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E-mail address: blueking hotmail. Email: blueking hotmail. Use the link below to share a full-text version of this article with your friends and colleagues. Learn more. The robust receding horizon control RHC synthesis approach is developed in this paper, for the simultaneous tracking and regulation problem STRP of wheeled vehicles with bounded disturbances. Considering the bounded disturbances, we firstly provide a robust positively invariant RPI set and associated feedback controller for the perturbed vehicles, which contribute to the foundation of the robust RHC synthesis approach.
The actual control input is implemented by correcting the solved nominal input with the designed feedback controller. Additionally, due to the elaborate control design, the effect of disturbances can be completely nullified to achieve better tracking performance. The effectiveness and advantage of the proposed approach are illustrated by two simulation examples. Due to the advantages in handling physical constraints and optimizing control performance, the receding horizon control RHC , also referred to as model predictive control MPC , has been widely employed in various applications for constrained systems, eg, chemicals, food processing, automotive, and aerospace applications, surveyed by Qin and Badgwell.
Therefore, the RHC strategy is preferred by the researchers to others eg, sliding mode control, 11 dynamic feedback linearization, 12 backstepping technique, 13 etc , and more and more approaches emerge to be effective solutions for these problems. Some of these emerged approaches are proposed in synthesis framework ie, both the recursive feasibility and stability are guaranteed for the control of unmanned vehicles.
More details for nonholonomic mechanics and control of complex vehicles can refer to the work of Bloch. As reported by Lee et al, 13 TPSC covers regulation ie, the specified reference is stationary and tracking ie, the specified reference is persistently dynamic cases and is usually solved for each case separately. For the tracking case, it is easy to establish the three key ingredients for wheeled vehicle by linearization, hence the RHC synthesis approaches can be achieved following the common practice, eg, the nominal RHC approaches proposed by Gu and Hu 22 and by Sun and Xia 23 and the robust RHC approach proposed by Sun et al.
Focused on the regulation case of wheeled vehicle, many researchers attempt to present RHC approach, eg, the controllers designed for nominal vehicle 6 , 10 , 25 and for perturbed vehicle. As the robust positively invariant terminal constraint set and associated control law are successfully developed, the approaches proposed by Fontes 6 , 26 can fall in synthesis category.
To avoid the challenge of developing positively invariant terminal constraint set, an alternative RHC synthesis approach can be presented without stabilizing constraints and costs, 10 where the asymptotic stability depends much on the prediction horizon. Unfortunately, the aforementioned synthesis approaches cannot be applied for tracking case because their associated control laws or control input candidates are designed only for stationary reference.
In fact, it is hard to develop a robust positively invariant set and associated control law for the simultaneous tracking and regulation problem STRP , ie, tracking a dynamic reference with its speeds approaching to zeros eg, the parallel parking. Aiming at the STRP of nominal vehicle, a terminal constraint set is defined by Gu and Hu, 7 whereas its positive invariance cannot be ensured. In the work of Wang and Ding, 8 a positively invariant terminal constraint set with associated controller is successfully developed for the STRP of nominal multiple vehicles.
However, its positive invariance is no longer guaranteed for perturbed vehicle, which hinders its adaption to robust case.
Motivated by this situation, this paper intends to propose a robust RHC synthesis approach for the STRP of perturbed wheeled vehicle, where an RPI set and associated feedback control law designed in our conference work 28 serve as the fundamental contribution here. To clearly distinguish the contributions of the aforementioned works, a brief comparison is provided in Table 1 , with more details concretely clarified in the following.
This is the main contribution of this work. Through incorporating the disturbance estimation scheme into controller design, 20 , 39 the perturbed effect can be estimated and compensated to enhance the disturbance resistance. Without adopting any of the aforementioned auxiliary techniques, this paper directly utilize the deviation between actual and nominal states which can be seen as the perturbed effect to correct the solved control input for better disturbance rejection.
This contributes to the performance advantage of the proposed approach in this work. A partial and preliminary version of this paper has been presented in our conference work. The rest of this paper is organized as follows. In Section 2 , we introduce the perturbed wheeled vehicle system, describe the desired control objective, and formulate the cost function accordingly. Section 4 provides the simulation examples to demonstrate the effectiveness and advantage of the proposed approach.
