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Impulse-to-Peak Control of a Space Launch Vehicle
European Journal of Control (2006)12:57–70 # 2006 EUCA
Multi-objective H2 =H1 /Impulse-to-Peak Control of a Space Launch Vehicle Denis Arzelier1,, Benoıˆ t Cle´ment2, and Dimitri Peaucelle1, 1
LAAS-CNRS, 7 Avenue du Colonel Roche, 31 077 Toulouse Cedex 4, France; 2CNES Launcher directorate, Rond Point de l’Espace 91003 Evry Cedex, France
A multi-objective synthesis problem involving H2 , H1 and impulse-to-peak performances is investigated. In general, multi-objective control problems are hard problems to solve and do not have exact solutions. Here, an LMI formulation is proposed for the mixed H2 /impulse-to-peak optimization under an H1 constraint for LTI discrete-time systems. This framework is then used to control the attitude of a space launcher. A particular control structure is defined and a multiobjective H2 =H1 /impulse-to-peak synthesis problem is solved to tackle specific specifications. A systematic synthesis procedure including the tuning of design parameters is defined and results from simulations are presented. Keywords: Attitude Control; Launch Vehicles; LMI Optimization; Multi-objective Control; Robust Synthesis
1. Introduction It is well-known that H1 synthesis guarantees robust stability in the face of worst-case disturbances while H2 synthesis is more adapted to deal with nominal performance. When imposing transient specifications (overshoot, settling time) a less-known approach is to guarantee a bound on the peak impulse response [3,15,18,19,23]. Applications involving sharp and Correspondence to: P. Arzelier E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
variouvs specifications naturally result in considering a mixed design framework that can integrate optimal transient performance and robustness in a single controller. In particular, the last developments of space launchers in terms of structure (optimized composite) and versatile payloads make it necessary to take hard design constraints into account in the pilot design phase. In such an application, the controller should counter the effects of uncertainties and dispersions affecting the launcher parameters as well as the disturbances (winds shear and gusts) while ensuring a high level of reliable performance (low level of consumption). Here, the reliability means that the angle of attack and the angle of deflection of the actuator (thrust) should stay below a pre-specified level during all the atmospheric flight of the launcher. During the last 10 years, some attempts have been made to tackle the complex problem of robust attitude control of a space launch vehicle via modern control synthesis methods [4,9,10]. Those efforts lead to the definition of the working group PIROLA (PIlotage RObuste des LAnceurs). 1.1. The Working Group PIROLA PIROLA was a three years research working group on robust control of launchers. This group, with financial support from CNES (French Space Agency) was aimed at improving control loop
Received 23 December 2004; Accepted 6 September 2005 Recommended by C.W. Scherer, M. Steinbuch.
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robustness and at reducing the time for tuning the control laws for this type of application. More specifically, the designer wishes to enlarge the controllable configurations and allow the use of low cost sensors and actuators. The reduction of the time spent in tuning the control law is obtained via the development of automatic procedures while keeping connection with physics. The group includes academic research partners: LAAS-CNRS (government research center) [1,2,17], ONERA (French aerospace research center) [8,20], Supaero, [21,22], Supelec [4,5] (engineering schools) and an industrial partner: EADS ST (Space Transportation) [10]. Robust synthesis as well as robust analysis methodologies have been proposed in the frame of PIROLA. The former ranged from stationary control methodologies (H1 design based on the cross standard form, -synthesis, multi-objective control) to linear parameter-varying techniques and multi-model modal self-scheduled control while the last have mainly included techniques based on tools [6] and SemiDefinite Programming (SDP) tools. The interested reader may consult the reference [8] for a more detailed information concerning the design methods. The goal of this article is to detail the LMI-based multi-objective control strategy proposed by LAAS-CNRS.
the Lyapunov shaping paradigm [1,12–15]. Here, an LMI formulation based on those ideas is proposed for the mixed H2 /impulse-to-peak optimization under an H1 constraint for LTI discrete-time systems. The relevance of the approach is illustrated by its application to the robust control of the attitude of a launch vehicle. An associated systematic synthesis procedure including the tuning of design parameters is defined and results from simulations are presented. These simulations are based on a very complete model including bending modes effects as well as time-varying behavior and non-linear (rate and position limitations of actuators) effects. Future directions of research are then proposed in the concluding section. Note that the work proposed in this paper is based on real industrial data provided by EADS Space transportation and CNES. We are therefore not allowed to make parameters of the launcher model available nor to give exact information about the specifications X dB, imax , max , _max or the sample time. Those numbers being strictly confidential, there was no other way but to remove those numbers and units from the proposed figures. The same policy has been followed by authors of reference [22] who have worked in the same industrial context.
1.2. A Multi-objective Controller for Launch Vehicle Control
1.3. Notations
Our approach consists in translating engineering requirements into automatic control specifications. These last specifications are converted into a particular multi-objective control problem involving H2 , H1 and impulse-to-peak performance criteria. As it turns out, despite the great number of papers on the subject, multi-objective synthesis problems are in general open problems except for some particular cases for which an exact convex formulation is given using LMI formalism. In our case, the best we can hope is to find a relaxed convex formulation as tight as possible since no exact convex characterization for the impulse-to-peak performance exists in the literature [15,18,19,23]. Here, a new characterization of impulse-to-peak performance is given for LTI discrete-time systems. It involves additional variables and allows a decoupling between the Lyapunov function and the system matrix in the spirit of references [7,11,14]. This property is particularly useful in robust and multi-objective control problems. Indeed, it paved the way for the reduction of inherent conservatism of robust or multi-objective control methods based on
Notation is standard. The transpose of a matrix A is denoted A0 . For symmetric matrices, > ( ) denotes the Lo¨wner partial order, i.e. A > ðÞ B iff A B is positive (semi) definite. I stands for the identity matrix and 0 for the zero matrix with the appropriate dimensions. The symmetric part of a square matrix A is denoted sym[A], i.e. sym[A]=A þ A0 .
2. Problem Formulation 2.1. Ariane Control Loop We are interested in the control of the launch vehicle during its atmospheric flight and particularly in the design of the autopiloting inner loop for the yaw axis of the launcher as illustrated by Fig. 1. The main function of this loop is to hold the launcher around its center of gravity which follows the guidance reference trajectory. G is the center of gravity, V and Vr are the absolute and relative velocity and W is the wind velocity. i is the angle of attack and is the deviation of the launcher from the yaw axis with respect to the guidance attitude
59
Multi-objective Control of a Space Launcher
Wind velocity w Launcher
Angle of attack i . Angular velocity Sensors
Attitude Thrusters
Noise
Commanded angle of deflection
Fig. 1.
Controller
+
c
Guidance attitude reference
ARIANE control loop.
reference (see Fig. 2). The control variable is the thruster angle of deflection . A thorough description of the plant may be found in the references [4,5,9].
Nominal Trajectory
x Vr
i w V
2.2. Model of the Launcher
G
To get a tractable analytical model for the plant, some simplifying assumptions have to be made. Thrust
.
Assumptions 1 Axis are decoupled. All angles remain small. Parameters are slowly time-varying.
Fig. 2. Model of the launcher.
The last assumption is motivated by the mildly nonstationary nature of the launcher. Moreover, it implies that an uncertain parametric LTI model is sufficiently realistic for design and analysis of robust autopilot for the whole range of the flying envelope. Indeed, as will be seen later, most of the design requirements are of time-invariant nature (flexible mode attenuation, gain/phase margins) and robust LTI controllers may therefore be evaluated during the atmospheric flight phase. Under these assumptions, the linearized dynamics of the launcher include a rigid model (without bending modes and actuator dynamics) where _ is the angular velocity and z_ the lateral drift rate. €ðtÞ ¼ A6 ðtÞ þ z_ðtÞ WðtÞ þ K1 ðtÞ V ð1Þ z€ðtÞ ¼ a1 ðtÞ þ a2 ðz_ðtÞ WðtÞÞ þ a3 ðtÞ:
Those equations depend on known aerodynamic coefficients ai , i ¼ 1, 2, 3 and time-varying coefficients A6 , the aerodynamic efficiency and K1 , the thruster efficiency along the flight envelope. _ z_ 0 , the Defining the state vector as x ¼ perturbation input as w and the controlled output as i, we get the following third-order state-space model for the rigid part of the launcher. 2
0
6 x_ ðtÞ ¼ 6 4 A6 a1 2 6 þ4
3 2 3 0 0 6 7 A6 7 7xðtÞ þ 6 A6 7W 0 4 V5 V5 a2 0 a2 3 0 7 K1 5uðtÞ 1
a3 zðtÞ ¼ iðtÞ ¼ 1
ð3Þ
1 1 xðtÞ W: 0 V V
Moreover, the angle of attack equation reads: z_ðtÞ WðtÞ : iðtÞ ¼ ðtÞ þ V
ð2Þ
Note that the launcher is aerodynamically unstable. The sensors dynamics (second order for and _ ) and the actuators dynamics (second order) are added
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Fig. 3. Bending modes of the ARIANE launcher.
Fig. 4. Illustration of the margins in the Nichols plot.
35
30
25
Wind velocity W
to the previous model to compose the complete linearized dynamics of the rigid launcher. Owing to lack of tests and system complexity, internal uncertainties and dispersions have to be faced. They mainly concern the propulsion system, aerodynamic coefficients, mass model and inertia, flexible modes (elastic and sloshing), actuators and sensors modelling. The wind (shear and gusts) is considered as an external disturbance. The first five bending modes are taken into account into the complete model (Fig. 3) and their characteristics are considered to be not exactly known (four uncertain parameters per mode) leading to the definition of a discrete-time uncertain LFT simulation model of order 17 and 2 R2222 .
