Transonic Flutter Suppression Control Law Design, Analysis and Wind Tunnel Results

IFA 1999 TRANSONIC FLUTTER SUPPRESSION CONTROL LAW DESIGN, ANALYSIS AND WIND TUNNEL RESULTS Vivek Mukhopadhyay NASA Langley Research Center, Hampton, VA,USA Abstract The benchmark active controls technology and This paper describes flutter suppression control law design processes using classical and unified linear- wind tunnel test program at NASA Langley Research Center was started with the objective to investigate the quadratic Gaussian minimax techniques. A unified nonlinear, unsteady aerodynamics and active flutter general formulation for the linear quadratic Gaussian and suppression of wings in transonic flow. The paper will minimax methods based on the steady state differential game theory is presented. Lessons learned in evaluating present the flutter suppression control law design process, numerical nonlinear simulation and wind tunnel test and improving the singular value based multi-input multi- results for the NACA 0012 benchmark active control output system robustness are described. Design considerations for digital implementation are outlined.

wing model. The flutter suppression control law design processes using (1) classical, (2) linear quadratic Gaussian Numerical simulation of the control law performance, and (LQG), and (3) minimax techniques are described. A wind-tunnel test results for flutter suppression, are also presented.

unified general formulation and solution for the LQG and minimax approaches, based on the steady state differential game theory is presented. Design considerations for improving the control law robustness and digital implementation are outlined. It was shown that simple control laws when properly designed based on physical principles, can suppress flutter with limited control power even in the presence of transonic shocks and flow separation. In wind tunnel tests in air and heavy gas medium, the closed-loop flutter dynamic pressure was increased to the tunnel upper limit of 200 psf. The control law robustness and performance predictions were verified in highly nonlinear flow conditions, gain and phase perturbations, and spoiler deployment. A non-design plunge instability condition was also successfully suppressed.

Fig. 1 BACT model test setup in wind tunnel Introduction The benchmark active controls technology Wind Tunnel Model Description (BACT) and wind tunnel test program at NASA Langley A perspective view of the BACT model test set Research Center was started with the objective to up on the Pitch and Plunge Apparatus (PAPA) in the wind investigate the nonlinear, unsteady aerodynamics and tunnel is shown in Fig. 1. Fig. 2 shows the control surface active flutter suppression of wings in transonic flow.

and sensor locations. The rigid wing section has pitch and Under the initial wind tunnel test program, a NACA 0012 plunge degrees of freedom. The accelerometer sensors are airfoil rectangular wing, equipped with pressure located near the section leading edge (zle) and trailing transducers, active trailing edge control surface, and two edge (zte) at the section inboard. An identical pair of spoilers were constructed for active flutter suppression sensors is located at the section outboard as a spare. The tests. The model was mounted on a pitch and plunge partial span spoilers are located on the upper and lower apparatus in the NASA Transonic Dynamics Tunnel in surfaces, just ahead of the trailing edge control surface.

order to test flutter suppression control laws and measure Each of the control surfaces stretched over 30% of the unsteady pressure distributions in nonlinear flows with span and 25% of the chord. The bending and torsion oscillating shocks and boundary layer separation. It was frequencies of the PAPA mounted NACA 0012 wing necessary to develop a flutter suppression system that model were 3.3 Hz and 5.2 Hz respectively.

would be stable under these flow uncertainties.

diverges at the rate of 6 lbs/sec. The moment diverges at a rate of 1 lb/sec.

and Plunge Apparatus) omn toeo'rE _. _ mt OpenLoopLE _. 22Sp_ 0.0( 0.04 PAPA

o otlA !ll

D 0.02 Flow 0 OS 1 1.5 2 Control IS Surfaces _NACA 0012 ._ Accelerometers Fig. 2 NACA 0012 BACT wing on PAPA.

°o o's i ;'s 2 0 05 1 15 "nme, m¢ T'rne, iic Preliminary Analysis Fig. 3 Open loop transient responses in air at 225 psf.

The preliminary analysis, control surface sizing, and flutter suppression control law design were based on Freo_uency Responses the analytical state-space equations of motion of the The open loop frequency responses were studied BACT wing modelJ "4 These equations were developed using this 14 th order plant model, to select a possible analytically, using structural dynamic analysis and candidate for feedback signal in the flutter suppression unsteady doublet lattice aerodynamics with rational control law design. The Bode diagram of the trailing edge polynomial approximations s. These linear state space and leading edge accelerometers (zte) and (zle) and their equations consisted of 14 States (plunge, pitch, plunge difference (zte-zle)due to the trailing edge control rate, pitch rate, 3 aerodynamic states for plunge, 3 surface excitation (&e) in air at 225 psf at Math 0.5, are aerodynamic states for pitch, 2 trailing edge flap actuator shown in Fig. 4. The magnitude plots indicate states, 2 Dryden gust states), 2 inputs (actuator command predominant plunge response at 3.3 Hz excitation and gust input noise) and 7 outputs (zte and zle frequency. At 4.2 Hz excitation, the motion is a acceleration, flap command, flap deflection, rate, combination of pitch and plunge with pitch motion acceleration, and gust velocity). This 14 th order state leading the plunge. The (zte -zle) represents a signal space equation was used for classical control law design proportional to the pitch acceleration and can be and for performance simulation and verification purposes.

