Modal control theory and application to aircraft lateral handling qualities design

section. It will illustrate that certain procedural idiosyncrasies, as simple as selecting the sequence of synthesizing eigenvectors, may induce problems which could be resolved by inspection for low order systems, but do require an algorithmic process for more complex systems.

Suppose the plant dynamics are given by

i o 6" 1 0

~*r V l_ _

U1 • = X + 1 5 6 0 1 "2 X -1 -1 0 _0 0_ _ 3_ -*3_ with open-loop eigenvalues at 1, 2, and 3. Suppose further that design specifi- cations are X^ = -1, X2 = -2, X3 = -1, and an eigenvector structure 1 -1 -1 1 1 V = 0 with the w elements computed to meet the desired pole specifications.

To complete the design, equation (8) can be used immediately (where F = 0, S = 0, and G = [-1 -l] ) to yield v l = -1 v = -1 V It is immediately obvious that the resulting modal matrix V will be singular, since V2 and V3 are identical. For a third-order problem such as this, equation (8) may be examined directly for each eigenvalue, and a sound decision made as to which specifications should be relaxed and in what manner to give a system performance very close to that desired. Of course, for sub- stantially higher order systems or for an automated on-line controller, a pre- cisely defined algorithm is required. Such an algorithm is now presented, and this problem is reexamined as an illustration of its utility.

SPECTRAL SYNTHESIS ALGORITHM As the discussions in the previous sections revealed, the central question in the new formulation of the pole-placement problem is the guaranteed genera- tion of the nonsingular matrix V satisfying equation (8) . An examination of the eigenvector constraints shows that there is an m-dimensional subspace asso- ciated with each eigenvalue: Thus, the problem reduces to selecting a nonsingu- lar set of n eigenvectors with each vector included from a subspace associated with a particular eigenvalue. However, this method is an inefficient way to synthesize since the designer loses direct control of the selection of arbitrary elements of the modal matrix (z-vectors) . Also, any algorithm would become computationally intractable since it would involve pairing n vectors from a set of n . m vectors until a nonsingular set resulted.

Alternatively, if the z-vectors are allowed to be chosen arbitrarily, then it is important to keep track of the linear independence of the eigenvectors as they are sequentially generated. Unfortunately, this procedure is poten- tially susceptible to the generation of a singular set of eigenvectors since it is very likely that no closed-loop matrix with the precise set of eigenvalues and eigenvectors as selected would exist for any choice of control law. This is exactly what occurred in example 2 of the section entitled "Numerical Examples." However, is it possible to detect the occurrence of such a situa- tion by selecting the eigenvectors sequentially as follows.

Let N = Fvi : V2 : • • • : v^-il be the matrix of k-1 linearly independent eigenvectors already generated. Then the projector of the sub- space spanning these eigenvectors is given by Selecting the kth eigenvector v^ so that insures the linear independence of the kth eigenvector. In the numerical example 2, equation (15) cannot be satisfied for the third eigenvalue and eigen vector specification. This condition implies a slight perturbation in the eigenvalue and/or eigenvector specification must be made to insure the linear independence of the eigenvectors.

An algorithm is now presented which incorporates the condition of equa- tion (15) without having to compute the projection matrices R(k~l) explicitly.

The sequential synthesis of the eigenvectors is accomplished in a special canonical form in which the testing of the linear independence between the eigenvectors is straightforward. Further, the procedure allows the designer to insure the linear independence between the synthesized eigenvectors to a degree determined by a prespecified tolerance parameter. For clarity of pre- sentation, the algorithm is detailed for real eigenvalues. The extension to complex conjugate pairs follows as a natural extension of the real eigenvalue problem.

The following notations are used throughout the algorithm presentation: (1) e is an n-vector with rth entry equal to unity and all other r entries zero.

T T T (2) V]< = (_Zk : Wfc J is the kth eigenvector, where z^ is the designer specified m-vector.

where Q(°) = I and Q<i), ,i t 0, will be (3) v^- n defined in notation (6) .

(4) is the rth element of and for convenience of notation (5) is an n x n matrix of the form (16) where the vectors m M and mfc,^ are computed so that the vector a is transformed to a canonical form as (k) (k-l) M Vk r where A is a subset of integers {l, 2,..., n} containing the indices not 1 2 already used in the construction of the matrices M^ ), M^ ' and A(l) is the complete set {l,2,...,n}. Note M^) can be constructed, if and only if, o / 0.

k (6) Q(k-D = (k-D (k-2) . . . (l) (k > 1) M M M The algorithm now proceeds as follows: Step 1: For k = l,2,...,n, do steps 2 to 5.

Step 2: For X = X , compute det £x l _ - Fj.

k k n m (a) If det = 0, perturb X to (X + 6X ) and repeat step 2.

k k k (b) If det 7* 0, go to step 3.

Step 3: For X = X , compute C (eg. (10)) as k k 6 + X s C = Okln-m - Fl^f k ] (17) k Step 4: For some r (a) Compute the m-vector T [g (k)] = (k-l) r f r : 4 where fr~ hj- " is the rth row of the transformation g(k-l) (notation ( 6 ) ) and h (k-1) is a (n - m)-row vector.

r (b) Compute k a = [ g ( ) ] z ( 1 8 ) k r k where z is the arbitrarily specified m design vector.

k k (i) If a ^ 0, compute w (eg. (10)) and M< > (eg. (16)) k k and go to step 1.

k (ii) If a = o, select another r e A( ) and return to k step 4 (a) .

(iii) If a = 0 for all r e ( A ( k ) } , go to step 5.

k Step 5: For some r (a) If g ^ 7* 0, perturb z to (z + 6z ) to make a ? o r k k k k k (eg. (18)), compute w and M< ), and go to step 1.

k k (b) If g ( ) = 0, select another r and repeat step 5 (a).

r k , perturb X to (X + 6X ) and (c) If g < ) = 0, all r k k k r go to step 2 .

Step 6: Compute the feedback gains by using equations (12).

Step 7: Stop.

