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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

A novel approach to the modelling and control of flexible robot arms

A general dynamic model of a two-link Euler-Bernoulli beam flexible robot arm is presented in the form of partial-differential-integral equations. Observations are made on important properties of the dynamic model. The resulting infinite-dimensional system is then input-output decoupled and partially linearized by a diffeomorphic state transformation and nonlinear state feedback. The local stability issue is addressed for a one-link flexible robot arm.

Ding, Xuru↗
NASA NTRS1988

Controller synthesis for infinite dimensional systems based on a passivity approach

The author discusses a generalization of hyperstability to systems in Hilbert space and its application to finite-dimensional stabilizing compensator design. The basic idea is to characterize tolerable perturbations in terms of the passivity of the nominal closed-loop system. Controllers achieving the required closed-loop passivity property can then be designed using H(infinity)-optimization. In particular, the author presents a design procedure for a stabilizing a finite-dimensional compensator for a given infinite-dimensional system.

Wen, John T.↗
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.↗
NASA NTRS1988

Parameter estimation in nonlinear distributed systems - Approximation theory and convergence results

An abstract approximation framework and convergence theory is described for Galerkin approximations applied to inverse problems involving nonlinear distributed parameter systems. Parameter estimation problems are considered and formulated as the minimization of a least-squares-like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. The theory applies to systems whose dynamics can be described by either time-independent or nonstationary strongly maximal monotonic operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. It is demonstrated that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin approximation assumption holds, then solutions to the approximating problems exist and approximate a solution to the original infinite-dimensional identification problem.

Banks, H. T.↗
NASA NTRS1989

A general dynamic model of flexible robot arms for control

Hamilton's principle is used to derive the dynamic model for a large class of flexible robot arms. The resultant dynamic model consists of a system of partial differential-integral equations and the dynamic boundary conditions associated with it. Some properties of the model are observed, and its application to control is discussed. This model represents an infinite-dimensional nonlinear dynamic system and yet can be turned into a finite-dimensional system that could be obtained by modal expansion, if it is desired. This provides more flexibility for control purposes as well as for the analysis of the system.

Ding, X.↗
NASA NTRS1990

Robust Adaptive Control In Hilbert Space

Paper discusses generalization of scheme for adaptive control of finite-dimensional system to infinite-dimensional Hilbert space. Approach involves generalization of command-generator tracker (CGT) theory. Does not require reference model to be same order as that of plant, and knowledge of order of plant not needed. Suitable for application to high-order systems, main emphasis on adjustment of low-order feedback-gain matrix. Analysis particularly relevant to control of large, flexible structures.

Wen, John Ting-Yung↗
NASA NTRS1989

Convergence of infinite dimensional sampled LQR problems - Theory and numerical results

A theory is developed for the convergence of the closed-loop solution to infinite-dimensional discrete-time linear-quadratic regulator (LQR) problems on the infinite time interval to the solution of a corresponding continuous-time LQR problem as the length of the sampling interval tends toward zero. Convergence of solutions to the operator algebraic Riccati equation and corresponding optimal feedback control gains is guaranteed under appropriate uniform stabilizability and detectability conditions and consistent sampling. Also presented are numerical results involving the optimal LQ control of a heat or diffusion equation, a hereditary or delay differential equation, and a hybrid system of ordinary and partial differential equations describing the transverse vibration of a cantilevered Voigt-Kelvin viscoelastic beam with tip mass.

Rosen, I. G.↗
NASA NTRS1990

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗
NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗
NASA NTRS1991

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

A hybrid method for computing the feedback gains in linear quadratic regulator problem is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

Banks, H. T.↗
NASA NTRS1991

The route to chaos for the Kuramoto-Sivashinsky equation

The results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of period-doublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609 .... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.

Papageorgiou, Demetrios T.↗
NASA NTRS1992

Minimax confidence intervals in geomagnetism

The present paper uses theory of Donoho (1989) to find lower bounds on the lengths of optimally short fixed-length confidence intervals (minimax confidence intervals) for Gauss coefficients of the field of degree 1-12 using the heat flow constraint. The bounds on optimal minimax intervals are about 40 percent shorter than Backus' intervals: no procedure for producing fixed-length confidence intervals, linear or nonlinear, can give intervals shorter than about 60 percent the length of Backus' in this problem. While both methods rigorously account for the fact that core field models are infinite-dimensional, the application of the techniques to the geomagnetic problem involves approximations and counterfactual assumptions about the data errors, and so these results are likely to be extremely optimistic estimates of the actual uncertainty in Gauss coefficients.

Stark, Philip B.↗
NASA NTRS1992

Stability of a class of interconnected evolution systems

Stability conditions for a class of interconnected systems modeled by linear abstract evolution equations and a memoryless nonlinearity are derived. These conditions are stated in terms of the passivity of each of the subsystems and can be considered as a partial generalization of the hyperstability theorem. A Liapunov function approach is used in the proof without requiring the positive definiteness of the Liapunov function. Application to the robustness analysis of the infinite-dimensional linear quadratic regulator is also discussed.

