Computation Of Optimal Control Research Articles

The Use of Slack Variables in the Adjoint Method Handling Inequality Constraints in Optimal Control and the Application to Tumor Drug Dosage

Abstract In this article a modified gradient approach, based on the adjoint method, is introduced, to deal with optimal control problems involving inequality constraints. So far, the only way to incorporate inequality constraints in the adjoint approach, is to introduce additional penalty terms in the cost functional. However, this may distort the optimal control due to weighting factors required for these terms and raise serious concerns about the magnitude of the weighting factors. The method in this article avoids penalty functions and can be used for the iterative computation of optimal controls. In order to demonstrate the key idea, first, a static optimization problem in the Euclidean space is considered. Second, the presented approach is applied to the tumor anti-angiogenesis optimal control problem in medicine, which addresses an innovative cancer treatment approach that aims to inhibit the formation of the tumor blood supply. The tumor anti-angiogenesis optimal control problem with free final time involves inequality and final constraints for control and state variables and is solved by a modified adjoint gradient method introducing slack variables. It has to be emphasized that the novel formulation and the special use of slack variables in this article delivers high accurate solutions without distorting the optimal control.

VAN-DAMME: GPU-accelerated and symmetry-assisted quantum optimal control of multi-qubit systems

We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains. Program summaryProgram Title: VAN-DAMMECPC Library link to program files::https://doi.org/10.17632/zcgw2n5bjf.1Licensing provisions: GNU General Public License 3Programming language: C++ and CUDANature of problem: The VAN-DAMME software package utilizes GPU-accelerated routines and new algorithmic improvements to compute optimized time-dependent magnetic fields that can drive a system from a known initial qubit configuration to a specified target state with a large (≈1) transition probability.Solution method: Quantum control, GPU acceleration, analytic gradients, matrix exponential, and gradient ascent optimization.

Modeling, analyzing and simulating the Measles transmission dynamics through efficient computational optimal control technique

Modeling, analyzing and simulating the Measles transmission dynamics through efficient computational optimal control technique

MISTER-T: An open-source software package for quantum optimal control of multi-electron systems on arbitrary geometries

We present an open-source software package, MISTER-T (Manipulating an Interacting System of Total Electrons in Real-Time), for the quantum optimal control of interacting electrons within a time-dependent Kohn-Sham formalism. In contrast to other implementations restricted to simple models on rectangular domains, our method enables quantum optimal control calculations for multi-electron systems (in the effective mass formulation) on nonuniform meshes with arbitrary two-dimensional cross-sectional geometries. Our approach is enabled by forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency. We provide several examples of the versatility and efficiency of the MISTER-T code in handling complex geometries and quantum control mechanisms. The capabilities of the MISTER-T code provide insight into the implications of varying propagation times and local control mechanisms to understand a variety of strategies for manipulating electron dynamics in these complex systems. Program summaryProgram Title: MISTER-TCPC Library link to program files:https://doi.org/10.17632/psymy4ddnw.1Licensing provisions: GNU General Public License 3Programming language: MATLABSupplementary material: animated movies of total electron densities under the influence of optimal control fields for (1) an asymmetric double-well potential for long propagation times, (2) an asymmetric double-well potential for short propagation times, and (3) a triple-well potential with a position-dependent effective mass.Nature of problem: The MISTER-T code solves quantum optimal control problems for interacting electrons within a time-dependent Kohn-Sham formalism. It can handle two-dimensional systems with arbitrary cross-sectional geometries within the effective mass formulation. The user-friendly code uses forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency.Solution method: iterative solution of the quantum optimal control equations using finite element methods, effective mass formulation, pseudoskeleton decomposition, sparse matrix linear algebra, and nonuniform fast Fourier transforms.

Determining Fast Battery Charging Profiles Using an Equivalent Circuit Model and a Direct Optimal Control Approach

This work used an electrical equivalent circuit model combined with a temperature model and computational optimal control methods to determine minimum time charging profiles for a lithium–ion battery. To effectively address the problem, an optimal control problem formulation and direct solution approach were adopted. The results showed that, in most cases studied, the solution to the battery’s fast-charging problem resembled the constant current–constant voltage (CC-CV) charging protocol, with the advantage being that our proposed approach optimally determined the switching time between the CC and CV phases, as well as the final time of the charging process. Considering path constraints related to the terminal voltage and temperature gradient between the cell core and case, the results also showed that additional rules could be incorporated into the protocol to protect the battery against under/over voltage-related damage and high-temperature differences between the core and its case. This work addressed several challenges and knowledge gaps, including emulating the CC-CV protocol using a multi-phase optimal control approach and direct collocation methods, and improving it by including efficiency and degradation terms in the objective function and safety constraints. To the authors’ knowledge, this is the first time the CC-CV protocol has been represented as the solution to a multi-phase optimal control problem.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

Online Inverse Optimal Control for Time-Varying Cost Weights.

