22] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, SIAM Rev. 49 (2007) 545–592. [23] W. Xiang, J. Xiao, H 1 finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance, J. Frankl. Inst. 348 (2) (2011) 331–352. [24] W. Xiang, J. Xiao, Convex sufficient conditions on asymptotic stability and ℓ2 gain performance for uncertain discrete-time switched linear systems, IET Control Theory Appl. 8 (2014) 211–218. [25] C. Diakaki, M. Papageorgiou, T. McLean, Application and evaluation of the integrated traffic-responsive urban corridor control strategy IN-TUC in Glasgow in: Preprints CD-ROM of the 78th Annual Meeting of the Transportation Research Board (Paper No. 990310), Washington, DC, USA, 1999. [26] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994.
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach Weiming Xianga,n, Jian Xiaob, Yangsheng Jianga a
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China b School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China
Received 12 February 2014; received in revised form 17 July 2014; accepted 27 September 2014
Abstract In this paper, the traffic-responsive signalization problem for an oversaturated intersection is addressed. A static state feedback control strategy is presented to relieve the waiting vehicles in lanes. By modeling the intersection as a discrete-time switched system and with the aid of quadratic Lyapunov function method, the green time of each phase in signalization is determined by a set of linear state feedback controllers, and moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the boundary of waiting queue length is estimated and further minimized by the state feedback control. Then, linear matrix inequality (LMI) technique is employed to numerically tackle the design problem through solving a set of LMI optimization problems. Furthermore, the design results are extended to the case with unknown disturbance, and H1 controller is designed to achieve the disturbance attenuation performance. In the end, a simulation study in comparison with traditional Webster method is presented to illustrate the effectiveness of state feedback control in relieving the oversaturated situation for an intersection. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Traffic signal control is a long-lasting research problem in urban transportation network systems, e.g. in [1–3]. The effectiveness of a traffic signal system can reduce the incidence of delays, stops, fuel consumption, emission of pollutants, and accidents. Moreover, due to the n
Corresponding author. E-mail address: [email protected] (W. Xiang).
http://dx.doi.org/10.1016/j.jfranklin.2014.09.017 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Nomenclature Ai;gp number of vehicles joining the movement i during the green time gp ε boundary distinguishing oversaturated between unsaturated situation Di;gp number of vehicles departing the movement i during the green time gp gp(k) green time in phase state k gmin , gmax lower limit and upper limit of green time I identity matrix Kp state feedback control gain for phase p ωi ðkÞ unknown disturbance of movement i in phase state k ωðkÞ vector of disturbance in phase state k P > ðx > Þ transposition of matrix P (vector x) P g0ðP≽0Þ matrix P is positive definite (semi-positive definite) Pb0 all the elements pij are non-negative where pij is the element in (i, j) position of P qi input flow rate for movement i si saturation flow rate for the movement i up(k) control variables for phase p in phase state k xi(k) queue length of lane or movement i in phase state k xðkÞ vector of queue length in phase state k Jx J Frobenius norm of vector x y(k) output variable in phase state k
rapid growth of traffic congestion, an effective traffic signalization plays an important role of relieve the oversaturated situation such as related papers [4–11] and references cited therein. Most of the signal control strategies are based on fixed-time signal control [1,4–7], which are mainly based on historical data rather than real-time data. In a few recent papers, some on-line signalization methods have been proposed [2,8–13], which are more adaptive to the real-time traffic conditions. As for oversaturated condition, some significant results have been reported. The result in [14], which is probably the first result proposing the two-stage timing method, attempts to find an optimal switch-over point during the oversaturated period to interchange the timing of the approaches. Recently, a discrete delay type model is introduced in [8] to improve the results of [14]. In [9], an optimal traffic light switching scheme was presented. Generally, it resorts to a minimization problem over a set of an extended linear complementarity problem, which is not an easy task when switching cycles are large. And concerned with the recent notable result [10], a dissipative idea is applied into traffic signal design problem, a state feedback controller based on dissipativity-based control is derived. From the control point of view, the traffic-responsive control strategies are supposed to be able to achieve better control performance than fixed-time strategies. As to the real-time responsive control, the static state feedback scheme is the prevailing method for various controlled systems. In particular to linear dynamic systems in state space form, the static linear state feedback control strategy has been widely used in numerous applications, however, few results have been devoted to apply this simple but effective control rule to traffic control issues, e.g. in [15]. But the dynamics of vehicle movements in [15] is modeled as a single discrete-time state space model which is not completely controllable, thus the traffic-responsive control strategy is only applicable to controllable variables. In this paper, Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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we are going to introduce the static state feedback control strategy applicable to all variables in an intersection by means of modeling it as a switched system. Switched system can be efficiently used to model many practical systems which are inherently multi-model in the sense that several dynamical systems are required to describe their behaviors. For more details about switched systems, the reader is referred to some recent papers [16–24] and the references cited therein. Due to the essential nature of multi-mode contained in a signalized intersection, which results from the multiple phases decided by traffic lights, the intersection can be modeled as a switched system composed by several linear subsystems describing the multiple phases in signalization. Based on the switched system model, the key point is to set the green time appropriately to relieve the waiting vehicles within the oversaturated intersection. By considering the green time as the control input of the system, the signalization design is transformed to design a set of state feedback controllers ensuring the state of closed-loop system being able to decrease below a certain bound, which implies the oversaturated situation is relieved. Moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the maximal number of waiting vehicles during the relieving period needs to be considered and less of it would be better and required, thus the optimal state feedback controller guaranteeing the minimal bound of system state is required. The Lyapunov function approach is employed to fulfill above design task. By constructing a Lyapunov function in the quadratic form, sufficient conditions for the existence of state feedback controller are given. Then, to make the design procedure numerically tractable, the linear matrix inequality (LMI) technique is introduced to solve the optimization problem, and to obtain the optimal state feedback control scheme. Furthermore, as an extension of our results, the robust signalization subjected to unknown disturbance is considered. In presence of unknown disturbance, the influence caused by disturbance needs to be attenuated by the control strategy. In this paper, the disturbance attenuation is considered as H1 performance, and the H1 feedback controller is designed. At last, a simulation study is carried to illustrate our results. In comparison to the traditional Webster method, it shows the state feedback control strategy is an effective trafficresponsive approach to release the oversaturated situation. The remainder of this paper is organized as follows. The system modeling and control problem formulation for an oversaturated intersection are presented in Section 2. The main results on the state feedback controller design and LMI formulation are given in Section 3. As an extension to robust signalization, the H1 control strategy design is studied in Section 4. Simulation study is in Section 5, and conclusions are given in Section 6.