The kth linearly independent eigenvector v can be synthesized pro- k vided a ^ Q. This can be seen clearly as follows. Without loss of gen- k erality, let the first (k - 1) eigenvectors be generated with indices r = 1,2,..., (k - 1). Then these vectors are transformed into a canonical form under Q(k-l) as Diag v _i] = k with (i = 1,2,...,k - 1) and the projector spanning the subspace of these transformed eigenvectors has the simple form Now choosing the kth eigenvector so that its transformed vector becomes l)v causes a (r e {k,k+l,...,n>) k and insures the linear independence of v since the constraint (eq. (15)) is k clearly satisfied.

The following observations can be made regarding the algorithm outlined: (1) The algorithm can be directly extended to complex pairs in quasi- diagonal form as described. Two eigenvectors are synthesized in one iteration and equation (A5) is used instead of equation ( 8 ) . Further, the test condi- tion 0^ ^ 0 in step 4(b(i)) is modified, to testing the nonsingularity of a 2x2 matrix Z^. This matrix is constructed by selecting two rows of the transformation matrix Q^-l) obtained in the previous iteration and develop- k ing a condition similar to equation ( 1 8 ) . The transformation matrix M< ) for the complex pair is then obtained as a product of two transformations corre- sponding to the real eigenvectors Tv^ : v^+il associated with p^. This pro- cedure will insure the linear independence between v^ and (2) Explicit evaluation of the eigenvalues of matrix F is not needed to detect coincident mode assignment. It is sufficient to determine the appropri- ate determinant in step 2, this determinant being available as a byproduct in the synthesis of the kth eigenvector when Cfc is evaluated. If a mode is coincident with the spectrum of F, then a perturbation of the mode is required F s to insure [^l^n-m ~ l i- nonsingular and hence that equation (8) has a solu- tion for any arbitrary z^. The degree of perturbation needed depends on the numerical tolerance set on the evaluation of the determinant. Also since the eigenvalue shift in step 2 is a designer's choicer system stability is always insured. Alternatively, if the eigenvalue assigned coincides with the spectrum of F, special eigenvector structures can be derived by noting that equa- tion (8) has a solution for z^ = 0, and also for [|G + XjSjzi = 0 if (n - m) < m. (See ref. 6.)

(3) The iterative procedure in step 4(b(ii)) attempts to meet exact eigenvalue /eigenvector specifications. In step 5(b) an attempt is made to meet exact eigenvalue specifications with slightly relaxed eigenvector specifica- tions (z-vector) . The test in step 5{c) indicates that the eigenvalue specifi- cation implies that the corresponding eigenvector will lie in the eigensubspace already synthesized; this condition demands a perturbation in the eigenvalue specification.

(4) Since the matrices M have only one nontrivial column, coordinate transformations Q(k-l) reduce to simple vector multiplications. Further, the inverse of V required in equation (10) to evaluate the feedback gains in equations (12) is easily evaluated by noting that Q(°)V has the general form g(n) = oiag [a a , . . . ,a ]L v lf 2 n The matrix L is an elementary permutation matrix dependent on the sequence of generating the indices r e {A(k)\ in steps 4 and 5, the a^ are the nonzero pivotal elements, and i i ! 1 1 1 1 Q(n) V = IT Diag a a CT l 2 n Also notice that n det

TT

k=l since det and det [L] are unity.

Thus, the numbers QI provide a good measure of the linear independence between the eigenvectors provided the eigenvector entries are scaled to a stan- T dard basis as Vi v^ = 1 for example. Thus, the numerical ill-conditioning of the modal matrix V for inversion is controlled by setting a tolerance on a (k = l,2,...,n) to pass the test a ^ 0. The important bearing of this k k modal matrix numerical condition on system sensitivity properties is discussed in the section entitled "Modal Sensitivity Analysis."

(5) A noteworthy feature of the algorithm is that the eigenvectors do not explicitly undergo any change in the sequence of transformations Q(k-l). thus, the mode coupling characteristics of C are kept transparent during synthesis, k a very desirable feature for an off-line synthesis problem.

(6) The algorithm outlined can be conveniently programed as an iterative, interactive multivariable synthesis procedure.

Example 2 given in the section entitled "Numerical Examples" is now used to highlight the features of the spectral synthesis algorithm. Applying the algorithm step by step to the problem of example 2 yields the following synthesis sequence: (1) First mode (k = 1) : T

Z! = (i 0)

Ad) = {1,2,3} Choose r = 1. Then G! = (1 1) T Since a^ ^ 0, the first eigenvector can be synthesized as V]_ = (1 0 1) ; and the transformation matrix Q(!) becomes 1 0 0 Q(D 0 1 0 -1 0 1 (2) Second mode (k = 2) X = -2 z T = (-1 1) A<2) = {2,3} Choose r = 2. Then C = (0.5 0.5) T g (2) = (0 1) a = 1 = (-1 1 0)T; Since O ^ 0, the second eigenvector can be synthesized as v 2 2 and the transformation Q(2) becomes 1 1 Q(2) 0 1

- 1 1

(3) Third mode (k = 3) X, = -1 T z - (-1 A<3) = {3} Thus r = 3. Then C = (1 1) T g (3) = [0 0] (73 = 0 Since 03 = 0, the third eigenvector cannot be synthesized. Further, since T g2<3) = [0 0] , the eigenvalue specification cannot be met. This result is obvious since the "basis" vectors spanning the two-dimensional subspace corre- sponding to X^ = -1 are 1 0 0 1 1 1 and {vi,V2) already span this subspace. Thus, a second eigenvector associated with X = -1 cannot be synthesized as revealed by the null vector g^^. This result implies X must be perturbed slightly. Let = (-1 - e) ; e > 0.

Then (3) = and again 03 = 0; thus, the third eigenvector still cannot be synthesized. In this case, since g3^) i not the null vector, the eigenvalue specification s can be met but the eigenvector specification requires slight perturbation. Let T z = (-1 1+6); 6 f 0. Then -eS 03 = — 7* 0 J 1 + e and can now be synthesized as v = -1 1+6 1 + e, The closed-loop modal matrix V is "l -1 -1 0 1 1 +.6 V = 1 0 1 + e_ -1 - eT| . Further and A = Diag [-1 -2 6.e det [v] 1 = a 1 + e since a^ = 02 = 1. Thus, in this case, setting a tolerance on the value of 03 would directly control the numerical ill-conditioning of V and conse- quently influence the choice of the perturbations e and 6.