For the optimal control law design purposes and for integrated to provide a pitch-rate signal. Feedback of this presentation of the design data in a concise form, the 14 th signal with proper gain can provide maximum pitch damping at the flutter frequency.

order state-space equations were reduced to 4 'h state-space equations, using residualization and Schur's balanced reduction method 6"_. First, it was reduced to an 8th order 025 , , , , , system using residualization technique, in which only the static part of all modes above 15 Hz were retained. The .O.lS ...... i...... " ........ : ...... _...... : ......

resulting 8th order system was then balanced and the four states of the system with largest balanced singular values t o.I ...... : ............ _ ...... _...... : ......

o ---_ were retained. A sample of the 4 '_ order model design data 1 2 $ 4 $ 6 7 $ is presented in the Appendix.

2_ .....

Ope_ n-loop Responses I_ ...... : ...... _- _ ...... : ...... _ ...... : ......

The analytical open-loop flutter dynamic pressure in air was 128 pounds per square feet (psi) at a flutter frequency of 4.5 Hz. Fig. 3 shows the response of , , .......

the wing trailing edge and leading edge accelerometers 1 2 3 4 S $ 7 8 due to a 1 degree step input of the trailing edge control r-_mmy. _ surface in air at 225 psf dynamic pressure. The primary Fig. 4 The Bode diagrams of zte and zle and (zte -zle) plunge motion mixed with small pitch diverges rapidly.

due to &e excitation in air at 225 psf, Mach 0.5.

The unsteady lift forces oscillate about 8 lbs mean lift and Classical Control Law Design moves against higher aerodynamic loads. The Nyquist Based on this Bode plot, a classical flutter diagram of the difference between trailing edge and suppression scheme using pitch-rate proportional leading edge accelerometers (zte - zle) with lO/(s + 10) feedback from the zte and zle accelerometers was first lag filter and a gain KR = 500 due to the trailing edge devised by studying the Nyquist diagrams. The Nyquist control surface excitation (_Ste) in air at 200 psf, Much 0.5, diagram of the difference between trailing edge and is shown in Fig. 5(b). The unit circle is also shown.

leading edge accelerometers (zte -zle) due to the trailing Because the Nyquist contour encircled the -1 point, the edge control surface excitation (_Ste) in air at 200 psf, is unit feedback closed loop system would be stable. As shown in Fig. 5(a). The arrow indicates increasing desired, the phase margin at the plant input above the frequency of excitation from 2 Hz to 6 Hz, with each * flutter frequency was about 60 degrees, but the phase representing frequency increment of I radian/second.

margin below the flutter frequency was only 20 degrees.

Since the open-loop plant had a pair of complex unstable Preliminary analysis indicated that this basic simple control law 1 poles, and the Nyquist contour did not encircle the -1 point, the unit feedback closed-loop system would be unstable. However, if the (zte - zle) signal was integrated &e=500 10 (zte-zle) s+10 to provide a 90 degree phase lag and then used for feedback with sufficient gain, the Nyquist contour would rotate 90 degrees clockwise and then expand to encircle can suppress the flutter instability in the dynamic pressure the -1 point to achieve stability. A washout filter of type range from 0 to over 225 psf, both in air and in heavy gas s/(s + a) was also required, to remove any static bias that medium. However, the closed loop transient responses and stability margins required substantial improvement.

would otherwise be amplified by the integration. The series connection of integrator 1/s and washout filter was zte, g equivalent to a first order lag filter a/(s + a), where s is the Laplace operator.

dte, deg plant r_ _) _ _00 air 200 psf zte, g zle, g air 200 psf dte, deg plant _+ 15[_ imaginar_art, (zte-zle) 24.5 / deg/deg zle, g 24.5 part,g/deg 23.

tt._25.5 _'--'_ .0 .C_ imaginary I rad/s_ I ('1 I I I real part, - "5 -.02 g/deg real part, o, o' '1to o, -15 _ 5 deg/deg racl/s _ 26.5 Fig. 5(b) Nyquist diagram of (zte -zle) with lO/(s+lO) . D lag filter and a gain KR = 500, due to &e excitation in air at 200 psf, Much 0.5.

Fig. 5(a) Nyquist diagram of (zte-zle) due to &e excitation in air at 200 psf, Mach 0.5 Root Locus Analysis of the root locus with pitch Gain Selection acceleration (zte - zle) feedback through a lO/(s + 10) lag Two types of lag filters, namely 5/(s + 5) and filter with increasing gain KR = O, 500, .... 2500 is shown lO/(s + 10) were examined. The latter was selected to in Fig. 6(a). The stabilization was achieved by increasing achieve a higher phase margin at the plant input above the the pitch model damping and lowering the plunge mode flutter frequency. Higher phase margin was desirable for frequency. An additional feedback of the pitch rate two reasons 7. First, the 25 Hz antialiasing filter and the proportional signal through a 5/(s + 5) lag filter with KR 1/200 seconds computational delay contribute about 20 = 500 and increasing gain KP =0,500, .... 2500 was used degrees of phase lag at the flutter frequency. Secondly, to increase the damping and frequency separation further, with increasing dynamic pressure, the actuators may have as indicated by the root-locus diagram shown in Fig. 6(b).

additional unknown phase lag, as the control surface This design strategy was equivalent to pitch-angle and equivalent to _+20 degrees phase and +3 dB gain margins at the plant inputs. The singular value 1/_[K(I+OK) "m] is pitch-rat_ proportional feedback that increased the pitch mode frequency and plunge mode damping. close to 0.005 g/degrees near 2 Hz. This means that the plant has very little tolerance to an additive perturbation A imaginary part. imaginary part.

to the plant. The complex determinant locus of (I+KG) rad/sec -30 and its distance from the origin is a measure of its closeness to singularity.

