Adaptive Control: Introduction, Overview, And Applications
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsAdaptive Control: Introduction,Overview, and ApplicationsEugene Lavretsky, Ph.D.E-mail: eugene.lavretsky@boeing.comPhone: 714-235-7736E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsCourse Overview Motivating ExampleReview of Lyapunov Stability Theory–––– Model Reference Adaptive Control–––– Nonlinear systems and equilibrium pointsLinearizationLyapunov’s direct methodBarbalat’s Lemma, Lyapunov-like Lemma, Bounded StabilityBasic concepts1st order systemsnth order systemsRobustness to Parametric / Non-Parametric UncertaintiesNeural Networks, (NN)– Architectures– Using sigmoids– Using Radial Basis Functions, (RBF) E. LavretskyAdaptive NeuroControlDesign Example: Adaptive Reconfigurable Flight Control usingRBF NN-s2
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsReferences J-J. E. Slotine and W. Li, Applied Nonlinear Control,Prentice-Hall, New Jersey, 1991 S. Haykin, Neural Networks: A ComprehensiveFoundation, 2nd edition, Prentice-Hall, New Jersey, 1999 H. K., Khalil, Nonlinear Systems, 2nd edition, PrenticeHall, New Jersey, 2002 Recent Journal / Conference Publications, (availableupon request)3E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMotivating 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:mm– ( Lp , Lδ ) are desired damping, control effectiveness– δ ( t ) is a reference input, (pilot stick, guidance command)e p ( t ) ( p ( t ) pm ( t ) ) 0– roll rate tracking error: Adaptive Roll Control: Kˆ γ p ( p p ) ppm, (γ p , γ δ Kˆ δ γ δ δ ( t )( p pm )ailE. Lavretskyδ ail Kˆ p p Kˆ δ δail) 04parameter adaptation laws
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMotivating Example: Roll Dynamics(Block-Diagram)desired flying qualities modelδ (t )unknown plantK̂δLmδs LmppmLδ ailproll trackingerrorep 0s LpKˆ pparameter adaptation loop Adaptive control provides Lyapunov stability Design is based on Lyapunov Theorem (2nd method) Yields closed-loop asymptotic tracking with all remainingsignals bounded in the presence of system uncertainties5E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Stability Theory
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsAlexander Michailovich Lyapunov1857-1918 Russian mathematician and engineer wholaid out the foundation of the StabilityTheory Results published in 1892, Russia Translated into French, 1907 Reprinted by Princeton University, 1947 American Control Engineering CommunityInterest, 1960’s7E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsNonlinear Dynamic Systems andEquilibrium Points A nonlinear dynamic system can usually berepresented by a set of n differential equationsin 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 thesystem is said to be autonomous: x f ( x ) A state xe is an equilibrium if once x(t) xe, itremains equal to xe for all future times: 0 f ( x )8E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Equilibrium Points of aPendulumM R 2 θ bθ M g R sin (θ ) 0 System dynamics: State space representation, ( x1 θ , x2 θ)x1 x2Rbgx2 x sin ( x1 )2 2MRRθ Equilibrium points:0 x20 bgsin ( x1 )x 2 2MRRx2x1E. Lavretskyx2 0, sin ( x1 ) 0 π k xe , 0 M( k 0, 1, 2, )9
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Linear Time-Invariant(LTI) Systems LTI system dynamics:x Ax– has a single equilibrium point (the origin 0) ifA is nonsingular– has an infinity of equilibrium points in the nullspace of A: A xe 0 LTI system trajectories: x ( t ) exp ( A ( t t0 ) ) x ( t0 ) If A has all its eigenvalues in the left halfplane then the system trajectoriesconverge to the origin exponentially fast10E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsState Transformation Suppose that xe is an equilibrium pointIntroduce a new variable: y x - xeSubstituting for x y xe into x f ( x )New system dynamics: y f ( y x )New equilibrium: y 0, (since f(xe) 0)Conclusion: study the behavior of the newsystem in the neighborhood of the origine11E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsNominal Motion Let x*(t) be the solution ofx f ( x)– the nominal motion trajectory corresponding to initialconditions x*(0) x0 Perturb the initial condition x ( 0 ) x0 δ x0 Study the stability of the motion error: e ( t ) x ( t ) x ( t ) The error dynamics:– non-autonomous!()()e f x ( t ) e ( t ) f x ( t ) g ( e, t )e ( 0 ) δ x0 Conclusion: Instead of studying stability of thenominal motion, study stability of the errordynamics w.r.t. the originE. Lavretsky12
