Centralized, Decentralized and Distributed Nonlinear Model Predictive Control of a Tractor-Trailer System: A Comparative Study Erkan Kayacan, Joshua M. Peschel and Erdal Kayacan, Abstract— This paper presents centralized, decentralized and distributed nonlinear model predictive controllers design for a tractor-trailer system. Several comparisons are made in terms of their performances and computation time. The experimental results show that the centralized nonlinear model predictive controller has ability to let a tractor-trailer system follow trajectories with the lowest tracking error, while the decentralized one has the lowest computation time.
I. I NTRODUCTION In agricultural field work, machines must be accurately navigated to achieve an optimal result. The entire field should be covered with minimal overlap during tillage, fertilizing and spraying. However, this is a challenging task because of having considerable overlap and variation in plant distances observed in the field. Apart from navigating the machine, the operator must also supervise the work performed by the machine. Switching between paying attention to the steering and paying attention to the machine control results in an increase in the deviation from the optimal path in practice. For the automation of agricultural machineries, advanced control methods are needed to achieve higher control accuracy for the trailer for both straight and curved target paths under varying soil conditions and to make tractor control subordinate to the trailer control. Therefore, different advanced model predictive control (MPC) structures are elaborated in this study and implemented on an autonomous tractor-trailer system to investigate their potential in practice. A centralized nonlinear model predictive controller (CeNMPC) has been designed in which a full mathematical model of the tractor-trailer system was used including all interactions in the subsystems. This centralized controller managed to let the system follow the target trajectory with high accuracy. However, it requires a relatively high computational cost due to the complexity of the optimization problem up to 7.2 ms. To reduce the computational cost, a fully decentralized NMPC (DeNMPC) was also designed and implemented. Thanks to dividing the optimization problem into two smaller problems, the computation time could be reduced by a factor of 5. However, the trajectory error increased by 20% to 50% by ignoring the interactions. As both the centralized and decentralized approaches had their Erkan Kayacan and Joshua M.Peschel are with the Department of Civil and Environmental Engineering, University of Illinois at UrbanaChampaign, 205 N. Mathews Ave. Urbana, IL 61801. e-mail: {erkank,
peschel}@illinois.edu Erdal Kayacan is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, 639798, Singapore. e-mail:
[email protected]
merits and limitations, a compromise was sought in the form of a distributed NMPC (DiNMPC) in which the interactions are partially taken into account. These distributed controllers were between the centralized and decentralized controllers in terms of both computation time and tracking error [1]–[3]. This paper is organized as follows: The system is described in Section II. The formulations of the centralized, decentralized and distributed NMPC structures are given in Section III. The real-time experimental results are presented in Section IV. Finally, a brief conclusion is given in Section V. II. D EFINITION OF THE U NMANNED T RACTOR -T RAILER S YSTEM The main goal of the experiments is to track a spacebased trajectory in which the yaw angles of the tractor and trailer are controlled separately. Meanwhile, the longitudinal speed is fixed. The experimental setup is a small agricultural tractor-trailer (or tractor-implement) system which is shown in Fig. 1 [1]–[7]. The autonomous tractor-trailer system model is a kinematic model neglecting the dynamic force balances in the equations of motion [8]–[10]. The equations of motion of the system are written as follows: x˙t
=
t
=
˙t
=
i
x˙
=
i
=
y˙ ψ
µv cos (ψ t )
(1)
t
µv sin (ψ ) µv tan (κδ t ) Lt µv cos (ψ i )
µv sin (ψ i ) µv Ld ψ˙ i = sin (ηδ i + β ) + t tan (κδ t ) cos (ηδ i + β ) i L L where x and y indicate the position, and ψ is the yaw angle of the system where the superscripts show that the related symbols t and i are for the tractor and trailer, respectively. Besides, the steering angle of the front wheel of the tractor is shown by δ t , β is the angle between the tractor and drawbar, and δ i is the steering angle between the trailer and drawbar. The longitudinal speed of the system is shown with v. The physical parameters, which are easily measured, are as follows: The distance between the front axle and rear axle of the tractor Lt = 1.4m, the distance between the connection point of the drawbar at the trailer and the rear axle of the trailer Li = 1.3m) and the distance between the rear axle of the tractor and the connection point of the drawbar at the trailer Ld = 1.1m), respectively. y˙
In this section, we considered an CeNMPC formulation in the following form at each sampling time t: V (x(t), u(t))
min x(.),u(.)