20
15
2.3. Control Objectives and Performance Specifications During the atmospheric flight phase, the physical constraints and objectives defining the requirements that the autopilot must fulfill are the following. (1) Guidance demand tracking. (2) Robustly stabilize the launcher, i.e. the controller must be stabilizing for all parameters in a prespecified set . (3) Limit the aerodynamical loads or equivalently limit angle of attack deviations. (4) Limit the energy/power consumption of the controller. The physical constraints and the objectives become automatic control requirements shaping the autopilot. Closed-loop stability with sufficient stability margins (for a given flight time): Gain margins (low and high frequencies) must be respectively over LF and HF. Delay margin must be greater than one sampling period. In addition to the stability margins on the rigid part of the model, a specified attenuation of bending modes (X dB) has to be insured. The first flexible mode may be alternatively controlled with respect to the previously defined attenuation X dB or with respect to its phase (see Fig. 4).
10
5
0
0
20
40
60
80
100
120
t/Ts140 = 1950
Fig. 5. Wind profile.
The autopilot has to reject disturbances (wind and gusts) influence on the angle of attack i. In particular, angle of attack peaks must not exceed the limits imax in response to a typical wind profile presented in Fig. 5. This typical wind profile is composed of observed worst-case gust and worstcase wind speed drift. The previous performance specifications must be achieved without an excessive thruster consumption expressed as cumulated deflections. C¼
Tend X
jkþ1 k j:
ð4Þ
k¼Tini
The autopilot must make sure that robustness requirements with respect to uncertainties on the rigid and bending modes are fulfilled. This means that all previous control objectives have to be achieved for all possible configurations of the 22 uncertain parameters.
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Multi-objective Control of a Space Launcher
3. Multi-objective H2 =H1 /Impulse-to-Peak
w31
Design
w32
Σ aug w2
Keeping in mind the high heterogeneity of the performance specifications, we adopt a multi-objective strategy for the control design procedure. Even if the problem of the synthesis of a robust autopilot achieving the different specifications may be naturally recast as a multi-objective control problem, it is first necessary to translate the control objectives and the performance specifications in terms of a set of design constraints involving adequate system norms. One therefore imposes the following: A bound on the impulse-to-peak performance of the closed-loop transfer W i with an additional filter modeling a typical wind profile to limit the angle of attack. A bound on the H1 performance of the sensitivity S to get a minimum modulus margin and therefore good gain margins. A bound on the H2 performance of the transfer between measurement noises and _ to reduce the consumption. A mixed roll-off and lead filter to attenuate the bending modes and control in phase the first one. The first step of the design procedure consists in choosing a particular model for synthesis purpose. Owing to the high complexity of the complete model of the launcher, a simplified model is usually utilized for design purpose. Here, this simplified model is composed of the rigid model defined by the equation of the torque equilibrium of the yaw axis of the launcher. The synthesis model also includes the sensors dynamics (second order) but is free of the bending modes and actuator dynamics. 3.1. Structure of the Controller As will be seen in the next section, the controller is composed of two parts. One part of the controller is tuned (mixed filter) and the other part is computed via multi-objective optimization leading to the particular structure for the controller detailed in Fig. 6. Note 2 B1 ¼ ½ B1
D21 ¼ ½ Dy1
Bi2p
Dyi2p
C1
3
6 7 B2 C1 ¼ 4 Ci2p 5
D11
C2 2 3 D1u 6 7 Dy2 D ¼ 4 Di2pu 5: D2u
w1
Wind gust model
+
z1 = z3 = β
Actuator
Rigid Launcher with sensors
z2 = i ψ . ψ
– Controller Mixed Filter
+ + + + delay delay
Multiobjective Controller
Fig. 6. Structure of the multi-objective controller.
also that the wind gust model acts like a weighting function which has to be tuned in the controller synthesis procedure. The complete synthesis model is then defined by aug . An H2 =H1 =impulse–to–peak multi-objective control problem is therefore set on aug to enforce the three first specifications. 3.2. The Multi-objective Control Problem 3.2.1. Definition Let the LTI discrete plant aug be given by its statespace minimal realization: 32 3 2 3 2 xk A B1 B xkþ1 4 zk 5 ¼ 4 C1 D11 D 54 wk 5, ð5Þ yk uk C D21 0 where x 2 Rn is the state vector, w 2 Rmw is the disturbance vector, u 2 Rmu is the input vector, z 2 Rrz is the controlled output vector and y 2 Rry is the measured output vector. The z and w vectors are partitioned as indicated in Fig. 7. 2 3 2 3 w1 z1 ð6Þ z ¼ 4 zi2p 5 w ¼ 4 wi2p 5 z2 w2
The associated matrices are therefore consequently partitioned. 2 3 D1 D1i2p D12 6 7 ¼ 4 Di2p1 Di2p Di2p2 5 D21 D2i2p D2 ð7Þ
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As in the continuous-time case, a conservative LMI characterization may be used to compute a bound on the peak of the impulse response of a discrete LTI system. Theorem 1. If there exists a matrix Pcl 2 Sþ and a scalar satisfying ¼ min Pcl , Fig. 7. Standard model for multi-objective control.
s:t: Acl Pcl A0cl Pcl < 0
The controller K is given by its state-space realization: kþ1 ¼ AK k þ BK yk
? K is given by its stateThe closed-loop system space matrices: A þ BDK C BCK B1 þ BDK D21 Bcl ¼ Acl ¼ BK C AK BK D21 aug:
Ccl ¼ ½ C1 þ DDK C
DCK
Dcl ¼ ½D11 þ DDK D21 : ð9Þ
The multi-objective control problem is then defined in the following way. Problem 1. (multi-objective H2/H1/i2p control problem). Find a controller K in the set of internally stabilizing controllers K such that min K2K
Dcl D0cl I < 0
ð8Þ
u k ¼ C K k þ D K y k :
i wi þ c cons
s:t: ð10Þ
2 kaug: mod ? Kk1 mod
kaug: Wi
? Kk2i2p wi 2 kaug: cons ? Kk2 cons :
ð12Þ
Bcl B0cl Pcl < 0 Ccl Pcl C0cl I < 0 then kzk k2
8k 0.
Proof 1. First note that by Schur complement, all the inequalities defining the realizable set of (12) may be equivalently written with respect to Xcl ¼ P1 cl : A0cl Xcl Acl Xcl < 0
ð13Þ
B0cl Xcl Bcl < I
ð14Þ
1 C0cl Ccl Xcl < 0
ð15Þ
1 D0cl Dcl I < 0:
ð16Þ
Let x and z be the solutions of (11) when the impulsive input wki is applied with x(0) ¼ 0. wki ¼ ei k , jj 1, i ¼ 1, . . . , mw where ei 2 Rmw is the basis vector with 1 at the position i, 0 elsewhere and k is the unit pulse. Then 8 i: x0 ¼ 0
z0 ¼ Dcl ei
x1 ¼ Bcl ei
z1 ¼ Ccl x1 ¼ Ccl Bcl ei
xk ¼
Aclk1 Bcl ei
zk ¼ Ccl xk ¼
If performance specifications are often expressed in terms of H2 and H1 norms, impulse-to-peak performance is not so usual and deserve a thorough description.
From (13), we get
3.2.2. The Impulse-to-Peak Performance
So, 8 k 1 and 8 i, from (14) and (15),
At our knowledge, only references dealing with the continuous-time case may be found in the literature [3,15,18,19,23]. Let the closed-loop discrete-time plant be given by its minimal realization: ?K
xkþ1 ¼ Acl xk þ Bcl wk zk ¼ Ccl xk þ Dcl wk
xð0Þ ¼ 0
ð11Þ
ð17Þ
Ccl Aclk1 Bcl ei
x0kþ1 Xcl xkþ1 < x0k Xcl xk < < x01 Xcl x1 :
1 > 2 e0i B0cl Xcl Bcl ei
¼ x01 Xcl x1 > x0k Xcl xk > 1 x0k C0cl Ccl xk ¼ 1 z0k zk
So, kzk k2 < . Using the last inequality (16) leads to kz0 k2 < 2 :
&
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Multi-objective Control of a Space Launcher
This LMI characterization of the impulseto-peak performance is new for discrete-time systems. As mentioned above, this characterization of the impulse-to-peak performance is conservative as may be a convex relaxation of a non convex problem. The evaluation of this conservatism is a difficult and open problem for which, at our knowledge, no direct approach exists so far. Here, we are interested in an equivalent alternative LMI characterization of a bound on the impulse-topeak performance of the closed-loop system (11). A generalization of the method in [13] where an extra matrix G is introduced for the test of H2 and H1 performances leads to the equivalent LMI optimization problem. Theorem 2. Let the following semidefinite programming problem be given: G
¼
min
Pcl , , G, F
s.t. Pcl Acl G <0 G0 A0cl Pcl G G0 Pcl Bcl <0 B0cl I I Ccl F <0 F0 C0cl Pcl F F0 I Dcl <0 D0cl I
Problem 2. (worst-case i2pk synthesis). Find a fullorder dynamic output-feedback controller K in the set of internally stabilizing controllers K. min kaug: ?Kki2p ¼ min sup kzkL1 , K2K
K2K w2I
ð19Þ
where w I ¼ fw 2 lm 2 jðwi Þk ¼ k , jj 1,
k ¼ 1, 2, . . . ,
for i 2 f1, 2, . . . , mw g
and ðwj Þk ¼ 0, k ¼ 1, 2, . . . for i 6¼ jg To address the impulse-to-peak performance synthesis problem, we need to relax the previous optimization problem (18) by setting G ¼ F. This relaxation is known as the generalized shaping paradigm [13]. Theorem 3. If there exist X 2 Rnn , Y 2 Rnn , ^ 2 Rmn , ^ 2 Rnn , ^ 2 Rnr , ^ 2 Rmr , B C D A þ þ nn Q 2 S n , H 2 S n , J 2 R , and a scalar G 2 Rþ such that: G ¼
min X, Y, J, H, Q, S
s:t ð18Þ
then ¼ G :
2
2
So far, the new proposed characterization (18) of impulse-to-peak performance seems not to bring any novelty. In fact, we will see later that it allows to reduce the conservatism of the proposed bound when dealing with multi-objective control problem. The reference [1] gives another application of this condition in a robust control context.
