- Email: [email protected]
Gain Scheduling for an Aerospace Launcher With Bending Modes
GAIN SCHEDULING FOR AN AEROSPACE LAUNCHER WITH BENDING MODES
Benoit Clement*, Gilles Due*, Sophie Mauffrey**, Arnaud Biard***
•
Ecole Superieure d'Electricite, Service Automatique 3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France benoit.clement@SUpelecJr and gil/es. duc~upelecJr
**
EADS Launch Vehicles BP 2, 78133 Les Mureaux cedex, France [email protected]
••• CNES, Direction des Lanceurs Rond Point de l'Espace, 91023 Evry cedex, France arnaud. [email protected]
Abstract: An interpolation method with guaranteed stability is first presented. It can be perfonned with an LTV (Linear Time Varying) system which parametric trajectory is off-line known. An important point of this method is that it can turn into linear interpolation with some special conditions. This is then perfonned considering an aerospace application. It concerns a non stationary launcher during the atmospheric phase; the control law has to reject any wind disturbance. Copyright © 2001 IFAC Keywords: gain scheduling, interpolation method, convex optimization, discrete-time systems, Youla parameterization, aerospace launcher.
1. INTRODUCTION The gain scheduling approach is a very classical nonlinear control technique which first appears in industrial fields (in particular aeronautic and military applications) and have been theoretically investigated since the '90 (Rugh, 1991). Roughly speaking, it can be described as a six-step procedure: 1. 2. 3. 4. 5. 6.
get a linear parameter-varying model; choose the parameters to schedule; choose a family of operating points compute controllers for each point with linear design methods; interpolate these controllers; check the performance assessment.
Though differing technical avenues are available in each step, the scheme stays the same. Among papers that have investigated gain scheduling (Rugh & Shamma, 2000), very little focus in the interpolation
problem, which is a fundamental step in the synthesis of a scheduled control law. In the literature, a number of ad hoc approaches have been presented. Most of the time linear interpolation is used, either for poles, zeros and gains (Nichols et al., 1993), either for the solutions of Riccati equations (Reichert 1992), either on state feedback and observer gains (Hyde & Glover, 1993). An other point of view consists in a collection of linear time invariant (LTI) observers which scheduling is based on the measurement of the parameters (Buschek, 1997). All these approaches have an intuitive appeal and have satisfying performances for the considered applications but no proof of stability guarantee is given. These considerations are the main motivation of our work about interpolation methods for discrete-time systems, more precisely sampled systems. We propOse, in this paper, an interpolation method that guarantees the stability of the nonlinear system along its trajectory with LMIs conditions. The organization is the following: in section 2, the SESSION 9
475
LTV (Linear Time Varying) problem and the notations are described. The main result is given in section 3 as a theorem. In section 4, it is perfonned for the control of an aerospace launcher with a realistic model and hard design objectives.
2. NOTATIONS AND DEFINITIONS All plants considered in the paper are LTV finite dimensional, and described by a discrete-time statespace representation.
{
X(k + 1) = A(k )x(k) + B(k ;U(k) y(k) = C(k )x(k)
x(k + 1) = A(k)x(k) + B(k;U(k)
(3)
Let two state feedback gains KI and K2 such that A(k) + B(k)Kt et A(k) + B(k)K 2 are Schur-Cohn for each frozen value of k in a discrete temporal neighborhoods [a c] and [b d] of kl and k2 respectively (figure I). Assume that b < c, such that KI and K2 are simultaneously stabilizing for each frozen kin [b c].Moreover, stability intervals (with a frozen time) intersection is a closed interval. The a, b, c and d temporal (discrete) instants are defined on figure I :
(1)
K,
I
K,
Without loss of generality, direct transfer from u to y is zero. All state-space parameters are time varying and the following function f is supposed to be continuous; that means that the system evolution is continuous along his trajectory.
