AIAA 2010-8194
AIAA Guidance, Navigation, and Control Conference 2 - 5 August 2010, Toronto, Ontario Canada
Robust LPV control design for a RLV during reentry Nicolas Fezans ONERA / ISAE, 31055 Toulouse, FRANCE
∗
-
CNES, 91023 Evry, FRANCE
Daniel Alazard†
Nicole Imbert‡
Unversit´e de Toulouse - ISAE, 31055 Toulouse, FRANCE
ONERA, 31055 Toulouse, FRANCE
Benjamin Carpentier§ CNES, 91023 Evry, FRANCE In this paper a methodology is proposed to design reduced order Linear Parameter Varying (LPV) controllers for LPV mechanical systems. This methodology is based on a H∞ standard problem weighting the acceleration sensitivity function to take into account performance specifications. This standard problem is then augmented into an LFR (Linear Fractional Representation) -based problem to handle parametric dependency of both model and specifications. This non convex problem is relaxed and solved using LMI based DGZDGKH iteration procedure. The overall methodology is applied to a Reusable Launch Vehicle (RLV) whose model parameters are uncertain and depend on the Mach number and the dynamic pressure.
Nomenclature n q M, K, D F ξi ωi X(s) diag(Fi (s)) Fl (P, K) σmax (M ) α, β, µ p, q, r γ, χ δ. Cl , C m , C n I Va ρ g Sl
Number of degrees of freedom (d.o.f.) in the system Vector of the n d.o.f. Mass matrix, stiffness matrix, damping matrix Input matrix Specified damping ratio and pulsation for the i-th d.o.f.; i = 1, · · · , n Laplace transform of time-domain signal x(t) Diagonal transfer matrix with transfer functions F (i); i = 1, ·, n Lower Linear Fractional Transformation (LFT) of plant P and controller K Maximal singular value of matrix M Angle of Attack, sideslip angle and aerodynamic bank angle Roll, pitch and yaw rates Aerodynamic flight path angle, aerodynamic azimuth Variations of variable (.) with respect to linearization point roll, pitch and yaw aerodynamic coefficients Vehicle inertia matrix in body frame Aerodynamic velocity of vehicle’s center of mass Air density Gravitational acceleration Reference surface and length
∗ PhD Student, ONERA / CNES / ISAE at the time this work was made. Is now Research Scientist at the DLR - German Aerospace Center - Institute for Flight Systems, Lilienthalplatz 7, 38108 Braunschweig, GERMANY - Email:
[email protected] - Tel: +49 531.295.2653 † Professor, ISAE, 10 av. Edouard Belin - 31055 Toulouse Cedex, FRANCE - Email:
[email protected] - Tel: +33 5.61.33.80.94 ‡ Research Scientist, ONERA, 2 av. Edouard Belin - 31055 Toulouse Cedex, FRANCE - Email:
[email protected] - Tel: +33 5.62.25.27.86 § Scientist, CNES, Rond Point de l’Espace - 91023 Evry Cedex, FRANCE - Email:
[email protected] - Tel: +33 1.60.87.74.45
1 of 22 American Institute of Aeronautics and permission. Astronautics Copyright © 2010 by ISAE. Published by the American Institute of Aeronautics and Astronautics, Inc., with
I.
Introduction
In references1 ,2 and3 a general H∞ design based on the acceleration sensitivity function was proposed to control mechanical systems taking into account specifications on disturbance rejection performance and dynamic decoupling between degrees of freedom (d.o.f.). This particular H∞ standard problem will be called the SOTAS (Second-Order Template on Acceleration Sensitivity function) problem in the sequel. Basic principle of this standard problem was presented and was applied on Reusable Launch Vehicle (RLV) during atmospheric reentry to determine the worst-case flight instant along the reentry trajectory where the trade-off, between performance and robustness to high-frequency unstructured uncertainties, is the most difficult to tune.1 Some extensions were proposed to take an integral term (for input reference tracking performance) into account in this pure disturbance rejection problem.2 Lastly, an H2 /H∞ problem was derived to minimize energy consumption.3 In this paper, the robust LPV (Linear Parameter Varying) control design for an LPV and uncertain model of an RLV during reentry is considered. This approach is also based on the SOTAS H∞ problem. The parameters of the RLV during reentry are subject to strong variations according to dynamic pressure and Mach number and require the controller to depend on the flight conditions. A practical approach consists in scheduling various LTI (Linear Time Invariant) controllers designed in various flight conditions.4, 5 More recent developments address the direct design of a LPV controller from an LFR of the parameter dependant model.6, 7 These approaches are based on eigen-structures assignment and their parametric extensions. As the SOTAS problem is a frequency-domain disturbance rejection problem, LPV approaches based on robust H∞ design8 are more suitable but provide full-order controllers. Finally, a reduced-order robust LPV controller design method, called DGZ - DGKH iteration, is used in this paper.9, 10 In the next section, the SOTAS problem and its extensions are recalled. A particular focus is proposed on the way to reduce the set of solutions meeting the H∞ SOTAS constraint to the unique solution saturating the frequency domain template at each frequency. This extension allowes the energy consumption to be indirectly minimized and is an alternative of the H2 /H∞ problem proposed in3 but works better in the LPV context. Section III details the application to the RLV during reentry, that is: the problem considered, the LPV modelling, the robust LPV control design and the simulations results.
II.
Second-Order Template on Acceleration Sensitivity function (SOTAS) and extensions Basic SOTAS H∞ standard problem
II.A.
