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Robust flight control design with respect to delays, control efficiencies and flexible modes
ControlEng. Practice, Vol. 3, No. 10, pp. 1373-1384, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/95 $9.50 + 0.00
Pergamon 0967-0661 (95)00140-9
ROBUST FLIGHT CONTROL DESIGN WITH RESPECT TO DELAYS, CONTROL EFFICIENCIES AND FLEXIBLE MODES T. Livet*, F. Kubica* and J.F. Magni** *AEROSPATIALE, Aircraft Division, 316 route de Bayonne, 31060 Toulouse, France **ONERA-CERT, 2 avenue Edouard Belin, 31400 Toulouse, France
(ReceivedJuly 1994; infinal form June 1995)
Abstract: This paper presents an iterative technique which makes it possible to design a
robust flight control system. The methodology described in this paper has multiple applications. Two different applications are considered: the first consists of computing a robust feedback with respect to parameter uncertainties by considering a bank of models, while the second provides a robust flight-control design for a highly flexible aircraft with significant cross-coupling between flight mechanics and structural dynamics modes. The proposed method will solve these problems via an iterative quadratic optimization under linear inequality constraints. Finally, the iterative algorithm is applied to a civil aircraft control system in lateral flight. Key Words: Flight Control; Parameter Robustness; Stability Robustness; Flexible Aircraft;
Eigenstructure Assignment; Control Algorithms.
1. I N T R O D U C T I O N In practice, aircraft control laws are designed by eigenstructure assignment. This assignment is performed in such a way that natural decoupling of aircraft is preserved or even reinforced, see (Mudge and Patton, 1988; Sobel and Shapiro, 1985). The fact that natural behavior is not modified leads to good robustness. However, in the design procedure, robustness ideas are not explicity considered. This paper shows that robustness can be improved by using a very simple multi-model approach (but one might also consider more sophisticated techniques such as that based on the relationship between eigenvectors and eigenvalue variation induced by model variations, see (Magni and Manouan, 1994). The technique proposed is some kind of parameter optimization technique, without guarantee of convergence, but which has been shown to be very efficient in the field of interest here. It utilizes a bank of models, and simultaneously shifts the poles of all these models inside a given trapezium. Basically, the first-order approximation proposed in (Xu et a l , 1986) is used, but pole shifting is performed in such a way that maximum freedom is offered to the motion of the poles in order to avoid singularities. At each step of the algorithm, the pole is shifted toward the required trapezium by the solution of a quadratic problem under linear constraints. The area for the motion of the pole is a sector of the complex plane that is as large as possible.
Flight control law design for aircraft is a very attractive problem, considering the robustness properties which are required. In this field of interest, robustness appears to be a specific problem which has not received much attention in the literature because generally feedback gains must be constant matrices (no dynamics except dynamics with physical significance as filters); moreover, some gains might be fixed in advance. Using the techniques existing in the literature, which offer a guarantee of global convergence or directly give a control law, it is impossible to satisfy these kinds of design specifications. The translation Of practical specifications in terms of convex problems may lead to the solution to the problem, see (Geromel et al, 1993; D o y l e , 1989), but for constant output feedback, the optimization problem is not convex, so there is no guarantee of global convergence. Moreover the results are quite conservative from the robustness point of view, on account of the use of quadratic stability ideas. Other iterative techniques have been proposed in the literature; for instance, the vector optimization technique presented in (Kreisselmeier and Steinhauser,1983), see also (Maciejowski, 1989). This technique is very useful for dealing with a tradeoff between objectives from a Pareto optimization point of view. Standard optimization is considered in this approach, and the problem of finding local minima is often encountered. 1373
1374
T. Livet et al.
Firstly, the design technique is described, followed by a discussion of the problem of finding a control law which does not depend on calibrated airspeed. The same technique is used for dealing with uncertain delays (delays can be considered as random variables between 0 and 300 ms) and finally for dealing with uncertainties due to changes in the efficiency of the control surfaces. After applying the design technique to a rigid-body aircraft for parameter uncertainties, it will be extended to a highly flexible aircraft in order to guarantee aeroelastic stabilization in a second application.
Superscripts:
r. rigid e: elastic Consider a multivariable linear system including the delays, the actuators and the aircraft modeling as follows: {~
+ Bu Cx + Du
= Ax
(I)
(1) It can be assumed that the system has
I
n
states x e ]~n
minputsue
2 . P R E L I M I N A R I E S AND NOTATIONS Vectors:
Rm
p outputs y E RP
with u = Ky + Hzc
(2)
ui: leR eigenvector vi: right eigenvector wi: closed-loop input direction u: control input x: state vector
The system can thus be drawn as follows (see Fig. 1), where P is a feedforward which ensures zero steady state error.
Matrices:
A: open-loop state matrix B: control input matrix C: measurement matrix I: identity matrix UR: real part of the left eigenvector matrix ui: imaginary part of the left eigenvector matrix VR: real part of the fight eigenvector matrix vi: imaginary part of the right eigenvector matrix
Scalars:
s: Laplace operator ki: ith eigenvalue 13: sideslip (deg) p: roll rate (deg/sec) r: yaw rate (deg/sec) ~b:roll attitude (deg) 5p: aileron deflection (deNs) ~. redder deflection (deg/s) ki.R: ith eigenvalue associated with the rigid aircraft model ki-F: ith eigenvalue associated with the flexible aircraft model Xr: response time which delimits the R-f~ domain Gr: damping which delimits the R-f~ domain %: response time which delimits the F-fi domain Ge: damping which delimits the F-~ domain
Fig. 1. Flight Control System Note that the stability robusmess of the control law will be evaluated by the tt-analysis: gain and phase MIMO margins will be computed with the following relations: GM + > 1
(3)
1 - ao
[PM[ > 2 arcsin ( ~ )
w,h
,i.
(4)
o
0w,,)'
K is the feedback of the considered system whose transfer matrix is G. For more details about the structured singular value analysis, see (Doyle, 1982). 3. DESCRIPTION OF T H E METHOD
3.1 Principle of the proposed method Consider the example of Fig. 2:
Robust Flight Control Design k3,i
Jm
1375
[Real (•i)
Vi
-> 3/'17d
t Real (~'i)
I~'i
-> o d
where "Cd is the response time and G d the damping which delimit the trapezium. w
Re
3.3 Initialization Consider a flight case roughly centered in the flight envelope D. The feedback K 0 associated with this case will be computed by eigenstrucmre assignment: it consists of assigning a triplet {Xi, v i, wi} verifying: K o C vi = w i (5) where v i and w i are the solution of:
g4, i
Fig. 2. Mode migration example It is desired that the poles of the i th model should migrate into a trapezium. The migration will be processed step by step; at each step the required constraints on the gain are computed, For the example corresponding to Fig. 2, there will be no constraint on X2, i. For k 1, i, x3, i, )~4, i, there would be three inequality constraints: the f~rst one concerns the real part to make the mode migrate towards the trapezium. The other constraints concern the gains of feedback K, which must not take too large values. For that, the algorithm requires the definition of a quadratic criterion, an initialization of the criterion, linear inequality constraints and the considered "perfomumce trapezium".
[EA-~.iI B] [viJ w
=0
(6)
where E represents the desired decoupling.
3.4 Choice of quadratic criterion Let K0 be the initial gain previously defined. The variation AK i of the feedback will be determined at each iteration. 1
Assume that
+X
K = I~
AKi
(7)
i=l 1-1
¢:,
K=I~+
~AK i+AK 1 i=l v
3.2 Definition of performance trapezium
¢:~
It is important to define a trapezium in which all the considered modes must be found after convergence of the algorithm. This trapezium has the following characteristics:
Assuming that the vector G is composed of the elements of AK 1 at the Ith iteration:
K = K~I + AK 1 at the lth iteration (8)
...
AKml...
i~-t will also be expressed as: fT: (K~- ~ b._ y
Re
1""
KI-1 1-1 1-1 1-1 01p K02I"" K02p"" K0ml""
k
The final feedback must not take too large values. To meet this constraint, it is suggested that the Fig. 3. Definition of performance trapezium
norm of the gain --Ko1 + AK 1 should be minimized; i.e., the following criterion must be minimized:
with sin~0 = G d
J = ~ trace
+ AKI) T
+ AK 1) (9)
T. Livet et al.
1376
¢ . . I = ~1 trace (AKIT + AKI) + trace (K~-IT AKI) + constant. B u t i f A = [aij] andB = [bij]: trace (AB) ffi ~ aij bji {ij}
J=~1 GT HG + fT G
(I0)
with H the identitymatrix (np x mp).
for the eigenvalues which do not belong to the "performance trapezium" ft. If all the eigenvalues are within fi, stop the algorithm; otherwise go to step 3. Step 3: Make the eigenvalues outside fl migrate towards £2 by minimizing the quadratic criterion J under the previous constraints on AK. Step 4: The optimization of J provides a feedback variation AK. The new feedback to be considered is K = K + AK. Then go to step 2.