subject to x(tk ) = x(t ˆ k) x(t) ˙ = f x(t), u(t), p
(7)
xmin ≤ x(t) ≤ xmax umin ≤ u(t) ≤ umax ∀t ∈ [tk ,tk + th ] where V is the plant objective function, x(t ˆ k ) is the estimated state vector by the nonlinear moving horizon estimator (NMHE), xmin , xmax , umin and umax represent the upper and lower constraints on the state and input. The first sample of u(t) is applied to the plant: Fig. 1.
u(t, x(t)) = u∗ (tk )
The tractor-trailer system
III. D IFFERENT NMPC S TRUCTURES A. Centralized Nonlinear Model Predictive Control A nonlinear system model can be described with the following equation: x(t) ˙ = f (x(t), u(t))
where X ⊆ Rn is closed, U ⊆ Rm is compact and each set contains the origin in its interior point. The feasible regions of the inputs do not influence on each other so that the constraints for each input are uncoupled. Formulations of the stage cost and terminal penalty are respectively denoted as follows: VSC (x(t), u(t)) = kxr (t) − x(t)k2Q + kur (t) − u(t)k2R (4) VT P (x(t)) = kxr (tk + th ) − x(tk + th )k2S Rn×n ,
Rm×m
and the nonlinear optimization problem is solved again over a shifted horizon for the next sampling time. Considering the constraints on the inputs given in (9), the CeNMPC optimization problem is solved at each sampling time: −35 deg ≤ δ t (t) ≤ 35 deg −25 deg ≤ δ i (t) ≤ 25 deg
(2)
where x ∈ Rn is the state vector, u ∈ Rm is the control input, f (·, ·, ·) : Rn × Rm −→ Rn is the continuously differentiable state update function and f (0, 0) = 0 ∀t. At each time instant, the states and inputs have to satisfy the following: x ∈ X, u ∈ U (3)
(5)
Rn×n
where Q ∈ R∈ and S ∈ are symmetric and positive definite weighting matrices, xr and ur denote respectively the state and input references, x and u denote repectively the states and inputs, tk and th denote respectively the current time and the prediction horizon. The terminal penalty is used to enforce the stability of NMPC [11]. By combining (4) and (5) the objective function is formulated as follows Z tk +th V (x(t), u(t)) = VSC (x(t), u(t)) dt +VT P (x(t)) tk
∀t ∈ [tk ,tk + th ] (6) where u(t) = [u(tk ), . . . , u(tk + th )] is the input sequence over time horizon th , x(t) for t = u(tk ), . . . , u(tk + th ) is the state trajectory obtained by applying the control sequence u(t) to the system (2). It is assumed VSC (0, 0) = 0.
(8)
(9)
The state and input references for the system are changed online as follows: xr
= (xtr , ytr , ψrt , xri , yir , ψri , vr )T
ur
= (δrt , δri )T
(10)
The input references, which are the recent measured steering angles of the tractor and trailer, and hydrostat position, are used in the objective function to ensure a possibility to penalize the variation of the inputs from time-step to timestep. Equation (1) shows that the input to the trailer does not have any effect on the tractor yaw dynamics. On the other hand, there is a dynamic effect from the input to the yaw dynamics of the tractor towards the trailer yaw dynamics. Therefore, an inevitable conclusion is that while the trailer input will try to minimize the trailer position error only, the input to the tractor will try to minimize the position errors of both the tractor and trailer simultaneously. The weighting matrices Q, R and S have been defined as follows: Q
= diag(1, 1, 0, 0.001, 0.001, 0)
R
= diag(10, 0.01)
S
= diag(10, 10, 0, 0.01, 0.01, 0, 0)
(11)
The weights for the trailer are chosen smaller than the ones for the tractor. The reason is that when the coefficients for tractor and trailer were set to the same value or the ones for the trailer were chosen bigger than the ones for the tractor, an oscillatory behavior on the steering mechanism of the tractor was observed. The reason for such a behaviour was that the trailer actuator is slower than the tractor actuator.