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Stability Definition: The equilibrium state x 0 ofautonomous nonlinear dynamic system is said tobe stable if: R 0, r 0, { x ( 0 ) r} { t 0, x ( t ) R} Lyapunov Stability means that the systemtrajectory can be kept arbitrary close to the originby starting sufficiently close to itx ( 0)0StableRE. Lavretskyrx ( 0)0UnstableRr13
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsAsymptotic Stability Definition: An equilibrium point 0 isasymptotically stable if it is stable and if inaddition: r 0, { x ( 0 ) r} {limx ( t ) 0}t Asymptotic stability means that the equilibrium isstable, and that in addition, states started closeto 0 actually converge to 0 as time t goes toinfinity Equilibrium point that is stable but notasymptotically stable is called marginally stable 14E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExponential Stability Definition: An equilibrium point 0 isexponentially stable if:{} r , α , λ 0, x ( 0 ) r t 0 :x ( t ) α x ( 0 ) e λ t , The state vector of an exponentially stablesystem converges to the origin faster thanan exponential function Exponential stability implies asymptoticstability15E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLocal and Global Stability Definition: If asymptotic (exponential) stabilityholds for any initial states, the equilibrium pointis called globally asymptotically (exponentially)stable. Linear time-invariant (LTI) systems are eitherexponentially stable, marginally stable, orunstable. Stability is always global. Local stability notion is needed only for nonlinearsystems. Warning: State convergence does not implystability!16E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov’s 1st Method Consider autonomous nonlinear dynamicsystem: x f ( x ) Assume that f(x) is continuously differentiable Perform linearization: x f ( x ) x f h.o.t . ( x ) A x x x 0higher-order terms TheoremA– If A is Hurwitz then the equilibrium is asymptoticallystable, (locally!)– If A has at least one eigenvalue in right-half complexplane then the equilibrium is unstable– If A has at least one eigenvalue on the imaginary axisthen one cannot conclude anything from the linear17approximationE. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov’s Direct (2nd) Method Fundamental Physical Observation– If the total energy of a mechanical (orelectrical) system is continuously dissipated,then the system, whether linear or nonlinear,must eventually settle down to an equilibriumpoint. Main Idea– Analyze stability of an n-dimensional dynamicsystem by examining the variation of a singlescalar function, (system energy).18E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov’s Direct Method(Motivating Example) Nonlinear mass-spring-damper systemmm x b x x k0 x k1 x3 0dampingspring termx Question: If the mass is pulled away andthen released, will the resulting motion bestable?– Stability definitions are hard to verify– Linearization method fails, (linear system isonly marginally stableE. Lavretsky19
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov’s Direct Method(Motivating Example, continued) Total mechanical energyx()11112322V ( x ) m x k0 x k1 x dx m x k0 x k1 x 422240kineticpotential Total energy rate of change along thesystem’s motion:V ( x ) m x x ( k x k x ) x x ( b x x ) b x 03031 Conclusion: Energy of the system isdissipated until the mass settles down: x 020E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov’s Direct Method(Overview) Method– based on generalization of energy concepts Procedure– generate a scalar “energy-like function(Lyapunov function) for the dynamic system,and examine its variation in time, (derivativealong the system trajectories)– if energy is dissipated (derivative of theLyapunov function is non-positive) thenconclusions about system stability may bedrawnE. Lavretsky21
