Control of Aerothermoelastic Effects in Hypersonic Vehicles

1. Introduction

Aeroservoelasticity considers the interaction between aerodynamics, inertial, structural, actuation, and control system dynamics. Flight controllers are usually designed for a rigid-body model and any aeroservoelastic issues are eliminated by including notch filters to eliminate observability of structural modes. This approach may not be acceptable for future aircraft that have lightweight and flexible components with low natural frequencies. Thus, modeling and control of both rigid-body and structural dynamics must be used.

Structural dynamics has been extensively studied for modeling and control at transonic and high dynamic pressure regimes for limited applications such as flutter and buffet suppression; however, there are several aircraft that must consider aeroservoelastic dynamics within a standard flight envelope. The first-bending mode of the SR-71 fuselage has a natural frequency near 3 Hz that is easily excited by pilot maneuvers. Several uninhabited aerial vehicles such as Theseus and APEX with lightweight and high aspect-ratio wings have several structural modes less than 2 Hz that affect flight characteristics. Also, the proposed high-speed civil transport anticipates a fuselage bending mode near 1.5 Hz that must be actively controlled to attain acceptable ride quality.

Modeling and control of aeroservoelasticity must also be considered for a proposed class of hypersonic aerospacecraft. These vehicles are being investigated for economic competitiveness of access to space missions such as payload delivery to low Earth orbit, reconnaissance, and cruise flight. The main aeroservoelastic feature is a coupling between the wedge-shape body and an air-breathing propulsion system that causes aerodynamic, inertial, structural, control, and even thermal dynamics to interact.

Aeropropulsive and aeroelastic models and associated controllers were extensively developed for several models that generally describe the proposed aerospacecraft. The strong interactions between the coupled dynamics of these models present many challenges for control design to achieve acceptable closed-loop properties. These interactions are a direct result of the scramjet engine which essentially uses the fuselage as part of the propulsion system. Thus, the aerodynamics and elastic dynamics can affect and respond to the propulsive dynamics. Several control synthesis methodologies have been considered for hypersonic models including classical control, H-infinity, mu, and a linear parameter-varying approach.

The effects of aerothermoelasticity were studied for a hypersonic vehicle known as the National AeroSpace Plane (NASP) and shown to significantly affect the open-loop dynamics; however, these effects were generally not considered for controller synthesis. A computational analysis was performed to show a large range of surface temperatures that can be reached during a typical ascent profile. Correspondingly, the structural dynamics show large variations in modal parameters and mode shapes as a result from temperature-induced stiffness changes.

These aerothermoelastic effects were considered in designing controllers for active flutter suppression and ride quality augmentation. The controller synthesis utilized a classical linear quadratic regulator design that directly used information from only the worst-case temperature model with lowest structural damping. The resulting controller was applied over the entire ascent and assumed to be sufficient for any temperature level.

This project considers the modeling and control of structural dynamics using a linear parameter-varying (LPV) framework. The LPV framework is used to represent systems whose dynamics depend on a set of norm-bounded and time-varying operating parameters. Specifically, the LPV framework considers linear systems with state-space matrices that are affine functions of the operating parameters. One of the benefits to using the LPV framework is the simplicity of representing the dynamics as a single model. Another benefit is the ability to design controllers that include that same affine dependency on the operating parameters. In essence, the dependency ensures a gain-scheduled controller is synthesized that inherently accounts for the time-varying nature of the operating parameters.

The linear parameter-varying nature of the structural dynamics can be used to account for several types of in-flight variations; however, this project will particularly focus on modeling and control of aerothermoelastic effects for hypersonic vehicles. The aerothermoelastic effects are described by variations in modal parameters such as natural frequency and damping. The structural model accounts for these variations by formulating the state-space matrices as affine functions of the temperature. An active structural controller is then formulated that is gain scheduled over the temperature.

