Terminal State Constraints Research Articles

Highly constrained cooperative guidance for flexible landing on asteroids

Flexible lander, composed of multiple nodes connected by flexible material, can reduce the bouncing and overturning during the asteroid landing. To satisfy the complex constraints in the node cooperation of the flexible landing, an intelligent cooperative guidance method is proposed. The method consists of a double-layer cooperative guidance structure, a guidance parameter determination approach, and an action priority strategy. The double-layer contains a basic guidance used to satisfy the terminal state constraints, and a compensatory guidance used to satisfy the lander’s attitude constraint. For the compensatory guidance, the parameters are determined by multi-agent system, which are trained according to the performance index of flexible landing trajectories. The action priority strategy is used to reduce the detrimental effect of parameter inconsistency on the node cooperation. The simulation of flexible landing shows that the cooperative guidance method is effective in improving the landing accuracy while satisfying the constraints. Meanwhile, the method is robust to the disturbance in the navigation and control.

Two-person zero-sum game with terminal state constraints

In this paper, the discrete-time two-person zero-sum game with terminal state constraints is considered. By applying the variational method, the maximum principle for zero-sum game with terminal constraint is derived along with the forward and backward difference equations (FBDEs). The main contributions of this paper are two-fold. On the one hand, by decoupled solving the FBDEs, the analytical solution to the zero-sum game with terminal state constraint is given. On the other hand, the necessary and sufficient conditions for the reachability of the system are obtained. In this way, the solvability of the zero-sum game with terminal state constraint is re-expressed by the reachability of the system.

A novel model predictive controller for the drifting vehicle to track a circular trajectory

ABSTRACT In rally racing, a high sideslip cornering technique known as drifting is employed to minimise the lap time. However, for autonomous vehicles, tracking the trajectory while drifting is a challenging control problem due to the conflict between the two objectives. This paper proposes a novel model predictive controller (MPC) with two operation modes to address this conflict. In Mode 1, fuzzy expert strategies combined with the lateral error are designed to calculate the drift equilibrium point. Subsequently, the linear MPC controller is utilised to achieve quick adjustments of the lateral error. In Mode 2, terminal state constraints and error constraints are added to the MPC optimisation problem. The stability of the MPC controller is guaranteed to keep the vehicle within the expected error range. Finally, a comparison between the existing expert controller and the proposed controller is carried out on both the CarSim platform and a 1:10 scale race car. The results demonstrate that the proposed controller outperforms the existing expert controller in terms of tracking performance and drift stability.

The maximum principle for optimal control of mean‐field FBSDE driving by Teugels martingales with terminal state constraints

AbstractThis article studies the problem of optimal control with state constraints for mean‐field type stochastic systems, which is governed by a fully coupled forward‐backward stochastic differential equation with Teugels martingales. In this system, the coefficients contain not only the state processes but also its expectation value, and the cost function is of mean‐field type as well. We use an equivalent backward formulation to deal with the terminal state constraint, and then we obtain a stochastic maximum principle by Ekeland's variational principle. In addition, we discuss a stochastic linear‐quadratic control problem with state constraints.

Lagrangian dual method for solving stochastic linear quadratic optimal control problems with terminal state constraints

A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed. The strong convergence of the iterative sequence generated by ALM is proved. In addition, some sufficient conditions for the surjectivity of the linear constraint mapping are obtained.

Nonlinear Terminal-Free MPC on Multitype Bend Tracking With Discontinuous Reference Paths for Autonomous Vehicles

In this paper, multitype bend tracking with discontinuous reference paths is researched by a nonlinear terminal-free model predictive control (MPC) with an artificial reference path for an autonomous vehicle. The MPC strategy without terminal penalty and terminal state constraints is proposed to deal with the discontinuous reference paths, which improves solvability on an optimization problem effectively. The artificial reference path is designed to realize replanning and smoothing for multitype bend paths. Both recursive feasibility and stability are discussed for the autonomous vehicle by a lower bound of prediction horizon and a controlled forward invariance set. Effectiveness of the nonlinear terminal-free MPC strategy is shown by experimental results on multitype bend path for the autonomous vehicle.

