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At least 145 records · Page 8

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DOE OSTI2021

A Convex Data-Driven Approach for Nonlinear Control Synthesis

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1973

Asymptotic stability and instability of large-scale systems

The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations.

Grujic, L. T.↗

NASA NTRS1977

Comparison of continuous and discrete adaptive identification algorithms

Discretization of a popular continuous-time control algorithm is effected and an equivalent discrete-time identification law developed and compared to a published discrete identification algorithm developed from Lyapunov Theory. Results are compared as regards asymptotic stability as insured using Lyapunov theory. Some analysis and design guidelines are proposed as regards implementation and practical utility.

Colburn, B. K.↗

NASA NTRS1985

Calculating Closed-Loop Sensitivity to Parameter Variations

Performance sensitivity calculated by back substitution. Original Lyapunov equation differentiated two times with respect to parameter p. Each differential turns out to yield Lyapunov equation for first and second derivatives with respective to parameter p of steady-state covariance of state variables. Derivatives desired closed-loop performance sensitivities.

Schaechter, D. B.↗

NASA NTRS1983

Stability and capture of asteroids

The problem of stability of asteroids is treated from the point of view of Hill's stability-concept and using Lyapunov's Characteristic Numbers. The quantitative measure of stability (S) introduced earlier is evaluated for over 300 asteroids and a surprisingly simple relation is established between the semi-major axes of some of the asteroids' orbits and S. A detailed analysis is presented of the Lyapunov Characteristic Numbers for two minor planets and the time-variation of these numbers is discussed. The technology of capture of asteroids is vitally dependent on their orbital stability, therefore, these two problems, i.e., capture and stability, are closely related. In fact, some predictable instabilities may be properly utilized to capture and/or change asteroidal orbits to accomplish collisions with the Earth.

Szebehely, V.↗

NASA NTRS1988

New class of control laws for robotic manipulators. I - Nonadaptive case. II - Adaptive case

A new class of exponentially stabilizing control laws for joint level control of robot arms is discussed. Closed-loop exponential stability has been demonstrated for both the set point and tracking control problems by a slight modification of the energy Lyapunov function and the use of a lemma which handles third-order terms in the Lyapunov function derivatives. In the second part, these control laws are adapted in a simple fashion to achieve asymptotically stable adaptive control. The analysis addresses the nonlinear dynamics directly without approximation, linearization, or ad hoc assumptions, and uses a parameterization based on physical (time-invariant) quantities.

Wen, John T.↗

NASA NTRS1988

Robustness analysis based on passivity

A technique of robust stability analysis for multivariable control systems is introduced. Existing tools in the literature are mostly based on small gain type of conditions such as the H(infinity)-norm or mu measure. In contrast, the approach presented is motivated by passivity theory. The natural energy Lyapunov function associated with a passive system allows robustness to be analyzed with Lyapunov stability theory. Consequently, nonlinear and time-varying perturbations can be considered within the same framework. For unstructured uncertainties, the robustness margin is characterized in terms of an index related to the Hermitian part of a transfer function. If the uncertainties contain additional diagonal structure, stability margins are obtained for variations of each uncertain block in both positive and negative directions. Applications of this robustness analysis technique are illustrated.

Wen, John T.↗

NASA NTRS1988

Chaos in a spatially-developing plane mixing layer

A spatially-developing plane mixing layer was analyzed for chaotic behavior. A direct numerical simulation of the Navier-Stokes equations in a 2-D domain infinite in y and having inflow-outflow boundary conditions in x was used for data. Spectra, correlation dimension and the largest Lyapunov exponent were computed as functions of downstream distance x. When forced at a single (fundamental) frequency with maximum amplitude, the flow is periodic at the inflow but becomes aperiodic with increasing x. The aperiodic behavior is caused by the presence of a noisy subharmonic caused by the feedback between the necessarily nonphysical inflow and outflow boundary conditions. In order to overshadow this noise the flow was also studied with the same fundamental forcing and added random forcing of amplitude upsilon prime sub R/delta U = 0.01 at the inlet. Results were qualitatively the same in both cases: for small x, spectral peaks were sharp and dimension was nearly 1, but as x increased a narrowband spectral peak grew, spectra decayed exponentially at high frequencies and dimension increased to greater than 3. Based on these results, the flow appears to exhibit deterministic chaos. However, at no location was the largest Lyapunov exponent found to be significantly greater than zero.

