Open-loop Optimal Control Research Articles

Anti-disturbance attitude control of combined spacecraft with enhanced control allocation scheme

In this paper, we propose a novel anti-disturbance attitude control law for combined spacecraft with an improved closed-loop control allocation scheme. More specifically, a saturated approach is adopted to guarantee the global asymptotic stability under control input saturation. To enhance the robustness of the system, a nonlinear disturbance observer is constructed to compensate the disturbances caused by inertial parameter uncertainty and unmodeled dynamics. Next, the quadratic programming algorithm is used to obtain an optimal open-loop control allocation scheme, where both energy consumption and actuator saturation have been considered in the allocation of the virtual control command. Then, a modified closed-loop control allocation scheme is proposed to reduce the allocation error under the actuator uncertainty. Finally, stability analysis of the closed-loop system with the proposed allocation scheme is provided. Simulation results confirm the effectiveness of the proposed control scheme.

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.

Data-driven surrogates of value functions and applications to feedback control for dynamical systems

Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since the discretizations typically suffer from the curse of dimensionality. In this paper we introduce a novel approximation technique for the value function of an infinite horizon optimal control. The method is based on solving optimal open loop control problems on a finite horizon with a sampling of the global value function along the generated trajectories. For the interpolation we choose a kernel orthogonal greedy strategy, because these methods are able to produce extreme sparse surrogates and enable rapid evaluations in high dimensions. Two numerical examples prove the performance of the approach and show that the method is able to deal with high-dimensional feedback control problems, where the dimensionality prevents the approximation by most existing methods.

Variational integrators for open‐loop and closed‐loop optimal control of mechanical systems

AbstractMechanical systems have structural properties, e.g. symplecticity, symmetry, and a specific energy behavior, which get lost in standard integration methods. Therefore, symplectic integration methods are used in simulation and control of mechanical systems. This paper combines two methods of the class of structure‐preserving control methods, namely a recently developed feedback control method and open loop optimal control based on variational integrator discretization. The combination is applied to the benchmark example of a cart pendulum system. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards Riccati‐Feedback Control of Complex Flows with Moving Interfaces

AbstractProblems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi‐phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality.We consider the two‐phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection‐diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking‐type cost functional as a target of a control input.We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear‐quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain.Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal open-loop strategies in a debt management problem

The paper studies optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present, which increases with the size of the debt. This induces a pool of risk-neutral lenders to charge a higher interest rate, to compensate for the possible loss of part of their investment. Solutions are interpreted as Stackelberg equilibria, where the borrower announces his repayment strategy [Formula: see text] at all future times, and lenders adjust the interest rate accordingly. This yields a highly non-standard problem of optimal control, where the instantaneous dynamics depend on the entire future evolution of the system. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as [Formula: see text].

Separation of motion techniques for trajectory optimization in flexible redundant robot manipulators

This paper presents a computational method for optimization of trajectory in redundant robot manipulators. For this purpose, all possible answers are acquired based on rigid conditions and redundancy of the robot. Using open loop optimal control method, the trajectory which minimizes the objective function will be obtained. The objective function is considered as an integral index that will be minimized in the entire trajectory. The objective function and constraints of optimization problem will be selected based on conditions of motion. Dynamic equations of the system are constraints of optimization problem in point-to-point motion. For motion conditions in the specified path, kinematic equations will be added. Also, unequal constraints are applied for limiting the velocity and torque. By selecting the state and control signal vectors which are obtained by assuming rigid motion of the robot, the objective function and constraints will be changed to standard form of an optimization problem. Pontryagin’s maximum principle is used to solve equations. So, the equations of classical form will be changed into two-point boundary value problem. Suggested method is applied to point-to-point motion and movement in the specified path. Results demonstrate accuracy and efficiency of suggested method.

