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All sources NASA NTRS DOE OSTI

DOE OSTI2025

Convex relaxation for Fokker–Planck equation

We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics by solving the high-dimensional Fokker–Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. In a specific setting, we show the proposed method can recover a mean-field-type equilibrium density. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

Chen, Yian↗

DOE OSTI2023

Convex Q-Learning in Continuous Time with Application to Dispatch of Distributed Energy Resources

Convex Q-learning is a recent approach to reinforcement learning, motivated by the possibility of a firmer theory for convergence, and the possibility of making use of greater a priori knowledge regarding policy or value function structure. This paper explores algorithm design in the continuous time domain, with a finite-horizon optimal control objective. The main contributions are (i) The new Q-ODE: a model-free characterization of the Hamilton-Jacobi-Bellman equation. (ii) A formulation of Convex Q-learning that avoids approximations appearing in prior work. The Bellman error used in the algorithm is defined by filtered measurements, which is necessary in the presence of measurement noise. (iii) Convex Q-learning with linear function approximation is a convex program. It is shown that the constraint region is bounded, subject to an exploration condition on the training input. (iv) The theory is illustrated in application to resource allocation for distributed energy resources, for which the theory is ideally suited.

Lu, Fan↗

DOE OSTI2022

Distributionally Robust Decision Making Leveraging Conditional Distributions

Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision-making with some auxiliary information, and DRO in the context of conditional distributions is not straightforward. We propose a conditional kernel distributionally robust optimization (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). In particular, we consider problems where there is a correlation between the unknown variable y and an auxiliary observable variable x. Given past data of the two variables and a queried auxiliary variable, CKDRO represents the conditional distribution P(y|x) as the conditional mean operator in the RKHS space and quantifies the ambiguity set in the RKHS as well, which depends on the size of the dataset as well as the query point. To justify the use of RKHS, we demonstrate that the ambiguity set defined in RKHS can be viewed as a ball under a metric that is similar to the Wasserstein metric. The DRO is then dualized and solved via a finite dimensional convex program. The proposed CKDRO approach is applied to a generation scheduling problem and shows that the result of CKDRO is superior to common benchmarks in terms of quality and robustness.

Chen, Yuxiao↗

DOE OSTI2022

Real-Time On-Ramp Merging Control of Connected and Automated Vehicles using Pseudospectral Convex Optimization

Highway on-ramp merging can be a challenging task for human drivers due to the complex vehicle negotiations and interactions in limited time and space. Connected and automated vehicles (CAVs) have great potential to address the problem and offer many benefits in terms of safety, traffic efficiency, and fuel economy. However, real-time optimal control of CAVs still faces many challenges, including nonlinear dynamics, complex inter-vehicle interactions, and a highly dynamic and uncertain traffic environment. To address these challenges, we develop a novel control approach that balances the solution optimality and computational efficiency to determine optimal merging speed profiles in real time. Specifically, by employing a pseudospectral method and a sequential convex programming approach, two algorithms are proposed and implemented within the model predictive control (MPC) framework to enable real-time generation of optimal solutions for potential on-vehicle applications. The convergence and optimality of the proposed algorithms are validated by comparing with a general-purpose solver under different traffic scenarios.

Shi, Yang↗

NASA NTRS1988

A methodology for the synthesis of robust feedback systems

A new methodology is developed for the synthesis of linear, time-variant (LTI) controllers for multivariable LTI systems. The resulting closed-loop system is nominally stable and exhibits a known level of performance. In addition, robustness of the feedback system is guaranteed, i.e., stability and performance are retained in the presence of multiple unstructured uncertainty blocks located at various points in the feedback loop. The design technique is referred to as the Causality Recovery Methodology (CRM). The CRM relies on the Youla parameterization of all stabilizing compensators to ensure nominal stability of the feedback system. A frequency-domain inequality in terms of the structured singular value mu defines the robustness specification. The optimal compensator, with respect to the mu condition, is shown to be noncausal in general. The aim of the CRM is to find a stable, causal transfer function matrix that approximates the robustness characteristics of the optimal solution. The CRM, via a series of infinite-dimensional convex programs, produces a closed-loop system whose performance robustness is at least as good as that of any initial design. The algorithm is approximated by a finite dimensional process for the purposes of implementation. Two numerical examples confirm the potential viability of the CRM concept; however, the robustness improvement comes at the expense of increased computational burden and compensator complexity.

Milich, David Albert↗

NASA NTRS2017

Autonomous Guidance Navigation and Control for Agile Quadrotors Using Polynomial Trajectory Planning and L1 Adaptive Control

We address the challenge to allow efficient autonomous flight in real world environments, both indoor and outdoor. We use a straight-line SE-SCP (Spherical Expansion and Sequential Convex Programming) [algorithm] to find an initial route through the environment and minimum snap trajectory generation using piecewise polynomials. Then, we implement an adaptive robust control able to address some robustness issues for quadrotors in outdoor flight, such as mass variation and wind disturbances. Coupling these techniques we allow high-speed and aggressive autonomous flight through obstacle-dense indoor environments, as well as address outdoor disturbances.

