## TEMPUS: A Methodology For Model-Based Robust-Optimal Design Of Time-Dynamic System Identification Experiments Using Variational Asymptotic Expansions

##### Abstract

The development of appropriate flight tests has proven to be a critical element in the development process of many revolutionary next-generation aerospace vehicles. For example, in the case of hypersonic vehicles with air-breathing SCRAMjet engines, sophisticated computational analyses have been developed which require extensive validation and calibration with physical test data. The current state of hypersonic ground testing facilities has not yet been able to accommodate these demands due to the inability to replicate hypersonic flow conditions with sufficient accuracy. These deficiencies have put increased demand and pressure on hypersonic flight testing experiments which have historically proven to produce the highest quality results but at the potential price of extreme complexity and expense. In the case of hypersonic flight testing for SCRAMjet vehicles, the combination of high expense, high complexity, and high modeling uncertainties has led to conservative, risk-averse experiments. These efforts have historically yielded little gain in knowledge, observing only marginal improvements to prediction confidence in the computational models.
There is an entire discipline devoted to the process of design and information extraction from aerospace-type experiments known as aircraft system identification (SysID) which combines three interdependent topics: (i) computational modeling and simulation, (ii) experimental design methods, and (iii) statistical estimation techniques. Essentially, SysID attempts to develop time-dynamic experiments so that statistical estimation techniques can most effectively be used to identify high-confidence physics-based models. An implicit limitation to this process lies within the topic of dynamic experiment design, often posed as a mixed parameter optimization/optimal control problem for the concurrent design of aircraft maneuver inputs, instrumentation system parameters, flight conditions, test duration, etc. Here, Fisher information-based optimality criterion are sought to be used for the quantification of information quality; however, these metrics can only be accurately computed if the true values of the unknown model parameters (e.g. SCRAMjet aero-propulsive-elastic stability and control coefficients, vehicle mass/inertia parameters, etc.) are known prior to conducting an actual experiment, which is often not the case. This is commonly referred to as the circulatory problem in statistics literature, suggesting that dynamic optimal experiment design (DOED) requires an augmented robust-optimization approach (DROED) to account for modeling uncertainties.
This research focuses on the design of flight-dynamic experiments from the perspective of an integrated system for the concurrent design of information-dense flight experiments which are robust with respect to model parameter uncertainties. The proposed methodology is called TEMPUS, which stands for Time-dynamic Experiment design using a Model-based approach to Propagate Uncertainty for System identification. By using the top-down design decision support process within the Georgia Tech Integrated Product/Process Development methodology (GT-IPPD), TEMPUS fuses elements from two existing experiment design methodologies to enable a systems engineering approach to the design of large-scale robust-optimal dynamic system identification experiments (such as the design of SCRAMjet-powered flight tests). Within this method the generation of feasible design alternatives is achieved via a sizing and synthesis method, providing for the concurrent design of measurement system parameters, control system architecture and parameters, probabilistic uncertainty models, aero-thermal-fluids models, design constraints, and even vehicle geometry and mission-level parameters. To assess the performance of a given experiment design, a variety of different information quality metrics are able to be calculated from a dynamic high-order sensitivity analysis, providing for an a priori estimate of expected goodness-of-fit quality in the a posteriori parameter estimators. To evaluate feasible alternatives, a virtual experimentation strategy is utilized to assess information performance metrics of a given alternative via nondeterministic techniques (e.g. Monte Carlo methods).
Implementation of TEMPUS depends on the capability to perform a high-order dynamic sensitivity analysis on nonlinear industrial-sized aerospace flight-dynamic models (including guidance, navigation, and control logic) in a fashion that is both automatable and easily implementable by flight test designers and control systems engineers, all the while without introducing computational uncertainties. To address this challenge, an automatic differentiation tool specialized for use in dynamic experiment design was developed, providing for the ability to automatically compute robust-optimal Fisher information performance metrics by constructing variational asymptotic expansions (i.e. time-dynamic arrays of multivariate Taylor series expansions, parameterized by design and uncertainty perturbation variables). In general, these variational asymptotic expansions (VAEs) allow for a number of desirable capabilities for SysID applications, because they can essentially be considered as asymptotically accurate surrogate models to solutions of dynamic systems, including: (i) the construction of nominal Fisher information metrics (requiring at least 1st-order output-to-parameter sensitivity trajectories to be computed); (ii) the construction of arbitrarily high-order robust-optimal Fisher information metrics using both (deterministic and nondeterministic approaches to calculate robustness); (iii) rapid exploration of neighboring solutions to optimal control problems; and (iv) the implementation of arbitrarily high-order optimization algorithms (e.g. high-accuracy nonlinear parameter estimators in SysID) (not considered in this work).
