Modal Isolation and Damping for Adaptive Aeroservoelastic ... - AIAA Info

AIAA Atmospheric Flight Mechanics (AFM) Conference August 19-22, 2013, Boston, MA

AIAA 2013-4743

Modal Isolation and Damping for Adaptive Aeroservoelastic Suppression Brian P. Danowsky,* Peter M., Thompson,† Dongchan Lee‡ Systems Technology, Incorporated, Hawthorne, CA, 90250 and

Downloaded by David Klyde on August 26, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4743

Marty Brenner§ NASA Dryden Flight Research Center, Edwards, CA, 93560

Adverse aeroservoelastic interaction is a problem on aircraft of all types causing repeated loading, enhanced fatigue, undesirable oscillations and catastrophic flutter. This adverse response is traditionally suppressed using notch and/or roll off filters in the primary flight control system architecture. This solution has pitfalls; rigid body performance is degraded due to resulting phase penalty and the filter may not be robust to off nominal behavior. An adaptive approach has been developed that determines optimal blends of both multiple outputs and multiple inputs which effectively isolate and suppress problematic lightly damped modes via negative feedback while minimizing adverse effects on the remaining modal response. Additional emphasis is given towards minimizing adverse effects on aircraft rigid body modes so that low frequency behavior is unchanged with minimal phase penalty. A subspace system identification solution has been incorporated to rapidly identify a large order model of the aircraft from multiple measurement sensors. This identified model is used to synthesize the controller, demonstrating the solution to be completely adaptive. Due to successful isolation of problematic modes, this solution can be applied independent of any primary flight control solution. Algorithm validation was performed via real-time piloted simulation of large order aeroelastic F/A-18C aircraft models.

I. Introduction DVERSE aeroservoelastic (ASE) interaction is a problem on new and existing aircraft of all types (e.g., low speed subsonic transports, high speed transonic and supersonic fighter aircraft, hypersonic test vehicles, etc.). ASE mode excitation can cause the aircraft to experience repeated loading, leading to enhanced fatigue and undesirable oscillations for pilots. In some cases the ASE modes may be unstable at certain flight conditions, causing divergent oscillations leading to catastrophic flutter and limit cycle oscillations (LCO). Specifically, LCO is used to describe “sustained, periodic, but not catastrophically divergent” aeroelastic motions. Unlike flutter, the LCO phenomenon results from a nonlinear coupling of aircraft structural response and the unsteady aerodynamics. Due to the detrimental effect of ASE interaction, active feedback control has been utilized to add damping and nullify the effect of these undesirable lightly damped modes. Traditionally, notch and/or roll off filters have been utilized in the primary flight control system architecture to effectively “cancel out” problematic frequencies that will potentially excite the ASE dynamics.1,2,3 Notch and roll off filters present a solution that has proven to be effective but this approach has pitfalls. One problem is the penalty taken in phase due to the filter. This phase penalty can be significant at lower frequencies, making the aircraft less responsive than desirable and can inversely affect rigid body aircraft flying and handling qualities, often requiring adjustment to the primary flight control system. Approaches have been employed that are aimed at minimizing this phase penalty.4 Another concern is that the filter frequencies are fixed. If the ASE frequencies (which vary as a function of flight condition, orientation, configuration, etc.) lie outside of the bound of the preset frequency band, the filter will be ineffective. The filter

A

*

Principal Research Engineer, Senior Member AIAA. Chief Scientist, Senior Member AIAA. ‡ Principal Specialist § Aerospace Research Engineer, Member AIAA †

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approach is not robust to uncertainties and unexpected or un-modeled system changes. Research has been conducted on adaptive notch filtering to provide this robustness.5,6 There is a need for a practical and robust solution to ASE suppression that is adaptive to aircraft system changes and has minimal adverse effect on low frequency (rigid body) dynamics. These changes can result from many factors including flight condition changes (e.g., orientation, speed, altitude, etc.), different stores configurations, unmodeled dynamics and system failures (e.g., battle damage, skin separation and ineffective or “frozen” control surfaces). Several adaptive approaches to ASE suppression and flutter boundary expansion have been developed and documented. Nam describes an approach using the Modified Model Following Reconfigurable (MMFR) control algorithm coupled with Modal-Based Parameter Estimation (MPE) for system identification.7 An adaptive spatial filter approach has been developed by Keas.8 Zeng has developed an adaptive flutter boundary expansion method using iterative estimates of the aeroelastic system.9 An adaptive approach has also been developed by Karpel that accounts for modeling uncertainties using the  synthesis and analysis technique from standard robust control theory.10 Other approaches utilizing dynamic inversion and adaptive dynamic inversion have been researched by Gregory11 and Calise.12 These modern approaches show innovative promise and provide analytical verification but provide little validation with flight tests on actual systems. The traditional non-adaptive notch and roll off filter has been the only extensively proven technique for aeroservoelastic suppression supported by abundant flight test data. There is a strong need for practical adaptive approaches to aeroservoelastic suppression geared toward application to real-world flexible aircraft.