Broze, J. G.↗

NASA NTRS1989

One-degree-of-freedom motion induced by modeled vortex shedding

The motion of an elastically supported cylinder forced by a nonlinear, quasi-static, aerodynamic model with the unusual feature of a motion-dependent forcing frequency was studied. Numerical solutions for the motion and the Lyapunov exponents are presented for three forcing amplitudes and two frequencies (1.0 and 1.1 times the Strouhal frequency). Initially, positive Lyapunov exponents occur and the motion can appear chaotic. After thousands of characteristic times, the motion changes to a motion (verified analytically) that is periodic and damped. This periodic, damped motion was not observed experimentally, thus raising questions concerning the modeling.

Yates, L. A.↗

NASA NTRS1989

A Study of Strong Stability of Distributed Systems

The strong stability of distributed systems is studied and the problem of characterizing strongly stable semigroups of operators associated with distributed systems is addressed. Main emphasis is on contractive systems. Three different approaches to characterization of strongly stable contractive semigroups are developed. The first one is an operator theoretical approach. Using the theory of dilations, it is shown that every strongly stable contractive semigroup is related to the left shift semigroup on an L(exp 2) space. Then, a decomposition for the state space which identifies strongly stable and unstable states is introduced. Based on this decomposition, conditions for a contractive semigroup to be strongly stable are obtained. Finally, extensions of Lyapunov's equation for distributed parameter systems are investigated. Sufficient conditions for weak and strong stabilities of uniformly bounded semigroups are obtained by relaxing the equivalent norm condition on the right hand side of the Lyanupov equation. These characterizations are then applied to the problem of feedback stabilization. First, it is shown via the state space decomposition that under certain conditions a contractive system (A,B) can be strongly stabilized by the feedback -B(*). Then, application of the extensions of the Lyapunov equation results in sufficient conditions for weak, strong, and exponential stabilizations of contractive systems by the feedback -B(*). Finally, it is shown that for a contractive system, the first derivative of x with respect to time = Ax + Bu (where B is any linear bounded operator), there is a related linear quadratic regulator problem and a corresponding steady state Riccati equation which always has a bounded nonnegative solution.

Cataltepe, Tayfun↗

NASA NTRS1992

Stochastic stability properties of jump linear systems

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented.

Feng, Xiangbo↗

NASA NTRS1992

Passivity/Lyapunov based controller design for trajectory tracking of flexible joint manipulators

A passivity and Lyapunov based approach for the control design for the trajectory tracking problem of flexible joint robots is presented. The basic structure of the proposed controller is the sum of a model-based feedforward and a model-independent feedback. Feedforward selection and solution is analyzed for a general model for flexible joints, and for more specific and practical model structures. Passivity theory is used to design a motor state-based controller in order to input-output stabilize the error system formed by the feedforward. Observability conditions for asymptotic stability are stated and verified. In order to accommodate for modeling uncertainties and to allow for the implementation of a simplified feedforward compensation, the stability of the system is analyzed in presence of approximations in the feedforward by using a Lyapunov based robustness analysis. It is shown that under certain conditions, e.g., the desired trajectory is varying slowly enough, stability is maintained for various approximations of a canonical feedforward.

