Robust Adaptive Control with Improved Transient Performance

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Robust Adaptie Control with Improed ransient Performance Eugene Laretsky MI May 03, 2012

2 Presentation Oeriew Engineering, Operations & echnology Boeing Research & echnology Introduction Adaptie Control Boeing ransient Dynamics in Adaptie Control Motiating Example ransient Analysis with All States Accessible Adaptie Output Feedback Design Extension Conclusions, Comments, and Future Research Directions 2

3 Introduction Engineering, Operations & echnology Boeing Research & echnology Classical Model Reference Adaptie Control (MRAC) Originally proposed in 1958 by Whitaker et al., at MI Main idea: Specify desired command-to-output performance of a serotracking system using reference model Defines ideal response of the system due to external commands Later called explicit model following MRAC First proof of closed-loop stability using Lyapuno theory was gien in by Butchart, Shackcloth, and Parks External Command Reference Model Ref. Model Output System Response Adaptie Law Controller Control Command Process System Response 3

Robust and Adaptie Flight Control echnology ransitions: Adanced Aircraft and Weapon Systems Engineering, Operations

Improements Focus on System ID, Implementation, and Actuator Saturation Issues Design Retrofits onto Existing

ransitioned to JDAM production programs Robust Adaptie Control echnology ransition imeline 93 94 95 96 97 98 99 00

(NASA/Boeing) F-15 ACIVE Adaptie Control For Munitions (AFRL-MN/GS//Boeing) MK-84 Boeing IRAD/CRAD MK-82 L-JDAM

Control For ailless Fighters (AFRL-VA/Boeing) X-36 MK-84 JDAM X-36 RESORE MK-82 JDAM MK-82 Laser Seeker

4 Robust and Adaptie Flight Control echnology ransitions: Adanced Aircraft and Weapon Systems Engineering, Operations & echnology Boeing Research & echnology echnology Maturation & ransitions Extended to Munitions (00-02) Boeing IRAD Improements Focus on System ID, Implementation, and Actuator Saturation Issues Design Retrofits onto Existing Flight Control Laws Flight Proen on X-36, MK-84, MK-82, MK-82L, MK-84 IDP 2000, Boeing Phantom Ray, NASA AirStar ransitioned to JDAM production programs Robust Adaptie Control echnology ransition imeline X-45C 04 X-45A J-UCAS & Phantom Ray 05 Phantom Ray Intelligent Flight Control System (NASA/Boeing) F-15 ACIVE Adaptie Control For Munitions (AFRL-MN/GS//Boeing) MK-84 Boeing IRAD/CRAD MK-82 L-JDAM MK-84 IDP 2000 Gen I, flown 1999, 2003 Gen II, flight test 4th Q 2005 Gen III, 2006 Reconfigurable Control For ailless Fighters (AFRL-VA/Boeing) X-36 MK-84 JDAM X-36 RESORE MK-82 JDAM MK-82 Laser Seeker heoretically heoretically justified, justified, numerically numerically efficient, efficient, and and flight flight proen proen technology technology 4

5 Motiating Example Engineering, Operations & echnology Boeing Research & echnology Adaptie Seromechanism for Scalar dynamics Global asymptotic closed-loop stability Bounded tracking in the presence of constant unknown parameters Process : Ref. Model : Controller : x ax bu x a x b r t ref ref ref ref u kˆ xkˆ r t x r ˆ kx x x xxref Adaptie Law : kˆ r r r xx Lyapuno-based ref Benefits : lim et lim xtx t 0 x x r t t ref External Command ref 5

6 Motiating Example (continued) Engineering, Operations & echnology Boeing Research & echnology uning MRAC Increase adaptation gains x, r to get desired (fast) tracking performance Design radeoff Large adaptation gains lead to oscillations (undesirable transients) Cause and effect Reference and transient (error) dynamics hae the same time constant Reference Dynamics : x a x b r ref ref ref ref 1 e aref ransient Dynamics : e aref e b kx xkr r Bounded Signal Need transient dynamics to be faster than reference model Similar to state obserer design separation between controller and obserer poles reduces transients 6