The designed CeNMPC compensates both errors coming from the tractor and the trailer, because it steers the tractor accurately on the desired trajectory, and then steers the trailer. Moreover, since the tractor-trailer system dynamics are slow, the weighting matrix R is selected larger than the weighting matrix Q so as to obtain well damped closed-loop system response. Furthermore, inasmuch as the last state error value in the prediction horizon is so crucial for the stability issues, the weighting matrix S is adjusted to 10 times larger than the weighting matrix Q. The prediction horizon and control horizon th have been set to 3 seconds which was found by trial-and-error. If the prediction and control horizons are increased, the computation time for CeNMPC will increase dramatically so that solving the optimization problem will be infeasible. Moreover, if the prediction and control horizons are too small, the stabilization of the system cannot be achieved. As reported in [12], the prediction and control horizons of the NMPC must be larger for a stable performance when the velocity of the vehicle increases. Since the velocity of the tractor-trailer system is quite low, the prediction and control horizons do not have to be very large in this study. B. Decentralized Nonlinear Model Predictive Control A disadvantage of the CeNMPC is that all subsystems are linked to the central agent, which must have information related to every single state and control in a system [13]. This architecture may not be feasible/reliable for control of complex systems. An alternative design is the DeNMPC scheme in which the system to be controlled is divided into a set of local subsystems of smaller size, neglecting possible dynamical coupling or interactions, and each subsystem is controlled by a local NMPC agent without communication between the local controllers [2]. Hence, the control problem breaks down into a set of local NMPC agents of smaller size. Each NMPC controller is based on a local model of a subsystem, and provides an independent control input. The effect of the tractor steering angle on trailer yaw dynamics is neglected, and the new equation for the trailer yaw dynamics is obtained as: µv (12) ψ˙ i = i sin (ηδ i + β ) L A nonlinear model with N subsystems is given for each subsystem as: x˙i (t) = fi (xi (t), ui (t)) + gi (x(t), u(t)) + di (t), i ∈ I1:N (13) where xi ∈ Rni , ui ∈ Rmi , and di ∈ Rni are respectively the state, input and disturbance of the ith subsystem, and fi and gi , nonlinear and continuously differential equations, denote respectively the ith subsystem dynamics and the other subsystems dynamics on the ith subsystem. The states, the inputs and the disturbance have to satisfy at each time-instant: xi ∈ Xi , ui ∈ Ui , di ∈ Di
(14)
where Xi ⊆ Rni is closed, Ui ⊆ Rmi and Di ⊆ Rni are compact and each set contains the origin in its interior point. The
feasible regions of the inputs do not influence on each other so that the constraints for each input are uncoupled. From (13), the nominal system model for each subsystem is formulated by disregarding the subsystem interaction gi (x(t), u(t)) and the disturbance di (t) as follows: x˙¯i (t) = fi (x¯i (t), u¯i (t)), i ∈ I1:N Rni
(15)
Rm i
where x¯i ∈ and u¯i ∈ are respectively the nominal state and input. Formulations of the stage cost and terminal penalty are respectively denoted for each subsystem i ∈ I1:N : ViSC (x¯i , u¯i ) = kx¯ir (t) − x¯i (t)k2Qi + ku¯ir (t) − u¯i (t)k2Ri(16) ViT P (x¯i ) = kx¯ir (tk + th ) − x¯i (tk + th )k2Si
(17)
where Qi ∈ Rni ×ni , Ri ∈ Rmi ×mi and Si ∈ Rni ×ni are symmetric and positive definite weighting matrices, xr and ur denote respectively the state and input references, x and u denote repectively the states and inputs, tk and th denote respectively the current time and the prediction horizon. The objective function for each subsystem i ∈ I1:N is given as follows: Z tk +th Vi (x¯i , u¯i ) = ViSC (x¯i , u¯i ) dt +ViT P (x¯i ) (18) tk ∀t ∈ [tk ,tk + th ] The plant objective function is written as follows: Vi (x¯i , u¯i )
min
x¯i (.),u¯i (.)