II. Overview Description of the MIDAAS Algorithm An algorithm has been developed that is entitled Modal Isolation and Damping for Adaptive Aeroservoelastic Suppression (MIDAAS, Figure 1). The primary goal of this algorithm is to adaptively suppress adverse aeroservoelastic response in aircraft platforms via active feedback. This distributed control input and sensor output technique, determines an optimal blend of multiple sensor outputs that effectively isolates a problematic lightly damped mode and simultaneously determines an optimal blend of multiple control inputs to suppress the problematic mode via negative feedback while minimizing adverse effects on remaining modal response. Emphasis is placed on minimizing adverse effects on aircraft rigid body modes so that rigid body dynamic behavior is unchanged with minimal phase penalty. A primary flight control system for rigid body stability augmentation can therefore be designed completely independently of MIDAAS and MIDAAS will have minimal to no effect on the primary flight controller and subsequent aircraft rigid body performance. Moreover, MIDAAS has been shown to be adaptive by utilizing short duration input and output data and an efficient subspace system identification approach so that a data-based model can be utilized to automatically synthesize the MIDAAS controller gains online. The MIDAAS control architecture is shown in Figure 2. This technique has been validated against linear dynamic models of representative aeroelastic high-speed fighter aircraft trimmed and operating in the transonic flight regime. Validation has been performed with these models via piloted simulation. The method is general in that it is applied to a generic linear time invariant (LTI) system. A high speed fighter aircraft LTI system exhibits specific dynamic qualities. In general, the eigenvalues (or system poles) of the LTI system are grouped into three sets: 1) low frequency rigid body poles that have various levels of damping, 2) higher frequency aeroelastic structural dynamic poles that are lightly damped and 3) very high frequency unsteady aerodynamic poles that are heavily damped. For systems that exhibit this type of behavior, the MIDAAS algorithm performs very well in that it successfully and adaptively suppresses the adverse lightly damped aeroelastic structural dynamic poles while minimally affecting the low frequency rigid body poles. Although developed specifically for a high-speed fighter aircraft, the success of MIDAAS with these systems suggests a much broader application to aircraft that may not have significant frequency separation between rigid body and aeroelastic dynamics (i.e. highly flexible HALE type aircraft). Although not shown in the results presented herein, some recent investigations with highly flexible aircraft have shown that MIDAAS is successful for aircraft models with flexible structural dynamics that are highly coupled with the rigid body dynamics. Due to the generality of the algorithm, MIDAAS may also be effective with various other dynamic systems (e.g., rotorcraft, non-aircraft structures). The MIDAAS algorithm utilizes least squares constrained optimization to isolate adverse modal response by solving for gains that comprise an output blend from multiple sensor outputs, producing a single signal that represents the problematic mode. As part of this constrained least squares optimization, the output space is projected onto a full rank subspace so that a solution is guaranteed. Aeroelastic modes are complex and are accompanied by a complex conjugate. This fact allows flexibility in definable coefficients that can be used to effectively “tune” the output mode signal for certain desirable feedback characteristics. These coefficients are ideally chosen based on the desired closed loop response, which is to suppress the mode while minimally affecting the low frequency response.


To select optimal input gains and to suppress the problematic aeroservoelastic mode(s), MIDAAS utilizes an output feedback Linear Quadratic Regulator (LQR) with weighting targeted at damping the problematic mode(s) and requiring low control authority. The two methods (optimal output blending and optimal input blending) assume a solution from the other, therefore the input and output blends are determined in an iterative manner until solution convergence, which is determined rapidly with minimal computational overhead. MIDAAS is efficient and therefore very effective for systems with many distributed sensors and many control inputs. Moreover, it can be utilized to suppress multiple adverse modes by synthesizing separate gains for each mode in a systematic manner. High Fidelity ASE Fighter Model Flight Condition (Mach, altitude, , )

Damage and Failures (skin seperation, loss of control inputs)


Adaptive ASE Suppression Controller

Configuration Changes (stores configs)

cos( t )

1 sa

T ( )

1 sa sin(t )

Assessment of Flying Qualities, Handling Qualities and Ride Quality Real-Time Pilot in the Loop Simulation Capability

Real-Time Methods for Adaptive Suppression of Adverse Aeroservoelastic Dynamics

Figure 1: Modal Isolation and Damping for Adaptive Aeroservoelastic Suppression. Controls used in example are  elev ,  diffail ,  rudders Rigid body measurements



All sensors



K

Gain K is result of LQ optimization

L Intermediate signal is blended output approximating desired mode shape

Figure 2: The MIDAAS Control Architecture. By incorporating an efficient subspace system identification solution, MIDAAS has been shown to be effective utilizing a model identified from only input and output time series data signals. By applying this system identification enhancement, MIDAAS has been shown to be fully adaptive, where on-line measured data is used directly to adaptively suppress adverse aeroelastic response.