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsPositive Definite Functions Definition: A scalar continuous functionV(x) is called locally positive definite ifV ( 0 ) 0 { x 0 x R} V ( x ) 0 If V ( 0 ) 0 { x 0} V ( x ) 0 then V(x) isglobally positive definite Remarks– a positive definite function must have aV ( x ) V ( xmin ) Vminunique minimum minx B– if Vmin / 0 or xmin / 0 then useW ( x ) V ( x xmin ) VminRE. Lavretsky22
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Functions Definition: If in a ball BR the function V(x)is positive definite, has continuous partialderivatives, and if its time derivative alongany state trajectory of the system x f ( x ) isnegative semi-definite, i.e., V ( x ) 0 then V(x)is said to be a Lyapunov function for thesystem. Time derivative of the Lyapunov function V ( x) V ( x) n V ( x ) V ( x ) f ( x ) 0, V ( x ) R xn x1E. Lavretsky23
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Function(Geometric Interpretation)V V1V V2V V3V ( x (t ))x2x20x (t )x1x (t )V1 V2 V3 Lyapunov function is a bowl, (locally) V(x(t)) always moves down the bowl System state moves across contourcurves of the bowl towards the originE. Lavretskyx124
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Stability Theorem If in a ball BR there exists a scalar functionV(x) with continuous partial derivativessuch that x B : V ( x ) 0 V ( x ) 0 then theequilibrium point 0 is stableR– If the time derivative is locally negativedefinite V ( x ) 0 then the stability is asymptoticV ( x) , If V(x) is radially unbounded, i.e., limx then the origin is globally asymptotically stable V(x) is called the Lyapunov function of thesystem25E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Local Stability Pendulum with viscous damping: θ θ sin θ 0 State vector: x (θ θ )θVx 1 cosθ ) Lyapunov function candidate: ( ) (2T2– represents the total energy of the pendulum– locally positive definite– time-derivative is negative semi-definite V ( x) V ( x)V ( x) θ θ θ sin θ θ θ θ 2 0 θ θ θ sin θ Conclusion: System is locally stable26E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Asymptotic Stability( System Dynamics: x x x() 2) 4 xx1 x1 x12 x22 2 4 x1 x222221 x22 Lyapunov function candidate:21x2V ( x1 , x2 ) x12 x22– positive definite– time-derivative is negative definite in the 222x xdimensional ball defined by 1 2 2V ( x1 , x2 ) 2 ( x12 x22 )( x12 x22 2 ) 0 Conclusion: System is locallyasymptotically stableE. Lavretsky27
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Global AsymptoticStability Nonlinear 1st order systemc ( x)xx c ( x ) , where: x c ( x ) 0 Lyapunov function candidate:V ( x ) x2– globally positive definite– time-derivative is negative definiteV ( x ) 2 x x 2 x c ( x ) 0 Conclusion: System is globallyasymptotically stable Remark: Trajectories of a 1st order system aremonotonic functions of time, (why?)E. Lavretsky28
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLa Salle’s Invariant Set Theorems It often happens that the time-derivative ofthe Lyapunov function is only negativesemi-definite It is still possible to draw conclusions onthe asymptotic stability Invariant Set Theorems (attributed to LaSalle) extend the concept of Lyapunovfunction29E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: 2nd Order NonlinearSystem System dynamics:x b ( x) c ( x) 0– where b(x) and c(x) are continuous functions verifyingthe sign conditions: x b ( x ) 0, for x 0x c ( x ) 0, for x 0 Lyapunov function candidate:x1 2V ( x, x ) x c ( y ) dy20– positive definite– time-derivative is negative semi-definiteV x x c ( x) x x b ( x) 0 system energy is dissipatedx b ( x ) 0 x 0 x c ( x ) xe 0 system cannot get “stuck” at a non-zero equilibrium30 Conclusion: Origin is globally asymptotically stableE. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov Functions for LTISystems LTI system dynamics: x A x Lyapunov function candidate: V ( x ) xT P x– where P is symmetric positive definite matrix– function V(x) is positive definite Time-derivative of V(x(t)) along the systemtrajectories: V ( x ) xT P x xT P x xT ( AT P P A) x xT Q x 0 Q– where Q is symmetric positive definite matrix– Lyapunov equation: AT P P A Q Stability analysis procedure:E. Lavretsky– choose a symmetric positive definite Q– solve the Lyapunov equation for P– check whether P is positive definite31