L25 .G(l+zz)-> -_bm)G-> -20 -20 -25 -15 -15

KR=_ i :iiiiiii

lo 10 _,_.1_, KR= 0.500, I(XX) Ot 1500.2000,2500 0 5 10 0 5 10 Fnq. HZ r-r.q, _ 5 5 D_(t+KG) LAW2 225 I_¢ | ;- _ I I I I -20 -15 -10 -5 5 10 -20 -15 -10 -5 5 f ......... :: .........

real part. mQ/sec real part, r4d/se¢ :i_ - . , ": ......

Fig. 6(a) Root-locus with (zte -z/e) feedback through a _ o.,[ ......... i.........

IO/(s + 10) lag filter, with increasing gain KR (at left).

6(b) Root-locus with additional pitch-rate feedback "._ . s through a 5/(s + 5) lag filter with KR = 500 and increasing gain KP. The arrows indicate increasing gain.

Fig. 7 Singular value plots for analysis of multivariable stability margins to a perturbation A, at the plant input or Pilqh and Pitch-Rate Feedback Control Law output, with classical control law 2, at 225 psf, in air.

From the root-locus study, the feedback gains were selected as KR = 500 and KP = I. Thus, the second O" Minimium singular valuQ order state space equations of the initial pitch and pitch- rate feedback 99ntrol law 2 is given by

{::};[7

-5]lx_ j o Jl_J 0.8 .;t,oo ,1{::}.

0.6 1.0_ The control law inputs are zte and zle in g unit and the output 8re is in degrees. The high feedback gain was required because the maximum (zte - tie) signal was o, I..........

0,2 rgin _+0 only of the order 0.1 g/deg. The response exhibited 2% 0.0t ........ _f0: ........ I settling time in 1.5 seconds. However, the high gain -10 -8 -6 ,-4 -2 0 2 4 6 8 10 Gain margin,dB resulted in a severe sensitivity with respect to plant Fig. 8 Universal gain and phase margin analysis diagram.

perturbation and individual sensor uncertainty, as indicated by the corresponding singular value plots in Fig.

Final Pitch and Pitch-Rate Feedback Control Law 7. Here G, K and A denote plant, controller and This lack of robustness associated with this pitch uncertainty block, respectively _'9. This figure indicates and pitch rate feedback control law was alleviated by that the minimum singular value __(I+KG) is 0.3 at plant choosing a feedback of a proper linear combination of the input and o(I+GK) is only 0.01 at plant output. This two sensors with lower gains for KR, instead of using the means that at 225 psf dynamic pressure, the closed-loop difference (zte -zle). The linear combination of zt_ and system has very little robustness to multiplicative zle, which is equivalent to feeding back both pitch perturbation s'9 at the plant output.

acceration (zte - zle) and plunge acceleration (zte + zIe) in the ratio 0.7(zte-zle) + 0.3(zte + zle), appeared to These singular value __ plots can be related to provide a superior control law. The final classical multivariable gain and phase margins using the universal feedback control law, using this combination that is gain and phase margin diagram s shown in Fig. 8. For equivalent to (zte - 0.4 zle) feedback, along with reduced example, minimum singular value __(I+KG) of 0.3 is gains of KR = 50 and KP = 1, was analyzed and solution only requires an eigen-solver. The corresponding implemented. The basic control is shown here in state- Matlab script is presented in the Appendix.

space form and is denoted by classical control law 3.

_.5 c!°_ k"e a" t'Y,3 15 cao_. t_ .t_3

[;o5°]{X}x2 • [,Oo

o Jle,,J

1 xl 0 0.5 1 1.5 2 0 0.5 1 1.5 Response and Robustness Analysis The closed-loop transient responses due to 1 degree step deflection of _te, in air at 225 psf, at Mach 0.5, is shown in Fig. 9. The trailing edge control surface

t t '' 5 I .0.6t_ii ,, v , "

0 -08 L_' shows only 0.25 degrees overshoot with a maximum rate 0 0.5 1 1.5 2 0 0.5 I 1.5 2 of 12 degrees/sec. The lift and moment forces indicate Tune. see Trine, s_c about 20% load alleviation compared to the open-loop Fig. 9 Closed-loop responses: control surface deflection initial transient values shown in Fig. 3. Figure 10 shows and rate, lift and pitching moment, due to step input &e the singular-value plots for analyzing the system stability with control law 3, at 225 psf, in air, at Mach 0.5 (open margins 8'9 with law 3 at 225 psf dynamic pressure. Here loop qnu--r =128 psf).

