Daniel P. Giesy

Brief paper: Application of multiobjective optimization in aircraft control systems design

Multiobjective optimization techniques are applied in the design of an aircraft lateral control system. A large manned reentry vehicle and a fighter aircraft are considered. An algorithm suggested by Lins Proper Inequality Constraints method, is implemented in the numerical computation of Pareto-optimal solutions. Subsequently, a trade-off analysis of several Pareto-optimal solutions is conducted.

The NASA Langley Multidisciplinary Uncertainty Quantification Challenge

NASA missions often involve the development of new vehicles and systems that must be designed to operate in harsh domains with a wide array of operating conditions. These missions involve high-consequence and safety-critical systems for which quantitative data is either very sparse or prohibitively expensive to collect. Limited heritage data may exist, but is also usually sparse and may not be directly applicable to the system of interest, making uncertainty quantification extremely challenging. NASA modeling and simulation standards require estimates of uncertainty and descriptions of any processes used to obtain these estimates. The NASA Langley Research Center has developed an uncertainty quantification challenge problem in an effort to focus a community of researchers towards a common problem. This challenge problem features key issues in both uncertainty quantification and robust design using a discipline-independent formulation. While the formulation is indeed discipline-independent, the underlying model, as well as the requirements imposed upon it, describes a realistic aeronautics application. A few high-level details of this application are provided at the end of this document. Additional information is available at: http://uqtools.larc.nasa.gov/nda-uq-challenge-problem-2014/.

Multicriteria Optimization Methods for Design of Aircraft Control Systems

In the design of airplane control systems, many disparate objectives must be considered. The pilot desires rapid, precise, and decoupled response to his control inputs, so that natural objective functions for computer-aided design (CAD) are computable functions that are useful measures of the speed, stability, and coupling of the responses. These response properties are often referred to as the handling qualities or flying qualities of the airplane. The military has developed a set of specifications for a number of handling quality functions, and the CAD research described in this paper uses objective functions based on these military handling qualities criteria. Additional design objective functions have been developed to avoid control limiting, since there are always limits on available control in any real system, and limiting can be destabilizing in an automatic control system. Another important property of a good design is that it be “robust”; that is, the design objectives should be insensitive to significant uncertainties in system parameters. In fact, such insensitivity is an essential property of any well-designed feedback system. Therefore, a vector of “stochastic sensitivity” functions is defined as the vector of probabilities that each “deterministic” objective violate specified requirement limits, and decreasing sensitivity is considered a design objective. If both the deterministic objectives (the nominal or expected values) and their sensitivities are considered in the design process, the number of objective functions is doubled. Moreover, modern airplanes operate over a wide range of speed and altitude, and the linearized differential equations that are used to describe the response to controls (the plant dynamic models) are different at each flight condition.

A Verification-driven Approach to Control Analysis and Tuning

This paper proposes a methodology for the analysis and tuning of controllers using control verification metrics. These metrics, which are introduced in a companion paper, measure the size of the largest uncertainty set of a given class for which the closed-loop specifications are satisfied. This framework integrates deterministic and probabilistic uncertainty models into a setting that enables the deformation of sets in the parameter space, the control design space, and in the union of these two spaces. In regard to control analysis, we propose strategies that enable bounding regions of the design space where the specifications are satisfied by all the closed-loop systems associated with a prescribed uncertainty set. When this is unfeasible, we bound regions where the probability of satisfying the requirements exceeds a prescribed value. In regard to control tuning, we propose strategies for the improvement of the robust characteristics of a baseline controller. Some of these strategies use multi-point approximations to the control verification metrics in order to alleviate the numerical burden of solving a min-max problem. Since this methodology targets non-linear systems having an arbitrary, possibly implicit, functional dependency on the uncertain parameters and for which high-fidelity simulations are available, they are applicable to realistic engineering problems.

Interval predictor models with a formal characterization of uncertainty and reliability

This paper develops techniques for constructing empirical predictor models based on observations. By contrast to standard models, which yield a single predicted output at each value of the models inputs, Interval Predictors Models (IPM) yield an interval into which the unobserved output is predicted to fall. The IPMs proposed prescribe the output as an interval valued function of the models inputs, render a formal description of both the uncertainty in the models parameters and of the spread in the predicted output. Uncertainty is prescribed as a hyper-rectangular set in the space of models parameters. The propagation of this set through the empirical model yields a range of outputs of minimal spread containing all (or, depending on the formulation, most) of the observations. Optimization-based strategies for calculating IPMs and eliminating the effects of outliers are proposed. Outliers are identified by evaluating the extent by which they degrade the tightness of the prediction. This evaluation can be carried out while the IPM is calculated. When the data satisfies mild stochastic assumptions, and the optimization program used for calculating the IPM is convex (or, when its solution coincides with the solution to an auxiliary convex program), the models reliability (that is, the probability that a future observation would be within the predicted range of outputs) can be bounded rigorously by a non-asymptotic formula.