A novel re-entry trajectory design strategy enforcing inequality and terminal constraints in height-velocity plane

A novel re-entry trajectory design approach for a winged reusable launch vehicle that satisfies path and terminal state constraints in the height velocity plane is proposed in this paper. A two-phase re-entry trajectory is proposed. A height-velocity plane entry corridor is designed using all path and quasi-equilibrium glide constraints. The proposed method creates the reference trajectory inside the re-entry corridor using three piecewise polynomials, such as linear and logarithmic. A two-parameter search optimization problem minimizes terminal range-to-go error for this reference trajectory generation problem. Improved Search Space Reduction solves this optimization problem. The Improved Search Space Reduction algorithm is compared to the Grey Wolf Optimizer and Particle Swarm Optimisation algorithms to prove its efficacy and superiority. The reference bank angle is modified using feedback linearization and bank reversal logic to create an entry trajectory that meets all path and terminal constraints. The effectiveness of the proposed method has been validated through Monte Carlo simulations that incorporate uncertainties such as variations in initial state, uncertainties in aerodynamics, and atmospheric perturbations. The controller OPAL-RT OP4510 provides bank angle command at every instant based on the current state of the vehicle parameters associated with real-time simulator OP4510. The real-time hardware-in-loop simulation results show that the obtained re-entry trajectory satisfies all path and terminal constraints.

Stochastic linear quadratic optimal control problem with terminal state constraints

AbstractThis paper addresses the stochastic linear quadratic (LQ) control problem with first‐ and second‐order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative‐noise stochastic system rather than an additive‐noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two‐point boundary‐value problem (TPBVP). We provide the optimal closed‐loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results.

Distributed Dual Closed-Loop Model Predictive Formation Control for Collision-Free Multi-AUV System Subject to Compound Disturbances

This paper focuses on the collision-free formation tracking of autonomous underwater vehicles (AUVs) with compound disturbances in complex ocean environments. We propose a novel finite-time extended state observer (FTESO)-based distributed dual closed-loop model predictive control scheme. Initially, a fast FTESO is designed to accurately estimate both model uncertainties and external disturbances for each subsystem. Subsequently, the outer-loop and inner-loop formation controllers are developed by integrating disturbance compensation with distributed model predictive control (DMPC) theory. With full consideration of the input and state constraints, we resolve the local information-based DMPC optimization problem to obtain the control inputs for each AUV, thereby preventing actuator saturation and collisions among AUVs. Moreover, to mitigate the increased computation caused by the control structure, the Laguerre orthogonal function is applied to alleviate the computational burden in time intervals. We also demonstrate the stability of the closed-loop system by applying the terminal state constraint. Finally, based on a connected directed topology, comparative simulations are performed under various control schemes to verify the robustness and superior performance of the proposed scheme.

Relative Controllability for a Class of Linear Impulsive Systems

Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.

Rapid Suboptimal Low-Thrust Space Trajectory Design

The finite Fourier series shape-based approach for fast initial trajectory design satisfies the equations of motion and other problem constraints at discrete points by using a nonlinear programming solver to design the Fourier coefficients. In this paper, the finite Fourier series approach is extended to account for the necessary conditions for optimality during the Fourier coefficient design. The proposed method uses the approximate trajectories to analytically estimate the costates at these discrete points by employing a finite difference technique on the adjoint differential equations. Lagrange multipliers associated with any terminal state constraints are determined using an auxiliary function. Residual errors in the stationarity condition are reduced through the objective function, with the Legendre–Clebsch condition enforced as a nonlinear inequality constraint; applying these two conditions finds suboptimal solutions without an excessive loss of computational efficiency. Test cases use steering angle and thrust acceleration magnitude controls and span Earth-escape spirals, geostationary spacecraft rendezvous, and phasing maneuvers. The presented results demonstrate the proposed method’s ability to achieve trajectories closer to the optimal solution.

Economic Predictive Control with Periodic Horizon for Water Distribution Networks

This paper deals with control of pumps in large-scale water distribution networks with the aim of minimizing economic costs while satisfying operational constraints. Finding a control algorithm in combination with a model that can be applied in real-time is a challenging problem due to the nonlinearities presented by the pipes and the network sizes. We propose a predictive control algorithm with a periodic horizon. The method provides a way for the economic operation of large water networks with a low-complexity linear model. Economic Predictive control with a periodic horizon and a terminal state constraint is constructed to keep the state trajectories close to an optimal periodic trajectory. Barrier terms are also included in the cost function to prevent constraint violations. The proposed method is tested on a state-of-the-art EPANET model of the water network of a medium size Danish town (Randers) and shown to perform as intended under varying conditions.