Landolfi, Mattia↗

NASA NTRS1990

Design of cryogenic tanks for launch vehicles

During the period since January 1990, work was concentrated on the problem of the buckling of the structure of an ALS (advanced launch systems) tank during the boost phase. The primary problem was to analyze a proposed hat stringer made by superplastic forming, and to compare it with an integrally stiffened stringer design. A secondary objective was to determine whether structural rings having the identical section to the stringers will provide adequate support against overall buckling. All of the analytical work was carried out with the TESTBED program on the CONVEX computer, using PATRAN programs to create models. Analyses of skin/stringer combinations have shown that the proposed stringer design is an adequate substitute for the integrally stiffened stringer. Using a highly refined mesh to represent the corrugations in the vertical webs of the hat stringers, effective values were obtained for cross-sectional area, moment of inertia, centroid height, and torsional constant. Not only can these values be used for comparison with experimental values, but they can also be used for beams to replace the stringers and frames in analytical models of complete sections of tank. The same highly refined model was used to represent a section of skin reinforced by a stringer and a ring segment in the configuration of a cross. It was intended that this would provide a baseline buckling analysis representing a basic mode, however, the analysis proved to be beyond the scope of the CONVEX computer. One quarter of this model was analyzed, however, to provide information on buckling between the spot welds. Models of large sections of the tank structure were made, using beam elements to model the stringers and frames. In order to represent the stiffening effects of pressure, stresses and deflections under pressure should first be obtained, and then the buckling analysis should be made on the structure so deflected. So far, uncharacteristic deflections under pressure were obtained from the TESTBED program using two types of structural elements. Similar results were obtained using the ANSYS program on a mainframe computer, although two finite element programs on microcomputers have yielded realistic results.

Copper, Charles↗

DOE OSTI2021

Ensemble Learning Based Convex Approximation of Three-Phase Power Flow

Though the convex optimization has been widely used in power systems, it still cannot guarantee to yield a tight (accurate) solution to some problems. To mitigate this issue, this paper proposes an ensemble learning based convex approximation for alternating current (AC) power flow equations that differs from the existing convex relaxations. The proposed approach is based on three-phase quadratic power flow equations in rectangular coordinates. To develop this data-driven convex approximation of power flows, the polynomial regression (PR) is first deployed as a basic learner to fit convex relationships between the independent and dependent variables. Then, ensemble learning algorithms such as gradient boosting (GB) and bagging are introduced to combine learners to boost model performance. Based on the learned convex approximation of power flow, optimal power flow (OPF) is formulated as a convex quadratic programming problem. The simulation results on IEEE standard cases of both balanced and unbalanced systems show that, in the context of solving OPF, the proposed data-driven convex approximation outperforms the conventional semi-definite programming (SDP) relaxation in both accuracy and computational efficiency, especially in the cases that the conventional SDP relaxation fails

Convex approximation↗

DOE OSTI2021

An infeasible-start framework for convex quadratic optimization, with application to constraint-reduced interior-point and other methods

A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating rule. Under mild assumptions, the framework is proved to converge on CQPs with both inequality and equality constraints and, at a negligible additional cost per iteration, produces an infeasibility certificate, together with a feasible point for an (approximately) ℓ 1 -least relaxed feasible problem, when the given problem does not have a feasible solution. The framework is applied to a feasible-start constraint-reduced interior-point algorithm previously proved to be highly performant on problems with many more inequality constraints than variables (“imbalanced”). Numerical comparison with popular codes (OSQP, qpOASES, MOSEK) is reported on both randomly generated problems and support-vector machine classifier training problems. The results show that the former typically outperforms the latter on imbalanced problems. Finally, application of the proposed infeasible-start framework to other feasible-start algorithms is briefly considered, and is tested on a simplex iteration.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Regional surrogates for predictive control of digital twins

Digital twins of complex systems must involve a model that is fast, generalizable, and usable for real-time control. For example, high-fidelity nonlinear multiphysics simulations can capture laser-material interactions, but are too slow for optimization or model predictive control (MPC). Reduced-order models, used to accelerate such computation, frequently fail to generalize to unseen inputs or control states. We show theoretically that this failure is intrinsic, i.e., that a learned model is non-unique outside the sampled subspace when its low-rank structure arises from limited excitation and clustered eigenvalues, rather than from a user-imposed truncation alone. Motivated by this result, we propose a control-ready regional surrogate-construction framework for both autonomous and nonautonomous dynamics; it employs Koopman lifting to represent nonlinearities, while preserving spatial locality. We illustrate our approach by constructing a control-ready surrogate for the digital twin of a thermal component of additive-manufacturing process. Our surrogate, localized in space through a von Neumann stencil, is learned from noisy high-fidelity simulations that emulate thermal-camera images collected during the manufacturing. It is linear in thermo-physically augmented states so that MPC reduces to a convex quadratic program. The surrogate requires no online correction, generalizes to unseen scan paths and power profiles of the laser, and is more than three orders of magnitude faster than a finite-difference solver. Furthermore, when the MPC sequence computed on the digital twin is applied to this solver, closed-loop temperature regulation is recovered, showing that the surrogate preserves control-relevant input-output behavior.

