Abstract
In this paper, a new approach to the decentralized control design for vehicle platooning for uncertain automated highway systems is proposed. The uncertainty in the system, which is nonlinear and (possibly) fast time-varying, is bounded. The bound is assumed to be within a prescribed fuzzy set. A creative transformation is made to the system, which converts a local problem to a global problem. Based on the fuzzy description of the uncertainty bound and the transformation, a class of decentralized control is proposed in which each vehicle only needs the knowledge of its preceding vehicle in the platoon. No acceleration feedback or the information of the leading vehicle is required. Both the vehicle platooning system and the control are deterministic, hence not if-then fuzzy rule-based. The performance of the resulting controlled system is twofold. First, the collision avoidance performance is guaranteed under any safe initial conditions regardless of the value of the uncertainty. Second, the minimization of a fuzzy-based performance index is guaranteed based on an optimal choice of a control design parameter. Numerical simulations are conducted to validate the efficiency of the proposed algorithm.
1. Introduction
The Automated Highway System (AHS) is a proposed intelligent transportation system (ITS) technology designed to increase capacity and safety with the increasingly severe traffic congestion since the last two decades. The basic idea is by grouping vehicles into platoons at closer spacing under automatic control, which cannot be achieved by human drivers alone, to increase the capacity on highways [1, 2]. With the distances between vehicles becoming smaller, safety (or collision avoidance) is the first and foremost concern, which must be guaranteed for the AHS. From the control design point of view, stability is often the primary system performance. However, a platoon of automated vehicles is an interconnected system in which stability of each component system per se is not sufficient to guarantee the boundedness of the spacing errors for all the vehicles [3], which is directly related to safety.
For this reason, the alternative string stability has been proposed, which implies the boundedness of all the states of the interconnected system if the initial states of the vehicles are within some neighborhood of the equilibrium states [4]. Efforts are then devoted to linearize the system through control with required platoon information (acceleration feedback and/or information of the leading vehicle) and to use transfer function analysis to investigate string stability [5–8]. Many important contributions have been made (for a survey, see, for example, [9–12]). However, as it was pointed out in [13], the string stability alone does not provide a warranty for global collision avoidance of the platoon. Rather, the initial conditions should be within a neighborhood around the equilibrium position in order to prevent vehicle collision [14, 15], hence a local performance. Earlier research studies regarding local collision avoidance can be found in [16–18]. The restriction on the initial conditions reduces the range of applications. This promotes a need to further design a controller for the platoon which guarantees global collision avoidance under any safe initial condition (that is, as long as the collision does not occur initially).
One salient feature of the paper is that the uncertainty in the system, which may come from parameter variations (the mass variation and the movement of the passengers and/or the stowage) and aerodynamics (the unmeasurable side wind), as well as external disturbances (the rolling resistance and slope of road), is described using fuzzy set theory [19–21], instead of fuzzy if-then inference rules [22, 23]. Neither the platoon model nor the controller is fuzzy if-then rule-based, hence not the Takagi–Sugeno–Kang (TSK) fuzzy system [24, 25] nor the Mamdani-type fuzzy controller [26]. This distinguishes the current work from some others in fuzzy systems and control research. For example, the uncertainty bound may be close to 2. Here, “close to” is a linguistic variable which is associated with a fuzzy set. Earlier works related to fuzzy if-then rule-based approach for platooning can be found in [27, 28].
The main contributions of this paper are fourfold. First, a creative state transformation is made on the error dynamics of each vehicle in the platoon. This converts a local problem to a global problem. Second, fuzzy set instead of fuzzy if-then rules is used to describe the uncertainty bound of the system. A class of decentralized robust controllers is proposed based on the fuzzy description of the uncertainty bound. Each vehicle in the platoon only needs the information of its preceding vehicle for the control. No acceleration feedback or information of the following or leading vehicle or other vehicle is required. The control is proven to render each transformed error dynamics globally practically stable regardless of the uncertainty. Third, it is shown that the global practical stability leads to global collision avoidance and the spacing error will converge to a small region. Fourth, an optimal design problem associated with the fuzzy-based performance index and the control cost is formulated, which enables us to choose an optimal gain for a control parameter. It is proven that the global solution to the optimal problem exists and is unique. Furthermore, the closed-form (that is, the analytic) expression of the optimal solution is shown. This completely solves the problem.
