Naira HovakimyanDepartment of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University

This talk was originally given as a plenary atSIAM Conference on Control and Its ApplicationsSan Francisco, CA, June 2007A. M. Lyapunov1857-1918G. Zames1934-1997

Lyapunov Memorial Conference

Outline Historical Overview Limitations and the V&V Challenge of Existing Approaches Paradigm Shift Speed of Adaptation Performance and Robustness Rohrs Example Various aerospace applications Future Directions of Research

Motivation Early 1950s design of autopilots operating at a wide range of altitudes and speeds Fixed gain controller did not suffice for all conditions Gain scheduling for various conditions Several schemes for self-adjustment of controller parameters Sensitivity rule, MIT rule 1958, R. Kalman, self-tuning controller Optimal LQR with explicit identification of parameters

1950-1960 - flight tests X-15 (NASA, USAF, US Navy) bridge the gap between manned flight in the atmosphere and space flight Mach 4 - 6, at altitudes above 30,500 meters (100,000 feet) 199 flights beginning June 8, 1959 and ending October 24, 1968 Nov. 15, 1967, X-15A-3

First Flight Test in 1967The Crash of the X-15A-3 (November 15, 1967) X-15A-3 on its B52 mothershipX-15A-3Crash site of X-15A-3Crash due to stable, albeit non-robust adaptive controller!

Objectives of Feedback Stabilization and/or Tracking

Historical Background Sensitivity Method, MIT Rule, Limited Stability Analysis (1960s) Whitaker, Kalman, Parks, et al. Lyapunov based, Passivity based (1970s) Morse, Narendra, Landau, et al. Global Stability proofs (1970-1980s) Morse, Narendra, Landau, Goodwin, Keisselmeier, Anderson, Astrom, et al. Robustness issues, instability (early 1980s) Egard, Ioannou, Rohrs, Athans, Anderson, Sastry, et al. Robust Adaptive Control (1980s) Ioannou, Praly, Tsakalis, Sun, Tao, Datta, Middleton,et al. Nonlinear Adaptive Control (1990s) Adaptive Backstepping, Neuro, Fuzzy Adaptive Control Krstic, Kanelakopoulos, Kokotovic, Zhang, Ioannou, Narendra, Ioannou, Lewis, Polycarpou, Kosmatopoulos, Xu, Wang, Christodoulou, Rovithakis, et al. Search methods, multiple models, switching techniques (1990s) Martenson, Miller, Barmish, Morse, Narendra, Anderson, Safonov, Hespanha, et al.

Landmark Achievement: Adaptive Control in Transition Air Force programs: RESTORE (X-36 unstable tailless aircraft 1997), JDAM (late 1990s, early 2000s) Demonstrated that there is no need for wind tunnel testing for determination of aerodynamic coefficients (an estimate for the wind tunnel tests is 8-10mln dollars at Boeing) Lessons Learned: limited to slowly-varying uncertainties, lack of transient characterization Fast adaptation leads to high-frequency oscillations in control signal, reduces the tolerance to time-delay in input/output channels Determination of the best rate of adaptation heavily relies on expensive Monte-Carlo runsBoeing question: How fast to adapt to be robust?

Stability, Performance and Robustness Absolute Stability versus Relative Stability

Linear Systems Theory Absolute stability is deduced from eigenvalues Relative stability is analyzed via Nyquist criteria: gain and phase margins Performance of input/output signals analyzed simultaneously

Nonlinear Systems Theory Lack of availability of equivalent tools Absolute stability analyzed via Lyapunovs direct method Relative stability resolved in Monte-Carlo type analysis No correlation between the time-histories of input/output signals

The Theory of Fast and Robust Adaptation Main features

guaranteed fast adaptation guaranteed transient performance

for systems both signals: input and output

guaranteed time-delay margin

uniform scaled transient response dependent on changes in initial condition value of the unknown parameter reference input

Suitable for development of theoretically justified Verification& Validation tools for feedback systems

System:Nominal controller in MRAC:Desired reference system:This is the nominal controller in the L1 adaptive control paradigm!One Slide Explanation of the New Paradigm

Two Equivalent Architectures of Adaptive Control

Transient Performance and Robustness in Both Schemes Lyapunov stability Transient performance

Implementation DiagramsMRACState predictor based reparameterizationReference systemNo more reference system upon filtering!!!u cannot be low-pass filtered directly.