Inverse optimal control is a method for recovering the cost function used in an optimal control problem in expert demonstrations. Most studies on inverse optimal control have focused on building the unknown cost function through the linear combination of given features with unknown cost weights, which are generally considered to be constant. However, in many real-world applications, the cost weights may vary over time. In this study, we propose an adaptive online inverse optimal control approach based on a neural-network approximation to address the challenge of recovering time-varying cost weights. We conduct a well-posedness analysis of the problem and suggest a condition for the adaptive goal, under which the weights of the neural network generated to achieve this adaptive goal are unique to the corresponding inverse optimal control problem. Furthermore, we propose an updating law for the weights of the neural network to ensure the stability of the convergence of the solutions. Finally, simulation results for an example linear system are presented to demonstrate the effectiveness of the proposed strategy. The proposed method is applicable to a wide range of problems requiring real-time inverse optimal control calculations.

Identification of the object as a system

It has been found that the influence of living systems on a technical system within a complex system reduces the accuracy of controlling such a system. The article proposes a method for identifying an object to improve the accuracy of managing a complex system. The calculation of optimal control for a complex system is performed.

APPROACH TO EFFECTIVE METHODS FOR SOLUTION OF MULTIDIMENSIONAL EICONAL EQUATIONS

The article presents an extended method of solving the eiconal equation in four-dimensional space with weak deformations. The eiconal equation combines wave optics with geometric optics and has various physical interpretations, including the task of finding the shortest paths and calculation of electromagnetic or gravitational potentials. The proposed method extends the sphere tracing technique to spaces of many dimensions with deformations and demonstrated for the problem in the space of four dimensions. The method uses implicit functions for describing the boundaries of objects built from a finite or an infinite number of multidimensional primitives. Nonlinear sphere tracing is achieved by generation at each trace step ordinary (multidimensional) differential equations of the first order using a hybrid solution method combining the Euler method, when the sphere is close to the boundary, with higher order methods, when the sphere is far from the boundaries. Influence of non-linear transformations the tracing process is implemented using the Jacobian deformation matrix. The approach is implemented as a shader program in the GLSL language, and the effect of nonlinear transformations is determined using of the transformation parameter that affects the Jacobian matrix. The computational performance of the method is evaluated through average and maximum frame rates for different parameter values. The proposed approach can find application in such fields, as computer graphics, time-dependent computer tomography, seismic tomography, astrophysical modeling and optimal control.

TRAVOLTA: GPU acceleration and algorithmic improvements for constructing quantum optimal control fields in photo-excited systems

We present an open-source software package, TRAVOLTA (Terrific Refinements to Accelerate, Validate, and Optimize Large Time-dependent Algorithms), for carrying out massively parallelized quantum optimal control calculations on GPUs. The TRAVOLTA software package is a significant overhaul of our previous NIC-CAGE algorithm and also includes algorithmic improvements to the gradient ascent procedure to enable faster convergence. We examine three different variants of GPU parallelization to assess their performance in constructing optimal control fields in a variety of quantum systems. In addition, we provide several examples with extensive benchmarks of our GPU-enhanced TRAVOLTA code to show that it generates the same results as previous CPU-based algorithms but with a speedup that is more than ten times faster. Our GPU enhancements and algorithmic improvements enable large quantum optimal control calculations that can be efficiently and routinely executed on modern multi-core computational hardware. Program summaryProgram Title: TRAVOLTACPC Library link to program files:https://doi.org/10.17632/grwppm37rn.1Licensing provisions: GNU General Public License 3Programming language: C++, openBLAS, and CUDASupplementary material: Brief review of LU decomposition, raw numerical values used to generate Fig. 6 in the main text, and input examples for the TRAVOLTA software package.Nature of problem: The TRAVOLTA software package utilizes GPU accelerated routines and new algorithmic improvements to compute optimized electric fields that can drive a system from a known initial vibrational eigenstate to a specified final quantum state with a large (≈1) transition probability.Solution method: Quantum control, GPU acceleration, analytic gradients, Crank-Nicolson propagation, and gradient ascent optimization.

