Multiobjective H 2 /H /impulse-to-peak synthesis: Application to the control of an aerospace launcher

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1 Multiobjective H /H /impulse-to-peak synthesis: Application to the control of an aerospace launcher D. Arzelier, D. Peaucelle LAAS-CNRS, 7 Avenue du Colonel Roche, 3 77 Toulouse, Cedex 4, France s: arzelier@laas.fr, peaucell@laas.fr October 3, 3 Keywords: Impulse-to-peak performance, Multiobjective control, LMI optimization, Aerospace launcher. Problem formulation Let the LTI discrete plant Σ be given by its state-space imal realization: x k+ A B B x k z k = C D D w k () y k C D u k where x R n is the state vector, w R mw is the disturbance vector, u R mu is the input vector, z R rz is the controlled output vector and y R ry is the measured output vector. The z and w vectors are partitioned woo wip w Σ zoo zip z u y K as indicated in figure. Figure : Standard model for multiobjective control z = z z ip z w = w w ip w () The associated matrices are therefore consequently partitioned. B = D D ip D B B ip B D = D ip D ip D ip D D ip D D = [ D y D yip D y ] C = C C ip C D = D u D ipu D u The controller K is given by its imal state-space realization: η k+ = A K η k + B K y k (4) u k = C K η k + D K y k The closed-loop system Σ K is given by its state-space matrices: A + BDK C BC A cl = K B + BD B B K C A cl = K D K B K D C cl = [ (5) ] C + DD K C DC K D cl =[D + DD K D ] (3)

2 Problem (multiobjective H /H /ip control problem) Find a controller K in the set of internally stabilizing controllers K such that: K K α ip γ ip + α γ Σ K γ Σ ip K ip γ ip Σ K γ (6) LMI formulation of the impulse-to-peak synthesis problem. The impulse-to-peak performance Let the closed-loop discrete-time plant be given by its imal realization: x k+ = A cl x k + B cl w k x() = (7) z k = C cl x k + D cl w k Problem (worst-case ipk synthesis) Find a controller K in the set of internally stabilizing controllers K. K K sup w i i m w z L (8) where z is the performance output response to the impulsive input w i k = wi δ k, w l k =(l i) andδ k is the unit pulse. Theorem If there exists a matrix P S + satisfying: then z k L ξ k. A cl PA cl P< B clb cl <P C cl PC cl ξ < D cl D cl <ξ (9) Proof The solutions of (7) may be written as (k : x = z = D cl x = B cl z = C cl x = C cl B cl () x k = A k cl B cl z k = C cl A k cl B cl = C cl x k Then working on the dual of (7) (A cl,c cl,b cl,d cl ), we get x k+px k+ <x k Px k < <x Px = C cl PC cl So, k x k Px k C cl PC cl and k x k Px k <ξ. Finally, from z k z k = x k B clb cl x k,weget k z k z k <ξ. Using the last inequality leads to z k <ξ k As in continuous-time case [], [], the conservative LMI characterization (9) may be used to compute a bound on the peak of the impulse response of (7). γ = γ P cl,γ A cl P cl A cl P cl < B ipcl B ipcl P cl < () C ipcl P cl C ipcl γ < D ipcl D ipcl γ < Here, we are interested in an equivalent alternative LMI characterization of a bound on the impulse-to-peak performance of the closed-loop system (7. A generalization of the method in [8] where an extra matrix G is introduced for the test of H and H performances leads to the equivalent LMI optimization problem.

3 Theorem Let the following semidefinite programg problem be given: γg = P cl,γ,g γ Pcl A cl G G A cl P cl G G < Pcl B ipcl B ipcl < [ ] γ C ipcl G G C ipcl P cl G G < γ Dipcl < D ipcl () then γ γ G Proof Any feasible solution to (9) is a feasible solution of (). Indeed, using adequate Schur complement on all inequalities, if there exists a feasible pair (P cl, γ) S n + R + for (9) then the triplet (P cl, G = P cl, γ) is a feasible triplet for (). We are now in position to address the impulse-to-peak performance synthesis problem. Theorem 3 If there exist X R n n, Y R n n, Â R n n, ˆB R n r, Ĉ R m n, ˆD R m r, Q S n +, H S n +, J R n n, and a scalar γ G R+ such that: γ G = X, Y, γ G J, H, Q, S γ then a full-order controller reconstructed as follows: V U = S YX D K = ˆD C K =(Ĉ ˆDCX)U B K = V T (ˆB YB ˆD) A K = V T Q J AX + BĈ A + B ˆDC H Â YA + ˆBC Q X X J S H Y Y Q J B ip + B ˆDD ipy H YB ip + ˆBD ipy γ C ipx + D ipu Ĉ C ip + D ipu ˆDC Q X X J S H Y Y γ Dip + D ipu ˆDD ipy < [Â Y(A + B ˆDC)X V B K CX YBC K U ] leads to bound the peak value of impulse response of the system: U (3) (4) z L < γg 3