It is also interesting to note that if the sequence of assignment of modes = -1 an< = tnen were changed to ^i = -1, X ' * ^3 ~^' both eigenvectors corre- sponding to \j_ = -1 could be synthesized as -1 and \3 = -2 could still be assigned without perturbation since T g^(3) = (-0.5 -0.5) . However, the eigenvector specification could not be T met since 03 = 0. Thus, a perturbation in Z3 would allow completion of the synthesis.

MODAL SENSITIVITY ANALYSIS One of the primary objectives of a feedback control design is to insure that system responses remain well behaved under variations in plant parameters.

It is possible to derive quantitative robustness measures for eigenvalue and eigenvector perturbations due to plant variations using the following sensi- tivity analysis detailed in reference 7.

The closed-loop system satisfies Avi = XiVj[ (i = l,2,...,n) (19) Then for first-order differential changes in equation (19) , dA vi + A dvi = \i dvi + d\i Vi (20) For real eigenvalues let tj be a dual vector so that (21) T By premultiplying equation (20) by ti and using equation (21) the real eigen- value perturbation can be expressed as T ti dA vi d\i = - - (22) The corresponding eigenvector change is given by n-1 d ijVi (23) with 0 (i = j) Similar expressions can be easily developed for complex conjugate pairs. (See ref. 6.) From equations (22) and (23), bounds can be established for the eigenvalue/eigenvector changes as , dXi| S . . = TTidA (24) where (25) where Tr^ is defined as the "mode condition number" corresponding to the eigen- value X^ and clearly i\i £ 1. If the eigenvectors V£ are normalized so that v Tv Tv i i = ti i = ! (26) = and Then TTj^ lUillo |dXi| S TTildA^ ( 2 7 , ) T T where lltjJL = ti ti and ||dA||2 = J Maximum eigenvalue of [_dA . dAJ.

Similarly/ the corresponding bound for eigenvector perturbation is given by n |dA« J-l The condition numbers ir^ in equation (27) provide quantitative robust- ness measures for the nominal eigenvalues and eigenvectors of the closed-loop system A. It is also clear that for plant perturbations not known a priori, the system has best robustness properties if TT^ = 1 (i = l,2,...,n), which implies that the nominal modal matrix V is orthonormal with t^ = v^ (i = l,2,...,n).

From equation (8) it is apparent that sufficient eigenvector freedom is generally unavailable to construct an orthonormal modal matrix. Thus, from a practical design perspective it is better to seek a mode-decoupled design, with each state variable dominantly displaying one mode (as determined by a dominant eigenvector entry). This design, in the limit when total decoupling^is possi- ble, results in an orthonormal V. Further, if the perturbations dA are known a prigri, then the appropriate zero sensitivity eigenvectors span the null space of dA, and if such an eigenvector can be synthesized by using state ^ feedback, the corresponding eigenvalue will be invariant to the specified dA.

The mode condition numbers ifi derived as the norm of the dual vectors fc i (eq. (27)) also directly relate to the numerical ill-conditioning of the modal matrix V for inversion (linear dependence of eigenvectors). This fact becomes apparent by considering the relation (28) : v : = Ir and noting that as the eigenvectors v^ tend toward being linearly dependent, the norms of the corresponding dual vectors t^ increase in order to satisfy the constraint equation (28). Thus, large values of 11^ indicate that the system has poor robustness properties and also reinforce the fact that the corresponding modal matrix is ill-conditioned. It should be recalled that the spectral synthesis algorithm described earlier allows the designer to effi- ciently control this system robustness property by setting a numerical toler- ance on the a of equation (18) to pass the test cr ^ 0 in steps 4 and 5 k k of the algorithm. Since the determinant of V is obtained as the product of the a^ (k = 1,2,...,n), it is clear that the design process effectively con- trols the ill-conditioning of V for inversion.

AIRCRAFT LATERAL CONTROL SYSTEM SYNTHESIS In this section, the lateral performance objectives of an aircraft are formulated as an eigenvalue/eigenvector assignment problem and the utility of the spectral synthesis algorithm is demonstrated by designing a feedback con- trol system for a fighter aircraft. Major qualitative design objectives for lateral aircraft dynamics are (a) Fast roll rate response with minimum overshoot (b) Good Dutch roll damping (c) Low sideslip and peak lateral acceleration in response to roll stick command, for good turn coordination (d) Low roll response to sideslip gust inputs.

Performance Specification The quantitative performance specifications can be briefly summarized as follows (ref . 8) : (1) The frequency and damping of the Dutch roll mode shall satisfy 0-19 <% > 1.0 d °- (2) If the spiral mode is unstable, the time to double shall be greater than 20 seconds.

(3) The roll-subsidence time constant T shall be less than 1.0 second.

R (4) After a rudder-pedals-free step aileron command, the ratio of sideslip to the parameter 9 shall be less than that specified below. The aileron command shall be held fixed until the bank angle has changed at least 9 0 ° .

(a) Category A (rapid maneuvering phase) 3/9 < 6° for adverse sideslip 3/9 < 2° for proverse sideslip (b) Category C (take off/landing phase) 3/0 < 10° for adverse sideslip 3/0 < 3° for proverse sideslip The parameter 0 is given by 4>t 6 = —- (t = 1.3) 90° for category A and 4>t

0 = —- (t = i.o)

30° for category C, where <j> is the bank angle achieved at t seconds after the t step input.

(5) For a step aileron command held until the bank angle has changed at least 90°, the roll rate at the first minimum following the first peak shall be at least 60 percent of the peak value.

(6) For category A turn coordination the lateral acceleration at the pilot shall be less than 0.15g for a 60°/sec roll to a 60° bank.