G, K and A denote plant, controller and uncertainty block -6_)-Q-> -,t_l*A)-> transfer function, respectively. This figure indicates that the minimum singular value __I+KG) is increased to 0.8 at plant input and at plant output _(I+GK) is increased to 0.3 from the corresponding values with law 2 presented in

Fig. 7. The minimum singular value 6__(I+KG) of 0.8 is i!'

equivalent to +45 degrees phase and -5 dB to 12 dB gain 5 I0 FM¢I, i_ margins at the plant inputs. These gain and phase margins ..{G+_)-> Det(I+KG) LA_J 225 !_1 are determined from Fig. 8 as previously described. The 3 minimum singular value I/_o[K(I+GK) 1] is also increased

o,[

0.15 from 0.005 g/degree to 0.04 g/degree near flutter frequency, thus increasing the plant's tolerance to additive plant perturbation. The complex determinant loci of 0.05 _°" I (I+KG) ideally should be outside the unit circle to achieve +6dB gain margins and _+60 degrees phase margins. The computational delay and antialiasing filters added 20 Fig. 10 Singular value plots for analysis of multivariable degrees phase lag. Hence, the system nearly attained these stability margins to perturbation A, with classical control margins. The singular value plots indicate that the system law 3, at 225 psf, in air.

is stable with adequate singular value based multivariable stability margins even at this high design dynamic Unified Minimax Formula_ola pressure of 225 psf. This pressure is 97 psf above the Consider the state space Eqs.(1-3) representing open-loop flutter dynamic pressure qflutter 128 psf, the nth order plant, control input u(t), disturbance w(t), representing a 75% increase.

design output Yd and sensor output Ys, where all necessary rank, controllability and observability conditions are Unified Optimal Design assumed to be satisfied.

Flutter suppression control law design using an unified (1) linear quadratic Gaussian (LQG) and (2) Plant state-space equations Minimax methodl0,11 is presented next. The Minimax dx(t)/dt = F x(t) + G u(t) + Gw w(t) approach is analogous to the time domain H-infinity and x(0) -- xo (l) designl2 and is based on the steady state differential game Design output formulation. The unified formulation of these optimal design techniques provide a basic understanding of the ydO = Ha x(O + Edu u(O (2) relation between them. The derivation from basic Sensor output principles using variational principles are provided. The ys(t) = H s x(t) + Esw w(t) (3) State-Feedback Minimax Re malator Problem The Minimax problem is to determine the plant input u(t) which would minimize the quadratic [ dt J performance index J , and find the worst plant disturbance w(t) and initial condition x o, which would withx(O) = xo and Moo)= 0 (12) maximize J defined in Eq. (4), State-Feedback Regulatql" Substituting 2(t) = S(t)x(t), in Eqs.(10-12), leads (4) J = ½_[(x_Gx + x_Q=u + u_Gu)at to Eqs.(13-15). The general Riccati Equation (15) is then solved for the unknown n by n matrix S.

subject to the constraint Eq.(1) with xoTxo = 1 and specified W defined by, u(O = "Qu-I(GTs + Qxu T) x(t) (13) I _"_ W T w = rt ( R.w)dt (5) w(t) = ? '-2 Rw-lGwTS x(t) (14) JO Usually, the constant weighting matrices Qx, Qu are dS/dt + SF + FTs + Qx-(SG + Qxu)Qu-l(sG + Qxu) T unity, and Qxu = [0] in a H-infinity exposition. These are + s(¢ -20wRw-lCwT)S = o (15) included herein to derive a unified general time-domain formulation. The cross weighting matrix Qxu originates if The positive definite symmetric solution for S is obtained one uses Yd from Eq. (2) in the performance index J to from the (2n x n) eigenvectors of the n stable eigenvalues replace x. Then, Qx = HdrQydHa, Qxu = HdrQydE&, and of the Hamiltonian matrix inside the square bracket [ ] in Eq.(12). For the steady state problem (i.e. dS/dt = 0 ), Qu is replaced by [ Qu + Ea.rQyaEa. 1. The significance only the steady part of the Riccati Equation (15) is solved of the cross weighting matrix Qxu and how it can be in order to obtain the symmetric positive-definite matrix selected for pole-placement of the state regulator will be S. If the eigenvectors are partitioned into two n x n shown later in the state-feedback regulator subsection.

matrices X and A which represent the stable subspace eigenvectors of x and :t, then S = X_-1_A. The constant The minimax solution is given by the stationary optimal feedback gains, Co and Cw, and the closed-loop condition of the augmented performance index J system matrix are given by, j _- __[._z " (xrQxx + 2xrQ_u + u r Q_u - yZwrp_w)dt Co = "Qu-I(GTs + Qxu T) (16) +/],rf:(Fx+Gu+Gww--_)dt+z2W (6) where, y is a scalar parameter. Using the calculus of Cw = 7 '-2 Rw-IGwTS (17) variation with respect to x(O, u(t), w(t) and the vector Lagrange multiplier 2(t), the conditions for OJ=O are dx/dt = IF + GwCw + GCo]x. (18) given by Eq.(1) and Eqs. (7) to (9).

Using Eqs.(1),(15) and (18), it can be shown I1 that d_t/dt = - Qx x - FT :c - QxuU (7) optimal J and W defined in Eqs.(4) and (5) are given by, Quu = - GT,_ - QxuTx f8) J = 0.5 Trace [S] (19) 7/2Rw TM = GwT_. (9) W = 0.5 Trace [CwTRwCwX].