3.5 Definition of constraints Direct linear constraints on the ~ain coefficients. For each gain Kij equality and inequality constraints can be written, for example, K 11 = 2 and 0,5 < K12 < 1, that is, AKll = 2 - KO11 and 0,5 - KO12 < AK12 < 1 - KOl 2. The following notation can be used:
[1 0l , , '0 "'"
G=
o
I
2
,
I
5. APPLICATION:
SYNTHESIS OF A L A T E R A L ROBUST A U T O P I L O T ON LANDING
K0111
I
:::1 I G-<
Remark: note that the convergence of the algorithm depends on the characteristics of the "performance trapezium". In fact, if the required performance is too demanding, it will be impossible to find a feedback gain K which assigns all the eigenvalues to the "performance trapezium".
1 - K01221 -0,5 + K01
Remark: The usefulness of such a constraint is that it prevents any major departure from the initial gain, so as to remain in a domain where the properties obtained after linearization about K0 still hold. Constraints on the feedback ~ain in order to make l J a g , . l ~ g . I I ~ Assuming that pole X must vary by Ar + jAi, it is easy to show, see (Magni and Manouan, 1994), that there is the following linear constraint on AK: (U R + j u I ) B A K C ( v R + j v i ) = ~ + j A i (11) with
UR = Real (u) ui = Imag (u) v R = Real (v) vi = Imag (v) One can also write: trace {C (V R UR - v I UI) B AK} =At
(12)
trace {C (v R u I - v I u R) a AK} =Ai
(13)
If 0tij denote elements of the matrices C (vR u R v I ui) B or C (v R uI + vI UR) B, the constraints become: E cdj AKji = Ar or Ai.
The considered structure is that of Fig. 1, where Zc = (¢e 13c.)T x = ([3 p r ~ 8p 8r) T y = ([~ p r ~b)T u = (Spc 8rc) T where [3 is the sideslip, p the roll rate, r the yaw rate, 0 the roll attitude, 8p the aileron deflection and 5r the rudder deflection, zc is the reference input.
5.1 Multimodel synthesis Consider that the feedback gains on landing are computed with 12 flight cases and then interpolated. It is assumed that the gains do not depend on the weight. So, it would be sufficient, at a constant altitude in lateral flight, for the gains to depend on calibrated airspeed V c only. It is therefore desirable to compute a unique feedback K which will not depend on V c for the flight envelope under consideration. Table 1 Closedqoop _vole characteristics with Kg_
Flight Flight Flight Flight
case case case case
no. no. no. no.
1 2 3 4
Weight (0
Center of gravity (%)
(~)
Ve
Altitude
152 152 152 152
30 30 30 30
129 146 163 180
1500 1500 1500 1500
(n)
4. A L G O R I T H M Step 1: Initialize the feedback K by eigenstructure assignment. K 0 is the feedback associated with the flight case of the flight envelope D being considered. Define the performance trapezium ~.
Step 2: Compute
the eigenvaiues of all the flight cases belonging to the flight envelope D. Search
The nominal feedback K 0 that will be used corresponds to the flight case number 2, and is computed by assigning the modes in the following way: - Dutch roll has a response time of 9 s and its damping is 0.8, - the response time of the other 2 real modes are 5 s (for spiral mode) and 4 s (for roll mode).
Robust Flight Control Design
1377
and 13are decoupled: the mode/output decoupling is made by eigenvector assignment, whereas the feedforward P ensures the input/output decoupling at low frequencies. Note that the feedfotward P can ensure the transient input/mode decoupling if the zero steady-state error is made by an integral control design ( Livet et al., 1994a). Remark: The choice of K0 corresponding to flight case number 2 makes it possible to obtain better performance than that obtained with the other flight cases.
K0=
3.11 ~0.12
0.09 -0.50
2.47 0.53'~ 2.15 - 0 . 1 6 , /
The "performance trapezium" is defined by: T d = 9 s and G d = 0.6. The constraints to make the modes migrate are: Ar < - 0.01. (There is no constraint on Ai but, after convergence of the algorithm, all the complex modes will have an acceptable damping _>0.6.) The constraints on feedback coefficients will allow them a ± 2 % gain variation at each iteration. Note that this constraint is useful in order to induce variations of the gain AK that are small enough to ensure that the fLrst-order relationship of section 3.5 remains valid.
II
o
-02.
~-0.4 Q
,n -0.1
20
40
Fig. 4. Simulations of flight case. n°l
With these constraints, the algorithm converges after 76 iterations and the final feedback Kfina 1
25
is :
Kfinal=
(~ 3.14 - 0 . 0 0 1.94 0.65~ 0.15 0.24 2.54 - 0 . 0 4 , / "
Note that if the feedback Kfinal is unique for the four flight cases, it is necessary to compute, for each flight case, feedforward P which ensures the non-interactivity of the system and depends on the aircraft modeling. In fact, the feedforward P is computed in order to invert the system for s = 0 (where s is the Laplace parameter). Another possibility is to consider an integral control design in order to ensure a null error in the steady state from the regulated outputs (~ and 13in lateral flight) with a unique feedback K, and to compute a unique static feedforward for the 4 flight cases in order to provide transient input/mode decoupling (Livet et al., 1994a). The simulations of the two extreme flight cases, 1 and 4, are shown in Figs 4 and 5 (time units in s e c o n d s ) , where ~ is the sideslip, pc is the sideslip command, t~ the roll attitude, ~c the roll attitude command, p the roll rate, r the yaw rate, fp the aileron deflection, fpc the aileron command, fir the rudder deflection, frc the rudder command, nycg the lateral acceleration at the center of gravity and nyav the lateral acceleration 6.5 m from the nose of the aircraft.
.... 7 / .... t ,
...... i .................... . . . . . . . . . i ....................
,o
iiiiiiiiiiiiiiiii
?
-0.5
41.
Q
20
Fig. 5. Simulations of flight case n°4
40
1378
T. Livet et al.
At the system input, there is a 20 ° square input with a 3 deg/sec slope.A transient peak of [~ < 1.5 ° can be observed for each figure, which is very acceptable, considering that the iterative optimization does not take the decoupling into account (the input/output decoupling is considered only by the algorithm initialization).
be seen that the performance is downgraded, whereas the input stability margins have been improved (see Fig. 9): GM + > 3.2 PM[ > 4 0 °
A multimodel synthesis is not particularly necessary, but there are a lot of possible applications. For example, in a flight case with delay uncertainties, it is possible to use a multimodel approach by considering the same flight case with different delay values. The same reasoning can be made with control efficiency uncertainties.
.~
0
ts ' I
.........
O
F
5"
..........
i ...........
....... i...........
~
: ..................... " .........
5.2 Delay uncertainties
;o Consider a nominal flight case at 1500 ft with a 152 t weight. Since the delays can vary from 0 to 300 ms, 4 cases will be considered here: - Case no. 1: nominal flight case with no delay. - Case no. 2: nominal flight case with 100 ms delay - Case no. 3: nominal flight case with 200 ms delay - Case no. 4 : nominal flight case with 300 ms
d~lay
0.0,5 CII "ID
'
The algorithm will be initialized with the feedback associated with case no. 2: 3.11 0 . 0 9 2.47 0.53"~ K 0 = : 0.12 - 0.50 2.15 - 0 . 1 6 2 The "performance trapezium" is defined by: "17d = 9 s and G d = 0.6. The algorithm converges after 14 iterations. The final feedback is: ( S 2.57 0.13 1.6656 0.40~ K f i n a l = _ 0.10 - 0.65 2.58 - 0 . 2 1 1 The constraints are the same as previously. With K 0, the most unfavorable response time associated with the modes is about 12 s. After algorithm convergence, it is 8.8 s with a very good damping of all the modes (G > 0.67).
,o
~t .0
0
-0.1; -0.15
" i .....................
10
20
IPM[ ___35°
40
Fig. 6. Simulations obtained with K0
I~[bt ..............
i~i!~- i~iiii!~ ~iii ::"
Observe the simulations obtained with feedback K 0 associated with the nominal flight case with 100 ms delay (K 0 being computed by eigenstructure assignment with respect to the desired I~-q~ decoupling, response time and damping, see Fig. 6). Good performance is obtained with the following stability margins at inputs: GM ÷ _>2.6
30
i i!iiii|
m ~
,
i i-i: .[~-~
i iiiii
i-...-.i i!
,L~-.-~--, ____.. ~_:::,_ r~ .......... ~'"::.
:: ~i~:.d.:'.