subject to
x¯i (tk ) = xˆi (tk ) x¯˙i (t) = fi x¯i (t), u¯i (t)
(19)
x¯imin ≤ x¯i (t) ≤ x¯imax u¯imin ≤ u¯i (t) ≤ u¯imax ∀t ∈ [tk ,tk + th ] where Vi is the objective function. Furthermore, constraints on the state and input are denoted by x¯imin , x¯imax , u¯imin and u¯imax . The stability of DeNMPC has been proved in [14]– [16]. The functions fi and gi in (13) are respectively given for the tractor and trailer where the subscript 1 refers to the tractor and the subscript 2 refers to the trailer: µv cos (θ ) µv cos (ψ) µv sin (ψ) f1 = µv sin (θ ) , f2 = µv i +β) µv tan (κδ t ) sin (ηδ t Li L 0 0 (20) 0 g1 = 0 , g2 = 0 − Llt tan (κδ t ) cos (ηδ i + β ) The DeNMPC problems for the two subsystems are solved at each sampling time with the constraints on the inputs in (9). The inputs references are the last measured steering angles and hydrostat position while the states references are relied on the reference trajectory to be tracked. x1r
= (xtr , ytr , θr )T ,
u1r = (δrt )
x2r
= (xri , yir , ψr )T ,
u2r = (δri )
(21)
Moreover, the weighting matrices Qi , Ri and Si are given as: Qi = diag(1, 1, 0),
Ri = 10,
Si = diag(10, 10, 0)
(22)
C. Distributed Nonlinear Model Predictive Control Combining DeNMPC with additional communication and coordination between the local NMPC controllers results into a DiNMPC. For the formulation of DiNMPC, a decoupled nonlinear model of the system is formulated in the following form: x˙ = f (x, u) (23) where x1 u1 f1 (x1 , u1 , u2 ) x= , u= , f (x, u) = x2 u2 f2 (x2 , u1 , u2 ) Rn ,
Rm ,
Rn × Rm
where V is the plant objective function, x1 and x2 are the states, and u1 and u2 are the inputs, Furthermore, the constraints on the states and input are denoted by x1min , x1max , x2min , x2max u1min and u1max . The input of the second subsystem u∗2 denotes the prediction values receiving from the cDiNMPC for the second subsystem. In a similar way, the cDiNMPC formulation for the second subsystem is: min
x1 (.),x2 (.),u∗1 (.),u2 (.)
subject to
−→ Rn .
for which x ∈ u∈ and f : The inputs have to satisfy at each time-step, u 1 ∈ U1 , u 2 ∈ U2
(26)
where Qi ∈ Rni ×ni , Ri ∈ Rmi ×mi and Si ∈ Rni ×ni are symmetric and positive definite weighting matrices, xr and ur denote respectively the state and input references, x and u denote respectively the states and inputs, tk and th denote respectively the current time and the prediction horizon. The objective function for each subsystem i ∈ I1:2 is written as follows: Z tk +th ViSC (xi , ui ) dt +ViT P (xi ) Vi (xi , u1 , u2 ) = (27) tk ∀t ∈ [tk ,tk + th ] 1) Formulation of cooperative DiNMPC: For the cDiNMPC case, the states xi are the functions of the inputs u1 and u2 . Thus, the objective function Vi becomes a function of the inputs u1 and u2 . Therefore, the plant objective function is written in the following form: (28)
where ρ1 , ρ2 > 0 are the coefficients. The cDiNMPC formulation of the first subsystem is formulated: min
x1 (.),x2 (.),u1 (.),u∗2 (.)
