Final State Constraints Research Articles

From boundary lines to boundary surfaces for dynamic programming with final state constraints

This article addresses the generalization of the boundary lines method for Dynamic Programming, called here boundary surface. This method can be applied on any system with 2 states as long as there are border constraints (initial or final). The method has been applied to two different problems. The first one is a simplified convex acceleration problem with final constraints on speed and position. The second problem is the energy minimization of a parallel hybrid electric vehicle along a trip in a non linear and non convex formulation. Results show a non negligible accuracy improvement despite the increase in calculation time. This can yield more accurate optimal results than those found by any possible mesh grids on an average computer for a reasonable calculation time.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.

Dynamic Posture Prediction Model in Informationized Combat Environment

The several local wars that broke out at the turn of the century have shown to the world that the rapid development of high and new technology with information technology as the core is effectively promoting the pace of world military reform, and its most essential feature is information dominance. In local wars under modern high-tech conditions, rapid response and multiarms joint and coordinated operations have become the key factors to win, which requires multiarms, multiweapon platforms, and each combat unit software to work autonomously and dynamically. This research mainly discusses the dynamic posture prediction model in the informationized combat environment. It designs the organization of command elements, related data (products and tools), competency, personnel matching checklists, and activities in joint operational planning and relies on activities to model the binding of groups, capabilities, and related data. Finally, it forms process segments through the combined design of activities and then sets the final state and background knowledge constraints on the process segments to construct the process knowledge, turning each combat unit into an agent unit, so that the combat software unit has the performance characteristics of the agent’s autonomy, reactivity, and sociality. At the same time, it has good communication performance and autonomous negotiation ability, so it can realize the autonomous connection, communication, and cooperative work among multiagent systems. During the process execution or business processing, the seat staff can input the goal and background of the activity according to the current operating environment. It relies on the background constraints of the process knowledge to quickly match the corresponding process knowledge, insert the process fragments in the process knowledge into the original process, realize the rapid and efficient decomposition of the process redefinition, and effectively ensure the orderly development of the combat command. Techniques and methods for automating integration testing of combat systems are provided. These technologies are further applied to the process of combat system integration tests to improve the efficiency of combat system integration. In addition, through the research on the integration testing technology of combat systems, the problems of low generality of integration testing software and many structural repetitive codes can also be solved and the existing integration testing technology can be further improved. In the research, the time-consuming of the script interpretation module accounts for a relatively low proportion of about 5% of the simulation. The article embodies the platform to explore the realization of multiagent-based dynamic cooperative operations, to achieve the purpose of simulating the battlefield situation, rehearsing the combat plan, and evaluating the strike effect. This study will improve the efficiency of overall combat system integration.

Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

In this paper, we consider a class of optimal control problems governed by a differential system. We analyze the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well.

Time-Optimal Trajectory Generation for Landing a Quadrotor Onto a Moving Platform

We propose a time-optimal trajectory generation algorithm for landing a quadrotor onto a translating (heaving) and tilting (pitching) platform. The proposed algorithm exploits the differential flatness of the system dynamics and optimizes the trajectory in the space of system flat outputs, i.e., system states and their derivatives. This reformulation of the problem improves computational efficiency and can incorporate input and state constraints during the landing maneuver, such as collision avoidance and time-varying final state constraints. In order to verify the optimality of the solution obtained from the proposed algorithm, a switching time search mechanism is also proposed for a subclass of time-optimal trajectory generation problems. Experimental results demonstrating the landing of a quadrotor onto a moving platform are presented to verify the effectiveness of the proposed trajectory generation algorithm.

Free finite horizon LQR: A bilevel perspective and its application to model predictive control

We consider linear-quadratic optimal control problems with free final time and terminal state constraints and propose a solution procedure that is particularly useful for online feedback control in a model-predictive control (MPC) framework. The procedure avoids the standard time transformation, which transforms the problem into an equivalent but non-convex optimal control problem on a fixed time horizon. The transformed problem typically suffers from many local minima, which might cause instabilities in online optimization tasks like LQR or model-predictive control. To avoid this drawback of the time transformation we develop a method from the viewpoint of bilevel optimal control, which is beneficial especially in online control tasks. The novelty of the approach is the optimal final time tracking procedure, for which we exploit a property of the Hamiltonian function. To this end we show that within the continuous-time closed-loop controlled system the optimal final time linearly decreases in time, as one could intuitively expect. Finally, numerical experiments support the effectiveness of the proposed algorithm.

Hybrid Aeronautical Propulsion: Control and Energy Management

In this paper, a control strategy for energy management of a parallel hybrid electric UAV (Unmanned Air Vehicle) powertrain is presented. A Simulink-based model of the system is first presented, consisting of an internal combustion engine, a gearbox (which includes a planetary gear and a continuously variable transmission), an electric motor (which can also work as a generator), an electric drive (Inveter) and a Li-Po battery pack. The proposed control strategy consists of a quasi-real-time iterative algorithm based on Dynamic Programming, which allows to optimize power management and torque-split of the powertrain with final state constraints on state variable. Aim of the study is to investigate new flight capabilities that derive from the use of a hybrid architecture, e.g., silent mode using only electric motor (useful for military and civil purpose), and the capacity of reducing fuel consumption. Simulation studies are performed, based on data of an existing UAV and a real flight mission.