Accelerating quantum optimal control of multi-qubit systems with symmetry-based Hamiltonian transformations

We present a novel, computationally efficient approach to accelerate quantum optimal control calculations of large multi-qubit systems used in a variety of quantum computing applications. By leveraging the intrinsic symmetry of finite groups, the Hilbert space can be decomposed and the Hamiltonians block diagonalized to enable extremely fast quantum optimal control calculations. Our approach reduces the Hamiltonian size of an n-qubit system from 2n×2n to O(n×n) or O((2n/n)×(2n/n)) under Sn or Dn symmetry, respectively. Most importantly, this approach reduces the computational runtime of qubit optimal control calculations by orders of magnitude while maintaining the same accuracy as the conventional method. As prospective applications, we show that (1) symmetry-protected subspaces can be potential platforms for quantum error suppression and simulation of other quantum Hamiltonians and (2) Lie–Trotter–Suzuki decomposition approaches can generalize our method to a general variety of multi-qubit systems.

Optimization of Control and Parameters in Systems with Phase Constraints

For an optimal control problem with constraints on phase coordinates, an iterative method for finding a numerical solution is considered, based on reduction to a finite-dimensional problem and applying to it a sequential linearization algorithm using a modified Lagrange function. To solve linear auxiliary problems at iterations of the method, the reduced gradient method is used. The efficiency of taking into account phase constraints in the calculation of optimal control is illustrated by the numerical solution of problems from the field of aerodynamics and robotics.

Iterative Maneuver Optimization in a Transverse Gust Encounter

This paper presents a framework based on either iterative simulation or iterative experimentation for constructing an optimal, open-loop maneuver to regulate the aerodynamic force on a wing in the presence of a known flow disturbance. The authors refer to the method as iterative maneuver optimization and apply it in this paper to regulate lift on a pitching wing during a transverse gust encounter. A candidate maneuver is created by performing an optimal control calculation on a surrogate model of the wing–gust interaction. Execution of the proposed maneuver in a high-fidelity simulation or experiment provides an error signal based on the difference between the force predicted by the surrogate model and the measured force. The error signal provides an update to the reference signal used by the surrogate model for tracking. A new candidate maneuver is calculated such that the surrogate model tracks the reference force signal, and the process repeats until the maneuver adequately regulates the force. The framework for iterative maneuver optimization is tested on a discrete vortex model as well as in experiments in a water towing tank. Experimental results show that the proposed framework generates a maneuver that reduces the magnitude of lift overshoot by 92% for a trapezoidal gust with peak velocity equal to approximately 0.7 times the freestream flow speed.

Analysis of Optimal Deep Learning Approach for Battery Health Condition Monitoring inElectric Vehicle

Compared with other commonly used batteries, lithium-ion batteries are featured by high energy density, high power density, long service life and environmental friendliness and thus have found wide application in the area of consumer electronics. The narrow area in which lithium-ion batteries operate with safety and reliability necessitates the effective control and management of battery management system. This present paper, through the analysis of literature and in combination with our practical experience, gives a brief introduction to the composition of the battery management system (BMS). First-principles models that incorporate all of the key physics that affect the internal states of a lithium-ion battery are in the form of coupled nonlinear PDEs. While these models are very accurate in terms of prediction capability, the models cannot be employed for on-line control and monitoring purposes due to the huge computational cost. A reformulated model is capable of predicting the internal states of battery with a full simulation running in milliseconds without compromising on accuracy. This paper demonstrates the feasibility of using this reformulated model for control-relevant real-time applications. The reformulated model is used to compute optimal protocols for battery operations to demonstrate that the computational cost of each optimal control calculation is low enough to be completed within the sampling interval in model predictive control (MPC). Observability studies are then presented to confirm that this model can be used for state-estimation-based MPC. A moving horizon estimator (MHE) technique was implemented due to its ability to explicitly address constraints and nonlinear dynamics. The MHE uses the reformulated model to be computationally feasible in real time. The feature of reformulated model to be solved in real time opens up the possibility of incorporating detailed physics-based model in battery management systems (BMS) to design and implement better monitoring and control strategies

Computational Optimal Impact Angle Control Guidance Laws Weighted by Arbitrary Functions

This paper presents a computational impact angle control guidance law based on the energy cost weighted by arbitrary functions in order to shape the acceleration command as desired. The optimal guidance problem is established in the impact angle frame and is solved by the Gauss orthogonal collocation method. The proposed guidance law is formulated from the generalized optimal control framework, thus a new guidance law that allows achieving a specific guidance goal can be easily obtained in the way of devising a proper weighting function (smooth, piecewise, nonsmooth, or even discontinuous). This property provides more degrees of freedom in the guidance law design to accomplish a specified guidance objective. A hardware experiment is conducted to evaluate the real-time computational capacity of the proposed computational guidance law. Illustrative examples with several types of weighting functions, including smooth, piecewise, nonsmooth, and discontinuous functions, are provided to demonstrate the advantages and capacity of the proposed guidance law.