III. MIDAAS Gain Synthesis: Optimal Input and Output Blending A. Optimal Output Blending for Mode Isolation 1. General Problem Definition Beginning with a multiple-input-multiple-output (MIMO) LTI system representing the aeroelastic aircraft, it is desired to construct a multiple-input-single-output (MISO) or single-input-single-output (SISO) system where the 3 American Institute of Aeronautics and Astronautics Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

single output “best” represents the mode to control. For a single-input, multi-output (SIMO) system, the SISO system that performs this is a linear combination of the multiple outputs. This can be found by minimizing an objective function. First, assume the SIMO system is described in continuous time LTI state space form in Eq. (1). x  Ax  Bu (1) y  Cx  Du The system mode shapes and frequencies are the eigenvectors (z) and eigenvalues () of A, where  z  Az . The system states (x) can be described as linear combinations of the modal states (z) by: x  Xz


z  X 1 x  Yx X   x1 x2  xn 

 y1  y  Y   2      yn 

(2)

In the above equation, X is a matrix composed of right eigenvectors as columns, Y is a matrix composed of left eigenvectors as rows and z are the modal coefficients (or modal states). By this similarity transformation, the state space system in Eq. (1) can be represented in the modal domain (Eq. (3)). z  z  YBu , where   YAX (3) y  CXz  Du Now, assume that a desired single output (yd) is a linear combination of inputs according to Eq. (4).

yd  Ly, where L  [l1 l2  l p ]

(4)

The goal is to determine the gains in L that will best measure a particular system mode. 2. No Direct Input Feed Through Applying Eq. (4) to Eq. (3) results in Eq. (5). Also, for now it is assumed that D = 0 (no direct input feed through). The approach for systems where D  0 will be shown in the following subsection. z  YAXz  YBu (5) yd  LCXz For this context, it is desired to measure a certain mode: zi, which is one of the states in the modal form above. Setting yd equal to zi results in the following condition for LCX.

LCX  eîT

(6)

In the above equation, eî is defined as a column vector where the ith element (cooresponding to zi) is 1 and the remaining elements are zero. Solving the above equation for LC results in Eq. (7).

LC  eîT X 1  eîT Y  yi

(7)

Assuming C is either square or has more columns that rows (usually the case), the result is in an over-determined system of linear equations for the unknowns L. L can then be solved for in a least squares sense (Eq. (8)).



min LC  yi , then L  yi C T CC T L



1

(8)

The resulting vector L is composed of the coefficients of the outputs that produce the “best” measure of the mode zi. This method is only concerned with the output equations and is independent of the inputs; therefore it is not restricted to SIMO systems. L best measures the mode zi, therefore it should effectively “best filter” the other modes. This includes the rigid body modes if they are included as part of the system. As described, it is desired to keep the rigid body modes unaffected. 3. Eliminating Direct Input Feed-Through Using Equality Constraints It is common in aircraft systems with common sensors for the input to feed through directly to the output, in other words, for the D matrix to be non-zero. In this case it is not sufficient to only project the outputs. The same projection should also zero out the direct feed through. Adding a constraint on zero direct feed through can be critical. 4 American Institute of Aeronautics and Astronautics Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

When the D matrix is nonzero Eq. (5) becomes: z  YAXz  YBu yd  LCXz  LDu

(9)

The desired output is the mode zi. The above output equation can be rewritten:

yd  eîT z  LCXz  LDu For this equation to be true, the following holds:

(10)

LCX  eîT subject to LD  0 Equations (7) and (8) still hold with the addition of an equality constraint:

(11)

min LC  yi such that LD  0

(12)


L

If there are more outputs than inputs (usually the case of concern), the equality constraint equation (LD = 0) is underdetermined and restricts solutions for L to be contained on a fixed hyperplane. This problem can be solved in the standard constrained linear least squares sense. For instance, the Matlab function lsqlin can be used. 4. A Real Solution and Further Isolating the Rigid Body Structural modes are complex valued, and so the left eigenvector associated with this mode is also complex valued. A complex value will produce a complex solution for L, meaning that the optimal blending coefficients will be complex. A solution with real coefficients is desired for simplicity and practical implementation. The eigenvectors occur in complex conjugate pairs, and a real solution can be obtained by a linear combination of the complex conjugates vectors with complex conjugate coefficients (Eq. (13)).  eîjT   0  0 

  a  bj,

 