In a previous article,1 a H∞ standard problem weighting the acceleration sensitivity function was proposed for the control of mechanical systems represented by the following generalized second order differential equation: M q¨ + Dq˙ + Kq = F u, with: F invertible . (1) This H∞ standard problem is depicted in Figure 1a where (ωi )i∈J1,nK and (ξi )i ∈ J1, nK are the tuning parameters allowing to select closed-loop modes and disturbance rejection properties. Figure 1b represents a minimal realization of this SOTAS H∞ standard problem P (s). +
w
diag(s2 + 2 ξi ωi + ωi2) s
u
F
+
M
+
+
-1
+ +
z
w diag(2 ξi ωi) u
1/s
z +
2
1/s
x
D
F
+
M
+
+
-1
+
diag(ωi²)
1/s
1/s
+
x
D
+
K
K (a) full order
(b) minimal realization Figure 1. SOTAS standard problem: P (s).
Among all static feedbacks K satisfying the closed-loop specification: γ = kFl (P, K)k∞ ≤ 1, the partic2 of 22 American Institute of Aeronautics and Astronautics
ular solution K0 defined by: K0 = F −1 K − M diag(ωi2 )
D − M diag(2ξi ωi ) ,
leads to the following closed-loop properties:
(2)
• all the d.o.f are decoupled, ¨i (s) Q Wi (s)
• each d.o.f i has a second order dynamic behavior:
=
s2 , s2 +2ξi ωi s+ωi2
• the frequency-domain template on the acceleration sensitivity function is saturated at each frequency: σmax (Fl (P, K0 )(jω)) = 1 ∀ω. II.B.
SOTAS H∞ standard problem with state vector mapping
The previous SOTAS problem involves the state vector x = [q T , q˙T ]T . For some applications (like RLV), it could be interesting to use a first order state-space realization: " #" # " # A11 A12 x1 0 x˙ = + u, with: A12 invertible . (3) A21 A22 x2 Fe Although such a system can be described by the general second order form (1) with: q = x1 ,
M = In ,
D = −A11 − A12 A22 A−1 12 ,
K = A12 A22 A−1 12 A11 − A12 A21 ,
F = A12 F˜ ,
it could be easier to express the parametric dependency on the matrix Aij rather than on the matrix M ,D and K. Indeed, the RLV linear model described in Eq.(11) is under this form (3) and the parametric uncertainties on aerodynamics coefficients can be easily modeled in this form. So, the standard-form of Fig. 1b can be reformulated into the one of Fig. 2. This standard control problem weights x¨1 but allows to work with x1 and x2 . The main advantage of this formulation is to keep a realization based on physical variables of the system which improves control design engineers understanding and simplifies robust / parameter-varying realizations. +
A12
w
+
+
A12-1 u
F
1/s
+ + +
x2
A12 +
+
A22
+
diag(ωi²)
A11
+ +
z
diag(2 ξi ωi)
1/s A11
+
x1 x x2
A21 Figure 2. Block-diagram of the SOTAS standard-form based on the realization of equation (3)
II.C.
SOTAS H∞ standard problem with integral term
In, the SOTAS H∞ standard problem is augmented to take into account an integral term in order to ensure robust tracking performance in response to an input reference qc . This augmented standard problem is depicted in Fig. 3a and involves a third order weight on the acceleration sensitivity function. Note that this standard problem does not involve the input reference qc . From a practical point of view, if the optimal static state feedback provided by H∞ design is denoted [KR q , Kq , Kq˙ ], then the control law (taking into account qc ) which will be implemented is: 2
U (s) = (KR q
In ˙ + Kq )(Q(s) − Qc (s)) + Kq˙ Q(s) . s
By this way, the tuning of the integral term cannot degrade the disturbance rejection performance as it could be the case when the integral term is tuned in a second phase. Fig. 3b) represents the minimal realization of this standard form which will be used by the H∞ solver. 3 of 22 American Institute of Aeronautics and Astronautics
+
diag(2 ξi ωi + λi)
w
2
2
diag((s + 2 ξi ωi + ωi )(s + λi))
z
w
diag(λi ωi²)
s3 u
F
+
M
+
-1
+ +
1/s 1/s
1/s
x
u
F
+
M
-
D
+
+
diag(2 ξi ωi λi + ωi2)
+
+
K
-1
1/s
+
1/s
+
1/s
z
+
+
x
D K
(a) Full-order
(b) Minimal realization
Figure 3. SOTAS standard-form augmented with integrators
SOTAS H∞ standard problem with a template saturation criterion
II.D.
In its basic formulation, the SOTAS problem suffers from the fact that there is an infinity of solutions K meeting the specification (γ = kFl (P, K)k∞ = 1). Some of these solutions may correspond to undesirable behaviors of the closed-loop (very-fast modes for instance). Although classical H∞ solvers do provide satisfying solutions in practice, it is relevant to restrict the set of optimal solutions to the nominal solution K0 given by Eq.(2) by modifying the control design problem. In order to illustrate the physical meaning of the SOTAS criterion and the type of solutions that should be rejected, the relation between closed-loop eigenvalues and H∞ -norm on the SOTAS criterion (γ) is displayed in Fig. 4a in the 1 d.o.f. case. In Fig. 4, three different static controllers Ki i = 1, 2, 3 are analyzed from: • closed-loop dynamic (eigenvalues of Fl (P, Ki )): Figure 4a • closed-loop singular value (σmax (Fl (P, Ki )(jω))): Figure 4b. Although each controller meets the spec: kFl (P, Ki )k∞ = 1, only the controller marked with black stars saturates the template for all frequencies. This controller corresponds to the nominal one shown in Eq.(2). Fig. 4 also shows the iso-H∞ -norm curves depending on the poles of the system: curves are drawn with increments of 0.4 dB. Singular Value 3
1.2
2
1
1
0.8
imag
0.6
0.4
0.2
Singular velur (dB)
0
−1
−2
−3
−4
−5
−6
−1.4
−1.2
−1
−0.8 real
−0.6
−0.4
−0.2
0 0
−7 −1 10
0
1
10
10
2
10
Frequency (rad/sec
(a) iso-H∞ -norm in the s-plane plotted with a step of 0.4 dB (first one is 0.4 dB, level 0 dB = 1 corresponds to the white lower-left region).
