Open-loop Optimal Control Problem Research Articles

Asynchronous switching and sparse discretization with time scaling for control vector parameterization optimal control

This work introduces a novel approach for solving open-loop Optimal Control Problems (OCPs) using multiple time grids, sparse discretization and a time scaling transformation. Control Vector Parameterization (CVP) method parameterizes control variables to create a finite-dimensional problem. The Variable Time Nodes Control Vector Parameterization (VTNCVP) discretization strategy allows each control component an independent time grid, offering enhanced input design flexibility and potentially yielding improved outcomes. A novel time scaling technique for OCPs with multiple time grids is presented that is easier to comprehend than existing methods and can circumvent some numerical difficulties. Additionally, Multiple Grid (MG)–Sparse Variable Time Nodes (MG-SVTN) is presented. This novel discretization strategy employs asynchronous switching with user-specified numbers of subdivisions for input controls, potentially yielding comparable objective function values to VTNCVP while reducing the dimensionality of decision variables. Two example problems with path constraints are solved using the methods.

Control Co-Design Optimization of Spacecraft Trajectory and System for Interplanetary Missions

This paper develops a control co-design (CCD) framework to simultaneously optimize the spacecraft’s trajectory and onboard system (rocket engine) and quantify its benefit. An open-loop optimal control problem (two-finite burn Mars missions) is used as the benchmark, and the engine design considers the combustion equilibrium and nozzle geometry. The objective function is the fuel burn. The design variables are the trajectory control parameters (such as burn times, burn directions, and time of flight), initial fuel mass, and engine design parameters (such as throat area, mixture ratio, and chamber pressure). The constraints include the final velocities and positions of spacecraft. Single-point optimizations are conducted for three departure dates in May, July, and September 2020. A multipoint optimization is also performed to balance the engine performance for these dates with 49 design variables and 20 constraints. It is found that the CCD optimizations exhibit 22–28% more fuel burn reduction than the trajectory-only optimization with fixed engine parameters and 16–20% more fuel burn reduction than the decoupled trajectory-engine optimization. The proposed CCD optimization framework can be extended to more spacecraft trajectory control parameters and onboard systems and has the potential to design more efficient spacecraft missions.

Energy- and time-optimal reconfiguration of spacecraft clusters with collision avoidance

Spacecraft cluster reconfiguration is one of the enabling technologies to ensure non-traditional attributes of distributed space systems. This work treats energy- and time-optimal reconfiguration problems for both circular and eccentric reference orbits. Furthermore, typical local and coupling constraints have been taken into account. Particularly, focus is given to two coupling constraints: final configuration constraints and collision avoidance constraints. For final configuration constraints, a parameterization method is applied to ensure that the reconfiguration problem can be solved as only one optimization problem, rather than a large number of optimization problems resulting from the traditional discretization method. A generalized formulation is proposed for non-convex collision avoidance constraints, which are then convexified via linearization and convex restriction technology. This method provides the affine approximation as a special case. After incorporating above constraints, the reconfiguration problem is formulated as an open-loop optimal control problem, which is solved via the Gauss pseudospectral method (GPM). By virtue of elegant features of GPM, those solutions can serve as a counterpart and stepping stone for a distributed implementation of reconfiguration algorithms. Various simulations demonstrate that minimum-energy/time cluster reconfiguration problems with collision avoidance for circular and eccentric reference orbits can be solved effectively and efficiently using GPM.

Data-driven dynamic relatively optimal control

We show how the recent works on data-driven open-loop minimum-energy control for linear systems can be exploited to obtain closed-loop control laws in the form of linear dynamic controllers that are relatively optimal. Besides being stabilizing, they achieve the optimal minimum-energy trajectory when the initial condition is the same as the open-loop optimal control problem. The order of the controller is N−n, where N is the length of the optimal open-loop trajectory, and n is the order of the system. The same idea can be used for obtaining a relatively optimal controller, entirely based on data, from open-loop trajectories starting from up to n linearly independent initial conditions.

