The Role of Zero Dynamics in Aerospace Systems

Transcription

1 The Role of Zero Dynamics in Aerospace Systems A Case Study in Control of Hypersonic Vehicles Andrea Serrani Department of Electrical and Computer Engineering The Ohio State University

2 Outline q Issues in Control of Hypersonic Vehicles (HSVs) q Trajectory Tracking for a Longitudinal HSV Model q Control by Model Inversion q The Zero Dynamics of HSVs q Pitfalls of Approximate Linearization q Shaping the Zero Dynamics: Output Redefinition q Simulation Results q Conclusions 2

3 Air-breathing Hypersonic Vehicles Two StageArtist s torendering Orbit Concept of X-51. Image courtesy of NASA X-51 Reference Vehicle for technology evaluation Focus of Level 4 tool development Hypersonic Two-Stage-To-Orbit Vehicle Concept (NASA) q Oxygen taken from the atmosphere no need to carry oxidant on board TBCCand firstmilitary stage and q Increased payload for civilian applications rocket powered second q Part of Two-Stage-to-Orbitstage concept (current version) q Rocket booster or combined ramjet-scramjet cycle required. 3

4 Issues in HSV Dynamics: Aerodynamics X-43,M 1 T Propulsion system integrated in the airframe Fuselage provides compression at the inlet and serves as expansion nozzle Scramjet engine below the CG generates thrust / pitching moment coupling Thrust produced by the scramjet engine affected by the inflow of air Bow shock and spillover of airflow depend on angle-of-attack and Mach no. Structural modes significantly affect aerodynamic and propulsive forces Flexibility effects produce significant changes in lift and pitching moment Elevator-to-Lift coupling generates loss of lift when climbing Non-minimum phase behavior that complicates control system design 4

5 Longitudinal Vehicle Model V = 1 T ( ) cos m D( q, ) mg sin( ) ḣ = V sin( ) = Q 1 L( q,, e)+t ( ) sin mv mg cos( )] = Q Q = 1 M( q,, e)+z T T ( ) I yy Elevator-to-Lift Coupling D( q, ) = qscd( ), L( q,, e) = qs CL ( )+C L M( q,, e) = c qs C M ( )+C M e, e, C D( ) =C 2 D 2 + C D + C 0 D C L ( ) =C L + C 0 L C M ( ) =C 2 M 2 + C M + C 0 M This term is responsible for the non-minimum phase behavior: the input appears too soon in the equations e 5

6 Output Trajectory Tracking y ref x ref inverse model u ref feedback controller u x plant y x =[V,h,,,Q] u =[, e] tracking controller y =[V,h] The control action embeds the inversion of the plant model How do we compute the inverse? What is the resulting dynamics when? y = y ref y ref inverse model u ref plant model y ref x = x ref 6

7 The Zero Dynamics of Control Systems x(0) x ref (t) (t) = ref (t) (t) =q( (t), ref (t)) e(t) =0 x(t) Z The set of all forced trajectories of the system compatible with zero tracking error A fundamental concept for a myriad control problems: Non-interacting control and disturbance decoupling with stability Linearization of the input-output and input-state map Tracking and regulation Limit of performance of nonlinear control systems 7

8 Non-minimum Phase Behavior of HSVs The system has unstable zero dynamics when y =[V,h] Feedback transformation ( V = u1 8 >< >: ḧ = u 2 = Q Imaginary Axis Flexible effects Q = 1 I yy M( q,, u1,u 2 ), Real Axis Pitch Dynamics Zeros, one is nonminimum M( q,, 0, 0) > 0 hyperbolic saddle Zeroes Controlling altitude via model inversion (linearization by feedback) results in an unstable closed-loop system (even if the tracking error is regulated) 8

9 Naïve Approach: Ignoring the Coupling Approximate Linearization: Feedback linearization with NMP coupling strategically ignored to achieve full relative degree (no zero dynamics) Outer-loop compensator achieves stable tracking for the rigid-body model Results in instability when flexible dynamics are included in the model (closed-loop system not robust to dynamic uncertainty) 9

