Mark Milman

A new algorithm for L 2 optimal model reduction

In this paper the quadratically optimal model reduction problem for single-input, single-output systems is considered. The reduced order model is determined by minimizing the integral of the magnitude-squared of the transfer function error. It is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and, on this basis, a two-step iterative algorithm is developed combining a least squares solver with a gradient minimizer. Convergence of the proposed algorithm to stationary values of the quadratic cost function is proved. The formulation is extended to handle the frequency-weighted optimal model reduction problem. Three examples demonstrate the optimization algorithm.

Automated on-orbit frequency domain identification for large space structures

Abstract Recent experiences in the field of flexible structure control in space have indicated a need for on-orbit system identification to support robust control redesign to avoid in-flight instabilities and maintain high spacecraft performance. This paper highlights an automated frequency domain system identification methodology recently developed to fulfill this need. The methodology is focused to support (1) the estimation of system quantities useful for robust control analysis and design, (2) experiment design tailored to performing system identification in a typically constrained on-orbit environment and (3) the automation of operations to reduce “human in the loop” requirements. A basic overview of the methodology is presented first, followed by an experimental verification of the approach performed on the JPL/AFAL test-bed facility.

Combined control-structural optimization

ConclusionsAn approach for combined control-structure optimization keyed to enhancing early design trade-offs has been outlined and illustrated by numerical examples. The approach employs a homotopic strategy and appears to be effective for generating families of designs that can be used in these early trade studies.Analytical results were obtained for classes of structure/control objectives with LQG and LQR costs. For these, we have demonstrated that global optima can be computed for small values of the homotopy parameter. Conditions for local optima along the homotopy path were also given. Details of three numerical examples employing the LQR control cost were given showing variations of the optimal design variables along the homotopy path. The results of the second example suggest that introducing a second homotopy parameter relating the two parts of the control index in the LQG/LQR formulation might serve to enlarge the family of Pareto optima, but its effect on modifying the optimal structural shapes may be analogous to the original parameter λ.

Mode shape expansion techniques for prediction - Experimental evaluation

Mode shape expansion techniques fall under four broad categories. Spatial interpolation methods use geometric information to infer mode shapes at unmeasured locations. Direct methods use the dynamic equations of motion to obtain closed-form solutions to the expanded eigenvectors. These methods can be interpreted as constrained optimization problems. Projection methods use a least-squares formulation that also can be formulated through constrained optimization. Error methods use a formulation that can account for uncertainties in the measurements and in the prediction. This includes penalty methods and the new expansion techniques based on least-squares minimization techniques with quadratic inequality constraints (LSQI). Some of these expansion techniques are selected herein for evaluation using the full set of experimental data obtained on the microprecision interferometer test bed. Both a pretest and an updated analytical model are considered in the trade study. The robustness of these methods is verified with respect to measurement noise, model deficiency, number of measured degrees of freedom, and accelerometer location. It is shown that the proposed LSQI method has the best performance and can reliably predict mode shapes, even in very adverse situations.

Eigenvalue error analysis of viscously damped structures using a Ritz reduction method

The efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed. Eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with Ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem. An analysis of this technique in the case of a single damper is presented. A priori and a posteriori error estimates are generated and tested on numerical examples. Comparison theorems with modally truncated models and a Markov parameter matching reduced-order model are derived. These theorems corroborate the heuristic that residual flexibility methods improve low-frequency approximation of the system. The analysis leads to other techniques for eigenvalue approximation. Approximate closed-form solutions are derived that include a refinement to eigenvalue derivative methods for approximation. An efficient Newton scheme is also developed. A numerical example is presented demonstrating the effectiveness of each of these methods.

A note on the solution to a common thermal network problem encountered in heat-transfer analysis of spacecraft

Abstract A widely used discretization method for modeling thermal systems is the thermal network approach. The network approach is derived from energy balance equations and is equivalent to a particular finite difference discretization of the underlying heat-transfer equation. The steady-state problems that arise in the analysis of spacecraft systems using network models are frequently dominated by radiative transfer, which introduces quartic nonlinearities in the equations. Although these systems are routinely encountered, there has not appeared any detailed analysis of these equations. Questions of existence and uniqueness of solutions and numerical methods for solving the systems have never been addressed in any generality. In this paper, general existence and uniqueness properties of the network equations are established. Globally convergent methods for solving the systems are developed and insight into the relative success of existing methods is given. Numerical examples are presented illustrating the methods. The perspective adopted here is also useful in interdisciplinary applications. A simple example involving thermal control is used to demonstrate this.

Error sources and algorithms for white-light fringe estimation at low light levels

We address the problem of highly accurate phase estimation at low light levels, as required by the Space Interferometry Mission (SIM). The most stringent SIM requirement in this regard is that the average phase error over a 30-s integration time correspond to a path-length error of approximately 30 pm. Most conventional phase-estimation algorithms exhibit significant enough bias at the signal levels at which the SIM will be operating so that some correction is necessary. Several algorithms are analyzed, and methods of compensating for their bias are developed. Another source of error in phase estimation occurs because the phase is not constant over the integration period. Errors that are due to spacecraft motion, the motion of compensating optical elements, and modulation errors are analyzed and simulated. A Kalman smoothing approach for compensating for these errors is introduced.

On a class of operators on Hilbert space with applications to factorization and systems theory

Abstract A class of operators is defined in a Hilbert resolution space setting that offers a new perspective on problems of causal invertibility, special factorization, and the theory of quadratic cost optimization problems for dynamical systems. The major results include an extension of the special factorization to a class of noncompact operators and the definition of an abstract state space. These results are then used to obtain an optimal feedback solution to an abstract linear regular-quadratic cost problem.

Accuracy and covariance analysis of global astrometry with the Space Interferometry Mission

Construction of an all-sky microarcsecond astrometric grid is an essential preliminary step toward global astrometry with the Space Interferometry Mission (SIM). The general setup of the global astrometry problem is given, and fast, robust algorithms to solve the problem are described. The results of extensive numerical simulations are used to outline the conditions necessary to achieve the performance goal of absolute astrometric accuracy. Rigorous computation of the global covariance matrix makes it possible to correctly interpret and utilize the SIM data. We formulate and resolve the new issue of the basic performance uncertainty of a single astrometric mission.

A spatial operator algebra for manipulator modeling and control

A spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The operators themselves are elements in the algebra of linear bounded operators. The effect of these operators when operating on elements in the domain is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of spatially recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for developing corresponding control and trajectory design algorithms. Expressions interpreted within the operator algorithm framework led to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the high-level operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms has been greatly simplified. The analytical formulation of the operator algebra and its implementation in Ada are discussed.<<ETX>>