Principle For Optimal Control Research Articles

Model for discrete optimal control of the enterprise’s financial processes

We consider a problem of financial resources planning where the processes of financial flows coordination of an enterprise under uncertainty are described with the discrete optimal control approach and methods of dynamic programming. We give a critical survey of theoretical and methodological approaches for financial resources planning problem. The financial flows dynamics is modelled with using the methods of system analysis, control theory, optimal control and methods of data processing under uncertainty. The model for the financial flows distribution has been developed. It uses the principles of optimal dynamic control under the criterion of cumulative non-payment risks and transaction and opportunity costs minimizing. The practical significance of the developed model is in its application for the financial planning in the enterprises that allows to improve its financial planning quality and to increase its management and operational efficiency.

Optimal Control and Maximum Principle in (B)-Spaces. Examples for PDE in (H)-Spaces and ODE in $${{\mathbb{R}}^{n}}$$

Observation and control problems in Banach (B)-spaces are investigated. A criterion for controllability and optimal controllability is formulated on the basis of the BUME method and the monotone mapping method. An inverse controllability problem is introduced, and an abstract maximum principle is formulated in (B)-spaces. For PDE in Hilbert (H)-spaces and for ODE in $${{\mathbb{R}}^{n}}$$, an integral maximum principle is proved and an optimality system is presented.

Improving the efficiency of a variable frequency asynchronous electric drive

Asynchronous electric drives hold leading positions in the global structure of electricity consumption. That’s why the development and implementation of an energy-efficient asynchronous electric drive are always economically feasible and important. The aim of this article is to develop an algorithm to minimize losses of asynchronous electric drive power. The article offers methods to optimize the level of losses in the winding of an electric motor, powered in a steady-state mode by a frequency converter with a scalar control. The substitution scheme for an asynchronous electric drive is considered as a consistent set of conductivities for the stator and rotor wirings. The research tasks were solved based on theoretical and experimental methods, including mathematical and physical simulation of studied processes using modern measuring and computation devices as well as statistic treatment of experimental data. The article offers methods to minimize the losses of power in the winding of an electric motor, powered in a steady-state mode by a frequency converter with a scalar control. The offered method is significantly different in that it uses the substitution for an asynchronous electric drive expressed as a consistent set of conductivities for the stator and motor wirings and the use of the energy efficiency ratio which determines the losses of power in an asynchronous drive for assessing the efficiency of a drive. The main principle for optimal frequency control of an asynchronous engine in terms of energy efficiency was formulated. The laboratory bench for studying the modes of frequency-controlled electric drives was built. The tested drive AIR100S4 showed an increase in energy efficiency ratio from 0.59 to 0.67.

Designing an Adaptive Prosthetic Socket Controller Using H-Infinity Loop Shaping Synthesis.

Amputees often find wearing a prosthetic limb for a long period of time uncomfortable. Prosthetic sockets that adjust the socket's fit automatically, or adaptive sockets, would encourage amputees to wear their prosthesis more frequently. In this work, we simulate the control system design of a Multiple-Input, Multiple-Output (MIMO) adaptive socket using principles of optimal control and robust control. A data-driven model of the socket-limb interface is first obtained by applying regression to open-loop recordings of the socket interacting with the limb during a simulated grasping task. A MIMO controller is then designed to maintain a desired uniform socket fit. An $H_{\infty}$ controller, obtained from loop shaping synthesis using the Glover-McFarlane method, is shown to perform comparably to a Linear Quadratic Gaussian (LQG) controller while maintaining robustness to uncertainties in the socket-limb interface model. This work then outlines a potential procedure on how to develop the control system for a real adaptive prosthetic socket with multiple sensors and actuators.

Indirect Robotic Movement Shaping through Motor Cost Influence.

Movement patterns are commonly disrupted after a neurological incident. The correction and recovery of these movement patterns is part of therapeutic practice, and should be considered in the development of robotic device control strategies. This is an area which has limited exploration in rehabilitation robotics literature. This work presents a new strategy aiming at influencing the cost associated with a movement, based on the principle of optimal motor control. This approach is unique, in that it does not directly modify the movement pattern, but instead encourages this altered movement. This 'Indirect Shaping Control' is applied in a preliminary experiment using an end-effector based device with 5 healthy subjects. The study concludes that such an approach may encourage changes in movement patterns which do persist to out-of-robot reaching actions, but this was not consistent over all subjects and further experiments are required.

Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator

The field of sub-Riemannian geometry has flourished in the past four decades through the strong interactions between problems arising in applied science (in areas such as robotics) and questions of a pure mathematical character about the nature of space. Methods of control theory, such as controllability properties determined by Lie brackets of vector fields, the Hamilton equations associated to the Maximum Principle of optimal control, Hamilton-Jacobi-Bellman equation etc. have all been found to be basic tools for answering such questions. In this paper, we find a useful role for the vantage point of sub-Riemannian geometry in attacking a problem of interest in non-equilibrium statistical mechanics: how does one create rules for operation of micro- and nano-scale systems (heat engines) that are subject to fluctuations from the surroundings, so as to be able to do useful things such as converting heat into work over a cycle of operation? We exploit geometric optimal control theory to produce such rules selected for maximal efficiency. This is done by working concretely with a model problem, the stochastic oscillator. Essential to our work is a separation of time scales used with great efficacy by physicists and justified in the linear response regime.

Maximum principle for optimal control of McKean‐Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law

SummaryIn this paper, we study stochastic optimal control problem for general McKean‐Vlasov–type forward‐backward differential equations driven by Teugels martingales, associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our main result is based on the differentiability with respect to probability law and a corresponding Itô formula.

An optimal control theory for nonlinear optimization

The Karush–Kuhn–Tucker conditions for a given nonlinear programming problem are generated as the transversality conditions of an optimal control problem. The directional derivatives of the objective- and constraint-functions supply the vector fields for the optimal control problem with the search vector as the control variable. Zero-Hamiltonian trajectories along the steepest descent of control Lyapunov functions provide optimal optimization algorithms. The optimality of the algorithm also depends upon the choice of a metric for the finite dimensional control space. Many well-known algorithms – such as Newton’s method, the first-order Lagrangian method, the steepest descent method and Richardson’s method, to name a few – are derived by minimizing the Lie derivative of a quadratic control Lyapunov function. Merit functions in optimization may also be generated using the concept of a control Lyapunov function. These results suggest that optimal control principles hold the potential for a unified theory for optimization.

On Peng's type maximum principle for optimal control of mean-field stochastic differential equations with jump processes

In this paper, we investigate the Peng's type optimal control problems for stochastic differential equations of mean-field type with jump processes. The coefficients of the system contain not only the state process but also its marginal distribution through their expected values. We assume that the control set is a general open set that is not necessary convex. The control variable is allowed to enter into both diffusion and jump terms. We extend the maximum principle of Buckdahn et al. (2011) to jump case.

The Discrete-Time Geometric Maximum Principle

We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. These results are organized around a new theorem on critical and approximate critical points for discrete-time geometric control systems. We show that this theorem can be used to derive Lie group variational integrators in Hamiltonian form; to establish a maximum principle for control problems in the absence of state constraints; and to provide sufficient conditions for exact penalization techniques in the presence of state or mixed constraints. Exact penalization techniques are used to study sensitivity of the optimal value function to constraint perturbations and to prove necessary conditions for optimality, including in the form of a maximum principle, for discrete-time geometric control problems with state or mixed constraints.

Interaction of GEOTABS and secondary heating and cooling systems in hybridGEOTABS buildings: towards a sizing methodology

GEOTABS, a combination of TABS with a geothermal heat pump, is a promising heating and cooling system for decreasing greenhouse gas emissions in the building sector. However, TABS has a time delay when transferring energy from the pipes to the room. So, when the heat demand changes fast, TABS cannot properly compensate the heat demand. In order to solve this problem and maintain thermal comfort in the room, the concept of hybridGEOTABS proposes using a fast secondary system to assist the TABS. Yet, there is no integrated method for sizing both systems in a hybridGEOTABS building, considering the interaction between the secondary system and GEOTABS. This study will provide an integrated sizing methodology for hybridGEOTABS buildings. To that purpose, in this paper the interaction between the secondary system and TABS is investigated for two different scenarios by using a preference factor between the TABS and the secondary system. The methodology starts from heat demand curves, an analytic model for TABS, and optimal control principles for TABS to minimize the total energy use while providing thermal comfort. Finally, the method is used for 4 case studies in different scenarios with different secondary systems. Preliminary results of this research indicate that the secondary system type doesn’t have effect on the strategy of sizing. Therefore, designer can decide about secondary system type with investment and operating cost analysis.

Precise Contour Control of Biaxial Motion System Based on MPC

In order to realize accurate and fast contour tracking of biaxial motion system and ensure the smoothness of its path, this paper proposes a model predictive contour control strategy suitable for biaxial system based on the idea of unified modeling. The two permanent magnet synchronous motors driving the biaxial system are considered as a whole to build a unified model. Based on the unified model, a compact model predictive controller is designed, where the contour error, the tracking error, the control increment and the position increment are introduced into the cost function simultaneously, realizing the multivariable collaborative optimization control. In addition, to obtain the better dynamic response and steady-state performance, a method of real-time adaptive adjustment of the position increment weight coefficient and the control increment weight coefficient is proposed. At the same time, considering the constraints, a posteriori constraint limitation network with the current control increment is proposed. Furthermore, the closed-loop stability analysis of the model predictive contour control system is carried out according to the optimal control principle. The experimental results show that the proposed control strategy can not only track the trajectory accurately at steady state, but also improve the transient contour tracking performance.