7 Motiating Example (continued) Engineering, Operations & echnology Boeing Research & echnology Reference Model in MRAC Similar to Open-Loop Obserer x a x b r ref ref ref ref Add Obserer-like Error feedback erm to Reference Model Similar to Closed-loop Obserer Error Feedback Gain x a x b r k xx ref ref ref ref e ref Error Feedback erm ref e x r e a k eb k xk r Properties Error feedback regulates transients Conerges to ideal reference model No changes to control input Retains stability and tracking Main Benefit Control of transients External Command k e Reference Model Ref. Model Output System Response Adaptie Law Controller Control Command Process System Response 7

8 Motiating Example (continued) Engineering, Operations & echnology Boeing Research & echnology Simulation Data racking step-inputs 3 2 Process : x x 3u Ref. Model : x 10 x 10 rk xx ref ref e ref Controller : u kˆ ˆ x xkr r ˆ kx 10 xxx Adaptie Law : kˆ r 10rxx ref ref k e = 0 k e = 10 k e = 80 racking Error e = x - x ref k e = 80 k e = 10 k e = 0 (MRAC) Increasing Obserer Feedback Gain Reduces ransient Oscillations ime, sec Need: Formal Analysis of ransient Dynamics 8

9 Engineering, Operations & echnology Boeing Research & echnology 30 Motiating Example (continued) Simulation Data racking performance and control input System State x Command k e = 0 k e = k e = 80 k e = 0 (MRAC) Control Input u k e = 0 k e = ime, sec 9

10 ransient Analysis for Scalar Dynamics Engineering, Operations & echnology Boeing Research & echnology Obserer-like ref model Error Dynamics System State ransient Dynamics x e aref ke ebkx xkr r ref aref xref bref r ke xxref Reference Model Error Dynamics Error Feedback erm k e k0 t max From Lyapuno analysis k 0 e aref e t o(1) and uniformly bounded et e e t et e k0 O o 1 O t k0 max 0 k0 t xt xref t e k0 O o 1 O System State Reference State ransient Dynamics Asymptotic racking 10

11 ransient Analysis for Scalar Dynamics An Alternatie Approach Engineering, Operations & echnology Boeing Research & echnology Error Feedback Gain in Obserer-like Ref Model ransient Dynamics Singular Perturbation Model ref 0 x r e a k eb k xk r Fast (Boundary Layer) Dynamics = ransients Stretched ime t de k e k 0 o1,as t, fixed 0 From Lyapuno Analysis Positie constant Small Parameter ref e x r e a k eb k xk r Boundary Layer Dynamics d Slow Dynamics : e 0 x xref x a xb r k e ref 0 ref ref o1 0 xt x t+o 1 O e System State t k Asymptoticracking Assume to be uniformly continuous and bounded ransient Dynamics 11

12 MIMO Generalization: State Feedback Engineering, Operations & echnology Boeing Research & echnology What Adaptie state feedback seromechanism design for MIMO dynamical systems in the presence of matched uncertainties K Why Improes and streamlines adaptie design tuning How Model Reference Adaptie Control Obserer-like reference model Reduced and Quantifiable ransients 12

13 System Dynamics and Control Engineering, Operations & echnology Boeing Research & echnology Open-Loop Dynamics Uncertain Control Effectieness Command Integrated racking Error Plant State Regulated Output Matched Uncertainty e zi 0m m p zi 0 I C e mm mm u x z x 0 p Bp KI Ap Bp Kp xp B p np m x z 0mm Cp x C z Aref Hurwitz x B Unknown B Parameters Known Regressor ref cmd Regulated Output Hurwitz z Cz x Control Objectie Design control input u such that the regulated output z tracks bounded time-arying command z cmd with bounded errors, reduced transients, and while operating in the presence of matched uncertainties x A xb u x B z ref ref cmd Matched Uncertainties Command 13