Stability Solving: Closed-loop

Closed-loop Reference System

Guaranteed Transient Performance System state: Control signal:

LTI System for Control Specifications Independent of the unknown parameterReference system achieved via fast adaptationDesign system for defining the control specs

Achieving Desired SpecificationsSufficient condition for stabilityPerformance improvement

Guaranteed Performance Bounds Design C(s) to render sufficiently small Use large adaptive gain Desired design objective

Time-delay Margin and Gain MarginPoint Time-delay occursPlant Lower bound for the time-delay marginProjection domain defines the gain margin

Fast adaptation, in the presence of , leads to guaranteed transient performance and guaranteed gain and time-delay margins:

where is the time-delay margin of .Main Theorem of L1 Adaptive Controller

Design Philosophy Fast adaptation ensures arbitrarily close tracking of the virtual reference system with bounded away from zero time-delay marginAdaptive gain as large as CPU permits (fast adaptation)Increase the bandwidth of the filter: The virtual reference system can approximate arbitrarily closely the ideal desired reference system Leads to reduced time-delay marginLow-pass filter: Defines the trade-off between tracking and robustnessFast adaptation leads to improved performance and improved robustness.

Robot Arm with Time-varying Friction and DisturbanceControl design parametersCompact set for ProjectionParametric uncertainties in state-space form:Disturbances in three cases:

Simulation Results without any Retuning of the ControllerSystem outputControl signal

Verification of the Time-Delay MarginWith time-delay 0.02

MRAC and L1 for a PI controllerL1The open-loop transfer functions in the presence of time-delayMRACTime-delay marginTime-delay margin

Miniature UAV in Flight: L1 filter in autopilotCourtesy of Randy Beard, BYU, UTAHThe Magicc II UAV is flying in 25m/h wind, which is ca. 50% of its maximum flight speed

Rohrs Example: Unmodeled dynamicsSystem with unmodeled dynamics:Unmodeled dynamicsplantdynamicsNominal values of plant parameters:Reference system dynamics:Control signal:Adaptive laws from Rohrs simulations:

Instability/Bursting due to Unmodeled DynamicsParametersSystem outputSystem outputParameter driftInstabilityBursting

Rohrs Explanation for Instability: Open-Loop Transfer FunctionAt the frequency 16.1rad the phase reaches -180 degrees in the presence of unmodeled dynamics,reverses the sign of high-frequency gain, the loop gain grows to infinityinstabilityAt the frequency 8rad the phase reaches -139 degrees in the presence of unmodeled dynamics, no sign reversal for the high-frequency gainthe loop gain remains boundedbursting-135-180

Rohrs Example with Projection for Destabilizing Reference InputImproved knowledge of uncertainty reduces the amplitude of oscillations

Rohrs Example with L1 Adaptive ControllerAdaptive laws:Design elements

No Instability/Bursting with L1 Adaptive Controller

Difference in Open-Loop TFsMRACMRAC cannot alter the phase in the feedforward loopAdaptation on feedforward gainContinuous adaptation on phase

Classical Control PerspectiveMRACL1Adaptation simultaneously on both loop gain and phase Adaptation only on loop gain No adaptation on phase

Stabilization of Cascaded Systems with Application to a UAV Path Following ProblemCascaded system(in collaboration with Isaac Kaminer, NPS)

Augmentation of an Existing Autopilot by L1 Controller

Flight Test Results of NPS (TNT, Camp Roberts, CA, May 2007)UAVAutopilotInner-loop Guidance and Control Without adaptationErrors 100m

Architecture for Coordinated Path FollowingPath following: follow predefined trajectories using virtual path length Coordination: coordinate UAV positions along respective paths using velocity to guarantee appropriate time of arrivalGuaranteed stability of the complete system: use L1 to maintainindividual regions of attractionNormalizedPath lengthsOnboard A/P+ UAV(Inner loop)Path following(Outer loop)Pitch rateYaw ratecommandsCoordination VelocitycommanddesiredpathL1adaptationPathGenerationNetworkinfoL1adaptation

Time-critical Coordination of UAVs Subject to Spatial Constraints: Hardware-in-the-Loop Flight Test ResultsPerformance comparison with and without adaptation for one vehicleMission scenarios: Sequential autolanding, ground target suppression, etc.Arrival time is the same, but is not specified a prioriAssumptions:Vehicles dynamically decoupled Underlying communication network topology is fixed in time (currently relaxed)Each vehicle communicates only with its neighborsSimultaneous arrival of two vehicles with L1 enabled

Vision-Based Guidance for Tracking Ground TargetsDesired 2D RangeActual 2D RangeAngle in Degree: Amplitude: Target TrajectoryUAV Trajectory2D Horizontal RangeEstimated Target Velocity2D TrajectoryObjective: Vision-based ground target tracking for small UAVs. Maintain a desired 2D horizontal range. Assumptions:Known relative altitude between UAV and target. Unknown time-varying target velocity.Solution:Fast estimator is applied for the estimation of targets velocity. PDOP metric:

X-45AElevon/yaw vectoring control mixingTailless Unstable AircraftFlight Control Design (Models provided by the Boeing Co.)Autonomous Aerial Refueling Uncertain dynamic environment Receiver dynamics due to aerodynamic influences from tanker Drogue dynamics due to aerodynamic influences from tanker and receiver Finite time to contact the drogue in highly uncertain dynamic environment. Compensation for pitch break uncertainty and actuator failures

Better margin but low damping!Collaborations: Boeing/WP AFRL (X-45A: Time-delay margins of MRAC and L1)

