Modeling, Analysis, and Specifications

Worst-Case Modeling and Analysis Approach

Developing a robust hierarchical control (Hi-C) system requires extensive modeling pre-requisites if some type of “meaningful performance guarantees” are desired. Many relevant modeling issues are addressed within [5]-[60]. Rather than treating time- or aero-variable-dependent model parameters (e.g. aero coefficients) stochastically (with simple, albeit typically unjustified probability distributions), we intend to systematically bridge the “theory-practice gap” between the “aero-thermo (fluid-mechanics/kintetic-theory) worlds (see control challenges and issues) and the control system design world.” More specifically, we intend to use the aforementioned knowledge [28]-[29], [43], [53] to determine worst-case (“ball-park”) characteristics for each of the following:

• Aero-Variations: lift-drag, aero-coefficient, pressure distribution, center of pressure (cp), and center of gravity (cg) variations, unsteady aerodynamic effects, and control surface effectiveness [28]-[30], [223]-[228];



• Longitudinal/Lateral Mode Movement and Coupling: phugoid, short period, spiral, Dutch roll stability, damping, and coupling [28]-[30], [53], [223]-[228];



• Elastic Mode Movement: body flexing and aero-servo-elastic modes [28]-[29], [43], [53]

for a generic mission involving the following:

• separation from assisting rocket (e.g. Pegasus for X-43A), altitude/attitude stabilization, scramjet ignition, ascent/climb pull-ups, trajectory following, altitude/attitude capture and hold, cruise, bank-to-turn, unpowered and powered descent.

As mentioned earlier, the X-43A mission profile in Figure 4 will be used as a baseline trajectory (X-43A flight conditions: Mach 7, q = 100 ± 200 psf, β = ±0.5◦, α = 2.5◦ [6]-[7]). A similar type of analysis will be conducted for a typical SOAREX unpowered descent.


Modeling Requirements for Separation Maneuver

It should be noted, for example, that accurate aero-coefficients - requiring time-accurate aero computational fluid dynamics (CFD, mesh-gridding-smoothing method) solutions - - requiring time-accurate unsteady-aero computational fluid dynamics (CFD) solutions - may be needed to adequately model coupled roll-yaw, lateral phugoid (roll-spiral) Dutch roll dynamics during the separation maneuver; e.g. see two-stage hypersonic application within [30] which

• models the carrier stage as a flat plate near the orbital stage and



• uses an advection upstream splitting method (AUSM) which treats convective and acoustic pressure waves as distinct processes that combine additively (i.e. principle of superposition).

Use of Worst-Case Aero-Thermo-Elastic-Propulsion Analysis

The goal then is to use this worst-case information to suitably adapt based on real-time (typically non-worst-case) measurements about the actual flight environment. Such an approach permits us to prepare for the worst without being overly conservative (e.g. sacrificing control bandwidth).

Crucial to our approach is determining suitable parameterized families of models that reflect typical and worst-case uncertainty. Also important is the idea of determining which model parameters combinations we are most sensitive to over specific flight regimes. This type of analysis, in general, is computationally intensive. However, by using linearized models at desired flight conditions, one can use convex optimization methods to assess sensitivity properties and fundamental performance limitations. This implies that there exists very efficient (polynomial-time) algorithms [218] for computing purposes [158]-[160]. In fact, such algorithms are readily available within the MathWorks computational suite. Such analysis (which can be rapidly performed) will be very useful for designing truly robust Hi-C systems and for minimizing the number of Monte Carlo runs that ultimately must be conducted.

Reduction of Computations for Worst-Case Analysis

To further reduce the number of computations needed, one can rely on simple predictive methods for hypersonic flows. For example, it is useful to note that a classical Newtonian flat plate analysis [28] reveals that at hypersonic speeds,

Design Specifications: Fundamental Performance Limitations

Following analysis, we have the determination of realistic design specifications. Too often, specifications are “handed down” without a sufficiently thorough dynamical analysis. To address this, we propose to exploit efficient convex optimization methods (as discussed above) based on linear (time invariant) models [158]-[160], [216] in order to assess fundamental performance limitations at key places throughout the flight envelop. Specifications which we will address include [211]-[216], [220]-[222]:

• Time Domain Specifications: settling time, rise time, overshoot/undershoot [220]-[222]


• Frequency Domain Specifications: bandwidth, classical stability robustness margins (i.e. gain, phase, delay margins) [220]-[222], peak frequency response measures (e.g. H∞ norm [158]-[160], [213]-[216])