: :i:~i.~
l
:. . . . . . .
i ~i'..'~
:. "":
" i~'"~ i i!i~:; } i~!i'--- :: iiii:'F~ ! ii"t j iii}i~2• - - ! Nii,Tt.--4..;.~4"" .....;..i.ii~= ~.;;.-3-~-.~,-; --i {-i.~ !~~"
""
which are confirmed by the plot of l/~t (I + KoG (jw)), see Fig. 7. Note that the MIMO margins are sufficient stability conditions. Now consider the same simulations as previously with the robust feedback KfinaI (see Fig. 8). It can
Fig. 7. Plot of l/(p. ( I + K 0 G ( j w ) ) )
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1379
20
5.3 Control efficiency uncertainties Now consider a cruise flight case whose characteristics are: weight: 171 - center of gravity: 30 % - calibrated airspeed: 232 kt - altitude: 1500 ft. The control efficiency can be varied by considering the following multimodel approach: Flight case no. 1 : nominal flight case with - 20 % on control efficiency values Flight case no. 2 : nominal flight case with - 10 % on control efficiency values Flight case no. 3 : nominal flight case Flight case no. 4 : nominal flight case with + 10 % on control efficiency values Flight case no. 5 : nominal flight case with + 20 % on control efficiency values The initial feedback K 0 will be associated with flight case no. 3. - 2.26 0.29 1.26 0.33"~ KO= 0.34 - 0.07 1.09 0.02,/ The "performance trapezium" is defined by: "l;d= 13 s and Gd = 0.7. The algorithm converges after 11 iterations. The final feedback is: -2.14 0.22 0.93 0 . 4 5 ~ Kfinal= 0.42 - 0.05 1.46 0 . 0 2 / The constraints are the same as previously. With K0, the most unfavorable response time associated with one of the modes is about 18 s. After algorithm convergence, Trmax = 12.9 s. -
10
¢D
20
30
40
0
t
-
-
-
-'0.4
~
~
-0.6 -0.8
-1
0
10
20
30
40
Fig. 8. Simulations obtained with Kfinal
- i . . ~ # ;~;.~ .---.~--" . . . . . .
..;-&b;;~
In section 5, a unique "performance trapezium" was considered. Section 6 will consider another algorithm application, which will use several "performance trapezia". 6. M E T H O D O L O G Y EXTENSION TO A HIGHLY FLEXIBLE AIRCRAFT
6.1 Presentation of the "flexible problem" Fig. 9. Plot of
1/(l~
( I + Kfina I G ( j w ) ) )
The previous simulations display a wadcoff between robustness and performance. In fact, the feedback Kfinal makes it possible to be robust at inputs for delay uncertainties with just acceptable performance whereas the feedback Ko is less robust with good performance.This tradeoff must be solved by the work specification. Remark: The simulations of Figs 6 and 8 are made with delays of 100 ms and provide acceptable performance. However, they are not necessarily realistic because of the delay uncertainties. More attention must be paid to the stability margin improvement which will make it possible to obtain acceptable performance, not only for a specific delay but also for delay variations.
The evolution of modern-day aircraft has made aeroservoelasticity (ASE) into a significant phenomenon. In fact, the flexible modes get nearer and nearer to the handling quality bandwidth. So, for future aircraft programmes, it will become impossible to exclude elastic modes by introducing conventional structural filters without downgrading handling qualities. It is also important to note that any method which conceives a rigid-body law without considering the flexible body can induce instability. To avoid this problem, two courses are generally possible for control-system design. One is to conceive the rigid-body and elastic control laws separately whereas the second one consists of designing a single integrated control system to achieve both rigid performance and elastic stability. This section of the paper will consider the second solution in the case of the lateral control of a highly flexible fly-by-wire aircraft. As was stated above, the eigenstructure assignment methodology
T. Livet et al.
1380
is particularly suitable in aeronauucs because use of this technique (Sobel and Shapiro, 1985; Farineau, 1989; Livet et al., 1994a), makes it easy to choose eigenvalues in order to obtain the desired damping and rise time, and to assign the right eigenvectors for the decoupling. Unfortunately, eigenstructure assignment using output feedback makes it possible to assign the rigid modes only, because of the degrees of freedom limitation. This feedback will have a direct impact on the flexible modes which can be destabilized. An interesting methodology will be to define a tradeoff between decoupling, performance, and flexible mode stabilization, such a methodology has already been successfully achieved using an LQR technique with an output feedback (Kubica and Livet, 1994). However, such a technique entails a difficulty: the choice of the LQ criterion weighting. Another method will be presented here, based on the previous iterative algorithm, adapted to the "flexible problem" (Livet et al., 1994b). The rigid-body model is given by the following state equations:
{
Xy= A X + Bu CX + Du
wl~
(18)
0) (19)
B=
c
=
(Cr Ce)
-D = Dr + De. Note that this is a simplified model without coupling terms in the matrix A, but these terms can be ignored.
6.2 Algorithm modifications for the "flexible problem" Definition of the new mieration domains. The method adapted to the flexible problem is similar to the one presented in section 3. However, it will be necessary to make the rigid and flexible modes migrate differently: For rigid modes, a trapezium will be defined, which will make it possible to obtain acceptable performance in terms of response time and damping. After convergence of the algorithm, all the modes of the rigid aircraft model will be found on or in the R-fl trapezium. - For flexible modes, the associated migration domain (F-fl domain) will be very simple (see Fig. 10) in order to make the flexible modes stable and well-damped. In fact, the most significant problems of the flexible modes are their damping, and of course their stability, when they are coupled with the rigid aircraft model. -
J~'= A t X r + BrUr At ~ l[ nr x nr, Br E 1~nr x mr with
(14)
Xr = [13 p r ~]T Ur = [Sp ~r] T The measurement vector is: yr = Cr Xr + Dr Ur Cr ~ ltPr x nr, Dr ~ RPr x mr where yr = [13p r q)lT.
(15)
Inl
The aeroelasticity model is represented by: :Ke = Ae Xe + Be Ue Ae ~ I n s x ns, Be ~ 11% x ms with
(16)
Xi-R F-~t
Xe= [ r l l ~11 •2 ~2"'" tins ~ns] T Ue = Ur, where vii is the generalized coordination of the ith elastic mode and n s the number of elastic modes considered. The outputs are defined as: Ye = Ce Xe + De Ue (17) Ce e I~Pr x ns, De E i1~ns x mr with Ye = [13e pe re (pe] T. The number of elastic modes taken into account is chosen to be 10 (so n s = 20) because it will give a realistic representation of the structural dynamics of the aircraft. These two models are coupled by connecting their respective measured outputs at the same structural points of the aircraft. So the complete model is given by:
),k-F
Re
Isin (Pc = Ge With (sin q)r Gr Fig. 10. Pole migration towards the suitable domains
Note that generally Ge -- 2 . 1 0 -2 0,65 < Gr < 0,75 z r < 15 s.
Robust Flight Control Design
1381
- with mean weight. Inidaiization of [he algorithm. There are several ways of initializing the algorithm but first the feedback K o associated with the desired flight case must be considered. K 0 is also obtained by eigenstructure assignment on the rigid aircraft model. Application of K 0 to the flight case with the rigid and flexible coupled aircraft model yields: - either an unstable system, - or a disturbed system with flexible mode damping of less than 10 -3 , which is not acceptable. The fwst algorithm convergence gives feedback K, providing good performance and a stable system. However, there may be some structural constraints which impose a gain of K equal to zero. In this case, it is possible to initialize the algorithm again, with all the feedback gains equal to those of K except for the one which must be equal to zero. The algorithm will allow a robust solution to be found by keeping the desired feedback gain to zero.
The feedback K0 associated with this case, obtained by eigenstructure assignment on the rigid aircraft model only, is: p r tp (-1.6 0.87 1.34 0.7 ) ~ p K0= - 0 . 4 7 - 0 . 1 6 2.83 0.15 5r The closed-loop time responses to a 15 ° (pc step command are shown in Fig. 11, considering the rigid model only. Very good performance is obtained, with excellent stability margins at the inputs (see Fig. 12): MG + > 3.6
IMPI->420.
Phi (dig) 20 15 10
6.3 Algorithm adapted to the "flexibleproblem" 5
S t e p 1: Choice of:
- the desired flight case, - the initial feedback K 0, - the convergence domain ( R - ~ domain or trapezium) for rigid modes, - the convergence domain (F-f~ domain) for flexible modes. S t e p 2: Computation of the desired flight case eigenvalues. Search for. the rigid modes which do not belong to the R-O domain, the flexible modes which do not belong to the F-fl domain. If all the eigenvalues are within R-O, go to step 5, otherwise go to step 3.
0 -5
0
5
10
15
20
0"21
•
°'l . . . . . . . 0
-0.1
J
..... t
. . . . . . . . . . . . . . .
....... ". . . . . . . . . :. . . . . . . . . i ........
--0.2
, . . . . . . . . . . . . . . . . . . . ..........
-O30
5
10
15
20
S t e p 3: Make the "outside eigenvalues" migrate
to R-£~ (for rigid modes) and to F-O (for flexible modes) by minimizing the quadratic criterion J under previous constraints on AK.
Fig. 11. Simulations with the rigid aircraft model
S t e p 4: The optimization of J provides a feedback
!Lil
variation AK. The new feedback to be considered is K=K+AK. S t e p 5: Compare the gain values of the feedback
K to the structural constraints. If these constraints are respected, stop the algorithm; otherwise, modify K with respect to the structural constraints. Once K has been modified, take K0 = K and go to step 1.
6.4 Application "~0 ,~I
Consider the following lateral flight case: - calibrated airspeed: 200 kt - mach: 0.5 - center of gravity: 32 %
i ,0.+
i i~1l
i 101 llllll
|~
101
Fig. 12. Singular values with the rigid aircraft model
1382
T. Livet et al.
By introducing the flexible model with the same feedback KO, an unstable system is obtained (see Fig. 13). The closed-loop modes are listed in Table 2.