10 Approach: Beyond (Approximate) Linearization Exploiting Control Input Redundancy Tool for Robust and Adaptive Stabilization Exploiting System Structure Robust Semi-global Design Decentralized Adaptive Nonlinear Control Shaping the Zero-Dynamics Dynamic Output Redefinition Tracking via Integral Control FLEXIBLE STATES RIGID BODY CONTROLLER The key is to achieve regulation indirectly by using another output. This new output must be selected such that: 1. The resulting zero dynamics is stable 2. Regulation of the new output implies regulation of the original tracking error. Model uncertainty makes it a daunting task. 10

11 Redefinition of the Zero-dynamics The zero-dynamics with respect to the error e =[V V 1,h? h? 1] have an unstable equilibrium at (,Q)=(, 0) where T (, ) cos D =0, M(, )=0 Redefining the set-point tracking error as and applying the new decoupling input e = C M ( ) C M z T T (, ) qs cc M yield the new 1-dim zero dynamics (the flight path angle dynamics) Trim condition e aux =[V V, ] = L(, ) mg cos mv which has an asymptotically stable equilibrium at =0 11

12 Regulation to an Unknown Setpoint Asymptotic stability of the new zero-dynamics suggests to trade with in the regulated output The problem is that is unknown (any discrepancy will lead to lim (t) =0, hence to a diverging altitude) t Integral augmentation: 1 = ref (to enforce equilibrium at level flight) Change of coordinates: (to remove inputs from the zero-dyn.) µ 1 (y ref ) := 1 V ref µ 2 (y ref ) := 1 V ref I yy C L cmc M z T C L 1 cc M cos ref(y ref ) 2 = + µ 1 (y ref )Q + µ 2 (y ref )Ṽ tan ref (y ref ) αr [deg] m =169,V r =7500 m =169,V r =9500 m =169,V r =11000 m =202,V r =7500 m =202,V r =9500 m =202,V r =11000 parameterizes the angle-of-attack along the reference ref(y ref ) x 10 ρ [slugs/ft 3 ] 5 12

13 Design with Redefined Zero-dynamics Letting 1 = k r, 2 = where 0 <k 1 < 1, yields the new stable zero-dynamics 1 = a 1 (x, y ref ) µ 1 (y ref ) Q + µ 2 (y ref )Ṽ + d 1 (x, y ref ) 2 = a 2 (x, y ref ) 1 a 3 (x, y ref ) 2 + a 4 (x, y ref ) + b 2 (x, y ref ) Q + b 3 (x, y ref )Ṽ + d 2(x, y ref ) Ṽ, Q, d The overall system is stabilized by the selection 1 2 Ṽ, Q, d cmd = k r = 1 + r + r (y ref, ẏ ref ) d Q cmd = k [ cmd ] k 1 [ cmd ] ref (Ṽ, Q, ) Ṽ, Q (, ) 13

14 Simulation Results (High-Fidelity Model) "#!!! V,Vr [ft/s] ""!!! "!!!! *!!! V, V ref Velocity, V Φ )!!! Reference, V r (!!! +! "!! #!! $!! %!! &!! '!! (!! )!! 12.8 h,hr [ft] x 10 5 h, h ref Altitude, h Reference, h r δe [deg] e Time [s] !*&& +!"*"#' (, ) FLEXIBLE STATES RIGID- BODY Velocity (x, u) η1, [ftslug 1/2 ]!*&!*%&!*%!*$& + 1, 2 1 st bending mode, η 1 2 nd bending mode, η 2!"*"#)!"*"$!"*"$#!"" #"" $"" %"" &"" '"" ("" )""!"*"$% Time [s] η2, [ftslug 1/2 ] u cmd e Altitude Flight-Path Angle Pitch Angle Pitch Rate CONTROLLER cmd Q cmd y x y ref 14

15 Benefits of Adaptation in the Loop flight path angle command 0.4 flight path angle command FPA Tracking [deg] FPA Tracking [deg] , ref, ref Non-adaptive controller Time [s] 0 Adaptive controller Sizable error in FPA means large error in altitude tracking 15

16 Conclusions The concept of Zero Dynamics plays a fundamental role in virtually all control problems of interest This is especially true for aerospace systems, where typically not all the degrees of freedom are directly actuated Other noticeable examples include: Helicopters and Rotorcrafts Vertical Take-Off and Landing (VTOL) Vehicles Flapping-Wing Micro Air Vehicles Under-actuated Satellites Fixed-Wing Unmanned Air Vehicles It is impossible to imagine today the field of aerospace without the pivotal contribution of Alberto Isidori to the theory and the practice of flight control system design. 16