(b) Singular values of the closed-loop system Fl (P, Ki ).
Figure 4. SOTAS modal and frequency-domain properties
This problem was addressed by means of a complementary H2 criterion, thus leading to a mixed H2 /H∞ control design problem3 . Analytical solution to this problem could be expressed in the one degree-of-freedom 4 of 22 American Institute of Aeronautics and Astronautics
case and a proposition was made for the other cases. This approach works quite well for stationary control design but is more difficult to tune on LPV problems. Here a new formulation allowing the set of optimal solutions to be drastically reduced is introduced by extending the original H∞ criterion. This formulation consists in adding a new controlled output (called ”template saturation criterion” ) which weights the difference between the disturbance w and the previous SOTAS output through a low pass filter W2 (s). The standard problem is then depicted in Fig. 5. + -
w
z
+
diag(2 ξi ωi) diag(ωi²)
u
W2(s)
F
+
M
+
+
-1
+ +
1/s
+
+
1/s
x
D K
Figure 5. SOTAS standard-form with template saturation criterion
Filter W2 (s) introduces some new parameters to be tuned, for instance τi and Gi for the choice: G1
1+τ1 W2 (s) =
s
..
. Gn 1+τn s
.
(4)
Fig. 6 illustrates the iso-H∞ -norm curves obtained for 4 combinations of these parameters with the same representation than Fig. 4a. It appears clearly on this figure that the set of optimal solutions of this SOTAS+template saturation criterion is drastically smaller as the one of the classical SOTAS criterion shown in Fig. 4a. Norm of the optimal solution is still equal to 1 and its interpretation in terms of disturbance rejection and modal properties has been kept.
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1 d.o.f weighting function spec : ω = 1 , ξ = 0.71 G = 5 , τ = 1/10 (rad/s)−1
1.2
1.2
1
1
0.8
0.8
imag
imag
1 d.o.f weighting function spec : ω = 1 , ξ = 0.71 G = 5 , τ = 1/2 (rad/s)−1
0.6
0.6
0.4
0.4
0.2
0.2
0
−1.2
−1
−0.8 −0.6 real
−0.4
−0.2
0
0
−1.2
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
−1.2
−1
−0.8 −0.6 real
−0.4
−0.8 −0.6 real
−0.4
−0.2
0
1 d.o.f weighting function spec : ω = 1 , ξ = 0.71 G = 10 , τ = 1/10 (rad/s)−1
imag
imag
1 d.o.f weighting function spec : ω = 1 , ξ = 0.71 G = 10 , τ = 1/2 (rad/s)−1
−1
−0.2
0
0
−1.2
−1
−0.8 −0.6 real
−0.4
−0.2
0
Figure 6. H∞ norm of the SOTAS performance channel with template saturation criterion of 4 cases (gain and pulsation of the W2 filter).
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III. III.A.
Application to Robust LPV control design for a Reusable Launch Vehicle during reentry Problem statement
Problem considered is the control design for a Reusable Launch Vehicle during atmospheric reentry phase. This section provides information on the control strategy, the flight domain and the control requirements that are considered in this application. III.A.1.
Flight domain
Thr flight domain of such an RLV is defined by: • a domain in the M ach, Qb (dynamic pressure) plane,
• a domain in the α, M ach plane, • a domain in the β, M ach plane.
In this application, the entire hypersonic phase of the reentry is considered. For simplicity the LPV control design will be performed in a reduced domain, but simulations are performed over the entire hypersonic envelope. This reduced domain for synthesis has been defined as an hypercube containing a part of a typical reentry trajectory for this vehicle. Fig. 7 shows the reentry trajectory parameters (α,M ach,Qb ) vs time, in blue. The dark-pink rectangle represents the domain taken into account for the LPV synthesis in this application. The light-pink rectangles represent the domain on which the designed controller has been tested: it is a simplified version of the real flight envelope. The flight domain considered in terms of sideslip β was defined as follows: −3◦ < β < 3◦ . III.A.2.
Control requirements
For all the flight points and all uncertainty cases, the requirements are the followings: • pulsations ωi must be greater or equal to 2 rad/s,
• damping ratios ξi must be greater or equal to 0.71, • and all degrees-of-freedom must be decoupled,
Note that control requirements used in this application are the same for all d.o.f. and constant along the trajectory in order to plot in the s-plane the evolution of closed-loop eigenvalues over the entire flight domain (see for instance Fig. 13). Uncertainties are all the uncertainties on model parameters and the modeling error that can be introduced during transformations of the model (see. section III.D). For most model parameters the uncertainty bounds were roughly equivalent to a 5 to 40% multiplicative error bound. III.B.