Wen, John T.↗
NASA NTRS1992

On the continuous dependence with respect to sampling of the linear quadratic regulator problem for distributed parameter system

The convergence of solutions to the discrete- or sampled-time linear quadratic regulator problem and associated Riccati equation for infinite-dimensional systems to the solutions to the corresponding continuous time problem and equation, as the length of the sampling interval (the sampling rate) tends toward zero(infinity) is established. Both the finite-and infinite-time horizon problems are studied. In the finite-time horizon case, strong continuity of the operators that define the control system and performance index, together with a stability and consistency condition on the sampling scheme are required. For the infinite-time horizon problem, in addition, the sampled systems must be stabilizable and detectable, uniformly with respect to the sampling rate. Classes of systems for which this condition can be verified are discussed. Results of numerical studies involving the control of a heat/diffusion equation, a hereditary or delay system, and a flexible beam are presented and discussed.

Rosen, I. G.↗
NASA NTRS1996

Robust Stabilizing Compensators for Flexible Structures with Collocated Controls

For flexible structures with collocated rate and attitude sensors/actuators, we characterize compensator transfer functions which guarantee modal stability even when stiffness/inertia parameters are uncertain. While the compensators are finite-dimensional, the structure models are allowed to be infinite-dimensional (continuum models), with attendant complexity of the notion of stability; thus exponential stability is not possible and the best we can obtain is strong stability. Robustness is interpreted essentially as maintaining stability in the worst case. The conditions require that the compensator transfer functions be positive real and use is made of the Kalman-Yakubovic lemma to characterize them further. The concept of positive realness is shown to be equivalent to dissipativity in infinite dimensions. In particular we show that for a subclass of compensators it is possible to make the system strongly stable as well as dissipative in an appropriate energy norm.

Balakrishman, A. V.↗
NASA NTRS2006

Algorithms for Maneuvering Spacecraft Around Small Bodies

A document describes mathematical derivations and applications of autonomous guidance algorithms for maneuvering spacecraft in the vicinities of small astronomical bodies like comets or asteroids. These algorithms compute fuel- or energy-optimal trajectories for typical maneuvers by solving the associated optimal-control problems with relevant control and state constraints. In the derivations, these problems are converted from their original continuous (infinite-dimensional) forms to finite-dimensional forms through (1) discretization of the time axis and (2) spectral discretization of control inputs via a finite number of Chebyshev basis functions. In these doubly discretized problems, the Chebyshev coefficients are the variables. These problems are, variously, either convex programming problems or programming problems that can be convexified. The resulting discrete problems are convex parameter-optimization problems; this is desirable because one can take advantage of very efficient and robust algorithms that have been developed previously and are well established for solving such problems. These algorithms are fast, do not require initial guesses, and always converge to global optima. Following the derivations, the algorithms are demonstrated by applying them to numerical examples of flyby, descent-to-hover, and ascent-from-hover maneuvers.

Acikmese, A. Bechet↗
NASA NTRS2022

Computing Sensitivities in Evolutionary Systems: A Real-time Reduced Order Modeling Strategy

We present a new methodology for computing sensitivities in evolutionary systems using a model-driven low-rank approximation. To this end, we formulate a variational principle that seeks to minimize the distance between the time derivative of the reduced approximation and sensitivity dynamics. The first order optimality condition of the variational principle leads to a system of closed form evolution equations for an orthonormal basis and corresponding sensitivity coefficients. This approach allows for the computation of sensitivities with respect to a large number of parameters in an accurate and tractable manner by extracting correlations between different sensitivities on the fly. The presented method requires solving forward evolution equations, sidestepping the restrictions imposed by the forward/backward workflow of adjoint sensitivities. For example, the presented method, unlike the adjoint equation, does not impose any input/output load and can be used in applications in which real-time sensitivities are of interest. We demonstrate the utility of the method for three test cases: (1) computing sensitivity with respect to model parameters in the Rössler system, (2) computing sensitivity with respect to an infinite-dimensional forcing parameter in the chaotic Kuramoto--Sivashinsky equation, and (3) computing sensitivity with respect to reaction parameters for species transport in a turbulent reacting flow.

Reduced order model↗
NASA NTRSDate not supplied

Hamiltonian Control for Coupled Bending-Torsion Motion of a Swept Wing With R. T. Jones' Approximation

This paper presents distributed Hamiltonian control design approach for a Lagrangian infinite-dimensional system describing the coupled bending-torsion motion of a swept wing in unsteady airflow, which is taken into account using Theodorsen's complex valued function of reduced frequency. An implementable flutter suppressing control surface deflection commands are obtained using modes separation and R.T. Jones's approximation, and were validated in desktop simulations for flight conditions beyond the flutter airspeed.

Vahram Stepanyan↗