Sustainable energy-water management for residential houses with optimal integrated grey and rain water recycling

South Africa is a semi-arid developing country facing water and energy insecurity. There are colossal challenges in reliably providing these resources amid growing population, increased urbanization and improved living standards causing increased demand for these resources. Development of new supply centralized systems comes at an exorbitant cost, whereas decentralized systems are touted as an attractive alternative. Grey water recycling and rain water harvesting at buildings level is such an alternative that can provide water for non-potable uses. However, there are technological challenges of optimally operating such systems while ensuring efficient use of associated energy. This paper introduces two control strategies; open loop optimal control and closed-loop model predictive control (MPC) strategies aimed at ensuring safe and reliable operation of the grey water recycling and rain water harvesting system while efficiently using associated energy. From the case study, the proposed system with either control strategy can save the cost of water and waste water by up to 32.3% and 29.5% respectively, while leading to 35.7% in energy cost savings and 31.5% in total operational cost savings in a month. Adoption of these systems would have a huge environmental effect in reducing demand for sewerage services, conservation of water hence reducing demand for potable water as well as increasing the energy efficiency. Furthermore, the system would increase the reliability and security of water supply. Despite the benefits, the system does not pay within its lifetime and therefore, government intervention is required so as to make it economically attractive. High cost of implementation coupled with low potable and waste water tariffs harbour adoption of these systems. Appropriate regulations, policies, incentives and public education are necessary to support such novel technologies in ensuring resource conservation, efficiency and security are achieved.

Real-time feasible multi-objective optimization based nonlinear model predictive control of particle size and shape in a batch crystallization process

This paper presents nonlinear model predictive control (NMPC) and nonlinear moving horizon estimation (MHE) formulations for controlling the crystal size and shape distribution in a batch crystallization process. MHE is used to estimate unknown states and parameters prior to solving the NMPC problem. Combining these two formulations for a batch process, we obtain an expanding horizon estimation problem and a shrinking horizon model predictive control problem. The batch process has been modeled as a system of differential algebraic equations (DAEs) derived using the population balance model (PBM) and the method of moments. Therefore, the MHE and NMPC formulations lead to DAE-constrained optimization problems that are solved by discretizing the system using Radau collocation on finite elements and optimizing the resulting algebraic nonlinear problem using Ipopt. The performance of the NMPC–MHE approach is analyzed in terms of setpoint change, system noise, and model/plant mismatch, and it is shown to provide better setpoint tracking than an open-loop optimal control strategy. Furthermore, the combined solution time for the MHE and the NMPC formulations is well within the sampling interval, allowing for real world application of the control strategy.

Batch-to-Batch Finite-Horizon LQ Control for Unknown Discrete-Time Linear Systems Via Stochastic Extremum Seeking

We employ our recent discrete-time stochastic averaging theorems and stochastic extremum seeking to iteratively (batch-to-batch) optimize open-loop control sequences for unknown but reachable discrete-time linear systems with a scalar input and without known system dimension, for a cost that is quadratic in the measurable output and the input. First, for a multivariable gradient-based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open-loop control sequence. Second, to remove the convergence rate's dependence on the Hessian matrix of the cost function, which is unknown since the system's model (the system matrices $(A,B,C)$ ) is unknown, we develop a multivariable discrete-time Newton-based stochastic extremum seeking method, design the Newton-based algorithm for the iteration of the input sequence, and prove local exponential convergence to the optimal input sequence. Finally, two simulation examples are given to illustrate the effectiveness of the two methods.

A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles

We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.

Offset Risk Minimization for Open-loop Optimal Control of Oil Reservoirs

Simulation studies of oil field water flooding have demonstrated a significant potential of optimal control technology to improve industrial practices. However, real-life applications are challenged by unknown geological factors that make reservoir models highly uncertain. To minimize the associated financial risks, the oil literature has used ensemble-based methods to manipulate the net present value (NPV) distribution by optimizing sample estimated risk measures. In general, such methods successfully reduce overall risk. However, as this paper demonstrates, ensemble-based control strategies may result in individual profit outcomes that perform worse than real-life dominating strategies. This poses significant financial risks to oil companies whose main concern is to avoid unacceptable low profits. To remedy this, this paper proposes offset risk mimimization. Unlike existing methodology, the offset method uses the NPV offset distribution to minimize risk relative to a competing reference strategy. Open-loop simulations of a 3D two-phase synthetic reservoir demonstrate the potential of offset risk minimization to significantly improve the worst case profit offset relative to real-life best practices. The results suggest that it may be more relevant to consider the NPV offset distribution than the NPV distribution when minimizing risk in production optimization.