"
The previous characterization may be used to tackle the problem of designing a controller minimizing a bound on the impulse-to-peak performance.
J
6 6 ? H 6 6 ? 4 ? ? ? 2 Q J 6 4 ? H
Proof 2. The proof comes from the application of elimination lemma [16] on the first and third inequality of (12) &
3.2.3. Impulse-to-Peak Performance Design via LMIs
Q
?
6 4 ?
Q X X0 ? ^ Di2py Bi2p þ BD
7 ^ Di2py 5 < 0 YBi2p þ B
?
I ^ Ci2p X þ Di2pu C 0 QXX
?
3 ^C A þ BD 7 ^C 7 YA þ B 7<0 7 J S0 I 5 H Y Y0 3
^ AX þ BC ^ A
3 ^C Ci2p þ Di2pu D 7 J S0 I 5 < 0
?
^ Di2py Di2p þ Di2pu D
?
I
then a follows:
full-order
V01 U1 ¼SYX
ð20Þ
H Y Y0 # <0
controller
reconstructed
as
^ D K ¼D
^ D ^ CXÞU1 BK ¼VT ðB ^ YBD ^Þ CK ¼ðC 1 1 h i ^ YðAþBD ^ CÞXV0 BK CXYBCK U1 U1 AK ¼VT A 1 1 1 ð21Þ
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D. Arzelier et al.
leads to bound the peak value of impulse response of the system: kzk1 <
2
pffiffiffiffiffiffi G :
B1cl B01cl P1cl 6 ? 4
Proof 3. From (18), it is easy to see that the matrix G is invertible and may be partitioned as follows: G¼
X U1
U2
H1 performance
G1 ¼
Y V1
V2 :
3 0 7 0 5
B1cl D01cl mod I þ D1cl D01cl
? 22
3 Acl 66 7 þ sym44 C1cl 5½ 0 0
?
P1cl
3
7 G 1 5 < 0
ð24Þ
I ð22Þ
Then applying the following similarity transformation on the closed-loop Lyapunov matrix, we get I 0 I Y Q J P ¼ : ð23Þ Y0 V01 cl 0 V1 J0 H I Y Multiplying the first three terms in (18) by 0 V1 and by its transpose, we get the final result by applying the change of variables (21). & This parameterization is a slight modification of the one proposed in Ref. [15] and an extension to the impulse-to-peak case of the one given for the first time in the context of H2 and H1 performances in [13]. Its main interest relies in the decoupling between the Lyapunov function and the computation of the controller. The controller is now built from the additional variable G allowing to use independent Lyapunov functions in the context of multi-objective control.
3.3. LMI Formulation of the Multi-objective Control Problem Let us come back to our original multi-objective control problem (10). Except in some particular cases, this problem is hard to solve analytically. Even if many sub-optimal numerical approaches have been developed, the reduction of the conservatism of the proposed solution is still a challenging problem, [15,18]. In the previous subsection, we have seen how to convert the original worst-case impulse-to-peak design problem to a convex optimization problems involving LMIs. The same technique may be applied to the H2 and H1 performance criteria as originally proposed in Ref. [13] and an extended LMI formulation may be formulated for H1 and H2 performances of respectively aug: mod: ? K and aug: cons: ? K.
H2 performance traceðTÞ < cons 2
I 6 4 B2cl
3 0 7 0 5 P2cl 3
B02cl P2cl
?
? 22
0 66 7 þ sym44 Acl 5½ 0
3 0
7 G2 5 < 0
ð25Þ
I
0 T þ D2cl D02cl ? P2cl C2cl ½ 0 G2 < 0: þ sym I
Defining the notations
I
Y0 I Y0 I
Y0
Q2 I Y ¼ P 2cl J02 V01 0 V1 Q1 0 I Y ¼ P 1cl 0 J01 V1 0 V1 " Qi2p 0 I Y P ¼ i2pcl 0 J0i2p V1 0 V1 0
J2
H2 J1 ð26Þ H1 # Ji2p Hi2p
a full-order output-feedback controller minimizing an upper bound for the multi-objective H2 =H1 =i2p synthesis problem may be computed using semidefinite programming and LMI formulation. Theorem 4. If the following semidefinite programming problem has a solution min X, Y, cons , wi , T, S Ji , Hi , Qi , i ¼ 1, i2p, 2
c cons þ i wi
65
Multi-objective Control of a Space Launcher
under 2
Q1 6 ? 6 6 6 ? 6 6 ? 6 4 ? ?
2
Qi2p 6 6 ? 6 4 ? ?
2
Qi2p 4 ? ?
2
wi 4 ? ?
wi ?
J1 H1 ? ? ? ?
^ AX þ BC ^ A Q1 X X 0 ? ? ?
^C A þ BD ^C YA þ B I S0 þ J1 H1 Y Y0 ? ?
^ AX þ BC ^ A Qi2p X X0 ?
Ji2p Hi2p ?
3 ^ Di2py Bi2p þ BD ^ Di2py 5 < 0 YBi2p þ B I
ð27Þ
3 ^C Ci2p þ Di2pu D Ji2p S0 I 5 < 0 Hi2p Y Y0
^ Di2py Di2p þ Di2pu D I
<0
TraceðT2 Þ < cons 2
T2 6 ? 6 4 ? ?
2
Q2 6 ? 6 6 ? 6 4 ? ?
^ C2 X þ D2u C Q2 X X 0 ? ?
J2 H2 ? ? ?
^C C2 þ D2u D 0 I S þ J2 H2 Y Y 0 ?
^ AX þ BC ^ A Q2 X X0 ? ?
3 0 7 0 7 7 0 0 0 0 ^ X C1 þ C D1u 7 7<0 7 ^ 0 D0 C01 þ C0 D 1u 7 ^ 0 D0 5 D01 þ D0y1 D 1u mod I
3 ^C A þ BD 7 ^C 7 YA þ B 7<0 Ji2p S0 I 5 Hi2p Y Y0
Ji2p Hi2p ? ?
^ Ci2p X þ Di2pu C Qi2p X X0 ?
^ Dy1 B1 þ BD ^ Dy1 YB1 þ B 0 0 I ?
3 ^ Dy2 D2 þ D2u D 7 0 7<0 5 0 I
^C A þ BD ^C YA þ B 0 I S þ J2 H2 Y Y0 ?
3 ^ Dy2 B2 þ BD ^ Dy2 7 YB2 þ B 7 7<0 0 7 5 0 I
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D. Arzelier et al.
then a controller K constructed from (21) is a suboptimal solution to the multi-objective synthesis problem (10).
optimization step. The complete synthesis procedure therefore reads as Algorithm 1
Proof 4. The proof is very simple and consists in creating a particular Gi matrix for each performance constraint. Applying a generalized shaping paradigm G1 ¼ Gi2p ¼ G2 ¼ G ¼
X
U2
U1
G
1
Y ¼ V1
V2
(1) Choose the tuning parameters and form the augAug: mented plant Aug: . Extract Aug: Wi , cons and Aug: mod . (2) Solve the convex optimization problem via LMI optimization and get the decision variables. min ði wi þ c cons Þ under
ð28Þ
and the linearizing change of variables
^, B ^, D ^, C ^ , mod Þ < 0 L1m ðQm , Jm , Hm , X, Y, S, A ^, B ^, D ^, C ^ , wi Þ < 0 Li2p ðQi2p , Ji2p , Hi2p , X, Y, S, A ^, B ^, D ^, C ^ , cons Þ < 0 L2c ðQ2 , J2 , H2 , T2 , X, Y, S, A
^ ¼ DK D
ð31Þ
^ ¼ CK U1 þ D ^ CX C 0
^ ^ ¼ V1 BK þ YBD B 1
(3) Reconstruct the controller with formulae (21). ð29Þ
^ ¼ V10 AK U1 þ YðA þ BB ^ CÞX A 1 þ
V01 BK CX
4. Simulation Results
þ YBCK U1
leads to the result.
&
Notice also that the proposed approach is always less conservative than the one using the Lyapunov Shaping paradigm from [15].