3 d.o.f nonlinear model
As the dynamics of the translational motion can be neglected for reentry applications, only the 3 degrees-offreedom corresponding to vehicle’s rotational motion were kept in the nonlinear model. Indeed, translation motion will be seen by the control loop as a very low-frequency disturbance: such a disturbance will easily be rejected by the control law. ˙ Cl p p p 1 2 −1 −1 Cm − I q ∧ I q q = ρ S Va l I 2 Cn r r r α˙ = −p cos(α) tan(β) + q − r sin(α) tan(β) − γ˙
cos(γ) sin(µ) cos(µ) − χ˙ cos(β) cos(β)
β˙ = p sin(α) − r cos(α) − γ˙ sin(µ) + χ˙ cos(γ) cos(µ)
sin(α) cos(α) µ˙ = p +r + γ˙ cos(µ) tan(β) + χ˙ (sin(γ) + cos(γ) sin(µ) tan(β)) . cos(β) cos(β) 7 of 22 American Institute of Aeronautics and Astronautics
(5) (6) (7) (8)
8000
Dynamic Pressure (Pa)
7000 6000 5000 4000 3000 2000 1000
0
200
400
600
800
1000
1200
1400
1600
1800
0
200
400
600
800
1000
1200
1400
1600
1800
0
200
400
600
800
1000
1200
1400
1600
1800
25
Mach
20
15
10
5
40
alpha (deg)
35
30
25
20
15
time(s)
Figure 7. Considered flight enveloppes for control design
III.C. III.C.1.
Trim and Linearization Trim
In order to obtain linearized models of attitude dynamics of the Reusable Launch Vehicle for specific flight points, the system must be trimmed at these points. Trim basically consists in finding a set of commands u that makes the desired state xe be an equilibrium point of the system at each considered flight point θe . 0 = x˙e = f (xe , θe , u)
(9)
The Reusable Launch Vehicle used for this application could easily be trimmed over the entire flight domain. For the simplicity of the application, we assume here that at least one actuator (or a combination of actuators) can generate pitch without generating neither roll nor yaw. This assumption leads to consider only the pitch axis for trimming. Indeed, we would proceed similarly if we consider also the other axes. Nevertheless, we will later ensure that the control law will robustly compensate any left-right un-symmetry of the vehicle or any trim error, which could result from model errors. " # u1 Without loss of generality, we assume here that the input vector u can be decomposed into u = , u2 where u1 are commands only acting on pitch axis and that can be used for trimming and u2 are all other
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commands. Practically, trim were performed by solving the following problem:
ue = argmin f
u1
#! 2
u1
xe , θe ,
u2 = 0 "
(10)
by means of a quasi-newton optimization algorithm (BFGS11 ) where gradients were numerically approximated at each step. III.C.2.
Linearization
Linearization of model equations can be performed analytically (11): δα, δβ, δµ, δp, δq, and δr represent variation of α, β, µ, p, q, and r with respect to their equilibrium values. For practical reasons in this application, linearization has been performed numerically.
Zα m Va
0 ˙ δα 0 δβ δµ Clα = δp 1 2 −1 Cmα δq ρ V S l I a 2 δr Cnα
Yβ m Va 0
g cos(γe ) cos(µe ) Va −g cos(γe ) sin(µe ) Va 0
Clβ
Clµ
l Clp Va
l Clq Va
Cmβ
Cmµ
l Cmp Va
l Cmq Va
Cnµ
l Cnp Va
l Cnq Va
0
Cnβ
+
1 ρ Va2 S l 2
"
0
1
0
sin(αc )
0
−cos(αc )
cos(αc )
0
03 I −1
#
Clδ 1 Cmδ 1 Cnδ 1
δα sin(αc ) δβ l Clr δµ Va δp l Cmr δq Va δr l Cnr
(11)
Va
··· ··· ···
Clδ n Clδ n Clδ n
δ1 . . . δn
Reading these linearized equations, model dependency on ρ, Va , αe , and µe is obvious. Of course also depends on altitude h and Mach number M ach. As control objectives tend to keep β = 0, numerical linearizations were all performed with βe = 0. Y Coefficients mZαVa , m βVa , g cos(γeV)a cos(µe ) , and −g cos(γVea) sin(µe ) are responsible for the effects of translational motion onto rotational motion. At this step, we can also verify that these terms could really be neglected. III.D.
LPV modelling
Linearized models of the Reusable Launch Vehicle during reentry depend on the current flight points, control surfaces deflections, and even state values. Unification of all these linearized models leads to write a quasi linear parameter-varying model (quasi-LPV). Mathematical representation of such system can be done with several formalisms. In general, these formalisms can also take uncertainties into account. A high amount of work for analysis or synthesis have been performed using the Linear Fractional Transformation (LFT). Some very efficient tools allow to manipulate this formalism and permit to use the main results of the literature. In this application, we used the LFR-toolbox12 to create and manipulate our LPV, quasi-LPV, and uncertain systems. There are 2 major ways of obtaining a quasi-LPV model from a nonlinear one: • direct computation of the model from the knowledge of the f function of the nonlinear parameterdependent differential equation x˙ = f (x, θ, u), • indirect computation by means of several linearizations all over the domain in which parameters are included followed by the approximation of all linearized models by a single quasi-LPV model.