Impact of Climate Model Parametric Uncertainty in an MPC Implementation of the DICE Integrated Assessment Model

Integrated assessment models (IAMs) are a key tool in studying the interdependence of the global economy and the climate system. For example, the dollar value of carbon dioxide emissions due to anthropogenic climate damages, known as the social cost of carbon dioxide (SC-CO2), can be computed via the widely used DICE (Dynamic Integrated model of Climate and the Economy) IAM by solving an open-loop optimal control problem. The results of such an open-loop decision-making strategy, however, do not fully reflect the impacts of uncertainty in the dynamic response of the global climatic system to radiative forcing. In this paper, we propose an implementation of the DICE IAM based on model predictive control (MPC). This MPC-based approach draws a clear distinction between the climate model used by DICE for mitigation planning purposes, and the “true” global climate herein captured by a low-order emulation of a model drawn from a state-of-the-art climate model ensemble (CMIP5, the fifth phase of the Coupled Model Intercomparison Project). The closed-loop control methodology in this paper thereby quantifies the impact of parametric climate model uncertainty (plant-model mismatch) on estimates of the SC-CO2 obtained from DICE.

Optimal energy-water management in urban residential buildings through grey water recycling

Energy and water are inseparable resources whose management in urban residential buildings is important. Continuing urbanization in developing nations is increasing the demand yet the supply is inadequate or non-existent. Decentralized urban water recycling systems are an alternative source of water that could relieve the demand from public utilities. However, there are social, economic, environmental and technological factors that affect the uptake of these systems. Although advanced water treatment technologies for decentralized systems have been developed, there are challenges in their optimal operation. This paper introduces the open loop optimal control and closed-loop model predictive control (MPC) strategies aimed at ensuring safe and reliable operation of a grey water recycling system at building level. Both controllers have shown their ability in efficiently operate the system leading to water conservation and energy cost savings. Each of these controllers has its strengths in terms of cost, ease of implementation and robustness and they should be adopted according to specific application. Their adoption can greatly improve energy and water security in urban households, reduce their demand and wastewater. Technology alone cannot solve resource insecurity, and therefore, appropriate policies, regulations, incentives and public awareness should be implemented to support such novel technologies.

Model predictive control of heat pump water heater-instantaneous shower powered with integrated renewable-grid energy systems

Energy and water are two inseparable resources that are crucial for human survival, yet, most developing nations are struggling to reliably provide them to the population especially in rapidly growing urban areas. Increasing demand is forcing governments, organizations and private sectors to encourage end-users to increase efficiency and conservation measures for these resources. Water heating is one of the largest energy users in residential buildings thus has a huge potential to improve the efficiency of both energy and water. In this regard, heat pump water heaters (HPWHs) have been found to improve energy efficiency while providing domestic hot water. However, impediments such as optimal operation, integration and high initial cost especially in developing nations hinder their uptake. Further, since they are normally centrally located in a house, there are water and associated energy losses during hot water conveyance to the end-use, as the once hot water in the pipes that cooled off has to be poured away while end-user awaits for hot water. Therefore, this paper advances the previously developed open loop optimal control model by using the closed-loop model predictive control (MPC) to operate a HPWH and instantaneous shower powered using integrated renewable energy systems. This control strategy has the benefit of robustly and reliably dealing with disturbances that are present in the system as well as turnpike phenomenon. It has the potential to save 32.24% and 19l of energy and water in a day respectively, while also promising lower energy and water bills to the end users. In addition, there is revenue benefit through the sale of excess renewable energy back to the grid through an appropriate feed-in tariff. Life cycle cost (LCC) analysis is conducted to determine the total cost of setting up and operating the system over its life, which shows that the benefits would pay back the cost of the system even before half of its life elapses. This control strategy of both hot water devices powered using integrated renewable systems is suitable for peri-urban home owners.