2. System description and problem formulation 2.1. System description and modeling The urban transportation system is composed by a network of intersections, and generally, a signalized intersection is operated by a traffic signal that decides the movements of vehicles to pass the intersections or have to stop to generate the queues. The movement may include vehicles going straight, turning left, turning right, or a combination of them. In general, a signalized intersection has n phases, where n Z 2, and each phase is allotted a certain amount of time (green time) during which a group of traffic lanes is allowed to proceed. Here, in order to show our control idea clearly, a typical single intersection with 8 lanes and the traffic signal has 4 phases is considered, which is illustrated in Fig. 1. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Fig. 1. An intersection with 8 lanes and 4 phases.
The discrete time state space system model for the intersection in Fig. 1 is going to be proposed to characterize the evolution of queue lengths between the phase switching instant k and k þ 1. Taking the Phase 1 for example, the queue length xi(k) evolves according to xi ðk þ 1Þ ¼ xi ðkÞ þ Ai;g1 Di;g1 . For movements 1 and 5 which have same characteristics, it is obtained that Ai;g1 ¼ qi g1 ðkÞ, i¼ 1,5 and Di;g1 ¼ si g1 ðkÞ, i¼ 1,5 , thus we have the following equation in Phase 1: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg1 ðkÞ;
i ¼ 1; 5
ð1Þ
And for other movements i ¼ 2; 3; 4; 6; 7; 8, since the movements are stopped to generate queues, which implies Ai;g1 ¼ qi g1 ðkÞ, i ¼ 2; 3; 4; 6; 7; 8 and Di;g1 ¼ 0, i ¼ 2; 3; 4; 6; 7; 8, it is therefore obtained as xi ðk þ 1Þ ¼ xi ðkÞ þ qi g1 ðkÞ;
i ¼ 2; 3; 4; 6; 7; 8
ð2Þ
Following the similar guideline in Phase 1, the rest of evolution equations for Phase 2–4 can be derived as follows: Phase2: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg2 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g2 ðkÞ;
i ¼ 2; 6
i ¼ 1; 3; 4; 5; 7; 8
ð3Þ ð4Þ
Phase3: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg3 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g3 ðkÞ;
i ¼ 3; 7
i ¼ 1; 2; 4; 5; 6; 8
ð5Þ ð6Þ
Phase4: xi ðk þ 1Þ ¼ xi ðkÞ þ ðqi si Þg4 ðkÞ; xi ðk þ 1Þ ¼ xi ðkÞ þ qi g4 ðkÞ;
i ¼ 4; 8
i ¼ 1; 2; 3; 5; 6; 7
ð7Þ ð8Þ
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Among the 4 subsystems concerned with 4 phases, there is a switching signal σðkÞ determining which subsystem is activated at each instant k. Define the phase indication function as 1 σðkÞ ¼ p ηp ðkÞ ¼ ð9Þ 0 otherwise Then, augmenting the dynamics in Phases 1–4, the above equations can be restated compactly in state space form as follows: 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ þ Bp up ðkÞ
ð10Þ
p¼1
where up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 and 3 x1 ðkÞ 7 6 6 x2 ðkÞ 7 7 6 6 x3 ðkÞ 7 7 6 7 6 6 x4 ðkÞ 7 7 xðkÞ ¼ 6 6 x5 ðkÞ 7; 7 6 7 6 6 x6 ðkÞ 7 7 6 6 x7 ðkÞ 7 5 4 x8 ðkÞ 2
3 q 1 s1 6 q 7 7 6 2 7 6 6 q3 7 7 6 7 6 6 q4 7 7 B1 ¼ 6 6 q 5 s5 7 ; 7 6 7 6 6 q6 7 7 6 6 q 7 7 5 4 q8 2
2
q1
3
6 q s 7 6 2 27 7 6 6 q3 7 7 6 6 q 7 7 6 4 7 6 7; q B2 ¼ 6 5 7 6 7 6 6 q6 s6 7 7 6 6 q7 7 7 6 7 6 4 q8 5
2
q1 q2
3
7 6 7 6 7 6 6 q3 s3 7 7 6 7 6 6 q4 7 7 B3 ¼ 6 6 q5 7; 7 6 7 6 6 q6 7 7 6 6 q s7 7 5 4 7 q8
2
q1 q2
3
7 6 7 6 7 6 6 q3 7 7 6 7 6 6 q4 s4 7 7 B4 ¼ 6 6 q5 7 7 6 7 6 6 q6 7 7 6 6 q 7 7 5 4 q8 s8
Obviously, system (10) is a typical discrete-time switched system, and since the phases work in turns as Phase 1-Phase 2-Phase 3-Phase 4-Phase 1-⋯ in the intersection system model, the switching signal σðkÞ is described as 8 1 σðk 1Þ ¼ 4 > > > > < 2 σðk 1Þ ¼ 1 ð11Þ σðkÞ ¼ 3 σðk 1Þ ¼ 2 > > > > : 4 σðk 1Þ ¼ 3 The switched system model (10) represents an easy way to interpret and reflect the real operation of a crossroads. An obvious advantage of this model is that it can be easily extended to the case of n phases and m lanes, that is to say, a general modeling method is provided here. With n phases and m lanes, the state space system is presented to be consisted of n subsystems with m variables in vector xðkÞ. Nonetheless, there are 8 state variables in model (10) due to the 8 lanes in the intersection. If more lanes are involved in the intersection, more dimension of system state will be added into the system model. Because of the difficulties in analyzing and design for system model with large dimension, the reduced-order system model is necessary to be developed based on model (10). Since the control objective for oversaturated intersection is to relieve all the waiting vehicles in the intersection, the summation of waiting vehicles in the lanes can be used as the new state variables. For example, the summation of waiting vehicles in horizontal direction in Fig. 1 can be defined as the new variable x~ 1 ðkÞ ¼ x1 ðkÞ þ x2 ðkÞ þ x5 ðkÞ þ x6 ðkÞ, and the summation of waiting vehicles in vertical direction in Fig. 1 as x~ 2 ðkÞ ¼ x3 ðkÞ þ x4 ðkÞ þ x7 ðkÞ þ x8 ðkÞ is defined as another new variable. Then, the reduced-order system whose dimension is reduced from 8 to 2 can be expressed as follows: Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Phase1:
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql ∑ sl g1 ðkÞ
ð12Þ
l ¼ 1;5
! x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
ql g1 ðkÞ
Phase2:
ð13Þ !