0  0

ith element

   jth element

 0  0 

(13)

   a  bj

Having two eigenvectors introduces a degree of freedom. The complex eigenvector yi is scaled by a complex number creating the complex vector  yi , and it is the real part Re[ yi ] that is used to compute the sensor blend L vector. If  = 1, the real part of the eigenvector is used. If  = –j, the imaginary part of the eigenvector is used. The extra degree of freedom can be used advantageously by minimizing the interaction between the structural mode and the rigid body modes. The response from the blended inputs to blended outputs is yd /u. The extra degree of freedom can be used to “place” a zero of this response. Interaction with the rigid body modes is minimized by placing the zero at the origin, because in this way low frequency gain is minimized. It can be shown that the coefficient that places the zero of yd /u at the origin is defined by Eq. (14).



jpi yi BK

(14)

It is interesting to note that the scale factor depends on the left eigenvector yi , the associated complex pole pi, and the blended input vector b=BK. Using this result will place a zero as near the origin as possible given the output sensors used. One caveat is that this ideal coefficient can only be found for cases involving one input. Optimal input blending, which is described in Section III.B, can be used to determine this blend, defined by the column vector K. 5. Ensuring No Cross Coupling: Additional Constraint on the Output Blend For conventional aircraft, control inputs are designed to provide aerodynamic moments to lateral and longitudinal axes where inputs will not adversely affect the response on other axes. For example, elevators will affect the longitudinal dynamic response while not affecting the lateral response (e.g., elevators should not excite the dutch roll mode). The optimal blend of inputs does not guarantee unwanted cross coupling. This can be alleviated by applying a further constraint to the optimal output blend (Eq. (15)). LCX rb  0

(15)

In the above equation, L is the optimal output blend coefficient vector, C is the system C matrix and Xrb is a matrix of the eigenvectors (as columns) associated with the low frequency rigid body modes. Applying the optimum 5 American Institute of Aeronautics and Astronautics Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


output blend equation with this additional constraint will force the L solution to be contained on a hyper plane that is orthogonal to the rigid body modes, meaning the solution will have little to no effect on these modes. The result being that the blended output signal will contain little or no information in these modes and feeding this signal back will produce minimal to no adverse cross coupling effect. As it turns out, this constraint has a positive side effect as it also further assures that the lower frequency rigid body dynamics are isolated. B. Optimal Input Blending for Modal Damping What remains is determining the optimal input blend to supply damping to the mode while avoiding phase penalty at the lower rigid body frequencies. Choosing an optimal input blend involves closing the feedback loop. If multiple inputs are considered and with a sensor blend (L) determined, the system will be MISO. Assuming that the feedback architecture involves the system and a simple gain, an optimal sensor input blend can be determined using a Linear Quadratic Regular (LQR) with output feedback and the optimal solution to the problem is a vector of gains that, when multiplied by the input effectiveness matrix (B), produces a SISO system. If negative output feedback is applied to this SISO system, the result will be a system that has sufficient damping at the mode of interest while leaving the remaining modes virtually unchanged. By this respect, the rigid body modes will remain unchanged and very minimal phase penalty will be present at those modes. A brief summary of LQR with output feedback, based on the detailed description from Lewis and Syrmos,13 is provided here. The assumed system has no direct feed-through (D = 0) and the feedback is a linear combination of the outputs: x  Ax  Bu, y  Cx (16) u   Ky

The performance index is: J

0  x 

T



Qx  uT Ru dt , where Q  0 and R  0

(17)

The optimal feedback gain K is the simultaneous solution of the following two Lyapunov equations,

0  ACT P  PAC  CT K T RKC  Q

(18)

0  AC S  SACT  X

(19)

where:



K  R 1BT PSC T CSC T AC  A  BKC





1



(20)

X  E x(0) xT (0) (using X  I suffices)

An initial, stabilizing K must be supplied, and unlike the full state feedback version of the LQR problem, a stable solution is not guaranteed to exist. The state weight as a function of the eigenvectors, Q  yiT yi , works well since the goal is to damp the mode. Since the optimal output blend already isolates the mode, the state weighting can be effectively defined by Q = CTC, where C is a row vector representative of the system with the optimal blend applied. The input weight can be set accordingly to R = I, where  can be tuned to limit control input energy. C. Iterative Approach for Determining Optimal Input and Output Blends The ideal optimal output blending solution assumes a single input (SIMO) and the ideal optimal input blending solution assumes a single output (MISO). Since these techniques basically assume a solution from the other technique, an iterative approach was developed to obtain an optimal input and output blend. The resulting system loop transfer function will be SISO from blended input to blended output. The approach is as follows:

1. 2. 3. 4.

Choose an initial guess for the complex conjugate coefficient for output blending ( in Eq. (13)). Solve for the optimal output blend (L) assuming all inputs available. With the L, apply optimal LQR output solution to determine the optimal input blend K. Using the optimal input blend, solve for the optimal output blend L applying the zero at origin condition (Eq. (14)). 6 American Institute of Aeronautics and Astronautics

Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

5.

