NDI-BASED STRUCTURED LPV CONTROL A PROMISING APPROACH FOR AERIAL ROBOTICS

Transcription

1 NDI-BASED STRUCTURED LPV CONTROL A PROMISING APPROACH FOR AERIAL ROBOTICS J-M. Biannic AERIAL ROBOTICS WORKSHOP OCTOBER 2014

2 CONTENT 1 Introduction 2 Proposed LPV design methodology 3 Applications to Aerospace Systems 4 SMAC Tools 5 Conclusions & Future work J-M. BIANNIC (ONERA-DCSD) 2 / 20 AERIAL ROBOTICS WORKSHOP 2014

3 INTRODUCTION HANDLING LARGE & NON-SQUARE FLIGHT ENVELOPES In many aerospace (and especially UAV) control applications, one has to cope with large and non-square operating domains. LTI controllers with fixed gains will then usually fail to complete the job. The latter must be tuned as a function of many (possibly correlated, measured or estimated, fixed or time-varying) parameters such as: Mach, Altitude, Airspeed,... Mass, cog location, configuration,... J-M. BIANNIC (ONERA-DCSD) 3 / 20 AERIAL ROBOTICS WORKSHOP 2014

4 INTRODUCTION LPV CONTROL SOLUTIONS Generally based on polytopic or LFT descriptions of the parameter varying plants to be controlled, various LPV design strategies have been proposed over the past twenty years 1,2. z 1 w P 1 P 1 5 z θ Θ(t) wθ P 2 P 3 P 4 z1 w1 P(s) y θ(t) Ω 5 Ω 4 Ω 1 Ω Ω 3 2 u y w kθ K(s) Θ(t) z kθ u POLYTOPIC DESIGN LFT DESIGN 1 A. Packard. Gain scheduling via linear fractional transformations. In: Systems and Control Letters 22.2 (Feb. 1994), pp C. Scherer. Gain-scheduled synthesis with dynamic generalized strictly positive real multipliers: A complete solution. In: Proceedings of the IEEE CDC. Florence, Italy, Dec J-M. BIANNIC (ONERA-DCSD) 4 / 20 AERIAL ROBOTICS WORKSHOP 2014

5 INTRODUCTION LPV CONTROL: FROM THEORY TO PRACTICE... Aerospace systems LFT modeling on large and non-square flight domains often leads to high-order LFT models: y = F u (M(s), Θ) u Θ = diag(θ 1 I n1,..., θ q I nq ) with q 1 and q i=1 n i 1. Despite recent advances, LFT-based LPV design techniques will then still suffer from conservatism or numerical complexity. Standard alternatives are: Interpolation-based gain-scheduling approaches, Low-order LFT modeling. In this talk, an NDI-inspired LPV control methodology is introduced. J-M. BIANNIC (ONERA-DCSD) 5 / 20 AERIAL ROBOTICS WORKSHOP 2014

6 NDI-BASED LPV CONTROL: BASIC PRINCIPLE Consider a parameter-varying plant: θ Θ, det(λ(θ)) 0, and rewrite A(θ) as follows: { ẋ = A(θ)x + BΛ(θ)u z = Lx (1) With the following control law: A(θ) = A 0 + BZ(θ) + B W (θ) (2) u = Λ(θ) 1 ( Z(θ)x + v) (3) an LTI perturbed plant is easily obtained: { ẋ = A0 x + B w + Bv z = Lx (4) where w = W (θ)x denotes a parameter-varying input perturbation. J-M. BIANNIC (ONERA-DCSD) 6 / 20 AERIAL ROBOTICS WORKSHOP 2014

7 SOME REMARKS For the following reasons: the control input signals u are realized by actuation systems with limited bandwidths and saturations constraints 3, the parameters θ may not be available all the time, the decomposition of A(θ) may be inaccurate: parametric uncertainties neglected dynamics flexible modes the compensation scheme (3) is no longer exact and the LTI plant (4) is then possibly perturbed as follows: ẋ = (A 0 + A (θ))x + B w w(θ) + (B + B (θ))v y = Cx z = Lx 3 J-M. Biannic et al. On dynamic inversion with rate saturations. In: Proceedings of the ACC. Montreal, June J-M. BIANNIC (ONERA-DCSD) 7 / 20 AERIAL ROBOTICS WORKSHOP 2014 (5)

