Interactive aircraft flight control and aeroelastic stabilization

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^ INTERACTIVE AIRCRAFT FLIGHT CONTROL AND I AEROELASTIC STABILIZATION NASA/Langley Research Center Grant-NAG-1-157 Semi-annual Report 1 November 1983 through 30 April 1984 W-4-31214 (SAS A -CR-173b66) Ih`Ia. '11VE A1RChAxl FLIGHT CGIN:RUL ANC AEFCEi.AS11C STAJI...IZAZIUN huv. 1S63 - 3U Apr.

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ULiv.) 13 N liC AC, 2/MF AU 1 Uaclas

1984 (Purdue

CS^_L 010 t;3/08 .,01013 Submitted by: J i i Dr. Terrence A. Weisshaar f Dr. David K. Schmidt Principal Investigators SCHOOL OF AERONAUTICS AND ASTRONAUTICS PURDUE UNIVERSITY

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WEST LAFAYETTE, INDIANA 47907 May 1984 ^^Z^i

TABLE OF CONTENTS Pace Section Introduction The Use of Structural Gains as Design Parameters An Elementary Model for Aeroservoelastic Optimization 5 Appendix 1. SDM Paper - A Survey of Aeroelastic Tailoring- Theory Practice, Promise 2. Development of Perturbation Equations for Structures/Control Optimization

Introduction The purpose of this study is to examine the potential benefits and costs of optimizing both the structural stiffness and the active control of aircraft in a rational manner. The ultimate goal of this effort is to arrive at a unified treatment of structural and active control design for the stability augmentation of flexible aircraft.

Three separate efforts have taken place during the past six months of effort. The first effort is an exhaustive literature evaluation in the area of passive tailoring for aircraft performance. During this effort, several valuable and previously unrecognized tailoring studies were uncovered. This survey was combined w~th similar work by Mssrs.

M.H. Shirk and T.J. Hertz of the Air Force Wright Aeronautical Labora- tories to produce a paper presented at the 25th AIAA Structures, Struc- tural Dynamics and Materials Conference in Palm Springs, California in May 1984.

The second effort involved the identification of a mathematical technique to be used for aeroservoelastic tailoring studies. A promising candidate method has been identified and is described in the following section.

Finally, two analytical models, one elementary, the other sophisti- cated, have been developed to illustrate the potential for aeroservo- elastic tailoring. Both models have essential features of "real-world" hardware, yet the physical understanding is not buried in a myriad of detail. These models are also described in the next section.

The Use of Structural Gains as Design Parameters There are two major obstacles to simultaneous treatment of the struc- tural stiffness design optimization problem and the active controls problem. The first difficulty arises because of the dissimilarity of design variables in the two problems. This difficulty has been overcome, at least at the elementary level, by the selection of a characteristic set of nondimensional parameters for beam-like and plate-like structures.

The state space model of an aeroelastic system can be written as: x = Ax + Bu (1) y = Cx (2) with x as the n-dimensional state vector, u is an m-dimensional control vector and y is the output vector, while A,B and C are constant coeffi- cient matrices. If a linear, full-state feedback control law exists, of the form, u = -Gx (3) then the modified system equations are: z = (A-BG)x (4) On the other hand, the equations for a structural system with passive control may be written as: z = Ax - ',Ax (5) where ^ is a nondimensional parameter related to stiffness cross-coupling provided by structural tailoring and A is a muc'ification to the A matrix provided by changes in the stiffness matrix.

Equation 5 resembles Eqn. 6 in that *Ax - BGx (6) If there are several variables, 0 i , corresponding to tailored bays of a wing for instance, Eqn. 6 becomes jo i A i x = BG x (7) Theoretically, one should be able to construct a structural modification in terms of * i A i to furnish the same equivalent (in terms of eigenvalues) system as the actively controlled system. A major problem arises, however, because the elements of A are not free parameters while the elements of G are. Thus, standard optimal control procedures (for instance, pole placement) do not have an obvious adaptation. Attempts at such adaptations over the past six months have not proved productive.

Fortunately, a methid developed by Newson and Gilbert offers at least a preliminary approach to the simultaneous design problem. If the cross-coupling parameter ^ is treated as a design parameter that is held fixed during the control design, its effect on the control system performance can be assessed by employing optimal sensitivity techniques.

With this technique, a cost functional, J, is minimized to obtain the "optimal" control law for the system. The parameter 4, is then treated as a design variable so that the change in J with respect to ^ can be computed usirg an adaptation of the Newson/Gilbert approach. This adaptation is described in the Appendix to this report. In addition, the sensitivity of other aspects of the control law design to * may be assessed.

The "important", or a a technique is available t upon active control design are described in the next

I

An Elementary Model for Aeroservoelastic Optimization To study the problem of aeroservoelastic optimization, one may begin either at an advanced or an elementary level. The operative term would be "state-of-the-art." Reviews of the literature and past experience have convinced us that a first look at aeroservoelastic optimization (ASEO) should begin with an example that is simplistic, but meaningful. The model chosen is shown in Figure 1. This model consists of a typical section free to pitch and plunge as a rigid body. The design variable * is, in this case, equal to e/b. The dimension e/b measures the distance between the static aerodynamic center at the quarter-chord and the plunge spring position on the airfoil. For a fixed ratio R = K a/Kh and with the airfoil c.g. position fixed, the divergence speed of the fixed root airfoil declines with increas- ing e/b. On the other hand, the flutter speed increases with increasing e/b. This provides a design trade-off situation for which an optimum value of e/b exists to maximize the aeroelastic stability of the system.