2. Vehicle Platooning Model
2.1. Platoon Configuration
Consider a platoon of automated vehicles traveling on a straight line of a highway and closely following one another, as shown in Figure 1. The position of the leading vehicle and the following vehicle in the platoon is denoted by and with respect to an inertial frame, respectively. The actual space between the vehicle and its preceding one is given bywhere is the length of the vehicle in the platoon. Henceforth, subscripts L and i are used to label the leader and the follower in the platoon, respectively.
Suppose the desired space between the and vehicle is denoted by a constant scalar . Then, the space error is given by
Notice that means a collision happens at time . A definition related to the collision avoidance performance is given as follows.
Definition 1 (collision avoidance). A platoon of controlled vehicles is collision avoidable if given any safe initial condition (that is, initially noncolliding or ):for all .
Remark. It is worth pointing out here that collision avoidance (3) is the foremost performance to be guaranteed for automated vehicles. From the practical design point of view, one should first guarantee that the platoon is collision avoidable.
2.2. Platoon Error Dynamics
The longitudinal equation of motion for the ith vehicle is determined by the Newton’s second law aswhere is the time, is the uncertain parameter representing all the uncertainty in the vehicle, and is the control input which is from the vehicle propulsive/braking effort. Moreover, we assume , , and are continuous, where represents the vehicle mass, represents the aerodynamic drag force, and represents the rolling resistance force and other external disturbances acting on the vehicle.
Differentiating (2) with respect to time, we get the space error velocity and space error dynamics as follows:
With (4) in (6) yields (henceforth, arguments of functions are sometimes omitted when no confusion is likely to arise),
The control objective for vehicle platooning is stated as follows: design the control in (4) for the ith vehicle in the platoon with available information (will be elaborated later) such that the resulting controlled platoon is collision avoidable under arbitrary safe initial conditions.
2.3. Bounding and Structure Conditions
We decompose , , and in (4) as follows:where , , and denote the “nominal” portions with (this is always feasible since the nominal portion is the designer’s discretion), while , , and are the uncertain portions. The functions , , , , , and are all continuous.
Denote
Hence, .
Assumption 1. (i) Let the initial state of the error dynamics of the ith vehicle in the platoon, which is uncertain, be represented by . For each entry of , namely, , , there exists a fuzzy set in a universe of discourse characterized by a membership function . That is,where is known and compact. (ii) For each entry of the vector , namely, , , the function is Lebesgue measurable. (iii) For each , there exists a fuzzy set in a universe of discourse characterized by a membership function . That is,where is known and compact.
Assumption 2. (1) There exists a (possibly unknown) parameter such that for all , ,(2)The unknown parameter belongs to a known fuzzy number.
Remark. In the special case that (i.e., no uncertainty in the vehicle mass), we get . Hence, one can choose . The assumption imposes the effect of uncertainty on the possible deviation of from to be within a unidirectional threshold.
2.4. State Transformation
We propose the following transformation for the space error :which means that
Differentiating (15) with respect to time, we get
Letyields
With (7) in (20), the transformed space error dynamics is in the form of
Remark. From (15), we have for all , the corresponding is . Furthermore, . This is in fact the main reason for the transformation from to . If both and are bounded, then space error , hence no collision.
Remark. The state given by (19) is selected to make the transformed space error dynamics in a lower triangular form so that the backstepping method [29] can be adopted for the control design, which will be explained in the next section.
3. Robust Control Design and Performance Analysis
We now propose a robust controller for the transformed system to realize the performance of uniform boundedness and uniform ultimate boundedness. Then, we show that the controlled platoon will have guaranteed collision avoidance in the error space.
3.1. Robust Control Design
Let
Assumption 3. (1) There exists a possibly unknown parameter and a known function such that for all , ,(2)The unknown parameter belongs to a known fuzzy number.
Remark. The fuzzy numbers and are relevant to . Their associated membership functions can be determined via , the fuzzy arithmetic and the decomposition theorem [30]. The extreme value of the fuzzy number can be evaluated since the universes of discourse are known.