14 a Glance: How is It Currently Done? Engineering, Operations & echnology Boeing Research & echnology System Dynamics x A xb u x B z ref ref cmd e Reference Model x A x B z Plant w/o uncertainties racking Error e x x ref ref ref ref ref cmd Adaptie Control Input u ˆ x racking Error Dynamics e Aref eb x Algebraic Lyapuno Equation Adaptie Laws Closed-Loop Stability PAref Aref PQ0 ˆ xe PB lim et 0 t Adaptie Control uning Cycle Q PB Ratesof adaptation 14

15 Few houghts Engineering, Operations & echnology Boeing Research & echnology Open-Loop Dynamics Reference Model ~ Luenberger Open-Loop Obserer racking Error Dynamics = ransient Error Dynamics Need to be faster than system dynamics minimizes unwanted transients Main Idea: Use Closed-Loop Luenberger Obserer as Reference Model x A xb u x B z ref ref cmd x A x B z ref ref ref ref cmd e Aref e B x Obserer Gain x A x B z L xx ref ref ref ref cmd ref Innoation erm 15

16 MRAC with Obserer like Reference Model Engineering, Operations & echnology Boeing Research & echnology Open-Loop Plant Obserer-like Reference Model racking Error x A xb u x B z ref ref cmd x A x B z L xx ref ref ref ref cmd ref e x x ref Obserer-like Gain e Adaptie Control Input Error Dynamics u ˆ Obserer Riccati Equation Adaptie Laws Closed-Loop Stability With prescribed degree of stability 1 x ref e A L eb x A Hurwitz P A I A I P P R P Q 0 ref n n ref n n Lyapuno function 1, trace V e e P e PA APPR PQ 1 P A A P R P Q P Adaptie Laws ˆ 1 x e P B lim et 0 t Global Asymptotic racking 16

17 Obserer-Like MRAC State Feedback Design Summary Engineering, Operations & echnology Boeing Research & echnology System Dynamics Regulated Output No Uncertainties LQR PI Ref Dynamics Baseline Closed-Loop System z C x, dimz dimu x AxB u xp Bref zcmd z A ref Klqr x AB R B P x B z 1 ref ref ref ref ref cmd Sole Obserer ARE, Compute Obserer Gain, and Form Ref model 1 P A I A I P P R P Q 0 ref n n ref n n Adaptie Control State-feedback ˆ x e P ˆ u x 1 B L P R 1 x x ref x A x B z L xx ref ref ref ref cmd ref z z ~ z ref cmd 1 Q, R L P R Design Cycle Ratesof adaptation System State Asymptotic Reference Model racking Bounded Command racking 17

18 ransient Analysis for MIMO Dynamics Engineering, Operations & echnology Boeing Research & echnology Closed-Loop ransient (Error) Dynamics R I 1 nn e Aref P R e B t x t Obserer Gain: L t Singular Perturbed System 1 Hurwitz Matrix = Uniformly Bounded Function of ime 1 e Aref 1 P e t ref 1 0 O e A P e t Stretched ime Boundary Layer Dynamics = ransients Positie Definite Symmetric P P0 O, as 0 tt x t xref t P x t x t 0, O o 1 exp Asymptotic racking t de Pe 0 d 0 0 ref 0 ransient Dynamics 18

19 Closed-loop Reference Model (CRM) in Adaptie Control rais E. Gibson (PhD Student, MI) Engineering, Operations & echnology Boeing Research & echnology Uncertain Plant Reference Model x Aref xb u x x A x Br l xx ref ref ref ref Control Input ˆ u r x Obserer-like Gain racking Error e x x ref Lyapuno Equation with prescribed degree of stability A li PP A li I ref nn ref nn nn Adaptie Law ˆ Proj ˆ, xe PB uning Knobs: Obserer Gain l and Adaptation Rate l Main Result ut () O, ut () = O 0 t 4 e 4et l Asymptotic Bounds on Control Rate ransients 19

