Terminal Cost Function Research Articles

Optimal Gliding Trajectories for Descent on Mars

In this paper, optimal gliding trajectories are analyzed, formulating an optimal control problem to be solved computationally via nonlinear programming. A multi-objective terminal cost function that minimizes the velocity, altitude, horizontal flight path angle, and the error of a desired terminal point is formulated. The results are validated via Monte Carlo simulations, analyzing the influence of the variations in the lift-to-drag ratios in the atmospheric entry. The results show feasible solutions for minimizing the distance to the desired final point, which have significant practical implications for the controlled gliding entry of a spaceplane on Mars. In this sense, the results also show that the longitudes of the landing points do not change too much for almost all trajectories, while the latitude of these points is in the interval of about 4 degrees for the majority of the trajectories. This suggests that it is possible to implement an optimal control approach with reasonable accuracy for the controlled gliding entry of a spaceplane on Mars, even in the presence of some uncertainties regarding the aerodynamic performance of the spacecraft, such as the lift-to-drag ratio.

Stability and Performance Analysis of NMPC: Detectable Stage Costs and General Terminal Costs

We provide a stability and performance analysis for nonlinear model predictive control (NMPC) schemes subject to input constraints. Given an exponential stabilizability and detectability condition w.r.t. the employed state cost, we provide a sufficiently long prediction horizon to ensure asymptotic stability and a desired performance bound w.r.t. the infinite-horizon optimal controller. Compared to existing results, the provided analysis is applicable to positive semi-definite (detectable) cost functions, provides tight bounds using a linear programming analysis, and allows for a seamless integration of general positive-definite terminal cost functions in the analysis. The practical applicability of the derived theoretical results are demonstrated with numerical examples.

Robust nonlinear MPC without terminal constraint for tracking changing setpoints in the presence of non‐additive unknown disturbance

AbstractThis paper develops a novel robust tracking model predictive control (MPC) without terminal constraint for discrete‐time nonlinear systems capable to deal with changing setpoints and unknown non‐additive bounded disturbances. The MPC scheme without terminal constraint avoids difficult computations for the terminal region and is thus simpler to design and implement. However, the existence of disturbances and/or sudden changes in a setpoint may lead to feasibility and stability issues in this method. In contrast to previous works that considered changing setpoints and/or additive slowly varying disturbance, the proposed method is able to deal with changing setpoints and non‐additive non‐slowly varying disturbance. The key idea is the addition of tightened input and state (tracking error) constraints as new constraints to the tracking MPC scheme without terminal constraints based on artificial references. In the proposed method, the optimal tracking error converges asymptotically to the invariant set for tracking, and the perturbed system tracking error remains in a variable size tube around the optimal tracking error. Closed‐loop input‐to‐state stability and recursive feasibility of the optimization problem for any piece‐wise constant setpoint and non‐additive disturbance are guaranteed by tightening input and state constraints as well as weighting the terminal cost function by an appropriate stabilizing weighting factor. The simulation results of the satellite attitude control system are provided to demonstrate the efficiency of the proposed predictive controller.

A Robust Dynamic Game-Based Control Framework for Integrated Torque Vectoring and Active Front-Wheel Steering System

Distributed drive electric vehicles (DDEVs) eliminate the complex drivetrain. The independently driven in-wheel motors also endow the vehicle with more ability for improving maneuverability. To this end, this paper proposes an integrated control framework of torque vectoring (TV) and active front-wheel steering system (AFS) to ensure the vehicle lateral motion stability performance. First, the polytope method with finite vertices is employed to deal with the system uncertainties and simplify the modeling structure, based on which a distributed model predictive control is adopted to construct a dynamic interactive model between agents. Then, through introducing the game theory, a distributed parallel control scheme is developed to obtain the cooperative strategy of agents. Such a design can also satisfy the modular and scalable requirement for integrated chassis control. To ensure the system asymptotic stability, the terminal input combined with the terminal cost function are treated as the constraints in the game paradigm and then transformed as the linear matrix inequalities. Furthermore, a robust H <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\infty $</tex-math> </inline-formula> compensation method is used to suppress the system disturbance. Finally, the hardware-in-the-loop (HIL) tests are conducted to assess the control performance. The results verify the proposed integrated control scheme is effective to enhance the vehicle handling stability.