LetWe propose the following control:where
Theorem 1. Let . Subject to Assumptions 1–3, the control (27) renders the transformed system (22) and (21) the following performance:(i)Uniform boundedness: for any , there is a such that if , then for all (ii)Uniform ultimate boundedness: for any with , there exists a such that for any as , where
Proof. Consider the Lyapunov function candidate:For a given uncertainty , the derivative of along the trajectory of the controlled system is to be evaluated. First, we haveNext, in view of (21), the first term on the RHS of (30) is given byThe second term on the RHS of (30) isBy the third term on the RHS of (32), after decomposing , , and by (9)–(11) and noting that , we getBy (23), it can be shown thatNext, by (24),By Assumption 3, we obtainBy (28), Assumption 2, and (26),With (33)–(37) in (32), we haveWith (31) and (38) in (30), we obtainSincewe have (noting that )where . This means that is negative definite for all such thatSince all universes of discourse are compact (hence, closed and bounded), is bounded. Noticing that is crisp, we conclude that is a negative definite for sufficiently large . Therefore, upon invoking the standard arguments as in [31], we conclude the solution to the transformed error dynamics of the vehicle is uniform boundedness withFurthermore, uniform ultimate boundedness also follows withQ.E.D.
Remark. Notice the control given by (23) only needs the information of the preceding vehicle. No acceleration feedback or information of the leader or other followers are required. One may argue that the control given by (23) needs the information of besides and of the preceding vehicle, which increases the burden of communication. However, a straightforward modification to the control will show that without the information of , (noting that is the acceleration of the vehicle), the control still works as long as this unknown acceleration is bounded by a known function .
Remark. In the special case when the system is without uncertainty, that is , we may choose and hence . The control will render . Therefore, we conclude that as . Notice that . Hence, as . Furthermore, if we choose , then . This means if initially, then for all .
Remark. Notice that the third term in the control scheme (27) is for compensating the uncertainty and is the control gain. From (43) and (44), it can be concluded that the gain can be used to manipulate the size of the uniform ultimate boundedness region. In particular, the larger the gain, the smaller the size. This stands for a trade-off between the performance and the cost. As a result, it may be interested in seeking an optimal choice of for a compromise, which will be elaborated later.
3.2. Guaranteed Collision Avoidance
Theorem 2. Consider a platoon of vehicles with the vehicle described by (4). Suppose that Assumptions 1–3 are met. The control (27) for the following vehicle guarantees the collision avoidance of the platoon under arbitrary safe initial condition.
Proof. For the ith follower in the platoon, if an initial condition and a desired space are specified, the corresponding initial condition in the transformed state of the same vehicle is given bySuppose control (27) is proposed, and hence the transformed state of the follower is uniform bounded. By the definition of uniform boundedness in Theorem 1 with (43), we conclude that for any , there is a such that ifthenwherefor all . From (48), we havewhich means that in the worst caseFrom (51), we getTherefore,Since , from (54), we getand from (55), we havefor all . Consequently, by Definition 1, we conclude that the platoon is collision avoidable. Q.E.D.
4. Optimal Robust Control
4.1. Fuzzy-Based Performance Index
In the previous analysis, we know that the system performance can be guaranteed by a deterministic control design. The size of the uniform ultimate boundedness region decreases as increases. Furthermore, the size approaches zero when approaches infinity. Therefore, both the performance and control effort are affected by the control gain . From the practical design point of view, one may be interested in seeking an optimal choice of for a compromise among various conflicting criteria.
We first explore more on the deterministic performance of the uncertain system. From (41), we know thatwhere . This is a differential inequality [32], whose analysis can be made according to [33]. The following is needed for our analysis of (58).
Definition 2 (see [32]). If is a scalar function of the scalars ψ and t in some open connected set D, we say a function , and is a solution of the differential inequalityon if is continuous on and its derivative on satisfies (59).
Theorem 3 (see [32]). Let be continuous on an open connected set such that the initial value problem for the scalar equationhas a unique solution. If is a solution of (60) on and is a solution of (59) on with , then for .
Instead of exploring the solution of the differential inequality, which is often nonunique and not available, the theorem suggests that it may be feasible to study the upper bound of the solution. The reasoning is, however, based on that the solution of (60) is unique.
Theorem 4 (see [33]). Consider the differential inequality (59) and the differential equation (60). Suppose that for some constant , the function satisfies the Lipschitz conditionfor all points . Then, any function that satisfies the differential inequality (59) for satisfies also the inequalityfor .
We consider the differential equation
The RHS satisfies the global Lipschitz condition with . We proceed with solving the differential equation (63). This results in
Therefore,orfor all . By the same argument, we also have, for any and any ,where . The time is when the control scheme (27) starts to be executed. It does not need to be .
Since , the RHS of (67) provides an upper bound of . This in turn leads to an upper bound of . For each , let
Notice that for each , , as .