Repeat steps 3 and 4 until the K and L blend coefficients converge (usually only 3 to 4 iterations are required for convergence)

D. MIDAAS for Multiple Adverse Modes The MIDAAS approach described thus far has focused on the suppression of a single mode. The process can be nested effectively to suppress multiple modes. Figure 3 demonstrates the application of two MIDAAS loops targeting two adverse modes. A/C and actuators

K1L1


K 2 L2

A/C and actuators

K1L1  K 2 L2

Figure 3: Application of MIDAAS for Multiple Modes. The gain matrix K1L1 is synthesized 1st for a specific mode. The gain matrix K2L2 is then synthesized for a different mode using a closed loop model with the aircraft and the original gains (dotted lines in Figure 3). It can be shown that the resulting equivalent MIDAAS gain matrix is simply the sum of the two individual gain matrices (K1L1 + K2L2) making implementation the same regardless of how many modes are being isolated and damped. This process can be repeated for three or more modes.

IV. Incorporation of a System Identification Solution A. Significance of Incorporating a System Identification Solution with MIDAAS Aircraft dynamics change during flight. The changes are due in part to changes in weight (fuel loss), operation in different parts of the flight envelope, or changes in geometry. Changes in geometry can be due to flap and landing gear extension, damage, or different stores configurations. The aircraft structure with current generation aircraft is designed so that changes in modal frequencies and mode shapes are small and a fixed set of structural isolation filters suffice over the normal range of operation. New aircraft designs particularly with lighter structures will undergo larger changes in structural dynamics and this motivates the need to adaptively change the control algorithms. Current high performance aircraft operating at the edge of their envelopes (such as the F/A-18C with heavy stores configurations) also motivates the need for adaptation. There are two types of changes to structural dynamics that are of interest – changes to the modal poles (frequencies and damping ratios) and changes to the mode shapes. A fixed MIDAAS solution is not sensitive to changes in frequency. This is one powerful advantage of the methodology. Basically wherever the mode is located, as long as the mode shape does not significantly change, the L and K matrices will isolate the mode and provide damping. There is a secondary dependence of the output blend on the pole location, included to minimize the interaction with the rigid body modes, and so it is possible that rigid body interaction will increase due to changes in the pole location. Significant changes in mode shapes will result in unpredictable changes in the stability and performance of fixed control gains. The unpredictability can be reduced through analysis. The alternative is to adaptively estimate changes in the structure and project the sensor blend onto the estimated mode shape. MIDAAS is adaptive to the system and is not sensitive to mode shape changes; hence the goal is to update MIDAAS gains repeatedly using continuously identified models. This requires an efficient system identification solution capable of accurately identifying large order MIMO models using time series measurements from many sensors.


B. Subspace Identification Input and output time series data can be used to directly estimate the A, B, C, and D matrices of a linear state space system. An early and ground-breaking version of state space estimation was developed by Juang and Papps and is called the Eigensystem Realization Algorithm (ERA).14 This method has evolved over time and the general approach is now commonly referred to as Subspace Identification (SSID).15,16 Recent innovations by Miller and de Callafon have been to estimate the initial condition along with the state space matrices, and to use the more noise tolerant correlation matrices rather than direct use of the time series.17,18 In general, SSID is performed in two steps: first the A and C matrices are estimated followed by the B and D matrices and the initial state vector x0. While there are a variety of SSID methods, the general principles and techniques are very similar. This section focuses on the general problem.


1. Computing A and C The first part of the SSID method is to isolate the observability Grammian, from which the A and C matrices can be computed. The output response with a non-zero initial condition can be written as follows in Eq. (21).  y0   h0  y1   h1  y   h  2  2  y3   h3

0 h0 h1 h2

0 0 h0 h1

0  u0   0  0  u1   1   x 0  u2   2  0     h0  u3   3 

(21)

In the above, measured output records for different time steps are yi and input records are ui. The output and right hand side can be arranged as shown in Eq. (22).  y0  y1 y  2  y3

y1 y2 y3 y4

y2   h0 y3   h1  y4   h2   y5   h3

0 h0 h1 h2

0 0 h0 h1

0   u0 0   u1 0   u2  h0   u3

u1 u2 u3 u4

u2   0  u3   1   x u4   2  0    u5   3 

x1

x2 

(22)

The matrix version can then be written as follows in Eq. (23).