3.4. Synthesis Procedure and Tuning Parameters As seen before, the whole controller is formed from a mixed filter in series with a multi-objective controller. We have therefore two different sets of tuning parameters. The first set is formed with the parameters defining the wind gust model and the mixed filter. The wind gust model is a second order with two tuning parameters while the mixed filter is composed of a low-pass filter multiplied by a lead filter: KW þ 2TW p þ T2W 1 þ ap Kro : wm ¼ 1 þ p 1 þ Tro p
wwind ðpÞ ¼
p2
ð30Þ
We have therefore six synthesis parameters: KW , TW , Kro , Tro , a and . The other set of tuning parameters is formed from the optimization parameters i =c and mod of the multi-objective LMI
The previous algorithm has been used to design a robust autopilot for the atmospheric flight of a space launcher. Some worst-cases parametric configuration where the combination of parameter extremal values is particularly critical has been identified. The synthesis has been performed considering an LTI worst-case configuration of the time-varying parameters of the launcher. During the atmospheric flight phase, the timevariant behavior of the launcher and physical behavior is validated by simulations using a SIMULINK# model where information about specifications fulfillment is provided to the user. The time-domain specifications are therefore verified by inspecting the time responses provided by time-varying simulations while the frequency domain specifications are checked by considering only some pre-defined worst-cases. Indeed, the robustness analysis of the frequency domain performances is limited to those worst-cases as the experience has shown that they are quite representative of the complete problem. As is clearly implied by the very different proposed specifications, different tradeoff have to be dealt with. The main ones are summarized as follows. Limitation of the angle of attack deviations against specified attenuation of bending modes and noise reduction. Delay margin and H.F. gain margin against B.F. gain margin.
67
Multi-objective Control of a Space Launcher Bode Diagram 40
.
30
Ψ
u
Magnitude (dB)
20 10 0 –10
Ψ
–20
u
–30 –40 90
Phase (deg)
0
–90
–180
–270
–360 –4
10
–3
10
–2
–1
10
0
10
1
10
10
Frequency (rad/sec) f/f s
Fig. 8.
Bode diagram of the multi-objective autopilot.
Specified attenuation (X dB) of higher order ending modes against the control of the first flexible mode with respect to its phase. As the energy/power consumption is directly related to the reduction of sensors noise, the limitation of the angle of attack deviations plays also against the reduction of consumption. Keeping in mind that all those specifications have to be fulfilled for every parametric configuration of the launcher, it is not surprising that the tuning of the controller is difficult and particularly sensitive to the associated trade-off. As mentioned in the introduction, other design techniques have been proposed by other members of the PIROLA group. In particular, concerning pure LTI techniques, an H1 design based on the cross standard form is presented in reference [22] and can be usefully compared to the results presented below. In addition, some elements of comparison are also given between the proposed technique (extended multi-objective autopilot) and the standard mixed design (a single Lyapunov matrix for all specifications). Note that a structure of control identical to the one used here has been defined for the standard mixed design. As will be seen in the following, owing to the very conservative nature of this kind of design, it is not possible to tune a standard mixed autopilot satisfying all requirements.
X db
–300
–180
–60
60
180
300
420
540
Fig. 9. Nichols plot of the open-loop.
First, the Bode plot of the multi-objective controller computed via algorithm 1 is presented in Fig. 8. The effect of the lead filter is obvious for the channel u. The open-loop frequency responses are given in Nichols charts in Figs 9 and 10 for several linearized models of specific configurations. This plot is obtained by freezing the model at the worst-case time
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configuration of the launcher (it coincides with the time when the wind gust is applied). The roll-off specification on bending modes is indicated by the X dB horizontal line. A zoom on a specific range of frequencies allows to verify gain margins imposed on the rigid model. Note also that the first flexible mode remains between two critical points for all worst-cases satisfying the phase control requirement for this mode. These plots may be compared with the one obtained with a standard mixed autopilot (Fig. 11) where the phase control requirement for the first flexible mode is not satisfied for some parametric
configurations. This implies that the requirement on the delay margin is not satisfied in this case. The gain margins (H.F. and B.F.) are also smaller than the ones of the extended multi-objective autopilot. Various time responses are also proposed. Note that all scales have been normalized with respect to the sample period Ts . The first figure shows the variation of the angle of attack when the typical wind profile in Fig. 5 is applied (Fig. 12). Note that the requirement on limitation of angle of attack deviation is hardly satisfied for the standard mixed autopilot (Fig. 13). The consumption C¼
Tend X
jðk þ 1Þ ðkÞj
k¼Tinit
M l db
Mh db
520
525
530
535
540
545
550
Fig. 10. Zoom on the Nichols plot.
555
560
is computed and plotted with respect to the maximum allowed consumption (Figs 14 and 15) . These plots show that the consumption is very good in both cases since they only need 50% of the maximum allowable consumption (usual consumption of a normal autopilot) to tackle the specifications. Note that in Refs [5] and [21], the consumption is 60% while the autopilot proposed in Ref. [4] is slightly better (44%) but with some violated specifications. Finally, the angle of deflection and its velocity are plotted below for the extended multi-objective autopilot (Figs 16 and 17). They are far from the
0 Xdb
180
180
540
Fig. 11. Nichols plot of the open-loop (standard mixed controller).
69
Multi-objective Control of a Space Launcher 100 90 80
Angle of attack i
70 60 50 40 30
i max
20 10
Fig. 12. Extended multi-objective autopilot.
0 10
20
30
40
50
60
70
80
90
100
Fig. 15. Standard mixed autopilot.
Angle of deflection β
Angle of attack i
β max
Tf
Fig. 13. Standard mixed autopilot.
− βmax
100
Fig. 16. Angle of deflection.
90
Cmax
70
60
. β max
50
40
30
20
10
0
Tf
Control velocity β
Consumption (C/Cmax )
80
Fig. 14. Extended multi-objective autopilot. . − β max
specified maxima because of the over dimensioning of the actuators. Note that the high frequency component of the response comes from the measuring noise modelled as a white noise.
Tf
Fig. 17. Velocity of the angle of deflection.
70
5. Conclusions The simulations have shown that this approach leads to controllers verifying all the specifications imposed on the launcher during the atmospheric flight. This black box-like method allows a great flexibility in the synthesis process. The tuning parameters are clearly identified with respect to the fundamental trade-off proposed by the performance specifications. Easy-touse macros of MATLAB based on convex semidefinite optimization solvers allow the designer to tune adequately the synthesis parameters. The main drawback of the proposed method comes from the fact that it can be used only with a simplified synthesis model (without bending modes) due to numerical considerations. Moreover, the parametric uncertainty of the model is not directly dealt with. In fact, parametric robustness concerns are inherited from time-varying nature of the model which is not taken into account here. Considering that the time spent to design a multi-objective controller is not prohibitive, the first step in order to take non stationarity into account would be to interpolate a finite number (5 or 6) of multi-objective controllers computed for a frozen configuration of the launcher all along the atmospheric flight envelope. Of course, this does not mean that finding an adequate interpolation algorithm is an easy task!
References 1. Arzelier D, Peaucelle D. Robust impulse-to-peak synthesis: application to the control of an aerospace launcher. In: Proceedings of the CCA/ISIC/CACSD, Taipei, Taiwan, September 2004, pp. 214–219 2. Arzelier D, Peaucelle D. Multiobjective H2/Hinfinity/ impulse-to-peak synthesis: Application to the control of an aerospace launcher. In: Proceedings of the 16th IFAC Symposium on Automatic Control in Aerospace, St Petersburg, June 2004 3. Boyd SP, El Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. SIAM Studies, Philadelphia 1994 4. Cle´ment B, Duc G. A multiobjective control algorithm: Application to a launcher with bending modes. In: Proceedings of the 8th IEEE Mediterranean Conference on Control and Automation, Rio Patras, Greece. 2000 5. Cle´ment B, Duc G, Mauffrey S, Biard A. Aerospace launch vehicle control: a gain scheduing approach. In: Proceedings of the 15th triennial World Congress, Barcelona, 2002 6. Gahinet P, Nemirovski A, Laub A, Chilali M. LMI Control Toolbox, LMI Control Toolbox User’s Guide, 1995 7. Geromel JC, de Oliveira MC, Hsu L. LMI Characterization of structural and robust stability. Linear Algebra Appl 1998; 285: 1–3 pp. 69–80, 1998.