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In the case of the Reusable Launch Vehicle during reentry, it appeared that we could neglect the fact that some parameters of the system are also part of the state-vector. Which leads to approximate the quasi-LPV model obtained by a LPV model. For practical reasons, in this application we have chosen the fore-mentioned indirect computation. Although all linearized models (LTI) had the same state-space realization, each coefficient of the state-space realization has been modelled separately. Reason for proceeding so is that the differences in terms of magnitudes make it indeed easier to model them separately than simultaneously. The method used to model each coefficient was the following one: • choice of a set of models Es from the global set of available models E, • choice of a polynomial or rational parameter dependency form, • optimization of the LFT realization of the considered term based on a least-square criterion evaluated by means of the Es set, • choice of a validation set of models Ev chosen within the set of model that are not used for the LFT optimization E\Es , • computation of error criteria between the values in the validation set Ev and the values obtained by evaluating the LFT at the same points, • if all computed errors are acceptable, then try to find simpler approximation that are still acceptable, otherwise try other forms of dependency possibly more complex. In such an approximation process, one always has to make some trade-off between the complexity of the LFT realization and its quality. As errors must be taken into account by means of uncertainties, the question is then to know whether it is more suitable to approximate a complex function by means of a high order polynomial or rational or to keep a quite simple one with quite large uncertainty bounds. This is illustrated by means of Fig. 8, where a function f (θ) is approximated as an affine function plus an uncertainty, using the inclusion described in Eq. (12). f(θ)
y=aθ+b ±δθ
θ
Figure 8. Affine uncertain representation of a nonlinear parameter dependancy
∀θ ∈ [θmin , θmax ], a θ + b − δθ ≤ f (θ) ≤ a θ + b + δθ .
(12)
Keeping the order of the LFT realization quite low is very important since it has strong effects onto computation time, numerical conditioning, and conservatism of some techniques that can later be applied to this model. It most cases, the coefficients of the realization are not all equally important and it is often very interesting to represent some of them with a very good accuracy and the others very roughly, rather than each one of them with a similar accuracy. In order to put this into relief, one may use some other criterion, such as frequency responses of the system. Fig. 9 illustrates a comparison between a linearized open-loop model and the open-loop model obtained by evaluating the LFT at the same flight point: the transfer between a perturbation on the acceleration and the states is analyzed here by means of the singular values plot. On such open-loop models the frequency response often has some peaks that make it difficult to validate automatically such approximations as any slight error on peaks frequency generates high errors 10 of 22 American Institute of Aeronautics and Astronautics
/ residuals. For instance on the example of Fig. 9, quite high error would have computed due to slight modification of peak frequencies, although the LFT model is representative enough. Improvements on LFT approximations validation methods allowing to efficiently handle such frequency responses with resonance peaks would be interesting for the quasi-automatic obtention of a good trade-off between complexity of the LFTs and representativeness of them. Model #2312
80
Singular Values (dB)
60 40 20 0
− 20 − 40 −1 10
0
10 Frequency (rad/sec)
1
10
Figure 9. Comparison in open-loop between the LTI model (blue line) and the LFT model (red crosses) using singular ¨ µ values of the transfer “disturbance on [α, ¨ β, ¨ ] → state [α, β, µ, p, q, r] of the system”
III.E. III.E.1.
Robust LPV control design Standard form
Standard-form used for the LPV control design is depicted in Fig. 10 and involves the various extensions proposed in previous section. One can recognize: • the SOTAS problem using physical state space vector [α, β, µ, p, q, r]T : section II.B, • the integral terms taken into account in zone 1: section II.C, • the template saturation criterion in zone 2: section II.D, The parametric variations and uncertainties were taken into account by means of the LFTs that represent F (θ), A21 (θ), and A22 (θ). Note that A11 is null here (and thus not shown on this figure) and that A12 is constant (as synthesis domain is based on a constant value of α). The algorithm that is used to solve this robust LPV control design problem is in fact an algorithm of robust synthesis. Thus it allows the synthesis of linear controller maximizing the worst performance index obtained over the whole uncertainty domain, which can be represented by finding K in the interconnection of Fig. 11a. Indeed, we are here looking for a LPV controller maximizing the robust performance of an uncertain LPV standard-form. Fig. 11b shows the type of problem to be solved: K(s, ∆c ) is the LPV controller (written as an LFT) and P (s, ∆) is the uncertain LPV system. By chosing the structure of controller’s ∆c block before performing the synthesis, the robust LPV control design problem can be rewritten as a robust control design problem of a static controller, as shown on Fig. 12a. III.E.2.
Robust LPV feedback synthesis
Control design is performed with a robust control algorithm proposed by Roos.9, 10 We will later refer to it with the name of (D,G,Z)-(D,G,K,H) iteration. This method uses iterative resolution of Linear Matrix
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Zone 2
+
-
diag(2 ξi ωi)
+
diag(ω ) 2 i
W2(s)
+
+ z
+
diag(2 ξi ωi+ λi)
w
A12
Ξ|
u
F
+
2 i
+
diag(λi ω )
-1 A12
Δf
+
+
+
-
+
diag(2 ξi ωi λi+ ωi2)
1/s
A12
1/s
1/s
Δ22 +
Zone 1
∫( ) α-αc β-βc μ-μc α-αc β-βc μ-μc
( )
y
p q r
( )
A22
+
Δ21 A21 Figure 10. LPV control design standard-form
P(s,Δ) Δ In s
P(s, Δ) w
Δ
w
z
A
B1
B2
C1
D11 D12
C2
D21 D22
z
K(s,Δc ) Δc
P(s)
Im s
u
K(s)
y
AK
B1
C1
D11 D12
C2
D21 D22
K
Fl ( P(s) , K(s) ) (a) Standard-form for robust control design
K
K
B2
K
K
K
K K
(b) Interconnection for scheduled control design
robust
self-
Figure 11. Differences between robust control design and robust self-scheduled control design.