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsStability of LTI Systems Theorem– An LTI system is stable (globallyexponentially) if and only if for any symmetricpositive definite matrix Q, the unique matrixsolution P of the Lyapunov equation issymmetric and positive definite Remark: In most practical cases Q ischosen to be a diagonal matrix withpositive diagonal elements32E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsBarbalat’s Lemma: Preliminaries Invariant set theorems of La Salle provideasymptotic stability analysis tools forautonomous systems with a negativesemi-definite time-derivative of aLyapunov function Barbalat’s Lemma extends Lyapunovstability analysis to non-autonomoussystems, (such as adaptive modelreference control)33E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsBarbalat’s Lemma Lemma– If a differentiable function f(t) has a finite limit as t and if f ( t ) is uniformly continuous, then lim f ( t ) 0 Remarkst – uniform continuity of a function is difficult to verifydirectly– simple sufficient condition: if derivative is bounded then function is uniformly continuous– The fact that derivative goes to zero does not implythat the function has a limit, as t tends to infinity. Theconverse is also not true, (in general)34– Uniform continuity condition is very important!E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: LTI System Statement: Output of a stable LTI system isuniformly continuous in time– System dynamics: x A x B u– Control input u is boundedy Cx– System output: Proof: Since u is bounded and the system isstable then x is bounded. Consequently, theoutput time-derivative y C x C ( A x B u ) isbounded. Thus, (using Barbalat’s Lemma), weconclude that the output y is uniformlycontinuous in time.E. Lavretsky35
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsLyapunov-Like Lemma If a scalar function V(x,t) satisfies thefollowing conditions– function is lower bounded– its time-derivative along the systemtrajectories is negative semi-definite anduniformly continuous in timeV ( x, t ) 0 Then: limt Question: Why is this fact so important? Answer: It provides theoretical foundations36for stable adaptive control designE. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: Stable Adaptation Closed-loop error dynamics of an adaptivesystem e e θ w ( t ) , θ e w ( t )– where e is the tracking error, θ is the parameter error,and w(t) is a bounded continuous function Stability Analysis– Consider Lyapunov function candidate: V ( e,θ ) e2 θ 2 it is positive definite its time-derivative is negative semi-definiteV ( e,θ ) 2 e ( e θ w ) 2θ ( e w ) 2 e 2 0 consequently, e and θ are bounded since V ( e,θ ) 4 e ( e θ w ) is bounded, V ( e,θ ) isuniformly continuous hence: lim 2 e 2 lim V ( e,θ ) 0 lim e ( t )t E. Lavretsky()t t 37
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsUniform Ultimate Boundedness Definition: The solutions of x f ( x, t ) starting atx ( t0 ) x0 are Uniformly Ultimately Bounded (UUB)with ultimate bound B if: C0 0, T T ( C0 , B ) 0:( x (t )0) ( C0 x ( t ) B, t t0 T) Lyapunov analysis can be used to show UUBV ( x ) VB V0V ( x ) V0x Bx0x C C0V ( x ) 0, C x C0x C0All trajectories starting in large ellipse enter small ellipse within finite time T(C0 ,B)E. Lavretsky38
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsUUB Example : 1st Order System The equilibrium point xe is UUB if thereexists a constant C0 such that for everyinitial state x(t0) in an interval x ( t ) C thereexists a bound B and a time T ( B, x ( t ) ) suchthat x ( t ) x B for all t t T0000exe C0xe Bxexe Bxe C0E. Lavretskyx ( t0 )bound Bx (t )t0 Tt0Tt39