It should be noted that this project only considers modeling and control of the effects of aerothermoelasticity but does not consider the actual computation of aerothermoelastic dynamics. The study of aerothermoelasticity is a complex field and is well beyond the intended scope here. This project utilizes a set of effects that have been noted in previous aerothermoelasticity research and assumes those effects are generally representative of the variations that may be noted in hypersonic flight.

2. Control Issues for Hypersonic Vehicles

The basic configuration of hypersonic vehicle that will be considered is similar to NASP and X-30 proposed vehicles and is shown in the figure.

The main characteristic of this vehicle affecting the dynamics is the integrated fuselage and propulsion system. The fuselage is actually designed as part of the engine system by using the forebody as a compressor and the aftbody as an external nozzle. This design introduces a significant amount of coupling between the aerodynamics and propulsion dynamics. Firstly, the airflow through the compressor introduces a lift force and a nose-up pitch moment while the airflow through the external nozzle introduces a lift force and a nose-down pitch moment so variations in propulsion performance alter the aerodynamic characteristics. Conversely, any variation in angle of attack and sideslip affects the engine inlet conditions so the propulsion performance is altered by variations in aerodynamic characteristics. Also, change in pitch angle results in a change in thrust angle so there is an especially strong and fast coupling between pitch and propulsion.

The vehicle can be controlled by commanded responses from the control surfaces and engine. The control surface commands include elevators for longitudinal control and rudders for lateral-directional control while the engine commands include diffuser area ratio and fuel flow rate. The coupling between the aerodynamics and propulsion systems introduces some redundancy among control effectors which can be exploited for control design.

A typical mission for this vehicle is to place some payload into low Earth orbit which requires the vehicle to operate in many flight regimes such as subsonic, transonic, supersonic, hypersonic and orbital. Each regime introduces control problems that must be alleviated for a successful mission. For example, the control surfaces will probably be small so as to minimize heating during hypersonic flight, but this may create difficulties for properly controlling the vehicle at low supersonic speeds. Another potential control problem may arise from the shocks generated by unsteady aerodynamics at transonic flight. Also, the issue of orbit transfers for payload delivery while in space is a control problem for this type of vehicle that introduces issues not usually affecting atmospheric flight.

The control problems in every flight regime are important; however, this report will limit consideration to the hypersonic regime while still in the atmosphere. One reason for limiting consideration to this regime is simply to concentrate on a smaller set of problems so that a useful solution can be formulated in a short time. Another valid reason to restrict attention to this limited flight regime is because it avoids the issue of choosing a single-stage or two-stage to orbit vehicle. Every air-breathing vehicle must pass through the atmospheric hypersonic regime regardless of whether a booster was used initially or a different on-board propulsion system placed the vehicle at low hypersonic speeds. Thus, this project will assume it is feasible to place the vehicle at hypersonic flight conditions and focus on the difficulties between hypersonic and orbital flight.

Several control issues have been identified for hypersonic flight through the atmosphere that must be investigated.
  • accounting for strongly coupled aerodynamics and propulsion dynamics
  • active modal vibration suppression
  • accounting for aerothermoelasticity
  • determining controller structure
  • determining gain-scheduling strategies
  • considering constant acceleration flight paths
  • determining coupled trajectory-control commands
  • determining control allocation schemes
  • control strategies for tailless hypersonic vehicles

3. Control Structure

Flight testing of aircraft invariably finds deficiencies in control designs that require correcting by altering gains; however, robust control methodologies typically derive an unstructured state-space controller that is difficult to intelligently alter. For this reason, classical control designs based on several proportional and integral gains are favored by flight test organizations because it is often easy to determine which gains should be varied to achieve the closed-loop characteristics desired by the pilot. Also, the process of proving a controller is flight-ready can be expensive and time-consuming so the ease with which classical controllers can be altered helps reduce the cost of flight testing.