Harmonizing the use of optimization and feedback in process operations and control

This paper addresses how to combine the two main engineering design tools, optimization and feedback, to obtain high performance process control or process operation systems. We discuss the historical development of process control, where feedback dominated the early practical designs, and optimization was then incorporated much later through technologies like model predictive control. In chemical production scheduling, on the other hand, optimization had a strong early influence on the problem formulation, and feedback has only recently made an appearance. The recent developments in both process control and scheduling are illustrated with specific examples emerging from this series of FOCAPO/CPC meetings. The paper next presents recent theoretical developments in nominal and stochastic model predictive control. The closed-loop properties that arise from these different open-loop optimal control problems are then compared. The paper closes with some discussion of when the improvements of the closed-loop properties are worth the added complexity of the stochastic optimal control problem.

Linear Tracking MPC for Nonlinear Systems—Part II: The Data-Driven Case

In this article, we present a novel data-driven model predictive control (MPC) approach to control unknown nonlinear systems using only measured input–output data with closed-loop stability guarantees. Our scheme relies on the data-driven system parameterization provided by the fundamental lemma of Willems <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> We use new input–output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference while satisfying polytopic input constraints. As intermediate results of independent interest, we extend the fundamental lemma to affine systems and we derive novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

Control Co-Design of Lithium-Ion Batteries for Enhanced Fast-Charging and Cycle Life Performances

Abstract As enablers of electric vehicles, lithium-ion batteries are drawing much attention for their high energy density and low self-discharge. However, “range anxiety” has remained a significant hindrance to its further development. As an alternative to increasing capacity, fast charging seems a reasonable solution. However, challenges remain due to the conflict between high charging rate and excessive capacity loss. In the past, enormous efforts have been carried out to resolve the dispute between high charging rates and large capacity losses by either improving the battery design or optimizing the charging/discharging protocols. In contrast, this study proposes a novel control co-design framework with adaptive surrogate modeling to address the challenges and to generate the systematic optimal battery design and the corresponding charging protocol simultaneously. The proposed method is ideal for lithium-ion battery systems to offer the improved performances as compared with traditional sequential optimization approaches due to the integration of strong coupling effects between electrode design and control optimization. The integrated adaptive surrogate modeling technique allows model reduction for efficient optimal control and simulation solutions. Meanwhile, it preserves an accurate mapping from the first-principle model to the reduced-order model. A hybrid model like this captures the multiscale nature of the cell, that is, micro-scale parameters affect the macro-scale behavior. It reduces the computational cost significantly. The battery co-design problem is formulated as a nested problem, where the inner-loop solves an open-loop optimal control problem and the outer-loop optimizes the plant design variables. The results show that system-level optimal design can be obtained for minimized charging time at various levels of health requirement.

Behavioral cloning for fatigue-oriented wind turbine optimal predictive control

As wind energy production keeps on increasing, it is necessary to optimize the fatigue/performance trade-off of operating wind turbines. However, considering fatigue directly in optimal control needs to be carefully done because its faithful model does not necessarily fit standard forms commonly required by general purpose solvers. It was recently shown that a signal variance and its induced fatigue can be statistically related. This suggests that a fatigue-related cost function can be expressed as a non-quadratic cost function, and used in an open-loop optimal control problem. However, it was also shown that such cost function becomes very efficient only for relatively long prediction horizons. This makes the on-line solution of the underlying open-loop optimal control problem computationally demanding and undermines the real-time applicability of associated MPC schemes. In this paper, a behavioral learning solution is proposed to imitate the optimal controller obtained through open-loop, long prediction horizon-based optimization. The behavioral cloning controller is compared to a finely tuned quadratic MPC regarding its ability to reduce fatigue. Preliminary results show that the proposed solution enables significant fatigue reduction on a broad range of realistic disturbances while being realtime implementable.

Eco-Driving Operation of Connected Vehicle With V2I Communication Among Multiple Signalized Intersections

Eco-driving through multiple intersections has significant fuel benefit for road transportation. Most existing studies assume that vehicles travel at a constant speed between intersections, which naturally leads to large errors in fuel prediction and trajectory planning. This article focuses on eco-driving operation through multiple signalized intersections considering more realistic powertrain dynamics, which directly contain the engine and the transmission as well as aerodynamic drag, rolling resistance, and so on. This feature enables explicit descriptions of dynamic behaviors and fuel characteristics in acceleration, deceleration, and constant speed driving. An open-loop optimal control problem (OCP) is formulated to minimize fuel consumption between two red-signalized intersections, and the two- or three-stage operation rules are then proposed to approximate its optimal solution. The eco-driving of passing multiple intersections is then numerically solved by combining the two- or three-stage driving rule and the Dijkstra algorithm. This method can consider the acceleration and deceleration processes at each intersection and achieve improved fuel economy.

