Robust Modal Damping Control for Active Flutter Suppression

Flutter is an unstable oscillation caused by the interaction of aerodynamics and structural dynamics. It is current practice to operate aircraft well below their open-loop ﬂutter speed in a stable ﬂight regime. For future aircraft, weight reduction and aerodynamically eﬃcient high aspect ratio wing design reduce structural stiﬀness and thus reduce ﬂutter speed. Active control of the ﬂutter phenomena can counter adverse aeroservoelastic eﬀects and allow operation of an aircraft beyond its open-loop ﬂutter speed. This paper presents a systematic robust control design method for active ﬂutter suppression. It extends the standard four block mixed sensitivity formulation by a means to target speciﬁc dynamic modes and add damping. This enables a control design to augment damping of critical ﬂutter modes with minimal impact on the rigid-body autopilots. Finally, the design scheme uses a manageably low number of tunable parameters with a clear physical interpretation. Tuning the controller is hence considerably easier than with standard approaches. The method is demonstrated by designing an active ﬂutter suppression controller for a small, ﬂexible unmanned aircraft and veriﬁed in simulation.


I. Introduction
Aeroelastic flutter involves the adverse interaction of aerodynamics with structural dynamics and potentially rigid-body dynamics. It produces an unstable oscillation that often results in structural failure as shown in the picture sequence in Figs. 1a-e. Conventional aircraft are designed in accordance with FAA/EASA certification specification 25.629 such that flutter does not occur within the range of their operating conditions. This is usually achieved through the use of stiffening structure and thus at the expense of additional mass. The use of active control systems to expand the flutter boundary could therefore lead to a decrease in structural mass and consequently increase fuel efficiency and performance for future aircraft [1]. The fundamental principles of active flutter suppression control are discussed in [2,3] and a comprehensive survey was recently provided in [4]. Early research on active flutter suppression relied to a large extent on what is known as collocated feedback.
Collocated feedback employs sensors and actuators in the same location, see e. g. [5]. Collocated controllers are, in general, easy to design using root-locus techniques and are extremely robust with respect to gain variation. A closely related approach, termed the concept of identically located force and acceleration in [6], was successfully applied to address the damping of structural modes on the B-1 aircraft [7,8]. A similar configuration was also used in the first flight test beyond flutter speed, conducted in 1973 on a modified B-52 aircraft [9]. The control system on that aircraft involved two single feedback loops that fed back filtered vertical acceleration signals, acquired on the wing, to control surfaces located nearby (outboard ailerons and flaperons). Collocated acceleration feedback is also proposed in various other publications concerned with flutter suppression, e. g. [10][11][12][13][14][15].
The other main branch of research on flutter suppression control considers multivariable optimal control techniques such as linear quadratic Gaussian (LQG) [16][17][18] as well as H ∞ -norm optimal and linear parameter-varying (LPV) control [19][20][21][22][23][24]. Recent flight test demonstrations of active flutter suppression, e. g., on Lockheed Martin's Body Freedom Flutter vehicle [23], make use of such techniques. Multivariable controllers can make optimal use of all available control surfaces and sensor signals without potentially restrictive pre-grouping of inputs and outputs. As a drawback, accurate models are required and formulating an adequate optimization objective that represents the actual control design objectives can become complicated. Another multivariable approach that has recently emerged can be classified as modal isolation. It seeks to target specific dynamic modes, such as the aeroelastic flutter mode, and avoid interaction with other dynamics. This is achieved by first calculating a signal that represents a modal state. Then, a feedback loop is closed with this signal [25][26][27].
This paper contributes a systematic H ∞ -control design framework for modal damping and mode attenuation.
Inspired by [28] and [25], specific dynamic modes are incorporated as a performance measure. Section II formulates the method as a generalization of the standard four-block problem for control design. The objective of adding damping to a specific mode is expressed as reducing peaks in the frequency response of a closed-loop transfer function. The H ∞ closed loop shaping method augments the controller with roll-off and wash-out filters such that control activity can be precisely confined to a specific frequency range. Doing so avoids undesired interaction with unmodeled or uncertain plant dynamics at high frequencies and with autopilot systems at low frequencies. The proposed design method is hence particularly suitable for flutter suppression control. This is demonstrated by applying the method to the flight-test-validated model of a small, remote-piloted aircraft, described in Section III. The model involves asymmetric wing stiffness, causing non-negligible coupling of longitudinal and lateral dynamics. Further, two different flutter modes are present, which challenges conventional control methods. A comprehensive control design study that addresses simultaneous stabilization of these two flutter modes and verification in simulation is provided in Sec. IV. This paper builds on prior work in [29,30] with three main extensions. Firstly, a novel parameterization for all involved weighting functions is provided. This parameterization has a manageably low number of tunable parameters with clear physical interpretations and is hence easy to apply. In addition, the paper provides specific guidelines on how these parameters are chosen. Secondly, an explicit formulation for including generalized velocities of dynamic modes as a performance measure is provided. Finally, significant engineering insight about the application of the method is added through the design example and verification in a simulation that includes actuator nonlinearities such as rate limits and backlash.