Data-driven model↗

DOE OSTI2022

Nonlinear burn control in ITER using adaptive allocation of actuators with uncertain dynamics

Abstract ITER will be the first tokamak to sustain a fusion-producing, or burning, plasma. If the plasma temperature were to inadvertently rise in this burning regime, the positive correlation between temperature and the fusion reaction rate would establish a destabilizing positive feedback loop. Careful regulation of the plasma’s temperature and density, or burn control, is required to prevent these potentially reactor-damaging thermal excursions, neutralize disturbances and improve performance. In this work, a Lyapunov-based burn controller is designed using a full zero-dimensional nonlinear model. An adaptive estimator manages destabilizing uncertainties in the plasma confinement properties and the particle recycling conditions (caused by plasma–wall interactions). The controller regulates the plasma density with requests for deuterium and tritium particle injections. In ITER-like plasmas, the fusion-born alpha particles will primarily heat the plasma electrons, resulting in different electron and ion temperatures in the core. By considering separate response models for the electron and ion energies, the proposed controller can independently regulate the electron and ion temperatures by requesting that different amounts of auxiliary power be delivered to the electrons and ions. These two commands for a specific control effort (electron and ion heating) are sent to an actuator allocation module that optimally maps them to the heating actuators available to ITER: an electron cyclotron heating system (20 MW), an ion cyclotron heating system (20 MW), and two neutral beam injectors (16.5 MW each). Two different actuator allocators are presented in this work. The first actuator allocator finds the optimal mapping by solving a convex quadratic program that includes actuator saturation and rate limits. It is nonadaptive and assumes that the mapping between the commanded control efforts and the allocated actuators (i.e. the effector model) contains no uncertainties. The second actuator allocation module has an adaptive estimator to handle uncertainties in the effector model. This uncertainty includes actuator efficiencies, the fractions of neutral beam heating that are deposited into the plasma electrons and ions, and the tritium concentration of the fueling pellets. Furthermore, the adaptive allocator considers actuator dynamics (actuation lag) that contain uncertainty. This adaptive allocation algorithm is more computationally efficient than the aforementioned nonadaptive allocator because it is computed using dynamic update laws so that finding the solution to a static optimization problem is not required at every time step. A simulation study assesses the performance of the proposed adaptive burn controller augmented with each of the actuator allocation modules.

Physics↗

DOE OSTI2021

Dynamic Interference Management for UAV-Assisted Wireless Networks

Here, we investigate a transmission mechanism aiming to improve the data rate between a base station (BS) and a user equipment (UE) through deploying multiple relaying UAVs. We consider the effect of interference incurred by another established communication network, which makes our problem challenging and different from the state of the art. We aim to design the 3D trajectories and power allocation for the UAVs to maximize the data flow of the network while keeping the interference on the existing communication network below a threshold. We utilize the mobility feature of the UAVs to evade the (un)-intended interference caused by (un)-intentional interferers. To this end, we propose an alternating-maximization approach to jointly obtain the 3D trajectories and the UAVs transmission powers. We handle the 3D trajectory design by resorting to spectral graph the power allocation through convex optimization techniques. We also approach the problem from the intentional interferer’s perspective where smart jammers chase the UAVs to effectively degrade the data flow of the network. We also extend our work to the case for multiple UEs. Finally, we demonstrate the efficacy of our proposed method through extensive simulations.

42 ENGINEERING↗

NASA NTRS1988

A method for designing robust multivariable feedback systems

A new methodology is developed for the synthesis of linear, time-invariant (LTI) controllers for multivariable LTI systems. The aim is to achieve stability and performance robustness of the feedback system in the presence of multiple unstructured uncertainty blocks; i.e., to satisfy a frequency-domain inequality in terms of the structured singular value. The design technique is referred to as the Causality Recovery Methodology (CRM). Starting with an initial (nominally) stabilizing compensator, the CRM produces a closed-loop system whose performance-robustness is at least as good as, and hopefully superior to, that of the original design. The robustness improvement is obtained by solving an infinite-dimensional, convex optimization program. A finite-dimensional implementation of the CRM was developed, and it was applied to a multivariate design example.