Inequalities (LMI) as an heuristic to address a Bilinear Matrix Inequality (BMI) derived from the KalmanYakubovic-Popov (KYP) lemma. It allows simultaneous robust and fixed-order synthesis of both a feedback controller and a feedforward controller under LTI and LTV uncertainties. In this application, neither the capability to design a feedforward controller nor the capability to handle LTV uncertainties were used. Control design problem is based the standard form of Fig. 10 with the reformulation of Fig. 12b. Resolution algorithm is initialized with the LPV form K0 (θ) of the nominal controllers K0 associated with the SOTAS problem at each flight point Eq.(13). These controllers were introduced in previous articles1–3 .
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Δ
Δ Δc
Δc
Pa(s) P(s)
Pa(s) P(s)
In
In
s
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s
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B1
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z
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(a) General case
K K
K
B2
K K K
(b) By means of the (D,G,Z)-(D,G,K,H) iteration: direct tranfer from input 1 to output 3 is null.
Figure 12. Interconnection for robust LPV control design by means of a robust control design algorithm.
These controllers ensure pole placement at specified pulsations (ω(θ)i )i∈J1,nK and damping ratio (ξ(θ)i )i∈J1,nK and decoupling of the degrees-of-freedom, for the nominal system at each flight point. In the simplified application that is considered here, specified pulsations and damping ratios do not depend on the parameter θ: Eq.(13-14) are written for the general case though. These 2 equations are equivalent: Eq.(13) is based on the second-order realization, whereas Eq.(14) is based on the realization of Eq.(3) assuming A11 = 0. K0 (θ) = ˜ 0 (θ) = K
F (θ)+ K(θ) − M (θ) diag(ωi (θ)2 ) D(θ) − M (θ) diag(2ξi (θ)ωi (θ)) , (13) + 2 F (θ) −A12 (θ)A21 (θ) − diag(ωi (θ) ) − A12 (θ)A22 (θ) − diag(2ξi (θ)ωi (θ))A12 (θ) ,(14)
By using the (D,G,Z)-(D,G,K,H) iteration procedure on this standard problem and with K0 (θ) as initial controller, we will robustify the rejection properties of the controller and its decoupling properties. As the use of constant scalings (D and G) induces some conservatism, the convergence to the 1 value is quite long and even uncertain. A practical “trick” can be used: it basically consists in slightly overspecifying the control design problem (slight increases of (ωi )i∈J1,nK , (ξi )i∈J1,nK , and (λi )i∈J1,nK ) and to stop the resolution before reaching the optimal value 1. In this application, this trick has been used and increased values of design parameters used were: • ∀i ∈ J1, nK, ωi = 2.2 rad/s (instead of 2 rad/s), • ∀i ∈ J1, nK, ξi = 0.73 (instead of 0.71),
• ∀i ∈ J1, nK, λi = 0.6 rad/s (instead of 0.4 rad/s).
Template saturation filter W2 (s) is chosen as a diagonal pass-band filter (∀i ∈ J1, nK, 1/τi < ai ) that reads: 1 s W2 (s) = diag Gi , (15) τi s + 1 s + ai and whose parameters are chosen as follows: 13 of 22 American Institute of Aeronautics and Astronautics
• ∀i ∈ J1, nK, Gi = 3,
• ∀i ∈ J1, nK, τi = 1/1.2 s/rad, • et ∀i ∈ J1, nK, ai = 8 rad/s.
One also must ensure that all assumptions of the (D,G,Z)-(D,G,K,H) iteration are valid for the considered problem. In particular in Proposition 4.2 of Ref.9 the matrices D31 , D32 , and D33 must be null. On the interconnection of Fig. 12a, we have D31 = In 6= 0. As a consequence a low-pass filter with high enough cut-off frequency Wlp is added to the standard-form as shown on Fig. 12b. Resolution will thus be made ˜ which is the concatenation of with the (D,G,Z)-(D,G,K,H) iteration algorithm seeking a static matrix K controller’s LFT realization matrices (16): full LFT realization of the controller can easily be recovered from ˜ and ∆c . K AK B1K B2K ˜ = K (16) C1K D11K D12K C2K D21K D22K After 10 iterations, the robust performance level of 1.957 has been reached on this problem. This performance index is an upper-bound of the real performance index. In order to compute the performance index reached on the real problem (not the slightly over-constrained one) by both initial and final controllers, a frequency-based performance analysis method that were introduced in13 is used: this method computes an upper-bound of the robust performance. Computed upper-bounds for these controllers are: • 2.276 for the initial controller, • 1.215 for the controller obtained by means of this robust LPV control design. Fig. 13 shows the poles of 150 LTI models corresponding to many flight points and uncertainty cases. On this figure, the domain defined by a minimal damping ratio of 0.71 and a minimal pulsation of 2 rad/s is shown by means of a black line on its border. As expected, the poles of the closed-loop have been shifted during the synthesis. Poles on the real axis are the one corresponding to the integrators. Pole−Zero Map 3
0.86
0.76
0.58
Pole−Zero Map 3
0.35
2 0.96 1.5 0.982 1
0.58
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2 −2
1 −1
2 0.96 1.5 0.982 1
0.5 0.996 6 0 −6
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Imaginary Axis
Imaginary Axis
2.5 0.92
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0.5 0.996 5 −5
4 −4
3 −3 Real Axis
2 −2
1 −1
(a) Initial (analytical) controller: γ ≤ 2.276
6 0 −6
0
5 −5
4 −4
3 −3 Real Axis
0
(b) After robust LPV control design: γ ≤ 1.215
Figure 13. Poles of 150 LTI closed-loop models taken randomly in the union of parameter and uncertainty spaces
Fig. 14 represents the weighted acceleration sensitivity functions of 50 LTI models corresponding to many flight points and uncertainty cases. Weight used is the classical SOTAS criterion. During the control design, worst case of disturbance rejection has improved as well as the robustness of the dynamical decoupling of the degrees-of-freedom. Initial controller was decoupling perfectly the nominal system at all flight points, but quite important couplings could be observed in the presence of uncertainties: robustness of the decoupling has significantly been improved with the final controller.