It should be recognized that these aircraft response characteristics must usually be achieved under feedback gain magnitude constraints arising because of sensor noise and other considerations. For the numerical aircraft example to be discussed in the section entitled "Fighter Aircraft Control Analysis," all the feedback gain magnitudes were constrained to be about unity. The air- craft performance in the numerical example is evaluated by using the linearized model at selected flight conditions.

Aircraft Model and Design Considerations The nonlinear equations of motion of the aircraft are used to generate linear perturbation models at various flight conditions. Accordingly, the state space representation, referenced to the stability axes, takes the form x = Ax + BiT (29) I y = Cx + Du_ where 1 1 Lp L 0 r

•*

N N; N3 P B Y Y Y g/v 3 0 P r _ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 D =

M

M

MS

r

P **

_ 0 0 0 T T <6 : 6 ); y = (p \p a $) with X = (p 4> 3 <t»; a r y For the aircraft control problem/ from sensor considerations, feedback from 3 variable is difficult to implement. Hence, the control laws are derived by using the output vector y of equation (29), where the lateral acceleration a sensor substitutes for the sideslip 3 sensor. The state y feedback law of equation (2) is modified as finding a control law of the form u = Ky + u (30) t such that the closed-loop system meets desired eigenvalue/eigenvector speci- fications, with K the output feedback matrix and Up the external pilot input. The closed-loop system after applying feedback law (30) takes the form x = [A + BKCj A + BKC x + BP u.

(31) C + DKC x + DP u r f ~ 1-1 where P = [_I - KDJ and K = PK is the equivalent output feedback m matrix obtained by setting D = 0. K exists provided P exists. Further, the feedback law (30) is still equivalent to the state feedback case since C is rank n for all flight conditions. Thus, for every state feedback law u = Kx + Up (eq. (2)) derived by using the spectral synthesis algorithm, there exists an equivalent output feedback matrix K (eq. (30) given by = K[C (32) Direct matrix manipulations show that the inverse of the matrix [C + DK] in equation ( 3 2 ) exists if P exists. Hence , the theory developed for the state variable feedback problem is directly applicable to the aircraft lateral dynamics model described by equation (29) .

Appendix B gives the numerical values for the (A,B,C,D) quadruple of equa tion (29) for a typical fighter aircraft at selected flight conditions. An examination of these models over the complete flight envelope reveals that the following system constraints contribute critically to the evolution of a satis factory closed-loop system.

The large Lp derived in most of the flight conditions indicates that significant roll motions are induced for sideslip gust inputs. A feedback augmentation to reduce this effect yields the following control equations: with and MR = Me + M k + M k 6a Ae 6r Rg with the circumflex (~) denoting closed-loop magnitudes.

It should be noted that if the gain magnitudes k^ and k^ are ay 3i-y strained, substantial reduction in the Lo magnitude are possible only if the magnitude of the control derivatives Lg , Lg , and Mo are large. Further, since the Mg magnitude is dependent on the a sensor location, a possibility y exists to optimally locate the acceleration sensor from a control viewpoint.

For satisfactory turn coordination, $ and ay excitation at the pilot station must be minimized for aileron input. This minimization implies reduc- tion of cross coupling between the roll dynamics and the sideslip/yaw rate.

This coupling arises not only because of mode cross coupling in the system matrix A, but also because of the input mixing through the control matrix B.

Whereas feedback is effective only in suppressing the intervariable cross cou- pling in A, some form of precompensation is necessary to minimize the input mix- ing of B. Thus, L , Lg, Np, and Y must be reduced by feedback. Again, r p the feedback magnitude constraints determine the limit of reduction. Thus, if the turn coordination specification cannot be met by feedback alone, a feed- forward compensation, in the form of an aileron/rudder interconnect, must be introduced to increase the ratio L<$ The resulting composite control law takes the form u = Ky + Hu (33) t T where Up = (6 : SR) is the external pilot input and H is a 2 x 2 non- A singular feedforward matrix representing aileron/rudder interconnect. The closed-loop system after applying feedback law (33) takes the form x = [A + BKC] x = A + BKC x + BPH u.

(34) = [c + DKc] y = IC + DKClx + DPH u.

where K and P are defined in equation (31) Eigenvalue/Eigenvector Assignment Formulation The aircraft handling characteristics postulated earlier can be summarized as lateral dynamics being composed of two weakly coupled subsystems. Roll rate and bank angle constitute the first subsystem and display predominantly the roll subsidence and spiral modes. The second subsystem is characterized by a well damped Dutch roll mode defining the yaw rate and sideslip motions. This results in the mode (eigenvalue) and associated response variable (eigenvector shape) assignments of table I.

TABLE I.- DESIRED MODAL SPECIFICATIONS Dominant Mode response variable Roll subsidence Roll rate, p Spiral Bank angle, <j> Dutch roll Yaw rate and sideslip, ^,8 TABLE II.- DESIRABLE CLOSED-LOOP MODAL STRUCTURE Dutch roll Spiral Eigenvector Roll subsidence components mode mode mode Z P 0 Z13 14 Z 0 1 1

4»

w w wi2 w 13 14

6 ll

w 2 w «12 «24 <t> 2 23 It should be emphasized that the structure in table II is only one of several configurations that can be selected. In practice, alternative struc- tures must be tried to effectively trade off different (often conflicting) per- formance specifications.

In this report, the rationale for the selection of the z^j components in table II was to meet the mode decoupled structure postulated in table I.

This structure, in addition to meeting the handling quality specification, would result in robust feedback designs.

The roll subsidence mode design vector selection is straightforward. The Dutch roll mode eigenvector design parameters were selected to keep only one parameter ( z ^ ) available for manipulation in the design iteration cycles. The parameter z^3 directly influences the closed-loop Lg magnitude and is selected to effect the desired reduction in L£ magnitude. The spiral mode eigenvector components z^4 and Z24 must be selected so that the bank angle response is dominated by the spiral mode.

The g and $ components (wji elements) of the eigenvectors in table II are computed by using constraint equation (8) or (A5), after specifying the appropriate eigenvalues. The eigenvalues must be selected to meet quantitative speed of response and at the same time yield desirable eigenvector forms. The mode coupling matrices Ci (eq. (10)) clearly define this constraint relation- ship, and aid in the selection of the appropriate eigenvalues.