20 Bounds on Control Rate ransients with CRM* rais E. Gibson (PhD Student, MI) Engineering, Operations & echnology Boeing Research & echnology Ref Model dominating eigenalue min Real i Aref i Main Result: Bounds on control rate l ut ( ) = O, ut ( ) = O 0 t 4 e 4et l Error time constant: 1 e l l l 10, , 10 ime constant associated with A ref : 1 ref u Inequality enforced by design 10 e ref 4 e 20 t *.E. Gibson, E. Laretsky and A.M. Annaswamy, Closed-Loop Reference Models in Adaptie Control: Stability, Robustness, and ransient Performance, CDC 12 submitted Copyright 2009 Boeing. All rights resered. 20

21 Engineering, Operations & echnology Boeing Research & echnology Can We Extend Obserer-like State Feedback MRAC Design o Adaptie Output Feedback? * E. Laretsky, Adaptie Output Feedback Design Using Asymptotic Properties of LQG / LR Controllers, IEEE ransactions on Automatic. Control, Jun,

What, Why, and How Engineering, Operations & echnology Boeing Research & echnology Problem

Applications Very Flexible Aerial (VFA) platforms.

not aailable online, hae low damping ratios, and must be actiely controlled / stabilized

22 What, Why, and How Engineering, Operations & echnology Boeing Research & echnology Problem Output feedback design for MIMO systems in the presence of unknown unknowns Aerospace Applications Very Flexible Aerial (VFA) platforms. System dynamics exhibit no frequency separation between primary and flex modes Flex modes are not aailable online, hae low damping ratios, and must be actiely controlled / stabilized POLECA Control Design Architecture Robust LQG/LR + Adaptie output feedback augmentation Based on asymptotic properties of LQG/LR regulators HELIOS 22

23 Problem Formulation Engineering, Operations & echnology Boeing Research & echnology Plant Dynamics Restrictions: Obserable, Controllable, Minimum-Phase Uncertain Control Effectieness Matched Uncertainty Command Integrated racking Error Plant State Measured Output dxp e 0 zi mm Cp ezi 0 I mm mm u x z x 0n 0 p p m A p xp B p np m x A x B y Cx, z 0mm Cp x C z d d p cmd Unknown Parameters Known Regressor B ref Controlled Output Control Problem Using output measurements y, design control input u such that the regulated output z tracks its bounded time-arying command z cmd with bounded errors, while operating in the presence of unknown unknowns 23

24 Reference Model Construction Engineering, Operations & echnology Boeing Research & echnology System Dynamics w/o Uncertainties x Ax Bu B z ref ref ref ref cmd Controller Algebraic Riccati Equation P A A P P BR B P Q 0 1 ref ref ref ref ref ref Reference / Baseline LQR PI Controller Reference Model u R B P x K x 1 ref ref ref ref lqr ref K lqr LQR Gain x AB R B P x B z 1 ref ref ref ref ref cmd Hurwitz A ref x A x B z ref ref ref ref cmd Satisfies Model Matching Conditions by Design 24

25 Open-Loop Dynamics Reformulation Engineering, Operations & echnology Boeing Research & echnology Using Reference Model Data x A xbuk x x B z x Kx d d xp x ref x d d p ref cmd Hurwitz y Cx, z C x x A xb u x B z ref ref cmd z Measured Regulated 25

26 Design Idea Engineering, Operations & echnology Boeing Research & echnology Sufficient Condition for Closed-Loop Stability P Aref Inn Aref Inn P 0 State Feedback Adaptie Law ˆ 1 x e P B y Cx ~SPR y Output Measurements Output error e yˆ y C xˆx Ce (State Output) Adaptie Feedback P B C W 1 ˆ x e C W y x e W 26