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8. Imbert N, Clement B. Launcher attitude control: some answers to the robustness issue. In: Proceedings of the 16th IFAC Symposium on Automatic Control in Aerospace, St Petersbourg, Russia, 2004 9. Mauffrey S, Schoeller M. Non-stationnary H1 control for launcher with bending modes. In: Proceedings of the 14th IFAC Symposium on Automatic Control in Aerospace, Seoul, Korea, 1998, pp. 433–438 10. Mauffrey S, Meunier P, Pigni G, Biard A, Rongier I. H1 control for the ARIANE 5 plus Launcher. In: Proceedings of the 52nd International Astraunotic Congress, Toulouse, France, 2001 11. de Oliveira MC, Bernussou J, Geromel JC. A new discrete-time robust stability condition. Sys Control Lett 1999; 37(4): 12. de Oliveira MC, Geromel JC, Hsu L. LMI Characterization of structural and robust stability: the discrete-time case. Linear Algebra Appl 1999; 296(1–3): 27–38 13. Oliveira MC, Bernussou J, Geromel JC. Extended H2 and H1 norm characterizations and controller parametrizations for discrete-time systems. Int J Control 2002; 75: 666–679 14. Peaucelle D, Arzelier D, Bachelier O, Bernussou J. A new robust D-stability condition for real
convex polytopic uncertainty. Syst. Control Lett 2000; 40: 1 15. Scherer CW, Gahinet P, Chilali M. Multiobjective output-feedback control via LMI optimization. IEEE Trans Autom Control 1997; 42(7) 896–911 16. Skelton RE, Iwasaki T, Grigoriadis K. A unified algebraic approach to linear control design. Taylor and Francis, London, 1998 17. Tarbouriech S, Garcia G, Langouet P. Anti-windup strategy with guaranteed stability for linear systems with amplitude and dynamics restricted actuator. In: 6th IFAC-symposium on nonlinear control systems (NOLCOS’2004), Stuttgart, Germany, 1–3 September 2004, pp 1373–1378 18. Tokunaga H, Iwasaki T, Hara S. Multi-objective robust control with transient specifications. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, 1996, pp. 3482–3483 19. Tokunaga H, Iwasaki T, Hara S. Analysis and synthesis of the robust impulse-to-peak performance. Automatica, 1998; 34(11): 1473–1477 20. Voinot O, Alazard D, Piquereau A. A robust multiobjective synthesis applied to launcher attitude control. In: Proceedings of the 15th IFAC Symposium on Automatic Control in Aerospace, Bologna, Italy, 2001 21. Voinot O, Apkarian P, Alazard D. Gain-Scheduling H1 control of the launcher in atmospheric flight via linear-parameter varying techniques. AIAA guidance, navigation and—control conference, Monterey, CA, USA, August 2002, pp. 1–10 22. Voinot O, Alazard D, Apkarian P, Mauffrey et B. Cle´ment S., Launcher attitude control: discrete-time robust design and gain-scheduling. Control engineering practice 2003; 11(11): 23. Wilson DA. Convolution and Hankel operator norms for linear systems. IEEE Trans Autom Control 1989; 34(1): 94–97
Multi-objective H2 =H1 /Impulse-to-Peak Control of a Space Launch Vehicle Denis Arzelier1,, Benoıˆ t Cle´ment2, and Dimitri Peaucelle1, 1
LAAS-CNRS, 7 Avenue du Colonel Roche, 31 077 Toulouse Cedex 4, France; 2CNES Launcher directorate, Rond Point de l’Espace 91003 Evry Cedex, France
A multi-objective synthesis problem involving H2 , H1 and impulse-to-peak performances is investigated. In general, multi-objective control problems are hard problems to solve and do not have exact solutions. Here, an LMI formulation is proposed for the mixed H2 /impulse-to-peak optimization under an H1 constraint for LTI discrete-time systems. This framework is then used to control the attitude of a space launcher. A particular control structure is defined and a multiobjective H2 =H1 /impulse-to-peak synthesis problem is solved to tackle specific specifications. A systematic synthesis procedure including the tuning of design parameters is defined and results from simulations are presented. Keywords: Attitude Control; Launch Vehicles; LMI Optimization; Multi-objective Control; Robust Synthesis
1. Introduction It is well-known that H1 synthesis guarantees robust stability in the face of worst-case disturbances while H2 synthesis is more adapted to deal with nominal performance. When imposing transient specifications (overshoot, settling time) a less-known approach is to guarantee a bound on the peak impulse response [3,15,18,19,23]. Applications involving sharp and Correspondence to: P. Arzelier E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
variouvs specifications naturally result in considering a mixed design framework that can integrate optimal transient performance and robustness in a single controller. In particular, the last developments of space launchers in terms of structure (optimized composite) and versatile payloads make it necessary to take hard design constraints into account in the pilot design phase. In such an application, the controller should counter the effects of uncertainties and dispersions affecting the launcher parameters as well as the disturbances (winds shear and gusts) while ensuring a high level of reliable performance (low level of consumption). Here, the reliability means that the angle of attack and the angle of deflection of the actuator (thrust) should stay below a pre-specified level during all the atmospheric flight of the launcher. During the last 10 years, some attempts have been made to tackle the complex problem of robust attitude control of a space launch vehicle via modern control synthesis methods [4,9,10]. Those efforts lead to the definition of the working group PIROLA (PIlotage RObuste des LAnceurs). 1.1. The Working Group PIROLA PIROLA was a three years research working group on robust control of launchers. This group, with financial support from CNES (French Space Agency) was aimed at improving control loop
Received 23 December 2004; Accepted 6 September 2005 Recommended by C.W. Scherer, M. Steinbuch.
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robustness and at reducing the time for tuning the control laws for this type of application. More specifically, the designer wishes to enlarge the controllable configurations and allow the use of low cost sensors and actuators. The reduction of the time spent in tuning the control law is obtained via the development of automatic procedures while keeping connection with physics. The group includes academic research partners: LAAS-CNRS (government research center) [1,2,17], ONERA (French aerospace research center) [8,20], Supaero, [21,22], Supelec [4,5] (engineering schools) and an industrial partner: EADS ST (Space Transportation) [10]. Robust synthesis as well as robust analysis methodologies have been proposed in the frame of PIROLA. The former ranged from stationary control methodologies (H1 design based on the cross standard form, -synthesis, multi-objective control) to linear parameter-varying techniques and multi-model modal self-scheduled control while the last have mainly included techniques based on tools [6] and SemiDefinite Programming (SDP) tools. The interested reader may consult the reference [8] for a more detailed information concerning the design methods. The goal of this article is to detail the LMI-based multi-objective control strategy proposed by LAAS-CNRS.
the Lyapunov shaping paradigm [1,12–15]. Here, an LMI formulation based on those ideas is proposed for the mixed H2 /impulse-to-peak optimization under an H1 constraint for LTI discrete-time systems. The relevance of the approach is illustrated by its application to the robust control of the attitude of a launch vehicle. An associated systematic synthesis procedure including the tuning of design parameters is defined and results from simulations are presented. These simulations are based on a very complete model including bending modes effects as well as time-varying behavior and non-linear (rate and position limitations of actuators) effects. Future directions of research are then proposed in the concluding section. Note that the work proposed in this paper is based on real industrial data provided by EADS Space transportation and CNES. We are therefore not allowed to make parameters of the launcher model available nor to give exact information about the specifications X dB, imax , max , _max or the sample time. Those numbers being strictly confidential, there was no other way but to remove those numbers and units from the proposed figures. The same policy has been followed by authors of reference [22] who have worked in the same industrial context.
1.2. A Multi-objective Controller for Launch Vehicle Control
1.3. Notations
Our approach consists in translating engineering requirements into automatic control specifications. These last specifications are converted into a particular multi-objective control problem involving H2 , H1 and impulse-to-peak performance criteria. As it turns out, despite the great number of papers on the subject, multi-objective synthesis problems are in general open problems except for some particular cases for which an exact convex formulation is given using LMI formalism. In our case, the best we can hope is to find a relaxed convex formulation as tight as possible since no exact convex characterization for the impulse-to-peak performance exists in the literature [15,18,19,23]. Here, a new characterization of impulse-to-peak performance is given for LTI discrete-time systems. It involves additional variables and allows a decoupling between the Lyapunov function and the system matrix in the spirit of references [7,11,14]. This property is particularly useful in robust and multi-objective control problems. Indeed, it paved the way for the reduction of inherent conservatism of robust or multi-objective control methods based on
Notation is standard. The transpose of a matrix A is denoted A0 . For symmetric matrices, > ( ) denotes the Lo¨wner partial order, i.e. A > ðÞ B iff A B is positive (semi) definite. I stands for the identity matrix and 0 for the zero matrix with the appropriate dimensions. The symmetric part of a square matrix A is denoted sym[A], i.e. sym[A]=A þ A0 .
2. Problem Formulation 2.1. Ariane Control Loop We are interested in the control of the launch vehicle during its atmospheric flight and particularly in the design of the autopiloting inner loop for the yaw axis of the launcher as illustrated by Fig. 1. The main function of this loop is to hold the launcher around its center of gravity which follows the guidance reference trajectory. G is the center of gravity, V and Vr are the absolute and relative velocity and W is the wind velocity. i is the angle of attack and is the deviation of the launcher from the yaw axis with respect to the guidance attitude
59
Multi-objective Control of a Space Launcher
Wind velocity w Launcher
Angle of attack i . Angular velocity Sensors
Attitude Thrusters
Noise
Commanded angle of deflection
Fig. 1.
Controller
+
c
Guidance attitude reference
ARIANE control loop.
reference (see Fig. 2). The control variable is the thruster angle of deflection . A thorough description of the plant may be found in the references [4,5,9].
Nominal Trajectory
x Vr
i w V
2.2. Model of the Launcher
G
To get a tractable analytical model for the plant, some simplifying assumptions have to be made. Thrust
.
Assumptions 1 Axis are decoupled. All angles remain small. Parameters are slowly time-varying.
Fig. 2. Model of the launcher.
The last assumption is motivated by the mildly nonstationary nature of the launcher. Moreover, it implies that an uncertain parametric LTI model is sufficiently realistic for design and analysis of robust autopilot for the whole range of the flying envelope. Indeed, as will be seen later, most of the design requirements are of time-invariant nature (flexible mode attenuation, gain/phase margins) and robust LTI controllers may therefore be evaluated during the atmospheric flight phase. Under these assumptions, the linearized dynamics of the launcher include a rigid model (without bending modes and actuator dynamics) where _ is the angular velocity and z_ the lateral drift rate. €ðtÞ ¼ A6 ðtÞ þ z_ðtÞ WðtÞ þ K1 ðtÞ V ð1Þ z€ðtÞ ¼ a1 ðtÞ þ a2 ðz_ðtÞ WðtÞÞ þ a3 ðtÞ:
Those equations depend on known aerodynamic coefficients ai , i ¼ 1, 2, 3 and time-varying coefficients A6 , the aerodynamic efficiency and K1 , the thruster efficiency along the flight envelope. _ z_ 0 , the Defining the state vector as x ¼ perturbation input as w and the controlled output as i, we get the following third-order state-space model for the rigid part of the launcher. 2
0
6 x_ ðtÞ ¼ 6 4 A6 a1 2 6 þ4
3 2 3 0 0 6 7 A6 7 7xðtÞ þ 6 A6 7W 0 4 V5 V5 a2 0 a2 3 0 7 K1 5uðtÞ 1
a3 zðtÞ ¼ iðtÞ ¼ 1
ð3Þ
1 1 xðtÞ W: 0 V V
Moreover, the angle of attack equation reads: z_ðtÞ WðtÞ : iðtÞ ¼ ðtÞ þ V
ð2Þ
Note that the launcher is aerodynamically unstable. The sensors dynamics (second order for and _ ) and the actuators dynamics (second order) are added
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Fig. 3. Bending modes of the ARIANE launcher.
