Interactive aircraft flight control and aeroelastic stabilization

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(NASA-CH -

176035) I11TEfilCTJYE AIRCRAFT NE5 -?1064 FLIGHT CONTROL AND LESOELASZIC !I'BILIZATICN Seaiann aal Report, 1 Nov. 1964 - 30 Apr.

1985 (Purdue Univ.) 20 F k1C A02/RF 101 unclas

CSCL 01C :3/08 21805

INTERACTIVE AIRCRAFT FLIGHT CONTROL AND AEROELASTIC STABILIZATION NASA/Langley Research Center Grant-NAG-1-157 Semi-annual Report 1 November 1984 through 30 April 1985 Submitted by: t Dr. Terrence A. Weisshaar Principal Ins=stigator SCHOOL OF AERONAUTICS AND ASTRONAUTICS PURDUE UNIVERSITY WEST LAFAYETTE, INDIANA 47907 May 1985 188 5

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RECEIVED NASA Sn FMM

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Summary This reloet covers research performed under sponsorship of NASA/Langley Grant NAG-1-157 during the period 1 November 1984 through 30 April 1985. An analytical model of a 3-D airfoil was used to study an optimization procedure formulated to enhance staLility of an airfoil through integrated structural and control synthesis. This procedure is 3iscussed in this report, together with preliminary results. These results show that a sensitivity derivative approach utilizing structural parameters, weighting matrix parameters and optimal control parameters (in this case, the design airspeed) is effective in determining the "best" structural/control design.

No trips were taken during this reporting period. The first trip

was taken 28-29 January 1985 tc visit Langley Research Center to present a research progress report. The second trip was taken from 1 April to 3 April 1985 to attend the Second International Conference of Aeroelasti- city and Structural Dynamics in Aachen, West Germany. A paper was presented at this conference. A trip to MBA Aircraft in Munich was also included from 4 April to 5 April 1985 to discuss common research efforts. A trip summary is included in this report.

Discussion The objective of the current work was to develop and to demonstrate a procedure to ensure aeroelastic stability of an airfoil up to and including a target airspeed. To accomplish a portion of this objective, optimal control techniques were used to design the active control sys- tem. In addition, sensitivity derivatives were computed to assess how changes in system parameters affect the optimal control law design. In

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addition to control parameters such as the elements of the state weight- ing matrix (the [Q] matrix) and the design speed at which the control law is formulated, denoted as UDes, the position of the shear center, expressed as a parameter a e , was used as a design variable. U is a non- dimensional airspeed, while a e is a ncndimensional coordinate described below.

The objective of the new procedure is to define a control law such that the aeroelastic system is stable at all speeds below a certsin airspeed, called U a . A problem that arises with the use of optimal con- trol techniques to define such a control law is that the system may be stable at the airspeed at which the control law is formulated, but may be unstable at lower airspeeds. As a result the design airspeed for tha control law does not necessarily correspond to the maximum airspeed to be reached before aeroelastic stability is encountered. Conversely, UDes may be used as a parameter in the stability augmentation problem.

Similarly, the elements of the state weighting matrix [ used l Q ij for optimal control synthesis are arbitrary. However, the choice of these Q ij elements affects the control law and the off-design perfor- mance of the active control. Setting these parameters in an "optimal" manner is also advantageous. Therefore, the Q ij elements become design parameters.

The structural design affects the behavior of the flexible struc- ture. The example chosen for the current research is a 3-degree of freedom, typical section model as shown in Figure 1. The degrees of freedom are airfoil pitch, a, airfoil plunge, h, (bending) and control surface rotation, In particular, for this 3-DOF model, the position B.

of the shear center affects the aeroelastic stability of the airfoil.

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Figure 1 - Three-degree--of-freedom typical section airfoil.

In reality, the airfoil skin thicknesses, span areas and locations all affect the shear center location. A parameter, a , is used to denote e shear center position. The parameter a is the nondimensional position i (position divided by semi-chord) of the airfoil shear center with respect to the airfoil midchord. As a result, a = 0 represents the e shear center position at the airfoil midchord, while a e = -0.50 corresponds to a shear center placement at the airfoil quarter chord.

f - For this study we have assumed that there is no weight variation (penalty) to be incurred when the shear center is moved. Furthermore, -^ the sectional center of mass is held fixed as a is changed. These res- trictions do not invalidate the results to be discussed nor th o pro- cedure used to generate tiese results.