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql ∑ sl g2 ðkÞ
ð14Þ
l ¼ 2;6
! x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
Phase3:
ql g2 ðkÞ
ð15Þ
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql g3 ðkÞ
ð16Þ !
x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
Phase4:
ql ∑ sl g3 ðkÞ
ð17Þ
l ¼ 3;7
!
x~ 1 ðk þ 1Þ ¼ x~ 1 ðkÞ þ
∑
l ¼ 1;2;5;6
ql g4 ðkÞ
ð18Þ !
x~ 2 ðk þ 1Þ ¼ x~ 2 ðkÞ þ
∑
l ¼ 3;4;7;8
ql ∑ sl g4 ðkÞ
ð19Þ
l ¼ 4;8
The above reduced-order system can be rewritten into the following compact form: 4
~ þ B~ p up ðkÞ ~ þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ xðk
ð20Þ
p¼1
~ ¼ ½x~ 1 ðkÞ x~ 2 ðkÞ > and, where ηp ðkÞ, up ðkÞ, p ¼ 1; 2; 3; 4 are same as in Eqs. (9) and (10), xðkÞ 2 3 2 3 ∑ ql ∑ s l ∑ ql ∑ s l l ¼ 2;6 l ¼ 1;2;5;6 l ¼ 1;5 l ¼ 1;2;5;6 6 7 6 7 B~ 1 ¼ 4 5; B~ 2 ¼ 4 5 ∑ ql ∑ ql 2 6 B~ 3 ¼ 4
l ¼ 3;4;7;8
∑
ql
l ¼ 1;2;5;6
∑
l ¼ 3;4;7;8
ql ∑ s l l ¼ 3;7
3
2
7 5;
6 B~ 4 ¼ 4
l ¼ 3;4;7;8
∑
ql
l ¼ 1;2;5;6
∑
l ¼ 3;4;7;8
ql ∑ s l
3 7 5
l ¼ 4;8
The reduced-order system (20) differs somewhat from the full-order system (10), for instance, the exact information of queue length in each lane is lost in Eq. (20), which is replaced by the summation of queue lengths of several lanes. But, since the oversaturated situation considered in Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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this paper is concerned with the number of all vehicles waiting in the intersection (see the problem formulation later), the reduced-order system is suitable to be used in the oversaturated intersection problems as well. The reduced-order system can simplify the problem significantly in numerical aspect, which is an obvious advantage. 2.2. Control problem formulation The first control purpose of oversaturated intersection is to relieve the intersection out of the oversaturated situation quickly, i.e. reduce the number of the waiting vehicles within the intersection quickly. Now, the fundamental question arises naturally. How to distinguish between the oversaturated situation and unsaturated situation for a signal intersection? In this paper, the oversaturated situation refers norm of vector of waiting vehicles xðkÞ in the to the Frobenius 1=2 intersection, i.e. JxðkÞ J ¼ ∑8l ¼ 1 x2l ðkÞ . Given a positive scalar ε40 denoting the boundary between oversaturated and unsaturated situation, the oversaturated situation and unsaturated situation are distinguished by the following rule:
If JxðkÞJ rε, the intersection is unsaturated; If JxðkÞJ 4ε, the intersection is oversaturated.
where ε can be chosen by empirical observation for an intersection. Thus, the control problem to relieve the intersection out of the oversaturated situation can be formulated as follows: Problem 1. Design the green time of each phase gp(k), p ¼ 1; 2; 3; 4, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 of system (10), which guarantees that there exists a time K such that JxðkÞJ r ε, 8k Z K. Sometimes, the time K which is called relieving time is preferred to be estimated, so in the rest of this paper, not only the existence of relieving time K is considered to solve Problem 1, but also the estimation of relieving time K is involved in our solution. In addition, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the maximal number of waiting vehicles during the relieving period needs to be considered and less of it would be better and required. Given the different significance of different lanes considered, the weighted matrix R g 0 is introduced to notify the significance of lanes. Hence, the value of weighted number of waiting vehicles x > ðkÞRxðkÞ during relieving period ½0; K is of interest. In particular, if R ¼ I, it is concerned with JxðkÞJ 2 . An improved problem on the basis of Problem 1 is presented, in which the risk of overflow of waiting vehicles is also taken into account. Problem 2. Design the green time of each phase gp(k), p ¼ 1; 2; 3; 4, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 of system (10) such that J xðKÞ J r ε, 8k Z K, and the x > ðkÞRxðkÞ r ρ2 , 8k A ½0; K, where ρ40 is the prescribed bound avoiding risk of overflow. In some practice, the prescribed bound ρ has to be minimized by the obtained optimal controller. Thus, an optimization problem for ρ is required, which is also an important issue considered in our control strategy design. The above problems are the main concerns in our paper. At last, to achieve the control objective in problems, the state feedback control scheme adaptive to the number of waiting vehicles is employed. Thus, the following necessary assumption should be satisfied. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Assumption 1. The queue length of all movements (number of vehicles) can be measured in real-time, i.e. the value of vector xðkÞ can be obtained instantaneously. The value of queue length xi(k) can be measured in real-time, which can be measured in realtime by a traffic detector installed at the corresponding lanes or be obtained when video detection systems are utilized, otherwise, the local occupancy measurements oi, collected in real time by traditional detector loops, can be transformed into (approximate) numbers of vehicles via suitable nonlinear functions xi ¼ f i ðoi Þ such as the approach proposed in [3,25]. In this section, the system model of an oversaturated intersection is constructed, and the concerned problem is formulated. In next sections, the solutions for Problems 1 and 2 will be presented. 3. State feedback signalization strategy based on quadratic Lyapunov function 3.1. Static state feedback signalization In this section, the real time signalization for oversaturated intersection is considered to solve Problems 1 and 2 with online state feedback control policy. Under Assumption 1, the linear static state feedback scheme is used to set the green time as gp ðkÞ ¼ up ðkÞ ¼ Kp xðkÞ;
p ¼ 1; 2; 3; 4
ð21Þ
where Kp are feedback gains for each phase, which needs to be determined in our signalization strategy design. Substituting feedback control input (21) into system model (10), the closed-loop system becomes 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½ðI þ Bp Kp ÞxðkÞ
ð22Þ
p¼1
For the closed-loop system (22), a natural idea is to study the evolution of JxðkÞ J directly. According to the statements of Problem 1, the strategy only needs to ensure J xðKÞJ r ε, 8 k Z K, one choice is to make JxðkÞJ strictly decrease, however, the monotonic property of JxðkÞJ is not necessarily required, and moreover it is not easy to find such control strategy letting J xðkÞJ always decrease. Thus, we resort to the well known Lyapunov approach, whose basic idea is to construct a positive scalar function to study the convergence of system state during its evolution time, without requiring the actual solution of state trajectories. Herein, the Lyapunov function candidate can be chosen in the following quadratic form: 4