Y (0, n, m)  ( n ) X ( m)  H l (0, n, n )U (0, n, m) Y  X  H lU

(23)

The simplified version is shown in the second equation above by eliminating the indices. To isolate  multiply the right hand side by the null space P of the U matrix (Eq. (24)). YP  XP (24) The null space can be computed from the SVD of the U matrix. The dimensions are selected so that U is a short, wide matrix, wide enough so that the null space has dimension at least equal to the state dimension. The X matrix does not need to be created, it suffices that the rank is high enough. The important thing is the multiplication on the left by , which allows both the state matrix A and the output matrix C to be extracted. The method is summarized as follows in Eq. (25). U (0, n, m )  [U1 U 2 ][ S 0; 0 0][V1 V2 ]T , where P  V2  basis for null-space of U Y (0, n, m ) P  ( n ) X ( n ) P (don't care about X ( n ), important part is ( n )) M  (YP )(1: nn y ,1: n ) =U m SmVmT N  (YP )( n y  (1: nn y ),1: n ) 1

CT

T T AT  ( Sm1/2U m ) N (Vm Sm1/2 ) 1/2  (U m Sm )(1: n y ,1: n ) (1st n y rows) 1/2 T  (U m Sm )

2. Computing B, D and x0 The output response can be written in a different way as shown in Eq. (26).


(25)

 y0   0  0  y1   1  1  u0  y     T x0   u  2  2  1  u2  y3   3 

0 0 u0 u1

0 0 0 u0

0   0   u0  u  0   1  1 T B 1D 0   2  u  2   0   3   u3 

(26)

This is a linear least squares problem that can be solved for x0, B, and D. For the MIMO case a similar expression is used with B and D “vectorized” by lining up the columns into one long column. The last steps are summarized as follows in Eq. (27). X u  [ ( n )T U ls (0, n, n ) ( n )T U (0, n,1)]

(27)

[T 1 x0 ; T 1B; D ]  X u1Y (0, n,1)


V. MIDAAS Validation A. The High Speed Fighter Aircraft Model The vehicle utilized for validation was the aeroelastic F/A-18C model with full stores operating at 10,000 ft altitude and traveling Mach 0.7. The model was created by combining a rigid body aircraft model and a flexible aeroelastic model (Figure 4). The rigid body aircraft model is based on a linear model of the F/A-18 HARV19 and the flexible model was derived from a linear reduced order model (ROM) of a high order Computational Fluid Dynamic/ Computational Structural Dynamic (CFD/CSD) model.20 F/A-18C Flex

Rigid : xr  Ar xr  Br u

Flexible : x f  A f x f  B f u

y  Cr xr  Dr u

y  C f xf  Df u

Combined :

F/A-18C Rigid

 x f   A f    xr   0 y  C f

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Figure 4: Combined F/A-18C flexible and rigid body model. Since multiple modes require damping, a large sensor set was utilized consisting of angular velocities, translational velocities and translational accelerations at 9 different aircraft sensor locations: near the c.g., near the left and right exhaust, near the left and right vertical tail tip trailing edges, near the left and right wing tips, near the nose tip and near the pilot. With 3 angular velocities, 3 translational velocities and 3 translational accelerations for each of these locations, the total number of output sensors is 81. The 3 control inputs consist of elevators (collective stabilators), differential ailerons and collective rudders. The model is also outfitted with a thrust input but this was not used for MIDAAS feedback. A close up of the system poles is displayed in Figure 5 where the problematic modes are indicated. As expected, the asymmetric modes were a problem during lateral axis maneuvers (e.g., rolling and/or yawing) and the symmetric modes were a problem during longitudinal axis maneuvers (e.g., pitching). B. Rigid Body Stability Augmentation System The rigid body dynamic model of the aeroservoelastic vehicle is based on bare-airframe dynamics and it requires a Stability Augmentation System (SAS) to augment the vehicle in normal flight for sufficient flying qualities. The SAS design is based on the Multi-Input/Multi-Output successive loop closure method. The basic idea of the method is to close one SISO loop at a time using the system that is closed previously. It is not intended to develop a new controller or replicate the control system for the fleet F/A-18, rather it is intended to provide basic stability to the vehicle so that the pilot can perform required tasks in real-time. The SAS was designed for three major axes: roll, pitch, and yaw. The sequence used for the design is to close the yaw loop first, followed by roll and pitch. The yaw loop uses simple gain compensation to provide increased dutch roll damping. The roll and pitch loops use a PI (Proportional and Integral) type controller to provide good flying and handling qualities requirements. Bandwidth requirements for level I flying and handling qualities for fighter aircraft are taken from Ref. 21, as 4.5 rad/sec for pitch and 2.5 rad/sec for roll. Considered also is good “k/s-like” open-loop frequency response at the cross-over frequency for good tracking performance and disturbance rejection. The SAS for each axis was successively designed based on the rigid body dynamics with actuators. It is noted that the compensator designed with the rigid 9 American Institute of Aeronautics and Astronautics

Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

body dynamics is not sufficient enough to suppress the flexible modes, justifying the need for MIDAAS gains to be applied. It is also noted that this SAS was designed completely independently of MIDAAS as MIDAAS does not affect the rigid body dynamics. Including an independently designed SAS provides a “flyable” aircraft and also helps to validate that MIDAAS does not inversely affect independently designed primary flight control laws. 90 0.036