Second-Order Necessary Conditions for a Strong Local Minimum in a Control Problem with General Control Constraints

We establish some second-order necessary optimality conditions for strong local minima in the Mayer type optimal control problem with a general control constraint $$U \subset \mathrm{I\! R}^m$$ and final state constraint described by a finite number of inequalities. In the difference with the main approaches of the existing literature, the second order tangents to U and the second order linearization of the control system are used to derive the second-order necessary conditions. This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of set-valued maps lead to fairly general statements. We illustrate the results by considering the case of control constraints defined by inequalities involving functions with positively or linearly independent gradients of active constraints, but also in the cases where the known approaches do not apply.

Modeling and Dynamic Optimization of Semi-Batch Acetone–Butanol–Ethanol Fermentation with In-situ Pervaporation Membrane Separations

To alleviate the severe impact of the substrate and product inhibitions leads to the design of the semi-batch bio-reactors, which aims at finding the optimal control of substrate feed rate through time. The consequence of this problem is challenging, which highlights the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these dynamic systems. In this work, following the control vector parameterization, a profile generator algorithm is utilized to produce functions in an exponential form that can represent path of the control space comprising elements of both linearity and curvature. A novel method of stochastic dynamic optimization using Simulated Annealing incorporating direct search through a specialized model predictive controller is used to tackle final volume point state constraints. A case study of glucose to acetone–butanol–ethanol production, in terms of structured kinetics model, with in situ pervaporation will be illustrated. Prior to optimization, we proposed revised models based on model identification for the published experimental data in the literature of batch bio-reactor system. Modeling of the pervaporation membrane separations for the system have also been presented.

Global fuel consumption optimization of an open-time terminal rendezvous and docking with large-eccentricity elliptic-orbit by the method of interval analysis

By defining two open-time impulse points, the optimization of a two-impulse, open-time terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit is proposed in this paper. The purpose of optimization is to minimize the velocity increment for a terminal elliptic-reference-orbit rendezvous and docking. Current methods for solving this type of optimization problem include for example genetic algorithms and gradient based optimization. Unlike these methods, interval methods can guarantee that the globally best solution is found for a given parameterization of the input. The non-linear Tschauner- Hempel(TH) equations of the state transitions for a terminal elliptic target orbit are transformed form time domain to target orbital true anomaly domain. Their homogenous solutions and approximate state transition matrix for the control with a short true anomaly interval can be used to avoid interval integration. The interval branch and bound optimization algorithm is introduced for solving the presented rendezvous and docking optimization problem and optimizing two open-time impulse points and thruster pulse amplitudes, which systematically eliminates parts of the control and open-time input spaces that do not satisfy the path and final time state constraints. Several numerical examples are undertaken to validate the interval optimization algorithm. The results indicate that the sufficiently narrow spaces containing the global optimization solution for the open-time two-impulse terminal rendezvous and docking with target spacecraft on large-eccentricity elliptical orbit can be obtained by the interval algorithm (IA). Combining the gradient-based method, the global optimization solution for the discontinuous nonconvex optimization problem in the specifically remained search space can be found. Interval analysis is shown to be a useful tool and preponderant in the discontinuous nonconvex optimization problem of the terminal rendezvous and docking than GA.

Second order analysis for the optimal control of parabolic equations under control and final state constraints

We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L 1 and $L^{\infty }$ topologies.

Dynamic Traffic Feedback Data Enabled Energy Management in Plug-in Hybrid Electric Vehicles

Recent advances in traffic monitoring systems have made real-time traffic velocity data ubiquitously accessible for drivers. This paper develops a traffic data-enabled predictive energy management framework for a power-split plug-in hybrid electric vehicle (PHEV). Compared with conventional model predictive control (MPC), an additional supervisory state of charge (SoC) planning level is constructed based on real-time traffic data. A power balance-based PHEV model is developed for this upper level to rapidly generate battery SoC trajectories that are utilized as final-state constraints in the MPC level. This PHEV energy management framework is evaluated under three different scenarios: 1) without traffic flow information; 2) with static traffic flow information; and 3) with dynamic traffic flow information. Numerical results using real-world traffic data illustrate that the proposed strategy successfully incorporates dynamic traffic flow data into the PHEV energy management algorithm to achieve enhanced fuel economy.