Table 2 Closed-loop pole characteristics with I ~ Closed loop poles
Mode
Damping
Dutch roll - 0.65 + 0.59i 0.74 Spiral mode - 0.75 Roll mode 1.5 Flexible mode n°l + 0.20 + 22.16i unstable Flexible mode n°2 0.036 + 18.8i 1.9. 10-3 Flexible mode n°3 1.2. 10-2 0.18 + 14.63i Flexible mode n°4 - 0.57 + 16.92i 3.36 . 10-2 Flexible mode n°5 - 0.81 + 12.39i 6.55.10-2 Flexible mode n°6 1.07 + 16.9i 6 . 3 2 . 1 0 -2 Flexible mode n°'/ 1.12 + 26.12i 4 . 2 9 . 1 0 -2 Flexible mode n°8 - 2.68 + 25.15i 0.1 Flexible mode n°9 - 2.77 + 35.98i 7.7. 10-2 Flexible mode n°10 - 3.22 + 32.14i 9.98. 10-2 1
-
I
-
The R-l') domain will have the following characteristics: G r = 0.5 Tr= 12s. Those of the F-Q domain will be: G r = 0.015. There is, of course, the constraint that the system must be stable after algorithm convergence. The algorithm converges after 96 iterations and provides: (-0.56211.10723.09470.9171) KI= 0.5266 - 0 . 1 5 1 5 2.73380.0619 Note that K gives the following stability margins: MG + _> 1.55
IMPI-> 2050
-
(kl~//ip kp/~ip kr/Sp kq~//ip "~ Consider that K 1 = ~,kl3//ir kp//ir kr/8r ktp//ir j
-
-
Pt~ (dq) 25 20
The same characteristics for the F-~ domain can be retained, but with the constraint that all the rigid modes should have a damping value at least equal to 0.7.
15 10 5 00
The constraints imposed by the structure of the aircraft me: kl3/8p = 0 However, remembering that kl3//ip = 0 in KI, the performance of the aircraft would be affected. So, it would be interesting to initialize the algorithm again by: ( 0 1.11 3.1 0 . 9 2 ) K0= -0.53 - 0 . 1 5 2.73 0.06
5
10
15
20
With such specifications, the algorithm converges after 11 iterations and provides the following feedback K2: ( 0 1.04 2.22 0.9 ) K2= - 0 . 5 6 - 0 . 2 4 3.65 0.01 ,
a ~ (dog) O.Ir
with excellent stability margins: MG ÷ _>2.52
0.4 0.=
Note that these margins can be increased by augmenting the value of kr/& and decreasing that of
0 -0.2
--0.4
0
5
10
15
20
Fig. 13. Simulations with the rigid and flexible aircraft model, and K 0 Now the iterative algorithm can be initialized with K 0.
kr/Sp. One can take kr//~r = 4.5 and kr//i p = 1.8 in K2: this feedback will initialize the algorithm again. Finally, ( 0 1.02 1.76 0 . 8 9 ) Kfinal= - 0.55 - 0.24 4.41 0.01 which provides excellent performance (see Fig. 14) and excellent stability margins (see Fig. 15): MG + _>3.07
4oo
Robust Hight Control Design
1383
Table 3 Closed-loon oole characteristics with Kfinal Closed-loop
Mode
Damping
poles
30 20 10 0 -10
.........
%.,
iiiiiiiiiiiiiiill
0
,
i
S
10
am
15
20
Dutch roll Spiral mode Roll mode Flexible mode n°l Flexible mode n°2 Flexible mode n°3 Flexible mode n°4 Flexible mode n°5 Flexible mode n°6 Flexible mode n°7 Flexible mode n°8 Flexible mode n°9 Flexible mode n°10
1.04i - 0.58 - 1.4 - 0.69 + 22.68i - 0.41 _+ 18.7i - 0.37 + 14.5i - 0.44 + 17.4i - 0.33 + 16.56i - 1.31 + 12.05i - 0.96 + 25.2i - 2.56 + 25.42i - 2.53 + 35.67i - 3.22 + 32.13i - 1.32 +
0.78 1 1
3 . 10 -2 2 . 2 . 1 0 -2 2.53. 10-2 2.52. 10 -2 1.98. 10-2 0.1 3.8. 10 -2 0.1 7 . 10 -2 9.98. 10 -2
(~)
0.4
7. C O N C L U S I O N The iterative algorithm presented in this paper has multiple applications. For example, the fact that the feedback gains are usually tabulated with calibrated airspeed is not a constraint; several gains can depend on two or three parameters. It is also possible to tabulate the gain according to a unique parameter, whereas the feedback is computed in order to be robust with respect to the other parameter variations. In a more general way, the methodology presented makes it possible to obtain a feedback that is robust with respect to parametric variations of the model.
0.,2
0
-0.2
0
5
10
15
20
Fig. 14. Simulations with the complete aircraft model and Kf.ma1
I
-?++++++PI+++P???+++P++P++++++ + .... +..÷.:.: ~÷.,:.....+..+. +++++ii++
+.....,..+.+ ÷+~....÷ ..÷.÷.~:+....+...:.+.+.+:+,
~++i~
++++++]++ +++++++i+
.... .+. -. +. . / 4 .~.ll "'i%--~-~(~+~.--a
: I' . '.~ .+. i~' " ' - - - + .. 4. .-.i. .. (. +. . i ~ -
•~$~J.i..i..[i.i+;L..!
+ ++++fii 4" 1".'I't'll., "R
.:.i~iJ!i,[....i...l.:.+]!!!
!!'!~iili~i
!!i!!i!!l
;;+ + +++++++"'"++"++'++'t+ i++iii::+ + +:+:+++++,.!!~++:,++ + + +++'+" + ++'"++:
~
........
i
le"
I IIIiili
I
I ......
~
,e'
+f!ii!i+iii!ii w'
10'
i
Another important application of this technique is the application of "the flexible problem" by using several "performance domains". In fact, in this paper a new methodology for a flexible aircraft conlrol system was developed, in order to avoid the inlroduction of structural filters. This new approach optimizes the eigenvalue assignment within a chosen domain in order not to excite the flexible modes. No change in the law structure was envisaged; the problem was solved by optimizing the feedback gains (i.e., the eigenvalue assignment). Note that no decoupling was considered during the optimization phase because of the degrees-of-freedom limitation. However, by initializing the algorithm with a feedback obtained by eigenstructure assignment (making the desired decoupling) and by limiting the feedback gain variations at each step, the feedback which was finally obtained, provides acceptable performance. For systems which must have perfect decoupling, it could be envisaged to "create" new degrees of freedom by adding sensors in order to introduce the decoupling during the optimization.
Fig. 15. Singular values with the complete aircraft model and Kfinal REFERENCES The closed-loop modes for Kfina I are given in table 3.
Doyle J.G., (1982). "Analysis of Feedback Systems With Structured Uncertainties".
1384
T. Livetet IEEE Proceedings, Vol. 129, pt. D, No. 6, pp 242 - 249.
Doyle J.G., Glover K., Khargonekar P. and Francis B., (1989). "State Space Solutions to standard H2 and H** Control Problems". IEEE Transactions on Automatic Control, Vol. 34, No. 8, pp 831 - 847. Farineau J., (1989). "Lateral Electric Flight Control Laws of a Civil Aircraft Based upon eigenstructure Assignment Technique". AIAA Guidance, Navigation and Control Conference, Boston, Massachusset, USA.
Geromel J.C., Garcia G., and Bernussou J., (1993). "H2 robust control design with pole placement". In Proc. 12tla IFAC World Congress, Sydney, Australia, II:283. Kreisselmeier G. and Steinhauser R., (1983). "Application of vector performance optimization to a robust control loop design for a fighter aircraft". Int. J. Control, 37:251-284. Kubica F. and Livet T., (1994). "Flight Control Law Synthesis for a Flexible Aircraft". AIAA Guidance, Navigation and Control Conference, Scottsdale, Arizona, USA. Livet T., Kubica F., Magni J.F. and Antonel L., (1994a). "Non-Interactive Control by Eigenstructure Assignment and Feedforward". In Proc., AIAA Guidance, Navigation and Control Conference, Scottsdale, Arizona, USA.
al.
Livet T., Magni J.F., Fabre P. and Kubica F., (1994b). "Robust Flight Control Design for a highly Flexible Aircraft by Pole Migration". In Proc., IFAC World Congress, Palo Alto, CA, USA. Maciejowski J.M., (1989). "Multivariable Feedback Design". Electronic Systems Engineering Series. Addisson-Wesley Publishing Company, Wokingham, England. Magni J.F. and Manouan A., (1994). "Robust Flight Control Design by Eigenstructure Assignment". Submitted for publication, 1994. Mudge S.K. and Patton R.J., (1988). "Analysis of the techniques of robust eigenstructure assignment with application to aircraft control". IEEE Proceedings, Part D, Control Theory and Applications, 135(4):275-281. Sobel
K.M. and Shapiro E.Y., (1985). "Eigenstructure Assignment for design of multimode flight control systems." IEEE Control Systems Magazine, pp 9-I5.
Xiao-ming Xu, Zhi-ming Wu, and Zhong-jun Zhanh., (1986). "A design method of generalized root-loci for mimo systems." In Proc. of the 25 th Conf. on Decision and Control, Athens, Greece, pp 713-717.