Towards Risk-aware Machine Learning Supported Model Predictive Control and Open-loop Optimization for Repetitive Processes

Abstract Many processes operate repetitively, for example batch processes in biotechnology or chemical engineering. We propose a method for risk-aware run-to-run optimization and model predictive control of repetitive processes with uncertain models. The goal is to increase the performance as the number of runs increases by improving the model despite limited measurements while considering model uncertainty and avoiding uncertain areas. The method uses a gray-box model, i.e. a model formed by a first principle and a machine learning component, in this case an artificial neural network. The model uncertainty might be large, particularly in the first runs, where only a few measurements are available. We propose to quantify this uncertainty using Bayesian inference. This is in turn reflected by a risk measure entering an open-loop optimal control problem and a shrinking-horizon Model Predictive Controller as a constraint to limit control and exploitation in high risk areas. We show that using this risk measure we are able to efficiently reach high process performance. The proposed method is tested in simulations on two biotechnological fed-batch processes.

Perfect Torque Compensation of Planar 5R Parallel Robot in Point-to-Point Motions, Optimal Control Approach

SUMMARYIn this paper, a new approach is presented for perfect torque compensation of the robot in point-to-point motions. The proposed method is formulated as an open-loop optimal control problem. The problem is defined as optimal trajectory planning with adjustable design parameters to compensate applied torques of a planar 5R parallel robot for a given task, perfectly. To illustrate the effectiveness of the approach, the obtained optimal path is used as the reference command in the experiment. The experimental outputs show that the performance index has been reduced by over 80% compared to the typical design of the robot.

Control Node Selection Algorithm for Nonlinear Dynamic Networks

The coupled problems of selecting control nodes and designing control actions for nonlinear network dynamics are fundamental scientific problems with applications in many diverse fields. These problems are thoroughly studied for linear dynamics. However, in spite of a number of open research questions, methods for nonlinear network dynamics are less developed. As observed by various studies, the prevailing graph-based controllability approaches for selecting control nodes might result in significantly suboptimal control performance for nonlinear dynamics. We present a new, intuitive, and simple method for simultaneous control node selection and control sequence design for complex networks with nonlinear dynamics. The method is developed by incorporating the control node selection problem into an open-loop optimal control problem and by approximately solving the resulting mixed-integer optimization problem using a mesh adaptive direct search method. The developed framework is numerically robust and can deal with stiff networks and with control and actuator constraints. Good numerical performance of the method is demonstrated by testing it on prototypical Duffing oscillator and associative memory networks. The developed codes that can easily be adapted to models of other complex systems are available online.

Trajectory Planning of Quadrotor Systems for Various Objective Functions

SUMMARYQuadrotors are unmanned aerial vehicles with many potential applications ranging from mapping to supporting rescue operations. A key feature required for the use of these vehicles under complex conditions is a technique to analytically solve the problem of trajectory planning. Hence, this paper presents a heuristic approach for optimal path planning that the optimization strategy is based on the indirect solution of the open-loop optimal control problem. Firstly, an adequate dynamic system modeling is considered with respect to a configuration of a commercial quadrotor helicopter. The model predicts the effect of the thrust and torques induced by the four propellers on the quadrotor motion. Quadcopter dynamics is described by differential equations that have been derived by using the Newton–Euler method. Then, a path planning algorithm is developed to find the optimal trajectories that meet various objective functions, such as fuel efficiency, and guarantee the flight stability and high-speed operation. Typically, the necessary condition of optimality for a constrained optimal control problem is formulated as a standard form of a two-point boundary-value problem using Pontryagin’s minimum principle. One advantage of the proposed method can solve a wide range of optimal maneuvers for arbitrary initial and final states relevant to every considered cost function. In order to verify the effectiveness of the presented algorithm, several simulation and experiment studies are carried out for finding the optimal path between two points with different objective functions by using MATLAB software. The results clearly show the effect of the proposed approach on the quadrotor systems.

