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NASA NTRS1976

An iterative decoupling solution method for large scale Lyapunov equations

A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

Athay, T. M.↗

NASA NTRS2011

A Globally Stable Lyapunov Pointing and Rate Controller for the Magnetospheric MultiScale Mission (MMS)

The Magnetospheric MultiScale Mission (MMS) is scheduled to launch in late 2014. Its primary goal is to discover the fundamental plasma physics processes of reconnection in the Earth's magnetosphere. Each of the four MMS spacecraft is spin-stabilized at a nominal rate of 3 RPM. Traditional spin-stabilized spacecraft have used a number of separate modes to control nutation, spin rate, and precession. To reduce the number of modes and simplify operations, the Delta-H control mode is designed to accomplish nutation control, spin rate control, and precession control simultaneously. A nonlinear design technique, Lyapunov's method, is used to design the Delta-H control mode. A global spin rate controller selected as the baseline controller for MMS, proved to be insufficient due to an ambiguity in the attitude. Lyapunov's design method was used to solve this ambiguity, resulting in a controller that meets the design goals. Simulation results show the advantage of the pointing and rate controller for maneuvers larger than 90 deg and provide insight into the performance of this controller.

Shah, Neerav↗

NASA NTRS1975

Control optimization, stabilization and computer algorithms for aircraft applications

Research related to reliable aircraft design is summarized. Topics discussed include systems reliability optimization, failure detection algorithms, analysis of nonlinear filters, design of compensators incorporating time delays, digital compensator design, estimation for systems with echoes, low-order compensator design, descent-phase controller for 4-D navigation, infinite dimensional mathematical programming problems and optimal control problems with constraints, robust compensator design, numerical methods for the Lyapunov equations, and perturbation methods in linear filtering and control.

Source record↗

NASA NTRS1975

Prediction of axial-flow instabilities in a turbojet engine by use of a multistage compressor simulation on the digital computer

A method of estimating the undistorted stall line for an axial-flow compressor by using the digital computer is presented. The method involves linearization of nonlinear dynamic equations about an operating point on a speed line, and then application of the first method of Lyapunov to determine the stability of the nonlinear system from the stability of the linear system. The method is applied to a simulation of the J85 compressor, which utilizes stage stacking and lumped volume techniques for the interstage regions to simulate steady-state and dynamic compressor performance. The stability boundary predicted by the digital simulation compares quite well with the stall line predicted by a dynamic simulation of the J85 compressor programmed on the analog computer. Since previous studies have shown that the analog-predicted stall line agrees well with the stall line of the compressor, the digital method presented is also a good means of estimating the stall line.

Daniele, C. J.↗

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 NTRS2005

Variable Structure Control of a Hand-Launched Glider

Variable structure control system design methods are applied to the problem of aircraft spin recovery. A variable structure control law typically has two phases of operation. The reaching mode phase uses a nonlinear relay control strategy to drive the system trajectory to a pre-defined switching surface within the motion state space. The sliding mode phase involves motion along the surface as the system moves toward an equilibrium or critical point. Analysis results presented in this paper reveal that the conventional method for spin recovery can be interpreted as a variable structure controller with a switching surface defined at zero yaw rate. Application of Lyapunov stability methods show that deflecting the ailerons in the direction of the spin helps to insure that this switching surface is stable. Flight test results, obtained using an instrumented hand-launched glider, are used to verify stability of the reaching mode dynamics.

Anderson, Mark R.↗

NASA NTRS2010

A Spectral Lyapunov Function for Exponentially Stable LTV Systems

This paper presents the formulation of a Lyapunov function for an exponentially stable linear timevarying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.

Zhu, J. Jim↗

NASA NTRS1989

Numerical solution of large Lyapunov equations

A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.

Saad, Youcef↗

NASA NTRS1989

Joint-space adaptive control of a 6 DOF end-effector with closed-kinematic chain mechanism

The development is presented for a joint-space adaptive scheme that controls the joint position of a six-degree-of-freedom (DOF) robot end-effector performing fine and precise motion within a very limited workspace. The end-effector was built to study autonomous assembly of NASA hardware in space. The design of the adaptive controller is based on the concept of model reference adaptive control (MRAC) and Lyapunov direct method. In the development, it is assumed that the end-effector performs slowly varying motion. Computer simulation is performed to investigate the performance of the developed control scheme on position control of the end-effector. Simulation results manifest that the adaptive control scheme provides excellent tracking of several test paths.

Nguyen, Charles C.↗

NASA NTRS1989

Design of an adaptive controller for a telerobot manipulator

The design of a joint-space adaptive control scheme is presented for controlling the slave arm motion of a dual-arm telerobot system developed at Goddard Space Flight Center (GSFC) to study telerobotic operations in space. Each slave arm of the dual-arm system is a kinematically redundant manipulator with 7 degrees of freedom (DOF). Using the concept of model reference adaptive control (MRAC) and Lyapunov direct method, an adatation algorithm is derived which adjusts the PD controller gains of the control scheme. The development of the adaptive control scheme assumes that the slave arm motion is non-compliant and slowly-varying. The implementation of the derived control scheme does not need the computation of the manipulator dynamics, which makes the control scheme sufficiently fast for real-time applications. Computer simulation study performed for the 7-DOF slave arm shows that the developed control scheme can efficiently adapt to sudden change in payloads while tracking various test trajectories such as ramp or sinusoids with negligible position errors.