VðxðkÞÞ ¼ ∑ ηp ðkÞx > ðkÞPp xðkÞ
ð23Þ
p¼1
where Pp g 0, 8 p ¼ 1; 2; 3; 4. Letting ΔVðxðkÞÞ ¼ Vðxðk þ 1ÞÞ VðxðkÞÞ, and q ¼ sðpÞ maps the phase p to the successive phase q, the following derivation can be obtained along with the state trajectory: ΔVðxðkÞÞ ¼ x > ðk þ 1ÞPq xðk þ 1Þ x > ðkÞPp xðkÞ ¼ x > ðkÞðI þ Bp Kp Þ > Pq ðI þ Bp Kp ÞxðkÞ Pp xðkÞ ¼ x > ðkÞ½ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ Pp xðkÞ
ð24Þ
where the phase q ¼ sðpÞ. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
W. Xiang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
9
Following the basic idea of Lyapunov method, it requires the value of Lyapunov function VðxðkÞÞ strictly decreasing, thus, we have to let ΔVðxðkÞÞo0, moreover, in order to estimate the decay of VðxðkÞÞ, it is further required as ΔVðxðkÞÞo ηVðxðkÞÞ
ð25Þ
where 0oηo1. With the aid of (24), the following condition is sufficient to guarantee Eq. (25) satisfying ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ ð1 ηÞPp !0
ð26Þ
and q ¼ sðpÞ. By what indicates in Eq. (25), and iterating Eq. (25), it straightforwardly arrives Vðxðk þ 1ÞÞoð1 ηÞVðxðkÞÞ ) VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ
ð27Þ
It is easy to be seen that there always exist scalars 0obmin obmax so that bmin I⪯Pp ⪯bmax I, 8p ¼ 1; 2; 3; 4. So, we have V ðxðkÞÞoð1 ηÞk V ðxð0ÞÞ ) bmin J xðkÞJ obmax ð1 ηÞk Jxð0ÞJ bmax ð1 ηÞk ) JxðkÞJ o Jxð0ÞJ bmin
ð28Þ
Given the ε40, the time of relieving oversaturated situation K should satisfy JxðkÞJ oε, 8k Z K, which can be ensured by the following condition: bmax ð1 ηÞK Jxð0Þ Joε bmin
ð29Þ
And the relieving time K can be estimated by ð1 ηÞK o
εbmin bmax J xð0ÞJ
εbmin bmax Jxð0ÞJ lnðbmin =bmax Þ þ lnðε=J xð0ÞJÞ ) K4 lnð1 ηÞ
) K lnð1 ηÞo ln
ð30Þ
Summarizing above discussion, the following statements can be concluded. Proposition 1. Consider the signalized oversaturated intersection (10), if there exist positive matrices Pp g0, 8 p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ ð1 ηÞPp !0;
8 p ¼ 1; 2; 3; 4
ð31Þ
where the phase q ¼ sðpÞ is the successive phase after phase p. Then, the oversaturated situation can be relieved by green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4. If the initial number of waiting vehicles are xð0Þ and the boundary between oversaturated and unsaturated situation is given as ε40, the relieving time can be estimated by K4
lnðbmin =bmax Þ þ lnðε=J xð0Þ JÞ lnð1 ηÞ
ð32Þ
where bmin and bmax satisfy bmin I⪯Pp ⪯bmax I, 8p ¼ 1; 2; 3; 4. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
10
W. Xiang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
Sometimes, the time K which is called relieving time is preferred to be estimated, and in this case the decay rate is often required to be prescribed before designing the control strategy. In Proposition 1, it is seen that η stands for the decay rate. If η ¼ 0, the existence of relieving tine K can be ensured, but the estimation of it stays unsolved. Thus, the decay rate 0oηo1 is usually prescribed in advance to obtain a quantitative estimation of relieving time K according to Eq. (32). In addition, though a value of η close to 1 would make a fast decay and quick relief, it may not find feasible solutions for Eq. (31). Hence, regarding to the decay rate 0oηo1, it should be carefully selected. In Eq. (32), the effect of boundary parameter ε and initial length of waiting queue Jxð0ÞJ can be observed. The first point is that the larger value of εleads to less time to relieve the oversaturated situation, this means a relaxed definition of oversaturated situation can be easier to be achieved, which meets the actual situation. The second point is that the larger Jxð0ÞJ which indicates the more waiting vehicles involved in the oversaturated intersection requires more time to release the waiting queue, this is consistent with the actual observation. The above Proposition 1 provides a state feedback control scheme to solve Problem 1, which can relieve the oversaturated situation of an intersection, and the relieving time can be estimated according to Eq. (32). Then, we attend towards Problem 2 to avoid the risk of overflow of waiting vehicles. Again considering the quadratic Lyapunov function (23) and assuming Eq. (26) holds, it therefore yields VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ
ð33Þ
Before solving Problem 2, some explicit facts are recalled. For a symmetric positive definite matrix R ARnn , it is easy to verify that R can be factorized according to R ¼ ðR1=2 Þ > R1=2 , where R1=2 A Rnn is a symmetric positive definite matrix. And for any positive definite matrix R A Rnn , there always exists R 1 A Rnn which is positive definite. Then, let Wp ¼ R 1=2 PR 1=2 , if the weighted matrix R≽0 is given for Problem 2, it is obtained that VðxðkÞÞ ¼ x > ðkÞR1=2 Wp R1=2 xðkÞZ inf λmin ðWp Þ x > ðkÞRxðkÞ ð34Þ p ¼ 1;2;3;4
Vðxð0ÞÞ ¼ x > ðk0 ÞR1=2 Wp R1=2 xðk0 Þ r
sup
p ¼ 1;2;3;4
λmax ðWp Þ x > ðk 0 ÞRxðk0 Þ
ð35Þ
where λmin ðWi Þ, λmax ðWi Þ stand for the smallest and the largest eigenvalue of matrix, respectively. Moreover, since Wp ¼ R 1=2 Pp R 1=2 g0, it always exists 0ocmin r cmax such that cmin I r Wp r cmax I;
8 p ¼ 1; 2; 3; 4
ð36Þ are constrained by cmin r inf p ¼ 1;2;3;4 λmin ðWp Þ and cmax Z supp ¼ 1;2;3;
where cmin , cmax 4 λmax ðWp Þ . Then, due to 0oηo1, combining Eqs. (34)–(36), it arrives that VðxðkÞÞoð1 ηÞk Vðxð0ÞÞ ) inf λmin ðWp Þ x > ðkÞRxðkÞr p ¼ 1;2;3;4
sup
p ¼ 1;2;3;4
λmax ðWp Þ x > ðk0 ÞRxðk 0 Þ
) cmin x > ðkÞRxðkÞr cmax x > ðk0 ÞRxðk 0 Þ Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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) x > ðkÞRxðkÞr ðcmax =cmin Þx > ðk 0 ÞRxðk0 Þ
11
ð37Þ
Thus, the condition cmax x > ðk0 ÞRxðk 0 Þ cmin ρ2 o0, where ρ40, can ensure x > ðkÞRxðkÞr ρ2 , 8k A ½0; K be satisfied. In conclusion, the solution of Problem 2 can be presented as what follows. Proposition 2. Consider the signalized oversaturated intersection (10), if there exist positive matrices Pp g0, 8p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that Eq. (31) is satisfied, and moreover, given R g0, if the following condition is satisfied: cmax x > ðk0 ÞRxðk0 Þ cmin ρ2 o0
ð38Þ
where ρ40, cmin Ir R 1=2 Pp R 1=2 r cmax I. Then, the green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees x > ðkÞRxðkÞ rρ2 , 8 k A ½0; K, where K is the relieving time estimated by Eq. (32).