(2O) Solving for u(t) and w(t) from Eqs. (8-9) and substituting where X is the solution of the Lyapunov Equation (21) them in Eqs. (1) and (7), the necessary stationary conditions for J are obtained as, [F+GwCw + GCo] X + X[F + GwCw + GCo] T + x(O)x(O) r = [Ol (21) u =--Qu-I(GT_ + QxuTx) (10) The worst x(O) that maximizes J is given by the w = y-2 Rw-IGwT_, (1 I) eigenvector of the maximum eigenvalue of S. The standard linear quadratic regulator (LQR) solution is Gw Hd T obtained when y= oo, (i.e. Cw = 0). As )2 is decreased, Qx GwRwGw T the worst response due to the disturbance w(t), measured Qu Rv by the maximum singular value of [xrQx 1/2 urO.ul/2], is S P reduced. The minimum value of )2 for which a stable Qxu Rwv solution of Eq. (15) exists provides the minimax state- (tf- t) (t - tO) feedback regulator that minimizes the maximum singular value of [xTQx 1/2 uTQul/'21.

Table 1. Duality relations between linear quadratic state- regulator and state-estimator equations.

The State-Estimator Eq.uation The derivation of coupled state-estimator The Controller Equation equations using linear quadratic minimax approach is still Substituting Eqs. (13), (14), (22) and (23) in Eq.

a subject of research. Here the equivalent state-space (26), the state-estimation feedback controller equations solutions 12 of the H-infinity problem are presented. The state estimator gain BoDo is obtained by finding the dz/dt = [F + GwKw + GCo + DoBoHsl z -DoBoYs (27) symmetric positive definite solution for P from the state estimator Riccati Eq. (24) which is dual to the state u = Coz (28) regulator Riccati Eq. (15).

are obtained. The standard LQG solution is obtained B o = -(PHsT+ Rwv)Rv -1 (22) when y = oo. The minimum value of ?2 for which a stable solution exists for P in Eqs. (22-24), provides the Do = (1- y--2ps)-I, p(PS) < 72 (23) minimax control law that minimizes the maximum singular value of the matrix [ysTQys 1/2 uTQul/21, for the closed loop system.

dP/dt = PF T + FP + GwRwGw T - (PHs T + Rwv) Rv -1 (PHs T + Rwv)T + p()r-2 Qx)P (24) Unified Design Procedure In this design, the output Yd was chosen to be the where Rv = EswRwEsw T and must be positive definite and linear combination of the trailing edge and leading edge Rwv = GwRwEsw T. In Eq. (23) the spectrum pIPS1 must accelerometer output (zte-0.4 zle), same as that used in the final classical design. One advantage of this choice me less than _ for Do to exist. The positive definite was that the plant had no transmission zeros in the open symmetric steady-state solution for P in Eq. (24) is right half plane. Usually in a frequency domain H-infinity obtained from the (2n x n) eigenvectors of the n stable design, the plant equations are augmented with weighting eigenvalues of the estimator Hamiltonian matrix, transfer functions. In this time domain formulation, the weights were chosen as constants. These weighting F -z r , ..r--l,, + -2Hr_-i H.a constants are chosen as inverse of the desired magnitude L-GwR_G2-R_R_-'t_ -(F- P_R_-'H2) J of the weighted quantities. The initial controller was designed with a large value of 72, using the plant Eqs. (1- (25) 3), at 225 psf, in air, at Mach 0.5, assuming Gw = G. The which is dual to the state regulator Hamiltonian matrix inside [ ] in Eq.(12). If the eigenvectors are partitioned block diagram for this unified design procedure is shown in Figure 11. The detailed design steps are described next.

into two (n x n) matrices X and A, then P = X-I_A. The state estimate vector z is given by the Eq. (26).

State-feedback Regulator Design Initially, the maximum output of the dz/dt = F z + Gww +Gu + DoBo(H2z -Y2) (26) accelerometer sensors were of the order 0. lg, (see figure 3), and control surface maximum root-mean-square The complete duality relations between the state regulator deflection was desired to be of the order I degrees. Thus, problem and the state estimator problem are presented in the initial values of the weighting matrices were chosen as Table 1.

follows: Qx = [ HsrQysHsl, Qys = [1001, and Qu = [11.

It was interesting to note that, instead of setting the cross state regulator state estimator

F Fr weighting matrix Qxu = [0] as usual practice, the cross

weighting matrix Qxu can be selected to place all state G Hs T regulator poles beyond a certain distance a to the left of lower values of 72 up to ?2> p(PS), below which the imaginary axis. This selection is accomplished by using disturbance authority exceeded the control authority. The 4th-order optimal control law was designed with 72=50 in Qxu = - aGu(GuTGu)-lQu (29) order to obtain a low bandwidth controller. Fig. 12 shows the key singular value plots for analysis of multivariable so that in Eq. (12), the eigenvalues of the diagonal matrix stability margins to multiplicative and additive block F are off-set by perturbation A at the plant input and output, with this minimax optimal control law, denoted as control l_w 4.