Sicard, Pierre↗

NASA NTRS1993

Large-scale isentropic mixing properties of the Antarctic polar vortex from analyzed winds

Winds derived from analyzed geopotential height fields are used to study quasi-horizontal mixing by the large-scale flow in the lower stratosphere during austral spring. This is the period when the Antarctic ozone hole appears and disappears. Trajectories are computed for large ensembles of particles initially inside and outside the main polar vortex. Mixing and transport are diagnosed through estimates of finite time Lyapunov exponents and Lagrangian dispersion statistics of the tracer trajectories. At 450 K and above prior to the vortex breakdown: Lyapunov exponents are a factor of 2 smaller inside the vortex than outside; diffusion coefficients are an order of magnitude smaller inside than outside the vortex; and the trajectories reveal little exchange of air across the vortex boundary. At lower levels (425 and 400 K) mixing is greater, and there is substantial exchange of air across the vortex boundary. In some years there are large wave events that expel small amounts of vortex air into the mid-latitudes. At the end of the spring season during the vortex breakdown there is rapid mixing of air across the vortex boundary, which is evident in the mixing diagnostics and the tracer trajectories.

Bowman, Kenneth P.↗

NASA NTRS2002

LPV Controller Interpolation for Improved Gain-Scheduling Control Performance

In this paper, a new gain-scheduling control design approach is proposed by combining LPV (linear parameter-varying) control theory with interpolation techniques. The improvement of gain-scheduled controllers can be achieved from local synthesis of Lyapunov functions and continuous construction of a global Lyapunov function by interpolation. It has been shown that this combined LPV control design scheme is capable of improving closed-loop performance derived from local performance improvement. The gain of the LPV controller will also change continuously across parameter space. The advantages of the newly proposed LPV control is demonstrated through a detailed AMB controller design example.

Wu, Fen↗

NASA NTRS2003

Parameter Transient Behavior Analysis on Fault Tolerant Control System

In a fault tolerant control (FTC) system, a parameter varying FTC law is reconfigured based on fault parameters estimated by fault detection and isolation (FDI) modules. FDI modules require some time to detect fault occurrences in aero-vehicle dynamics. This paper illustrates analysis of a FTC system based on estimated fault parameter transient behavior which may include false fault detections during a short time interval. Using Lyapunov function analysis, the upper bound of an induced-L2 norm of the FTC system performance is calculated as a function of a fault detection time and the exponential decay rate of the Lyapunov function.

Belcastro, Christine↗

NASA NTRS2004

Switching LPV Control of an F-16 Aircraft via Controller State Reset

In flight control, the design objective and the aircraft dynamics may be different in low and high angle of attack regions. This paper presents a systematic switching Linear Parameter-varying LPV method to determine if it is practical to use for flight control designs over a wide angle of attack region. The approach is based on multiple parameter-dependent Lyapunov functions a family of LPV controllers are designed, and each of them is suitable for a specific parameter subspace. The state of the controller is reset to guarantee the stability requirement of the Lyapunov function when the switching event occurs. Two parameter-dependent switching logics, hysteresis switching and switching with average dwell times are examined. The proposed switching LPV control scheme is applied to an F-16 aircraft model with different design objectives and aircraft dynamics in low and high angle of attack regions. The nonlinear simulating results using both switching logics are compared.

Lu, Bei↗

NASA NTRS2003

Extension of Liouville Formalism to Postinstability Dynamics

A mathematical formalism has been developed for predicting the postinstability motions of a dynamic system governed by a system of nonlinear equations and subject to initial conditions. Previously, there was no general method for prediction and mathematical modeling of postinstability behaviors (e.g., chaos and turbulence) in such a system. The formalism of nonlinear dynamics does not afford means to discriminate between stable and unstable motions: an additional stability analysis is necessary for such discrimination. However, an additional stability analysis does not suggest any modifications of a mathematical model that would enable the model to describe postinstability motions efficiently. The most important type of instability that necessitates a postinstability description is associated with positive Lyapunov exponents. Such an instability leads to exponential growth of small errors in initial conditions or, equivalently, exponential divergence of neighboring trajectories. The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

Zak, Michail↗

DOE OSTI2020

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Huang, Bowen↗