(2)
Note that f is defined on the whole set R even if t takes its values in N. The theoretical results (Shamma, 1988, Shamma & Athans, 1990, Fromion et al., 1996, Scorletti & El Ghaoui, 1998) concerning gain scheduling are hard to be used in a practical way because of the generality of the field of gain scheduling. The choice of a closer class of systems seems to be a way to particularize some general results and to justify rigorously practical applications. According to the space launcher problem, the presented results consider an LTV system following a known trajectory. It means that all launcher parameters are known as time functions. According to the general gain scheduling procedure, some operating points are chosen and corresponding controllers are designed. Then the interpolation problem can be considered; the proposed method is presented in the next section jointly with a sufficient condition to preserve stability with the nonlinear plant. For ease of understanding we focus on the state feedback case and a generalization will be given with the application.
I
K
K,
K,
a
d
b
Figure I. State feedback gain evolution Theorem 1 If there exist two symmetric positive definite matrices XI et X 2 and a strictly positive re R such that the following relations hold: for every k such that a
~
k
~
b,
for every k such that b ~ k :s; d , (5)
(6)
Then the scheduled state feedback gain
K(k):
K(k) = ~
I~
WaSkSb
C-k k-b IC-k k- b )- . - K . X . + - K 2X 2 - X . + - X 2 ifbSksc c-b c -b c- b c-b 2
WcSksd
is such that the closed-loop L TV system is exponentially stable. • A sketch of proof is given in appendix A.
3. INTERPOLATION CONDITIONS For brevity, we consider the state feedback case with only two models corresponding to different instants kt and k2 . Let consider the evolution and control L TV state equation:
476
SESSION 9
Remarks: The interpolation is perfonned only on the discrete time interval [b c] when both controllers are stabilizing. The constrains are LMIs which allow to ensure stability with convex optimization. A non stationary Lyapunov function associated
with the scheduled state feedback is:
V(x, k) =x(k)T X{k
t l x{k)
with
x c~k k-b X(k) = - - XI + - - X 2
j
c-b Xz
c-b
if a ~ k ~ b if b ~ k ~ c
(8)
~c~k~d
The condition (6) is original because no nonn is considered. This is made possible by the knowledge of the trajectory. This result allows the state feedback gain K(k) to vary continuously along the trajectory. Note that the interpolation turns into linear interpolation as a particular case. Indeed, if there is a solution such that XI = X Z = X , the gain becomes a linear interpolation: K
K(k)=
c~k k-b --KI +--K
j
c-b Kz
c-b
2
described in (Mauffrey & Schoeller 1998, Clement & Duc 2000b), launcher control objectives, during the atmospheric flight phase, are the following: closed-loop stability with sufficient margins: decreasing and increasing gain margins (!lG BF and !lGHF) have to stay higher than given specifications (see figure 5 for interpretation). control the destabilizing bending modes: the aim is to attenuate these modes under a gain limit (XdB ) except for the first one which can be controlled in phase with a sufficient delay margin (one sample period Ts). limit the angle of attack i in case of wind (with a typical wind profile given on figure 6). limit the angle of deflection f3 and its velocity
/3
the consumption C must be limited to Cmax where:
if a ~ k ~ b ifb~k~c
(9)
~cSk~d
This is possible when the two consecutive gains and frozen time plants are rather closed. In this case there exist a common Lyapunov function such that the inequalities (4) and (5) hold while (6) is obviously satisfied. A dual problem is the interpolation of a full order observer which is not presented for brievety. The last point is to connect this approach with the interpolation of non structured controller. Let n be the system (3) order and ne the controller order. In the case ne ~ n, it is known that the controller can be interpreted as an observed state feedback structure connected with a Youla Parameter (Alazard & Apkarian, 1999). Moreover, Clement & Duc (2000a) propose an efficient algorithm to obtain a common Youla parameter for different systems. Then the state feedback interpolation case can be extended for controllers designed with various methods by only interpolating observer gains of the equivalent observer / state feedback structure.
4. AEROSPACE LAUNCHER CONTROL
4.1. Control problem The application is developed for the yaw axis of an European space launcher (figure 2) whose dynamics include a rigid mode and two bending modes (sloshing modes are not considered). The automatic control of the launcher has the function of keeping the process stable around its center of gravity, fOllowing the guidance reference trajectory. As
all these objectives have to be robust against uncertainties (which affect rigid and bending modes)
x
Fig. 2 - Simplified launcher representation The closed-loop structure is given figure 3. The synthesis is perfonned along the guidance trajectory with several operating points: for each of them, a linear controller is perfonned according to the algorithm presented in (Clement & Duc, 2000a) which uses Youla parameterization and LMI optimization. Method and results for a particular operating point are described in (Clement & Duc, 2000b). The interpolation method is particularly adapted to the launcher problem because the simplified model is already LTV (only small angles are considered and saturations have to be avoided) and the parameters evolutions are known as continuous time functions. Moreover, the parameters are slowly varying, then a linear interpolation is perfonned in order to reduce the number of on-line calculus.