GQu.l Qxu T=_ GI. (30) The minimum singular value _O(I+KG) is increased to 0.9 at plant input and at plant output o'(I+GK) is increased to The control-weighting matrix Qu was 0.5 from the corresponding values of 0.8 and 0.3 for control law 3, shown in Fig. I0.

subsequently reduced to 0.01 after a few design cycles to improve the regulator performance. This process of 41-_)G-> reducing Qu is equivalent to the state estimator loop- -G(I*6)-:.

S ; i i transfer recovery technique at the plant output.

,l ..... ......... J

and a large value of 1'2 _ I Design state regulator o 5 lO 0 5 10 Fr_, Hz - I Check stabilityand -IG*6t-> Det(k-KG) LAW4 225 pit _ ,J perf°Tnce J [_ ..... ?,! ..........

I No Design state estimator?J No _ Does the solution exist j.tt. ..........

°O 5 lO I TM Freq. Hz ReeJ d_( I I Analyse stability,performance Fig. 12 Singular value plots for analysis of multivariable I with4th order plant I stability margins with minimax optimal control law 4, at 225 psf, in air.

Clcwd Loop, Law 2 225 pl¢ 0,00 0.04 p, with full order plant o 0.01 ! ii 002 I

l

j _. r,_ ^ I Nonlinear simulation I .o.o,_i/ V "'-J Fig.ll. Unified minimax control law design and -0021 'V , , 0 05 1 IS 2 • 0 0`5 1 1 `5 2 evaluation procedure block diagram.

11me. lec T'ene, W¢ Clomld Loop. LAW 3 + FIL 225 psi Closed Loop. LAW 4 ÷ FBL 225 i_f State-estimator Design 0._ The state estimator was designed as a dual to the state-regnlator with Rw = 1, each diagonal elements of

t

Rv = 0.01, and Rwv = [0]. After a few design cycles the performance of the combined full-order controller was "0 01_)' 0'5 t 1.5 f • 0 0.5 1 1.5 2 examined, and then Rw was increased to 36. Since we l"=r,e, =,: Time, _4¢ also choose Gw = G, this was equivalent to asymptotic Fig. 13 Open-loop and closed-loop responses due to step state-regulator loop-transfer recovery at the plant input.

input 8te, with classical control laws 2, and 3, and minimax optimal control law 4, at 225 psf, in air.

4th Order Optimal Control law Subsequent solutions to the state-regulator and Fig.13 shows the open-loop and closed-loop state-estimator were obtained with the same choice of responses due to unit step input 8te of trailing edge weighting matrices but for decreasing value of ?2, for control surface, at 225 psf, in air, at Mach 0.5, with initial which positive definite solutions for S and P could be control law 2, classical control law 3, and minimax obtained. Note that feasible solutions can be obtained for optimal control law 4. The transient responses indicate that the classical control law 3 provided better damping laws are comparable in performance, with control law 3 with lower control surface activity although the exhibiting higher stability margins.

minimax control law 4 provided better robustness properties. This is the traditional trade-off between performance and robustness. The classical control law 3 was implemented and tested in wind tunnel. These test results along with those using two optimal control laws designed by Waszak t3 are presented next.

Flutter Suppression Test Results The performance and robustness of the final design was tested using the original full plant state-space equations and filters required for digital implementation.

The 25 Hz antialiasing filters 157/(s+157) were added to Fig.14 Numerical simulation block diagram of the control the plant output. The washout filter 5s/(s+5)and system implementation using the final classical control computational delay were added to the controller output law 3 for flutter suppression.

equations. The 1/200 second computational delay was modeled by a (400-s)/(400+s) filter. Before the wind- Closed Loop Tut Points tunnel test entry, the digital implementation was also 22O numerically simulated. The numerical simulation block diagram of the control system using the final classical control law 3 is shown in Figure 14. This nonlinear el simulation also included the effects of a dead-band qO _4 0 • t80 present in the electro-hydraulic actuator. Application of -I the upper and lower spoiler for transonic flutter a.

suppression with the same digital control law was also o investigated using this simulation. E m w t- • O: e,t The active flutter suppression control-law using classical • TE Control design was successfully tested in air and in heavy gas

1 I

Flutter boundary medium at transonic speeds up to Mach 0.95. The tests in air indicated an increase in the flutter instability boundary 0.5 0.6 0.7 1.0 0.8 0.9 from the open-loop dynamic pressure of 158 psf (Mach Mach Number 0.38) to the tunnel limit of 200 psf. A summary of flutter Fig. 15 Open-loop flutter boundary and closed-loop flutter suppression test results in heavy gas is shown in Fig. 15.

suppression results from wind-tunnel tests in heavy gas.

The solid line indicates the experimental flutter boundary, with the transonic dip at Mach 0.8. The tests at Mach 0.8 0.04 loop unstable 1-1 Open loop open indicated an increase in the flutter stability boundary from [] TE cont! ol the open-loop dynamic pressure of 142 psf to the tunnel U) US contl o - 0.03 upper limit of 200 psf. A non-design plunge instability US+LS / tO condition was also successfully suppressed. Classical tO <[ control law 3 exhibited superior performance and was demonstrated to be stable with gain variation from 0.25 to 0.02 7, and phase variation from -90 to 60 degrees. A non- .Q _c design plunge instability condition was also successfully ILl suppressed. Comparison of open-loop and closed-loop _- 0.01 n loop _ al le root mean square (RMS) responses of trailing edge /I accelerometer and control surfaces using the present /1 classical control law 3 and two other control laws /1 0.00 M= 0.63 M= 0.71 M= 0.77 M= 0.83 designed by Waszak 13 are shown in Figs. 16 and 17, 125psf 150psf 178psf 195psf respectively. These two control laws used upper and lower spoilers as control surface, for flutter suppression.