II. Control Design Methodology
Shaping closed-loop transfer functions through H ∞ -norm optimization is a well-established approach for control design [31,32]. The H ∞ -norm of a dynamic system ( ) from input to output is defined as where (·) denotes the largest singular value. This norm measures the maximum gain of the transfer function ( ), i. e., the largest amplification of L 2 input signals over all frequencies and input/output directions.  [33][34][35]. These algorithms provide an optimal controller that is, however, of the same order as the generalized plant. If additional structure is imposed on the controller, e. g. a PID controller or fixed-order filter is required, non-smooth and non-convex optimization methods can be used instead [36][37][38]. Doing so only changes the controller synthesis step, not the design formulation. As such, the method presented in this paper is also applicable for structured H ∞ -controller design. With mature synthesis machinery for both full-order and structured controllers readily available, e. g., in the Matlab Robust Control Toolbox [39], the challenging part of any H ∞ -design is to provide meaningful performance specifications in terms of a generalized plant.

A. Generalized Plant for Closed-Loop Shaping
In the following, the closed-loop transfer functions that ought to be shaped by the H ∞ -controller and the weights that specify their desired shapes are introduced. Consider a dynamic system with input ( ) ∈ R , state vector ( ) ∈ R , a measurable output ( ) ∈ R , and an additional performance output   problem. An appropriate choice of can emphasize specific dynamics in the output, e. g., velocity states [41] and modal velocities as shown in Sec. II.C. The output thus provides additional means to specify control objectives such as damping augmentation. Note that a generalized formulation for two-degrees-of-freedom controllers and arbitrary disturbance models can be found in [42].
The generalized closed loop shown in Fig. 3 is used to formulate loop shape requirements. Its transfer function is The weights ( ), ( and Bryson's rule from LQG control [43]. The frequency-dependent weighting filters can be chosen independent of the plant model's units to represent closed-loop bandwidth and magnitude bounds.

B. Weight Selection
This subsection discusses a parameterization of the weights and how it is used for tuning. The discussion considers a representative example with two control inputs and two measured outputs. The first output shall be tracked, whereas the second output is used as an additional feedback signal. Further, only the first control input may be used in steady state.
The second shall be restricted to a specific frequency range. This example covers sufficient complexity to be adaptable The first entry ,1 ( ) in Eq. should not exceed two. Note that a peak sensitivity of less than two implies a gain margin of at least 6 dB, see e. g. [32].
The choice of determines cross-coupling and leaves the specifications on the diagonal unaffected. Consequently, ,1 can be selected as the maximum expected command value 1 for the tracked output and ,2 as the maximum acceptable cross-coupling due to this command. Further, Fig. 6 shows that the relative size of and specifies the level of disturbance attenuation. Frequency (rad/s)

Fig. 5 Implied loop shape requirements on the sensitivity o .
The weighting filter ( ) determines the shape of the control sensitivity o and hence distributes control authority across frequency. It is parameterized as Here, u,1 sets the upper available bandwidth for the control signals 1 , while u,2 and l,2 specify upper and lower limits for 2 . The corresponding frequency response is shown in Fig. 7. The weight on 1 has unit gain up to u,1 and bounds shown in Figs. 8 and 9. At high frequencies, a low magnitude in o is only achieved when rolls off, since o ≈ as → 0. Therefore, the roll-off requirement on o also imposes a roll-off on the controller and reduces the control activity beyond u,1 and u,2 for 1 and 2 , respectively. Such a roll-off is necessary to avoid excitation of unmodeled or uncertain dynamics at high frequencies and to reduce control action in response to noise. Similarly, the requirement that i = − i has low magnitude in the low frequency range implies that i ≈ and hence is only achieved with a low magnitude in . The wash-out requirement on i hence translates directly to and confines the control activity for 2 to frequencies above l,2 . A bandpass characteristic limits interaction with low-frequency dynamics and is particularly useful for flutter control cf. [21]. Figure 8 shows that the ratio of to bounds the maximum controller gain. The weight is consequently selected to represent available control action relative to the previously specified maximum errors. Thus, is often selected based on the actual saturation limits of the actuators. Figure 9 finally shows the transfer functions that represent the control effort in response to disturbances. Frequency (rad/s)