subject to
V (x1 , x2 , u1 , u∗2 ) x1 (tk ) = xˆ1 (tk ) x2 (tk ) = xˆ2 (tk ) x˙1 (t) = f1 x1 (t), u1 (t), u∗2 (t) x˙2 (t) = f2 x2 (t), u1 (t), u∗2 (t) x1min ≤ x1 (t) ≤ x1max x2min ≤ x2 (t) ≤ x2max u1min ≤ u1 (t) ≤ u1max
x˙1 (t) = f1 x1 (t), u∗1 (t), u2 (t) x˙2 (t) = f2 x2 (t), u∗1 (t), u2 (t)
(24)
ViSC (xi , ui ) = kxir (t) − xi (t)k2Qi + kuir (t) − ui (t)k2Ri(25)
V (x1 , x2 , u1 , u2 ) = ρ1V1 (x1 , u1 , u2 ) + ρ2V2 (x2 , u1 , u2 )
x1 (tk ) = xˆ1 (tk ) x2 (tk ) = xˆ2 (tk )
where U1 and U2 are compact and convex. The feasible regions of the inputs do not influence on each other so that the constraints for each input are uncoupled. Formulations of the stage cost and terminal penalty are respectively denoted for each subsystem i ∈ I1:2 : ViT P (xi ) = kxir (tk + th ) − xi (tk + th )k2Si
V (x1 , x2 , u∗1 , u2 )
(29)
(30)
x1min ≤ x1 (t) ≤ x1max x2min ≤ x2 (t) ≤ x2max u2min ≤ u2 (t) ≤ u2max where the constraints on the input of the second system are represented by u2min and u2max . The input of the first subsystem u∗1 denotes the prediction values receiving from the cDiNMPC for the first subsystem. Since the input of the trailer does not exist in the dynamics of the tractor, (30) cannot be formulated for our test-bed which enforces us to design an iDiNMPC. The related formulation is given in Section III-C.2 for the trailer. 2) Formulation for independent DiNMPC: For the iDiNMPC case, the plant objective function consists of only V2 (x2 , u1 , u2 ). The iDiNMPC formulation for the trailer is as follows: min
x2 (.),u∗1 (.),u2 (.)
subject to
V2 (x2 , u∗1 , u2 ) x2 (tk ) = xˆ2 (tk ) x˙2 (t) = f2 x2 (t), u∗1 (t), u2 (t)
(31)
x2min ≤ x2 (t) ≤ x2max u2min ≤ u2 (t) ≤ u2max As can be seen from (31), the objective function consists of only the states of the second subsystem (the trailer) and the inputs of the first and second subsystems. Since there are no states of the first subsystem (the tractor), this objective function can be minimized with respect to the second input to the second subsystem. In iDiNMPC, since each controller is responsible to minimize only its own cost function, the outcome of the optimization problem is supposed to find the Nash equilibrium point. On the other hand, in cDiNMPC, each local controller tries to optimize the plantwide cost function using only its own control which results in a Pareto optimum [13]. Remark: For all the different NMPCs mentioned above, ACADO code generation tool is used to solve the constrained nonlinear optimization problems. This software is an open source software package for optimization problems.
IV. E XPERIMENTAL R ESULTS , A NALYSIS AND D ISCUSSIONS In agricultural applications, the most important performance criterion is the lateral error of the vehicle for the straight lines in order to avoid damaging field crops. Also, the longitudinal speed must not fluctuate too much to avoid negative effects on field operations such as planting, weeding, harvesting and tillage. The accuracy in case of the curvilinear trajectories suffices when the vehicle does not cross the farm border and starts to track the straight lines from the beginning of every line. Therefore, the space-based trajectory approach in real-time has been used. An autonomous tractor-trailer is a complex mechatronic system, consisting of three subsystems with the following dynamics: yaw dynamics and longitudinal dynamics of the tractor, and yaw dynamics of the trailer. To control this system, four different approaches have been proposed: centralized, decentralized, independent distributed and cooperative distributed control structures. Each subsystem has its own control objective, and thus also its own best input. The best input for a subsystem may ruin the control performance of other subsystems due to interactions between different subsystems. Therefore, the applied input to a subsystem must be ideal not only for itself, but also for others. The aim in a CeNMPC is to find an optimal input value for each subsystem which must be as close as possible to the best input value for each subsystem. Theoretically, since the CeNMPC has one overall optimization formulation consisting of cost functions of all subsystems and a model knowing all interactions between subsystems, it is able to find the pareto optimum, i.e. the global minimum point of the optimization problem for the control of the system. On the other hand, in a DeNMPC, the models, and thus also the controller neglect the interactions between subsystems and each DeNMPC only tries to minimize its own subsystem cost function. Therefore, the solution is able to find a local minimum of the optimization problem for the control system. However, in an iDiNMPC, each subsystem model has information about interactions between subsystems, but the cost function for each iDiNMPC consists of only its own subsystem cost function. In this case, each iDiNMPC cannot benefit from other subsystems’ solutions by changing its solution while the other subsystems solutions are kept unchanged. This results in a Nash equilibrium for the solutions of all subsystems. An alternative distributed control structure is the cooperative distributed case. It is assumed that models neglect the interactions but, the cost function for each cDiNMPC consists of the cost functions of all subsystems. The aim of this approach is to find the best solution for all subsystems rather than competition between the subsystems. This results in finding a pareto optimum similar to the centralized case. Regarding their abilities to find the global optimum of optimization of each subsystem, these approaches can be theoretically compared with respect to their performances for the overall trajectory tracking performance.