Fig. 4. Illustration of the margins in the Nichols plot.
35
30
25
Wind velocity W
to the previous model to compose the complete linearized dynamics of the rigid launcher. Owing to lack of tests and system complexity, internal uncertainties and dispersions have to be faced. They mainly concern the propulsion system, aerodynamic coefficients, mass model and inertia, flexible modes (elastic and sloshing), actuators and sensors modelling. The wind (shear and gusts) is considered as an external disturbance. The first five bending modes are taken into account into the complete model (Fig. 3) and their characteristics are considered to be not exactly known (four uncertain parameters per mode) leading to the definition of a discrete-time uncertain LFT simulation model of order 17 and 2 R2222 .
20
15
2.3. Control Objectives and Performance Specifications During the atmospheric flight phase, the physical constraints and objectives defining the requirements that the autopilot must fulfill are the following. (1) Guidance demand tracking. (2) Robustly stabilize the launcher, i.e. the controller must be stabilizing for all parameters in a prespecified set . (3) Limit the aerodynamical loads or equivalently limit angle of attack deviations. (4) Limit the energy/power consumption of the controller. The physical constraints and the objectives become automatic control requirements shaping the autopilot. Closed-loop stability with sufficient stability margins (for a given flight time): Gain margins (low and high frequencies) must be respectively over LF and HF. Delay margin must be greater than one sampling period. In addition to the stability margins on the rigid part of the model, a specified attenuation of bending modes (X dB) has to be insured. The first flexible mode may be alternatively controlled with respect to the previously defined attenuation X dB or with respect to its phase (see Fig. 4).
10
5
0
0
20
40
60
80
100
120
t/Ts140 = 1950
Fig. 5. Wind profile.
The autopilot has to reject disturbances (wind and gusts) influence on the angle of attack i. In particular, angle of attack peaks must not exceed the limits imax in response to a typical wind profile presented in Fig. 5. This typical wind profile is composed of observed worst-case gust and worstcase wind speed drift. The previous performance specifications must be achieved without an excessive thruster consumption expressed as cumulated deflections. C¼
Tend X
jkþ1 k j:
ð4Þ
k¼Tini
The autopilot must make sure that robustness requirements with respect to uncertainties on the rigid and bending modes are fulfilled. This means that all previous control objectives have to be achieved for all possible configurations of the 22 uncertain parameters.
61
Multi-objective Control of a Space Launcher
3. Multi-objective H2 =H1 /Impulse-to-Peak
w31
Design
w32
Σ aug w2
Keeping in mind the high heterogeneity of the performance specifications, we adopt a multi-objective strategy for the control design procedure. Even if the problem of the synthesis of a robust autopilot achieving the different specifications may be naturally recast as a multi-objective control problem, it is first necessary to translate the control objectives and the performance specifications in terms of a set of design constraints involving adequate system norms. One therefore imposes the following: A bound on the impulse-to-peak performance of the closed-loop transfer W i with an additional filter modeling a typical wind profile to limit the angle of attack. A bound on the H1 performance of the sensitivity S to get a minimum modulus margin and therefore good gain margins. A bound on the H2 performance of the transfer between measurement noises and _ to reduce the consumption. A mixed roll-off and lead filter to attenuate the bending modes and control in phase the first one. The first step of the design procedure consists in choosing a particular model for synthesis purpose. Owing to the high complexity of the complete model of the launcher, a simplified model is usually utilized for design purpose. Here, this simplified model is composed of the rigid model defined by the equation of the torque equilibrium of the yaw axis of the launcher. The synthesis model also includes the sensors dynamics (second order) but is free of the bending modes and actuator dynamics. 3.1. Structure of the Controller As will be seen in the next section, the controller is composed of two parts. One part of the controller is tuned (mixed filter) and the other part is computed via multi-objective optimization leading to the particular structure for the controller detailed in Fig. 6. Note 2 B1 ¼ ½ B1
D21 ¼ ½ Dy1
Bi2p
Dyi2p
C1
3
6 7 B2 C1 ¼ 4 Ci2p 5
D11
C2 2 3 D1u 6 7 Dy2 D ¼ 4 Di2pu 5: D2u
w1
Wind gust model
+
z1 = z3 = β
Actuator
Rigid Launcher with sensors
z2 = i ψ . ψ
– Controller Mixed Filter
+ + + + delay delay
Multiobjective Controller
Fig. 6. Structure of the multi-objective controller.
also that the wind gust model acts like a weighting function which has to be tuned in the controller synthesis procedure. The complete synthesis model is then defined by aug . An H2 =H1 =impulse–to–peak multi-objective control problem is therefore set on aug to enforce the three first specifications. 3.2. The Multi-objective Control Problem 3.2.1. Definition Let the LTI discrete plant aug be given by its statespace minimal realization: 32 3 2 3 2 xk A B1 B xkþ1 4 zk 5 ¼ 4 C1 D11 D 54 wk 5, ð5Þ yk uk C D21 0 where x 2 Rn is the state vector, w 2 Rmw is the disturbance vector, u 2 Rmu is the input vector, z 2 Rrz is the controlled output vector and y 2 Rry is the measured output vector. The z and w vectors are partitioned as indicated in Fig. 7. 2 3 2 3 w1 z1 ð6Þ z ¼ 4 zi2p 5 w ¼ 4 wi2p 5 z2 w2
The associated matrices are therefore consequently partitioned. 2 3 D1 D1i2p D12 6 7 ¼ 4 Di2p1 Di2p Di2p2 5 D21 D2i2p D2 ð7Þ
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D. Arzelier et al.
As in the continuous-time case, a conservative LMI characterization may be used to compute a bound on the peak of the impulse response of a discrete LTI system. Theorem 1. If there exists a matrix Pcl 2 Sþ and a scalar satisfying ¼ min Pcl , Fig. 7. Standard model for multi-objective control.
s:t: Acl Pcl A0cl Pcl < 0
The controller K is given by its state-space realization: kþ1 ¼ AK k þ BK yk
? K is given by its stateThe closed-loop system space matrices: A þ BDK C BCK B1 þ BDK D21 Bcl ¼ Acl ¼ BK C AK BK D21 aug:
Ccl ¼ ½ C1 þ DDK C
DCK
Dcl ¼ ½D11 þ DDK D21 : ð9Þ
The multi-objective control problem is then defined in the following way. Problem 1. (multi-objective H2/H1/i2p control problem). Find a controller K in the set of internally stabilizing controllers K such that min K2K
Dcl D0cl I < 0
ð8Þ
u k ¼ C K k þ D K y k :
i wi þ c cons
s:t: ð10Þ
2 kaug: mod ? Kk1 mod
kaug: Wi
? Kk2i2p wi 2 kaug: cons ? Kk2 cons :
ð12Þ
Bcl B0cl Pcl < 0 Ccl Pcl C0cl I < 0 then kzk k2
8k 0.
Proof 1. First note that by Schur complement, all the inequalities defining the realizable set of (12) may be equivalently written with respect to Xcl ¼ P1 cl : A0cl Xcl Acl Xcl < 0
ð13Þ
B0cl Xcl Bcl < I
ð14Þ
1 C0cl Ccl Xcl < 0
ð15Þ
1 D0cl Dcl I < 0:
ð16Þ
Let x and z be the solutions of (11) when the impulsive input wki is applied with x(0) ¼ 0. wki ¼ ei k , jj 1, i ¼ 1, . . . , mw where ei 2 Rmw is the basis vector with 1 at the position i, 0 elsewhere and k is the unit pulse. Then 8 i: x0 ¼ 0
z0 ¼ Dcl ei
x1 ¼ Bcl ei
z1 ¼ Ccl x1 ¼ Ccl Bcl ei
xk ¼
Aclk1 Bcl ei
zk ¼ Ccl xk ¼
If performance specifications are often expressed in terms of H2 and H1 norms, impulse-to-peak performance is not so usual and deserve a thorough description.