Fig. 9 Implied loop shape requirements on the input complementary sensitivity i .
In summary, the proposed architecture has a manageably low total number of tunable parameters (9 or 10 for the present example, depending on whether = is used). One critique of ∞ -control, especially for signal-weighted design schemes, is that there are many parameters with opaque interdependence that must be chosen to get a reasonable controller [32]. In the proposed formulation, each of the parameters has a clear interpretation. It remains to discuss the selection of and . How these weights are chosen depends to a larger extend on the specific choice of the output variables , but often = is sufficient and is selected such that is normalized to a magnitude of 1. The subsequent section II.C details the case where is used to target a specific dynamic mode of the system and augment its damping.

C. Adding Modal Damping via Modes Targeting
It is shown in the following that damping can be effectively augmented by penalizing the generalized velocity of a dynamic mode. Dynamic modes are decoupled subsystems of order one (real) or two (complex) such that the system ( ) can be expressed as The rm real modes ( ) are not considered further in this section as they do not contribute to oscillations. The cm complex modes ( ) contain oscillatory system dynamics. They can be represented by state space realizations : where 0, is the natural frequency and is the damping ratio of the th dynamic mode. In this form, the modal state variables are a generalized displacement and a generalized velocity . The row vectors 1, ∈ R 1× und 2, ∈ R 1× describe how the input excites the th mode.
Assume that 1, = [0 · · · 0] which is the case, e. g., for ordinary mechanical systems [44]. The multi-input-singleoutput transfer functionˆ ( ) from to in this case iŝ It has a distinct peak at the frequency 0, and exhibits band-pass characteristics, see Fig. 10. For underdamped modes , the damping ratio is inversely proportional to the peak magnitude ˆ as Hence, reducing the peak magnitude directly increases damping. The transfer function (9) is an ideal description of damping requirements in the framework of H ∞ -norm optimization: Increasing damping simply translates to flattening the peak and other specifications remain largely unaffected due to the narrow frequency band. Adding the modal velocity as a performance output in Eq. (2) hence emphasizes, or "exposes", the specific dynamic mode within the minimization problem (4) [28,29]. In case 1, ≠ [0 · · · 0], the transfer function from to retains very similar properties, but has a non-vanishing steady-state gain.
It remains to briefly discuss how the matrix in Eq. (2) is selected to achieve = . Let = . That is, ∈ C 1× is a normalized left eigenvector and ∈ C is the corresponding eigenvalue for the th complex mode. The generalized displacement and the generalized velocity of the th mode can be calculated as This fact is readily verified by using Eq. (11)

III. Test Aircraft
The aircraft under consideration in this paper is the University of Minnesota's mini MUTT, specifically the third mini MUTT built named "Geri". It is a remote-piloted flying wing aircraft with a wing span of 3 m and an approximate total mass of about = 6.7 kg. The mini MUTT is a laser-scanned replica of Lockheed Martin's Body Freedom Flutter vehicle [45] and resembles NASA's X56 MUTT aircraft [18,46]. A schematic of the aircraft with sensor and flap positions is shown in Fig. 11. It is a low-speed swept-back flying wing with winglets on the wing tips for directional stability. An electric motor drives a pusher propeller mounted at the top rear of the rigid center body. The mini MUTT is specifically designed to exhibit strong coupling of rigid body and structural dynamics at low airspeed. Its wing structure consists of a solid spar and foam covered by a composite skin. The center body also has a composite structure. The entire trailing edge of the wing consists of eight control surfaces, two on the center body ( 1 , 1 ) and six on the wings ( 2 , . . . , 4 , 2 , . . . , 4 ). The aircraft is equipped with a rate gyro in the center body that measures the rotational rates .
Six accelerometers measure vertical accelerations in the center body (subscript C) and on both wing tips (subscripts L,R). These accelerometers are mounted in pairs, one fore (subscript f) and one aft (subscript a).