8 DESIGN MODEL #1 Σ ~ z + w( θ) P(s) zc K(s) v y z ε R(s) ˆK(s) = Arg min max (µ p T zc z ɛ (s), µ r T w zɛ (s) ) (6) K(s) K J-M. BIANNIC (ONERA-DCSD) 8 / 20 AERIAL ROBOTICS WORKSHOP 2014

9 DESIGN MODEL #2 Σ ~ w( θ) z c F(s) K(s) v P(s) z y + z ε R(s) ˆK(s) = Arg min max (µ p T zc z ɛ (s), µ r T w zɛ (s) ) (7) K(s) K J-M. BIANNIC (ONERA-DCSD) 9 / 20 AERIAL ROBOTICS WORKSHOP 2014

10 DESIGN MODEL #3 Σ ~ w( θ) z c F(s) K(s) v P 1 P 2 P 4 P 3 z y + z ε R(s) ˆK(s) = Arg min max (µ p T zc z ɛ (s), µ r T w zɛ (s) ) (8) K(s) K J-M. BIANNIC (ONERA-DCSD) 10 / 20 AERIAL ROBOTICS WORKSHOP 2014

11 DESIGN MODEL #4 ~ w( θ) z c F(s) K(s) v P 1(s) P 2(s) P 3(s) Σ z y + z ε R(s) ˆK(s) = Arg min max (µ p T zc z ɛ (s), µ r T w zɛ (s) ) (9) K(s) K J-M. BIANNIC (ONERA-DCSD) 11 / 20 AERIAL ROBOTICS WORKSHOP 2014

12 RESOLUTION ASPECTS In all cases, the problem is formulated as a multi-objective H design issue which is now efficiently solved via nonsmooth H optimization techniques 4,5 which also allows to consider: structured and constrained K(s) controllers (PID, filters,...). multiple models as illustrated by design model #4 additional transfers to cope with saturations via anti-windup loops 6,7. 4 P. Apkarian and D. Noll. Nonsmooth H Synthesis. In: IEEE Transactions on Automatic Control 51.1 (2006), pp S. Gumussoy and M.L. Overton. Fixed-Order H Controller Design via HIFOO, a Specialized Nonsmooth Optimization Package. In: Proceedings of the ACC. Seattle, USA, June J-M. Biannic and P. Apkarian. Anti-windup design via nonsmooth multi-objective H optimization. In: Proceedings of the ACC. San Francisco, CA. USA, June J-M Biannic. Limit-cycles prevention via multiple H constraints with an application to anti-windup design. In: 9th IFAC Symposium on Nonlinear Control Systems. Toulouse, France, Sept J-M. BIANNIC (ONERA-DCSD) 12 / 20 AERIAL ROBOTICS WORKSHOP 2014

13 BACK TO PARAMETER-VARYING CONTROL Once K(s) is obtained, the parameter-varying control law reads: u = Λ(θ) 1 Z(θ)x + K(s) w(θ) z c y [ ] zc = F u (C(s), Θ(t)) y (10) The LFT format is convenient for preliminary validation. Note that θ and w may not be directly available. In that case, the controller implements estimates 8 : u = F u (C(s), ˆΘ(t) ) [ z c y ] (11) 8 G. Ferreres. Adaptive gain-scheduled control with application to a transport aircraft. CNES Workshop on Adaptive Control. Toulouse, France, February J-M. BIANNIC (ONERA-DCSD) 13 / 20 AERIAL ROBOTICS WORKSHOP 2014