Strategic evaluation of different direct nucleation control approaches for controlling batch cooling crystallisation via simulation and experimental case studies

This paper presents a systematic evaluation of direct nucleation control (DNC) approach of controlling batch cooling crystallization systems. DNC, as named is aimed to control nucleation events to achieve the desired crystal size distribution (CSD) in a model free approach that requires no prior knowledge on nucleation or growth kinetics of the system. The performance and robustness of proposed DNC approaches were examined through computer simulations and laboratory experiments. Computer simulations were done in two phases. In the first phase, detailed mathematical models covering reaction kinetics and heat mass balances were developed for a batch cooling crystallization system of Paracetamol in water. Based on these models, rigorous simulation programs were developed in MATLAB®. This mechanistic model was used to solve an open loop optimal control problem of maximizing the mean crystal length within a fixed batch time. The successive quadratic programming solution provides the optimal temperature trajectory, corresponding number of counts and the total length. In the second phase, treating the MATLAB® model as the real processes, the performance of DNC was evaluated by setting the optimal number of counts as the target and applying different DNC strategies: changing DNC structures/algorithm; varying the acceptable range of counts limits; increasing and decreasing the limit of desired number of counts; and changing the heating and cooling rates. The results demonstrate that the fast/slow cooling-fast/slow heating approach yields the best result in terms of reduced number of temperature loops that produced the desired counts in smaller batch duration than the optimal one. Later, the simulated results were justified through laboratory experiments.

Integrated missile guidance and control using optimization-based predictive control

This paper focuses on the design and implementation of optimization-based predictive control for the problem of missile interception. Due to the inherent nonlinearities of the missile–target dynamics or even constraints, it is usually difficult to design a high-accuracy and high-efficiency control algorithm. A nonlinear receding horizon pseudospectral control (RHPC) scheme is constructed and applied to generate the optimal control command. The problem of state estimation, in the presence of measurement noise, is solved by implementing a moving horizon estimation (MHE) algorithm. Since the RHPC and MHE algorithms solve the online open-loop optimal control problem at each sampling instant, the computational cost associated with them can be high. In order to decrease the computational demand due to the optimization process, a recently proposed nonlinear programming sensitivity-based algorithm is used and embedded in the optimization framework. Numerical simulations and analysis are presented to demonstrate the effectiveness of the proposed control scheme.

Functional Nonlinear Model Predictive Control Based on Adaptive Dynamic Programming.

This paper presents a functional model predictive control (MPC) approach based on an adaptive dynamic programming (ADP) algorithm with the abilities of handling control constraints and disturbances for the optimal control of nonlinear discrete-time systems. In the proposed ADP-based nonlinear MPC (NMPC) structure, a neural-network-based identification is established first to reconstruct the unknown system dynamics. Then, the actor-critic scheme is adopted with a critic network to estimate the index performance function and an action network to approximate the optimal control input. Meanwhile, as the MPC strategy can effectively determine the current control by solving a finite horizon open-loop optimal control problem, in the proposed algorithm, the infinite horizon is decomposed into a series of finite horizons to obtain the optimal control. In each finite horizon, the finite ADP algorithm solves the optimal control problem subject to the terminal constraint, the control constraint, and the disturbance. The uniform ultimate boundedness of the closed-loop system is verified by the Lyapunov approach. Finally, the ADP-based NMPC is conducted on two different cases and the simulation results demonstrate the quick response and strong robustness of the proposed method.

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.

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.

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.

Research into 2D Dynamics and Control of Small Oscillations of a Cross-Beam during Transportation by Two Overhead Cranes

A new mathematical model of a 3DOF 2D mechanical system “transported cross-beam, two moving bridge cranes” has been proposed. Small system oscillations have been derived through the introduction of Lagrange equations. The numerical estimation of 3DOF system motion has been carried out with equation-based Modelica language. The present article uses the Lagrange method and numerical and optimization methods, realized with JModelica.org and Optimica freeware. The absolute swaying of the cross-beam with respect to the displacement of the two moving bridge cranes was estimated. The phase portraits of the 3DOF system for linear and angular coordinates were presented. An open loop optimal control problem was posed for the motion of the bridge cranes. A “bang-bang” control strategy was implemented for the derivation of an optimal control solution, which enables the travel of two bridge cranes at a prescribed distance for minimum time and minimum swaying of a heavy cross-beam. The derived results of the numerical simulation can be easily practically realized by crane operators with good agreement with simple engineering estimations. The proposed control strategy enables synchronous motion of two bridge cranes with a cross-beam that practically solves the posed problem of unwanted excessive oscillations of a heavy cross-beam during transportation.