Pergamon 0967-0661 (95)00140-9
ROBUST FLIGHT CONTROL DESIGN WITH RESPECT TO DELAYS, CONTROL EFFICIENCIES AND FLEXIBLE MODES T. Livet*, F. Kubica* and J.F. Magni** *AEROSPATIALE, Aircraft Division, 316 route de Bayonne, 31060 Toulouse, France **ONERA-CERT, 2 avenue Edouard Belin, 31400 Toulouse, France
(ReceivedJuly 1994; infinal form June 1995)
Abstract: This paper presents an iterative technique which makes it possible to design a
robust flight control system. The methodology described in this paper has multiple applications. Two different applications are considered: the first consists of computing a robust feedback with respect to parameter uncertainties by considering a bank of models, while the second provides a robust flight-control design for a highly flexible aircraft with significant cross-coupling between flight mechanics and structural dynamics modes. The proposed method will solve these problems via an iterative quadratic optimization under linear inequality constraints. Finally, the iterative algorithm is applied to a civil aircraft control system in lateral flight. Key Words: Flight Control; Parameter Robustness; Stability Robustness; Flexible Aircraft;
Eigenstructure Assignment; Control Algorithms.
1. I N T R O D U C T I O N In practice, aircraft control laws are designed by eigenstructure assignment. This assignment is performed in such a way that natural decoupling of aircraft is preserved or even reinforced, see (Mudge and Patton, 1988; Sobel and Shapiro, 1985). The fact that natural behavior is not modified leads to good robustness. However, in the design procedure, robustness ideas are not explicity considered. This paper shows that robustness can be improved by using a very simple multi-model approach (but one might also consider more sophisticated techniques such as that based on the relationship between eigenvectors and eigenvalue variation induced by model variations, see (Magni and Manouan, 1994). The technique proposed is some kind of parameter optimization technique, without guarantee of convergence, but which has been shown to be very efficient in the field of interest here. It utilizes a bank of models, and simultaneously shifts the poles of all these models inside a given trapezium. Basically, the first-order approximation proposed in (Xu et a l , 1986) is used, but pole shifting is performed in such a way that maximum freedom is offered to the motion of the poles in order to avoid singularities. At each step of the algorithm, the pole is shifted toward the required trapezium by the solution of a quadratic problem under linear constraints. The area for the motion of the pole is a sector of the complex plane that is as large as possible.
Flight control law design for aircraft is a very attractive problem, considering the robustness properties which are required. In this field of interest, robustness appears to be a specific problem which has not received much attention in the literature because generally feedback gains must be constant matrices (no dynamics except dynamics with physical significance as filters); moreover, some gains might be fixed in advance. Using the techniques existing in the literature, which offer a guarantee of global convergence or directly give a control law, it is impossible to satisfy these kinds of design specifications. The translation Of practical specifications in terms of convex problems may lead to the solution to the problem, see (Geromel et al, 1993; D o y l e , 1989), but for constant output feedback, the optimization problem is not convex, so there is no guarantee of global convergence. Moreover the results are quite conservative from the robustness point of view, on account of the use of quadratic stability ideas. Other iterative techniques have been proposed in the literature; for instance, the vector optimization technique presented in (Kreisselmeier and Steinhauser,1983), see also (Maciejowski, 1989). This technique is very useful for dealing with a tradeoff between objectives from a Pareto optimization point of view. Standard optimization is considered in this approach, and the problem of finding local minima is often encountered. 1373
1374
T. Livet et al.
Firstly, the design technique is described, followed by a discussion of the problem of finding a control law which does not depend on calibrated airspeed. The same technique is used for dealing with uncertain delays (delays can be considered as random variables between 0 and 300 ms) and finally for dealing with uncertainties due to changes in the efficiency of the control surfaces. After applying the design technique to a rigid-body aircraft for parameter uncertainties, it will be extended to a highly flexible aircraft in order to guarantee aeroelastic stabilization in a second application.
Superscripts:
r. rigid e: elastic Consider a multivariable linear system including the delays, the actuators and the aircraft modeling as follows: {~
+ Bu Cx + Du
= Ax
(I)
(1) It can be assumed that the system has
I
n
states x e ]~n
minputsue
2 . P R E L I M I N A R I E S AND NOTATIONS Vectors:
Rm
p outputs y E RP
with u = Ky + Hzc
(2)
ui: leR eigenvector vi: right eigenvector wi: closed-loop input direction u: control input x: state vector
The system can thus be drawn as follows (see Fig. 1), where P is a feedforward which ensures zero steady state error.
Matrices:
A: open-loop state matrix B: control input matrix C: measurement matrix I: identity matrix UR: real part of the left eigenvector matrix ui: imaginary part of the left eigenvector matrix VR: real part of the fight eigenvector matrix vi: imaginary part of the right eigenvector matrix
Scalars:
s: Laplace operator ki: ith eigenvalue 13: sideslip (deg) p: roll rate (deg/sec) r: yaw rate (deg/sec) ~b:roll attitude (deg) 5p: aileron deflection (deNs) ~. redder deflection (deg/s) ki.R: ith eigenvalue associated with the rigid aircraft model ki-F: ith eigenvalue associated with the flexible aircraft model Xr: response time which delimits the R-f~ domain Gr: damping which delimits the R-f~ domain %: response time which delimits the F-fi domain Ge: damping which delimits the F-~ domain
Fig. 1. Flight Control System Note that the stability robusmess of the control law will be evaluated by the tt-analysis: gain and phase MIMO margins will be computed with the following relations: GM + > 1
(3)
1 - ao
[PM[ > 2 arcsin ( ~ )
w,h
,i.
(4)
o
0w,,)'
K is the feedback of the considered system whose transfer matrix is G. For more details about the structured singular value analysis, see (Doyle, 1982). 3. DESCRIPTION OF T H E METHOD
3.1 Principle of the proposed method Consider the example of Fig. 2:
Robust Flight Control Design k3,i
Jm
1375
[Real (•i)
Vi
-> 3/'17d
t Real (~'i)
I~'i
-> o d
where "Cd is the response time and G d the damping which delimit the trapezium. w
Re
3.3 Initialization Consider a flight case roughly centered in the flight envelope D. The feedback K 0 associated with this case will be computed by eigenstrucmre assignment: it consists of assigning a triplet {Xi, v i, wi} verifying: K o C vi = w i (5) where v i and w i are the solution of:
g4, i
Fig. 2. Mode migration example It is desired that the poles of the i th model should migrate into a trapezium. The migration will be processed step by step; at each step the required constraints on the gain are computed, For the example corresponding to Fig. 2, there will be no constraint on X2, i. For k 1, i, x3, i, )~4, i, there would be three inequality constraints: the f~rst one concerns the real part to make the mode migrate towards the trapezium. The other constraints concern the gains of feedback K, which must not take too large values. For that, the algorithm requires the definition of a quadratic criterion, an initialization of the criterion, linear inequality constraints and the considered "perfomumce trapezium".
[EA-~.iI B] [viJ w
=0
(6)
where E represents the desired decoupling.
3.4 Choice of quadratic criterion Let K0 be the initial gain previously defined. The variation AK i of the feedback will be determined at each iteration. 1
Assume that
+X
K = I~
AKi
(7)
i=l 1-1
¢:,
K=I~+
~AK i+AK 1 i=l v
3.2 Definition of performance trapezium
¢:~
It is important to define a trapezium in which all the considered modes must be found after convergence of the algorithm. This trapezium has the following characteristics:
Assuming that the vector G is composed of the elements of AK 1 at the Ith iteration:
K = K~I + AK 1 at the lth iteration (8)
...
AKml...
i~-t will also be expressed as: fT: (K~- ~ b._ y
Re
1""
KI-1 1-1 1-1 1-1 01p K02I"" K02p"" K0ml""
k
The final feedback must not take too large values. To meet this constraint, it is suggested that the Fig. 3. Definition of performance trapezium
norm of the gain --Ko1 + AK 1 should be minimized; i.e., the following criterion must be minimized:
with sin~0 = G d
J = ~ trace
+ AKI) T
+ AK 1) (9)
T. Livet et al.
1376
¢ . . I = ~1 trace (AKIT + AKI) + trace (K~-IT AKI) + constant. B u t i f A = [aij] andB = [bij]: trace (AB) ffi ~ aij bji {ij}
J=~1 GT HG + fT G
(I0)
with H the identitymatrix (np x mp).
for the eigenvalues which do not belong to the "performance trapezium" ft. If all the eigenvalues are within fi, stop the algorithm; otherwise go to step 3. Step 3: Make the eigenvalues outside fl migrate towards £2 by minimizing the quadratic criterion J under the previous constraints on AK. Step 4: The optimization of J provides a feedback variation AK. The new feedback to be considered is K = K + AK. Then go to step 2.