However, Problem 2 has not been fully solved by above Proposition 2 regardless of consideration to minimize ρ. Thus, further investigation on obtaining optimal state feedback strategy subjected to ρ is required. Moreover, both Propositions 1 and 2 only provide sufficient conditions on the existence of state feedback controller to set proper green time relieving the waiting vehicles, however, the computation on the feedback gains are generally in non-convex form and known NP-hard, e.g. positive matrix Pq is coupled with the controller matrix Kp , it yields a typical bilinear matrix inequality (BMI) problems, which is rather unlikely to find a polynomial time algorithm for solving such general BMI problems. In order to obtain numerically tractable conditions, the conditions in Propositions 1 and 2 will be turned into convex forms mainly expressed in linear matrix inequality (LMI) problems in next subsection. 3.2. LMI formulation Firstly, we present the following technical lemma that will be essential for the proofs in our following results. Lemma 1 (Boyd et al. [26]). The linear matrix inequality " # S11 S12 S¼ T ⪯0 S12 S22 > > where S11 ¼ S11 and S22 ¼ S22 are equivalent to 1 > S22 ⪯0; S11 S12 S22 S12 ⪯0;
Lemma 1 is the well-known Schur complement lemma. Based on Lemma 1, the condition (31) in Proposition 1 can be transformed into " # ð1 ηÞPp ðI þ Bp Kp Þ > !0; 8p ¼ 1; 2; 3; 4 I þ Bp Kp Pq 1
ð39Þ
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Letting Qp ¼ Pp 1 , p ¼ 1; 2; 3; 4 and performing a congruence transformation to Eq. (39) via diagfQp ; Ig, we have " # ð1 ηÞQp Qp þ Xp> Bp> !0; 8p ¼ 1; 2; 3; 4 ð40Þ Qp þ Bp Xp Qq where Xp ¼ Kp Qp . The parameter 0oηo1 indicates the decay rate of waiting vehicles, which is assumed to be prescribed in the following discussion. Therefore, the state feedback control can be obtained by solving following LMI feasibility problem, if the 0oηo1 is fixed. Proposition 3. Consider the signalized oversaturated intersection (10), if there exist positive matrices Qp g 0, 8 p ¼ 1; 2; 3; 4 and a scalar 0oηo1 such that " # ð1 ηÞQp Qp þ Xp> Bp> !0; 8p ¼ 1; 2; 3; 4 ð41Þ Qp þ Bp Xp Qq where the phase q ¼ sðpÞ is the successive phase after phase p. Then, the oversaturated situation can be relieved by green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4, where Kp ¼ Xp Qp 1 . If the initial vector of waiting vehicles are xð0Þ and the boundary between oversaturated and unsaturated situation is given as ε40, the relieving time can be estimated by K4
lnðbmin =bmax Þ þ lnðε= Jxð0ÞJ Þ lnð1 ηÞ
ð42Þ
where bmin and bmax satisfy bmin I⪯Qp 1 ⪯bmax I, 8 p ¼ 1; 2; 3; 4. Then, Proposition 2 is considered. At first, it is assumed that there exist Qp g 0, 8 p ¼ 1; 2; 3; 4 such that Eq. (41) is established. Since in Eq. (38), the parameter ρ is coupled with cmin , we let cmin ¼ 1 and cmax ¼ μ, then Eq. (38) becomes ρ2 4μx > ðk0 ÞRxðk 0 Þ
ð43Þ
The main purpose is to formulate LMI-based conditions and these conditions are convenient to minimize parameter ρ, which is equivalent to minimize μ according to Eq. (43). To fulfill this aim, by using Lemma 1 again, it has " # Qp Qp ⪯0 ) Qp þ Qp RQp r 0 ) R r Qp 1 ) Ir R 1=2 Qp 1 R 1=2 Qp R1 "
μR
I
I
Qp
# ⪯0 ) μR þ Qp 1 r 0 ) R 1=2 Qp 1 R 1=2 r μI
Thus, the following optimization problem with a fixed 0oηo1 can be formulated for 8 p ¼ 1; 2; 3; 4: minμ
"
ð1 ηÞQp s:t: Qp þ Bp Xp
# Qp þ Xp> Bp> !0 Qq
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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" "
Qp
Qp
#
⪯0 R1 # I ⪯0 Qp
Qp μR I
13
ð44Þ
where the phase q ¼ sðpÞ is the successive phase after phase p. On the basis of optimization problem in the LMI form, the following result can be derived to get the optimal green time setting with respect to the minimized ρ. Proposition 4. Consider the signalized oversaturated intersection (10), the optimal feedback controller gains Kp ¼ Xp Qp 1 can be obtained by solving optimization problem (44) with μmin . Then, the green time of each phase p asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees x > ðkÞRxðkÞr ρ2min , 8 k A ½0; K, where ρmin ¼ μmin x > ðk 0 ÞRxðk0 Þ and K is the relieving time estimated by Eq. (42). Propositions 3 and 4 are the LMI formation of Propositions 1 and 2 with a prescribed decay rate η. Though the parameter η is fixed in above discussion, it is possible to set a variation and run optimization (44) with respect to all the values in ð0; 1Þ with discretized step Δη. All the LMI conditions in this subsection can be efficiently solved by existing software tools such as LMI toolbox in MATLAB. 4. Robust signalization subjected to unknown disturbances In the previous sections, the unknown disturbances are not involved in system model (10). However, since the disturbances inevitably exist in practice such as the uncertainties or disturbances in the input traffic flow to the intersection, the previous results should be extended to robust signalization subject to disturbances. At first, the system model with disturbances is presented. By adding the unknown disturbance term ωðkÞ into system (10), the system with unknown input disturbances can be expressed by 4
xðk þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ þ Bp up ðkÞ þ ωðkÞ
ð45Þ
p¼1
where ωðkÞ ¼ ½ω1 ðkÞ ω2 ðkÞ ω3 ðkÞ ω4 ðkÞ ω5 ðkÞ ω6 ðkÞ ω7 ðkÞ ω8 ðkÞ > is the vector of unknown input disturbances and xðkÞ, Bq , up(k), ηp ðkÞ are defined same as in Eq. (10). Furthermore, the reduced-order system can be presented as 4
~ þ 1Þ ¼ ∑ ηp ðkÞ½xðkÞ ~ þ B~ p up ðkÞ þ ωðkÞ ~ xðk
ð46Þ
p¼1