The shear center location has a substantial effect upon the "open- loop" aeroelastic stability of the 3-DOF airfoil. The shear center position also has a substantial effect upon the value of the "cost func- tion" used to generate control laws using full-state feedback with optimal, steady-state, linear-quadratic-regulator theory (LQR theory).

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a a.

Figure 2 - Constant optimal control cost contours for an actively controlled 3-DOF typical section as a function of UDes and ae.

In Figure 2 the effect of the parameter a upon the standard LQR U Des is indicated. The open- cost function J for different values of loop flutter and divergence speeds, as functions of a e , are indicated as bold solid and dashed lines, respectively, in Figure 2.

UDes - 6.0 an optimal control law may be designed If a - -0.20 and such that closed-loop stability is ensured at U - 6.0. As seen in Fig- ure 2, the cost, J, is approximately J - 10. If UDes is fixed at 6.0, but a is changed to equal -0.4 (a forward shear center movement), sta- e bility of the closed-loop system is still ensured at U 6.0. However, the "cost" has increased to J - 20. While the control cost can not be directly translated into real costs, the implication here is that the system requires more effort to control. This is due to the close prox- the open-loop stability boundary when a - -0.40.

imity of Next, let us examine the closed-loop behavior of this system. To generate Figure 3, active control laws were formulated, using LQR theory with UDes fixed at 6.0, and at several different values of a e . Both cpen-loop And closed-loop instability speeds are shown in Figure 3.

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Figure 3 - Closed-loop stability boundaries for an actively controlled 3-DOF typical section. The design airspeed is U - 6.0 while control laws are formulated as various values ofDas.

e With UDes fixed at 6.0 the closee-loop stability behavior may be degraded for one of two reasons. First of all, the open-loop stability boundary may be degraded at speeds above UDes m 6.0 when active control is added. This is seen to be the case when a e is in the vicinity of -0.30. In a second case, when a is less than -0.40, a low-speed insta- bility region appears due to the addition of feedback control.

Notice also, that if one were to fix UDes and maximize the airspeed at which closed-loop seroelastic instability occurred, this maximum I would occur near a - -0.20. This value is slightly less than the max- imum obtained using passive means (change in a e only). !•^.

i_ In summary, the choice of both a e and UDes has a substantial effect upon closed-loop system stability. In addition, low values of the LQR cost function, J, do not necessarily lead to acceptable dynamic response at all speeds below the instability speed. As a result of these obser- vations a nodified procedure was investigated to both take advantage of LQR theory, optimization theory and optimal sensitivity derivatives.

This method was proposed by Mr. T.A. Zeiler who has included work by Mr.

M.G. Gilbert to produce a hybrid scheme for selecting the best S

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structural/control design.

An Integrated Design Synthesis Scheme Let us define as our design objective the increase in the aeroelas- tic instability speed (either flutter or divergence) to a certain value, U a . This increase will be accomplished by utilizing two sets of design variables, those involving structural parameters (in the present case, only ae ) and those involving active control law parameters. The optimal control design airspeed UDes does not necessarily correspond to the We also require closed-loop system sta- design objective airspeed, U s .

bility at all airspeeds below U .

s At or at any other airspeed below U 8 , the closed-loop system V will yield eigenvalues of the form + jwi Ai - of (1) Our design objective requires that all values of o f be negative (in the left half-plane) at speeds below U s . A function, F., may be defined such that 1 N poi `' Fa ln[ E e ] (2) P i-1 F a is called a cumulative constraint function. In Eqn. 2, p is a con- stant chosen to scale the problem while N corresponds to the number of distinct values of of obtained from the closed-loop eigenvalue analysis.

It can be shown (see Sobieski, et al., AIAA Paper No. 83-0832-CP) that F is bounded as follows: s vi (max) c F a c a (max) 1n N (3)

+ p

Thus if at a given airspeed, U, F a is negative, then ai (max), the larg-

eat value of at this airspeed, U, is also negative and the system is a I stable.

What happens if, U or some other airspeed U j , the value of Fs at is fcund to be positive? How do we modify the system control law and structure to reduce the value of F s to an acceptable value in an "optimal" manner? Certainly we could rely exclusively upon active con- trol. However, situations such as shown in Figure 3 might arise, for which "sub-critical" instabilities appear.

Let us designate F s as being a function of a set of "design" param- The functional F s is also a eters, p i , in this case UDes and (l ij .

B e' function of the control, u.