Definition 3. Consider a fuzzy set:For any function , the D-operation is given by
Remark. In a sense, the D-operation takes an average value of over . In the special case that , this is reduced to the well-known center-of-gravity defuzzification method (see, e.g., [34]). If is crisp (i.e., for all ), then .
Lemma 1. For any crisp constant ,
Proof. By Definition 3,Q.E.D.
We now propose the following performance index. For any , letwhere . The performance index consists of three parts. The first part may be interpreted as the value of the overall transient performance (via the integration) from time . The second part may be interpreted as the value of the steady state performance. The third part may be interpreted as the value of the control effort. Both and are weighting factors. The weighting of is normalized to be unity.
For any , given , , and , our design problem is to choose in (28) such that the performance index is minimized.
Remark. A standard LQG (i.e., linear-quadratic- Gaussian) problem in stochastic control is to minimize a performance index which is the average (via the expectation value operation in probability) of the overall state and control accumulation. The current approach may be viewed, loosely speaking, as a parallel, though not equivalent, in fuzzy dynamical systems. However, one cannot be too careful in distinguishing the differences. For example, the Gaussian probability distribution implies that the uncertainty is unbounded (although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is always finite. Also, the standard LQG does not take parameter uncertainty into account.
One can show thatTaking the D-operation yields,The last equality is due to Lemma 1. Next, we analyze the cost . Again, by Lemma 1,With (76) and (77) in (74), we obtainwhere , , , and .
4.2. Formulation of the Optimal Design Problem
The optimal design problem is then equivalent to the following constrained optimization problem. For any ,
For any , taking the first order derivative of with respect to ,
That leads toorwhich is a quartic equation.
Theorem 5. Suppose . For any , given , , , and , the solution to (82) always exists and is unique, which globally minimizes the performance index (79).
Proof. Let . Then, and is continuous in . In view of and , we conclude that is strictly increasing in . Since , we have , , , , and, therefore, . Consequently, the solution to (82) always exists and is unique. For the unique solution that solves (82),Therefore, the positive solution of the quartic equation (82) solves the constrained minimization problem (68). Q.E.D.
Remark. In the special case that the fuzzy sets are crisp, , , etc. The current setting still applies. The optimal design can also be found by solving (82), the solution of which will be given in the next section.
4.3. Solution to the Optimal Problem
The solutions of the quartic equation (82) depend on the following cubic resolvent:where
Let and . The discriminant of the cubic resolvent is
Since , , , and , we have , thus , so that . The solutions of the cubic resolvent are given bywhere
The cubic resolvent has one real solution and two complex conjugate solutions. This in turn implies that the quartic solution has two real solutions and one pair of complex conjugate solutions. The maximum real solution, which is positive and is therefore the optimal solution to the constrained optimization problem, of the quartic equation is given by
With , , and into (91), the positive solution of the quartic equation is given bywhere
With (82), the cost J in (78) can be rewritten as
With (92), the minimum cost is given by
Remark. Combining the results of Sections 2–4, the robust control scheme (27) using the optimal design of renders the solution of the transformed error dynamic system uniformly bounded and uniformly ultimately bounded. In addition, the performance index given by (78) is globally minimized. Furthermore, the closed-form (i.e., analytic) solution is shown. The optimization problem is completely solved.
The optimal design procedure is summarized as follows: Step 1: for given , the control terms in (23) and in (24) are obtained. Step 2: obtain the fuzzy descriptions of and and the bounding function . The control is given by (28) with undetermined. Step 3: calculate based on the α-cuts of the membership functions, the fuzzy arithmetic, and the decomposition theorem. Step 4: for given , calculate the optimal gain in (92). The resulting minimal performance index can be obtained by (95). Step 5: the robust control scheme is given by (27).
5. Simulation Results
In this section, numerical simulations will be conducted to examine the behavior of a vehicle platoon traveling on a highway under the robust control law (27) developed in previous sections. Consider there are totally four vehicles in a platoon, that is, one leader with three followers. The following parameters are chosen for numerical simulations. The nominal vehicle masses () , the nominal aerodynamic coefficients , the nominal resistance forces () , and the vehicle lengths . The uncertainties in each vehicle which are (possibly) time-varying are given by , , , , , , , , , and . Notice that all uncertainties are bounded. Suppose the parameters , , and in Assumption 2 are all “close to 0”, where the associated membership functions are given by (all triangular)and the known functions in Assumption 3 are given bywith the uncertain parameters all “close to 1,” where the associated membership functions are given by (all triangular)
Hence, Assumptions 2 and 3 are met.