From (13), we get
3.2.2. The Impulse-to-Peak Performance
So, 8 k 1 and 8 i, from (14) and (15),
At our knowledge, only references dealing with the continuous-time case may be found in the literature [3,15,18,19,23]. Let the closed-loop discrete-time plant be given by its minimal realization: ?K
xkþ1 ¼ Acl xk þ Bcl wk zk ¼ Ccl xk þ Dcl wk
xð0Þ ¼ 0
ð11Þ
ð17Þ
Ccl Aclk1 Bcl ei
x0kþ1 Xcl xkþ1 < x0k Xcl xk < < x01 Xcl x1 :
1 > 2 e0i B0cl Xcl Bcl ei
¼ x01 Xcl x1 > x0k Xcl xk > 1 x0k C0cl Ccl xk ¼ 1 z0k zk
So, kzk k2 < . Using the last inequality (16) leads to kz0 k2 < 2 :
&
63
Multi-objective Control of a Space Launcher
This LMI characterization of the impulseto-peak performance is new for discrete-time systems. As mentioned above, this characterization of the impulse-to-peak performance is conservative as may be a convex relaxation of a non convex problem. The evaluation of this conservatism is a difficult and open problem for which, at our knowledge, no direct approach exists so far. Here, we are interested in an equivalent alternative LMI characterization of a bound on the impulse-topeak performance of the closed-loop system (11). A generalization of the method in [13] where an extra matrix G is introduced for the test of H2 and H1 performances leads to the equivalent LMI optimization problem. Theorem 2. Let the following semidefinite programming problem be given: G
¼
min
Pcl , , G, F
s.t. Pcl Acl G <0 G0 A0cl Pcl G G0 Pcl Bcl <0 B0cl I I Ccl F <0 F0 C0cl Pcl F F0 I Dcl <0 D0cl I
Problem 2. (worst-case i2pk synthesis). Find a fullorder dynamic output-feedback controller K in the set of internally stabilizing controllers K. min kaug: ?Kki2p ¼ min sup kzkL1 , K2K
K2K w2I
ð19Þ
where w I ¼ fw 2 lm 2 jðwi Þk ¼ k , jj 1,
k ¼ 1, 2, . . . ,
for i 2 f1, 2, . . . , mw g
and ðwj Þk ¼ 0, k ¼ 1, 2, . . . for i 6¼ jg To address the impulse-to-peak performance synthesis problem, we need to relax the previous optimization problem (18) by setting G ¼ F. This relaxation is known as the generalized shaping paradigm [13]. Theorem 3. If there exist X 2 Rnn , Y 2 Rnn , ^ 2 Rmn , ^ 2 Rnn , ^ 2 Rnr , ^ 2 Rmr , B C D A þ þ nn Q 2 S n , H 2 S n , J 2 R , and a scalar G 2 Rþ such that: G ¼
min X, Y, J, H, Q, S
s:t ð18Þ
then ¼ G :
2
2
So far, the new proposed characterization (18) of impulse-to-peak performance seems not to bring any novelty. In fact, we will see later that it allows to reduce the conservatism of the proposed bound when dealing with multi-objective control problem. The reference [1] gives another application of this condition in a robust control context.
"
The previous characterization may be used to tackle the problem of designing a controller minimizing a bound on the impulse-to-peak performance.
J
6 6 ? H 6 6 ? 4 ? ? ? 2 Q J 6 4 ? H
Proof 2. The proof comes from the application of elimination lemma [16] on the first and third inequality of (12) &
3.2.3. Impulse-to-Peak Performance Design via LMIs
Q
?
6 4 ?
Q X X0 ? ^ Di2py Bi2p þ BD
7 ^ Di2py 5 < 0 YBi2p þ B
?
I ^ Ci2p X þ Di2pu C 0 QXX
?
3 ^C A þ BD 7 ^C 7 YA þ B 7<0 7 J S0 I 5 H Y Y0 3
^ AX þ BC ^ A
3 ^C Ci2p þ Di2pu D 7 J S0 I 5 < 0
?
^ Di2py Di2p þ Di2pu D
?
I
then a follows:
full-order
V01 U1 ¼SYX
ð20Þ
H Y Y0 # <0
controller
reconstructed
as
^ D K ¼D
^ D ^ CXÞU1 BK ¼VT ðB ^ YBD ^Þ CK ¼ðC 1 1 h i ^ YðAþBD ^ CÞXV0 BK CXYBCK U1 U1 AK ¼VT A 1 1 1 ð21Þ
64
D. Arzelier et al.
leads to bound the peak value of impulse response of the system: kzk1 <
2
pffiffiffiffiffiffi G :
B1cl B01cl P1cl 6 ? 4
Proof 3. From (18), it is easy to see that the matrix G is invertible and may be partitioned as follows: G¼
X U1
U2
H1 performance
G1 ¼
Y V1
V2 :
3 0 7 0 5
B1cl D01cl mod I þ D1cl D01cl
? 22
3 Acl 66 7 þ sym44 C1cl 5½ 0 0
?
P1cl
3
7 G 1 5 < 0
ð24Þ
I ð22Þ
Then applying the following similarity transformation on the closed-loop Lyapunov matrix, we get I 0 I Y Q J P ¼ : ð23Þ Y0 V01 cl 0 V1 J0 H I Y Multiplying the first three terms in (18) by 0 V1 and by its transpose, we get the final result by applying the change of variables (21). & This parameterization is a slight modification of the one proposed in Ref. [15] and an extension to the impulse-to-peak case of the one given for the first time in the context of H2 and H1 performances in [13]. Its main interest relies in the decoupling between the Lyapunov function and the computation of the controller. The controller is now built from the additional variable G allowing to use independent Lyapunov functions in the context of multi-objective control.
3.3. LMI Formulation of the Multi-objective Control Problem Let us come back to our original multi-objective control problem (10). Except in some particular cases, this problem is hard to solve analytically. Even if many sub-optimal numerical approaches have been developed, the reduction of the conservatism of the proposed solution is still a challenging problem, [15,18]. In the previous subsection, we have seen how to convert the original worst-case impulse-to-peak design problem to a convex optimization problems involving LMIs. The same technique may be applied to the H2 and H1 performance criteria as originally proposed in Ref. [13] and an extended LMI formulation may be formulated for H1 and H2 performances of respectively aug: mod: ? K and aug: cons: ? K.
H2 performance traceðTÞ < cons 2
I 6 4 B2cl
3 0 7 0 5 P2cl 3
B02cl P2cl
?
? 22
0 66 7 þ sym44 Acl 5½ 0
3 0
7 G2 5 < 0
ð25Þ
I
0 T þ D2cl D02cl ? P2cl C2cl ½ 0 G2 < 0: þ sym I
Defining the notations
I
Y0 I Y0 I
Y0
Q2 I Y ¼ P 2cl J02 V01 0 V1 Q1 0 I Y ¼ P 1cl 0 J01 V1 0 V1 " Qi2p 0 I Y P ¼ i2pcl 0 J0i2p V1 0 V1 0
J2
H2 J1 ð26Þ H1 # Ji2p Hi2p
a full-order output-feedback controller minimizing an upper bound for the multi-objective H2 =H1 =i2p synthesis problem may be computed using semidefinite programming and LMI formulation. Theorem 4. If the following semidefinite programming problem has a solution min X, Y, cons , wi , T, S Ji , Hi , Qi , i ¼ 1, i2p, 2
c cons þ i wi
65
Multi-objective Control of a Space Launcher
under 2
Q1 6 ? 6 6 6 ? 6 6 ? 6 4 ? ?
2
Qi2p 6 6 ? 6 4 ? ?
2
Qi2p 4 ? ?
2
wi 4 ? ?
wi ?
J1 H1 ? ? ? ?
^ AX þ BC ^ A Q1 X X 0 ? ? ?
^C A þ BD ^C YA þ B I S0 þ J1 H1 Y Y0 ? ?
^ AX þ BC ^ A Qi2p X X0 ?
Ji2p Hi2p ?
3 ^ Di2py Bi2p þ BD ^ Di2py 5 < 0 YBi2p þ B I
ð27Þ
3 ^C Ci2p þ Di2pu D Ji2p S0 I 5 < 0 Hi2p Y Y0
^ Di2py Di2p þ Di2pu D I
<0
TraceðT2 Þ < cons 2
T2 6 ? 6 4 ? ?
2
Q2 6 ? 6 6 ? 6 4 ? ?
^ C2 X þ D2u C Q2 X X 0 ? ?
J2 H2 ? ? ?
^C C2 þ D2u D 0 I S þ J2 H2 Y Y 0 ?
^ AX þ BC ^ A Q2 X X0 ? ?
3 0 7 0 7 7 0 0 0 0 ^ X C1 þ C D1u 7 7<0 7 ^ 0 D0 C01 þ C0 D 1u 7 ^ 0 D0 5 D01 þ D0y1 D 1u mod I
3 ^C A þ BD 7 ^C 7 YA þ B 7<0 Ji2p S0 I 5 Hi2p Y Y0
Ji2p Hi2p ? ?
^ Ci2p X þ Di2pu C Qi2p X X0 ?
^ Dy1 B1 þ BD ^ Dy1 YB1 þ B 0 0 I ?
3 ^ Dy2 D2 þ D2u D 7 0 7<0 5 0 I
^C A þ BD ^C YA þ B 0 I S þ J2 H2 Y Y0 ?
3 ^ Dy2 B2 þ BD ^ Dy2 7 YB2 þ B 7 7<0 0 7 5 0 I
66
D. Arzelier et al.
then a controller K constructed from (21) is a suboptimal solution to the multi-objective synthesis problem (10).
optimization step. The complete synthesis procedure therefore reads as Algorithm 1
Proof 4. The proof is very simple and consists in creating a particular Gi matrix for each performance constraint. Applying a generalized shaping paradigm G1 ¼ Gi2p ¼ G2 ¼ G ¼
X
U2
U1
G
1
Y ¼ V1
V2
(1) Choose the tuning parameters and form the augAug: mented plant Aug: . Extract Aug: Wi , cons and Aug: mod . (2) Solve the convex optimization problem via LMI optimization and get the decision variables. min ði wi þ c cons Þ under
ð28Þ
and the linearizing change of variables
^, B ^, D ^, C ^ , mod Þ < 0 L1m ðQm , Jm , Hm , X, Y, S, A ^, B ^, D ^, C ^ , wi Þ < 0 Li2p ðQi2p , Ji2p , Hi2p , X, Y, S, A ^, B ^, D ^, C ^ , cons Þ < 0 L2c ðQ2 , J2 , H2 , T2 , X, Y, S, A
^ ¼ DK D
ð31Þ
^ ¼ CK U1 þ D ^ CX C 0
^ ^ ¼ V1 BK þ YBD B 1
(3) Reconstruct the controller with formulae (21). ð29Þ
^ ¼ V10 AK U1 þ YðA þ BB ^ CÞX A 1 þ
V01 BK CX
4. Simulation Results
þ YBCK U1
leads to the result.