SESSION 9
477
r-------,
Angle of artack
trajectory. Then the feasibility LMI problem becomes simpler:
j
LAUNCHER
Attitude 'I'
,nd angular velocity d'Y/dt
measurements commlnded angle of deflexion ~
Figure 3. Control structure
4.2. Gain scheduling procedure This section describes the six step of the gain scheduling procedure given in section I. I. Steps I and 2 are given by the choice of the model:Jdefined in (2) is known as a table giving all parameters of the launcher model at each instant k. 2. Step 3: the set of operating points is chosen in order to take into account the critical points; 10 points are chosen with constant intervals:
k. I
3.
=T
. + i Tend - T;nit mll 10
Step 4: 10 controllers are performed according to (Clement & Duc, 2000b). This method consists in 3 steps: a first H~ synthesis is performed; the second steps consists in the controller transformation into an observer based controller interconnected with a Youla parameter. This transformation requires some critical choices (which dynamics are from the observer and which ones are from the command) and they are done to insure the continuity of the closed loop dynamics partition as shown on figure 4. Third step is the synthesis of a common Y oula parameter for every model. Note that the synthesis structure is the same for all operating points.
5.
This LMI problem is then checked for every k between ki and ki+ 1 to insure stability for every frozen system. This is possible because it is known that the parameters are slowly varying. Note that the same interpolation is performed with the observer gains. Steps 6 is useless for stability because it is guaranteed a priori. But the .performances have to be checked; this is done in the next section with graphical interpretations.
This procedure success because of the parameters evolutions are smooth. Next paragraph shows the details of the final results.
4.3. Results The efficiency of the method is considered with the frequency open loop responses of the frozen systems in a Black-Nichols chart (figure 5); all specifications (gain margins, first mode phase, roll-off) are almost satisfied. Time responses of the closed loop plant: angle of attack i specification is almost satisfied (figure 7); the deflection angle {3 is far from the saturations (figure 8), and it is the same for the velocity of the deflection angle d~/dt (figure 9). The consumption constraints is clearly respected as shown on figure 10.
Figure 5. Nichols chart for every linearized point
Figure 4. Closed loop eigenvalues map 4.
Step 5: the main step of the procedure is to insure stability with linear interpolation (i.e. with a common X). For each interval and according to theorem I, the instants are chosen as a = b = k; and c = d = k;+1 which means that the interpolation is performed all along the
T••
Figure 6. wind profile
478
SESSION 9
5. CONCLUSION +i
.jl:=:=============±=====~
T...
Figure 7. angle of attack i
.....tF================,
_~t~=====================d
T....
Figure 8. angle of deflection {3 (control)
An original interpolation procedure has been presented for state feedback controllers. The fundamental tool is the exponential stability with time varying Lyapunov functions. The advantages of the method are: The stability is preserved during the interpolation phase The use of LMI optimization insures the convergence In some cases the interpolation can turn into linear interpolation. This last point is fundamental; indeed, the proposed method can be used to validate if a linear interpolation can be performed or not. Indeed, by imposing a common Lyapunov function, the interpolation is linear and guarantees the closed-loop stability for the non-linear plant. As a generalization this procedure can be performed for any controller, e.g. an H.. one. It is known that any full order (or more) controller has an observerbased structure if coupled with a Youla parameter [AA99]. The application of the method to an aerospace launcher gives efficient results. The difficult point is the synthesis which have to not break the continuity of the closed loop dynamics.