Time-delay margin further improved via the choice C(s) at the price of transient responseX-45A: Time-delay margins of MRAC and L1

Transient tracking not too sensitive to changes in adaptive gain beyond certain lower boundFilter impacts the transient trackingThe filter can be selected to improve the margin at the price of transient trackingVertical accelerationTransient tracking compared for MRAC and L1 with two different filtersThe performance of C1(s) (with better margins) still better than for MRACX-45A: Transient Performance vs RobustnessTDM=0.1TDM=0.045

Finite time to contact the drogue in highly uncertain dynamic environment. Control objective: An adaptive method without retuning adaptation gains for highly uncertain dynamic environment with guaranteed transient performance for finite time-to-contact Solution/Approach:Aerial Refueling Barron Associates Tailless Aircraft Model 6 DOF Flying-wing UAV, Statically unstable at low angle of attack Aerodynamic data primarily taken from [1] Vortex effect data from wind tunnel test [2] ICE model Tanker: KC-135R, Receiver: Tailless 65 degree delta wing UAV Aerodynamic coefficients change as a function of relative separation Varying more significantly with lateral and vertical separation[1] S.P Fears et. al., Low-speed wind-tunnel Investigation of the stability and control characteristics of a series of flying wings with sweep angles of 50 degree[2] W.B. Blake et. al., UAV Aerial Refueling Wind Tunnel Results and Comparison with Analytical Predictions

No retuning!Scaled Responsethrust channelaileron channelelevator channel- Twice the vortex model of a different tanker- Uniform transient and steady state response- Scaled control efforts- Starting from different initial positions- Scaled response

Stable Minimum PhaseUnstable Minimum PhaseStable Non-minimum PhaseUnstable Non-minimum Phasechoose m and w/o limitation is lower-bounded m = 3, = 10m = 10, = 500m = 10, = 500m = 10, = 10 is upper-bounded choose m and from a limited domain Within constraints choose largeFilter vs Reference System (software of Raytheon)

SensorCraft (Air Force / NASA / NG model) High flying, extremely high duration and range, ISR aircraft Light, inherently flexible wing Strong rigid body/flexible mode coupling (poorly modeled) L1 adaptive on & offL1 adaptive offL1 adaptive onDynamic pressure: 30 psfDynamic pressure: 70 psf Guaranteed transient via output feedback Unmatched uncertainties and non SPR Uniform performance for different unknown operational points>70 psf, unstable>70 psf, same desired transientDesired verticaldisplacementActual response

Air-Breathing Hypersonic Vehicle (Boman model WP AFRL) Control Challenge of Air-Breathing Hypersonic Vehicles Vibration due to high speed and structural flexibility Uncertainties caused by rapid change of the operating conditions Integration of airframe dynamics and propulsion system Coupling of attitude and velocity desired speedVelocity profileDesired path flight angleSystem responseControl signal: Elevator (deg) Control Signal: Equivalent ratio

Current Status and Open Problems Adaptive Output Feedback Theory

Partial results in this direction Numerical discretization issues due to fast adaptation Integration of stiff systems

Adaptive Observer Design

Control of distributed autonomous systems over ad-hoc communication network topologies

Performance evaluation on the interface of different disciplines

More work on its way towards final answers!!!

Conclusions What do we need to know?

Boundaries of uncertainties sets the filter bandwidth CPU (hardware) sets the adaptive gain

AcknowledgmentsMy group: Chengyu Cao (theoretical developments) Jiang Wang (aerial refueling) Vijay Patel (UCAV, flight test support) Lili Ma (vision-based guidance) Yu Lei (hypersonic vehicle) Dapeng Li (theoretical developments, vision-based guidance) Amanda Young (cooperative path following) Enric Xargay (cooperative path following) Zhiyuan Li (discretization issues for flexible aircraft) Aditya Paranjape (differential game theoretic approaches for VBG)

Collaborations:

The Boeing Co. (E. Lavretsky, K. Wise, UCAV model, aerial refueling) Wright Patterson AFRL, CerTA FCS program (ICE vortex) Raytheon Co. (R. Hindman, B. Ridgely) NASA LaRC (I. Gregory, SensorCraft) Wright Patterson AFRL (D. Doman, M. Bolender, hypersonic model)

Randy Beard (BYU) Isaac Kaminer (NPS)Sponsors: AFOSR, AFRL, ARO, ONR, Boeing, NASA

X-45A is an autonomous unmanned aircraft.The X-45A has an empty weight of 8,000 lb., a fuel volume of approximately 2700 lb., and a payload capability of 1500 lb. It was designed to nominally operate at 35,000 ft. altitude with a cruise Mach number of 0.8.

The X-45A flight control system uses six trailing edge elevons and yaw thrust vectoring to control the vehicleduring flight, as shown in Figure 3. The elevons are referred to as right or left outboard, midboard, and inboard.The outboard and midboard elevons are contained on the detachable wing.