Robust state-space controllers may not be optimal with respect to minimizing effort to prove flight-ready, but they have several advantages over classical controllers that must be considered. The main advantage of these designs is their level of robustness. Methods such as mu-synthesis generate controllers that inherently account for the amount of modeling uncertainty that is provided by the designer and attempt to maximize the robustness of the closed-loop design with respect to that uncertainty. Another advantage of robust methods is the formulation of automatic gain scheduling to account for parameter variations throughout a flight trajectory. Also, several tools have been developed that make designing robust controllers straightforward.

A multi-loop control structure is proposed for a hypersonic vehicle that provides some level of both structure and robustness. This multi-loop structure is closely related to the multi-element structure in which the plant may be formulated; namely, separated gains based essentially on aerothermoelastic dynamics and rigid-body dynamics. The multi-loop controller uses a set of inner-loop gains to achieve objectives associated with the aerothermoelastic dynamics and a set of outer-loop gains to achieve the remaining objectives.

The inner-loop controller is denoted as K_e with the subscript noting it is essentially an aerothermoelastic controller. The purpose of this controller is to actively augment structural damping in the aeroelastic modes. Thus, it can be viewed as a modal controller. K_e does not attempt to stabilize the rigid-body dynamics or achieve closed-loop levels of performance; rather, it is merely ensuring the structural dynamics are highly damped and are not easily excited by the outer-loop controller. This controller will act to minimize structural vibration and eliminate any local angle of attack variations resulting from fuselage elasticity that could effect the propulsion system.

The output-loop controller is denoted as K_r with the subscript noting it is essentially a rigid-body controller. This controller assumes the inner-loop controller is active and some desired set of modal parameters are associated with the structural dynamics. K_r works to stability the rigid-body dynamics and maximize the achievable performance in the presence of any modeling uncertainties.

A third control element, M, is also introduced as a mixer. This element interprets the controller commands as control surface and engine commands.

The structured closed-loop model with the separated open-loop dynamics and the multi-loop controller is shown in the block diagram.

Each of the controller elements is dependent on one of the parameter operators that are used to define the functional dependence of the open-loop elements. The controller K_e is dependent on the same parameters for which the aerothermoelastic dynamics, P_e, is dependent. This parameter is represented by the operator theta_e and, with the current model, it only accounts for temperature. The controller K_r is also dependent on a set of parameters. K_r is mainly concerned with rigid-body dynamics so it is a function of the parameter operator theta_r that accounts for Mach and angle of attack. The mixer is only associated with control allocation and so the parameter dependence of this operator is associated only the parameter dependence of the actuation dynamics. Thus, M only needs to vary with the operator theta_A.

4. Inner-Loop Controller

An LPV controller is synthesized to actively damp the aerothermoelastic modes for any value of surface temperature along an ascent trajectory. These gains are used as an inner-loop controller for a multi-loop compensator that is designed such that the closed-loop dynamics achieve a level of performance in following pilot commands. The purpose of this inner-loop controller is only to actively damp the structural modes despite aerothermoelastic effects. Thus, this controller is evaluated by considering the closed-loop damping properties with no consideration of handling qualities or tracking performance.

An explicit model-following approach is used for the control design such that the closed-loop dynamics approximate a desired model. This desired model is chosen as a highly-damped structural mode with a natural frequency near the open-loop structural frequency for the cold model. Specifically, the desired dynamics have natural frequency of 2.6 Hz and damping of 0.23. The open-loop transfer functions of the hot and cold plant models and the desired actively damped structural dynamics are shown in the figure.

The dynamics used for control synthesis are a single-input and single-output subset of the full open-loop model. The only measurement from the physical plant used for feedback is the angle of attack because this variable is strongly dependent on the elastic response and it provides sufficient information for control design. Also, the only control effector commanded from the inner-loop controller is the elevator deflection. The control surface is chosen because the bandwidth should be high enough to control the structural modes whereas it is doubtful the engine effectors can be effective.