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Bode Diagram From: In(1)
From: In(2)
From: In(3)
5
To: Out(1)
0
−5
−10
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−20 5
To: Out(2)
Magnitude (dB)
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10 10 Frequency (rad/sec)
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10
(a) Initial (analytical) controller: γ ≤ 2.276, strong couplings con be observed on the 3rd row, 2nd column transfer. Bode Diagram From: In(1)
From: In(2)
From: In(3)
5
To: Out(1)
0
−5
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Magnitude (dB)
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10 10 Frequency (rad/sec)
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10
(b) After robust LPV control design: γ ≤ 1.215 Figure 14. Magnitude Bode diagram of the weighted acceleration sensitivity functions of 50 LTI closed-loop models taken randomly in the union of parameter and uncertainty spaces. Weight is the basic SOTAS weighting function.
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III.E.3.
Feedforward
A very simple dynamic feed-forward controller has been manually tuned to satisfy time-domain templates. It basically consists of a low-pass filter W (s) on the references [δαc , δβc , δµc ]T : 1 τα s + 1 W (s) =
1 τβ s + 1
1 τµ s + 1
,
(17)
with tuning parameter where chosen as follows: • τα = 0.5 s/rad, • τβ = 0.5 s/rad, • τµ = 2 s/rad . III.F.
Simulation results
Some simulations results are presented in this section in order to show the effectiveness of the controller on the nonlinear simulator, over a flight envelope significantly larger than the one considered for the synthesis and with turbulence. The nonlinear model includes all the actuator dynamics (neglected in the linearized model) as well as a filter that splits pitch commands depending on command signal frequencies: low frequencies are applied by means of a body flap whereas high frequencies are applied by means of symmetrical deflection of the elevons. Saturations on control surfaces positions are also implemented in this model. III.F.1.
Along a reentry trajectory
This first simulation (see Fig. 15-17) is performed all along a reentry trajectory. This simulation illustrates the capacity of the designed LPV controller to control the entire trajectory. Deflection rate limitations are not satisfied at the beginning of the trajectory (see Fig. 16), but it is a consequence of the simplifications that were made for this application. Indeed, on the real vehicle, requirements on closed-loop modes on this part of the trajectory were not to have a pulsation greater or equal to 2 rad/s but 0.8 rad/s. Moreover, with the low dynamic pressures that are encountered, bounds on deflection rates were larger. Here, the choice of taking a single specified pulsation for the whole flight envelope has been made in order to able to show the pole shifting capability of the method (see Fig. 13). This method can handle specifications that vary over the flight envelope, without any increase in complexity. The behavior of the vehicle matches the expected one. Errors on α, β, and µ stay in a acceptable domain. Parameter adaptation of the controller permits to control the entire flight envelope with this single controller. III.F.2.
Bank maneuver
Time scale on previous simulation is not adequate to analyze in details the closed-loop behavior. We detail in Fig. 18-20 a bank maneuver. These maneuvers are typical of atmospheric reentry trajectories: they allow to manage the total energy of the vehicle. On this simulation and more precisely on Fig. 18, we observe that: • Degrees-of-freedom decoupling is satisfying. In particular the high amplitude of the maneuver on µ does not lead to significant increase of the errors on α and β. • At the very beginning of the simulation an oscillation can be observed though α and q: this oscillation is due to a wrong initialization of the states of the controller and of the filter that separates pitch actions onto the body flap and the elevons. A better initialization can be made, but this one was kept as it shows how well the closed-loop counteracts this initial condition. • The last plot in Fig. 19 shows how trim error are compensated through the integral terms. In order to show it, errors in the model used for trimming were voluntary introduced which then resulted in this error that the controller must compensate. As we also kept the wrong initial value for the controller 16 of 22 American Institute of Aeronautics and Astronautics
40 35 30 25 20 15 10 5
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600
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−2 −4 −6
p (deg/s) 400
600
40 20 0 −20
mu (deg)
−40 −60
reference (deg) 400
600
Figure 15. Long time-varying simulation with turbulence - States and references. Abscissa in seconds.
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elevons − symmetric deflection (deg) elevons − antisymmetric deflection (deg)
0 400
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external saturation limit 0 400
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600
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body flap deflection (deg)
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Figure 16. Long time-varying simulation with turbulence - Commands. Abscisse en secondes.