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Figure 5: F/A-18C model system poles at 10,000 ft., Mach 0.7. In addition to the primary SAS, an auto-throttle was also implemented to relieve the pilot’s workload and to aid in maintaining altitude during tasks. The augmented auto-throttle feedback command is based on a simple speed regulator. The throttle (or thrust) command is generated based on the speed deviation from the trim speed with a first order engine lag model and a proportional controller. The proportional gain was tuned to give good speed responses. C. Piloted Simulation Description The aeroservoelastic vehicle dynamics is implemented as a SimulinkTM model for testing and validation before generating C/C++ codes for real-time simulation. An overview diagram of the simulation model is shown in Figure 6. First, the model reads three pilot stick commands, aileron, elevator, and rudder. The model contains a throttle input as well but this is not currently linked for external pilot input and is only used internally for the auto-throttle. These pilot stick inputs are processed through the Command Shaping block where commanded control surface deflections are converted from stick inputs. Look-up tables, control surface vs. stick position, are used to convert stick inputs to control surface deflections assuming linear stick position gradients.22 The stick positions are normalized positions so that different stick characteristics can be easily incorporated. Maximum and minimum control surface deflections were also implemented. The Feedback block implements the rigid body SAS as described previously and MIDAAS feedback gains. These feedback commands pass through the Actuator block which assumes 2nd order actuator models for individual command path.23 The Aircraft Dynamics block consists of separated rigid- and flexible- body dynamics for dynamic simulation. The outputs of the block are the combined outputs from rigid and flexible body dynamics. The flexible body dynamics outputs are utilized for cueing the vibration of the flexible mode dynamics to the pilot via the graphics display. These outputs are fed through the Output block where all necessary signals for cueing and data recording are processed. It is noted that the vehicle dynamics is based on a linear model but all kinematics, the Euler angles and the position vector, are based on the full nonlinear relationships so that the cueing to the pilot is more reliable and accurate.

Figure 6: F/A-18C simulation model architecture displaying the available input signals.



The numerical integration scheme is based on fixed step size discrete system integration. The nature of the aeroservoelastic dynamics requires a small integration step size to capture all high frequency dynamics. Moreover, the complete system is on the order of 100+ states. A continuous time integration scheme imposes a great deal of computational burden when the integration time step is reduced to capture the high frequency dynamics, as a result real-time simulation is less feasible. Employment of discrete system simulation removes this constraint. The sampling time for the discrete system was set to 0.004 seconds, adequately capturing frequencies up to 125 Hz (Nyquist limit). For this model, the actual computation time for one time step integration is less than 15% of the sampling time and this confirms real-time capability. The rest of the computation time was allocated to the graphics display (Figure 7) which updates at 50 Hz. The outputs from the SimulinkTM model are recorded at the dynamics update rate (250 Hz).

Figure 7: Simulator graphics display showing HUD with tracking task command displayed. The simulation was conducted using a fixed base simulator. Since the simulator produces no motion, it was desired to provide a “cue” of the accelerations from the aeroelastic response by other means. The effective solution was to amplify the visual graphics orientation by augmenting the displayed pitch, roll and yaw angles. Since it was desired to effectively display the aeroelastic dynamics, a scale factor (SF) was applied to the flexible-only counterpart of these displayed Euler angles. The rigid body counterpart remained the same. These augmented Euler angles were only utilized for the graphics display to serve as a pilot cue. They were not used for sensor measurement and were not part of the aircraft dynamic equations. This visual amplification was necessary for this aircraft at this flight condition but may not be necessary at a flight condition closer to the flutter boundary. To assess performance of MIDAAS, tracking tasks were conducted. The task commands employed were the discrete tracking task described in Refs. 24 and 25. The task is very suitable for evaluating handling qualities in tight, closed-loop tracking tasks. These tracking task commands are displayed on the HUD (the orange bar in Figure 7) and the pilot is advised to track the displayed task commands aggressively and attempt to keep errors within the specified tolerances (within the green circles). These tasks provided maneuvers that adequately excited the adverse modes so that the effectiveness of MIDAAS can be evaluated. D. The MIDAAS Design From the perspective of MIDAAS, the aircraft model can be either an open loop model or a model with SAS implemented. Also, the SAS can be applied as an inner or outer loop. Since the SAS concentrates on the rigid body modes, the MIDAAS gain matrix will remain basically unchanged utilizing a system with or without a SAS. For the results presented here, the MIDAAS gains were synthesized using an open loop aircraft model with actuators. This was done using both the exact aircraft model and a model identified from measurement data.