14 PRELIMINARY VALIDATIONS: LFT MODELING Unlike "standard" LPV control design techniques (in the ideal case where θ is fully accessible), the proposed methodology does not offer any guarantee a priori, more specifically when: w(θ) becomes "too large" (although its effects is minimized), an estimation of θ is needed Here, both the initial LPV plant and LPV controller are assumed available in LFT format. Then, closed-loop LFT models of growing complexity are easily obtained. Θ (t) 1 Θ 2(t) Θ3(t) δ (t) Θ z c M 1(s) z ε zc M (s) 2 z ε z c M 3(s) zε actuators dynamics & saturations ( and its inverse do no longer cancel each other) Λ indirect adaptive scheme J-M. BIANNIC (ONERA-DCSD) 14 / 20 AERIAL ROBOTICS WORKSHOP 2014

15 ON COMPLEXITY OF LFT MODELS When the main parametric variations are cancelled by the proposed control structure, a simple closed-loop model is obtained: W (θ)x ẋ = A 0 x + BK(s) z c y + B W (θ)x (12) zc K(s) B C B T R(s) Θ(t) W I/s A 0 x W( θ) L M 1 (s) zε As is summarized on the previous slide, the LFT model complexity increases in the presence of: inexact cancellations control limitations unavailability of a few parameters J-M. BIANNIC (ONERA-DCSD) 15 / 20 AERIAL ROBOTICS WORKSHOP 2014

16 ANALYSIS TOOLS Once closed-loop LFT models are obtained, robust stability and performance tests can be performed to "pre-validate" the parameter-varying control laws. At ONERA, thanks to the SMAC Tools 9, such a robustness analysis process can be performed via: enhanced "grid-free" µ and skew-µ analysis tests, various IQC-based approaches: "KYP-free" frequency-domain techniques for high-order systems, "KYP-based" approaches in combination with appropriate LMI solvers enhanced optimization of parameter-dependent Lyapunov functions 9 SMAC. Systems Modeling Analysis & Control Toolbox. ONERA Research Project, J-M. BIANNIC (ONERA-DCSD) 16 / 20 AERIAL ROBOTICS WORKSHOP 2014

17 APPLICATIONS TO AEROSPACE SYSTEMS PAST AND ON-GOING PROJECTS During the past four years, the proposed methodology has been evaluated on various aerospace applications: A reentry vehicle control design problem 10 Fighter aircraft control on the whole subsonic domain 11 Q control law design for a civilian aircraft 12 From A330 autoland design to small UAVs control Mario Hernandez s PhD thesis, EDA-ICET, NICE Project, DTP-Cockpit project in cooperation with AIRBUS Industry. 13 Jeremy Lesprier s PhD thesis, J-M. BIANNIC (ONERA-DCSD) 17 / 20 AERIAL ROBOTICS WORKSHOP 2014

18 SMAC TOOLS A BRIEF OVERVIEW OF THE SMAC TOOLBOX J-M. BIANNIC (ONERA-DCSD) 18 / 20 AERIAL ROBOTICS WORKSHOP 2014

19 SMAC TOOLS A BRIEF OVERVIEW OF THE SMAC TOOLBOX The SMAC project ( ) involves 12 research scientists from ONERA/DCSD. The central objective of this project is not only to update existing software such as: the LFR modeling toolbox (which will now be compatible with uss objects) and its Simulink interface, the SMT Toolbox (which will become much easier to use and implement branch-and-bound techniques), the AWAST Toolbox to facilitate controller design with saturations but also to implement new analysis, modeling and control design tools. Examples analysis: enhanced IQC techniques, modeling: new approach for minimizing rational expressions, control design: tools in close connection with this talk. J-M. BIANNIC (ONERA-DCSD) 19 / 20 AERIAL ROBOTICS WORKSHOP 2014

20 In this talk, a general and highly adaptable framework has been presented to design controllers for LPV plants. Inspired by Dynamic Inversion strategies, the proposed approach is still applicable with numerous parameters. The gains of the control law are tuned via non-smooth H optimization algorithms whose flexibility is used to: structure the controllers, limit the order, introduce spectral constraints, consider multiple models Further work is needed to better understand the influence of weighting functions in this new context. It is then expected to better "automatize" the procedure for non-expert users. The proposed method may also be combined with adaptive control strategies and further evaluated on UAV systems. Thank you for your attention! J-M. BIANNIC (ONERA-DCSD) 20 / 20 AERIAL ROBOTICS WORKSHOP 2014