First, we select a finite set of airspeeds at intervals up to Us; call these airspeeds L1 j . These speeds are chosen to monitor system sta- bility at "sub-critical" airspeeds. At each airspeed except U s , the yield functionals F such actively controlled closed-loop eigenvalues X that (4) Fj<0 atU j where the form of F is the same as that indicated in Eqn. 2. At our heroelastic design airspeed U F s > 0. We wish to choosi a new set of design parameters, p i , in such a way that we minimize F S while keeping F < 0. In addition we require that the LQR cost function J be a minimum with respect to a choice of control, u at Changes in the UDes• will, in parameters p i to reduce Fa so as to stabilize the system at U S general, change the values of F at the intermediate design airspeeds.

This will occur because of reshaping of the root locus curves for the closed-loop system.

The procedure to be used to optimize the actively controlled system - is a modification of the multi level, linear, decomposition procedure suggested by Sobieski and c^-workers. In the present case the system is decomposed into a structural subsystem and an active control subsystem.

Let us first discuss the active control subsystem.

At the active control level, the subsystem optimi-ation is the - state LQR problem, solution to the optimal steady + (S) • min J f (X* C*QCX u*Ru]dt u o with X- AX+Bu , (6) The solution is: u - GX - -R 1 B*PX , (7) where P is the solution to the steady-state matrix Riccati equation.

The change in J with respect to design parameters is achieved through differentiation of the necessary conditions satisfied at the minimum J, known generally as Kuhn-Tucker conditions. The results of interest are derivatives of the optimal feedback gain matrix with respect to the design parameters, (assuming B * B(p)) -RB* ap - (8) a p That is, the derivatives are constrained to describe how the optimal imal gains change with changes in parameters. Thus, the control - augmented ♦ ) derivatives are written as: state matrix ( denoted as A - aA+ aA + B aG (9) ap ap ap Eigenvalue sensitivity derivatives are found from

3A -1 aA+ ap - DIAC[E El ap where, A i - diagonal matrix of system eigenvalues; E - eigenvector matrix of A+.

These derivatives describe how the eigenvalues of the optimally con- trolled system change with changes in the design parameters. The real parts of the eigenvalues and their sensitivity derivatives, aQ aA api - Re ( api) ._are.then used in the system level optimization. At this point we can change parameters p i such tl^at we still have an optimally controlled system, that is, at a r.inimum control "cost".

For LMs example there is no structural cost, that is, no addi- tional weight is associated with changes in ae.

At the system level, the optimization problem can be written in linearized form as: m aF F s so - F + g min Ap i i-1 pi subject to m aF (13) (a) F - F + E _ 1 - Ap < 0 i api j Jo i- 1 (b) J - min J U (c) x - Ax + Bu where m - the number of active design parameters. Note that constraints (b) and (c), representing the subsystem optimization, are implicitly satisfied since optimal sensitivity derivatives of the eigenvalue real parts are used in construction of the Taylor series approximations to

the cost function F s &ad constraints. The derivatives in Eqns. 12 and 13a are given as: N poi 3ai 1 e ^P 2Fs - i- 1 f,14 ) pal 8p N E e i-1 To test this procedure a "simplex" algorithm adapted from linear optimization was used. The parameters p 1 - a and p2 - UDes were chosen as design parameters. An initial design with a - -0.4 and U 6.0 Des n was chosen. In this case the closed loop system was unstable at U - - 7.0. The objective was to stabilize the system atU - 8.0 in an s optimal manner. To do this, the functional associate d with 'U 7.0 was first reduced. Then the functional associated with U - 8.0 became Us and it was reduced.

Figure 4 shows the history of a and UDes versus number of design After seven iterations, cycles, together with the values of F s and F j .

no meaningful stability improvements were possible. Figure 5 shows the system root locus for the final design. The final design appears to be a compromise between flutter of the closed-loop system in two different modes, that is, a cusp in the flutter boundaries such as shown in Figure 3.

Figure 6 shows a plot of pole - zero "migrations" with airspeed for a fixed value a - -0.4. At this value of a there is close pole-zero proximity ( not exact cancellation) for an unstable mode. This may account for the fact that the control designed at this point does not pro •'uce superior stability boundaries. However, the process of reducing F a moves the system away from this divergence region.