Data-driven multi-stage multi-objective motion planning of mobile robots, application to near minimum power fuzzy parking

In this paper, the problem of multi-objective trajectory planning is studied for a wheeled mobile robot (WMR) in a crowded environment using a hybrid data-driven neuro-fuzzy system. This approach is composed of two basic stages involving a hard computing through a state space detailed modeling of the system and then a data driven evolutionary neuro-fuzzy system is used for online planning. Typically, a first pre-processing step involves an offline planning generating a large dataset of multi-objective trajectories, optimizing a time-power criterion while including robot, task, and workspace constraints. The discrete augmented Lagrangian is implemented on a decoupled form of the robot dynamics to solve for the resulting non-linear multi-objective optimal control problem. The final state constraint is satisfied with a gradient projection technique. The outcomes of this pre-processing step allow building a genetic neuro-fuzzy inference system to learn and capture the robot multi-objective dynamic behavior. Once this system is trained and optimized, it is used in a generalization phase to achieve online planning. The approach was also applied to the near optimal power fuzzy parking problem. Simulation results showing the effectiveness of the proposal are presented and discussed.

Sensitivity analysis for relaxed optimal control problems with final-state constraints

In this article, we compute a second-order expansion of the value function of a family of relaxed optimal control problems with final-state constraints, parameterized by a perturbation variable. In this framework, relaxation with Young measures enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function. The sensitivity analysis is performed in a neighborhood of a local optimal solution of a reference problem. The local solution ū is assumed to be optimal with respect to the set of feasible relaxed controls having their support in a ball of a given radius R>‖ū‖∞ and having an associated trajectory very close to the reference trajectory, for the L∞-norm. We call such a solution a relaxed R-strong solution.

Implementation of Dynamic Programming for $n$-Dimensional Optimal Control Problems With Final State Constraints

Many optimal control problems include a continuous nonlinear dynamic system, state, and control constraints, and final state constraints. When using dynamic programming to solve such a problem, the solution space typically needs to be discretized and interpolation is used to evaluate the cost-to-go function between the grid points. When implementing such an algorithm, it is important to treat numerical issues appropriately. Otherwise, the accuracy of the found solution will deteriorate and global optimality can be restored only by increasing the level of discretization. Unfortunately, this will also increase the computational effort needed to calculate the solution. A known problem is the treatment of states in the time-state space from which the final state constraint cannot be met within the given final time. In this brief, a novel method to handle this problem is presented. The new method guarantees global optimality of the found solution, while it is not restricted to a specific class of problems. Opposed to that, previously proposed methods either sacrifice global optimality or are applicable to a specific class of problems only. Compared to the basic implementation, the proposed method allows the use of a substantially lower level of discretization while achieving the same accuracy. As an example, an academic optimal control problem is analyzed. With the new method, the evaluation time was reduced by a factor of about 300, while the accuracy of the solution was maintained.

ε-Subgame Perfectness of an Open-Loop Stackelberg Equilibrium in Linear-State Games

Open-loop Stackelberg equilibria in linear-state games are subgame perfect. This result holds under the hypothesis of unconstrained final state; whereas we need to take into account suitable final-state conditions in order to correctly formalize certain economic problems. A striking contribution of this paper is that it tackles the consistency problem for an open-loop Stackelberg equilibrium in linear-state games with a final-state constraint in the leader’s problem. In this paper, after proving that such a type of equilibrium is not subgame perfect, we introduce a weaker definition of subgame perfectness, which we call e-subgame perfectness. This new definition can be applied to the open-loop Stackelberg equilibrium of a constrained linear-state game. Finally, we present some explanatory examples to show how the definition of e-subgame perfectness can be meaningful.

A robust and efficient genetic algorithm for solving a chemical reactor problem: theory, application and convergence analysis

Solution of the chemical reactor problem, as an optimal control problem, using continuous genetic algorithms (CGAs) is presented in this paper. The proposed approach overcomes the drawbacks of the traditional approaches in terms of lack of efficiency, lack of accuracy and lack of robustness. The solution is based on the value of the performance index and the final system state constraints. Simulation results show clearly that the new technique outperforms the existing direct and indirect methods. Based on the convergence analysis, the solution of the optimal control problem is achieved without any limitation on the nature of the problem and regardless of the CGA tuning parameters.

Characterization of the value function of final state constrained control problems with BV trajectories

This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations.

Optimal Control of State Constrained Integral Equations

We consider the optimal control problem of a class of integral equations with initial and final state constraints, as well as running state constraints. We prove Pontryagin’s principle, and study the continuity of the optimal control and of the measure associated with first order state constraints. We also establish the Lipschitz continuity of these two functions of time for problems with only first order state constraints.

On Implementation of Dynamic Programming for Optimal Control Problems with Final State Constraints

In this paper we present issues related to the implementation of dynamic programming for optimal control of a one-dimensional dynamic model, such as the hybrid electric vehicle energy management problem. A study on the resolution of the discretized state space emphasizes the need for careful implementation. A new method is presented to treat numerical issues appropriately. In particular, the method deals with numerical problems that arise due to high gradients in the optimal cost-to-go function. These gradients mainly occur on the border of the feasible state region. The proposed method not only enhances the accuracy of the final global optimum but also allows for a reduction of the state-space resolution with maintained accuracy. The latter substantially reduces the computational effort to calculate the global optimum. This allows for further applications of dynamic programming for hybrid electric vehicles such as extensive parameter studies.