It is helpful at this stage to clarify these design concepts by using a numerical example. Consider the aircraft model corresponding to flight condi- tion 1 (appendix B). In particular, consider the synthesis of the spiral mode eigenvector. The spiral mode eigenvalue can lie in the range -0.01 to -0.1.

The design equations take the form (eq. (10)). For X = -0.01, -25.52 -7.16 Z w 14 14 (35) W -100 -6.02 24 Z24 Judicious choice of z^ and Z2 can now be made by examining the modal 4 4 coupling relations in equation (35) to meet the requirement that v?2 be very large compared with wj , z^ , and Z2 « One acceptable solution is obtained 4 4 4 by selecting Z2 = -3.5z^ ; as a result, the normalized eigenvector (with 4 4 T v v = 1) is obtained as 4 4 V = Z Z W W = (36) 4 ( 14 24 14 24> (-0.012 0.044 0.006 0.998) which has the desired bank angle dominant structure. Suppose, on the other hand, that if the spiral mode is chosen as X = 0.1, the design equations become W -4.52 -11.59 z (37) -10 -0.6 w Z24 and thus the eigenvector structure would be worse compared with that in equa- tion (36). Thus, a trade-off between speed of response and minimal influence of the slow spiral mode on the response of \j) and 3 would finally determine the spiral mode in the range -0.01 to -0.1. This analysis also highlights the fact that mode selection is not only dictated by speed of response but also by the eigenvector structure it generates through the mode coupling matrices C^ (eq. (10)). In the design of the spiral mode eigenvector, it sould be empha- sized that suppression of this slow mode from the 3 response is vital in limiting steady-state g excursions under the roll-stick command.

The synthesis procedure up to this stage can be summarized as consisting of the following steps: Step 1.- For each mode i compute the eigenvector coupling matrix (eq. (10)) for the range of assignable values.

Step 2.- Select the mode from step 1 which yields the eigenvector struc- ture closest to the desired one by appropriately choosing the z vectors as illustrated in the numerical example.

Step 3.- Complete the synthesis by specifying all the modes and design vectors and applying the spectral synthesis algorithm.

It is very likely that a design in step 3 may violate gain magnitude con- straints. The design parameters must then be iteratively modified to meet these constraints. The direction of parameter change is found by studying the open loop modal structure and identifying the design specifications that unduly violate the physics of the basic plant. For example, the zj^ parameter directly determines the reduction of the 1$ derivative and consequently con- trols the magnitude of the feedback gains k^ and k^ .

FIGHTER AIRCRAFT CONTROL ANALYSIS Twenty flight conditions describing the operational flight envelope of the aircraft are shown in figure 1. The design points are represented as a plot of Mach number against angle of attack. The altitude parameter is identified with different symbols. Linearized state variable models describing the lateral dynamics at each of these flight conditions were used for the synthesis anal- ysis. The flight conditions of figure 1 were classified into three catagories as shown in table III. The group numbers are arranged in decreasing order of control effectiveness as measured by the magnitudes of the control derivatives L§ and Lfi , with group III flight conditions posing the worst conditions for synthesis from a mode decoupling viewpoint.

TABLE III.- FLIGHT CONDITION CLASSIFICATION Flight group Flight conditions* I 1, 2, 3, 4, 6, 9, 10, 12, 13, 14, 19, 20 II 5, 7, 8, 11, 15, 16 III 17, 18 *See figure 1.

Feedback Design The basic modal matrix structure of table II was employed for the flight conditions in groups I and II. The design parameter 2^3 ranged from -0.2 to -1 for group I and -2 to -4 for group II to meet the gain constraint require- ments on the lateral acceleration sensor. Larger magnitudes of 2^3 implied increased roll response to 3 gust inputs. As mentioned earlier, significant changes in the basic aircraft characteristics could not be achieved for low altitude and low Mach number flight conditions, namely, flight conditions 17 and 18. Low control derivative magnitudes coupled with feedback gain limita- tions contributed to the problem. For these cases, the iterative design pro- cess was initiated with the Dutch roll mode eigenvector structure corresponding to that of the free aircraft. Final trimming of the design parameters and Dutch roll mode damping were effected to give a reasonable improvement in response with acceptable feedback gains. ' Precompensation Design As noted earlier, if the turn coordination specification cannot be achieved by mode decoupling alone (through feedback), additional feedforward compensation by the aileron/rudder interconnect is required. In order to meet the roll stick input specification, the following structure for the precompensation matrix H (eq. (33)) was found to be satisfactory H = (38) The interconnect gain k is determined to maximize the ratio of R CONTROL AUGMENTED AIRCRAFT PERFORMANCE The modal control procedure was used to obtain the gain matrices K and H (eq. (33)) at all 20 design flight conditions summarized in figure 1. The next objective is to evolve a control system which operates over the flight envelope based on these fixed point designs. This procedure usually takes the form of developing gain schedules using appropriate air data parameters as scheduling parameters. However, in this report the question of gain scheduling will not be pursued; instead, the improvement in system performance, based on fixed point designs is shown at three selected design flight conditions 1, 17, and 20. These flight conditions cover comprehensively the complete flight envelope from the synthesis complexity viewpoint. Flight condition 1 represents the nominal cruise condition with no significant synthesis constraints. Flight condition 20 is a high-angle-of-attack condition usually known to result in poor lateral control and turn coordination (ref. 8). Finally, flight condi- tion 17 covers the landing approach condition and represents the most diffi- cult flight regime from a mode decoupling viewpoint.

Tables IV to VI detail the eigenvalue/eigenvector modifications achieved at these flight conditions. In tables IV to VII, j = >/-!. The results are based on the control law given in the respective tables. It is worthwhile noting the modification achieved in the eigenvector pair corresponding to the Dutch roll mode in flight conditions 1 and 20. The desired decoupling of the roll rate variable from the Dutch roll mode could be achieved only by assign- ing an overdamped complex pair of eigenvalues (-1.5 ± J0.75) as the Dutch roll mode. Further, the design iterations revealed that the Dutch roll mode damp- ing could not be reduced (to improve ty and 3 responses) without violating feedback gain limits on the ay sensor.