Nguyen, Charles C.↗

NASA NTRS1990

Position control of redundant manipulators using an adaptive error-based control scheme

A Cartesian-space control scheme is developed to control the motion of kinematically redundant manipulators with 7 degrees of freedom (DOF). The control scheme consists mainly of proportional derivative (PD) controllers whose gains are adjusted by an adaptation law driven by the errors between the desired and actual trajectories. The adaptation law is derived using the concept of model reference adaptive control (MRAC) and Lyapunov direct method under the assumption that the manipulator performs non-compliant and slowly-varying motions. The developed control scheme is computationally efficient because its implementation does not require the computation of the manipulator dynamics. Computer simulation performed to evaluate the control scheme performance is presented and discussed.

Nguyen, Charles C.↗

NASA NTRS1992

Experimental investigation of adaptive control of a parallel manipulator

The implementation of a joint-space adaptive control scheme used to control non-compliant motion of a Stewart Platform-based Manipulator (SPBM) is presented. The SPBM is used in a facility called the Hardware Real-Time Emulator (HRTE) developed at Goddard Space Flight Center to emulate space operations. The SPBM is comprised of two platforms and six linear actuators driven by DC motors, and possesses six degrees of freedom. The report briefly reviews the development of the adaptive control scheme which is composed of proportional-derivative (PD) controllers whose gains are adjusted by an adaptation law driven by the errors between the desired and actual trajectories of the SPBM actuator lengths. The derivation of the adaptation law is based on the concept of model reference adaptive control (MRAC) and Lyapunov direct method under the assumption that SPBM motion is slow as compared to the controller adaptation rate. An experimental study is conducted to evaluate the performance of the adaptive control scheme implemented to control the SPBM to track a vertical and circular paths under step changes in payload. Experimental results show that the adaptive control scheme provides superior tracking capability as compared to fixed-gain controllers.

Nguyen, Charles C.↗

NASA NTRS1993

Attitude control of spacecraft using neural networks

This paper investigates the use of radial basis function neural networks for adaptive attitude control and momentum management of spacecraft. In the first part of the paper, neural networks are trained to learn from a family of open-loop optimal controls parameterized by the initial states and times-to-go. The trained is then used for closed-loop control. In the second part of the paper, neural networks are used for direct adaptive control in the presence of unmodeled effects and parameter uncertainty. The control and learning laws are derived using the method of Lyapunov.

Vadali, Srinivas R.↗

NASA NTRS2013

Explicit Low-Thrust Guidance for Reference Orbit Targeting

The problem of a low-thrust spacecraft controlled to a reference orbit is addressed in this paper. A simple and explicit low-thrust guidance scheme with constrained thrust magnitude is developed by combining the fundamental equations of motion for constrained systems from analytical dynamics with a Lyapunov-based method. Examples are given for a spacecraft controlled to a reference trajectory in the circular restricted three body problem.

low-thrust guidance↗

NASA NTRS2004

Intelligent, Robust Control of Deteriorated Turbofan Engines via Linear Parameter Varying Quadratic Lyapunov Function Design

A method for accommodating engine deterioration via a scheduled Linear Parameter Varying Quadratic Lyapunov Function (LPVQLF)-Based controller is presented. The LPVQLF design methodology provides a means for developing unconditionally stable, robust control of Linear Parameter Varying (LPV) systems. The controller is scheduled on the Engine Deterioration Index, a function of estimated parameters that relate to engine health, and is computed using a multilayer feedforward neural network. Acceptable thrust response and tight control of exhaust gas temperature (EGT) is accomplished by adjusting the performance weights on these parameters for different levels of engine degradation. Nonlinear simulations demonstrate that the controller achieves specified performance objectives while being robust to engine deterioration as well as engine-to-engine variations.

Turso, James A.↗

NASA NTRS1977

Exact solution of some linear matrix equations using algebraic methods

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Djaferis, T. E.↗

NASA NTRS2008

An Optimal Control Modification to Model-Reference Adaptive Control for Fast Adaptation

This paper presents a method that can achieve fast adaptation for a class of model-reference adaptive control. It is well-known that standard model-reference adaptive control exhibits high-gain control behaviors when a large adaptive gain is used to achieve fast adaptation in order to reduce tracking error rapidly. High gain control creates high-frequency oscillations that can excite unmodeled dynamics and can lead to instability. The fast adaptation approach is based on the minimization of the squares of the tracking error, which is formulated as an optimal control problem. The necessary condition of optimality is used to derive an adaptive law using the gradient method. This adaptive law is shown to result in uniform boundedness of the tracking error by means of the Lyapunov s direct method. Furthermore, this adaptive law allows a large adaptive gain to be used without causing undesired high-gain control effects. The method is shown to be more robust than standard model-reference adaptive control. Simulations demonstrate the effectiveness of the proposed method.

Nguyen, Nhan T.↗

NASA NTRS1972

An approximate method for the synthesis of optimal control of distributed systems.

An approximate method is presented for the synthesis of an optimal control of distributed parameter systems, based on a combination of the ideas of Lyapunov's direct method and those of Bellman's successive approximation in policy space. The method is such that the approximate equation is improved after each iteration in the sense of the performance index being used.

Park, K. E.↗