High-order VAEs can suffer from many of the same complications that often hinder high-order multivariate response surface equations (RSEs), such as: (i) poor numerical conditioning, (ii) diminishing returns on accuracy (e.g. slow rates of convergence, finite radii of convergence, etc.), and (iii) the curse of dimensionality (e.g. large computational times and memory requirements). Therefore prior to using VAEs for dynamic experiment design problems within TEMPUS, four developmental experiments were designed to study the adverse effects of diminishing returns on accuracy, the curse of dimensionality, and application of VAEs to create surrogates of optimal control problems on simple dynamic systems. These include: (i) investigating the potential improvements of using alternative sets of basis functions on problems where diminishing returns on accuracy are observed for the standard Taylor (monomial) basis; (ii) investigating the effects of diminishing returns on accuracy in dynamic uncertainty propagation using high-order VAEs and various probabilistic uncertainty models; (iii) investigating the computational time and memory complexities of high-order, high-dimensional VAEs for use in dynamic experiment design; and (iv) investigating how automatic differentiation can be used to generate high-order VAEs to solutions of optimal control problems. The objective of the fourth experiment is to overcome the limitations that many indirect numerical optimization methods possess, namely, being cumbersome, nonautomatable analyses which hinder the ability to perform design space exploration and uncertainty propagation analyses due to a human-in-the-loop dependency. The results of the first experiment suggest that the use of Chebyshev basis functions can alleviate problems where the diminishing returns on accuracy are observed when Taylor basis functions are used. In the second experiment, it was observed that even for uncertainty propagation with high-order VAEs that slow/poor convergence characteristics can result in adverse effects, such as artificial multi-modality in propagated uncertainty distributions. The results of the third experiment suggest that high-dimensional problems (such as experiment design problems) scale exponentially with increasing order, and therefore high-performance computing capabilities will be necessary to practically obtain robust-optimal dynamic experiment designs for large industrial-sized aerospace problems. In the final experiment, two high-order optimal control formulations were developed for computing VAE surrogates. Promising results were observed for a simple optimal control problem where VAE surrogates were successfully computed; however, more effort is needed before these formulations can be applied to larger dynamic experiment design problems.
In light of the results of the aforementioned experiments, the TEMPUS methodology was applied to two design problems: (i) a simple mass-spring-damper problem under sinusoidal forcing, and (ii) the Generic Hypersonic Vehicle (GHV) model to design information-dense SCRAMjet-powered flight tests at steady-level flight under multi-sine forcing. In the first study, the small problem size allowed for investigation of high-order VAEs without experiencing the adverse effects due to the curse of dimensionality. Here, it was observed that robust-optimal experiment designs did produce probabilistic information metric distributions with better robustness with respect to parameter uncertainties than designs using the traditional nominal information metrics, and all experiment designs were found to produce intuitive results, serving as a form of validation (e.g. the sinusoidal forcing frequency was designed to excite the system near the expected natural frequency to maximize output-to-parameter sensitivities). For the flight test design problems, a nonlinear robust-adaptive flight controller is required to ensure safe operation throughout flight, because the GHV open-loop dynamics possess unstable, non-minimum phase behavior in the aero-propulsive-elastic modes in addition to the parametric uncertainties within the aero-propulsive-elastic stability and control coefficients. As a result, the complexity of the overall closed-loop model is greatly increased; however, computation of high-order VAEs for this system does not require any special attention in regards to practical implementation, but a substantial increase in computation time and memory was observed. The objective of experiment designs for the SCRAMjet-powered flight tests was to generate data for the system identification of eight thrust force stability and control coefficients: CTPA3, CTPA2, CTPA, CTP, CTA3, CTA2, CTA, CT0. For the combination of the adaptive control architecture and multi-sine excitation maneuvers implemented here, this experimental objective proved difficult to obtain where adaptation is known to have a canceling effect on the open-loop dynamics, therefore, making it difficult to excite the system enough to generate sufficient amounts of the high angle of attack data for improving the information content of the high-order coefficients CTPA3 and CTA3. It is hypothesized that alternative control strategies, employing machine learning for real-time estimation of open-loop natural frequencies, may improve the information quality, but implementation of this is beyond the scope of this work. Nevertheless, TEMPUS does allow for the robust-optimal assessment of information quality for alternative flight test designs (by using the computation of variational asymptotic expansions to overcome the deficiencies of the circulatory problem), implying that trade-offs between alternative controls architectures, measurement systems, etc. is now an available capability to the flight test designer and controls system engineer.