Fig. 16 Open-loop and closed-loop RMS responses with Fig. 16 indicates that when the system is open loop stable, classical control law 3 and two control laws employing closing the loop actually reduces the response by 30%.

spoilers from wind-tunnel tests in heavy gas medium.

Fig. 17 indicates that that the classical control law 3 generally requires less control activity. All three control Conclusions 4Durham, M.H.; Keller, D.F.; Bennett, R. M.; and Simple classical control laws, when properly Wieseman, C. D.: A Status Report on a Model for designed based on physical principle, can successfully Benchmark Active Control Testing. AIAA Paper No. 91- suppress transonic flutter and provide significant stability 1011, 1991.

robustness in presence of shock and flow separation.

5Hoadley, S. T.; and Adams, W.M. Jr.: ISAC - Interaction Comparable robust optimal control laws can also be Structures, Aerodynamics, and Controls. Version 5.1 designed using a new generalized unified minimax Sample Cases with Guides for UNIX Operating Systems, formulation. Verification and improvement of the NASA TM-100667, January, 1992.

multivariable system stability robustness to unstructured 6Mukhopadhyay, V.: Flutter Suppression Control Law perturbations at the plant, input and output were important Design and Testing for the Active Flexible Wing, Journal steps in such a design process. Wind-tunnel tests in air of Aircraft (Special Adaptive Flexible Wing Issue) Vol.

and heavy gas indicated an increase in the transonic 32, No. 1, Jan-Feb. 1995, pp. 45-51.

flutter dynamic pressure to the tunnel limit upper limit of 7Perry, B. III; Cole, S. R.; and Miller, G. D.: A Summary 200 psf. The control law robustness and performance of an Active Flexible Wing Program," Journal of Aircraft predictions were verified in highly nonlinear flow , Vol. 32, No. (1995), pp.10-15.

conditions, gain and phase perturbations, and spoiler SMukhopadhyay, V.: Stability Robustness Improvement deployment.

Using Constrained Optimization Techniques. Journal of 0.58 Guidance, Control, and Dynamics, Vol. 10, No. 2, 0.48 March-April 1987, pp. 172-177.

m u m m 0.4 9Pototzky, A.S.; Wieseman, C.D.; Hoadley, S. T.; and [] Open Ioo _ o_ )p I n _ b o_ [] TE contrt Mukhopadhyay, V.: On-line Performance Evaluation of o _ US contn \ Multiloop Digital Control Systems, Journal of Guidance, "a _ US+L.S \ _0.3 Control, and Dynamics, Vol. 15, (1992), pp. 878-884.

\,- = Ol i Ioo \/ ]°Bryson, A. E. Jr.; and Carrier, A.: A comparison of

°

_e \ Control Synthesis Using Differential Games (H-Infinity) "6 0.2 _ \ and LQR. AIAA Paper 89-3598 CP, August 1989.

_]Ghaoui, E., L.; Carrier, A.; and Bryson, A. E. Jr.: Linear Quadratic Minimax Controllers. Journal of Guidance, "_ \/, o \ Control, and Dynamics, Vol. 15, No. 4, July-August o \ 1992, pp.953-951.

0.0

_2Doyle, J. C.; Glover K.; Khargonekar, P. P.; and Frances, B. A.: State-Space Solutions to Standard H2 and M= 0.63 M= 0.71 M= O.T/ M= 0.83 H-Infinity Control Problems. IEEE Transactions on 125psf 160psf 178psf 195psf Automatic Control, Vol. 34, No. 8, August 1989, pp.831- 847.

Fig. 17 RMS responses of the control surfaces with the _3Waszak, M. R.: Robust Multivariable Flutter classical control law 3 and two control laws from wind- Suppression for the Benchmark Active Control tunnel tests in heavy gas medium.

Technology (BACT) Wind-Tunnel Model, Proc. llth References Symposium on Structural Dynamics and Control, VPISU, _Bennett, R. M.; Eckstrom, C.V.; Rivera, J. A. Jr., Blacksburg,VA,1997.

Dansberry B. E.; Farmer M. G.; and Durham, M.H.: The Benchmark Aeroelastic Models Program - Description Appendix and Highlights of Initial Results. NASA TM 104180, Reduced 4 th order state space equations in air at 225 psf.

1991.

and Matlab script for unified minimax formulation and 2Rivera, J. A. Jr.; Dansberry B. E.; Farmer M. G.; solution Eckstrom, C.V.; Seidel, D. A.; and Bennett, R. M.: Experimental Flutter Boundaries with Unsteady Pressure F = [ -1.6073 21.0010 O. O.

Distribution for the NACA 0012 Benchmark Model.

-21.0010 -1.6073 O. O.

