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All sourcesNASA NTRSDOE OSTI
NASA NTRS2009

Control of Systems With Slow Actuators Using Time Scale Separation

This paper addresses the problem of controlling a nonlinear plant with a slow actuator using singular perturbation method. For the known plant-actuator cascaded system the proposed scheme achieves tracking of a given reference model with considerably less control demand than would otherwise result when using conventional design techniques. This is the consequence of excluding the small parameter from the actuator dynamics via time scale separation. The resulting tracking error is within the order of this small parameter. For the unknown system the adaptive counterpart is developed based on the prediction model, which is driven towards the reference model by the control design. It is proven that the prediction model tracks the reference model with an error proportional to the small parameter, while the prediction error converges to zero. The resulting closed-loop system with all prediction models and adaptive laws remains stable. The benefits of the approach are demonstrated in simulation studies and compared to conventional control approaches.

Stepanyan, Vehram↗
NASA NTRS2010

Optimal Control Modification Adaptive Law for Time-Scale Separated Systems

Recently a new optimal control modification has been introduced that can achieve robust adaptation with a large adaptive gain without incurring high-frequency oscillations as with the standard model-reference adaptive control. This modification is based on an optimal control formulation to minimize the L2 norm of the tracking error. The optimal control modification adaptive law results in a stable adaptation in the presence of a large adaptive gain. This study examines the optimal control modification adaptive law in the context of a system with a time scale separation resulting from a fast plant with a slow actuator. A singular perturbation analysis is performed to derive a modification to the adaptive law by transforming the original system into a reduced-order system in slow time. A model matching conditions in the transformed time coordinate results in an increase in the actuator command that effectively compensate for the slow actuator dynamics. Simulations demonstrate effectiveness of the method.

Nguyen, Nhan T.↗
NASA NTRS2012

Optimal Control Modification for Time-Scale Separated Systems

Recently a new optimal control modification has been introduced that can achieve robust adaptation with a large adaptive gain without incurring high-frequency oscillations as with the standard model-reference adaptive control. This modification is based on an optimal control formulation to minimize the L2 norm of the tracking error. The optimal control modification adaptive law results in a stable adaptation in the presence of a large adaptive gain. This study examines the optimal control modification adaptive law in the context of a system with a time scale separation resulting from a fast plant with a slow actuator. A singular perturbation analysis is performed to derive a modification to the adaptive law by transforming the original system into a reduced-order system in slow time. A model matching conditions in the transformed time coordinate results in an increase in the actuator command that effectively compensate for the slow actuator dynamics. Simulations demonstrate effectiveness of the method.

Adaptive Control↗
DOE OSTI2023

Data-driven time-scale separation of ODE right-hand sides using dynamic mode decomposition and time delay embedding

Multi-physics simulation often involves multiple different scales. The ARKODE ODE solver package in the SUNDIALS library addresses multi-scale problems with a multi-rate time-integrator that can work with a right-hand side that has fast scale and slow scale components. In this report, we use dynamic mode decomposition and time delay embedding to extract the fast and slow components of the right-hand sides of a simple ODE from data. We then use the extracted components to solve the ODE with ARKODE. Finally, to move towards a real-world use case, we attempt to extract fast and slow scale dynamics from synthetic seismic modeling data.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Separation of time scales in aircraft trajectory optimization

Two methods for analyzing the time-scale properties of aircraft trajectory optimization problems are presented. Time-scale properties must be identified before solutions can be obtained by using singular perturbation methods. Both methods only require a knowledge of the state equations, the aircraft characteristics, and the bounds on the state and control variables. Although these methods give only rough estimates of time-scale separation, they do not require that an 'exact' optimal trajectory be known, as do the more rigorous methods, and they are an improvement on the ad hoc methods currently in use. The two methods are applied to an example problem for a high performance aircraft.