Mean-Field stochastic Linear Quadratic optimal control problems: Open-loop solvabilities

This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Then, it is proved that the uniform convexity of the cost functional is equivalent to the solvability of two coupled differential Riccati equations and the unique open-loop optimal control admits a state feedback representation in the case that the cost functional is uniformly convex. Finally, some examples are presented to illustrate the theory developed.

Automated identification of a complex storage model and hardware implementation of a model-predictive controller for a cooling system with ice storage

Future sustainable energy systems could increase the share of energy converted from fluctuating renewable energy sources by intelligent model-based predictive control of cooling systems with thermal energy storage. This study investigated an experimental cooling system comprising a compression chiller and an ice storage. A runtime-efficient predictive model for partial charge and discharge of ice storage was derived. In addition, techniques for automatic model determination and adaptation were introduced and examined. The experimental setup involved the development and implementation of a model-predictive controller (MPC) to minimize operating expenses under dynamic electricity pricing based on a forward dynamic programming algorithm. The objective function included energy charges, compressor start-up costs, and terminal costs that depended on the state of charge and state of the chiller at the end of the optimization horizon. Three examples of cases validated and compared the advantages of the MPC over an open-loop (day ahead) optimal control concept. The cases examined the influence of temperature and load forecast inaccuracy, and investigated the coping mechanism of the system to sudden updates involving price and temperature predictions. The findings illustrated that the MPC achieved significant savings of operating expenses when compared with the open-loop optimal control concept.

Distributed Model Predictive Control for Heterogeneous Vehicle Platoons Under Unidirectional Topologies

This paper presents a distributed model predictive control (DMPC) algorithm for heterogeneous vehicle platoons with unidirectional topologies and a priori unknown desired set point. The vehicles (or nodes) in a platoon are dynamically decoupled but constrained by spatial geometry. Each node is assigned a local open-loop optimal control problem only relying on the information of neighboring nodes, in which the cost function is designed by penalizing on the errors between predicted and assumed trajectories. Together with this penalization, an equality based terminal constraint is proposed to ensure stability, which enforces the terminal states of each node in the predictive horizon equal to the average of its neighboring states. By using the sum of local cost functions as a Lyapunov candidate, it is proved that asymptotic stability of such a DMPC can be achieved through an explicit sufficient condition on the weights of the cost functions. Simulations with passenger cars demonstrate the effectiveness of proposed DMPC.

Approximation of closed-loop prediction for dynamic real-time optimization calculations

Abstract Dynamic real-time optimization (DRTO) is an extension of the traditional steady-state RTO paradigm to account for process dynamics in the RTO calculations. This paper presents methods for approximating closed-loop dynamic predictions within DRTO calculations for processes regulated under constrained model predictive control (MPC). Three approximation approaches are formulated and analyzed – hybrid, bilevel and input clipping formulations. The hybrid formulation involves application of rigorous closed-loop prediction over a limited DRTO horizon, followed by open-loop optimal control. In the bilevel formulation, only a single MPC optimization subproblem is embedded, whereas the input clipping approach is formulated using an unconstrained MPC algorithm with an input saturation mechanism applied over the DRTO horizon. The relative performance of the proposed approximation approaches is illustrated through two case study applications, the second of which involves economically optimal polymer grade transitions. Excellent closed-loop approximation is achieved without significant loss of prediction accuracy.

Stochastic Linear Quadratic Optimal Control Problems in Infinite Horizon

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with constant coefficients. It is proved that the non-emptiness of the admissible control set for all initial state is equivalent to the $L^2$-stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation (ARE, for short). Different from the finite horizon case, it is shown that both the open-loop and closed-loop solvabilities of the LQ problem are equivalent to the existence of a static stabilizing solution to the associated generalized ARE. Moreover, any open-loop optimal control admits a closed-loop representation. Finally, the one-dimensional case is worked out completely to illustrate the developed theory.