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Naira Hovakimyan Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University This talk was originally given as a plenary at SIAM Conference on Control and Its Applications San Francisco, CA, June 2007 A. M. Lyapunov 1857-1918 G. Zames 1934-1997

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Lyapunov Memorial Conference

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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

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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

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

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Objectives of Feedback Stabilization and/or Tracking Performance (comparison with desired system behavior) Robustness (measure of relative stability)

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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.

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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 runs Boeing question: How fast to adapt to be robust?

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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

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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

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System: Nominal controller in MRAC: Desired reference system: Implementable nom. cont. This is the nominal controller in the L 1 adaptive control paradigm! Small-gain theorem gives sufficient condition for boundedness:Result: fast and robust adaptation via continuous feedback! One Slide Explanation of the New Paradigm

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Two Equivalent Architectures of Adaptive Control System dynamics Hurwitz, unknown Ideal Controller Adaptive Controller Adaptive law: Bounded Barbalats lemma State predictor Same error dynamics independent of control choice Closed-loop state predictor with u

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Transient Performance and Robustness in Both Schemes Lyapunov stability Transient performance Robustness in case of PI controller

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Implementation Diagrams MRAC State predictor based reparameterization Reference system No more reference system upon filtering!!! u cannot be low-pass filtered directly. Enables insertion of a low-pass filter Low-pass filter: C(s)

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Stability Sufficient condition for stability via small-gain theorem Barbalats lemma Low-pass filter: C(s) Solving: bounded Closed-loop

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Closed-loop Reference System Filtered ideal controller: Closed-loop system with filtered ideal controller = non-adaptive predictor: Reference system Stability via small-gain theorem Reference system of MRAC Plant with unknown parameter

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Guaranteed Transient Performance System state: Control signal: (MRAC)

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LTI System for Control Specifications Independent of the unknown parameter Reference system achieved via fast adaptation Design system for defining the control specs Plant with unknown parameter Virtual Plant with clean output

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Achieving Desired Specifications Sufficient condition for stability Performance improvement

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Guaranteed Performance Bounds Design C ( s ) to render sufficiently small Use large adaptive gain Desired design objective Large adaptive gain Smaller step-size Faster CPU

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Time-delay Margin and Gain Margin Point Time- delay occurs Plant Lower bound for the time-delay margin Projection domain defines the gain margin

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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 L 1 Adaptive Controller

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Design Philosophy Fast adaptation ensures arbitrarily close tracking of the virtual reference system with bounded away from zero time-delay margin Adaptive 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 margin Low-pass filter: Defines the trade-off between tracking and robustness Tracking vs robustness can be analyzed analytically. The performance can be predicted a priori. Fast adaptation leads to improved performance and improved robustness.

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Robot Arm with Time-varying Friction and Disturbance Control design parameters Compact set for Projection Parametric uncertainties in state-space form: Disturbances in three cases:

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Simulation Results without any Retuning of the Controller System output Control signal

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Verification of the Time-Delay Margin With time-delay 0.02 With time-delay 0.1

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Application of nonlinear L 1 theory MRAC and L 1 for a PI controller L1L1 The open-loop transfer functions in the presence of time-delay MRAC Time-delay margin

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Miniature UAV in Flight: L 1 filter in autopilot Courtesy of Randy Beard, BYU, UTAH The Magicc II UAV is flying in 25m/h wind, which is ca. 50% of its maximum flight speed

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

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Instability/Bursting due to Unmodeled Dynamics Parameters System output Parameter drift Instability Bursting

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Rohrs Explanation for Instability: Open-Loop Transfer Function At 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 infinity instability At the frequency 8rad the phase reaches -139 degrees in the presence of unmodeled dynamics, no sign reversal for the high-frequency gain the loop gain remains bounded bursting -135 -180

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

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Rohrs Example with L 1 Adaptive Controller Predictor model System Adaptive Law Control Law Adaptive laws: Design elements