3.5 Definition of constraints Direct linear constraints on the ~ain coefficients. For each gain Kij equality and inequality constraints can be written, for example, K 11 = 2 and 0,5 < K12 < 1, that is, AKll = 2 - KO11 and 0,5 - KO12 < AK12 < 1 - KOl 2. The following notation can be used:
[1 0l , , '0 "'"
G=
o
I
2
,
I
5. APPLICATION:
SYNTHESIS OF A L A T E R A L ROBUST A U T O P I L O T ON LANDING
K0111
I
:::1 I G-<
Remark: note that the convergence of the algorithm depends on the characteristics of the "performance trapezium". In fact, if the required performance is too demanding, it will be impossible to find a feedback gain K which assigns all the eigenvalues to the "performance trapezium".
1 - K01221 -0,5 + K01
Remark: The usefulness of such a constraint is that it prevents any major departure from the initial gain, so as to remain in a domain where the properties obtained after linearization about K0 still hold. Constraints on the feedback ~ain in order to make l J a g , . l ~ g . I I ~ Assuming that pole X must vary by Ar + jAi, it is easy to show, see (Magni and Manouan, 1994), that there is the following linear constraint on AK: (U R + j u I ) B A K C ( v R + j v i ) = ~ + j A i (11) with
UR = Real (u) ui = Imag (u) v R = Real (v) vi = Imag (v) One can also write: trace {C (V R UR - v I UI) B AK} =At
(12)
trace {C (v R u I - v I u R) a AK} =Ai
(13)
If 0tij denote elements of the matrices C (vR u R v I ui) B or C (v R uI + vI UR) B, the constraints become: E cdj AKji = Ar or Ai.
The considered structure is that of Fig. 1, where Zc = (¢e 13c.)T x = ([3 p r ~ 8p 8r) T y = ([~ p r ~b)T u = (Spc 8rc) T where [3 is the sideslip, p the roll rate, r the yaw rate, 0 the roll attitude, 8p the aileron deflection and 5r the rudder deflection, zc is the reference input.
5.1 Multimodel synthesis Consider that the feedback gains on landing are computed with 12 flight cases and then interpolated. It is assumed that the gains do not depend on the weight. So, it would be sufficient, at a constant altitude in lateral flight, for the gains to depend on calibrated airspeed V c only. It is therefore desirable to compute a unique feedback K which will not depend on V c for the flight envelope under consideration. Table 1 Closedqoop _vole characteristics with Kg_
Flight Flight Flight Flight
case case case case
no. no. no. no.
1 2 3 4
Weight (0
Center of gravity (%)
(~)
Ve
Altitude
152 152 152 152
30 30 30 30
129 146 163 180
1500 1500 1500 1500
(n)
4. A L G O R I T H M Step 1: Initialize the feedback K by eigenstructure assignment. K 0 is the feedback associated with the flight case of the flight envelope D being considered. Define the performance trapezium ~.
Step 2: Compute
the eigenvaiues of all the flight cases belonging to the flight envelope D. Search
The nominal feedback K 0 that will be used corresponds to the flight case number 2, and is computed by assigning the modes in the following way: - Dutch roll has a response time of 9 s and its damping is 0.8, - the response time of the other 2 real modes are 5 s (for spiral mode) and 4 s (for roll mode).
Robust Flight Control Design
1377
and 13are decoupled: the mode/output decoupling is made by eigenvector assignment, whereas the feedforward P ensures the input/output decoupling at low frequencies. Note that the feedfotward P can ensure the transient input/mode decoupling if the zero steady-state error is made by an integral control design ( Livet et al., 1994a). Remark: The choice of K0 corresponding to flight case number 2 makes it possible to obtain better performance than that obtained with the other flight cases.
K0=
3.11 ~0.12
0.09 -0.50
2.47 0.53'~ 2.15 - 0 . 1 6 , /
The "performance trapezium" is defined by: T d = 9 s and G d = 0.6. The constraints to make the modes migrate are: Ar < - 0.01. (There is no constraint on Ai but, after convergence of the algorithm, all the complex modes will have an acceptable damping _>0.6.) The constraints on feedback coefficients will allow them a ± 2 % gain variation at each iteration. Note that this constraint is useful in order to induce variations of the gain AK that are small enough to ensure that the fLrst-order relationship of section 3.5 remains valid.
II
o
-02.
~-0.4 Q
,n -0.1
20
40
Fig. 4. Simulations of flight case. n°l
With these constraints, the algorithm converges after 76 iterations and the final feedback Kfina 1
25
is :
Kfinal=
(~ 3.14 - 0 . 0 0 1.94 0.65~ 0.15 0.24 2.54 - 0 . 0 4 , / "
Note that if the feedback Kfinal is unique for the four flight cases, it is necessary to compute, for each flight case, feedforward P which ensures the non-interactivity of the system and depends on the aircraft modeling. In fact, the feedforward P is computed in order to invert the system for s = 0 (where s is the Laplace parameter). Another possibility is to consider an integral control design in order to ensure a null error in the steady state from the regulated outputs (~ and 13in lateral flight) with a unique feedback K, and to compute a unique static feedforward for the 4 flight cases in order to provide transient input/mode decoupling (Livet et al., 1994a). The simulations of the two extreme flight cases, 1 and 4, are shown in Figs 4 and 5 (time units in s e c o n d s ) , where ~ is the sideslip, pc is the sideslip command, t~ the roll attitude, ~c the roll attitude command, p the roll rate, r the yaw rate, fp the aileron deflection, fpc the aileron command, fir the rudder deflection, frc the rudder command, nycg the lateral acceleration at the center of gravity and nyav the lateral acceleration 6.5 m from the nose of the aircraft.
.... 7 / .... t ,
...... i .................... . . . . . . . . . i ....................
,o
iiiiiiiiiiiiiiiii
?
-0.5
41.
Q
20
Fig. 5. Simulations of flight case n°4
40
1378
T. Livet et al.
At the system input, there is a 20 ° square input with a 3 deg/sec slope.A transient peak of [~ < 1.5 ° can be observed for each figure, which is very acceptable, considering that the iterative optimization does not take the decoupling into account (the input/output decoupling is considered only by the algorithm initialization).
be seen that the performance is downgraded, whereas the input stability margins have been improved (see Fig. 9): GM + > 3.2 PM[ > 4 0 °
A multimodel synthesis is not particularly necessary, but there are a lot of possible applications. For example, in a flight case with delay uncertainties, it is possible to use a multimodel approach by considering the same flight case with different delay values. The same reasoning can be made with control efficiency uncertainties.
.~
0
ts ' I
.........
O
F
5"
..........
i ...........
....... i...........
~
: ..................... " .........
5.2 Delay uncertainties
;o Consider a nominal flight case at 1500 ft with a 152 t weight. Since the delays can vary from 0 to 300 ms, 4 cases will be considered here: - Case no. 1: nominal flight case with no delay. - Case no. 2: nominal flight case with 100 ms delay - Case no. 3: nominal flight case with 200 ms delay - Case no. 4 : nominal flight case with 300 ms
d~lay
0.0,5 CII "ID
'
The algorithm will be initialized with the feedback associated with case no. 2: 3.11 0 . 0 9 2.47 0.53"~ K 0 = : 0.12 - 0.50 2.15 - 0 . 1 6 2 The "performance trapezium" is defined by: "17d = 9 s and G d = 0.6. The algorithm converges after 14 iterations. The final feedback is: ( S 2.57 0.13 1.6656 0.40~ K f i n a l = _ 0.10 - 0.65 2.58 - 0 . 2 1 1 The constraints are the same as previously. With K 0, the most unfavorable response time associated with the modes is about 12 s. After algorithm convergence, it is 8.8 s with a very good damping of all the modes (G > 0.67).
,o
~t .0
0
-0.1; -0.15
" i .....................
10
20
IPM[ ___35°
40
Fig. 6. Simulations obtained with K0
I~[bt ..............
i~i!~- i~iiii!~ ~iii ::"
Observe the simulations obtained with feedback K 0 associated with the nominal flight case with 100 ms delay (K 0 being computed by eigenstructure assignment with respect to the desired I~-q~ decoupling, response time and damping, see Fig. 6). Good performance is obtained with the following stability margins at inputs: GM ÷ _>2.6
30
i i!iiii|
m ~
,
i i-i: .[~-~
i iiiii
i-...-.i i!
,L~-.-~--, ____.. ~_:::,_ r~ .......... ~'"::.
:: ~i~:.d.:'.