TABLE IV.- MODAL CHARACTERISTICS (FLIGHT CONDITION 1)

&•

Characteristic for free aircraft Characteristic for augmented Eigenvector eigenvalue of - aircraft eigenvalue of - components -0.03 -0.34 ± J2.66 -6.00 -0.01 -1.5 ± jO.75 -3.70 -0.964 -0.032 -0.403 0.829 0.986 0.013 0. -0.117 P -0.041 0.044 -0.096 -0.131 0. -0.045 0.586 0.586 0.002 0.002 0.069 -0.034 -0.008 -0.0004 0.137 0.527 B 0.261 0.998 0.314 0.112 -0.164 -0.999 -0.041 0.034 CONTROL LAW (eq. ( 3 3 ) ) -0.204 -0.491 -0.799 0.017 K = -0.152 0.318 -0.504 -0.021 1 0 H = 0.325 1 TABLE V.- MODAL CHARACTERISTICS (FLIGHT CONDITION 17) Characteristic for free aircraft Characteristic for augmented Eigenvector eigenvalue of - aircraft eigenvalue of - cumpuiien ts -0.063 -1.97 -0.168 ± jl.594 -4.5 -0.07 -0.5 ± jO.9 -0.89 -0.078 -0.63 0.556 P 0.976 -0.086 -0.49* 0.49 -0.039 0.139 -0.03 -0.075 0. 0.140 -0.074 -0.098 * -0.001 0.017 0.122 -0.005 B -0.017 -0.004 0.139 -0.173 0.454 0.987 0.381 0.357 <t> -0.217 0.986 0.643 0.198 CONTROL LAW (eq. (33)) -1.07 0.595 -1.259 -0.294 -0.179 0.362 -0.171 -0.084 H -0.069 TABLE VI.- MODAL CHARACTERISTICS (FLIGHT CONDITION 20) Characteristic for free aircraft Characteristic for augmented Eigenvector eigenvalue of - aircraft eigenvalue of - components -0.77 -0.096 -1.034 ± J2.898 -6.0 -0.01 -1.5 ± jO.75 0.588 0.102 -0.046 0.946 0.986 -0.022 0. -0.116 P 0.11 -0.022 -0.022 0.005 0. 0.045 0.578 0.578 * B -0.022 -0.007 0.077 -0.028 -0.044 0.0004 0.113 0.544 -0.801 -0.995 0.296 -0.087 -0.164 -0.073 -0.066 4> 0.999 CONTROL LAW (eq. (33)) -0.357 -0.305 -0.847 -0.001 K = -0.083 0.581 0.051 -0.033 1 Q" H = -0.02 1 Table VII depicts the improvement in the aircraft robustness character- istics in terms of the mode condition numbers introduced in the section entitled "Modal Sensitivity Analysis." Significant improvement in system sensitivity characteristics has been achieved in flight conditions 1 and 20. However, the improvement in sensitivity performance in flight condition 17 is marginal and is due to the inability to appreciably decouple Dutch roll mode from p and <|> variables. In table VII the determinant of the modal matrix V with eigen- vectors normalized as v^v^ = 1 is also included. It is evident from table VII that the determinant of V reflects the aggregate effect of all the mode condition numbers (ir^). The determinant value decreases as the mode condition numbers increase.

Finally, time response histories for roll stick input and sideslip gust inputs are compared in figures 2 and 3. From the response curves it is appar- ent that control augmentation has reduced the cross coupling between the roll axis (p,4>) and the yaw axis (^,B). It is also important to note that in fig- ures 2(a) and 2(c), feedback augmentation has reduced the peak roll rate capa- bility for pilot aileron input. This reduction can be offset by introducing appropriate gain factors in the forward loop (H matrix (eq. (33))). It should be noted that the roll response to sideslip gust input has been significantly reduced in figures 3(a) and 3(c). Finally, for flight condition 20 the quan- titative 8/9 performance specification could not be met but was significantly improved compared with the free aircraft performance. In all other flight con- ditions the feedback system met all the lateral handling criteria postulated earlier.

TABLE VII.- AIRCRAFT MODAL SENSITIVITY CHARACTERISTICS Free aircraft Augmented aircraft Flight condition Mode condition Mode condition Eigenvalue Eigenvalue number , 1^ number , TT^ 1 -3.70 10.39 -6.0 1.065 -0. 03 2.27 -0.01 1.027 -0.34 ± J2.66 14.09 -1.5 ± jO.75 4.337 17 -1.97 6.76 -4.5 3.51 -0.06 3.52 -0.07 3.17 -0.17 ± jl.59 9.68 -0.5 ± jO.9 6.817 20 -0.77 8.62 -6.0 1.069 6.36 -0.096 -0.01 1. 021 -1.03 ± J2.9 14.32 -1.5 ± jO.75 4.0 Flight Augmented aircraft Free aircraft condition Det [v] Det [v] -0.0146 -0.225 17 -0.0202 0.0412 20 -0.0086 0.242 CONCLUDING REMARKS A new multivariable synthesis procedure for aircraft handling qualities design has been developed. The synthesis procedure is computationally simple in that it only involves solution of linear systems of equations. This results in an efficient computer-aided interactive design tool for flight control sys- tem synthesis. It is shown that the quantitative lateral design objectives can be conveniently formulated as an eigenvalue/eigenvector assignment problem.

By noting the freedom available in the selection of eigenvalues and eigenvectors under state variable feedback, a systematic design methodology is evolved to meet the performance specifications. The inherently mode-oriented synthesis procedure provides significant insight into the design process and allows optimizing the response of the aircraft under often conflicting requirements.

The study also revealed that the closed-loop modal matrix (matrix of eigen- vectors) plays a pivotal role in characterizing system sensitivity to plant parameter variations. New robustness measures related to the numerical ill- conditioning of the modal matrix for inversion, termed mode condition numbers, are also developed.