An ADMM-based algorithm for stabilizing distributed model predictive control without terminal cost and constraint

The stability assurance is not straightforward for distributed model predictive control (DMPC), since the well-known terminal cost and constraint technique cannot be readily adapted to the distributed schemes. An alternative method to the complicated procedure of splitting the terminal cost function and constraint set to achieve stability, is to calculate a minimum length for the DMPC prediction horizon. It is known that the DMPC cost will then have a relationship to the infinite-horizon cost, which is determined by a so-called performance factor. Moreover, the stopping condition of the distributed optimization algorithm, which for the sake of smooth handling of the coupling constraints is usually a duality-based algorithm, can be formulated using two scalars: an upper bound for the cost of the next sampling period and a lower bound for the optimal cost of the current one. For the duality-based algorithms, feasibility can only be guaranteed in the limit of iterations, thus the constraint tightening technique should be employed to determine the former scalar. The tightening technique, however, impinges upon calculation of the latter one. This problem is already addressed in the literature for gradient-based methods. By contrast, the consensus form of alternating direction method of multipliers (ADMM) is employed in this paper to stabilize the DMPC scheme. Simulation results reveal that this method outperforms the existing gradient-based approaches in terms of the required number of communications.

Variational Problems Involving a Generalized Fractional Derivative with Dependence on the Mittag–Leffler Function

In this paper, we investigate the necessary conditions to optimize a given functional, involving a generalization of the tempered fractional derivative. The exponential function is replaced by the Mittag–Leffler function, and the kernel depends on an arbitrary increasing function. The Lagrangian depends on time, the state function, its fractional derivative, and we add a terminal cost function to the formulation of the problem. Since this new fractional derivative is presented in a general form, some previous works are our own particular cases. In addition, for different choices of the kernel, new results can be deduced. Using variational techniques, the fractional Euler–Lagrange equation is proved, as are its associated transversality conditions. The variational problem with additional constraints is also considered. Then, the question of minimizing functionals with an infinite interval of integration is addressed. To end, we study the case of the Herglotz variational problem, which generalizes the previous one. With this work, several optimization conditions are proven that can be useful for different optimization problems dealing with various fractional derivatives.

Efficient self-driving strategy with preceding car following and road slopes based on learning predictive control

Self-driving vehicles has been being popular since it not only improves traffic safety and flow but also the efficiency of energy consumption. Technical advance of self-driving vehicles significantly becomes requiring the energy efficiency of the electric vehicles (EVs) since the most car has very limited power capacity which is based on sole battery source. In this research, a hybrid learning predictive control architecture is proposed to reduce the energy consumption of the EVs. This paper combines the model predictive control (MPC) to the deep reinforcement learning (DRL) networks to solve the optimization problems with constrained environments to maximize energy efficiency of the EVs. Especially, the computed cost of the predictive horizon is transferred into the DRL networks as the state value and the reward is inversely fed into MPC controller to find the optimal control strategy. Resultingly, the proposed architecture can reduce the terminal control cost function using precedent state value. The proposed algorithm was tested in the PGDrive simulation environment and the experimental results show the efficiency of the proposed energy saving architecture.

Efficient min–max MPC: Achieving a large domain of attraction with short horizon

AbstractThis article considers the regulation problem of a constrained linear system with bounded disturbance. The objective is to overcome the offline and online problems of the standard dual‐mode min–max model predictive control (MPC). The offline problem is related to the construction of a polyhedral terminal set, which is prohibitively complex especially for high order systems. We propose to replace the polyhedral set with the ellipsoidal one, which is much easier to construct. The online problem is related to the computational burden, which grows exponentially with the prediction horizon . We show how to employ a new terminal cost function, that results in a new min–max MPC with a large domain of attraction even with or . Hence the computational complexity is drastically reduced. Two numerical examples with comparison to earlier solutions from the literature illustrate the effectiveness of the proposed approach.

Model predictive control with implicit terminal ingredients

This paper proposes a new formulation of model predictive control for linear discrete time systems under state and control constraints. The main novelty of the proposed formulation is the introduction, and a computationally adequate utilization, of the implicit terminal constraint set and the implicit terminal cost function. From a numerical point of view, the deployment of the implicit terminal ingredients enables a computationally tractable use of model predictive control for higher dimensional control problems. From a practical point of view, the design of the proposed model predictive control algorithm is very constructive for a class of linear systems with polyhedral constraints. From a structural point of view, the proposed model predictive control retains a priori guarantees of the highly desirable positive invariance and exponential stability properties.