A Pontryagin maximum principle for terminal state-constrained optimal control problems of Volterra integral equations with singular kernels

<abstract><p>We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $ \alpha \in (0, 1) $. Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature.</p></abstract>

Optimal Control with Terminal State Constraints for Multichannel Input Linear Systems

Optimal Control with Terminal State Constraints for Multichannel Input Linear Systems

Sequential adaptive switching time optimization technique for optimal control problems

The control parameterization method used together with time-scaling transformation is an effective approach to transform optimal control problems into optimal parameter selection problems. The approximate problems can then be solved by gradient-based optimization methods. However, the conventional time-scaling transformation requires that all the control component functions switch simultaneously, which can be undesirable in practice. In this paper, we introduce a novel technique where the switching times for each of the control component functions can be adaptively selected. To demonstrate the effectiveness and flexibility of the proposed approach, two example problems with control input and terminal state constraints are solved using the method proposed. Numerical results show that the new method achieves much better objective values with some increase in computation time compared to the conventional time-scaling transformation technique.

A two-player portfolio tracking game

We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank et al. (Math Financial Economics 11(2):215–239 2017). Specifically, both agents track their own stochastic running trading targets while interacting through common aggregated temporary and permanent price impact à la Almgren and Chriss (J Risk 3:5–39 2001). The resulting stochastic linear quadratic differential game with terminal state constraints allows for a unique and explicitly available open-loop Nash equilibrium. Our results reveal how the equilibrium strategies of the two players take into account the other agent’s trading targets: either in an exploitative intent or by providing liquidity to the competitor, depending on the relation between temporary and permanent price impact. As a consequence, different behavioral patterns can emerge as optimal in equilibrium. These insights complement and extend existing studies in the literature on predatory trading models examined in the context of optimal portfolio liquidation games.

Aerogravity-assist capture into the three-body system: A preliminary design

Aero-gravity assist (AGA) is a low-cost maneuver for an interplanetary vehicle since it can potentially replace thrust with aerodynamic force to save fuel consumption. In this study, an AGA-maneuver capture scheme to deliver a vehicle into a three-body system is presented, and a successive approximation-based method is proposed to optimize atmospheric trajectories. Based on the analysis of the aerodynamic deceleration process, the AGA capture maneuver is first formulated as a coplanar atmospheric flight process with fixed time and strict endpoint-state constraints. Then, the lossless convexification and successive approximation techniques are applied to construct the solving process of iteratively optimizing the AGA trajectories. Finally, the relaxation strategy to the strict terminal state constraints and the heavily penalized-function method are developed to avoid the artificial infeasibility of the problem and improve the convergence. Numerical simulations, demonstrated by the AGA-capture into Sun-Mars L2 Halo orbits and Sun-Mars backward stable orbits, show the effectiveness and reliability of the proposed method. It is the first time that the AGA-maneuver is combined with the dynamic property of the three-body system, and the proposed method can potentially enrich the possibility of the low-energy interplanetary transfer.

Analysis and trajectory planning for anti-swing on a one-stage cart pendulum

In order to achieve smooth and anti-swing positioning process, a smooth trajectory planning scheme, which is mainly the linear combination of trigonometric functions, is proposed for a one-stage cart pendulum. Based on the internal dynamics of the cart pendulum system, the planning trajectory of the cart is derived from the anticipated swing angle of the pendulum bar, which satisfies the initial and terminal state constraints. Experiments are conducted to demonstrate the performance of anti-swing with the introduced planned trajectory.

Stabilizing terminal constraint-free nonlinear MPC via sliding mode-based terminal cost

It is well-known that terminal state constraints play an instrumental role in ensuring the feasibility and stability of the nonlinear model predictive control (MPC). Yet, they inherently and largely limit the size of the feasible region and restrict the use of MPC to practical applications. In this paper, a terminal cost characterized by an implicit sliding mode control (SMC) law is proposed for developing a stabilizing constrained MPC scheme. This SMC law developed for the linearized model helps compensate the model mismatch between the linearization and the original nonlinear system model. Thanks to it, the proposed MPC strategy can stabilize the constrained nonlinear system of which the corresponding linearization around the equilibrium is non-stabilizable. Moreover, by appropriately tuning the sliding mode parameters, the conventional terminal constraints and large prediction horizon typically used in the literature are no longer required. We establish the conditions of ensuring the recursive feasibility and asymptotic stability of the closed-loop system. Finally, numerical comparison results on two examples of dynamic systems are reported to demonstrate the effectiveness of the developed strategy.

Continuity of the value function for deterministic optimal impulse control with terminal state constraint

Deterministic optimal impulse control problem with terminal state constraint is considered. Due to the appearance of the terminal state constraint, the value function might be discontinuous in general. The main contribution of this paper is the introduction of an intrinsic condition under which the value function is proved to be continuous. Then by a Bellman dynamic programming principle, the corresponding Hamilton-Jacobi-Bellman type quasi-variational inequality (QVI, for short) is derived. The value function is proved to be a viscosity solution to such a QVI. The issue of whether the value function is characterized as the unique viscosity solution to this QVI is carefully addressed and the answer is left open challengingly.