The synthesis technique is illustrated by using the linearized lateral dynamics model of a typical fighter aircraft. The simulation studies revealed that the feedback-augmented aircraft exhibited improved Dutch roll damping, good turn coordination for pilot roll stick command, and reduced roll response to sideslip gusts over the flight envelope.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 April 13, 1978

APPENDIX A COMPLEX CONJUGATE PAIR IN REAL ARITHMETIC FORM In order to perform real arithmetic computations with complex conjugate pair eigenvalues, the following transformation is established for reference.

Consider the eigenvalue/eigenvector relation A(q + js) = (a + jb)(q + js) (Al) where a + jb is a complex eigenvalue of A and q + js the associated eigen- vector (with j = \pl). Equation (Al) can be solved for real and imaginary parts as (A2) with the same relation holding for the conjugate eigenvalue a - jb. Equa- tion (A2) can be written as -b\ A(q : s) = (q : s) (A3) where q and s are real vectors corresponding to the real and imaginary part of the complex eigenvector and the complex conjugate pair of eigenvalues a ± jb can be written in equivalent real form as (A4) P = -b By using the real arithmetic representation for a pair of complex eigenvalues a 6 i * Jbi' th eigenvector constraints of equation (8) take-the form

APPENDIX A • • F G + afS . -biS Wi zi sjJn-m ~ * -bjJn-m • • (A5) * z b I F b>iS . G + aiS Wi+l i+l i n-m • M^-m ~ • • where : v ] = i+1 are the real eigenvector pair associated with a± ± jbi-

APPENDIX B AIRCRAFT STATE VARIABLE MODEL AT SELECTED PLIGHT CONDITIONS The lateral dynamics of the aircraft modeled in state space takes the form x = Ax + Bu (Bl) y = Cx + Du T T where X = (p \|) B 4>) ; U = (6 : 6 ) ; and y = ay. For brevity of doc- a r umentation the state variables p, ij;, and <j) are omitted from the output set in equation (Bl). The respective quadruples (A, B, C, 6) for selected flight conditions are listed in tables Bl to B3.

TABLE Bl.- FLIGHT CONDITION 1 (NOMINAL CRUISE) [~Mach number, 0.67; altitude, 6096 m (20 000 ft); |_ angle of attack, 3.45° .

A matrix: -3.79E+00 4.06E-02 -5.20E+01 0.

-1.34E-01 -3.59E-01 4.24E+00 0.

6.02E-02 -9.97E-01 -2.72E-01 4.62E-02 l.OOE+00 6.03E-02 0.

0.

B matr ix: 2.50E+01 9.83E+00 1.42E+00 -4.20E+00 5.01E-03 5.03E-02 0. 0.

C matrix: -1.25E-01 -6.12E-02 -3.41E+00 -1.50E-03 D matrix: 1.03E+00 -2.66E-01

APPENDIX B TABLE B2.- FLIGHT CONDITION 17 (LANDING APPROACH) [kach number, 0.19; altitude, 30 m (98.43 ft);] |_ angle of attack, 6.72° J A matrix: -1.99E+00 9.21E-01 -1.65E+02 0.

-7.97E-02 -1.77E-01 5.68E-01 0.

1.19E-01 -9.92E-01 -2.01E-01 1.51E-01 l.OOE+00 1.18E-01 0. 0.

B matrix: 2.98E+00 1.79E+00 2.47E-02 -7.92E-01 3.50E-03 3.74E-02 0. 0.

C matrix: -6.35E-02 5.93E-03 -9.79E-01 -4.38E-03 D matrix: 9.01E-02 -2.89E-02

APPENDIX B TABLE B3.- FLIGHT CONDITION 20 (HIGH ANGLE OF ATTACK) fMach number, 0.6; altitude, 6096 m (20 000 ft); load!

|_ factor, g units, 4; angle of attack, 15.44° J A matrix: -2.25E+00 7.58E-01 -3.63E+01 0.

-7.03E-02 -4.06E-01 -4.50E-02 0.

2.66E-01 -9.63E-01 -2.83E-01 4.98E-02 l.OOE+00 2.76E-01 0.

0.

B matr ix: 1.61E+01 8.21E+00 7.25E-01 -3.43E+00 1.90E-04 4.76E-02 0. 0.

C matrix: -1.04E-01 3.13E-01 -3.95E+00 -2.07E-02 D matrix: 5.76E-01 -1.93E-01 REFERENCES 1. Harvey, C. A.; Stein, G.; and Doyle, J. C.: Optimal Linear Control (Charac- terization of Multi-Input Systems). ONR CR 215-238-2, U.S. Navy, Aug.

1977. (Available from DDC as ADA043771.)

2. Porter, Brian; and Crossley, Roger: Modal Control: Theory and Applications.

Taylor & Francis (London), 1972.

3. Anderson, B. D. O.; and Luenberger, D. G.: Design of Multivariable Feed- back Systems. Instn. Elec. Engrs. - Proc., vol* 114, no. 3, Mar. 1967, pp. 395-399.

4. Chidambara, M. R.; Broen, R. B.; and Zaborszky, J.: A Simple Algorithm for Pole Assignment in a Multiple Input Linear Time Invariant Dynamic System.

Trans. ASME, Ser. G: J. Dynamic Systems, Measurement, and Control, vol. 96, no. 1, Mar. 1974, pp. 13-18.

5. Srinathkumar, S.; and Rhoten, R. P. : Eigenvalue/Eigenvector Assignment for Multivariable Systems. Electronic Lett., vol. 11, no. 6, Mar. 1975, pp. 124-125.

6. Srinathkumar, S.: Spectral Characterization of Multi-Input Dynamic Systems.

Ph. D Diss., Oklahoma State Univ., 1976.

7. Fadeev, D. K.; and Faddeeva, V. N. (Robert C. Williams, trans.): Computa- tional Methods of Linear Algebra. W. H. Freeman and Co., c.1963.

8. Hartmann, Gary L.; Hauge, James A.; and Hendrick, Russell C.: F-8C Digital CCV Flight Control Laws. NASA CR-2629, 1976.