4 Proof 3 From (), it is easy to see that the matrix G is invertible and may be partitioned as follows: X U Y G = G U V = (5) V Then applying the following similarity transformation on the closed-loop Lyapunov matrix, we get: Y Q J Y V P cl = V J (6) H Y Multiplying the first three terms in () by and by its transpose, we get the final result by applying V the change of variables (4). The same technique may be applied to the H and H performance criteria as originally proposed in [8]. A full-order output-feedback controller imizing an upper bound for the multiobjective H /H /ip synthesis problem may be computed using semidefinite programg and LMI formulation. Theorem 4 If the following semidefinite programg problem has a solution X, Y, γ G, T, S J i, H i, Q i,i=,ip, α γ + α ip γ ip Q J AX + BĈ A + B ˆDC B + B ˆDD y H  YA + ˆBC YB + ˆBD y Q X X S + J X C + Ĉ D u H Y Y C + C ˆD D u D + D y ˆD D u γ Q ip J ip AX + BĈ A + B ˆDC H ip  YA + ˆBC Q ip X X J ip S H ip Y Y Q ip J ip B ip + B ˆDD ipy H ip YB ip + ˆBD ipy γ ip C ip X + D ipu Ĉ C ip + D ipu ˆDC Q ip X X J ip S H ip Y Y γ Dip + D ipu ˆDD ipy < Trace(T) < γ T C X + D u Ĉ C + D u ˆDC D + D u ˆDD y Q X X S + J H Y Y Q J AX + BĈ A + B ˆDC B + B ˆDD y H  YA + ˆBC YB + ˆBD y Q X X S + J H Y Y < then a controller K constructed from (4) is a suboptimal solution to the multiobjective synthesis problem (6). Proof 4 The proof is very simple and consists in creating a particular G i matrix for each performance constraint. Applying a generalized shaping paradigm G = G ip = G and the linearizing change of variables leads to the result. (7) 4

5 3 Application to the multiobjective control of an aerospace launcher 3. Model of the launcher and specifications The description of the application is mainly borrowed from [3]. The yaw axis of the space launcher with the associated variables is presented in figure. G is the center of gravity, V and V r are the absolute and relative velocity and W is the wind velocity. i is the angle of attack and Ψ is the deviation of the launcher from the yaw axis with respect to the guidance attitude reference. One has to design an automatic controller keeping the launcher around its center of gravity which follows the guidance reference trajectory where the control variable is the thruster angle of deflection β. The linearized dynamics of the launcher include seven rigid modes with mode for sensors and mode for actuator and 5 bending modes. During the atmospheric W Vr V x θ ψ X i Pc G Z β flight phase, the specifications for the controller are: Figure : Model of the launcher - to insure closed-loop stability with sufficient stability margins - to insure good attenuation of bending modes - to reject disturbance due to wind and gusts - to insure robustness with respect to uncertainties on the rigid and bending modes 3. Performance specifications and structure of the controller Our approach consists in choosing a particular model (including sensors dynamics but without bending modes and actuator dynamics) of the yaw axis of the launcher. The specifications are then translated into control objectives involving adequate system norms as follows: - a bound on the impulse-to-peak performance of the closed-loop transfer W i with an additional filter modelling a typical wind profile to limit the angle of attack - a bound on the H performance of the sensitivity S to get a imum modulus margin and therefore good gain margins - a bound on the H performance of the transfer between measurement noises and β to reduce the consumption - a mixed roll-off and lead filter to attenuate the bending modes and control in phase the first one The control problem for the launcher is naturally set as a multiobjective control problem of the form (6). K K α i γ W i + α c γ cons Σ mod K γ mod Σ W i K ip γ ip Σ cons K γ cons (8) 5