Ardema, M. D.↗
DOE OSTI2025

Multi-scale, Multi-disciplinary, and Multi-agent Explainable AI with Koopman-Undergirded Learning, Prediction, and Analysis (M3EA KULPA) (Project Closeout Report)

The goal of this project was to develop and use domain-aware machine learning formulations, based on the Koopman Operator (KO), for modelling multi-scale, multi-disciplinary (e.g., multi-physics), and/or multi-agent systems. The project developed these formulations for the following cases: • Systems with dynamics at two separate time scales, • Systems with a bi-level hierarchical control structure, • Systems with bi-level hierarchical control and dynamics at two separate time scales (the lower level controls operating at the faster time scale), and • Systems with n separate but interacting agents/disciplines (with/without control, respectively); the controls for each agent could include bi-level hierarchical control and dynamics at two separate time scales as described above. The project then defined a set of dynamical systems consisting of different nonlinear oscillators that could be used to test these different formulations and then subsequently learned the KO models for those systems. With the KO models, we were able to do the following: • Quantify system stability, including both long-term and transient behavior, • Quantify the effects of feedbacks between the different time scales and agents/disciplines in terms of those feedbacks’ effects on system stability, • Replace a standard Proportional-Integral (PI) control in the hierarchical control structure with a KO-based Linear-Quadratic Regular (LQR), a form of optimal control, • Calculate optimal supervisory control policies a) with and without time scale separated dynamics at the lower level control levels and b) with both PI and KO-based LQR lower level control policies, and • Calculate dynamic Nash equilibria for multi-agent systems where each agent makes its own control decisions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2013

Atomic Force Microscope Mediated Chromatography

The atomic force microscope (AFM) is used to inject a sample, provide shear-driven liquid flow over a functionalized substrate, and detect separated components. This is demonstrated using lipophilic dyes and normal phase chromatography. A significant reduction in both size and separation time scales is achieved with a 25-micron-length column scale, and one-second separation times. The approach has general applications to trace chemical and microfluidic analysis. The AFM is now a common tool for ultra-microscopy and nanotechnology. It has also been demonstrated to provide a number of microfluidic functions necessary for miniaturized chromatography. These include injection of sub-femtoliter samples, fluidic switching, and sheardriven pumping. The AFM probe tip can be used to selectively remove surface layers for subsequent microchemical analysis using infrared and tip-enhanced Raman spectroscopy. With its ability to image individual atoms, the AFM is a remarkably sensitive detector that can be used to detect separated components. These diverse functional components of microfluidic manipulation have been combined in this work to demonstrate AFM mediated chromatography. AFM mediated chromatography uses channel-less, shear-driven pumping. This is demonstrated with a thin, aluminum oxide substrate and a non-polar solvent system to separate a mixture of lipophilic dyes. In conventional chromatographic terms, this is analogous to thin-layer chromatography using normal phase alumina substrate with sheardriven pumping provided by the AFM tip-cantilever mechanism. The AFM detection of separated components is accomplished by exploiting the variation in the localized friction of the separated components. The AFM tip-cantilever provides the mechanism for producing shear-induced flows and rapid pumping. Shear-driven chromatography (SDC) is a relatively new concept that overcomes the speed and miniaturization limitations of conventional liquid chromatography. SDC is based on a sliding plate system, consisting of two flat surfaces, one of which has a recessed channel. A fluid flow is produced by axially sliding one plate past another, where the fluid has mechanical shear forces imposed at each point along the channel length. The shear-induced flow rates are very reproducible, and do not have pressure or voltage gradient limitations. SDC opens up a new range of enhanced separation kinetics by permitting the sample confinement with submicron dimensions. Small, highly confined liquid is advantageous for chromatographic separation because the separation rate is known to scale according to the square of the confined sample diameter. In addition, because shear-driven flows are not limited by fluid velocity, shear-driven liquid chromatography may provide up to 100,000 plate efficiency.

Anderson, Mark S.↗
DOE OSTI2024

Structural Impact of Grid-Forming Inverters on Power System Coherency

This paper addresses the following fundamental research question: how does the integration of grid-forming inverters (GFMs) replacing conventional synchronous generators (SGs) impact the slow coherent eigen-structure and the low frequency oscillatory behavior of future power systems? Due to time-scale separated dynamics, generator states inside a coherent area synchronize over a fast time-scale due to stronger coupling, while the areas themselves synchronize over a slower time scale. Our mathematical analysis shows that due to the large-scale integration of GFMs, the weighted Laplacian structure of the frequency dynamics is preserved, however, the entries of the Laplacian may be significantly modified based on the location and penetration levels of the GFMs. This can impact and potentially significantly alter the coherency structure of the system. We have validated our findings with numerical results using the IEEE 68-bus test system.