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No Instability/Bursting with L 1 Adaptive Controller

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Difference in Open-Loop TFs MRAC r(t) P(s)P(s) y(s) - u P ref (s) + + + P um (s) MRAC cannot alter the phase in the feedforward loop Adaptation on feedforward gain L1L1 r(t) P(s)P(s) y(s) - u P pr (s) + + + P um (s) kgkg + + + Continuous adaptation on phase

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Classical Control Perspective MRAC L1L1 Adaptation simultaneously on both loop gain and phase Adaptation only on loop gain No adaptation on phase

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

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Augmentation of an Existing Autopilot by L 1 Controller Path following kinematics Path following controller Exponentially stable Very poor performance Path following kinematics Path following controller UAVAutopilot Output feedback L 1 augmentation L 1 adaptive controller Path following kinematics Path following controller UAVAutopilot

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Flight Test Results of NPS (TNT, Camp Roberts, CA, May 2007) UAV Autopilot Inner-loop Guidance and Control L 1 Adaptive Controller L 1 Adaptive Controller Augmented Outer-loop path-planning, navigation Without adaptation Errors 100m With adaptation, errors reduced to 5m

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Architecture for Coordinated Path Following Path following: follow predefined trajectories using virtual path length Coordination: coordinate UAV positions along respective paths using velocity to guarantee appropriate time of arrival Guaranteed stability of the complete system: use L 1 to maintain individual regions of attraction Normalized Path lengths Onboard A/P + UAV (Inner loop) Path following (Outer loop) Pitch rate Yaw rate commands Coordination Velocity command desired path L 1 adaptation Path Generation Network info L 1 adaptation

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Time-critical Coordination of UAVs Subject to Spatial Constraints: Hardware-in-the-Loop Flight Test Results Performance comparison with and without adaptation for one vehicle Mission scenarios: Sequential autolanding, ground target suppression, etc. Arrival time is the same, but is not specified a priori Assumptions: Vehicles dynamically decoupled Underlying communication network topology is fixed in time (currently relaxed) Each vehicle communicates only with its neighbors Simultaneous arrival of two vehicles with L 1 enabled

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Vision-Based Guidance for Tracking Ground Targets Desired 2D Range Actual 2D Range Angle in Degree: Amplitude: V t t Target Trajectory UAV Trajectory 2D Horizontal Range Estimated Target Velocity2D Trajectory Objective: 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:

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X-45A Elevon/yaw vectoring control mixing Tailless Unstable Aircraft Flight 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

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Better margin but low damping! Collaborations: Boeing/WP AFRL (X-45A: Time-delay margins of MRAC and L 1 )

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Time-delay margin further improved via the choice C ( s ) at the price of transient response X-45A: Time-delay margins of MRAC and L 1

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Transient tracking not too sensitive to changes in adaptive gain beyond certain lower bound Filter impacts the transient tracking The filter can be selected to improve the margin at the price of transient tracking Vertical acceleration Transient tracking compared for MRAC and L 1 with two different filters The performance of C 1 (s) (with better margins) still better than for MRAC MRAC X-45A: Transient Performance vs Robustness TDM=0.1 TDM=0.045

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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

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No retuning! Scaled Response thrust channel aileron 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

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

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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) L 1 adaptive on & off L 1 adaptive off L 1 adaptive on Dynamic pressure: 30 psf Dynamic 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 transient Desired vertical displacement Actual response

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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 speed Velocity profile Desired path flight angle System response Control signal: Elevator (deg) Control Signal: Equivalent ratio

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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!!!

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Conclusions What do we need to know? Boundaries of uncertainties sets the filter bandwidth CPU (hardware) sets the adaptive gain Fast adaptation no more trial-error based gain tuning! performance can be predicted a priori robustness/stability margins can be quantified analytically Theoretically justified Verification & Validation tools for feedback systems at reduced costs With very short proofs!

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Acknowledgments My 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