:f the airfoil is attached to a fuselage element that is, in turn, free .. i to pitch and plunge, the situation becomes more interesting because the value of a now determines the attitude stability of the aircraft and values of a that maximize the stability of the fuselage/wing combination may differ significantly from those which were found for the wing alone.

The addition of the control surface to the model provides additional design options. With R fixed the control effectiveness is unchanged by

rt

changes in e/b. Thus any design benefit or degradation is unrelated to control effectiveness in this idealization.

For fixed values of the system structural and inertial parameters, an optimal control law may be generated. A sensitivity analysis will then

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f

W a CJ .-r

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J be performed to assess the effect of a change in e/b (de/b) on the active control of the vehicle. From this information, a new value of a will L.: selected, together with new control parameters. One possible limiting case of this procedure is that the active control could disappear entirely, meaning that passive control is sufficient to handle the stability problem.

Ris model has most of the structural dynamic characteristics of an actual vehicle. The potential for strong rigid body/wing interaction exists, as does '6he capability of studying the differences between control- ling the stability of the wing itself (in a fixed fuselage condition) and the wing/fuselage combination. The most serious limitation of this model is the limited number of degrees of freedom.

This analytical model does have advantages. It is a valuable learning tool, uncluttered by a myriad of numbers. Each step of the ASEO procedure is easily understood and interpreted in light of the substantial amount of information available on 2-D sections.

Because of the limitations of the 2-D model, the kU procedure will t be exterided to a realistic wing with a structurally tailored span. In this case the cross-coupling parameter is the design variable. As p before, the objective will be to improve overall performance in light of lessons learned with the 2-D model.

k

e Appendix Optimal Control and Sensitivity Derivatives for the Redesign Problem The aeroelastic equations of motion may be written (B.1) X = AX + Bu .

If the control is a linear, measurement feedback control, then (B.2) u = GMX M = state measurement matrix (i.e. z = MX) where, and G = feedback gain matrix.

Then the control-augmented system matrix is A+ = A + BGM, so (B.3a) X = A+X . (6.3b) The subscript "plus" sign denotes augmentation.

A quadratic cost function, used in linear regulator design, is (21) (X*C*QCX J - + u*Ru]dt (B.4)

f'

C = output matrix (i.e. y = CX), where, Q = output weighting matrix, and " = control weighting matrix The well known solution for the optimal control that minimizes J, subject to the constraint, eqn. B.I. is k

'16

t R -1 B*PX u (B.5s) or, GM - -R -1 B*P (B.5b) where P is the solution to the steady-state matrix Riccati equation, PA + A*P + C*QC - PBR -1 B*P = 0 (B.6) For computation of sensitivity derivatives, it is assumed that the optimal control, eqn. B.5a, has been determined for a baseline configur- ation and that the weightin g natrices used in this determination, Q and R, are "frozen" (i.e. insensitive to the design parameters, p i , so that - B B p - 0, and ap i — = 0). Also, the control input matrix, B, is considered > to be dependent upon the type and geometry of the control being used, and BB not upon the design parameters. So, = 0, also. Since the cost Bpi function, defined in eqn. B.4, is what determines the optimality of the control design, it will also be the measure by which subsequent redesigns are Judged.

First, the cost function is decomposed into its re g ulation and control parts, J = + J u (B.7)

j

Now, ix = X*S X X o (B.8a) where S satisfies SX A + + A+S x + C*QC - 0 (B.8b) X*Su X o and Ju = (B.9a) fh

I# where S u satisfies (B.9b) S u + + A u + PBR -1 B*P = 0 A + S x is the initial condition (time, t, is zero) on the state vector.

is found by i The regulation cost sensitivity with respect to p differentiating egns. B.8a and B.8b so that

aJ as

(B.lOa) X Xo X o api B i p i as where p x satisfies i aA aA* as as (B. 10b) + C*Q ac ) 0 apx A+ + A+ apx + (Sx .L QC ap i p i i i i Similarly, the control cost sensitivity can be found from aJu aSu (B. 11a) = X* X o ap i o ap i as satisfies where api as ? A u A + A* aSu + ( aP ) p S a + + + PBR-1B* 0 + S + a BR -1 B*P ap i + + ap i u i ap i u ap i api ap (B.11b) Then, in general, any desired change in the costs can be effected within the theoretical limits of the parameters, p i (and provided there are a sufficient number. NP, of parameters), as NP as

ap i Xo (B.12n)

eJx ep i

iI1Xo

A NP DS s u Xoepi(B.12b) ! , X* and eJu a pi Since only first order derivatives are being used, it would be wise if the Bp i 's are kept small throughout the redesign iterations.

To complete this derivation, it is necessary to obtain expressions for ap+ and ep , found in eqns. B.10b and B.11b, which are as yet i i undetermined. By first different-sting the Riccati equation, eqn. B.6, and using the definition for A+ , eqn. B.3a, and the solution for GM, eqn. B.5b, an equation that car be solved for can be obtained, namely, ap t aP A aP + aC* + aC + P aA + A* + A* P) - 0 (B._3) ( . LC* - QC C*Q + op. ap ap ap. + ap ap t i i i t t Equation B.13 is similar to one derived in (22) except that the Q, R.

and B matrices are assumed insensitive to p i , and the output matrix, f,

2L

is included explicitly. Note that is known (see sect'ons 2 and 3).

api aA+ aC It is assumed that ap is known.* Then, can be found from ap i i aA aP+ = ap - BR-1B* (B.14) aP i t i T The output matrix, C, can either be insensitive (i.e. 2p = 0), or be i some other known function of the parameter. For instance, if the output to be regulated, y = CX, consists of internal structural loads, then C will resemble some portion of the structural stiffness matrix.