Mukherjee, Sayak [BATTELLE (PACIFIC NW LAB)]↗
DOE OSTI2026

SODAs: sparse optimization for the discovery of differential and algebraic equations

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by time-scale separation, conservation laws and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods for DAE systems with unknown constraints and time scales. We introduce sparse optimization for differential-algebraic systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems. It has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves since SODAs is singular numerical stability when handling high correlations between library terms, caused by near-perfect algebraic relationships, by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

DAE↗
NASA NTRS2009

Optimal Control Modification for Robust Adaptation of Singularly Perturbed Systems with Slow Actuators

Recently a new optimal control modification has been introduced that can achieve robust adaptation with a large adaptive gain without incurring high-frequency oscillations as with the standard model-reference adaptive control. This modification is based on an optimal control formulation to minimize the L2 norm of the tracking error. The optimal control modification adaptive law results in a stable adaptation in the presence of a large adaptive gain. This study examines the optimal control modification adaptive law in the context of a system with a time scale separation resulting from a fast plant with a slow actuator. A singular perturbation analysis is performed to derive a modification to the adaptive law by transforming the original system into a reduced-order system in slow time. The model matching conditions in the transformed time coordinate results in increase in the feedback gain and modification of the adaptive law.

Nguyen, Nhan T.↗
NASA NTRS1982

Nonlinear zero-sum differential game analysis by singular perturbation methods

A class of nonlinear, zero-sum differential games, exhibiting time-scale separation properties, can be analyzed by singular-perturbation techniques. The merits of such an analysis, leading to an approximate game solution, as well as the 'well-posedness' of the formulation, are discussed. This approach is shown to be attractive for investigating pursuit-evasion problems; the original multidimensional differential game is decomposed to a 'simple pursuit' (free-stream) game and two independent (boundary-layer) optimal-control problems. Using multiple time-scale boundary-layer models results in a pair of uniformly valid zero-order composite feedback strategies. The dependence of suboptimal strategies on relative geometry and own-state measurements is demonstrated by a three dimensional, constant-speed example. For game analysis with realistic vehicle dynamics, the technique of forced singular perturbations and a variable modeling approach is proposed. Accuracy of the analysis is evaluated by comparison with the numerical solution of a time-optimal, variable-speed 'game of two cars' in the horizontal plane.

Sinar, J.↗
DOE OSTI2024

How Does Escherichia coli Allocate Proteome?

Microorganisms are shown to actively partition their intracellular resources, such as pro- teins, for growth optimization. Recent experiments have begun to reveal molecular com- ponents unpinning the partition; however, it remains unclear quantitatively how individual parts orchestrate to yield precise resource allocation that is both robust and dynamic. Here we developed a coarse-grained mathematical framework that centers on guanosine pentaphosphate (ppGpp)-mediated regulation, and used it to systematically uncover the design principles of proteome allocation in Escherichia coli. Our results showed that cellular ability of resource partition lies in an ultrasensitive, negative feedback control- ling topology with the ultrasensitivity arising from zero-order amino acid kinetics and the negative feedback from ppGpp-controlled ribosome synthesis. In addition, together with the time-scale separation between slow ribosome kinetics and fast turnovers of ppGpp and amino acids, the network topology confers the organism an optimization mechanism which mimics sliding mode control, a nonlinear optimization strategy that is widely used in man-made systems. We further showed that such a controlling mechanism is robust against parameter variations and molecular fluctuations, and is also efficient for biomass production over time. Furthermore, this work elucidates the fundamental controlling mechanism of E. coli proteome allocation, thereby providing insights into quantitative microbial physiology as well as the design of synthetic gene networks.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2010

A Singular Perturbation Approach for Time-Domain Assessment of Phase Margin

This paper considers the problem of time-domain assessment of the Phase Margin (PM) of a Single Input Single Output (SISO) Linear Time-Invariant (LTI) system using a singular perturbation approach, where a SISO LTI fast loop system, whose phase lag increases monotonically with frequency, is introduced into the loop as a singular perturbation with a singular perturbation (time-scale separation) parameter Epsilon. First, a bijective relationship between the Singular Perturbation Margin (SPM) max and the PM of the nominal (slow) system is established with an approximation error on the order of Epsilon(exp 2). In proving this result, relationships between the singular perturbation parameter Epsilon, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time-domain for SISO LTI systems using the SPM with a standardized testing system called "PM-gauge," as demonstrated by examples. PM is a widely used stability margin for LTI control system design and certification. Unfortunately, it is not applicable to Linear Time-Varying (LTV) and Nonlinear Time-Varying (NLTV) systems. The approach developed here can be used to establish a theoretical as well as practical metric of stability margin for LTV and NLTV systems using a standardized SPM that is backward compatible with PM.