CO w 5 —.. —.. C TH •£?• ^ ^ O -H o O O .2 o CO O O O O o fl CO O O C 3 0.3 "2 o °o CD esi x: W £ O O 2 •pf' CO ^ CO CD. N

E

c •H

a

co c co CO •o co l°v

<] IH

o & CM

I> 0)

co U rt (0 1-1 O M •H CO (V U D O> •H co (M co Sap ' jo 0)

<

be o> T3 oT

\_

15 B

Si O K -10 O 0)

•s

oT

•s V

!H B -2 .1 be o> CQ 0) T3 • |H CQ -.1 B bo T3 V -50 0 1 2 3 4 5 Time, sec (a) Flight condition 1 (nominal cruise).

Figure 2.- Comparison of free aircraft and augmented aircraft response to roll stick step input. 5^(0) = 1°; A indicates free aircraft response and B cor- responds to augmented aircraft response.

.2 0) CO 4) -(-> rt 0) o 0)

? Sf

73 -Q G O 0) i—i •i-i < .5 o •PH -4-> U <u 5n bfl 0) 0) 73 T3 O T5 I K -.5 0 1 2 3 Time, sec (a) Concluded.

Figure 2.- Continued.

u o>

tf

•

O -2 K .5 u <U o

•a

I

0)

•

-B cn 0) I I -2 0 1 2 3 4 Time, sec (b) Flight condition 17 (landing approach).

Figure 2.- Continued.

tf

5 .0155

— IH B •—I 03 ~~ /~ ^^ «4J ^^^ /

uni

o

g

A

Lateral ace

b i-i CJl

, en

\^ /-B ^A

»-i

'

o

deg

:on deflection, M O) i—i •rH — . • - . - . . .

<: -i

—

d- '

O 4-> O A

A

"3 be

/

•S -S o

*» —

0)

'^B

I I 1

PH — 5

i 3 . 1 2 3 4 5

Time, sec

(b) Concluded.

Figure 2.- Continued.

9) bD 9) •C oT

•a

O K -5 O 0) be 0) oT -M rt

I

-2 .5 bO 0) T5 B CO <u •o -.5 be

•s

w 0)

•a

-25 0 . 1 2 3 4 Time, sec (c) Flight condition 20 (high angle of attack) Figure 2.- Continued.

.06.— o •l-l <L> —I CQ o b O ^ p ct ' ri o •i-< •«-> o o> <4-^ 0) bJD C O t-> 0)

-1

O 0) bD

r

TJ ^ 0)

_ ^B

s

1 1 1 1 1

-.5

3 4 E

3 1

Time, sec (c) Concluded.

Figure 2.- Concluded.

u (U

•a

-10 O O) Ml V T3 oT -u rt ^

I

bn CD •a co 0) -o •i-» CO bfl 0> -B <D r—I

!

I I -5 0 1 2 3 4 5 Time, sec (a) Flight condition 1 (nominal cruise).

Figure 3.- Comparison of free aircraft and augmented aircraft response to side- slip step disturbances. 3(0) = 1°. A indicates free aircraft response and B corresponds to augmented aircraft response.

ef o

.155

•i-t •M Oj f-l 0) -.155U eT o

B

O ^H 0) tf o •rH -4-> O 0) fac o> 0) T3 TS

Time, sec

(a) Concluded.

Figure 3.- Continued.

bD o> TJ oT

•s

f-, (1) B be 0) •o

I

-1

I

B bJD 0) -O w 0)

s

w -1 bC O> •O oT -5 I I I 0 1 2 3 4 Time, sec (b) Flight condition 17 (landing approach) Figure 3.- Continued.

o •l-l

•8

SH O -rt o G oS 3 --J fafi 0)

-.0311

o • rH 4-> O faJD QJ

I

o 0) ?, bfi 0)

•= • "

\

0)

^A

—

1 I 1 1 1

-1

3 4 5

0 1 2

Time, sec

(b) Concluded.

Figure 3.- Continued.

o> o> S-,

I

•B be to 0) CO bO

•8

B oT -5 0 1 2 3 4

Time, sec

(c) Flight condition 20 (high angle of attack) Figure 3.- Continued.

.155 o •l-l 0) S o e § * bJD 1—I rt B u 0) C O t-l 0) 1—I •|H

•<

cf o fcJD

<D : V

•O

A

i 1 1 1 1

-.5 0 1 2 3 4 5

Time, sec

(c) Concluded.

Figure 3.- Concluded.

1. Report No. 2. Government Accession No.

3. Recipient's Catalog No.

NASA TP-1234 4. Title and Subtitle 5. Report Date June 1978 MODAL CONTROL THEORY AND APPLICATION TO AIRCRAFT 6. Performing Organization Code LATERAL HANDLING QUALITIES DESIGN 7. Author(s) 8. Performing Organization Report No.

L-12177 S . Srinathkumar _ 10. Work Unit No.

9. Performing Organization Name and Address 505-07-33-04 NASA Langley Research Center 11. Contract or Grant No.

Hampton, VA 23665 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Paper National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, DC 20546 15. Supplementary Notes Center.

S. Srinathkumar: NASA-NRC Associate, Langley Research 16. Abstract A multivariable synthesis procedure based on eigenvalue/eigenvector assign- ment is reviewed and is employed, to develop a systematic design procedure to meet the lateral handling qualities design objectives of a fighter aircraft over a wide range of flight conditions. The study reveals that the closed-loop modal characterization developed provides significant insight into the design process and plays a pivotal role in the synthesis of robust feedback systems. The simplicity of the synthesis algorithm yields an efficient computer-aided inter- active design tool for flight control system synthesis .

17. Key Words (Suggested by Author(s)) 18. Distribution Statement Eigenvalues Eigenvectors Unclassified - Unlimited Modal control Aircraft control systems Handling qualities Subject Category 08 19. Security CJassif. (of this report) 20. Security Classif. (of this page) 21. No. < Jf Pages 22. Price* Unclassified Unclassified 6 0 $5.25 * For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Langley, 1978 THIRD-CLASS BULK RATE, Postage and Fees Paid National Aeronautics and National Aeronautics and Space Administration Space Administration NASA-451 Washington, D.C.

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