Experimental results for the tractor-trailer system for the space-based trajectories were shown in Figs. 2 and 3. The tractor-trailer system faced varying soil conditions represented by a bumpy and wet grass field in real time. In spite of these difficult soil conditions, it is to be noted that no inclinometer was used to correct the GPS measurements. For a fair comparison, the same reference trajectory has been used among these chapters which is an 8-shaped trajectory consisting of two straight lines and two smooth curves. With this trajectory, the performance of the designed controllers for both straight line and curvilinear geometries has been investigated. The Euclidean errors in real-time for the CeNMPC, the DiNMPC, and the DeNMPC for the tractor and trailer have been summarized for the space-based trajectory in Table I. The curvature of the smooth curves was equal to 0.1. For straight lines, the mean values of the Euclidean error of the tractor for the CeNMPC, the DiNMPC and the DeNMPC were approximately equal to 0.065 m, 0.071 m and 0.079 m, respectively. Also, the mean values of the Euclidean error of the trailer for the CeNMPC, the DiNMPC and the DeNMPC were approximately equal to 0.036 m, 0.049 m and 0.054 m, respectively. Moreover, for the curved lines, the mean values of the Euclidean error of the tractor for the CeNMPC, the DiNMPC and the DeNMPC were approximately equal to 0.50 m, 0.55 m and 0.60 m, respectively. Also, the mean values of the Euclidean error of the trailer for the CeNMPC, the DiNMPC and the DeNMPC were approximately equal to 0.41 m, 0.49 m and 0.56 m, respectively. As can be seen from the experimental results, the tracking performance of the CeNMPC is the best, while the DeNMPC is the worst, as theoretically expected. However, the differences are rather small. The execution times in real-time for the CeNMPC, the cDiNMPC for the tractor, the iDiNMPC for the trailer, the DeNMPCs for the tractor and trailer have been summarized in Table II for the space-based trajectory. As can be seen from Table II, the largest computation time is for the CeNMPC. The reason is that the CeNMPC has one big optimization problem, while the DiNMPC and the DeNMPC have two smaller ones. The purpose of having distributed and decentralized cases is to divide one overall optimization problem into smaller pieces to reduce the computational burden. Since cDiNMPC takes models and cost functions of all subsystem into account similar to the CeNMPC case, it is
TABLE I E UCLIDEAN ERRORS
CeNMPC Straight lines Curved lines DiNMPC Straight lines Curved lines DeNMPC Straight lines Curved lines
Tractor error (m)
Trailer error (m)
0.065 0.50
0.036 0.41
0.071 0.55
0.049 0.49
0.079 0.60
0.054 0.56
the second largest optimization problem. Moreover, since the optimization problem in the iDiNMPC is relatively simpler than the one in the cDiNMPC, the computation time is 40% lower. Finally, the computation time needed in the DeNMPC to solve the optimization problem was the lowest, and always below 1.5 ms for both the tractor and the trailer. As can be seen, the needed computation time decreases more than linear when the optimization problem becomes smaller. TABLE II E XECUTION TIMES IN REAL - TIME
CeNMPC cDiNMPC for the tractor iDiNMPC for the trailer DeNMPC for the tractor DeNMPC for the trailer
Preparation 6.6632 5.9236 3.8407 1.1816 1.2540
Feedback 0.1345 0.0326 0.0818 0.0313 0.0541
Overall 6.7977 5.9562 3.9225 1.2129 1.3081
0.8 0.7
Tractor − Euclidean error (m)
0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1
Centralized Decentralized Distributed
−0.2 0
20
40
60
80 100 Time (s)
120
140
160
Fig. 2. Euclidean error of the tractor to the space-based reference trajectory
0.8 0.7
Trailer − Euclidean error (m)
0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1
Centralized Decentralized Distributed
−0.2 0
20
40
60
80 100 Time (s)
120
140