1. MIDAAS Synthesis Using the Exact Model This section summarizes the MIDAAS synthesis results when utilizing the exact model of the aircraft as opposed to a model identified from sensor measurements. This serves as a benchmark for comparison of results using identified models. MIDAAS gains were synthesized for one mode at a time and a composite gain effective for all modes was formulated by summing the individual gains (see Section III.D). The modes were damped in order of increasing 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

frequency. First, the asymmetric mode near 5 Hz was isolated and damped. With this loop closed, the asymmetric mode near 8 Hz was isolated and damped next. Finally, the symmetric mode near 10 Hz was isolated and damped using the closed loop system from the previous two MIDAAS passes. Figure 8 below displays a survey where the blended input and output pair is from the final MIDAAS pass targeting the 9.8 Hz mode. Root Locus: blended output/blended input 100

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2. MIDAAS Synthesis using an Identified Model Actual piloted simulation data was also utilized to synthesize MIDAAS gains. For this case, 20 seconds of data was utilized. A piloted simulation tracking task was conducted where a random excitation signal was applied and summed with the pilot commands to provide sufficient input data for identification. Referring to Figure 6, this random input was applied just before the measurement at signal 2. The amplitude of the random excitation was such that it provided no discomfort for the pilot and did not affect the ability to conduct the task. A model was identified using the SSID method developed by Miller and de Callofon.17,18 The measured input data were the commands to the actuator (signal 4 in Figure 6) and the measured output data were the 81 sensor signals described above (the output signal of the “Aircraft Dynamics” block in Figure 6). These signals will likely be measureable on an actual aircraft and when used they will provide an identified model of the bare airframe with actuators. This is operationally advantageous as the resulting identified system is exactly the system that the MIDAAS controller is 12 American Institute of Aeronautics and Astronautics Copyright © 2013 by Systems Technology, Inc.. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

compensating. The identified system was assumed to be 30th order. The identified poles are compared to the exact (true) poles in Figure 9 below. Map of Poles and Zeros (no input or output defined, only poles displayed) 90

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1. Qualitative Assessment In the low frequency range, the aircraft behaves well with and without the active MIDAAS gains. This is in large part due to the independently designed SAS. The visual motion “cue” (described above) was very effective and created an artificial sense of motion with this fixed base simulator. The aeroelastic response was observed as high frequency oscillations visible in the heads up display (HUD) as well as the graphics scene. This visual cue provided a noticeable difference when MIDAAS was active and the increased damping of the aeroelastic response was obvious when compared to the MIDAAS “off” case. In summary, the aircraft “felt” generally the same with and without the active MIDAAS controller but was notably easier to fly with “MIDAAS ON” since the visible oscillations were noticeably suppressed. Also, there was no observed phase lag or gain change in the rigid body response when the MIDAAS controller was active. By these standards, the goal was achieved: adequate suppression of adverse aeroelastic dynamics with no negative effect on the aircraft flying and handling qualities.


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2. Quantitative Assessment The stick inputs from the simulation are shown in Figure 11. The units are Volts, as measured directly from the simulator hardware. One Volt is approximately equivalent to one inch of stick deflection. The Mach number is shown in Figure 12 which follows the same general trend for both “off” and “on” cases. This is expected since the tracking task was the same and the rigid body dynamics are essentially equivalent for both cases.


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VI. Conclusions The MIDAAS solution can be applied independent of any primary flight control solution and has virtually no adverse effect on lower frequency aircraft rigid body performance, making it an ideal alternative to traditional methods (i.e., notch and/or roll off filters). Also, the solution from MIDAAS is simply a matrix of gains that constitute each output sensor fed – via negative feedback – to each control input. Utilizing more information (i.e., more sensor outputs and more control inputs) increases the likelihood of success in suppressing the mode while maintaining that low frequency behavior is unchanged. Also, utilizing more control inputs keeps the control authority low (another goal of the algorithm) since more control inputs are available to provide suppression, leaving ample control authority for primary flight objectives (i.e., rigid body stability augmentation, pilot commanded maneuvers, etc.) and maintaining clearance from rate and saturation limits. The control authority can also be


effectively tuned by adjusting algorithm parameters. The solution was shown to be determined relatively rapidly using un-optimized code (In the Matlab environment on a Dell Precision T1600, Intel® Quad Core Xeon® CPU E31270 @ 3.4 GHz., the total time to solve for the optimal input and output blends targeting one mode was less than 3 seconds for a transonic high-speed aeroservoelastic aircraft system with 106 states, 3 control inputs and 81 sensor outputs). The algorithm efficiency suggests that MIDAAS can be embedded on an operational aircraft platform for continuous gain updates providing adverse aeroelastic suppression subject to a time varying system. Since there is virtually no adverse effect at lower frequencies, an embedded MIDAAS controller will have no effect on an existing flight control system and can be designed and incorporated completely independently as an inner or outer control loop. In addition to maintaining no adverse effect on low frequency modes, MIDAAS has been shown to successfully isolate and damp adverse modes while minimally affecting other modes in very close frequency vicinity, suggesting further success with systems that do no exhibit large frequency separation between adverse modes (e.g., aeroservoelastic modes in this context) and performance modes (e.g., rigid body modes in this context).

References Downloaded by David Klyde on August 26, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-4743

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