Application of Interval Predictor Models to Space Radiation Shielding

This paper develops techniques for predicting the uncertainty range of an output variable given input-output data. These models are called Interval Predictor Models (IPM) because they yield an interval valued function of the input. This paper develops IPMs having a radial basis structure. This structure enables the formal description of (i) the uncertainty in the models parameters, (ii) the predicted output interval, and (iii) the probability that a future observation would fall in such an interval. In contrast to other metamodeling techniques, this probabilistic certi cate of correctness does not require making any assumptions on the structure of the mechanism from which data are drawn. Optimization-based strategies for calculating IPMs having minimal spread while containing all the data are developed. Constraints for bounding the minimum interval spread over the continuum of inputs, regulating the IPMs variation/oscillation, and centering its spread about a target point, are used to prevent data over tting. Furthermore, we develop an approach for using expert opinion during extrapolation. This metamodeling technique is illustrated using a radiation shielding application for space exploration. In this application, we use IPMs to describe the error incurred in predicting the ux of particles resulting from the interaction between a high-energy incident beam and a target.

Sampling-based Strategies for the Estimation of Probabilistic Sensitivities

This note presents the mathematical background and numerical evaluation of several approaches for calculating probabilistic sensitivities. In particular, we use the finite difference method and the Leibniz integral rule to derive two formulations for approximating derivatives of the mean, the variance, and the failure probability of a dependent variable with respect to means, and variances of the independent variables. These derivatives not only indicate the sensitivity to the uncertainty model assumed but also allow for the identification of the most dominant uncertain parameters. Deterministic sampling techniques that eliminate the random character of the numerical approximation are used to evaluate the resulting expressions. Examples admitting closed-form expressions for the sensitivities are used to validate the efficiency and accuracy of the approximations and to perform convergence analyses as a function of the discretization parameters. Remarks on the advantages and limitations of each method as well as our take on the best practices are also presented.

Strict Constraint Feasibility in Analysis and Design of Uncertain Systems

This paper proposes a methodology for the analysis and design optimization of models subject to parametric uncertainty, where hard inequality constraints are present. Hard constraints are those that must be satisfied for all parameter realizations prescribed by the uncertainty model. Emphasis is given to uncertainty models prescribed by norm-bounded perturbations from a nominal parameter value, i.e., hyper-spheres, and by sets of independently bounded uncertain variables, i.e., hyper-rectangles. These models make it possible to consider sets of parameters having comparable as well as dissimilar levels of uncertainty. Two alternative formulations for hyper-rectangular sets are proposed, one based on a transformation of variables and another based on an infinity norm approach. The suite of tools developed enable us to determine if the satisfaction of hard constraints is feasible by identifying critical combinations of uncertain parameters. Since this practice is performed without sampling or partitioning the parameter space, the resulting assessments of robustness are analytically verifiable. Strategies that enable the comparison of the robustness of competing design alternatives, the approximation of the robust design space, and the systematic search for designs with improved robustness characteristics are also proposed. Since the problem formulation is generic and the solution methods only require standard optimization algorithms for their implementation, the tools developed are applicable to a broad range of problems in several disciplines.

Robust Control Design for Uncertain Nonlinear Dynamic Systems

Robustness to parametric uncertainty is fundamental to successful control system design and as such it has been at the core of many design methods developed over the decades. Despite its prominence, most of the work on robust control design has focused on linear models and uncertainties that are non-probabilistic in nature. Recently, researchers have acknowledged this disparity and have been developing theory to address a broader class of uncertainties. This paper presents an experimental application of robust control design for a hybrid class of probabilistic and non-probabilistic parametric uncertainties. The experimental apparatus is based upon the classic inverted pendulum on a cart. The physical uncertainty is realized by a known additional lumped mass at an unknown location on the pendulum. This unknown location has the effect of substantially altering the nominal frequency and controllability of the nonlinear system, and in the limit has the capability to make the system neutrally stable and uncontrollable. Another uncertainty to be considered is a direct current motor parameter. The control design objective is to design a controller that satisfies stability, tracking error, control power, and transient behavior requirements for the largest range of parametric uncertainties. This paper presents an overview of the theory behind the robust control design methodology and the experimental results.

Application of Dimensionality Reduction to a Large-Order Uncertain Dynamic System

This paper investigates dimensionality reduction techniques for linear, time-invariant systems subject to general nonlinear dependencies on uncertain parameters. In the context of this paper, dimensionality reduction refers to simultaneous reductions in both the number of model states and the number of uncertain parameters. Two complementary approaches will be reviewed, one for estimating the total worst case H-infinity norm error associated with both model state and parameter order reductions, and another, which is essentially the inverse problem, that determines the largest allowable parameter bounds for a given total Hinfinity norm error for the dimensionally reduced problem. Application of these methods will be demonstrated on a large-order finite element model of a flexible spacecraft.