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f -0.4 .8 -0.3 7 UAKWS INL PAGE IS OF. POOR QUALITY ae 6.015 -0.2 6 UDes

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-0.189 -0.1 5 U Des 0 0

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Number of Design Cycles

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Fs Fj - 0 0 - 4K -10 -10 - 2 4 6 8 10 12 Nus ier of Design Ly ► cles Figure 4 - Values of a and UDe versus design cycle iteration number. Alto shown Ire the values of the cumulative constraint functions F and F for each design cycle. Integer numbers refer

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to associatld values of Uj.

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Figure 6 - Plots of actively controlled system poles and zeroes as a function of airspeed for a 3--DOF model with a - -0.4. Real axis poles and zeroes are not shown.

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Figure 7 shows the pole-zero migration as a function of U for a that the configuration close to the final design. This figure shows problem of unstabilizability (uncontrollability of an unstable mode) is avoided by stabilization of the unstable mode through structural modifi- cation. There does appear to be some additional pole-zero separation, representing enhancement of controllability (as opposed to avoidance of incontrollability).

In addition to the above study, an investigation was made of the effects of changing the diagonal elements of the output weighting to be an identity matrix, Q ij . All previous studies had taken [Q ij ] matrix. Figure 8 shows stability boundaries versus a for U Des - 6.0 and airfoil pitch weighting of 0 - 100. With this weighting, the .

closed-loop divergence region for large negative vall!cs A a that exists when [Q ij ] is an identity matrix is eliminated. (Compare Figure 8 with Figure 3.) However, the flutter boundaries and the divergence boundary when a is near zero are unaffected by this change. The closed-loop divergence boundary appears to be associated with the con- trol deflection and, in fact, merges with the flutter boundary just i above it. Figure 9 shows stability boundaries for the control deflec- tion weighting, Q S - 100, with all other diagonal elements equal to unity. The divergence boundary for large negative a is again elim- e inated while the main flutter boundary is only slightly affected. The divergence and flutter boundaries fora near zero are shown to be more e heavily influenced by changes in QS.

Summary of Results and Future Work During the past six months a procedure has been developed to

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a- -0.5 0 CJaC Figure 7 - Plots of actively controlled system poles and zeroes as a function of airspeed for a 3-DOF model with a - -0.2. Real axis poles and zeroes are not shown.

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Qa = /00 S -o.s ° -04 -0. 1+ -0.3 -o

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ue Figure 8 - Open-loop and closed-loop stability boundaries with Q - 1, Qgg - 1, Q - 100, Q - 0, i s j for an actively cbntrolle g system ai j various values of a .

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- .-U .+ _V. 17 - v•-a - V.L -p.l -U.d Figure 9 - Open-loop and closed-loop stability boundaries with Q 1, Q = 100, Q = 0, 1 * j for an actively controlled sfstem at vVious values of a .

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redesign the structure and the control system to augment the aeroelastic stability of a idealized aeroelastic system. A number of problems remain to be resolved and some procedures need to be formalized. How- ever, it appears that a major step toward the integration of the structural/control optimization has been accomplished.

Future efforts will be directed towards exercising the method further on the 3-DOF model as well as the 4-DOF model with "fuselage" pitch freedom.

Work will also begin on more realistic models that incorporate multi-mode, laminated composite structures. A Master's degree student, Mr. V.J. Sallee will begin a 10 week residency at Langley Research i Center in mid - May 1985 to learn to operate the ISAC code so that these new thrusts can begin.

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Trip Report Second International Conference on Aeroelasticity and Structural Dynam- ics, Aachen, West Germany, 1 April through 3 April 1985.

A trip was made to this conference to present a paper entitled "Tailoring for Aeroelastic Stability and Lateral Control Enhancement".

The conference was well-attended by European specialists in this area of research and technology. In the field of aeroelastic tailoring the presentations were for the most part not state - of-the-art as we know it in the USA. Personal discussions with engineers from the German air- craft establishment lead one to believe that their capabilities and interest are much greater than conference presentations would indicate.

Some mention was made of the aeroservoelastic tailoring problem although ideas about how to approach the subject were not forthcoming. The Israeli aircraft engineers and researchers seem also to have a submerged interest in tailoring.

A visit to MBB, Munich was made on April 4 and 5. Mr. Otto Sensburg was the host for this visit. During this visit there was a keen interest expressed by the Germans in innovative technology. I was briefed on their design for a tailored composite vertical stabilizer and planned scale-model tests. They also plan active control testing. A lasting impression was that of a highly trained and qualified group of German engineers with many new ideas, not all of which they were willing to discuss.