+satf=================j 6. REFERENCES
~t~==============d T,,,J/
Figure 9. velocity of the angle of deflection ~ c".., -
-
- . - - - - - - -- - - - - j
~~_s:~
______.__.______.___ .____________
o~~------------~
TiJlit
Figure 10. Consumption
Alazard D., Apkarian P. (1999), Exact observerbased structures for arbitrary compensators, International Journal of Robust & Non Linear Control, 9, pp. 101-118. Buschek H. (1997), Robust autopilot design for future missile systems, AIAA Guidance. Navigation and Control Conference, New Orleans. Clement B., Duc G. (2000a), Multiobjective Control Youla parameterization and LMI via optimization: application to a flexible arm, 3rd !FAC Symposium on Robust Control and Design, Prague. Clement B., Duc G. (2000b), A Multi-Objective Control Algorithm: application to a launcher with bending modes, 8th IEEE Mediterranean Conference on Control and Automation, Patras. Fromion V., Monaco S., Normant-Cyrot D., (1996), Robustness and stability of LPV plants through frozen system analysis, International Journal Robust and Non Linear Control, 6, pp. 235-248. Hyde A.R., Glover K., (1993) The application of scheduled H.. controllers to a VSTOL aircraft, IEEE Transactions on Automatic Control, 38, pp. 1023-1039. Mauffrey S., Schoeller M. (1998), Non-Stationary H~ Control for Launcher with Bending Modes; 14th IFAC Symposium on Automatic Control in
SESSION 9
479
Aerospace, Seoul. Reichert R, (1992), Dynamic scheduling of modem robust control autopilot designs for missiles, IEEE Control Systems Magazines, 12, pp.35-42. Rugh WJ. (1991), Analytical framework for gain scheduling, IEEE Control System Magazine, 12, pp. 79-84. Nichols RA., Reichert RT., Rugh W.1. (1993), Gain scheduling for H-infinity controllers: a flight example, IEEE Transactions on Control Systems Technologies, 1, pp. 69-79. Rugh WJ., Shamma 1.S. (2000), Research on gain scheduling, Automatica, 36, pp. 1401-1425. Scorletti G., El Ghaoui L., (1998), Improved LMI conditions for gain scheduling and related problems, International Journal of Robust and Non Linear Control, 8, pp. 845-877. Shamma 1.S., Athans M. (1990), Analysis of gain scheduled control for nonlinear plants, IEEE Transactions on Automatic Control, 35, pp. 898907. Shamma S. (1988), Analysis and design of gain scheduled control system, PhD thesis with the Massachusset Institute of Technologie (MIT).
(point I) 2. The variation of V is given by
AV{k) = V{x{k + I~k + 1)- V{x(k~k) =x(k)T(Ac/(k)f X(k + ItI Ac/(k)- X{kt l )x(k) Let us prove that for every le:
The Schur lemma gives the equivalence with:
- X(k + 1) Ac/(k)) ()-I <0 fork=a, .. . d T ( Ac/k ( ) -Xk which can be reinterpreted with a congruence relation for k = a, .. . d :
It can be proved with the given interpolation that for any k, the following inequality is hold: 7. APPENDIX A The closed-loop system is defined with the following equation:
Moreover, according to the definition of X:
x{k + 1) = Ac/{k )x{k) = (A{k) + B{k )K{k ))x(k) Let prove that this system is globally quadratically stable. The demonstration then consists in the quadratic stability of the LTV system using Lyapunov stability theorem from (Shamma, 1988) with 3 points to demonstrate. First define the following function V: V:
H
V{x,k) is V{O,k) =O.