6 A conservative solution for this problem is given in the previous section. problem: It amounts to solve the LMI (α i γ W i + α c γ cons ) L m (Q m, J m, H m, X, Y, S, Â, ˆB, Ĉ, ˆD,γ mod ) < L ip (Q i, J i, H i, X, Y, S, Â, ˆB, Ĉ, ˆD) < L c (Q c, J c, H c, T c, X, Y, S, Â, ˆB, Ĉ, ˆD) < (9) where L ( ) are LMIs defined in (4) with respect to the decision variables in boldface. Of course, we need to design a particular structure for the controller which is detailed in figure 3. and defining the particular models Σ i. One part of the controller is tuned (mixed filter) and the other part is computed via multiobjective optimization. Note also that the wind gust model acts like a weighting function which has to be tuned in the controller synthesis. w w 3 3 z = z = β w w 3 Σ aug Wind gust model + Actuator Mixed Filter Rigid Launcher with sensors ψ ψ. Controller Multiobjective Controller delay delay z = i Figure 3: Structure of the multiobjective controller 3.3 Synthesis procedure and tuning parameters As seen before, the whole controller is then composed of the mixed filter in series with a multiobjective 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: K W W W (p) = p +T W p + TW W m = +aτp +τp K ro +T ro p () We have therefore 6 synthesis parameters: K W, T W, K ro, T ro, a and τ. The other set of tuning parameters is formed with the optimization parameters α i /α c and γ mod of the LMI optimization. Algorithm - Choose the tuning parameters and form the augmented plant Σ. Extract Σ W i,σ cons and Σ mod. 3- Solve the convex optimization problem via LMI optimization and get the decision variables. 4- Reconstruct the controller with formulae (4). 3.4 Simulation results The previous algorithm has been used to design a robust autopilot for the atmospheric flight of a space launcher. The synthesis has been performed considering an LTI worst-case configuration of the launcher. During the atmospheric flight phase, the time-variant behavior of the launcher is simulated with SIMULINK c where information about specifications is provided to the user. The typical wind profile is presented in figure 3.4. More details about the application are presented in the companion and collective paper [?] and in [5]. 6

7 Wind velocity W s t/t = 95 Figure 4: Wind profile The Bode plot of the multiobjective controller is first presented in figure 3.4 where the effect of the lead filter is obvious. 4 3 Ψ. u Bode Diagram Magnitude (db) 3 Ψ u 4 9 Phase (deg) Frequency (rad/sec) f/f s Figure 5: Bode diagram of the pilot The open-loop frequency responses of the worst-cases are given in Nichols charts in figure 3.4. This plot is obtained by freezing the model of the launcher when the wind gust is applied. The roll-off specification on bending modes is indicated by the red horizontal line. A zoom on a specific range of frequencies allows to verify gain margins imposed on the rigid model. X db M db l M db h

8 Various time responses are then proposed. Note that all scales have been normalized with respect to the sample period T s. The left figure shows the variation of the angle of attack when the typical wind T end profile is applied. The consumption C = β(k +) β(k) is computed and plotted with respect k=t init to the maximum allowed consumption. The right plot shows that the consumption is very good since the multiobjective controller needs only 5 % of the maximum allowable consumption to tackle the specifications. 9 8 C max Angle of attack i Consumption (C/C max ) i max T f T f Finally, the angle of deflection and its velocity are plotted below. They are far from the specified maxima β max β. max Angle of deflection β Control velocity β β max. β max T f 4 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. The definition of easy-to-use macros of MATLAB based on convex semidefinite optimization solvers allows the designer to tune adequately the synthesis parameters. On the contrary, the main drawback of the proposed method relies on the fact that it can be used only with a simplified synthesis model (without bending modes) due to numerical considerations. Indeed, the parametric uncertainty of the model is not directly dealt with. In fact, parametric robustness concerns comes from the inherent time-varying nature of the model which is not taken into account here. References [] S.P. Boyd, L.E. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies, 994. [] B. Clement, G. Duc, S. Mauffrey, A. Biard, Aerospace launch vehicle control: a gain scheduing approach, 5th triennial World Congress, Barcelona,. [3] B. Clement, G. Duc, A multiobjective control algorithm: Application to a launcher with bending modes, Proceedings of the 8th IEEE Mediterranean Conference on Control and Automation, Rio Patras, Greece,. [4] J.C. Geromel, M.C. de Oliveira, L. Hsu, LMI Characterization of structural and robust stability, Linear Algebra and its Applications, vol. 85, -3, pp. 69-8, December

9 [5] S. Mauffrey, M. Schoeller, Non-stationnary H control for launcher with bending modes, 4th IFAC Symposium on Automatic Control in Aerospace, Seoul, Korea, 998. [6] M.C. de Oliveira, J. Bernussou, J.C. Geromel, A new discrete-time robust stability condition, Systems & Control Letters, Vol. 37, No. 4, July 999. [7] M.C. de Oliveira, J.C. Geromel, L. Hsu, LMI Characterization of structural and robust stability: the discrete-time case, Linear Algebra and its Applications, Vol. 96, -3, pp.7-38, July 999. [8] M.C. Oliveira, J. Bernussou, J.C. Geromel, Extended H and H norm characterizations and controller parametrizations for discrete-time systems, International Journal of Control, Vol. 75, pp ,. [9] D. Peaucelle, D. Arzelier, O. Bachelier, J. Bernussou, A new robust D-stability condition for real convex polytopic uncertainty, Systems & Control Letters, vol. 4,, may. [] C.W. Scherer, P. Gahinet, M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Transactions on Automatic Control, Vol. 4, No. 7, pp , 997. [] R.E. Skelton, T. Iwasaki, K. Grigoriadis, A unified algebraic approach to linear control design, Taylor and Francis,