> ~ where ωðkÞ ¼ ∑l ¼ 1;2;5;6 ωl ðkÞ ∑l ¼ 3;4;7;8 ωl ðkÞ . In the presence of disturbances, besides the relief of oversaturated situation of interest, the disturbance attenuation performance of control strategy also draws great attention by us. In this paper, the disturbance attenuation performance is considered. With respect to the output as a convex combination of waiting vehicles in each lanes as yðkÞ ¼ CxðkÞ
ð47Þ
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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14
where C A R1n , the definition of H1 disturbance attenuation performance describing the relationship between disturbance and output is given as follows:
If under zero initial condition, i.e. xð0Þ ¼ 0, the H1 disturbance attenuation problem is defined by the following inequality: K
K
k¼0
k¼0
∑ JyðkÞ J 2 r γ 2 ∑ J ωðkÞJ 2 ;
8 K40
ð48Þ
where γ is the H1 performance level. Obviously, the control strategy is required to reduce performance level γ as small as possible to achieve better disturbance attenuation performance. Combining the ρ in Problem 2 which is also required to be minimized, the control objective here is formulated as to minimize the value of convex combination of γ2 and ρ2. The objective function to be minimized can be described as Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2
ð49Þ
where 0oθo1. Problem 3. Design the green time of each phase, i.e. the input up ðkÞ ¼ gp ðkÞ, p ¼ 1; 2; 3; 4 such that (1) For ωðkÞ ¼ 0: There exists a time K such that JxðKÞJ r ε, 8k Z K, where the estimated K is the relieving time; x > ðkÞRxðkÞr ρ2 , 8k A ½0; K, where ρ40 is the prescribed bound avoiding risk of overflow; For ωðkÞa 0: (2) Minimize objective function Hðρ; γÞ in (49).
On the basis of quadratic Lyapunov function approach, the following derivation can be obtained. Considering J ¼ JyðkÞ J γ JωðkÞJ þ ΔVðxðkÞÞ, we have J ¼ JyðkÞJ 2 γ 2 JωðkÞ J 2 þ ΔVðxðkÞÞ ¼ ½x > ðkÞC > CxðkÞ γ 2 ωðkÞ > ðkÞωðkÞ þ ΔVðxðkÞÞ " # h i xðkÞ > > ¼ x ðkÞ ω ðkÞ Ξ p;q ωðkÞ where Ξ p;q ¼
"
ðI þ Bp Kp Þ > Pq ðI þ Bp Kp Þ Pp þ C > C
ðI þ Bp Kp Þ > Pq
Pq ½ðI þ Bp Kp Þ
Pq γ 2 I
# :
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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If we have Ξ p;q !0, it arrives J yðkÞJ 2 γ 2 J ωðkÞ J 2 þ ΔVðxðkÞÞo0, which leads to K
∑
k¼0
J yðkÞJ 2 γ 2 J ωðkÞJ 2 þ VðxðK þ 1ÞÞ Vðxð0ÞÞo0
Due to the fact that VðxðK þ 1ÞÞ40 and Vðxð0ÞÞ ¼ 0, it has K
K
k¼0
k¼0
∑ J yðkÞJ 2 o ∑ γ 2 JωðkÞ J 2 ;
8 K40
Therefore, the H1 disturbance attenuation performance can be ensured by Ξ p;q ! 0, q ¼ sðpÞ, 8p ¼ 1; 2; 3; 4. Then, in order to apply Ξ p;q ! 0, q ¼ sðpÞ, 8 p ¼ 1; 2; 3; 4 into controller design, Lemma 1 is used to equivalently transform Ξ p;q !0, q ¼ sðpÞ, 8 p ¼ 1; 2; 3; 4 into what follows. 2 3 0 ðI þ Bp Kp Þ > C > Pp 6 0 γ2I I 0 7 6 7 ð50Þ 6 7 !0; q ¼ sðpÞ; 8 p ¼ 1; 2; 3; 4 1 4 I þ Bp Kp I Pq 0 5 C
0
0
I
By letting Qp ¼ Pp 1 , p ¼ 1; 2; 3; 4 and performing a congruence transformation to Eq. (39) via diagfQp ; I; I; Ig, we have 2 3 Qp 0 Qp þ Xp> Bp> Qp C > 6 7 6 0 γ2 I I 0 7 6 7 ! 0; q ¼ sðpÞ; 8p ¼ 1; 2; 3; 4 ð51Þ 6 Q þ Bp Xp I Qq 0 7 4 p 5 CQp 0 0 I So, Eq. (51) can guarantee the H1 disturbance attenuation performance with prescribed level γ. Moreover, to estimate relieving time K such that JxðKÞJ r ε, 8 k Z K, the decay rate 0oηo1 is added into Eq. (51), and the following condition is obtained: 3 2 ð1 ηÞQp 0 Qp þ Xp> Bp> Qp C > 7 6 6 0 γ2I I 0 7 7 ! 0; q ¼ sðpÞ; 8p ¼ 1; 2; 3; 4 ð52Þ 6 6 Q þ Bp Xp I Qq 0 7 5 4 p CQp 0 0 I It is obvious that Eq. (51) is a particular case of Eq. (52) with η ¼ 0, and Eq. (51) can be always established if Eq. (52) holds. And when ωðkÞ ¼ 0, it implies that there exists a time K such that JxðKÞJ r ε, 8 k Z K, where the relieving time K is estimated by Eq. (42). Then, by the same steps in deriving Proposition 2, if the following conditions are satisfied: " # Qp Qp ⪯0; 8p ¼ 1; 2; 3; 4 ð53Þ Qp R1 "
μR I
# I ⪯0; Qp
8 p ¼ 1; 2; 3; 4
ð54Þ
We can conclude that x > ðkÞRxðkÞr ρ2 , 8 k A ½0; K, where ρ2 Z μx > ðk0 ÞRxðk 0 Þ. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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By summarizing above procedures, the convex optimization problem to minimize the objective function Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2 with a fixed decay rate 0oηo1 can be formulated as follows: minθρ2 þ ð1 θÞγ 2 2 0 Qp þ Xp> Bp> ð1 ηÞQp 6 6 0 γ2 I I s:t:6 6 Q þ Bp Xp I Qq 4 p CQp 0 0 " # Qp Qp ⪯0 Qp R1 " # μR I ⪯0 I Qp μx > ðk0 ÞRxðk 0 Þ ρ2 r 0
Qp C > 0 0 I
3 7 7 7!0 7 5
ð55Þ
As to the improved optimization (55) concerned with both H1 disturbance attenuation performance γ and state boundary ρ, the LMI-based solution to Problem 3 is proposed based on Eq. (55). Proposition 5. Consider the signalized oversaturated intersection (45) with disturbances ωðkÞ, the optimal feedback controller gains Kp ¼ Xp Qp 1 can be obtained by solving optimization problem (55) with H min . Then, the green time of each phase as gp ðkÞ ¼ Kp xðkÞ, p ¼ 1; 2; 3; 4 guarantees Hðρ; γÞ r H min , 8 k A ½0; K, where Hðρ; γÞ ¼ θρ2 þ ð1 θÞγ 2 and K is the relieving time estimated by Eq. (42). In this section, an extension from system without disturbances to the one with unknown disturbances is presented. By Proposition 5, it is seen that the green time setting via state feedback strategy not only provides the oversaturated situation relieving performance as in the previous section, the disturbance attenuation in the sense of H1 performance is also considered to minimize the affect caused by the disturbances to the traffic flow. 