AIAA Paper No. 91-1010, 1991 (NASA TM-104072, O. O. 0.7515 25.1670 1991) O. O. -25.1670 0.7515] 3Rivera, J. A. Jr.; Dansberry B. E.; Bennett, R. M.; Durham, M.H. and Silva, W. A.: NACA 0012 Benchmark [G Gw] = [ -3.8259 0.0597 Model Experimental Flutter Results with Unsteady 12.7130 0.2720 Pressure Distribution. AIAA Paper No. 92-2396, April -2.2202 -0.1107 1992.

4.1351 -0.1745 ] H d = [-0.0517 -0.0132 -0.0668 0.0063 error('Can"t order eigenvalues'), end -0.0542 -0.0090 -0.0482 -0.0016 % select vectors with negative eigenvalues 0. 0. 0. 0.

chi = v(1 :n,index(1 :n)); 0. 0. 0. 0.

lambda = v((n+ 1):(2*n),index(1 :n)); 0. 0. 0. 0.

S = real(lambda/chi); 10.3780 0.6369 7.9924 0.0671 70 ...................................................

0.3897 -0.9373 -2.7625 0.9909 ] % Positive feedback gain ku ku=-Q22\(g'*S+Q12'); [E,. Ed.] = [0.0440 0.0002 kw=[rw\gw']*S/mu; 0.0421 0.0004 1.0000 0.

% closed loop plant f = fcl 50.0000 0.

fcl=f+g*ku+gw*kw; O. 0.0968 X=lyap(fcl,eye(n)) % assume xo*xo'=I 0.5758 -0.0004 Xg=lyap(fcl,gw*gw'*36) 0.0358 -0.0003 ] W=0.5*trace(kw'*rw*kw*Xg) Jo=0.5*trace(S) % Matlab script for unified formulation and solution Uu=trace(ku'*ku*Xg) %xdot=Fx+ Gww +Gu Ycov=Hd*Xg*Xg'*Hd ° 17_ ...................................................

% yd = Hd x + Edw w + Edu u Design output % H-infESTIMATOR DESIGN % ys= H x +Esww+Esuu Sensor output % define Hamiltonian Jam % 7 design output [zte zle dte ddte gust lift moment] ...............................................

Ru = [gw*rw*gw'+nu*g*g']; Q11 = h'*qh*h+hd'*qhd*hd; J = [ (f-[Rwv/Rv]*h)' [-h'/Rv]*h+[hd'/qhd]*hd/mu Q22 = [q2]+[Edu'*qhd*Edu]; -Ru-[Rwv/Rv]*Rwv' -(f-[Rwv/Rv]*h) ]; % pole placement using cross weight Q 12 [v,d]=eig(J); O_ ..................................................

alpha = 3.0; Q12=-alpha*g*((g'*g)\Q22)+[hd'*qhd*Edu]; % sort eigenvector of stable eigenvalues WW = [Qll Q12 ; Q12' Q22 ]; d = diag(d); [d,index] = sort(real(d)); % Generalized LQR Weights % sort on real part of eigenvalues Ru = [gw*rw*gw'+nu*g*g']; if (-((d(n)<0) & (d(n+l)>0))) Rv = [rv]+[Esw*rw*Esw']; error('Can"t order eigenvalues'), end ..................................................

Rwv = [gw*rw*Esw']; % generalized EST Weights % select vectors with negative eigenvalues VW=[Ru Rwv; Rwv' Rv]; chi = v(1 :n,index(1 :n)); % Generalized DESIGN with Edw--0 lambda = v((n+l):(2*n),index(l:n)); [af, bf, cf, df]=lqg(f,g,h,Edu,WW,VW); P = real(lambda/chi); I_ .............................................

ky = -(P*h'+Rwv)/Rv; % STATE REGULATOR % check positive definiteness of kd % Check if Q11 is positive semi-definite and symmetric % Is this matrix (I - P*S/mu) nonsingular ?

if any(eig(Q11) < -10*eps) I (norm(Ql I'-Q11,1)/norm mumin=max(abs(eig(P*S))) (Qll,1) > eps) if (mu > mumin), error('Ql 1 must be symmetric positive semi-def), end kd = inv(eye(n)-(P*S)/mu);, else % Check if Q22 is positive definite and symmetric error('spectrum(P*S) > mu, increase mu'), end I_ ................................................

if any(eig(Q22) <= 0) I (norm(Q22'-Q22,1)/norm(Q22,1) > eps) eve=eig(f+kd*ky*h); % controller structure error('Q22 must be sym. positive definite'), end ...................................................

Ao=f+g*ku+gw*kw+kd*ky*h; % Construct Hamiltonian evc=eig(Ao); Hm=[(f-g*[Q22\Q12']) -g*[Q22\g']+gw*[rw\gw']/mu % H-infinity controller -Q I 1-Q 12* [Q22\Q 12'] -(f-g* [Q22\Q 12'] )']; % Kop = [At -kd*ky -ku zeros(nc,ns)]; [v,d]=eig(Hm); [fc, gc ,he, ec] -- feedback(f, g, h, e, At, -kd*ky ,-ku, O_ ...................................................

zeros(nc, ns)); 17_ ...........................................

% Sort eigen vector of neg eigenvalues d = diag(d); [d,index] = sort(real(d)); if (-((d(n)<0) & (d(n+l)>0)))