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsUUB by Lyapunov Extension Milder form of stability than SISL More useful for controller design in practicalsystems with unknown bounded disturbances:x f ( x) d ( x) Theorem: Suppose that there exists a functionV(x) with continuous partial derivatives such thatfor x in a compact set S R n– V(x) is positive definite: V ( x ) 0, x 0– time derivative of V(x) is negative definite outside of S:V ( x ) 0, x R, ( x R ) ( x S )– Then the system is UUB and x ( t ) R, t t0 TE. Lavretsky40
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsExample: UUB by Lyapunov Extension System:(x1 x1 x22 x1 x12 x22 9)(x2 x12 x2 x2 x12 x22 9)x2x ( t0 )Rx (T ) Lyapunov function candidate:x1V ( x1 , x2 ) x x21 Time derivative:22(V ( x1 , x2 ) 2 ( x1 x1 x2 x2 ) 2 x12 x22)( x21 x22 9) Time derivative negative outside compact set{}V ( x1 , x2 ) 0, x : x12 x22 9 Conclusion: All trajectories enter circle of radius41R 3, in a finite timeE. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsAdaptive Control
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsIntroduction Basic Ideas in Adaptive Control– estimate uncertain plant / controllerparameters on-line, while using measuredsystem signals– use estimated parameters in control inputcomputation Adaptive controller is a dynamic systemwith on-line parameter estimationE. Lavretsky– inherently nonlinear– analysis and design rely on the LyapunovStability Theory2
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsHistorical Perspective Research in adaptive control started in theearly 1950’s– autopilot design for high-performance aircraft Interest diminished due to the crash of atest flight– Question: X-? aircraft tested Last decade witnessed the development ofa coherent theory and many practicalapplications3E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsConcepts Why Adaptive Control?– dealing with complex systems that have unpredictable parameterdeviations and uncertainties Basic Objective– maintain consistent performance of a system in the presence ofuncertainty and variations in plant parameters Adaptive control is superior to robust control in dealingwith uncertainties in constant or slow-varying parameters Robust control has advantages in dealing withdisturbances, quickly varying parameters, andunmodeled dynamics Solution: Adaptive augmentation of a Robust Baseline4controllerE. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsModel-Reference Adaptive Control(MRAC)reference modelrcontrollerθˆuplantymyeadaptation law Plant has a known structure but the parameters areunknown Reference model specifies the ideal (desired) responseym to the external command r Controller is parameterized and provides tracking Adaptation is used to adjust parameters in the control law5E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsSelf-Tuning Controllers (STC)rcontrollerθˆplantyestimator Combines a controller with an on-line (recursive) plantparameter estimator Reference model can be added Performs simultaneous parameter identification andcontrol Uses Certainty Equivalence Principle– controller parameters are computed from the estimates of theplant parameters as if they were the true ones6E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsDirect vs. Indirect Adaptive Control Indirect– estimate plant parameters– compute controller parameters– relies on convergence of the estimated parameters totheir true unknown values Direct– no plant parameter estimation– estimate controller parameters (gains) only MRAC and STC can be designed using bothDirect and Indirect approaches We consider Direct MRAC designE. Lavretsky7
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems System Dynamics:x a x b (u f ( x))– a, b are constant unknown parametersN– uncertain nonlinear function: f ( x ) θ i ϕi ( x ) θ T Φ ( x )i 1 vector of constant unknown parameters: θ (θ1 θ N )T vector of known basis functions: Φ ( x ) (ϕ1 ( x ) ϕ N ( x ) )T Stable Reference Model: x a Control Goal– find u such that: lim ( x ( t ) x ( t ) ) 0mt mxm bm r ,( am 0 )m8E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Control Feedback:u kˆx x kˆr r θˆT Φ ( x )– (N 2) parameters to estimate on-line: Closed-Loop System: ()(kˆx , kˆr , θˆx a b kˆx x b kˆr r θˆ θ Desired Dynamics: x a x b r Matching Conditions Assumptionmmm)TΦ ( x ) m– there exist ideal gains ( k x , kr ) such that:a b k x amb kr bm– Note: knowledge of the ideal gains is not required,only their existence is needed– consequently: a b kˆx am a b kˆx a b k x b ( kˆx k x ) b k x()b kˆr bm b kˆr b kr b kˆr kr b krE. Lavretsky9