0 a~k~b X(k)-X(k+l)= __I_{X 2 -XI ) b~k~c
1o
c-b
V{x,k) = x{k)T X{ktl x(k)
obviously
positive
definite
and
c~k~d
where XI and X 2 satisfy the theorem inequalities. These two considerations lead to:
V{x{k + I~ k + I) - V{x(k ~ k) ~ -ellx{k ~12
R"XN ~R +
(x,k)
(A(k) + B(k )K(k ))x(k)) - X(k) ( X(kXA(k) + B(k)K(k)f _ X(k) < -r I
(point 2)
3. The derivative of V with respect to x is bounded in a neighbourhood of 0, and as V is continuous with x, V is locally Lipschitz in x, upon the finite interval a ~ k ~ d . Then there exists a positive A. such that:
1. Define the following variables:
!l = min
C-k k-b - XI + -X2 [( c-b c-b
)-1
1(C-k k-b 2 )-1 7J=m --XI +--X c-b
c-b
the following inequalities hold:
480
SESSION 9
/ b~k ~
c
1
(point 3)
Benoit Clement*, Gilles Due*, Sophie Mauffrey**, Arnaud Biard***
•
Ecole Superieure d'Electricite, Service Automatique 3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France benoit.clement@SUpelecJr and gil/es. duc~upelecJr
**
EADS Launch Vehicles BP 2, 78133 Les Mureaux cedex, France [email protected]
••• CNES, Direction des Lanceurs Rond Point de l'Espace, 91023 Evry cedex, France arnaud. [email protected]
Abstract: An interpolation method with guaranteed stability is first presented. It can be perfonned with an LTV (Linear Time Varying) system which parametric trajectory is off-line known. An important point of this method is that it can turn into linear interpolation with some special conditions. This is then perfonned considering an aerospace application. It concerns a non stationary launcher during the atmospheric phase; the control law has to reject any wind disturbance. Copyright © 2001 IFAC Keywords: gain scheduling, interpolation method, convex optimization, discrete-time systems, Youla parameterization, aerospace launcher.
1. INTRODUCTION The gain scheduling approach is a very classical nonlinear control technique which first appears in industrial fields (in particular aeronautic and military applications) and have been theoretically investigated since the '90 (Rugh, 1991). Roughly speaking, it can be described as a six-step procedure: 1. 2. 3. 4. 5. 6.
get a linear parameter-varying model; choose the parameters to schedule; choose a family of operating points compute controllers for each point with linear design methods; interpolate these controllers; check the performance assessment.
Though differing technical avenues are available in each step, the scheme stays the same. Among papers that have investigated gain scheduling (Rugh & Shamma, 2000), very little focus in the interpolation
problem, which is a fundamental step in the synthesis of a scheduled control law. In the literature, a number of ad hoc approaches have been presented. Most of the time linear interpolation is used, either for poles, zeros and gains (Nichols et al., 1993), either for the solutions of Riccati equations (Reichert 1992), either on state feedback and observer gains (Hyde & Glover, 1993). An other point of view consists in a collection of linear time invariant (LTI) observers which scheduling is based on the measurement of the parameters (Buschek, 1997). All these approaches have an intuitive appeal and have satisfying performances for the considered applications but no proof of stability guarantee is given. These considerations are the main motivation of our work about interpolation methods for discrete-time systems, more precisely sampled systems. We propOse, in this paper, an interpolation method that guarantees the stability of the nonlinear system along its trajectory with LMIs conditions. The organization is the following: in section 2, the SESSION 9
475
LTV (Linear Time Varying) problem and the notations are described. The main result is given in section 3 as a theorem. In section 4, it is perfonned for the control of an aerospace launcher with a realistic model and hard design objectives.
2. NOTATIONS AND DEFINITIONS All plants considered in the paper are LTV finite dimensional, and described by a discrete-time statespace representation.
{
X(k + 1) = A(k )x(k) + B(k ;U(k) y(k) = C(k )x(k)
x(k + 1) = A(k)x(k) + B(k;U(k)
(3)
Let two state feedback gains KI and K2 such that A(k) + B(k)Kt et A(k) + B(k)K 2 are Schur-Cohn for each frozen value of k in a discrete temporal neighborhoods [a c] and [b d] of kl and k2 respectively (figure I). Assume that b < c, such that KI and K2 are simultaneously stabilizing for each frozen kin [b c].Moreover, stability intervals (with a frozen time) intersection is a closed interval. The a, b, c and d temporal (discrete) instants are defined on figure I :
(1)
K,
I
K,
Without loss of generality, direct transfer from u to y is zero. All state-space parameters are time varying and the following function f is supposed to be continuous; that means that the system evolution is continuous along his trajectory.
(2)
Note that f is defined on the whole set R even if t takes its values in N. The theoretical results (Shamma, 1988, Shamma & Athans, 1990, Fromion et al., 1996, Scorletti & El Ghaoui, 1998) concerning gain scheduling are hard to be used in a practical way because of the generality of the field of gain scheduling. The choice of a closer class of systems seems to be a way to particularize some general results and to justify rigorously practical applications. According to the space launcher problem, the presented results consider an LTV system following a known trajectory. It means that all launcher parameters are known as time functions. According to the general gain scheduling procedure, some operating points are chosen and corresponding controllers are designed. Then the interpolation problem can be considered; the proposed method is presented in the next section jointly with a sufficient condition to preserve stability with the nonlinear plant. For ease of understanding we focus on the state feedback case and a generalization will be given with the application.