&
Notice also that the proposed approach is always less conservative than the one using the Lyapunov Shaping paradigm from [15].
3.4. Synthesis Procedure and Tuning Parameters As seen before, the whole controller is formed from a mixed filter in series with a multi-objective controller. We have therefore two different sets of tuning parameters. The first set is formed with the parameters defining the wind gust model and the mixed filter. The wind gust model is a second order with two tuning parameters while the mixed filter is composed of a low-pass filter multiplied by a lead filter: KW þ 2TW p þ T2W 1 þ ap Kro : wm ¼ 1 þ p 1 þ Tro p
wwind ðpÞ ¼
p2
ð30Þ
We have therefore six synthesis parameters: KW , TW , Kro , Tro , a and . The other set of tuning parameters is formed from the optimization parameters i =c and mod of the multi-objective LMI
The previous algorithm has been used to design a robust autopilot for the atmospheric flight of a space launcher. Some worst-cases parametric configuration where the combination of parameter extremal values is particularly critical has been identified. The synthesis has been performed considering an LTI worst-case configuration of the time-varying parameters of the launcher. During the atmospheric flight phase, the timevariant behavior of the launcher and physical behavior is validated by simulations using a SIMULINK# model where information about specifications fulfillment is provided to the user. The time-domain specifications are therefore verified by inspecting the time responses provided by time-varying simulations while the frequency domain specifications are checked by considering only some pre-defined worst-cases. Indeed, the robustness analysis of the frequency domain performances is limited to those worst-cases as the experience has shown that they are quite representative of the complete problem. As is clearly implied by the very different proposed specifications, different tradeoff have to be dealt with. The main ones are summarized as follows. Limitation of the angle of attack deviations against specified attenuation of bending modes and noise reduction. Delay margin and H.F. gain margin against B.F. gain margin.
67
Multi-objective Control of a Space Launcher Bode Diagram 40
.
30
Ψ
u
Magnitude (dB)
20 10 0 –10
Ψ
–20
u
–30 –40 90
Phase (deg)
0
–90
–180
–270
–360 –4
10
–3
10
–2
–1
10
0
10
1
10
10
Frequency (rad/sec) f/f s
Fig. 8.
Bode diagram of the multi-objective autopilot.
Specified attenuation (X dB) of higher order ending modes against the control of the first flexible mode with respect to its phase. As the energy/power consumption is directly related to the reduction of sensors noise, the limitation of the angle of attack deviations plays also against the reduction of consumption. Keeping in mind that all those specifications have to be fulfilled for every parametric configuration of the launcher, it is not surprising that the tuning of the controller is difficult and particularly sensitive to the associated trade-off. As mentioned in the introduction, other design techniques have been proposed by other members of the PIROLA group. In particular, concerning pure LTI techniques, an H1 design based on the cross standard form is presented in reference [22] and can be usefully compared to the results presented below. In addition, some elements of comparison are also given between the proposed technique (extended multi-objective autopilot) and the standard mixed design (a single Lyapunov matrix for all specifications). Note that a structure of control identical to the one used here has been defined for the standard mixed design. As will be seen in the following, owing to the very conservative nature of this kind of design, it is not possible to tune a standard mixed autopilot satisfying all requirements.
X db
–300
–180
–60
60
180
300
420
540
Fig. 9. Nichols plot of the open-loop.
First, the Bode plot of the multi-objective controller computed via algorithm 1 is presented in Fig. 8. The effect of the lead filter is obvious for the channel u. The open-loop frequency responses are given in Nichols charts in Figs 9 and 10 for several linearized models of specific configurations. This plot is obtained by freezing the model at the worst-case time
68
D. Arzelier et al.
configuration of the launcher (it coincides with the time when the wind gust is applied). The roll-off specification on bending modes is indicated by the X dB horizontal line. A zoom on a specific range of frequencies allows to verify gain margins imposed on the rigid model. Note also that the first flexible mode remains between two critical points for all worst-cases satisfying the phase control requirement for this mode. These plots may be compared with the one obtained with a standard mixed autopilot (Fig. 11) where the phase control requirement for the first flexible mode is not satisfied for some parametric
configurations. This implies that the requirement on the delay margin is not satisfied in this case. The gain margins (H.F. and B.F.) are also smaller than the ones of the extended multi-objective autopilot. Various time responses are also proposed. Note that all scales have been normalized with respect to the sample period Ts . The first figure shows the variation of the angle of attack when the typical wind profile in Fig. 5 is applied (Fig. 12). Note that the requirement on limitation of angle of attack deviation is hardly satisfied for the standard mixed autopilot (Fig. 13). The consumption C¼
Tend X
jðk þ 1Þ ðkÞj
k¼Tinit
M l db
Mh db
520
525
530
535
540
545
550
Fig. 10. Zoom on the Nichols plot.
555
560
is computed and plotted with respect to the maximum allowed consumption (Figs 14 and 15) . These plots show that the consumption is very good in both cases since they only need 50% of the maximum allowable consumption (usual consumption of a normal autopilot) to tackle the specifications. Note that in Refs [5] and [21], the consumption is 60% while the autopilot proposed in Ref. [4] is slightly better (44%) but with some violated specifications. Finally, the angle of deflection and its velocity are plotted below for the extended multi-objective autopilot (Figs 16 and 17). They are far from the
0 Xdb
180
180
540
Fig. 11. Nichols plot of the open-loop (standard mixed controller).
69
Multi-objective Control of a Space Launcher 100 90 80
Angle of attack i
70 60 50 40 30
i max
20 10
Fig. 12. Extended multi-objective autopilot.
0 10
20
30
40
50
60
70
80
90
100
Fig. 15. Standard mixed autopilot.
Angle of deflection β
Angle of attack i
β max
Tf
Fig. 13. Standard mixed autopilot.
− βmax
100
Fig. 16. Angle of deflection.
90
Cmax
70
60
. β max
50
40
30
20
10
0
Tf
Control velocity β
Consumption (C/Cmax )
80
Fig. 14. Extended multi-objective autopilot. . − β max
specified maxima because of the over dimensioning of the actuators. Note that the high frequency component of the response comes from the measuring noise modelled as a white noise.
Tf
Fig. 17. Velocity of the angle of deflection.
70
5. Conclusions The simulations have shown that this approach leads to controllers verifying all the specifications imposed on the launcher during the atmospheric flight. This black box-like method allows a great flexibility in the synthesis process. The tuning parameters are clearly identified with respect to the fundamental trade-off proposed by the performance specifications. Easy-touse macros of MATLAB based on convex semidefinite optimization solvers allow the designer to tune adequately the synthesis parameters. The main drawback of the proposed method comes from the fact that it can be used only with a simplified synthesis model (without bending modes) due to numerical considerations. Moreover, the parametric uncertainty of the model is not directly dealt with. In fact, parametric robustness concerns are inherited from time-varying nature of the model which is not taken into account here. Considering that the time spent to design a multi-objective controller is not prohibitive, the first step in order to take non stationarity into account would be to interpolate a finite number (5 or 6) of multi-objective controllers computed for a frozen configuration of the launcher all along the atmospheric flight envelope. Of course, this does not mean that finding an adequate interpolation algorithm is an easy task!
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convex polytopic uncertainty. Syst. Control Lett 2000; 40: 1 15. Scherer CW, Gahinet P, Chilali M. Multiobjective output-feedback control via LMI optimization. IEEE Trans Autom Control 1997; 42(7) 896–911 16. Skelton RE, Iwasaki T, Grigoriadis K. A unified algebraic approach to linear control design. Taylor and Francis, London, 1998 17. Tarbouriech S, Garcia G, Langouet P. Anti-windup strategy with guaranteed stability for linear systems with amplitude and dynamics restricted actuator. In: 6th IFAC-symposium on nonlinear control systems (NOLCOS’2004), Stuttgart, Germany, 1–3 September 2004, pp 1373–1378 18. Tokunaga H, Iwasaki T, Hara S. Multi-objective robust control with transient specifications. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, 1996, pp. 3482–3483 19. Tokunaga H, Iwasaki T, Hara S. Analysis and synthesis of the robust impulse-to-peak performance. Automatica, 1998; 34(11): 1473–1477 20. Voinot O, Alazard D, Piquereau A. A robust multiobjective synthesis applied to launcher attitude control. In: Proceedings of the 15th IFAC Symposium on Automatic Control in Aerospace, Bologna, Italy, 2001 21. Voinot O, Apkarian P, Alazard D. Gain-Scheduling H1 control of the launcher in atmospheric flight via linear-parameter varying techniques. AIAA guidance, navigation and—control conference, Monterey, CA, USA, August 2002, pp. 1–10 22. Voinot O, Alazard D, Apkarian P, Mauffrey et B. Cle´ment S., Launcher attitude control: discrete-time robust design and gain-scheduling. Control engineering practice 2003; 11(11): 23. Wilson DA. Convolution and Hankel operator norms for linear systems. IEEE Trans Autom Control 1989; 34(1): 94–97