Finally, a conclusion summarizes this paper in Section 5. Here, we assume that the actual state z t is measurable in real time. By neglecting the disturbance d t , the nominal dynamics can be obtained as , with the nominal state and control input. In this paper, the control objective is to steer the perturbed vehicle 1 to track a prespecified reference, without violating the constraints 2 and 3 , ie, to solve a robust TPSC. Accordingly, the nominal tracking errors can be defined as and. Similarly, the error between actual and nominal states, say perturbed state error, is defined as.
For both parts of trajectories, the desired constraints need to be handled against the disturbances to guarantee recursive feasibility. Based on recursive feasibility, some additional properties are further required to achieve robust stability. Include the initial nominal state as a decision variable and impose the current actual state lying in an RPI set around it ie, , hence the feasible domain can be enlarged see proposition 2 in the work of Mayne et al Tighten the original constraints based on a mapped set of ie, and , hence the recursive feasibility of the optimization problem and the admissibility of actual control input can be guaranteed.
After optimization, the solved nominal control input is corrected for actual implementation as 9 with , where the predictive nominal state is evolved along the nominal dynamics 8c from under the solved control input. Generally, the development of the robust positively invariant set for nonlinear system is based on linearization near the equilibrium point.
Hence, in the following, we firstly focus on tackling this issue for the STRP of perturbed vehicle, then develop the actual initial and tightened constraints based on the designed RPI set and associated control law. The terminal constraint set will also be built with choosing a proper terminal cost. The terminal constraint set is desired to be positively invariant for the nominal tracking state error.
For ease of expression, the RPI set of general structure is designed based on the tracking error state, with the following definition. Definition 1. Remark 1. Although a positive invariant set with auxiliary control law has been designed in our previous work 8 see equations 16 and 17 therein for STRP of wheeled vehicles without considering the state constraint, it cannot guarantee robust positive invariance for perturbed vehicle and may lead to infeasibility of optimization problem 8.
Hence, an RPI set 11 with associated feedback control law 13 is designed specially for the robust STRP of perturbed vehicles subject to both state constraint 2 and input constraint 3. Theorem 1. The satisfaction of the admissibility requirements 10b and 10c is also guaranteed by the conditions 12 and For the satisfaction of conditions 10a and 10c , the detailed proof and a simulation demonstration have been provided in our conference work.
Hence, the robustly positive invariance of the set defined in 11 guarantees the satisfaction of requirement 10b. To additionally reduce the effect of unmatched disturbance, sliding mode control with compensating the disturbance by estimation is recommended. Remark 2. This provides much convenience to specify an RPI set for miscellaneous requirements eg, a smaller set for better robust convergence, or a bigger set for larger feasible domain.
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Note in the optimization problem 8 that both the initial and tightened constraints are dependent on the set , which is desired to be RPI for the perturbed state error. Then, following the definition of Minkowski sum and formulation 17 , is equal to. As the actual state z t k can be measured in real time, we can reformulate the initial constraint to constrain the decision variable as 18 with 19a 19b 19c 19d 19e It deserves to note that, in both cases of constraint 19e , the reference linear speed v r t should also be replaced by the nominal predictive one for robustly positive invariance.
Similar to the procedure in proof of Theorem 1 , it can be easily derived that, if in 17 , then the constraint is guaranteed for any , satisfying 20 where. Hence, the actual tightened state constraint can be developed as 21 where is the actual tightened state constraint set. According to formulation 13 , conditions 14c and 14f , the mapped set can be determined as 22 Then, following the definition of Pontryagin difference, we can develop the actual tightened input constraint as 23 where is the actual tightened input constraint set.
By slightly adapting the usual axioms in the work of Mayne et al 15 see p. According to the procedure in Section 3. Now, by substituting the actual initial and tightened constraints, the built terminal constraint set, and the chosen terminal cost into the conceptual optimization problem 8 , an actual optimization problem can be formulated as 34a 34b and the robust RHC algorithm is provided as follows. Then, the adapted algorithm is presented as follows. The relationships between the both constraint sets and between different trajectories are illustrated in Figure 2.
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