I
K
K,
K,
a
d
b
Figure I. State feedback gain evolution Theorem 1 If there exist two symmetric positive definite matrices XI et X 2 and a strictly positive re R such that the following relations hold: for every k such that a
~
k
~
b,
for every k such that b ~ k :s; d , (5)
(6)
Then the scheduled state feedback gain
K(k):
K(k) = ~
I~
WaSkSb
C-k k-b IC-k k- b )- . - K . X . + - K 2X 2 - X . + - X 2 ifbSksc c-b c -b c- b c-b 2
WcSksd
is such that the closed-loop L TV system is exponentially stable. • A sketch of proof is given in appendix A.
3. INTERPOLATION CONDITIONS For brevity, we consider the state feedback case with only two models corresponding to different instants kt and k2 . Let consider the evolution and control L TV state equation:
476
SESSION 9
Remarks: The interpolation is perfonned only on the discrete time interval [b c] when both controllers are stabilizing. The constrains are LMIs which allow to ensure stability with convex optimization. A non stationary Lyapunov function associated
with the scheduled state feedback is:
V(x, k) =x(k)T X{k
t l x{k)
with
x c~k k-b X(k) = - - XI + - - X 2
j
c-b Xz
c-b
if a ~ k ~ b if b ~ k ~ c
(8)
~c~k~d
The condition (6) is original because no nonn is considered. This is made possible by the knowledge of the trajectory. This result allows the state feedback gain K(k) to vary continuously along the trajectory. Note that the interpolation turns into linear interpolation as a particular case. Indeed, if there is a solution such that XI = X Z = X , the gain becomes a linear interpolation: K
K(k)=
c~k k-b --KI +--K
j
c-b Kz
c-b
2
described in (Mauffrey & Schoeller 1998, Clement & Duc 2000b), launcher control objectives, during the atmospheric flight phase, are the following: closed-loop stability with sufficient margins: decreasing and increasing gain margins (!lG BF and !lGHF) have to stay higher than given specifications (see figure 5 for interpretation). control the destabilizing bending modes: the aim is to attenuate these modes under a gain limit (XdB ) except for the first one which can be controlled in phase with a sufficient delay margin (one sample period Ts). limit the angle of attack i in case of wind (with a typical wind profile given on figure 6). limit the angle of deflection f3 and its velocity
/3
the consumption C must be limited to Cmax where:
if a ~ k ~ b ifb~k~c
(9)
~cSk~d
This is possible when the two consecutive gains and frozen time plants are rather closed. In this case there exist a common Lyapunov function such that the inequalities (4) and (5) hold while (6) is obviously satisfied. A dual problem is the interpolation of a full order observer which is not presented for brievety. The last point is to connect this approach with the interpolation of non structured controller. Let n be the system (3) order and ne the controller order. In the case ne ~ n, it is known that the controller can be interpreted as an observed state feedback structure connected with a Youla Parameter (Alazard & Apkarian, 1999). Moreover, Clement & Duc (2000a) propose an efficient algorithm to obtain a common Youla parameter for different systems. Then the state feedback interpolation case can be extended for controllers designed with various methods by only interpolating observer gains of the equivalent observer / state feedback structure.
4. AEROSPACE LAUNCHER CONTROL
4.1. Control problem The application is developed for the yaw axis of an European space launcher (figure 2) whose dynamics include a rigid mode and two bending modes (sloshing modes are not considered). The automatic control of the launcher has the function of keeping the process stable around its center of gravity, fOllowing the guidance reference trajectory. As
all these objectives have to be robust against uncertainties (which affect rigid and bending modes)
x
Fig. 2 - Simplified launcher representation The closed-loop structure is given figure 3. The synthesis is perfonned along the guidance trajectory with several operating points: for each of them, a linear controller is perfonned according to the algorithm presented in (Clement & Duc, 2000a) which uses Youla parameterization and LMI optimization. Method and results for a particular operating point are described in (Clement & Duc, 2000b). The interpolation method is particularly adapted to the launcher problem because the simplified model is already LTV (only small angles are considered and saturations have to be avoided) and the parameters evolutions are known as continuous time functions. Moreover, the parameters are slowly varying, then a linear interpolation is perfonned in order to reduce the number of on-line calculus.