Ambiguity tube MPC

This paper is about a class of distributionally robust model predictive controllers (MPC) for nonlinear stochastic processes, which evaluate risk and control performance measures by propagating ambiguity sets in the space of state probability measures. A framework for formulating such ambiguity tube MPC controllers is presented using methods from the field of optimal transport theory. Moreover, an analysis technique based on supermartingales is proposed, leading to stochastic stability results for a large class of distributionally robust controllers. In this context, we also discuss how to construct terminal cost functions for stochastic and distributionally robust MPC that ensure closed-loop stability and asymptotic convergence to robust invariant sets. The corresponding theoretical developments are illustrated by tutorial-style examples and a numerical case study.

Robust MPC for LPV systems via a novel optimization-based constraint tightening

We propose a novel approach to design a robust Model Predictive Controller (MPC) for constrained uncertain linear systems. The uncertain system is modeled as linear parameter varying with an additive disturbance. Set bounds for the system matrices and the additive uncertainty are assumed to be known. We formulate a novel optimization-based constraint tightening strategy around a predicted nominal trajectory which utilizes these bounds. With an appropriately designed terminal cost function and constraint set, we prove robust satisfaction of the imposed constraints by the resulting MPC in closed-loop with the uncertain system, and Input to State Stability of the origin. We highlight the efficacy of our proposed approach via a numerical example.

Trajectory-tracking control of an underactuated unmanned surface vehicle based on quasi-infinite horizon model predictive control algorithm

The method of a designed trajectory tracking for an underactuated unmanned surface vehicle (USV) in the presence of ocean disturbances is addressed in this paper; the differential flatness theory is applied to get the reference inputs and speed states at the reference position trajectory. Second, a transition process is arranged for the reference trajectory to reduce the overshoot of the actuator, which caused by the large deviation in the initial tracking. Third, the nonlinear disturbance observer is designed to obtain the estimated values of unknown disturbances in the ocean. Then, a controller-based model predictive control (MPC) and terminal cost function is designed for the nominal system. The inherent robustness of the controller and estimates of the observer are used to resist and compensate disturbances. Finally, the simulation experiments of linear trajectory and sinusoidal trajectories are carried out to prove the effectiveness and reliability of the control algorithm designed.

Позиционный принцип минимума: вариационное усиление понятий экстремальности в оптимальном управлении

Existing maximum principles of Pontryagin’s type and related optimality conditions, such as, e.g., the ones derived by F. Clarke, B. Kaskosz and S. Lojasiewicz Jr., and H.J. Sussmann, can be strengthened up to global necessary optimality conditions in the form of so-called feedback minimum principle. This is possible for both classical and non-smooth optimal control problems without terminal constraints. The formulation of the feedback minimum principle (or related extremality conditions) remains within basic constructions of the mentioned maximum principles (the Hamiltonian or Pontryagin function, the adjoint differential equation or inclusion, and its solutions –– co-trajectories). At the same time, the actual maximum condition –– maximization of the Hamiltonian –– takes a variational form: any optimal trajectory of the addressed problem should be optimal for a specific “accessory” problem of dynamic optimization. The latter is stated over all tubes of Krasovskii-Subbotin constructive motions generated by feedback strategies, which are extremal with respect to a certain supersolution of the Hamilton-Jacobi equation. Such a supersolution can be represented explicitely in terms of the co-trajectory of a reference control process and the terminal cost function. In a general version, the feedback minimum principle operates with generalized solutions of the proximal Hamilton-Jacobi inequality for weakly decreasing (𝑢-stable) functions

Risk-sensitive optimal stopping with unbounded terminal cost function

In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller--Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.

Synchronization Control of Dynamic Positioning Ships Using Model Predictive Control

Underway replenishment is essential for ships performing long-term missions at sea, which can be formulated into the problem of leader-tracking configuration. Not only the position and orientation but also the velocities are required to be controlled for ensuring the synchronization; additionally, the control inputs are constrained. On this basis, in this paper, a novel synchronization controller on account of model predictive control (MPC) for dynamic positioning (DP) ships is devised to achieve underway replenishment. Firstly, a novel synchronization controller based on MPC is devised to ensure the synchronization of not only the position and orientation but the velocities; furthermore, it is a beneficial solution for its advantages in handling the control input constraints ignored in most studies of underway replenishment. Secondly, a neurodynamic optimization system is applied to the implementation of MPC, which can improve the computational efficiency and shorten the simulation time. Thirdly, stability, frequently neglected by traditional MPC, is ensured by the means of adding a terminal cost function exported from the Lyapunov equation into the objective function. Finally, the effectiveness and advantages of the proposed control design are illustrated by extensive simulations.