5. Simulation study The basic mechanism of our approach is that the green time of each phase is capable of adjusting adaptively to the length of waiting queue, namely the state feedback control scheme. Compared with some methods with fixed time setting, our approach is supposed to have better control performance. In this section, a comparison through simulation will be presented to illustrate the advantages of approach proposed in this paper. By the signalization strategy proposed in the previous sections, the oversaturated situation will be relieved by the state feedback control scheme, and the risk of overflow can be avoided as well. However, in practical point of view, it is worth noting that, in order to fit physically reasonable signalization, the controller must respect the boundary conditions on green time gp(k). Generally, two boundary values gmin and gmax have to be selected so that gmin r gp ðkÞ r gmax , p ¼ 1; 2; 3; 4. The values of gmin and gmax must be well selected. For instance, too short effective green lights are impractical and too long effective green lights are unacceptable to the stopped drivers of the other approach. In addition, the approach proposed in this paper mainly focuses on the Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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oversaturated situation, if the intersection works in the unsaturated situation, it remains choosing some traditional signalization methods such as Webster method to determine green time gnp , p ¼ 1; 2; 3; 4. Without loss of generality, it is assumed that gmin r gnp r gmax , p ¼ 1; 2; 3; 4. In conclusion, with respect to the boundary condition on the control light and signalization for unsaturated situation, the operation of the controller leads to the following cases: 8 n if JxðKÞJ r ε g > > > p > < Kp xðkÞ if JxðkÞJ 4ε and gmin ogp ðkÞogmax ð56Þ gp ðkÞ ¼ gmax ifgp ðkÞ Z gmax > > > > : gmin ifgp ðkÞ r gmin The crucial point in designing controller (56) is to determine state feedback gain Kp , i ¼ 1; 2; 3; 4, which can be obtained by solving LMI feasibility problem in Proposition 3, 4 or 5. Here, one additional constraint must be considered in practical green time setting, that is the obtained green time gp(k), 8 p ¼ 1; 2; 3; 4 has to be positive. Since the length of waiting queue is always non-negative which implies the variables defined by state vector xðkÞ are always confined to the positive orthant, i. e. xðkÞ c 0. Therefore, the positivity of green time gp(k), 8 p ¼ 1; 2; 3; 4 is equivalent to Kp c 0, 8p ¼ 1; 2; 3; 4. By the design results in previous sections where the controller gains are determined by Kp ¼ Xp Qp 1 , herein we further choose positive matrices Qp ¼ diagfqp1 ; qp2 ; …; qpn g and confine matrices Xp c 0, 8p ¼ 1; 2; 3; 4 in our LMI. Thus, it can be seen that Qp 1 ¼ diagf1=qp1 ; 1=qp2 ; ⋯; 1=qpn g and Xp Qp 1 c 0, which directly leads to the positivity of green time gp(k), 8 p ¼ 1; 2; 3; 4. At first, the relief of an oversaturated intersection is considered. It is assumed that the intersection is depicted by Fig. 1 which involves 8 lanes and 4 phases and the simulation will be carried out using the following data:
Input flow rate: q1 ¼ q3 ¼ q5 ¼ q7 ¼ 0:35 veh=s, q2 ¼ q4 ¼ q6 ¼ q8 ¼ 0:3 veh=s. Saturation flow rate : s1 ¼ s3 ¼ s5 ¼ s7 ¼ 1:5 veh=s, s2 ¼ s4 ¼ s6 ¼ s8 ¼ 1:3 veh=s. Minimal and maximal green time: gmin ¼ 15 s, gmax ¼ 120 s.
At first, the green time of each phase by Webster method is obtained as gn1 ¼ gn3 ¼ 17:5069, g1 ¼ gn3 ¼ 17:3145. Then, the state feedback gains Kp , p ¼ 1; 2; 3; 4 are determined by optimization (44). In order to find feasible solution, we consider the reduced-order system form (20). With the prescribed decay rate η ¼ 0.9 and R ¼ I, the following feedback gains are obtained: n
K1 ¼ ½0:3203 0:1080;
K2 ¼ ½0:0610 0:0740;
K3 ¼ ½0:1080 0:3203;
K4 ¼ ½0:0740 0:0610
~ with the x~ > ðkÞRxðkÞr ρ2min , where ρmin ¼ 549:2136. Assuming the initial waiting vehicles > x ð0Þ ¼ ½70 50 60 50 60 50 80 70 and the boundary between oversaturated and unsaturated situation ε ¼ 100, the relieving process by our approach are shown in Fig. 2, in which it can be observed that the waiting vehicles can be effectively relieved by our approach. ~ To show the advantages of our feedback control strategy, the evolution of J xðkÞJ by Webster method and our approach are shown in Fig. 3. Comparing the simulation results in Fig. 3, the oversaturated situation can be relieved more quickly by our approach. The relieving time is given in Table 1. The main reason is that the green Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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time set by feedback scheme can adjust dynamically with respect to the length of waiting queue, which is more adaptive to the number of waiting vehicles and relieve them quickly. Then, the unknown disturbance input is considered. In this example, the output vector is selected as C ¼ ½1 1 and the concerned output is therefore standing for the total waiting vehicles ~ ¼ ∑8l ¼ 1 xl . The parameter θ in objective function is chosen as θ ¼ 0.5. To as yðkÞ ¼ CxðkÞ achieve the optimal control performance involving H1 disturbance attenuation performance γ and state boundary ρ, optimization (55) is executed to get the following state feedback gains: K1 ¼ ½0:5869 0:0309;
K2 ¼ ½0:7278 0:0150;
K3 ¼ ½0:0309 0:5869;
K4 ¼ ½0:0150 0:7278
450 x1+x2+x5+x6
400
x +x +x +x 3
4
7
8
350 300
x
250 200 150 100 50 0
0
20
40
60
80
100
k Fig. 2. Evolution of waiting vehicles relieved by our control method.