SESSION 9
477
r-------,
Angle of artack
trajectory. Then the feasibility LMI problem becomes simpler:
j
LAUNCHER
Attitude 'I'
,nd angular velocity d'Y/dt
measurements commlnded angle of deflexion ~
Figure 3. Control structure
4.2. Gain scheduling procedure This section describes the six step of the gain scheduling procedure given in section I. I. Steps I and 2 are given by the choice of the model:Jdefined in (2) is known as a table giving all parameters of the launcher model at each instant k. 2. Step 3: the set of operating points is chosen in order to take into account the critical points; 10 points are chosen with constant intervals:
k. I
3.
=T
. + i Tend - T;nit mll 10
Step 4: 10 controllers are performed according to (Clement & Duc, 2000b). This method consists in 3 steps: a first H~ synthesis is performed; the second steps consists in the controller transformation into an observer based controller interconnected with a Youla parameter. This transformation requires some critical choices (which dynamics are from the observer and which ones are from the command) and they are done to insure the continuity of the closed loop dynamics partition as shown on figure 4. Third step is the synthesis of a common Y oula parameter for every model. Note that the synthesis structure is the same for all operating points.
5.
This LMI problem is then checked for every k between ki and ki+ 1 to insure stability for every frozen system. This is possible because it is known that the parameters are slowly varying. Note that the same interpolation is performed with the observer gains. Steps 6 is useless for stability because it is guaranteed a priori. But the .performances have to be checked; this is done in the next section with graphical interpretations.
This procedure success because of the parameters evolutions are smooth. Next paragraph shows the details of the final results.
4.3. Results The efficiency of the method is considered with the frequency open loop responses of the frozen systems in a Black-Nichols chart (figure 5); all specifications (gain margins, first mode phase, roll-off) are almost satisfied. Time responses of the closed loop plant: angle of attack i specification is almost satisfied (figure 7); the deflection angle {3 is far from the saturations (figure 8), and it is the same for the velocity of the deflection angle d~/dt (figure 9). The consumption constraints is clearly respected as shown on figure 10.
Figure 5. Nichols chart for every linearized point
Figure 4. Closed loop eigenvalues map 4.
Step 5: the main step of the procedure is to insure stability with linear interpolation (i.e. with a common X). For each interval and according to theorem I, the instants are chosen as a = b = k; and c = d = k;+1 which means that the interpolation is performed all along the
T••
Figure 6. wind profile
478
SESSION 9
5. CONCLUSION +i
.jl:=:=============±=====~
T...
Figure 7. angle of attack i
.....tF================,
_~t~=====================d
T....
Figure 8. angle of deflection {3 (control)
An original interpolation procedure has been presented for state feedback controllers. The fundamental tool is the exponential stability with time varying Lyapunov functions. The advantages of the method are: The stability is preserved during the interpolation phase The use of LMI optimization insures the convergence In some cases the interpolation can turn into linear interpolation. This last point is fundamental; indeed, the proposed method can be used to validate if a linear interpolation can be performed or not. Indeed, by imposing a common Lyapunov function, the interpolation is linear and guarantees the closed-loop stability for the non-linear plant. As a generalization this procedure can be performed for any controller, e.g. an H.. one. It is known that any full order (or more) controller has an observerbased structure if coupled with a Youla parameter [AA99]. The application of the method to an aerospace launcher gives efficient results. The difficult point is the synthesis which have to not break the continuity of the closed loop dynamics.