The leading vehicle is under the control as follows:
For comparison purpose, we consider two types of control for the followers. The first is the robust controller (27):
The second is the PD controller:where and are the proportional and derivative control gains.
Suppose the desired space is given by . For the PD controller, we choose . Two cases of initial conditions are considered (notice that all initial conditions are crisp): first, the zero initial condition case with initial positions and initial velocities . We find that the position errors and velocity errors . Recalling that , we have , By using the fuzzy arithmetic and the decomposition theorem, we obtain , , , and , and the quartic equation (82) is given by
We choose five sets of weighting factors and for demonstration. Their values and the corresponding and are summarized in Table 1.
Next, the critical case with initial positions and initial velocities . Similarly, it can be found that and (these mean the vehicles are only 1m apart and they are approaching initially). Recalling that , we have . By using the fuzzy arithmetic and the decomposition theorem, we obtain , , , and . By following (84) to (95), the quartic equation (82) is given by
Again, five sets of weighting factors and are chosen for demonstration. Their values and the corresponding and are summarized in Table 2.
With these parameters obtained, numerical simulations are conducted. The final results are shown in Figures 2–13. Figures 2–9 show the performance of the platoon under zero initial conditions. Figures 2–5 show the position and velocity histories along x-direction of each vehicle in the platoon under the PD controller and robust controller (, when using ) with zero initial condition, respectively. All trajectories are smooth and no abrupt changes occurred. Figures 6 and 7 show the space error histories for the vehicles under the PD controller and robust controller, respectively. It is shown that the space errors of the followers under PD control oscillate and the maximum magnitude increases. The collisions occur () for the second and third vehicles at around . This is not surprising since this control does not guarantee stability. On the other hand, the space errors of the followers under robust control always stay in a region smaller than 0.3 m and hence no collisions.
It is interesting to see that in Figure 7, the third vehicle is the one with least upper bound of the space error, which means the space error is well suppressed under the robust controller. Figures 8 and 9 show the control effort history under two different controllers. It appears that the magnitude of the robust controller is smaller than that of the PD controller.
Figures 10–13 show the performance of the platoon under critical initial conditions. Figure 10 shows the space error histories for each vehicle under the robust controller. Despite initially being very close to each other, which means a very dangerous proximity, the space error for under robust control is always under 5m (i.e., ) and hence no collision. Moreover, it shows that the space error of each follower enters a small region (less than 0.3) in less than and stays there, thereafter, regardless of the acceleration changes of the leader at the time periods and . Figure 11 shows the control effort history under critical initial conditions. It can be shown that after the initial position and velocity error is suppressed (after ), the effort of the robust controller is smaller than that of the PD controller. For comparison, Figures 12 and 13 show the space error history and its corresponding control effort of the first follower under five , respectively. It appears that a higher optimal gain renders a faster settling of the space error at the beginning. This is because more weighting was assigned to the transient-state performance .
6. Conclusion
A new approach is proposed to the decentralized control design for vehicle platooning for uncertain automated highway systems. The uncertainty in the system is nonlinear and (possibly) fast time-varying. The only information about the uncertainty is that it is assumed to be within a prescribed fuzzy set. Based on a creative transformation, a class of decentralized control is proposed for the vehicle platoon. Each vehicle in the platoon only needs the knowledge of the preceding vehicle. No acceleration feedback or the information of the lead vehicle is required. The resulting controlled platoon is global collision avoidance. The resulting performance of the controlled platoon is two folds: one deterministic and one fuzzy. The deterministic performance assures that under the worst case we still have collision avoidance under arbitrary safe initial conditions; this assures the bottom line, while the fuzzy information allows us further consider optimization problem. A control design parameter is selected to minimize a fuzzy-based performance index. Both the analytic forms (i.e., closed forms) of the control design parameter and the resulting minimum performance index are obtained. The optimization problem is completely solved. The simulation results compare the performance of the system under the proposed control with PD control. It shows that even under a very critical initial state, where collision appears to be imminent, the robust control was able to pull out and avoid collision.
Data Availability
The simulation data used to support the findings of this study are included within the article and can be made freely available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was sponsored in part by the NSFC Program (Nos. 61872217, U1701262, and U1801263), Guangdong Provincial Key Laboratory of Cyber-Physical Systems, and the National and Local Joint Engineering Research Center of Intelligent Manufacturing Cyber-Physical Systems, as well as be sponsored in part by the Industrial Internet Innovation and Development project of Ministry of Industry and Information Technology.