A method for constructing an optimal control strategy in a linear terminal problem

This paper deals with an optimal control problem for a linear discrete system subject to unknown bounded disturbances, where the control goal is to steer the system with guarantees into a given terminal set while minimising the terminal cost function. We define an optimal control strategy which takes into account the state of the system at one future time instant and propose an efficient numerical method for its construction. The results of numerical experiments show an improvement in performance under the optimal control strategy in comparison to the optimal open-loop worst-case control while maintaining comparable computation times.

Safe polyhedral tubes for locally convexified MPC

The note considers optimal and model predictive controls for linear systems with strictly convex quadratic stage and terminal cost functions, and under closed polyhedral stage constraints, finitely many open spherical exclusion stage constraints, and closed polyhedral terminal constraints. The computational complexity of exact nonconvex optimal and model predictive controls is alleviated through the local convexification of the nonconvex exclusion constraints, which is obtained via safe polyhedral tubes. The safe polyhedral tubes are constructed via simple algebraic operations and are of paramount importance for the design of locally convexified optimal and model predictive controls via strictly convex quadratic programming.

A Distributed Integrated Control Architecture of AFS and DYC Based on MAS for Distributed Drive Electric Vehicles

Reconstitution of control architecture creates a great challenge for distributed drive electric vehicles (DDEV), due to the emergence of a new distributed driving strategy. To this end, a novel distributed control architecture is proposed in this paper for integrated control of active front steering (AFS) system and direct yaw moment (DYC) system. First, a multi-agent system (MAS) is employed to construct a general framework, where AFS and DYC act as agents that work together to improve vehicle lateral stability and simultaneously reduce workloads of drivers during path tracking. The cooperative control strategies of two agents are obtained through Pareto-optimality theory to ensure optimal control performance of AFS and DYC. Then, on the basis of dynamic interaction between agents, terminal constraints, including terminal cost function and terminal input with local static feedback, are designed to guarantee the asymptotic stability of the close-loop system. Finally, virtual simulations are conducted to evaluate the proposed controller. The results indicate that the proposed control architecture can effectively preserve vehicle stability and reduce workloads of drivers, especially for the inexperienced driver. Furthermore, the hardware-in-loop (HIL) test results also demonstrate the feasibility of the proposed controller.

Composing an Assistive Control Strategy Based on Linear Bellman Combination From Estimated User's Motor Goal

In assistive control strategies, we must estimate the user's movement intentions. In previous studies, such intended motions were inferred by linearly converting muscle activities to the joint torques of an assistive robot or classifying muscle activities to identify the most likely movement from pre-designed robot motion classes. However, the assistive performances of these approaches are limited in terms of accuracy and flexibility. In this study, we propose an optimal assistive control strategy that uses estimated user movement intentions as the terminal cost function not only for generating movements for different task goals but to precisely enhance the motion with an exoskeleton robot. The optimal assistive policy is derived by blending the pre-computed optimal control laws based on the linear Bellman combination method. Coefficients that determine how to blend the control laws are derived based on low-dimensional feature values that represent the user's movement intention. To validate our proposed method, we conducted an assisted basketball-throwing task and showed that the performances of our subjects significantly improved.

Nonlinear MPC with time-varying terminal cost for tracking unreachable periodic references

In this paper, nonlinear model predictive control is investigated for a nonlinear system using a time-varying terminal cost function. An artificial reference and a control input are added as decision variables for tracking an unreachable periodic reference. Some constraints are given in an optimization problem to guarantee feasibility if the periodic references change suddenly for the reason of that the constraints are independent on the periodic references. A time-varying periodic polytopic linear difference inclusion is proposed to guarantee behavior locally around the artificial reference of the nonlinear system. A time-varying weight matrix is designed for the time-varying terminal cost function to ensure stability of the nonlinear system. Effectiveness of the developed tracking control scheme is shown by a simulation example on a mobile robot system.