600
Feedback method Webster method
500 Estimated boundary ρmin
||x||
400
300
200 124
100 25 0
0
50
100
150
200
k ~ J with two methods and estimated boundary. Fig. 3. Evolution of vehicles J xðkÞ Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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Given the same initial waiting vehicles and randomly generated disturbance in Fig. 4, and the outputs by both feedback control and Webster method are shown in Fig. 5. By comparing the curves in Fig. 5, the feedback control has better disturbance attenuation performance, since the Table 1 Relieving time of two methods. Method
Feedback method
Webster method
Relieving time K
25
124
40 w (k) 1
w (k)
30
2
20
w(k)
10 0 −10 −20 −30 −40
0
50
100
150
200
k
Fig. 4. Randomly generated disturbance.
600 Feedback method Webster method 500
y
400
300
200
100
0
0
50
100
150
k
Fig. 5. Output responses subjected to unknown disturbance. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
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output response under feedback scheme is much less sensitive to the unknown disturbance method during the relieving period. 6. Conclusions This paper presents a static state feedback control for a signal oversaturated intersection. By employing Lyapunov function method, sufficient conditions for the existence of state feedback controller relieving the oversaturated situation are obtained, and moreover, in order to avoid the risk of overflow of waiting vehicles which could lead to the traffic congestion, the boundary of waiting queue length is estimated and further minimized by state feedback control. As an extension to the case with unknown disturbance, robust signalization via state feedback control is presented. All the design procedure can be executed through a set of LMI optimization problems. Through simulation study, the effectiveness and advantages of state feedback control are shown by comparison with traditional Webster method. In this paper, the state feedback control for a single intersection is investigated, but the network of intersections has not been considered here. It is worthwhile to extend our results to signalization for a network of several intersections.
References [1] M. Kaltenbach, H. Koivo, Modelling and control of urban traffic flow, in: Proceedings of the Joint Automatic Control Conference, University of Texas, Houston, 1974, pp. 147–154. [2] Y. Tangsheng, On Line Traffic Signalization Using Robust Feedback Control (Ph.D. dissertation), Faculty of the Virginia Polytechnic Institute and State University, Virginia, 1997. [3] C. Diakaki, M. Papageorgiou, K. Aboudolas, A multivariable regulator approach to traffic-responsive network-wide signal control, Control Eng. Pract. 10 (2002) 183–195. [4] D. Gazis, Optimal control of a system of oversaturated intersections, Oper. Res. 12 (1964) 815–831. [5] D. Gazis, R. Potts, The oversaturated intersection, in: Proceedings of the Second International Symposium on the Theory of Road Traffic Flow Organization for Economic Cooperation and Development, Paris, 1965, pp. 221–237. [6] D. Green, Control of oversaturated intersections, Oper. Res. Q. 18 (2) (1968) 161–173. [7] G. Dans, D. Gazis, Optimal control of oversaturated store-and-forward transportation networks, Transport. Sci. 1 (1976) 1–19. [8] T. Chang, J. Lin, Optimal signal timing for an oversaturated intersection, Transport. Res. B (2000) 471–491. [9] B. Schutter, B. Moor, Optimal traffic light for a single intersection, Eur. J. Control 4 (3) (1998) 260–276. [10] F. Motawej, R. Bouyekhf, A. El Moudni, A dissipativity-based approach to traffic signal control for an oversaturated intersection, J. Frankl. Inst. 348 (2011) 703–717. [11] F. Boillot, J.M. Blosseville, J.B. Lesort, V. Motyka, M. Papageorgiou, S. Sellam, Optimal signal control of urban traffic networks, in: Proceedings of the Sixth IEE International Conference on Road Traffic Monitoring and Control, London, 1992, pp. 75–79. [12] F. Dion, S. Yagar, Real-time control of signalized networks different approaches for different needs, in: Proceedings of the Eighth IEE International Conference on Road Traffic Monitoring and Control, Savoy Place, London, 1996, pp. 56–60. [13] P.B. Hunt, D.L. Robertson, R.D. Bretherton, M.C. Royle, The SCOOT on-line traffic signal optimisation technique, Traffic Eng. Control 23 (1982) 190–199. [14] P. Michalopoulos, G. Stephanopolos, Optimal control of oversaturated intersections theoretical and practical consideration, Traffic Eng. Control (1978) 216–221. [15] H. Zhou, B. Rachid, M. Adbellah, Modelling and H 1 control of urban transportation network, in: 13th IFAC Symposium on Control in Transportation Systems, The International Federation of Automatic Control, September 12–14, 2012. Sofia, Bulgaria, pp. 72–77. [16] R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilization of hybrid systems, Proc. IEEE 88 (2000) 1069–1082. Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017
W. Xiang et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
21
[17] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control 54 (2009) 308–322. [18] W. Xiang, J. Xiao, M.N. Iqbal, Asymptotic stability, l2 gain boundness analysis and control synthesis for switched systems: a switching frequency approach, Int. J. Adapt. Control Signal Process. 26 (4) (2012) 350–373. [19] W. Xiang, J. Xiao, Stability analysis and control synthesis of switched impulsive systems, Int. J. Robust Nonlinear Control 22 (2012) 1440–1459. [20] W. Xiang, J. Xiao, Stabilization of switched continuous-time system with all modes unstable via dwell time switching, Automatica 50 (2014) 940–945. [21] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (1998) 475–482. [22] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, SIAM Rev. 49 (2007) 545–592. [23] W. Xiang, J. Xiao, H 1 finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance, J. Frankl. Inst. 348 (2) (2011) 331–352. [24] W. Xiang, J. Xiao, Convex sufficient conditions on asymptotic stability and ℓ2 gain performance for uncertain discrete-time switched linear systems, IET Control Theory Appl. 8 (2014) 211–218. [25] C. Diakaki, M. Papageorgiou, T. McLean, Application and evaluation of the integrated traffic-responsive urban corridor control strategy IN-TUC in Glasgow in: Preprints CD-ROM of the 78th Annual Meeting of the Transportation Research Board (Paper No. 990310), Washington, DC, USA, 1999. [26] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994.
Please cite this article as: W. Xiang, et al., Real-time signalization for an oversaturated intersection via static state feedback control: A switched system approach, Journal of the Franklin Institute. (2014), http://dx.doi.org/ 10.1016/j.jfranklin.2014.09.017