: :i:~i.~
l
:. . . . . . .
i ~i'..'~
:. "":
" i~'"~ i i!i~:; } i~!i'--- :: iiii:'F~ ! ii"t j iii}i~2• - - ! Nii,Tt.--4..;.~4"" .....;..i.ii~= ~.;;.-3-~-.~,-; --i {-i.~ !~~"
""
which are confirmed by the plot of l/~t (I + KoG (jw)), see Fig. 7. Note that the MIMO margins are sufficient stability conditions. Now consider the same simulations as previously with the robust feedback KfinaI (see Fig. 8). It can
Fig. 7. Plot of l/(p. ( I + K 0 G ( j w ) ) )
Robust Flight Control Design
1379
20
5.3 Control efficiency uncertainties Now consider a cruise flight case whose characteristics are: weight: 171 - center of gravity: 30 % - calibrated airspeed: 232 kt - altitude: 1500 ft. The control efficiency can be varied by considering the following multimodel approach: Flight case no. 1 : nominal flight case with - 20 % on control efficiency values Flight case no. 2 : nominal flight case with - 10 % on control efficiency values Flight case no. 3 : nominal flight case Flight case no. 4 : nominal flight case with + 10 % on control efficiency values Flight case no. 5 : nominal flight case with + 20 % on control efficiency values The initial feedback K 0 will be associated with flight case no. 3. - 2.26 0.29 1.26 0.33"~ KO= 0.34 - 0.07 1.09 0.02,/ The "performance trapezium" is defined by: "l;d= 13 s and Gd = 0.7. The algorithm converges after 11 iterations. The final feedback is: -2.14 0.22 0.93 0 . 4 5 ~ Kfinal= 0.42 - 0.05 1.46 0 . 0 2 / The constraints are the same as previously. With K0, the most unfavorable response time associated with one of the modes is about 18 s. After algorithm convergence, Trmax = 12.9 s. -
10
¢D
20
30
40
0
t
-
-
-
-'0.4
~
~
-0.6 -0.8
-1
0
10
20
30
40
Fig. 8. Simulations obtained with Kfinal
- i . . ~ # ;~;.~ .---.~--" . . . . . .
..;-&b;;~
In section 5, a unique "performance trapezium" was considered. Section 6 will consider another algorithm application, which will use several "performance trapezia". 6. M E T H O D O L O G Y EXTENSION TO A HIGHLY FLEXIBLE AIRCRAFT
6.1 Presentation of the "flexible problem" Fig. 9. Plot of
1/(l~
( I + Kfina I G ( j w ) ) )
The previous simulations display a wadcoff between robustness and performance. In fact, the feedback Kfinal makes it possible to be robust at inputs for delay uncertainties with just acceptable performance whereas the feedback Ko is less robust with good performance.This tradeoff must be solved by the work specification. Remark: The simulations of Figs 6 and 8 are made with delays of 100 ms and provide acceptable performance. However, they are not necessarily realistic because of the delay uncertainties. More attention must be paid to the stability margin improvement which will make it possible to obtain acceptable performance, not only for a specific delay but also for delay variations.
The evolution of modern-day aircraft has made aeroservoelasticity (ASE) into a significant phenomenon. In fact, the flexible modes get nearer and nearer to the handling quality bandwidth. So, for future aircraft programmes, it will become impossible to exclude elastic modes by introducing conventional structural filters without downgrading handling qualities. It is also important to note that any method which conceives a rigid-body law without considering the flexible body can induce instability. To avoid this problem, two courses are generally possible for control-system design. One is to conceive the rigid-body and elastic control laws separately whereas the second one consists of designing a single integrated control system to achieve both rigid performance and elastic stability. This section of the paper will consider the second solution in the case of the lateral control of a highly flexible fly-by-wire aircraft. As was stated above, the eigenstructure assignment methodology
T. Livet et al.
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is particularly suitable in aeronauucs because use of this technique (Sobel and Shapiro, 1985; Farineau, 1989; Livet et al., 1994a), makes it easy to choose eigenvalues in order to obtain the desired damping and rise time, and to assign the right eigenvectors for the decoupling. Unfortunately, eigenstructure assignment using output feedback makes it possible to assign the rigid modes only, because of the degrees of freedom limitation. This feedback will have a direct impact on the flexible modes which can be destabilized. An interesting methodology will be to define a tradeoff between decoupling, performance, and flexible mode stabilization, such a methodology has already been successfully achieved using an LQR technique with an output feedback (Kubica and Livet, 1994). However, such a technique entails a difficulty: the choice of the LQ criterion weighting. Another method will be presented here, based on the previous iterative algorithm, adapted to the "flexible problem" (Livet et al., 1994b). The rigid-body model is given by the following state equations:
{
Xy= A X + Bu CX + Du
wl~
(18)
0) (19)
B=
c
=
(Cr Ce)
-D = Dr + De. Note that this is a simplified model without coupling terms in the matrix A, but these terms can be ignored.
6.2 Algorithm modifications for the "flexible problem" Definition of the new mieration domains. The method adapted to the flexible problem is similar to the one presented in section 3. However, it will be necessary to make the rigid and flexible modes migrate differently: For rigid modes, a trapezium will be defined, which will make it possible to obtain acceptable performance in terms of response time and damping. After convergence of the algorithm, all the modes of the rigid aircraft model will be found on or in the R-fl trapezium. - For flexible modes, the associated migration domain (F-fl domain) will be very simple (see Fig. 10) in order to make the flexible modes stable and well-damped. In fact, the most significant problems of the flexible modes are their damping, and of course their stability, when they are coupled with the rigid aircraft model. -
J~'= A t X r + BrUr At ~ l[ nr x nr, Br E 1~nr x mr with
(14)
Xr = [13 p r ~]T Ur = [Sp ~r] T The measurement vector is: yr = Cr Xr + Dr Ur Cr ~ ltPr x nr, Dr ~ RPr x mr where yr = [13p r q)lT.
(15)
Inl
The aeroelasticity model is represented by: :Ke = Ae Xe + Be Ue Ae ~ I n s x ns, Be ~ 11% x ms with
(16)
Xi-R F-~t
Xe= [ r l l ~11 •2 ~2"'" tins ~ns] T Ue = Ur, where vii is the generalized coordination of the ith elastic mode and n s the number of elastic modes considered. The outputs are defined as: Ye = Ce Xe + De Ue (17) Ce e I~Pr x ns, De E i1~ns x mr with Ye = [13e pe re (pe] T. The number of elastic modes taken into account is chosen to be 10 (so n s = 20) because it will give a realistic representation of the structural dynamics of the aircraft. These two models are coupled by connecting their respective measured outputs at the same structural points of the aircraft. So the complete model is given by:
),k-F
Re
Isin (Pc = Ge With (sin q)r Gr Fig. 10. Pole migration towards the suitable domains
Note that generally Ge -- 2 . 1 0 -2 0,65 < Gr < 0,75 z r < 15 s.
Robust Flight Control Design
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- with mean weight. Inidaiization of [he algorithm. There are several ways of initializing the algorithm but first the feedback K o associated with the desired flight case must be considered. K 0 is also obtained by eigenstructure assignment on the rigid aircraft model. Application of K 0 to the flight case with the rigid and flexible coupled aircraft model yields: - either an unstable system, - or a disturbed system with flexible mode damping of less than 10 -3 , which is not acceptable. The fwst algorithm convergence gives feedback K, providing good performance and a stable system. However, there may be some structural constraints which impose a gain of K equal to zero. In this case, it is possible to initialize the algorithm again, with all the feedback gains equal to those of K except for the one which must be equal to zero. The algorithm will allow a robust solution to be found by keeping the desired feedback gain to zero.
The feedback K0 associated with this case, obtained by eigenstructure assignment on the rigid aircraft model only, is: p r tp (-1.6 0.87 1.34 0.7 ) ~ p K0= - 0 . 4 7 - 0 . 1 6 2.83 0.15 5r The closed-loop time responses to a 15 ° (pc step command are shown in Fig. 11, considering the rigid model only. Very good performance is obtained, with excellent stability margins at the inputs (see Fig. 12): MG + > 3.6
IMPI->420.
Phi (dig) 20 15 10
6.3 Algorithm adapted to the "flexibleproblem" 5
S t e p 1: Choice of:
- the desired flight case, - the initial feedback K 0, - the convergence domain ( R - ~ domain or trapezium) for rigid modes, - the convergence domain (F-f~ domain) for flexible modes. S t e p 2: Computation of the desired flight case eigenvalues. Search for. the rigid modes which do not belong to the R-O domain, the flexible modes which do not belong to the F-fl domain. If all the eigenvalues are within R-O, go to step 5, otherwise go to step 3.
0 -5
0
5
10
15
20
0"21
•
°'l . . . . . . . 0
-0.1
J
..... t
. . . . . . . . . . . . . . .
....... ". . . . . . . . . :. . . . . . . . . i ........
--0.2
, . . . . . . . . . . . . . . . . . . . ..........
-O30
5
10
15
20
S t e p 3: Make the "outside eigenvalues" migrate
to R-£~ (for rigid modes) and to F-O (for flexible modes) by minimizing the quadratic criterion J under previous constraints on AK.
Fig. 11. Simulations with the rigid aircraft model
S t e p 4: The optimization of J provides a feedback
!Lil
variation AK. The new feedback to be considered is K=K+AK. S t e p 5: Compare the gain values of the feedback
K to the structural constraints. If these constraints are respected, stop the algorithm; otherwise, modify K with respect to the structural constraints. Once K has been modified, take K0 = K and go to step 1.
6.4 Application "~0 ,~I
Consider the following lateral flight case: - calibrated airspeed: 200 kt - mach: 0.5 - center of gravity: 32 %
i ,0.+
i i~1l
i 101 llllll
|~
101
Fig. 12. Singular values with the rigid aircraft model
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T. Livet et al.
By introducing the flexible model with the same feedback KO, an unstable system is obtained (see Fig. 13). The closed-loop modes are listed in Table 2.