A. Aeroelastic Model
A mathematical model of the aircraft was developed based on a mean axis approach as described in [47]. Mean axes are an attractive and widely used approach to model aeroelastic systems, specifically for control design. They offer the following advantages: (1) It is easy to incorporate nonlinear rigid-body dynamics, (2) the aerodynamics can be readily written in terms of dimensional aerodynamic derivatives, and (3) the format of the model is similar to a classical rigid-body flight dynamics model. A detailed description of mean axis models can be found in [47] based on a Lagrangian derivation. More recently, the mean axis formulation was also developed using a Newtonian approach in [48].
Mean axes form a body-referenced frame that is defined in the following way: The mean axis frame has no internal translational and angular momentum. Moreover, the origin of the frame is located at the instantaneous center of mass.
Effectively, the mean axis are equivalent to the body frame for rigid bodies. However, it is important to emphasize that for a flexible body, the mean axis are a body-referenced frame but not a body-fixed frame. The following assumptions, commonly satisfied for aircraft [49], are further used to simplify the model: (1) The moments of inertia of the elastic body equal the moments of inertia of the undeformed body and are constant. (2) All elastic displacements occur primarlily in the same direction within the body frame, i.e., the cross product of displacements and displacement rates is almost zero.
In [50], this is refered to as the practical mean axis constraint.  is used. The aerodynamic forces and moments can be written as where the matrices rr , re , er , ee , rc and ec contain dimensional aerodynamic coefficients and represents the deflection angles of the eight flaps. Initial aerodynamic coefficients were obtained using strip theory [53]. These values were then refined in a system identification campaign [52,54]. Note that the dimensional aerodynamic coefficients are functions of the airspeed.

B. Linearized Model
For the following control design and linear analysis, the aircraft model is trimmed for steady-level flight at airspeeds ranging from 25 m/s to 45 m/s and an analytical linearization is performed. The result is a parametric linear system representation in dependence on the true airspeed ∞ which can be written as The state vector ( ) ∈ R 18 of the parametric model contains the six rigid-body states (three translational and three rotational) as well as six generalized displacements and six generalized velocities for the structural modes. The output ( ) ∈ R 9 represents the available sensor signals, i. e., the six vertical acceleration measurements and the three angular rates at the center of gravity. The input ( ) ∈ R 8 represents the eight control signals for deflecting the flaps. All inputs and outputs are shown in Fig. 11.
The dynamics of the airframe are depicted in Fig. 14 in the form of a root locus with respect to airspeed. Note that the poles correspond to dynamic modes of the aircraft, i. e., they represent dynamic behavior that involves both rigid-body and elastic state variables. The corresponding dynamic mode shapes are shown in Fig. 15  The signals provided by the sensors are processed by the mini MUTT's flight computer that executes the control algorithm within a 6.6 ms frame (≈ 151 Hz). The controller output is passed on to a microcontroller that runs asynchronously with a 3.3 ms frame rate (≈ 302 Hz) to generate a pulse width modulation (PWM) signal. This PWM signal is the input to a servo controller that runs, also asynchronously, with a 3.3 ms frame rate. The actuator used on the mini MUTT is a Futaba S9254 servo whose dynamics were identified using a chirp input signal and frequency-domain system