Milich, David Albert↗

NASA NTRS1988

A method for designing robust multivariable feedback systems

A new methodology is developed for the synthesis of linear, time-invariant (LTI) controllers for multivariable LTI systems. The aim is to achieve stability and performance robustness of the feedback system in the presence of multiple unstructured uncertainty blocks; i.e., to satisfy a frequency-domain inequality in terms of the structured singular value. The design technique is referred to as the causality recovery methodology (CRM). Starting with an initial (nominally) stabilizing compensator, the CRM produces a closed-loop system whose performance-robustness is at least as good as, and hopefully superior to, that of the original design. The robustness improvement is obtained by solving an infinite-dimensional, convex optimization program. A finite-dimensional implementation of the CRM was developed, and it was applied to a multivariate design example.

Milich, David A.↗

DOE OSTI2022

A computationally efficient algorithm for computing convex hull prices

Electricity markets worldwide allow participants to bid non-convex production offers. While non-convex offers can more accurately reflect a resource's capabilities, they create challenges for market clearing processes. For example, system operators may be required to execute side payments to participants whose costs are not covered through energy sales as determined via traditional locational marginal pricing schemes. Convex hull pricing minimizes this and other types of side payments while providing uniform (i.e., locationally and temporally consistent) prices. Computing convex hull prices involves solving either a large-scale linear program or the Lagrangian dual of the corresponding non-convex scheduling problem. Further, the former approach requires explicit descriptions of market participants' convex hulls. While linear programs for computing convex hull prices are large, their structure is naturally decomposable by generators. Here, in this work, we propose and empirically analyze a Benders decomposition approach to computing convex hull prices that leverages recent advances in convex hull formulations for thermal generating units. We demonstrate across a large set of test instances that our decomposition approach only requires modest computational effort, obtaining solutions at least an order of magnitude faster than the equivalent large-scale linear programming approach. Overall, we provide a computationally feasible method for computing convex hull prices for industrial scale market clearing problems, enabling the possibility of practical adoption of this advanced pricing mechanism.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

NASA NTRS1963

Mathematical Optimization Techniques

The papers collected in this volume were presented at the Symposium on Mathematical Optimization Techniques held in the Santa Monica Civic Auditorium, Santa Monica, California, on October 18-20, 1960. The objective of the symposium was to bring together, for the purpose of mutual education, mathematicians, scientists, and engineers interested in modern optimization techniques. Some 250 persons attended. The techniques discussed included recent developments in linear, integer, convex, and dynamic programming as well as the variational processes surrounding optimal guidance, flight trajectories, statistical decisions, structural configurations, and adaptive control systems. The symposium was sponsored jointly by the University of California, with assistance from the National Science Foundation, the Office of Naval Research, the National Aeronautics and Space Administration, and The RAND Corporation, through Air Force Project RAND.

Bellman, R.↗

DOE OSTI2025

Data-Conforming Data-Driven Control: Avoiding Premature Generalizations Beyond Data

Data-driven and adaptive control approaches face the problem of introducing sudden distributional shifts beyond the distribution of data encountered during learning. Therefore, they are prone to invalidating the very assumptions used in their own construction. This is due to the linearity of the underlying system, inherently assumed and formulated in most data-driven control approaches, which may falsely generalize the behavior of the system beyond the behavior experienced in the data. This article seeks to mitigate these problems by enforcing consistency of the newly designed closed-loop systems with data and slowing down any distributional shifts in the joint state-input space. This is achieved through incorporating affine regularization terms and linear matrix inequality constraints to data-driven approaches, resulting in convex semi-definite programs that can be efficiently solved by standard software packages. We discuss the optimality conditions of these programs and then conclude this article with a numerical example that further highlights the problem of premature generalization beyond data and shows the effectiveness of our proposed approaches in enhancing the safety of data-driven control methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

A Comparison of Control Allocation Methods in the Presence of Parametric Model Uncertainty

When allocating redundant effectors to virtual control commands, linear (generalized inverse) allocators have historically been used on aircraft and spacecraft. While simple to implement, generalized inverses are unable to realize a significant portion of the attainable moments. To address this drawback, the control allocation problem can also be formulated as a linear programming or quadratic programming problem and solved using convex optimization based solvers. These approaches have been shown to access a larger set of attainable moments, however, little work has been done to understand the performance of convex optimization-based control allocation in the presence of parametric model uncertainty. This paper seeks to compare the performance of several control allocation approaches, including two forms of generalized inverse allocators, the pseudo inverse and minimum-variance, and the linear programming and quadratic programming approach in the presence of parametric model uncertainty. The performance of these four allocators were tested on an aircraft model in the presence of realistic parametric model uncertainty and the convex optimization approaches were shown to outperform the generalized inverses.

Luke J Miller↗