+satf=================j 6. REFERENCES
~t~==============d T,,,J/
Figure 9. velocity of the angle of deflection ~ c".., -
-
- . - - - - - - -- - - - - j
~~_s:~
______.__.______.___ .____________
o~~------------~
TiJlit
Figure 10. Consumption
Alazard D., Apkarian P. (1999), Exact observerbased structures for arbitrary compensators, International Journal of Robust & Non Linear Control, 9, pp. 101-118. Buschek H. (1997), Robust autopilot design for future missile systems, AIAA Guidance. Navigation and Control Conference, New Orleans. Clement B., Duc G. (2000a), Multiobjective Control Youla parameterization and LMI via optimization: application to a flexible arm, 3rd !FAC Symposium on Robust Control and Design, Prague. Clement B., Duc G. (2000b), A Multi-Objective Control Algorithm: application to a launcher with bending modes, 8th IEEE Mediterranean Conference on Control and Automation, Patras. Fromion V., Monaco S., Normant-Cyrot D., (1996), Robustness and stability of LPV plants through frozen system analysis, International Journal Robust and Non Linear Control, 6, pp. 235-248. Hyde A.R., Glover K., (1993) The application of scheduled H.. controllers to a VSTOL aircraft, IEEE Transactions on Automatic Control, 38, pp. 1023-1039. Mauffrey S., Schoeller M. (1998), Non-Stationary H~ Control for Launcher with Bending Modes; 14th IFAC Symposium on Automatic Control in
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Aerospace, Seoul. Reichert R, (1992), Dynamic scheduling of modem robust control autopilot designs for missiles, IEEE Control Systems Magazines, 12, pp.35-42. Rugh WJ. (1991), Analytical framework for gain scheduling, IEEE Control System Magazine, 12, pp. 79-84. Nichols RA., Reichert RT., Rugh W.1. (1993), Gain scheduling for H-infinity controllers: a flight example, IEEE Transactions on Control Systems Technologies, 1, pp. 69-79. Rugh WJ., Shamma 1.S. (2000), Research on gain scheduling, Automatica, 36, pp. 1401-1425. Scorletti G., El Ghaoui L., (1998), Improved LMI conditions for gain scheduling and related problems, International Journal of Robust and Non Linear Control, 8, pp. 845-877. Shamma 1.S., Athans M. (1990), Analysis of gain scheduled control for nonlinear plants, IEEE Transactions on Automatic Control, 35, pp. 898907. Shamma S. (1988), Analysis and design of gain scheduled control system, PhD thesis with the Massachusset Institute of Technologie (MIT).
(point I) 2. The variation of V is given by
AV{k) = V{x{k + I~k + 1)- V{x(k~k) =x(k)T(Ac/(k)f X(k + ItI Ac/(k)- X{kt l )x(k) Let us prove that for every le:
The Schur lemma gives the equivalence with:
- X(k + 1) Ac/(k)) ()-I <0 fork=a, .. . d T ( Ac/k ( ) -Xk which can be reinterpreted with a congruence relation for k = a, .. . d :
It can be proved with the given interpolation that for any k, the following inequality is hold: 7. APPENDIX A The closed-loop system is defined with the following equation:
Moreover, according to the definition of X:
x{k + 1) = Ac/{k )x{k) = (A{k) + B{k )K{k ))x(k) Let prove that this system is globally quadratically stable. The demonstration then consists in the quadratic stability of the LTV system using Lyapunov stability theorem from (Shamma, 1988) with 3 points to demonstrate. First define the following function V: V:
H
V{x,k) is V{O,k) =O.
0 a~k~b X(k)-X(k+l)= __I_{X 2 -XI ) b~k~c
1o
c-b
V{x,k) = x{k)T X{ktl x(k)
obviously
positive
definite
and
c~k~d
where XI and X 2 satisfy the theorem inequalities. These two considerations lead to:
V{x{k + I~ k + I) - V{x(k ~ k) ~ -ellx{k ~12
R"XN ~R +
(x,k)
(A(k) + B(k )K(k ))x(k)) - X(k) ( X(kXA(k) + B(k)K(k)f _ X(k) < -r I
(point 2)
3. The derivative of V with respect to x is bounded in a neighbourhood of 0, and as V is continuous with x, V is locally Lipschitz in x, upon the finite interval a ~ k ~ d . Then there exists a positive A. such that:
1. Define the following variables:
!l = min
C-k k-b - XI + -X2 [( c-b c-b
)-1
1(C-k k-b 2 )-1 7J=m --XI +--X c-b
c-b
the following inequalities hold:
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SESSION 9
/ b~k ~
c
1
(point 3)