Vortex Model Based Adaptive Flight Control Using Synthetic Jets

Vortex Model Based Adaptive Flight Control Using Synthetic Jets Jonathan Muse, Andrew Tchieu, Ali Kutay, Rajeev Chandramohan, Anthony Calise, and Anthony Leonard Department of Aerospace Engineering Georgia Institute of Technology August 20, 2008 1/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Overview 1 Introduction 2 Wind Tunnel Traverse Wing and Actuators 3 Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s 4 Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection 5 Determining Model Parameters Model Validation Closed Loop Experiments

Outline 1 Introduction 2 Wind Tunnel Traverse Wing and Actuators 3 Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s 4 Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection 5 Determining Model Parameters Model Validation Closed Loop Experiments

Introduction Introduction Aerodynamic flow control. Enable highly-maneuverable flight for small UAVs (e.g., in confined spaces). No moving control surfaces. Maneuver on convective time scale (Dragon Eye scales: 20 m/s, c 30cm, tconv = 15 msec) Flight dynamics and flow dynamics are coupled. Flow develops forces and moments on convective time scales. Flow state is affected by both vehicle dynamics and actuation. 4/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Experiment Diagram Introduction Wind Tunnel Traverse Wing and Actuators 6/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Traverse Introduction Wind Tunnel Traverse Wing and Actuators 7/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Force Control Introduction Wind Tunnel Traverse Wing and Actuators Purpose: Simulation of longitudinal free flight in a wind tunnel. A force control technique was developed to accomplish this. Force control maintains prescribed force/moment on model. Removes effect of gravity. Hides traverse nonlinearities from model. Applies prescribed force commands to the traverse. Feedback of wing states alters dynamics of flying model. Force is applied by regulating the deflection of the springs in the traverse. Moment applied via torque motor. 8/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Traverse Mechanism Introduction Wind Tunnel Traverse Wing and Actuators Inner loop PID control laws regulate the carriage positions. Force control law commands accelerations to the carriages. Allows regulation of the spring deflection on the airfoil. 9/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Wind Tunnel Traverse Wing and Actuators Wing Model 1m span NACA 4415 wing section Chord length is 457 mm. Modular and comprised of interchangeable spanwise segments for sensors. Includes module of a circumferential array of 70 static pressure ports located at mid-span. Several modules of high-frequency integrated pressure sensors for measurements of instantaneous pressure. 10/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Wing Section Introduction Wind Tunnel Traverse Wing and Actuators 11/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Flow Control Actuators Wind Tunnel Traverse Wing and Actuators Synthetic jet type actuators. Array of jets mounted on trailing edge of wing. Actuators are amplitude modulated. Characteristic actuation rise time O(2-3tconv). Usable control authority up to 30 Hz in pitch. 12/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Wind Tunnel Traverse Wing and Actuators Hybrid actuators on opposite sides of the trailing edge allow CM to be varied bidirectionally without moving surfaces. Manipulates concentrations of trapped vorticity. PS actuator increases C M (nose-up). SS actuator decreases C M (nose-down). Significant changes in C M with minimal lift and drag penalty Changes in actuator C µ allow aerodynamic performance to be continuously varied 13/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