IV. Control Design and Evaluation
The following flutter suppression control design aims at maximizing the closed-loop flutter speed. This objective must be achieved while remaining within actuator limits. Given the catastrophic consequences of flutter, the controller must achieve sufficient robustness margins across all airspeeds. Finally, minimal interference with the autopilot is desired.
The autopilot uses the midboard flaps ( 3 , 3 ) like a classical elevator for pitch attitude control and the inboard flaps ( 2 , 2 ) like classical ailerons for roll attitude control and is similar to the one described in [55]. Consequently, the flutter suppression controller is assigned full authority over the body flaps and outboard flaps, i. e. 1 , 1 , 4 , and 4 (in rad). Keeping the flutter suppression control loop completely separate from the autopilot loops reduces the risk of saturating the control surfaces and facilitates a simple control design. The body flaps have almost no effect on pitch, but produce sufficient lift to affect bending and twist of the structure. The outboard flaps have a strong effect on wing bending and twist as well as on pitch and roll. This choice of control surfaces leads to high functional controllability with effectors that can be combined in ways that isolate the two modes. The measured center-body pitch rate and roll rate (both in rad/s), and the vertical accelerations (in ft/s 2 ) as measured by the six accelerometers are used as feedback signals for the flutter suppression controller.
In order to maximize the flutter envelope expansion, several design airspeeds were considered. It was observed that the controllers designed at different airspeeds tend to stabilize the system up to their respective design airspeed, but not far beyond. This observation is consistent with previous research which showed that the design speed is an important design parameter [56]. Thus, a design airspeed as high as possible should be used. The frequency of the unstable first aeroelastic mode increases with higher airspeed. The maximum airspeed is hence limited by the available actuator bandwidth, since the bandwidth must be sufficiently higher than the frequency of the unstable mode [32,57]. Keeping model accuracy and actuator limitations in mind, it is decided to restrict control authority to a maximum frequency of about 70 rad/s. Consequently, an airspeed ∞ = 41 m/s is selected as the design condition. At this airspeed, the frequency of the unstable first aeroelastic mode (1 st AE in Fig. 14) is 37 rad/s, a factor two below the dedicated control bandwidth.
The second aeroelastic mode (2 nd AE in Fig. 14) is stable, but lightly damped at a frequency of 28 rad/s.

A. Weight Design and Controller Tuning
An airframe model with reduced state vector is used to form the plant model ( ). Only the pitch and roll dynamics The weights for the generalized plant (4) are selected as follows. Tracking is not an objective for the controller, so no frequency-dependent weight ( ) is required and = 0.5 is selected. Similarly, = is used, as the definition of as modal velocities already targets the specific narrow frequency range of the aeroelastic modes. Next, 0.01 +70 is selected such that all control surfaces are assigned the frequency range 10-70 rad/s for activity. The wash-out minimizes the interaction with low-frequency dynamics and the autopilot. The roll-off provides robustness against inevitable model uncertainty at high frequencies.
Once these specifications are fixed, the static weights are used for controller tuning. The matrix is used to normalize the outputs such that an error of 1 rad/s pitch and roll rate is equally important as an acceleration error of 32 ft/s 2 ≈ 1 . Thus, = diag(1, 1, 32, 32, 32, 32, 32, 32) is selected. The control effort weight is initially selected to specify a maximum deflection of 3 • for all control surfaces in response to these errors, i. e., = diag( 3 180 , 3 180 , 3 180 , 3 180 ). To emphasize that robustness is a key concern, = is selected, representing maximum input uncertainty at all inputs. The desired increase in modal damping is set by the weight . Smaller values result in larger damping augmentation, since the weight encourages the controller to reduce the sharp peak in the transfer function from to more. Consequently, damping is increased in accordance with Eq. Besides the two parameters that determine the frequency range of control activity and the single parameter to weight input disturbances, there is one tuning parameter for each of the eight measurement signals, four control effectors, and two modal velocity outputs. The number can be reduced to 8, when the tuning parameters and are not specified for each sensor and control effector individually, but instead for a group (one for all accelerometers, one for all rate sensors, one for all flaps).
Matlab's hinfsyn routine was used to synthesize a controller with these weights. A robustness analysis (see Sec. IV.B) after the first controller synthesis showed unsatisfactory input margins at the right outboard flap ( 4 ). This observation was attributed to the smaller stiffness of the right wing which effectively increases the control effectiveness on this side and hence the loop gain. To tackle the issue, the maximum deflection of 4 in was iteratively adjusted to = diag( 3 180 , 3 180 , 3 180 , 1.6 180 ) in order to decrease the controller magnitude in the 4 channel until satisfactory margins were obtained. Finally, the controller order was reduced from 35 to 25. This optional step was performed to simplify implementation. The reduction was performed by transforming the controller to a modal form (7) and residualizing all dynamic modes with poles faster than 150 rad/s. Doing so almost exactly preserved the frequency response of the controller up to about 100 rad/s and therefore did not cause any degradation in performance and robustness.