27 Open-Loop Dynamics Assumptions and Squaring-up Method Engineering, Operations & echnology Boeing Research & echnology Controllable & Obserable y Cx, z C x Number of measured outputs p is no less than number of control inputs m Achieing Nonzero High Frequency Gain and Minimum-phase Dynamics x A xb u x B z Measured ref ref cmd z Regulated output embedded into system dynamics dim y p m dimu dim z si A B p m det C B B 0det 0, Re s 0 nn 2 C 0 p p B 1 det si A C si A B det Squaring-Up Problem Find B 2 such that Rosenbrock System Matrix is Nonsingular in the RHP No ransmission Zeros in the RHP Allows to control non-minimum phase dynamics with relatie degree greater than 1 27

28 Adaptie Output Feedback Engineering, Operations & echnology Boeing Research & echnology Open-Loop Dynamics y Cx, z C x Luenberger-type State Obserer Closed-Loop Plant Dynamics x A xb u x B z ref ref cmd z ˆ xˆ A xˆbˆ u xˆ L y yˆ B z ref ref cmd yˆ Cxˆ Control Input Closed-Loop Obserer Dynamics u ˆ xˆ xˆ A xˆb z L y yˆ ref ref cmd ˆ Obserer Gain ˆ x A xb z B x x ref ref cmd e xˆ x x Obserer Error Estimated Parameters 28

29 Adaptie Output Feedback (continued) Engineering, Operations & echnology Boeing Research & echnology Closed-Loop Obserer Dynamics xˆ A xˆb z L y yˆ Closed-Loop Plant Dynamics ref ref cmd ˆ ˆ x A xb z B x x ref ref cmd Obserer Error e xˆ x x Obserer Error Dynamics ˆ ˆ e A L C e B x x x ref x Obserer Gain Estimated Parameters Design ask Reduce Obserer Error Choose Obserer Gain ex 1 Adapt Parameters 29

30 Adaptie Output Feedback, (Obserer Design) Engineering, Operations & echnology Boeing Research & echnology Squaring-Up (if p > m) Nonzero High Frequency Gain 1 pm det CB 0 det si A det C si A B 0, Re s0 Choose parameter dependent () weights 1 Q Q BB, R R 1 Sole Filter Algebraic Riccati Equation P A I A I P P C R CP Q 0 ref n n ref n n Calculate Obserer Gain, (parameter-dependent) L B B B 2 Small Positie Parameter Positie constant Enforces prescribed degree of stability Free to Choose P C R 1 No ransmission Zeros in the RHP 30

31 Adaptie Output Feedback, (Obserer Design) Engineering, Operations & echnology Boeing Research & echnology Parameter Dependent Algebraic Riccati Equation, with prescribed degree of stability 1 1 PAref Inn Aref InnP Q0 PC R0 CP BB 0 heorem Inerse solution exists Symmetric, positie-definite 1 P P Asymptotic relations take place, as 1 1 Dominating term, for small P P0 O, as 0 0 Computable 1 2 PB C R0 WO PC BW R 2 P B B C R W 2 0 O 1 0 O Enable MRAC design with output feedback Defines output measurements 1 I mm 2 PB C R0 W O 0 pmm Computable uning knob 31

32 Adaptie Output Feedback, (Completed) Engineering, Operations & echnology Boeing Research & echnology Parameter Dependent Algebraic Riccati Equation 1 P A I A I P P C R CP Q 0 ref n n ref n n 1 Q Q BB, R R A Aref PC R C Hurwitz L Obserer Gain Asymptotic Relation for Stability Proofs PA APPC R CPQ2 P0 1 Inerse ARE Solution 1 2 PB C R0 WSO heorem Stability & Bounded racking Parameter Adaptation with Projection Operator 1 ˆ ˆ 2 Proj, xˆy yˆ R0 W S Adaptie Output Feedback Control Reference Model Small ˆ u xˆ xˆ A xˆb z L y yˆ Computable ref ref cmd Lyapuno-based Stability Proof 32