System, Concept Introduction Wind Tunnel Traverse Wing and Actuators 14/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Static Actuator Model of the Wing Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s The effect of an actuator is modeled as a static moment actuator. The lift and moment can be modeled as L = QS (C L0 + C Lα α + C L α α) ( ) M = QS c C M0 + C Mα α + c 2V C M α α + C Mδa δ a Modeling leads to a system model of the form ẏ 0 1 0 0 y ÿ α = 0 a 2,2 a 2,3 a 2,4 ẏ 0 0 0 1 α + α 0 a 4,2 a 4,3 a 4,4 α 0 0 0 b f,4 δ a 16/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Concept of Vortex Model Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s y U ẏ θ M(a) Γ 1 Γ 2... Γ i c 2 01 01 01 01 01 01 01 01 01 01 c a ξ 2... ξ i 2 ξ 1 Constant Strength Vortices x Control Vortex, Γ C, ξ C Nascent Vortex 17/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s From our previous work, we obtained the following lift and moment relations L = ρπ( c2 4 ÿ + Ucẏ) + ρπ [ ac 2 ρuc 2 N i=1 4 θ + U(a + c 2 )c θ + ( Uc 2 Γ i ξ 2 i c 2 /4 + ρuγ C ] 4 + U 2 c)θ and M(a) = al + ρπuc2 4 ẏ + ρπ + ρuc2 8 N i=1 [ c 4 θ ] 128 Uac2 4 θ U2 c 2 4 θ Γ i ξ 2 i c 2 /4 + ρuγ C ξ C 18/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s The shed vortex positions, ξ i, were given by dξ 1 = U (ξ2 1 c2 /4) dγ 1 dt ξ 1 Γ 1 dt dξ i = U (i 2) dt The vortex strengths, Γ i, were defined by ( ) ξ 1 c/2 N ξ i + c/2 Γ 1 = Γ 0 + ξ 1 + c/2 ξ i c/2 Γ i = Constant i=2 19/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Corrections for Thickness and Camber Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Corrections needed for accurate simulation. Corrections based on NASA legacy data. Effect of thickness and camber is to translate lift and moment curves. Lift changes as ( ) 1 L = L + 2 ρu2 c C L,0 Moment changes as ( ) 1 ( M = M 2 ρu2 c 2 C M,0 + a c ) ( ) 1 4 2 ρu2 c C L,0 In our experiments, the c.g. is close to quarter chord and M simplifies since ( a c 4) 0 20/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Linear Model Development Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Vortex model captures dynamics that are negligible on time scales of rigid body dynamics. We define a characteristic circulation as Γ W = c N i=1 Γ i ξ 2 i c 2 /4 We consider the lift and moment generated when impulsively started from rest dγ w /dt = 0. Only a single vortex is created. 21/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Linear Model Development (cont.) Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s This gives the lift as L = ρu ( Γ 0 1 ) 2 Γ W At t = t 0, Γ W Γ 0. When t, Lift terms should disappear as wake vortices move downstream. To model as linear, we propose the following model dγ W = dγ 0 βγ W dt dt where β is a constant and the initial condition of the differential is Γ W (t 0 ) = Γ 0 (t 0 ) 22/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

The Linear Model Introduction Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s This induces an exponential rise in lift (1 e βt ) for a constant Γ 0. This is contrary to the classical square root type growth for lift. This is contrary to the decay in lift that is geometric at best. One can compute the best fit for β at a given t. Hence, the linearized characteristic circulation is ( ( Γ W + βγ W = πc ÿ + a + c ) θ + U θ 4) with an initial condition of ( ( Γ W (t 0 ) = πc ÿ + a + c θ + U θ) 4) t=t 0 23/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Linear Lift/Moment Relationships Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s The lift and moment expressions simplify to: ( ) ( ) c 2 1 L = ρπ 4 ÿ + Ucẏ ρu 2 Γ W + Γ C [ ( ) ] ac 2 ρπ 4 θ ( + U a + c ) Uc c θ 2 + 2 4 + U2 c θ and [ M = al + ρπuc2 Uac 2 ẏ + ρπ θ + U2 c 2 ] 4 4 4 θ ac2 128 θ ( c ) + ρu 8 Γ W Γ C ξ C The above equations include added mass, quasi-steady lift, lift due to wake, and control terms. 24/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Coupled Model Assumptions Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Assume the rigid body dynamics are given by U mÿ + b y ẏ + k y y = L I θ + b θ θ + k θ θ = M(a) b y k y L is the lift. M(a) is the moment about the location a. Γ C, ξ C ξ 1 Neglect thickness and camber corrections for control design purposes. b θ k θ 01 01 Γ 1 01 01 25/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Redefining Lift and Moment as Matrix Equations The Linear Vortex Model can be written as ẋ = Ax + BΓ C where x = [y θ ẏ θ Γw ] T. How does Γ C relate to the physical world? Γ C can be related to applied moment as Γ C (u f, θ) = 1 2 Uc ( a + ξc c ) C M (u f, θ) C M (u f, θ) is determined from static experimental data. Hence, the model becomes nonlinear! Luckily, Γ C (u f, θ) is invertible for fixed θ. 26/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

s Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s The vortex model is nonlinear. Γ C (u f, θ) is invertible for fixed θ We employ an inversion technique to make the control design effectively linear. C C 1, u ẋ=a x B C, x Inversion of Γ C (u f, θ) is pre-computed in a lookup table. Now, one can use standard linear analysis tools to develop control laws based on the static actuator model and the vortex model. 27/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Linear Control Law Design Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Defining the tracking error e = y r We must design a control law to ensure e(t) 0 as t Using a modified robust servomechanism LQR like formulation, feedback gains, K e and K x, are computed. Results in a control law of the form t u = K e e(τ)dτ K x x + Zr 0 28/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Nominal Control Architecture Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Robust Servo LQR with feedforward element 29/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Avoiding State Estimation for Vortex Control Law State feedback is not possible for vortex model. Aerodynamic state is unmeasurable. We modify the nominal vortex design using projective control. Augmenting the model dynamics with the control law dynamics, the closed loop system is given by [ ] [ ] [ ] [ e 0 C e 1 = ẋ BK e Ā BK + X x BZ y = [ 0 C ] [ ] e x where C is a matrix that multiplied by x gives the position. ] r 30/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Actuation Modeled as a Static Device Linear Vortex Model Coupled Vortex/Rigid Body Model s Avoiding State Estimation for Vortex Control Law We can retain all but one of the closed loop eigenvalues. Let K = [K e K x ] and X y be the eigenvectors corresponding to the closed loop eigenvalues we wish to retain. The required output feedback gain is given by ( ) 1 K = KX y C measured X y where C measured corresponds to the rigid body states of x. New Output Feedback Vortex Control Law [ e u = K y measured ] + Zr 31/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Plant Dynamics/Reference Behavior Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection We assume that our plant can be expressed as ẋ(t) = Ax(t) + BΛ [Γ C (t) + f (x, Γ C )] y(t) = Cx(t) The nominal control law can be expressed as Γ C,n = K y y + K r r Assuming f (x, Γ C ) = 0, we form the desired behavior ẋ m (t) = A m x m (t) + B m r y m (t) = Cx m (t) where A m = A BK r is Hurwitz and B m = BK r. 33/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Approximating System Uncertainty Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection We want to design an adaptive signal Γ C,ad to approximately cancel the modeling error f (x, Γ C ). The total control effort becomes Γ C (t) = Γ C,n (t) Γ C,ad (t) We will try to approximate Λf (x, Γ C ) with a SHL neural network Λf (x, u) = W T σ(v T η(t)) + ɛ(x, u), (x, u) D x D u where ɛ, W, and V are unknown but bounded. We reconstruct the nonlinearity via delayed values of system outputs and inputs as inputs to the neural network (η(t)). 34/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Error Observer Introduction Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection Since all of the states are not observable, we need an error observer. ξ = A m ξ + L(y y ξ y m ) y ξ = Cξ where Ã = A m LC is Hurwitz and satisfies the following Lyapunov equation Ã T P + PÃ = Q, Q = Q T > 0, Q R nxn The observer allows us to estimate the error state, x m x, of the system. 35/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Adaptive Weight Update Laws Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection The adaptive update laws are ( ] Ŵ (t) = Γ W Proj [Ŵ (t), σ ˆV (t), η(t)) ξ(t) T PB [ ˆV (t) = Γ V Proj ˆV (t), η(t)ξ T PBH (Ŵ (t), ˆV (t), η(t))] [ δλ ˆ T (t) = Γ δ Proj δλ ˆ T ] (t), u(t)ξ T (t)pb where ( ) ( ) ( ) σ ˆV (t), η(t) = σ ˆV (t) T η(t) σ ˆV (t), η(t) ˆV T (t)η(t) ) H (Ŵ (t), ˆV ( ) (t), η(t) = Ŵ T (t) σ ˆV (t), η(t) These laws use parameter projection. See the paper for additional details. 36/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Compensating for Saturation Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection Hedged reference model ẋ m = A m x m + B m r + B h Γ C,h Scheduled Control Hedging Gain Map for Hedging 0.5 0.4 C, desired C 1, u achieved 0.3 0.2 Gain Scheduled C, desired C, achieved+ C, h + - Γ C 0.1 0 0.1 Maximum Γ C Minimum Γ C 0.2 0.3 0.4 15 10 5 0 5 10 15 20 θ (deg) 37/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