Generalized Predictive Contour Control of the Biaxial Motion System

In order to overcome the redundancy of the biaxial motion system's traditional control structure and the cumbersome nature of the controller parameter adjustment, this paper simplifies the traditional control structure and proposes a generalized predictive contour control strategy applicable to the biaxial motion system based on the idea of unified modeling. The two permanent magnet synchronous motors that drive the biaxial motion system are considered as a whole to establish a unified model. Through the idea of multistep prediction, rolling optimization, and feedback correction in generalized predictive control, the optimal control input signal is obtained by coordinating the tracking error and contour error. In addition, an actuating value limitation has to be taken into account to design a realizable generalized predictive contour control for the biaxial system. At the same time, the closed-loop stability analysis of the generalized predictive contour control system is carried out according to the optimal control principle and the state equation model. The experimental results show that the proposed control strategy cannot only accurately track the contour trajectory at steady state but also improve the transient contour tracking performance.

Inverse problem for a class of nonlinear elliptic equations with entropy solution

We study a nonlinear ill posed inverse problem arising from the identification of nonlinearities in elliptic partial differential equations with L1 data. Using duality regularization techniques and optimal control principle, we approach the ill posed inverse problem by a constraint optimization problem. Then, we establish the well posedness and sensibility of the regularized solutions with respect to the given data. We finish by an application in image restoration, precisely, we use our approach to find an optimal choice of the regularization function in a weighted total variation model.

Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching

In this article, we consider the partially observed optimal control problemfor forward-backward stochastic systems with Markovian regime switching. A stochastic maximum principle for optimal control is developed using a variational method and filtering technique. Our theoretical results are appliedto the motivating example of the risk minimization for portfolio selection.

An integrated multi-population genetic algorithm for multi-vehicle task assignment in a drift field

This paper investigates the task assignment problem for a team of autonomous aerial/marine vehicles driven by constant thrust and maneuvering in a planar lateral drift field. The aim is to minimize the total traveling time in order to guide the vehicles to deliver a number of customized sensors to a set of target points with different sensor demands in the drift field. To solve the problem, we consider together navigation strategies and target assignment algorithms; the former minimizes the traveling time between two given locations in the drift field and the latter allocates a sequence of target locations to each vehicle. We first consider the effect of the weight of the carried sensors on the speed of each vehicle, and construct a sufficient condition to guarantee that the whole operation environment is reachable for the vehicles. Then from optimal control principles, time-optimal path planning is carried out to navigate each vehicle from an initial position to its given target location. Most importantly, to assign the targets to the vehicles, we combine the virtual coding strategy, multiple offspring method, intermarriage crossover strategy, and the tabu search mechanism to obtain a co-evolutionary multi-population genetic algorithm, short-named CMGA. Simulations on sensor delivery scenarios in both fixed and time-varying drift fields are shown to highlight the satisfying performances of the proposed approach against popular greedy algorithms.

Maximum Principle of Optimal Stochastic Control with Terminal State Constraint and Its Application in Finance

This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland’s variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.

Optimal Energy Management and Velocity Control of Hybrid Electric Vehicles

An assessment study of a novel approach is presented that combines discrete state-space Dynamic Programming and Pontryagin's Maximum Principle for online optimal control of hybrid electric vehicles (HEV). In addition to electric energy storage, engine state and gear, kinetic energy, and travel time are considered states in this paper. After presenting the corresponding model using a parallel HEV as an example, a benchmark method with Dynamic Programming is introduced which is used to show the solution quality of the novel approach. It is illustrated that the proposed method yields a close-to-optimal solution by solving the optimal control problem over one hundred thousand times faster than the benchmark method. Finally, a potential online usage is assessed by comparing solution quality and calculation time with regard to the quantization of the state space.

On Peng's type maximum principle for optimal control of mean-field stochastic differential equations with jump processes

In this paper, we investigate the Peng's type optimal control problems for stochastic differential equations of mean-field type with jump processes. The coefficients of the system contain not only the state process but also its marginal distribution through their expected values. We assume that the control set is a general open set that is not necessary convex. The control variable is allowed to enter into both diffusion and jump terms. We extend the maximum principle of Buckdahn et al. (2011) to jump case.

Stochastic maximum principle for optimal control of partial differential equations driven by white noise

We prove a stochastic maximum principle of Pontryagin’s type for the optimal control of a stochastic partial differential equation driven by white noise in the case when the set of control actions is convex. Particular attention is paid to well-posedness of the adjoint backward stochastic differential equation and the regularity properties of its solution with values in infinite-dimensional spaces.