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2.5 0 -2.5 −5
400
error on alpha (deg) 600
800
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Figure 17. Long time-varying simulation with turbulence - Errors. Abscissa in seconds.
and the filter (see preceding remark) it is difficult to really compare the time response observed here with the one of a first-order system with a pole in −λi = −0.4 rad/s. Nevertheless, typical response time of such a first-order system (2.5 seconds for 63% and 7.5 seconds for 95%) are approximately the values that can be observed here. These observations validate that the closed-loop constituted of the nonlinear model and the LPV controller has the desired behavior: disturbance rejection until chosen pulsation, robust trim compensation and reference tracking, and degrees-of-freedom decoupling.
IV.
Conclusion
A complete methodology based on the acceleration sensitivity function was developed. This basic H∞ was extended to take into account: • the need to work with a physical meaning state vector to express easily the parametric uncertainties and variations, • an integral term to ensure robust tracking performance, • a template saturation criterion to isolate the control allowing the modal requirement to be met. This approach was applied to the RLV control design during atmospheric flight to design a robust LPV static state feedback using an LMI based solver. Non-linear model, LPV modeling for control design purpose and non-linear simulation results were presented. The obtained results are quite promising and further works will be focused on numerical aspects to improve the LMI based solver and apply it on the RLV problem in the dynamic output feedback case.
References 1 Fezans, N., Alazard, D., Imbert, N., and Carpentier, B., “H ∞ control design for multivariable mechanical system Application to RLV reentry,” Proceedings of the 2007 AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA, Hilton Head Island, SC, August 2007, pp. 1–13.
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34 33.5 33 32.5 32 1390
alpha (deg) reference (deg) 1400
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40 30 20 10 0 !10 !20 !30 !40 1390
p (deg/s) 1450
mu (deg) reference points (deg) reference interpolation (deg) 1400
1410
1420
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Figure 18. Bank manoeuver - States and references. Abscissa in seconds.
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1450
0.0 elevons − symmetric deflection (deg) 1390
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0 1390
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deflection rate limits 1430
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expected body flap trim deflection (deg) trim correction (deg)
1390
1400
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body flap deflection (deg) 1430
1440
Figure 19. Bank manoeuver - Commands. Abscissa in seconds.
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1450
0.6 0.4 0.2 0.0 !0.2 !0.4 !0.6 1390
error on alpha (deg) 1400
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error on beta (deg) 1400
1.0 0.5 0.0 !0.5 !1.0 !1.5 1390
error on mu (deg) 1400
Figure 20. Bank manoeuver - Errors. Abscissa in seconds.
2 Fezans, N., Alazard, D., Imbert, N., and Carpentier, B., “H ∞ control design for generalized second order systems based on acceleration sensitivity function,” Proceeding of the 16th Mediterranean Conference on Control and Automation, June 2008. 3 Alazard, D., Fezans, N., Imbert, N., and Carpentier, B., “Mixed H /H ∞ control design for mechanical systems: analytical 2 and numerical developments,” Proceedings of the 2008 AIAA Guidance, Navigation, and Control Conference and Exhibit, Hawa¨ı, USA, August 2008. 4 Cavallo, A. and Ferrara, F., “Atmospheric Re-Entry Control for Low Lift/Drag Vehicles,” Journal of Guidance, Control and Dynamics, Vol. 19, No. 1, January-February 1996, pp. 47–53. 5 Cavallo, A., Maria, G. D., and Ferrara, F., “Attitude Control for Low Lift/Drag Re-Entry Vehicles,” Journal of Guidance, Control and Dynamics, Vol. 19, No. 4, Jul-Aug 1996, pp. 816–822. 6 Torralba, J., Kron, A., and de Lafontaine, J., “Control of Mars Guided Entry (part I): Robust Self-Scheduling Modal Control,” Proceedings of the IFAC Symposium on Automatic Control in Aerospace 2007 , 2007. 7 Kron, A., de Lafontaine, J., and LePeuv´ edic, C., “Mars Entry and Aerocapture Robust Control Using Static Output Feedback and LPV Techniques,” Proceedings of the 6th International ESA Conference on Guidance, Navigation and Control Systems, Loutraki, Greece, 17-20 October 2005. 8 Apkarian, P. and Gahinet, P., “A convex characterization of gain-scheduled H ∞ controllers,” IEEE Transactions on Automatic Control, Vol. 40, No. 5, May 1995, pp. 853–864. 9 Roos, C. and Biannic, J.-M., “A positivity approach to robust controllers analysis and synthesis versus mixed LTI/LTV uncertainties,” Proceedings of the American Control Conference, Minneapolis, Minnesota, June 14-16 2006, pp. 3661–3666. 10 Roos, C., Contribution ` a la commande des syst` emes satur´ es en pr´ esence d’incertitudes et de variations param´ etriques - Application au pilotage de l’avion au sol, Ph.D. thesis, Universit´ e de Toulouse / Institut Sup´ erieur de l’A´ eronautique et de l’Espace, Toulouse, France, 2007. 11 Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming - Theory and Algorithms (3rd edition), Wiley-Interscience, 2006, ISBN-10: 0-471-48600-0 ISBN-13: 978-0-471-48600-8. 12 Magni, J.-F., “User Manual of the Linear Fractional Representation Toolbox Version 2.0,” Tech. Rep. Technical Report TR 5/10403.01F DCSD, ONERA - the French Aerospace Lab, 2005. 13 Ferreres, G. and Roos, C., “Robust feedforward design in the presence of LTI/LTV uncertainties,” International Journal of Robust and Nonlinear Control, Vol. 17, No. 14, 2007, pp. 1278–1293.
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