Zhu, J. Jim↗
NASA NTRS2006

Singular-Arc Time-Optimal Trajectory of Aircraft in Two-Dimensional Wind Field

This paper presents a study of a minimum time-to-climb trajectory analysis for aircraft flying in a two-dimensional altitude dependent wind field. The time optimal control problem possesses a singular control structure when the lift coefficient is taken as a control variable. A singular arc analysis is performed to obtain an optimal control solution on the singular arc. Using a time-scale separation with the flight path angle treated as a fast state, the dimensionality of the optimal control solution is reduced by eliminating the lift coefficient control. A further singular arc analysis is used to decompose the original optimal control solution into the flight path angle solution and a trajectory solution as a function of the airspeed and altitude. The optimal control solutions for the initial and final climb segments are computed using a shooting method with known starting values on the singular arc The numerical results of the shooting method show that the optimal flight path angle on the initial and final climb segments are constant. The analytical approach provides a rapid means for analyzing a time optimal trajectory for aircraft performance.

Nguyen, Nhan↗
DOE OSTI2023

Thermal equilibrium of collisional non-neutral plasma in a magnetic dipole trap

This paper discusses thermal equilibrium states of single-species plasmas, such as pure electron plasmas and pure positron plasmas, that are confined in a dipole trap. Thermal equilibrium states for such plasmas are routinely realized in the homogeneous magnetic field of Penning–Malmberg traps. We generalize the theory of these states to include inhomogeneous magnetic dipole fields. The approach to thermal equilibrium takes place in two stages with well separated time scales. On the collision time scale, thermal equilibrium is established along each magnetic field line. On the much longer transport time scale, heat conduction and viscosity bring the plasmas on different flux contours into thermal equilibrium, we call this a state of global thermal equilibrium. We present numerical results for local and global thermal equilibria. These results agree with the analytic predictions for charge collections that are large compared with the Debye length. There is, in principle, no limit to the confinement time of a single-species plasma in a global thermal equilibrium state. Experiments with hot electron–ion plasmas performed in the LDX and RT1 devices give us confidence that, in contrast to a Penning–Malmberg trap, a magnetic dipole field can also confine cold quasi-neutral electron–positron pair plasmas on the time scale of the phenomena of interest. Such pair plasmas are assumed to form in the magnetosphere of neutron stars but have so far not been realized in a laboratory. The creation of an electron–positron pair plasma is the main goal of the APEX collaboration.

Steinbrunner, P. (ORCID:0000000317693686)↗
DOE OSTI2024

Coherency-Aware Learning Control of Inverter-Dominated Grids: A Distributed Risk-Constrained Approach

Here, this letter investigates the importance of integrating the coherency knowledge for designing controllers to dampen sustained oscillations in wide-area power networks with significant penetration of inverter-interfaced resources. Coherency is a fundamental property of power systems, where time-scale separation in frequency dynamics leads to clustered behavior among generators of different groups. Large-scale penetration of inverter-driven low inertia resources replacing conventional synchronous generators (SGs) can lead to perturbation in the coherent partitioning; hence, integrating such information is of utmost importance for oscillation control designs. We present the coherency-aware design of a distributed output feedback-based reinforcement learning method that additionally incorporates risk constraints to capture the uncertainties related to net-load fluctuations. The use of domain-aware coherency information has produced improved training and oscillation performance than the coherency-agnostic control design, hence proving to be effective in controller design. Finally, we validated the proposed method with numerical experiments on the benchmark IEEE 68-bus test system.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS1979

Aircraft trajectory optimization using singular perturbation methods

Singular perturbation methods are used to derive nonlinear feedback control laws for aircraft minimum time long range interception under the assumption of complete time scale separation. The resulting algorithms are in feedback form and can be implemented on a microcomputer for on-line trajectory optimization.

Mehra, R. K.↗