160
Fig. 3. Euclidean error of the trailer to the space-based reference trajectory
V. C ONCLUSIONS The centralized, decentralized and distributed NMPCs been developed for the tracking of space-based trajectories by an autonomous tractor-trailer system. The experimental results have shown that controllers are capable to keep the system on-track and follow the trajectory accurately. As can be observed from the experimental results, although finding the optimum point is becoming easier when the optimization problem gets larger, solving the big optimization problem requires large computation times. Dividing the big optimization problem into smaller pieces allows to obtain solutions more easily, but decreases the possibility for finding the optimum point. This came at the price of a worse tracking error relatively. R EFERENCES [1] E. Kayacan, E. Kayacan, H. Ramon, and W. Saeys, “Learning in centralized nonlinear model predictive control: Application to an autonomous tractor-trailer system,” IEEE Transactions on Control Systems Technology, vol. 23, no. 1, pp. 197–205, Jan 2015. [2] ——, “Robust tube-based decentralized nonlinear model predictive control of an autonomous tractor-trailer system,” IEEE/ASME Transactions on Mechatronics, vol. 20, no. 1, pp. 447–456, Feb 2015. [3] ——, “Distributed nonlinear model predictive control of an autonomous tractor-trailer system,” Mechatronics, vol. 24, no. 8, pp. 926 – 933, 2014. [4] E. Kayacan, H. Ramon, and W. Saeys, “Robust trajectory tracking error model-based predictive control for unmanned ground vehicles,” IEEE/ASME Transactions on Mechatronics, vol. 21, no. 2, pp. 806– 814, April 2016. [5] E. Kayacan, E. Kayacan, H. Ramon, and W. Saeys, “Towards agrobots: Identification of the yaw dynamics and trajectory tracking of an autonomous tractor,” Computers and Electronics in Agriculture, vol. 115, pp. 78 – 87, 2015. [6] E. Kayacan, E. Kayacan, H. Ramon, O. Kaynak, and W. Saeys, “Towards agrobots: Trajectory control of an autonomous tractor using type-2 fuzzy logic controllers,” Mechatronics, IEEE/ASME Transactions on, vol. 20, no. 1, pp. 287–298, Feb 2015. [7] E. Kayacan, E. Kayacan, H. Ramon, and W. Saeys, “Nonlinear modeling and identification of an autonomous tractortrailer system,” Computers and Electronics in Agriculture, vol. 106, pp. 1 – 10, 2014. [8] M. Karkee and B. L. Steward, “Study of the open and closed loop characteristics of a tractor and a single axle towed implement system,” Journal of Terramechanics, vol. 47, no. 6, pp. 379 – 393, 2010. [9] M. Karkee, “Modeling, identification and analysis of tractor and single axle towed implement system,” Ph.D. dissertation, Iowa State University, 2009. [10] E. Kayacan, “Learning control for autonomous vehicle guidance,” Ph.D. dissertation, KU Leuven, 2014. [11] D. Mayne, J. Rawlings, C. Rao, and P. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789 – 814, 2000. [12] P. Falcone, F. Borrelli, J. Asgari, H. Tseng, and D. Hrovat, “Predictive active steering control for autonomous vehicle systems,” Control Systems Technology, IEEE Transactions on, vol. 15, no. 3, pp. 566 – 580, May 2007. [13] B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright, and G. Pannocchia, “Cooperative distributed model predictive control,” Systems & Control Letters, vol. 59, no. 8, pp. 460 – 469, 2010. [14] L. Magni and R. Scattolini, “Stabilizing decentralized model predictive control of nonlinear systems,” Automatica, vol. 42, no. 7, pp. 1231 – 1236, 2006. [15] T. Keviczky, F. Borrelli, and G. J. Balas, “Decentralized receding horizon control for large scale dynamically decoupled systems,” Automatica, vol. 42, no. 12, pp. 2105 – 2115, 2006. [16] D. Raimondo, L. Magni, and R. Scattolini, “Decentralized MPC of nonlinear systems: An input-to-state stability approach,” International Journal of Robust and Nonlinear Control, vol. 17, no. 17, pp. 1651– 1667, 2007.