33 LQG / LR Design Iterations Adaptie Output Feedback Design Summary Engineering, Operations & echnology Boeing Research & echnology System Dynamics Measured and Regulated Output Set Uncertainties to Zero, Design LQR PI Controller, and Create Reference Dynamics Baseline Closed-Loop System 1 Compute B 2 such that: Choose Small Parameter 0 Sole Filter ARE, Compute Kalman Gain and Form State Obserer Output Feedback Adaptie Laws Output Feedback Control 1 ˆ ˆ 2 Proj, xˆy yˆ R0 W S ˆ u x AxB u x B z y Cx, z C x, dimy dimz B det CB B2 0 sinn A B det 0, Re s 0 C 0 p p PAref Inn Aref InnP PC R0 CP Q0 BB 0 xˆ z d d p ref cmd x AB R B P x B z Klqr ref ref ref ref ref ref cmd A x ˆ A xˆb z L y yˆ ref ref cmd 33

34 Key Design Features Engineering, Operations & echnology Boeing Research & echnology Adaptie laws and Control Input Do Not explicitly depend of the tuning parameter 0 1 ˆ ˆ 2 Proj, xˆy yˆ R0 W S u ˆ xˆ System Dynamics Reformulated Imbeds Desired Reference Model x AxB u x B z d d p ref cmd LQG / LR obserer tuning leads to improed reference model tracking x Aref xbref zcmd B u x xˆ A xˆb z L y yˆ ref ref cmd xˆ xˆ x x xref x ref Squared-up LI Dynamics LQG/LR Obserer Output Feedback Adaptie Controller Reference Model racking 34

Conclusions Engineering, Operations & echnology Boeing Research & echnology Constructie Methods to Design

ransients Based on asymptotic properties of LQG / LR regulators Obserer-like reference model modification

Adaptie Control with Nonparametric Uncertainties State Limiter (keeps system state within bounded

35 Conclusions Engineering, Operations & echnology Boeing Research & echnology Constructie Methods to Design Adaptie State and Output Feedback Controllers for MIMO Systems with Matched Uncertainties and Quantifiable ransients Based on asymptotic properties of LQG / LR regulators Obserer-like reference model modification Ongoing Work Robust and adaptie control for Very Flexible Aerial Platforms Future Work Output Feedback Adaptie Control with Nonparametric Uncertainties State Limiter (keeps system state within bounded approximation set) Combined / Composite Output Feedback Adaptie Design Using tracking and prediction errors in adaptie laws K 35

36 Engineering, Operations & echnology Boeing Research & echnology Open-Loop Plant Obserer-like Reference Model racking Error A echnical Challenge Adaptie Control Input SOP RIGH HERE!!! his is a Cancelation-Based Design May hae 0 margins Recoering ideal control may lead to loss of robustness A Controersy?! Need Optimal / Robust Control Solutions Are NO cancellation-based Hae nonzero gain and time-delay margins Question: Can MRAC solutions be formulated using Optimal Control? x A xb u x B z ref ref cmd x A x B z L xx ref ref ref ref cmd ref u e x x ref ˆ x e 36

37 Boeing in Seattle Engineering, Operations & echnology Boeing Research & echnology 37

38 Phantom Ray First Flight,

Lecture 9. Introduction to Kalman Filtering. Linear Quadratic Gaussian Control (LQG) G. Hovland 2004

Lecture 9. Introduction to Kalman Filtering. Linear Quadratic Gaussian Control (LQG) G. Hovland 2004 MER42 Advanced Control Lecture 9 Introduction to Kalman Filtering Linear Quadratic Gaussian Control (LQG) G. Hovland 24 Announcement No tutorials on hursday mornings 8-9am I will be present in all practical