Learning optimal controllers: a dynamical motion primitive approach

Real-time computation of optimal control is a challenging problem and, to solve this difficulty, many frameworks proposed to use learning techniques to learn (possibly sub-optimal) controllers and enable their usage in an online fashion. Among these techniques, the optimal motion framework is a simple, yet powerful technique, that obtained success in many complex real-world applications. The main idea of this approach is to take advantage of dynamic motion primitives, a widely used tool in robotics to learn trajectories from demonstrations. While usually these demonstrations come from humans, the optimal motion framework is based on demonstrations coming from optimal solutions, such as the ones obtained by numeric solvers. As usual in many learning techniques, a drawback of this approach is that it is hard to estimate the suboptimality of learned solutions, since finding easily computable and non-trivial upper bounds to the error between an optimal solution and a learned solution is, in general, unfeasible. However, we show in this paper that it is possible to estimate this error for a broad class of problems. Furthermore, we apply this estimation technique to achieve a novel and more efficient sampling scheme to be used within the optimal motion framework, enabling the usage of this framework in some scenarios where the computational resources are limited.

Clipped DeepControl: Deep neural network two-dimensional pulse design with an amplitude constraint layer

Advanced radio-frequency pulse design used in magnetic resonance imaging has recently been demonstrated with deep learning of (convolutional) neural networks and reinforcement learning. For two-dimensionally selective radio-frequency pulses, the (convolutional) neural network pulse prediction time (a few milliseconds) was in comparison more than three orders of magnitude faster than the conventional optimal control computation. The network pulses were from the supervised training capable of compensating scan-subject dependent inhomogeneities of B0 and B1+ fields. Unfortunately, the network presented with a small percentage of pulse amplitude overshoots in the test subset, despite the optimal control pulses used in training were fully constrained. Here, we have extended the convolutional neural network with a custom-made clipping layer that completely eliminates the risk of pulse amplitude overshoots, while preserving the ability to compensate for the inhomogeneous field conditions.

A Convex Approach to Data-Driven Optimal Control via Perron–Frobenius and Koopman Operators

This article is about the data-driven computation of optimal control for a class of control affine deterministic nonlinear systems. We assume that the control dynamical system model is not available, and the only information about the system dynamics is available in the form of time-series data. We provide a convex formulation for the optimal control problem (OCP) of the nonlinear system. The convex formulation relies on the duality result in the dynamical system’s stability theory involving density function and Perron–Frobenius operator. We formulate the OCP as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the Koopman operator’s data-driven computation, which is dual to the Perron–Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Optimal control of a NASCAR – specification race car

This paper adapts the minimum-lap-time simulation technologies developed in Formula One to NASCAR. The adaptation process requires recognition of the variable lateral curvature of some tracks, as well as the asymmetries of NASCAR-specification race cars that have been configured for high-speed oval tracks that are dominated by left-hand cornering. These asymmetries include different suspension characteristics on the left- and right-hand ends of each axle, different left- and right-hand tyres on each axle, as well as asymmetries in the car's aerodynamic performance. The results compare optimal control calculations, industrial simulation-based predictions and race-captured telemetry data. For the most part, these various data sources are in good agreement, with the importance of variable lateral camber highlighted. It is demonstrated how race car drivers have to find the best compromise between the conflicting objectives of exploiting the additional camber available on the outside of variable-camber tracks and increasing the travelled distance and racing line cornering curvature.

Static and dynamic multi-obstacle avoidance and docking of ASVs using computational geometry and numerical optimal control

Docking operations for autonomous surface vehicles (ASVs) involve challenges related to low-speed maneuvering in confined harbour spaces while handling environmental forces and traffic. Recently, optimization-based docking approaches have demonstrated promising results in unobstructed convex harbor areas, however they are still limited in dealing with static obstacles, which adds the significant challenge of non-convex constraints. In this paper, we present two hybrid methods that addresses the problem of docking while avoiding both static and dynamic obstacles that obstruct the direct path to the quay. The first phase involves the utilization of computational geometry for partitioning the non-convex harbor area into convex polygons. Then, a search algorithm computes the optimal sequence of convex polygons that connect the starting pose to the docking pose in an obstacle-free manner. Finally, trajectory planning is achieved via numerical optimal control. In this way, the proposed methods helps avoiding local minima and preserves the benefits of optimization-based approaches with respect to vehicle kinematics and dynamics, as well as actuator and spatial constraints.