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THE THEORY OF FAST AND ROBUST ADAPTATION. 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.

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Naira hovakimyan



Naira Hovakimyan

Department of Aerospace and Ocean EngineeringVirginia 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


G. Zames


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  • 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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  • 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 mothership


Crash due to stable, albeit non-robust adaptive controller!

Crash site of X-15A-3

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  • Robustness (measure of relative stability)

  • Performance (comparison with desired system behavior)

Objectives of Feedback

  • Stabilization and/or Tracking

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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 Lyapunov’s 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 system’s 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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Implementable nom. cont.

Small-gain theorem gives sufficient condition for boundedness:

Result: fast and robust adaptation via continuous feedback!

One Slide Explanation of the New Paradigm


Nominal controller in MRAC:

Desired reference system:

This is the nominal controller in the L1 adaptive control paradigm!

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Hurwitz, unknown

System dynamics

Ideal Controller

Adaptive Controller

Adaptive law:

Bounded Barbalat’s lemma

State predictor

Same error dynamics

independent of control choice

Closed-loop state predictor with u

Two Equivalent Architectures of Adaptive Control

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Enables insertion of a low-pass filter

Low-pass filter: C(s)

Implementation Diagrams

Reference system


u cannot be low-pass filtered directly.

State predictor based reparameterization

No more reference

system upon


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Low-pass filter: C(s)


  • Sufficient condition for stability via small-gain theorem

Barbalat’s lemma


  • Closed-loop

  • Solving:

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  • Filtered ideal controller:

  • Closed-loop system with filtered ideal controller =

    non-adaptive predictor:

Stability via small-gain theorem

Plant with

unknown parameter


system of MRAC

Reference system

Closed-loop Reference System

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  • (MRAC)

Guaranteed Transient Performance

  • System state:

  • Control signal:

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Virtual Plant with

clean output

Plant with

unknown parameter

LTI System for Control Specifications

Design system

for defining the control specs

Reference system

achieved via fast adaptation

Independent of the

unknown parameter

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Achieving Desired Specifications

  • Sufficient condition for stability

  • Performance improvement

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Large adaptive gain Smaller step-size Faster CPU

Guaranteed Performance Bounds

  • Use large adaptive gain

  • Design C(s) to render sufficiently small

Desired design objective

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Time-delay Margin and Gain Margin

Point Time-delay occurs


Lower bound for the time-delay margin

Projection domain defines the gain margin

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Main Theorem of L1 Adaptive Controller

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 .

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Tracking vs robustness can be analyzed analytically.

The performance can be predicted a priori.

Design Philosophy

Adaptive gain – as large as CPU permits (fast adaptation)

  • Fast adaptation ensures arbitrarily close tracking of the virtual reference system with bounded away from zero time-delay margin

Fast adaptation leads to improvedperformance and improvedrobustness.

Low-pass filter:

  • Defines the trade-off between tracking and robustness

Increase the bandwidth of the filter:

  • The virtual reference system can approximate arbitrarily closely the idealdesiredreference system

  • Leads to reduced time-delay margin

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Robot Arm with Time-varying Friction and Disturbance

Control design parameters

Parametric uncertainties in state-space form:

Compact set for Projection

Disturbances in three cases:

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Simulation Results without any Retuning of the Controller

System output

Control signal

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With time-delay 0.1

Verification of the Time-Delay Margin

With time-delay 0.02

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Application of nonlinear L1 theory

MRAC and L1 for a PI controller



The open-loop transfer functions in the presence of time-delay

Time-delay margin

Time-delay margin

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Miniature UAV in Flight: L1 filter in autopilot

The Magicc II UAV is flying in 25m/h wind,

which is ca. 50% of its maximum flight speed

Courtesy of Randy Beard, BYU, UTAH

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Rohrs’ Example: Unmodeled dynamics

System with unmodeled 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


Parameter drift

System output



System output

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Rohrs’ Explanation for Instability: Open-Loop Transfer Function



At the frequency 16.1rad

the phase reaches -180degrees

in the presence of unmodeled dynamics,

reverses the sign of high-frequency gain,

the loop gain grows to infinity


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


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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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Predictor model


Adaptive Law

Control Law

Rohrs’ Example with L1 Adaptive Controller

Design elements

Adaptive laws:

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No Instability/Bursting with L1 Adaptive Controller

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Ppr (s)













Difference in Open-Loop TFs


MRAC cannot alter the phase in the feedforward loop

Adaptation on

feedforward gain

Continuous adaptation on phase

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Classical Control Perspective



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

UAV with


Path Following


Cascaded system

(in collaboration withIsaac Kaminer, NPS)

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Path following kinematics

Path following kinematics



Very poor performance

Exponentially stable

Path following controller

Path following controller

Path following kinematics



Output feedback

L1 augmentation

L1 adaptive controller

Path following controller

Augmentation of an Existing Autopilot by L1 Controller

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Flight Test Results of NPS (TNT, Camp Roberts, CA, May 2007)

L1Adaptive Controller Augmented

L1Adaptive Controller

Outer-loop path-planning, navigation

With adaptation, errors reduced to 5m



Inner-loop Guidance and Control

Without adaptation

Errors  100m

Architecture for coordinated path following

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 L1to maintain

individual regions of attraction





Onboard A/P

Path following


Pitch rate



(Outer loop)


Yaw rate

(Inner loop)







Path lengths





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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

Simultaneous arrival of two vehicles with L1 enabled

Mission scenarios:Sequential autolanding, ground target suppression, etc.

  • Arrival time is the same, but is not specified a priori


  • Vehicles dynamically decoupled

  • Underlying communication network topology is fixed in time (currently relaxed)

  • Each vehicle communicates only with its neighbors

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Flight Control Design (Models provided by the Boeing Co.)

Tailless Unstable Aircraft


Autonomous Aerial Refueling

  • Uncertain dynamic environment

    • Receiver dynamics due to aerodynamic influences from tanker

    • Drogue dynamics due to aerodynamic influences from tanker and receiver

Elevon/yaw vectoring control mixing

  • Compensation for pitch break uncertainty and actuator failures

  • Finite time to contact the drogue in highly uncertain dynamic environment.

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X-45A: Transient Performance vs Robustness

  • 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 L1 with two different filters

  • The performance of C1(s) (with better margins) still better than for MRAC

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Aerial Refueling

  • Control objective:

  • Finite time to contact the drogue in highly uncertain dynamic environment.

  • Solution/Approach:

  • An adaptive method withoutretuning adaptationgains

    • for highly uncertain dynamic environment

  • with guaranteed transient performance

    • for finite time-to-contact

  • 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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Scaled Response

- Starting from different

initial positions

- Scaled response

No retuning!

- Twice the vortex – model of a different tanker

- Uniform transient and

steady state response

- Scaled control efforts

thrust channel

elevator channel

aileron channel

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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 path flight angle

desired speed

Control signal: Elevator (deg)

Velocity profile

System response

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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  • 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!

  • What do we need to know?

    • Boundaries of uncertainties sets the filter bandwidth

    • CPU (hardware) sets the adaptive gain

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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)


  • 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

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