System Conceptual Review Plant Dynamics/Reference Behavior Adaptive Control Implementation Saturation Protection 38/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Model Validation Introduction Determining Model Parameters Model Validation Closed Loop Experiments Static actuator model parameters were determined from static tests. Γ C map was determined from static pitching moment measurements. Measured Γ C (u f, θ) Saturated Γ 1 C (u f, θ) 0.6 0.4 1 Γ C 0.2 0 0.2 0.4 1 0.5 0 0.5 Control Voltage (%) 1 10 5 15 10 5 0 Pitch Angle (deg) 20 Control Voltage (%) 0.5 0 0.5 1 1 10 0.5 0 0 10 0.5 1 20 Pitch Angle (deg) Γ C Saturation of Γ C ensures invertability. 40/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Model Validation Introduction Determining Model Parameters Model Validation Closed Loop Experiments Experiment response to open loop actuator excitation has been compared with simulation results. Flow Control Input Voltage Experiment 10 Simulation Pitch Response Comparison 20 Experiment Simulation 15 5 10 u f (V) 0 α (deg) 5 5 0 10 0 2 4 6 8 10 12 14 16 18 Time(s) 5 0 2 4 6 8 10 12 14 16 18 Time(s) The vortex ROM performs significantly better than the static actuator model. 41/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Torque Motor Case Introduction Determining Model Parameters Model Validation Closed Loop Experiments Lets look at the flight response using a torque motor for actuation. This indicates that the experiment is closely representing a free flying wing. 42/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Control Law Comparisons Determining Model Parameters Model Validation Closed Loop Experiments Square Wave Tracking: 5 Linear Model Failure 5 Vortex Model Failure Plunge (cm) 0 Plunge (cm) 0 5 0 2 4 6 8 10 12 5 0 2 4 6 8 10 12 10 10 Pitch (deg) 5 0 5 Pitch (deg) 5 0 5 10 0 2 4 6 8 10 12 10 0 2 4 6 8 10 12 1 1 Control (%) 0.5 0 0.5 Control (%) 0.5 0 0.5 1 0 2 4 6 8 10 12 Time 1 0 2 4 6 8 10 12 Time Command Adaptive Control Law Static Actuator Vortex ROM 43/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Rise Time Stability Barrier Determining Model Parameters Model Validation Closed Loop Experiments Rise time: 10% 90% Static actuator limit: 0.31 sec Linear vortex model limit: 0.19 sec 1.2 1 Plunge Position (m) 0.8 0.6 0.4 0.2 0 Vortex Model Stability Barrier Static Actuator Stability Barrier 0.2 0 0.5 1 1.5 Time (s) 44/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Disturbance Rejection Determining Model Parameters Model Validation Closed Loop Experiments 45/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Conclusions Introduction Determining Model Parameters Model Validation Closed Loop Experiments Demonstrated closed loop longitudinal control of a wing model using synthetic jet type actuation. As the wing moves faster, the actuators can no longer be considered static. Simple vortex model developed to allow linear control designs to reach higher bandwidth. Unmodeled dynamics destabilize linear control designs at a high enough bandwidth. Adaptive control is able to deal with unmodelled dynamics and maintain stability. 46/47 Jonathan Muse Vortex Model Based Adaptive Flight Control

Determining Model Parameters Model Validation Closed Loop Experiments Questions? 47/47 Jonathan Muse Vortex Model Based Adaptive Flight Control