A linear parameter-varying controller is synthesized for the inner-loop model. This controller is originally formulated with 27 states; however, reduced-order controllers may be derived using standard model reduction methods.

The angle of attack measurements in response to a command elevator deflection are shown in the figure to the right. The open-loop response clearly shows a large oscillatory component because the deflection excites the low damped structural mode. The closed-loop response does not show this oscillatory component because the controller actively damps the structural mode and thus the elevator deflection does not strongly excite structural motion. Also, this response is computed along a parameter-varying trajectory with temperatures ranging from cold to hot in 10 seconds. This trajectory is unrealistically fast; however, it shows the response is damped over the wide range of temperatures with arbitrarily fast time variation.

5. Outer-Loop Controller

An outer-loop controller is designed to stabilize the vehicle and achieve desired handling qualities for maneuvering. This controller is the flight mechanics controller that is commonly designed for aircraft that do not require active structural damping.

A robust design approach is chosen to compute the outer-loop gains and account for modeling errors of the rigid-body dynamics and performance errors of the inner-loop controller. This controller would ideally be gain scheduled over flight conditions such as Mach and dynamic pressure;however, the model was not sufficiently complex to account for a range of flight conditions. Thus, mu-synthesis is used to design the controller at a single-point flight condition.

A Mach regulation loop is included with the outer-loop dynamics to track commands to the Mach variable. The feedback law for this loop is chosen based on classical design arguments and is realized as a simple proportional controller. The commands from the regulator are changes to the fuel flow rate only because a preliminary study indicated this variable is convenient for Mach regulation. The change in fuel flow is simply the difference between the measured and desired Mach numbers.

The outer-loop controller is allowed to use all control effectors which are the elevator surface, the engine diffuser area ratio, and the engine fuel flow ratio; however, only the elevator surface is used. The engine fuel flow ratio is not commanded by the outer-loop controller because the Mach regulation commands this variable. The diffuser area ratio is not used because it is not clear that such a parameter of a physical engine would have an appreciable bandwidth without an unreasonable time delay. Also, all sensors are available as feedback measurements to the controller.

The model used for control design of the outer-loop gains includes some knowledge of the inner-loop dynamics. Some methods of multi-loop control design, such as robust dynamic inversion~\cite{reiner}, replace the actual inner-loop dynamics with the desired inner-loop dynamics so the synthesis model has linear elements; however, this approach may not be suitable here. The problem arises mainly because the rigid-body instability. The inner-loop design was formulated without explicit knowledge of this instability because the LPV synthesis algorithm would automatically try to stabilize the closed-loop system but the inner-loop controller was not supposed to affect the rigid-body dynamics. Thus, the effects of the inner-loop controller on the rigid-body instability are not known so the outer-loop design must consider the true unstable inner-loop system.

A representative model of the linear parameter-varying inner-loop dynamics must be chosen for outer-loop synthesis model. The $\mu$-synthesis approach used for control design requires the model to have linear time-invariant dynamics with no parameter-varying dependencies so the gain-scheduled nature of the inner-loop dynamics must be ignored. The model chosen for outer-loop synthesis is computed by isolating the dynamics for the parameter representing the extreme hot temperature. This model is chosen because it is essentially a worst-case model in the sense that the inner-loop controller provides less active damping at hot temperatures than at cold temperatures.

The figure on the right demonstrates the Mach number measured in response to a commanded filtered-step increase in Mach. The response is slightly faster than the commanded increase but the outer-loop dynamics track the command reasonably well. This response demonstrates the Mach regulation loop coupled with the outer-loop controller provides acceptable tracking.

The measured angle of attack is also computed in response to the step increase in Mach and presented on the right. The magnitude of the angle of attack is relatively small and indicates the outer-loop controller is able to limit the effects of the Mach command. Also, the outer-loop and inner-loop controllers can simultaneously generate commands for the elevator deflection to achieve rigid-body performance and active damp the structural dynamics.

6. Information