Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Adaptive Control: Introduction, Overview, and Applications Eugene Lavretsky, Ph.D E-mail: eugene.lavretsky@boeing.com Phone: 714-235-7736 E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Course Overview • • Motivating Example Review of Lyapunov Stability Theory – – – – • Model Reference Adaptive Control – – – – • Nonlinear systems and equilibrium points Linearization Lyapunov’s direct method Barbalat’s Lemma, Lyapunov-like Lemma, Bounded Stability Basic concepts 1st order systems nth order systems Robustness to Parametric / Non-Parametric Uncertainties Neural Networks, (NN) – Architectures – Using sigmoids – Using Radial Basis Functions, (RBF) • • E Lavretsky Adaptive NeuroControl Design Example: Adaptive Reconfigurable Flight Control using RBF NN-s Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications References • J-J E Slotine and W Li, Applied Nonlinear Control, Prentice-Hall, New Jersey, 1991 • S Haykin, Neural Networks: A Comprehensive Foundation, 2nd edition, Prentice-Hall, New Jersey, 1999 • H K., Khalil, Nonlinear Systems, 2nd edition, PrenticeHall, New Jersey, 2002 • Recent Journal / Conference Publications, (available upon request) E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Motivating Example: Roll Dynamics (Model Reference Adaptive Control) p = Lp p + Lδ ail δ ail • Uncertain Roll dynamics: – p is roll rate, – δ ail is aileron position – L p , Lδ ail are unknown damping, aileron effectiveness ( ) pm = Lmp pm + Lmδ δ ( t ) • Flying Qualities Model: m m – ( Lp , Lδ ) are desired damping, control effectiveness – δ ( t ) is a reference input, (pilot stick, guidance command) e p ( t ) = ( p ( t ) − pm ( t ) ) → – roll rate tracking error: • Adaptive Roll Control: ⎧ Kˆ = −γ p ( p − p ) ⎪ p p m , (γ p , γ δ ⎨ ⎪⎩ Kˆ δ = −γ δ δ ( t )( p − pm ) ail E Lavretsky δ ail = Kˆ p p + Kˆ δ δ ail )>0 parameter adaptation laws Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Motivating Example: Roll Dynamics (Block-Diagram) desired flying qualities model δ (t ) unknown plant Kˆ δ Lmδ s + Lmp pm Lδ ail p roll tracking error ep → s + Lp Kˆ p parameter adaptation loop • Adaptive control provides Lyapunov stability • Design is based on Lyapunov Theorem (2nd method) • Yields closed-loop asymptotic tracking with all remaining signals bounded in the presence of system uncertainties5 E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Lyapunov Stability Theory Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Alexander Michailovich Lyapunov 1857-1918 • Russian mathematician and engineer who laid out the foundation of the Stability Theory • Results published in 1892, Russia • Translated into French, 1907 • Reprinted by Princeton University, 1947 • American Control Engineering Community Interest, 1960’s E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Nonlinear Dynamic Systems and Equilibrium Points • A nonlinear dynamic system can usually be represented by a set of n differential equations in the form: x = f ( x, t ) , with x ∈ R n , t ∈ R – x is the state of the system – t is time • If f does not depend explicitly on time then the system is said to be autonomous: x = f ( x ) • A state xe is an equilibrium if once x(t) = xe, it remains equal to xe for all future times: = f ( x ) E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Example: Equilibrium Points of a Pendulum M R θ + bθ + M g R sin (θ ) = • System dynamics: • State space representation, ( x = θ , x2 = θ ) x1 = x2 R b g x2 = − x − sin ( x1 ) 2 MR R θ • Equilibrium points: = x2 0=− b g sin ( x1 ) x − 2 MR R x2 x1 E Lavretsky x2 = 0, sin ( x1 ) = ⎛π k ⎞ xe = ⎜ ⎟, ⎝ ⎠ M ( k = 0, ± 1, ± 2,…) Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Example: Linear Time-Invariant (LTI) Systems • LTI system dynamics: x = Ax – has a single equilibrium point (the origin 0) if A is nonsingular – has an infinity of equilibrium points in the nullspace of A: A xe = • LTI system trajectories: x ( t ) = exp ( A ( t − t0 ) ) x ( t0 ) • If A has all its eigenvalues in the left half plane then the system trajectories converge to the origin exponentially fast 10 E Lavretsky Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Task 3: Persistency of Excitation in Flight Mechanics • Information content from adaptation / estimation processes depends on parameter convergence – Requires persistent excitation (PE) of control inputs • • Need numerically stable / on-line verifiable PE conditions for flight mechanics and control Aircraft Example: Longitudinal dynamics ⎧  T cos α − D − g sin (θ − α ) ⎪V = m ⎪ ⎪α = q − T sin α + L + g cos (θ − α ) ⎪ mV ⎨ ⎪ M  q = ⎪ Iy ⎪ ⎪θ = q ⎩ Control Inputs ⎧T = q S CT ≅ q S CTδT δ T ⎪ ⎪⎪ L = q S CL ≅ q S CL (α , q ) ⎨D = q S C ≅ q S C α , q ) D D( ⎪ ⎪ ⎪⎩ M = q S c CM ≅ q S c CM (α , q ) + CM δe (α , δ e ) δ e ( ) Problem: ¾Estimate on-line unknown aerodynamic coefficients ¾Find sufficient conditions (PE) that yield convergence of the estimated parameters to their corresponding true (unknown) values 13 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Design Example: F-16 Adaptive Pitch Rate Tracker Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Aircraft Data Short-Period Dynamics • Trim conditions – CG = 35%, Alt = ft, QBAR = 300 psf, VT = 502 fps, AOA = 2.1 deg • Nominal system – statically unstable – open-loop dynamically stable, (2 real negative eigenvalues) • Control architecture – baseline / nominal controller • LQR pitch tracking design – direct adaptive model following augmentation • Simulated failures – elevator control effectiveness: 50% reduction – battle damage instability • static instability: 150% increase • pitch damping: 80% reduction – pitching moment modeling nonlinear uncertainty Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI Baseline Controller • Using LQR PI state feedback design – nominal values for stability & control derivatives – pitch rate step-input command – no uncertainties, no control failures ⎧e = q − q – system dynamics: ⎪ Z Z ⎪  q α α = + + – “wiggle” system in matrix form ⎨⎪ V V I q cmd α ⎞ ⎛0 ⎛ ⎞ e ⎛ eq ⎞ ⎜ ⎛ ⎞ ⎟ q ⎜ ⎟ Zα Z ⎜  ⎟ ⎜ ⎜ ⎟ δ ⎟ δe ⇔ ⎟ ⎜ α ⎟ + ⎜ ⎜α ⎟ = ⎜0 V ⎟⎜ ⎟ ⎜ V ⎟ N ⎜ q ⎟ u q ⎠ ⎜ ⎟ ⎟ ⎝N⎠ ⎜ M ⎝ M q ⎠ N ⎝ Mδ ⎠ α ⎝ N x x A B δ δe ⎪q = M α α + M q q + M δ δ e ⎩ x = A x + B u Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI Baseline Controller (continued) • LQR design for the “wiggle” system – Optimal feedback solution: u = − K x – Using original states: δebl = − ( K qI Kα ⎛ eq ⎞ ⎜ ⎟ K q ) ⎜ α ⎟ = − K qI eq − Kα α − K q q ⎜ q ⎟ ⎝ ⎠ bl δ – Integration yields LQR PI feedback: e δ ebl = − K qI eqI − Kα α − K q q K P = − Kx x δ ebl = −10 eqI − 3.2433α − 10.7432 q closed-loop eigenvalues ⎞ ⎛0 ⎛ ⎞ ⎛ 100 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A = ⎜ -1.0189 0.9051 ⎟ , B = ⎜ -0.0022 ⎟ , Q = ⎜ ⎟ ⎜ 0.8223 -1.0774 ⎟ ⎜ -0.1756 ⎟ ⎜ 0 100 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Eigenvalue -7.97e-001 + 3.45e-001i -7.97e-001 - 3.45e-001i -2.39e+000 Damping 9.18e-001 9.18e-001 1.00e+000 Freq (rad/s) 8.68e-001 8.68e-001 2.39e+000 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Short-Period Dynamics with Uncertainties • System: ⎞ I ⎛0 ⎛ ⎞ ⎛ eqI ⎞ ⎜ ⎛ ⎞ e ⎛ −1⎞ ⎟ q ⎜ ⎟ Zα ⎜ ⎟ ⎜ ⎟ ⎜ α ⎟ + ⎜ Zδ ⎟ Λ ( δ e + K (α , q ) ) + ⎜ ⎟ q cmd  = α ⎜ ⎟ ⎜ ⎜ ⎟ ⎟⎜ ⎟ ⎜ V ⎟ V ⎜0⎟ ⎜ q ⎟ ⎜ ⎜q⎟ ⎜ ⎝N⎠ ⎝N⎠ M α M q ⎟ ⎝N ⎠ Mδ ⎟ ⎝ ⎠ ⎝ ⎠ N B2  x x A • Reference model: B1 = B x = A x + B1 Λ ( δ e + K (α , q ) ) + B2 q cmd – no uncertainties – (Plant + Baseline LQR PI) xref = ( A + B1 K xT ) xref + B2 q cmd = Aref xref + Bref q cmd N  B Aref ref tracking error vector • Control Goal – Model following pitch rate tracking: x − xref → Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Adaptive Augmentation Design • Total elevator deflection: K (α , q )  ˆ T Φ (α , q ) = K qI eqI + Kα α + K q q + kˆqI eqI + kˆα α + kˆq q − Θ   ˆ δ e = δ ebl + δ ead δ ebl ( δ ead δ e = K x + kˆx • Adaptive laws: ( ( ) ⎧kˆ = Γ Proj kˆ , − x eT P B x x ⎪ x ⎨ ˆ = Γ Proj Θ ˆ , Φ ( x ) eT P B ⎪Θ p Θ ⎩ ) ) T ˆ T Φ (α , q ) x−Θ ⎧⎛ kˆ ⎞ ⎛ ⎛ kˆ ⎞ ⎛ ⎞⎞ ⎪⎜ α ⎟ ⎜⎜ α ⎟ ⎛ α ⎞ ⎜ ⎟⎟ Zδ ⎟ ⎟ ⎪⎜ kˆ ⎟ = Γ Proj ⎜ ⎜ kˆ ⎟ , − ⎜ q ⎟ q − q ref α − α ⎜ q − qref ) P ( I x ref ⎪⎜ q ⎟ ⎜ ⎜ q ⎟ ⎜⎜ ⎟⎟ I ⎜ V ⎟⎟ I ⎪⎜ ˆ I ⎟ ˆ q ⎜ ⎟ ⎜ ⎜ M ⎟⎟ k ⎝ I⎠ ⎪⎜⎝ kq ⎟⎠ ⎝ δ ⎠⎠ ⎝⎝ q ⎠ ⎨ ⎛ ⎪ ⎛ ⎞⎞ ⎜ ⎪ ⎜ ⎟⎟ Zδ ⎟ ⎟ ˆ = Γ Proj ⎜ Θ ˆ , Φ (α , q ) ( q − q ref α − α ⎪Θ ⎜ q − qref ) P I I ref Θ ⎜ ⎪ ⎜ V ⎟⎟ ⎜ ⎪ ⎜ M ⎟⎟ ⎝ δ ⎠⎠ ⎝ ⎩ Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Adaptive Augmentation Design (continued) • Free design parameters – symmetric positive definite matrices: ( Q, Γ x , ΓΘ ) • Need to solve algebraic Lyapunov equation T P Aref + Aref P = −Q • Using Dead-Zone modification and Projection Operator Ref Model x = ( qI xref α q) T e Adaptive δ ead + δ ebl qcmd δe Control Allocation Actuators F-16 Sensors x Baseline Inner-Loop Controller Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Adaptive Design Data • Design parameters − φi = e – using 11 RBF functions: – Rates of adaptation: (α −αi )2 σ2 , αi ∈[ −10:.1:10] Γx = 0, ΓΘ = – Solving Lyapunov equation with: Q = diag ([ 800]) • Zero initial conditions • Pitch rate command input • System Uncertainties – – – – 50% elevator effectiveness failure,( 0.5* Mδbl ) bl 50% increase in static instability, (1.5* Mα ) 80% decrease in pitch damping, ( 0.2* M qbl ) nonlinear pitching moment ⎛ 2π ⎞ ⎜α − ⎟ 180 ⎠ −⎝ 0.01162 M (α ) = 1.5* Mαbl + e M (α ) Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI: Tracking Step-Input Command Unstable Dynamics due to Uncertainties Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI + Adaptive: Tracking StepInput Command Adaptive Augmentation yields Bounded Stable Tracking in the Presence of Uncertainties 10 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI: Tracking Sinusoidal Input with Uncertainties LQR PI Tracking Performance Degradation in the Presence of Uncertainties 11 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications LQR PI + Adaptive: Tracking Sinusoidal Input with Uncertainties Adaptive Augmentation Recovers Target Tracking Dynamics in the Presence of Uncertainties 12 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Model Following Tracking Error Comparison Adaptive Augmentation yields Significant Reduction in Tracking Error Magnitude 13 Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Adaptive Design Comments • RBF NN adaptation dynamics ˆ ref , Θ = Γ Φ α q k q − q ( ) ( ) ( ( Θ ii 1i i i I I ) + k i (α − α ref ) + k3 i ( q − qref ) ) • Fixed RBF NN gains – simulation data k1i = 0, k2 i = −1.1266, k3i = −24.0516 • Projection Operator ⎛ ⎞ ⎛ k1i ⎞ ⎜ ⎟ Z ⎜ ⎟ δ ⎟ ⎜ = k P ( ΓΘ )ii ⎜ 2i ⎟ ⎜ V ⎟ ⎜ k3 i ⎟ ⎜M ⎟ ⎝ ⎠ ⎝ δ⎠ – keeps parameters bounded – nonlinear extension of anti-windup integrator logic • Dead-Zone modification dead-zone tolerance – freezes adaptation process if: x − xref ≤ ε – separates adaptive augmentation from baseline controller 14