B. Robustness Margins
Every individual control loop, including autopilot loops for pitch and roll attitude hold, was verified to satisfy at least 45 • phase margin and 6 dB gain margin up to the design airspeed of 41 m/s. For this analysis, both the inputs (before the control surface actuators) and outputs (after the sensors) were considered as loop break points. Such an analysis is stricter in its requirements than standard procedures in aerospace such as MIL-SPEC9490 or SAE-AS94900 that consider loop break points at the inputs, only. The lowest margins are 6.8 dB gain margin at input 4 and 50 • phase margin at input 4 . The lowest delay margins are 37 ms at input 1 and 14 ms at output L,a . Note that these margins are on top of all known parasitic dynamics and delays. However, considering only classical margins and single loops does not acknowledge the complexity of the problem at hand. Destabilizing combinations of gain, phase, and directional changes that independently are considered safe can easily be overlooked. Therefore, symmetric disk margins [58] are used as an additional metric for robustness verification. They account for simultaneous perturbation of gain and phase and can also be calculated with respect to simultaneous perturbations at multiple inputs and outputs. Hence, they avoid the pitfalls of classical single-loop gain and phase margins.
The lowest single-loop disk margin at the design speed is 6.5 dB | 39 • and is attained at the input 4 . Thus, no critical combination of gain and phase variation exists in any of the feedback loops. It was further verified that all margins increase with lower airspeed. Further, multivariable symmetric disk margins are calculated to also consider directionality.
These margins account for simultaneous perturbation at all outputs (multi-output disk margin), all inputs (multi-input disk margin), as well as at all inputs and outputs (multi-input-multi-output disk margin) [58]. Figure 17 shows these multivariable margins as a function of airspeed. As expected, their values are significantly lower than their single-loop counterparts. All margins decrease only slightly with increasing airspeed up until the design point 41 m/s. Therefore, no particular bottleneck can be identified. Beyond the design speed, the margins collapse and instability occurs at 42.5 m/s. 25

C. Results
First, the effect of including the modal velocities as performance outputs is evaluated at the design point ∞ = 41 m/s. The weighted load disturbance sensitivities of the first and second aeroelastic mode, i. e, the transfer functions from 2 to 3 , are depicted in Figs. 18 and 19. Stabilizing an unstable system inevitably leads to a net increase in sensitivity in accordance with Bode's sensitivity theorem [57]. Figure 18 shows this effect clearly for the first aeroelastic mode at the design airspeed. The magnitude of the frequency response of the stable closed-loop is increased compared to the unstable open-loop system. However, the increase is smallest at the mode's frequency such that the relative peak magnitude is actually decreased. Thus, a well-damped behavior of the closed loop can be expected. Figure 19 shows that the overall sensitivity degradation is much lower for the second aeroelastic mode. The peak magnitude is drastically decreased, both in absolute and relative terms. Thus, a large increase in damping compared to the open-loop system is achieved. The sensitivity degradation is well behaved and evenly spread out around the mode's frequency. In summary, the controller adds sufficient loop gain to stabilize the first mode without introducing a sensitivity peak and reduces the sensitivity peak in the second mode.    deflected significantly less than the left surface 4 as intended by the choice of in the control design. Saturation is of no concern and the rate limit of the servo actuators was never violated during the simulation.
Finally, the aeroelastic response to a discrete gust as defined in FAA/EASA certification specification CS-25.341(a) is simulated. Figure 25 shows the displacement and twist of the right wing tip in response to a vertical gust. The gust has the form 5.

V. Conclusion
This paper develops a systematic multivariable robust control design method that incorporates targeting of specific dynamic modes into the framework of closed-loop shaping. This is achieved by introducing additional performance outputs to the design model. These outputs represent generalized velocities in modal coordinates. Reducing the peaks in structural stiffness of the wings and large phase loss due to parasitic dynamics make this a particularly challenging control problem. The comprehensive design study demonstrates that the proposed method is ideally suited to design the flutter suppression controller in a way such that it targets the aeroelastic modes. The controller is tuned using a manageably low number of parameters that all have clear physical interpretations. It is shown that the design parameters indeed modify the controller in a transparent way, e. g., reduce the magnitude of a particular control input. Robustness analyses show that the controller achieves large classical gain and phase margins as well as reasonable multivariable disk margins across all airspeeds up to 41 m/s. That is, the feedback system is robust even close to the extended flutter speed.
The controller is verified in simulation with actuator nonlinearities such as rate limits and backlash. Representative pilot inputs and discrete gusts are used as inputs. The control signals issued by the controller are only significant enough to