Table 2 Closed-loop pole characteristics with I ~ Closed loop poles
Mode
Damping
Dutch roll - 0.65 + 0.59i 0.74 Spiral mode - 0.75 Roll mode 1.5 Flexible mode n°l + 0.20 + 22.16i unstable Flexible mode n°2 0.036 + 18.8i 1.9. 10-3 Flexible mode n°3 1.2. 10-2 0.18 + 14.63i Flexible mode n°4 - 0.57 + 16.92i 3.36 . 10-2 Flexible mode n°5 - 0.81 + 12.39i 6.55.10-2 Flexible mode n°6 1.07 + 16.9i 6 . 3 2 . 1 0 -2 Flexible mode n°'/ 1.12 + 26.12i 4 . 2 9 . 1 0 -2 Flexible mode n°8 - 2.68 + 25.15i 0.1 Flexible mode n°9 - 2.77 + 35.98i 7.7. 10-2 Flexible mode n°10 - 3.22 + 32.14i 9.98. 10-2 1
-
I
-
The R-l') domain will have the following characteristics: G r = 0.5 Tr= 12s. Those of the F-Q domain will be: G r = 0.015. There is, of course, the constraint that the system must be stable after algorithm convergence. The algorithm converges after 96 iterations and provides: (-0.56211.10723.09470.9171) KI= 0.5266 - 0 . 1 5 1 5 2.73380.0619 Note that K gives the following stability margins: MG + _> 1.55
IMPI-> 2050
-
(kl~//ip kp/~ip kr/Sp kq~//ip "~ Consider that K 1 = ~,kl3//ir kp//ir kr/8r ktp//ir j
-
-
Pt~ (dq) 25 20
The same characteristics for the F-~ domain can be retained, but with the constraint that all the rigid modes should have a damping value at least equal to 0.7.
15 10 5 00
The constraints imposed by the structure of the aircraft me: kl3/8p = 0 However, remembering that kl3//ip = 0 in KI, the performance of the aircraft would be affected. So, it would be interesting to initialize the algorithm again by: ( 0 1.11 3.1 0 . 9 2 ) K0= -0.53 - 0 . 1 5 2.73 0.06
5
10
15
20
With such specifications, the algorithm converges after 11 iterations and provides the following feedback K2: ( 0 1.04 2.22 0.9 ) K2= - 0 . 5 6 - 0 . 2 4 3.65 0.01 ,
a ~ (dog) O.Ir
with excellent stability margins: MG ÷ _>2.52
0.4 0.=
Note that these margins can be increased by augmenting the value of kr/& and decreasing that of
0 -0.2
--0.4
0
5
10
15
20
Fig. 13. Simulations with the rigid and flexible aircraft model, and K 0 Now the iterative algorithm can be initialized with K 0.
kr/Sp. One can take kr//~r = 4.5 and kr//i p = 1.8 in K2: this feedback will initialize the algorithm again. Finally, ( 0 1.02 1.76 0 . 8 9 ) Kfinal= - 0.55 - 0.24 4.41 0.01 which provides excellent performance (see Fig. 14) and excellent stability margins (see Fig. 15): MG + _>3.07
4oo
Robust Hight Control Design
1383
Table 3 Closed-loon oole characteristics with Kfinal Closed-loop
Mode
Damping
poles
30 20 10 0 -10
.........
%.,
iiiiiiiiiiiiiiill
0
,
i
S
10
am
15
20
Dutch roll Spiral mode Roll mode Flexible mode n°l Flexible mode n°2 Flexible mode n°3 Flexible mode n°4 Flexible mode n°5 Flexible mode n°6 Flexible mode n°7 Flexible mode n°8 Flexible mode n°9 Flexible mode n°10
1.04i - 0.58 - 1.4 - 0.69 + 22.68i - 0.41 _+ 18.7i - 0.37 + 14.5i - 0.44 + 17.4i - 0.33 + 16.56i - 1.31 + 12.05i - 0.96 + 25.2i - 2.56 + 25.42i - 2.53 + 35.67i - 3.22 + 32.13i - 1.32 +
0.78 1 1
3 . 10 -2 2 . 2 . 1 0 -2 2.53. 10-2 2.52. 10 -2 1.98. 10-2 0.1 3.8. 10 -2 0.1 7 . 10 -2 9.98. 10 -2
(~)
0.4
7. C O N C L U S I O N The iterative algorithm presented in this paper has multiple applications. For example, the fact that the feedback gains are usually tabulated with calibrated airspeed is not a constraint; several gains can depend on two or three parameters. It is also possible to tabulate the gain according to a unique parameter, whereas the feedback is computed in order to be robust with respect to the other parameter variations. In a more general way, the methodology presented makes it possible to obtain a feedback that is robust with respect to parametric variations of the model.
0.,2
0
-0.2
0
5
10
15
20
Fig. 14. Simulations with the complete aircraft model and Kf.ma1
I
-?++++++PI+++P???+++P++P++++++ + .... +..÷.:.: ~÷.,:.....+..+. +++++ii++
+.....,..+.+ ÷+~....÷ ..÷.÷.~:+....+...:.+.+.+:+,
~++i~
++++++]++ +++++++i+
.... .+. -. +. . / 4 .~.ll "'i%--~-~(~+~.--a
: I' . '.~ .+. i~' " ' - - - + .. 4. .-.i. .. (. +. . i ~ -
•~$~J.i..i..[i.i+;L..!
+ ++++fii 4" 1".'I't'll., "R
.:.i~iJ!i,[....i...l.:.+]!!!
!!'!~iili~i
!!i!!i!!l
;;+ + +++++++"'"++"++'++'t+ i++iii::+ + +:+:+++++,.!!~++:,++ + + +++'+" + ++'"++:
~
........
i
le"
I IIIiili
I
I ......
~
,e'
+f!ii!i+iii!ii w'
10'
i
Another important application of this technique is the application of "the flexible problem" by using several "performance domains". In fact, in this paper a new methodology for a flexible aircraft conlrol system was developed, in order to avoid the inlroduction of structural filters. This new approach optimizes the eigenvalue assignment within a chosen domain in order not to excite the flexible modes. No change in the law structure was envisaged; the problem was solved by optimizing the feedback gains (i.e., the eigenvalue assignment). Note that no decoupling was considered during the optimization phase because of the degrees-of-freedom limitation. However, by initializing the algorithm with a feedback obtained by eigenstructure assignment (making the desired decoupling) and by limiting the feedback gain variations at each step, the feedback which was finally obtained, provides acceptable performance. For systems which must have perfect decoupling, it could be envisaged to "create" new degrees of freedom by adding sensors in order to introduce the decoupling during the optimization.
Fig. 15. Singular values with the complete aircraft model and Kfinal REFERENCES The closed-loop modes for Kfina I are given in table 3.
Doyle J.G., (1982). "Analysis of Feedback Systems With Structured Uncertainties".
1384
T. Livetet IEEE Proceedings, Vol. 129, pt. D, No. 6, pp 242 - 249.
Doyle J.G., Glover K., Khargonekar P. and Francis B., (1989). "State Space Solutions to standard H2 and H** Control Problems". IEEE Transactions on Automatic Control, Vol. 34, No. 8, pp 831 - 847. Farineau J., (1989). "Lateral Electric Flight Control Laws of a Civil Aircraft Based upon eigenstructure Assignment Technique". AIAA Guidance, Navigation and Control Conference, Boston, Massachusset, USA.
Geromel J.C., Garcia G., and Bernussou J., (1993). "H2 robust control design with pole placement". In Proc. 12tla IFAC World Congress, Sydney, Australia, II:283. Kreisselmeier G. and Steinhauser R., (1983). "Application of vector performance optimization to a robust control loop design for a fighter aircraft". Int. J. Control, 37:251-284. Kubica F. and Livet T., (1994). "Flight Control Law Synthesis for a Flexible Aircraft". AIAA Guidance, Navigation and Control Conference, Scottsdale, Arizona, USA. Livet T., Kubica F., Magni J.F. and Antonel L., (1994a). "Non-Interactive Control by Eigenstructure Assignment and Feedforward". In Proc., AIAA Guidance, Navigation and Control Conference, Scottsdale, Arizona, USA.
al.
Livet T., Magni J.F., Fabre P. and Kubica F., (1994b). "Robust Flight Control Design for a highly Flexible Aircraft by Pole Migration". In Proc., IFAC World Congress, Palo Alto, CA, USA. Maciejowski J.M., (1989). "Multivariable Feedback Design". Electronic Systems Engineering Series. Addisson-Wesley Publishing Company, Wokingham, England. Magni J.F. and Manouan A., (1994). "Robust Flight Control Design by Eigenstructure Assignment". Submitted for publication, 1994. Mudge S.K. and Patton R.J., (1988). "Analysis of the techniques of robust eigenstructure assignment with application to aircraft control". IEEE Proceedings, Part D, Control Theory and Applications, 135(4):275-281. Sobel
K.M. and Shapiro E.Y., (1985). "Eigenstructure Assignment for design of multimode flight control systems." IEEE Control Systems Magazine, pp 9-I5.
Xiao-ming Xu, Zhi-ming Wu, and Zhong-jun Zhanh., (1986). "A design method of generalized root-loci for mimo systems." In Proc. of the 25 th Conf. on Decision and Control, Athens, Greece, pp 713-717.