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Tracking Error: e ( t ) x ( t ) x ( t ) Error Dynamics:m Te ( t ) x ( t ) xm ( t ) a b kˆx x b kˆr r θˆ θ Φ ( x ) am xm bm r am x θ am ( x xm ) a b kˆx am x b kˆr kr r b θ T Φ ( x )(())((( am e b k x x kr r θ T Φ ( x )))) Lyapunov Function Candidate:V ( e ( t ) , k ( t ) , k ( t ) , θ ( t ) ) e b (γ k γ2xE. Lavretskyr 1x2x 1r kr2 θ T Γθ 1 θ )– where: γ x 0, γ r 0, and Γ ΓT 0 is symmetric positive 10definite matrix
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Time-derivative of the Lyapunov function(V ( e, k x , kr , θ ) 2 e e 2 b γ x 1 k x kˆx γ r 1 kr kˆr θ T Γθ 1 θˆ( 2 e am e b ( k x x kr r ) θ T Φ ( x )( 2 b γ x 1 k x kˆx γ r 1 kr kˆr θ T Γθ 1 θˆ())) 2 am e2 2 b k x x e sgn ( b ) γ x 1 kˆx 2 b kr r e sgn ( b ) γ r 1 kˆr 2 b θ T Φ ( x ) e sgn ( b ) Γθ 1 θˆ ()())11E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Adaptive Control Design Idea– Choose adaptive laws, (on-line parameter updates)such that the time-derivative of the Lyapunov functiondecreases along the error dynamics trajectorieskˆx γ x x e sgn ( b )kˆr γ r r e sgn ( b )θˆ Γθ Φ ( x ) e sgn ( b ) Time-derivative of the Lyapunov functionbecomes semi-negative definite!2V ( e ( t ) , k x ( t ) , kr ( t ) , θ ( t ) ) 2 am e ( t ) 0 0E. Lavretsky12
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Closed-Loop System Stability Analysis– Since V 0 and V 0 then all the parameterestimation errors are bounded– Since the true (unknown) parameters areconstant then all the estimated parametersare bounded Assumption– reference input r(t) is bounded Consequently, xm andxm are bounded13E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(continued) Since x e xm then x is bounded Consequently, the adaptive controlfeedback u is bounded Thus, x is bounded, and e x xm isbounded, as well It immediately follows that V 4 am e ( t ) e ( t ) isbounded Using Barbalat’s Lemma we conclude that V ( t )is uniformly continuous function of time 14E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(completed) Using Lyapunov-like Lemma: lim V ( x, t ) 0 Since V 2 a e ( t ) it follows that: lim e ( t ) 0 Conclusionst 2mt – achieved asymptotic tracking: x ( t ) xm ( t ) , as t – all signals in the closed-loop system arebounded15E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsMRAC Design of 1st Order Systems(Block-Diagram)rkˆrbms amxmbs axef ( x)fˆ ( x )kˆx Adaptive gains: kˆx ( t ) , kˆr ( t )NT On-line function estimation: fˆ ( x ) θˆ ( t ) Φ ( x ) θˆi ( t ) ϕ i ( x )i 1E. Lavretsky16
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsAdaptive Dynamic Inversion(ADI) Control
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsADI Design of 1st Order Systems System Dynamics:x a x b u f ( x)– a, b are constant unknown parametersN– uncertain nonlinear function: f ( x ) θ i ϕi ( x ) θ T Φ ( x )i 1 vector of constant unknown parameters: θ (θ1 θ N )T vector of known basis functions:Φ ( x ) (ϕ1 ( x ) ϕ N ( x ) )T Stable Reference Model: x a Control Goal– find u such that: lim ( x ( t ) x ( t ) ) 0mt mxm bm r ,( am 0 )m18E. Lavretsky
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsADI Design of 1st Order Systems(continued) Rewrite system dynamics:() (x aˆ x bˆ u fˆ ( x ) ( aˆ a ) x bˆ b u fˆ ( x ) f ( x ) a) f ( x ) b Function estimation error: f ( x )(fˆ ( x ) f ( x ) θˆ θ)TΦ ( x) θ On-line estimated parameters: aˆ, Parameter estimation errors aE. Lavretskyaˆ a, b bˆ b, θθˆ θbˆ, θˆ19
Robust and Adaptive Control WorkshopAdaptive Control: Introduction, Overview, and ApplicationsADI Design of 1st Order Systems(continued) ADI Control Feedback:u 1am aˆ ) x bm r ) θˆT Φ ( x )((bˆ– (N 2) parameters to estimate on-line:– Need to protect b̂from crossing zeroClosed-Loop System: x a x b r a x b u θ Φ ( x )x a x b rDesired Dynamics:Tracking error: e x xTracking error dynamics: e a e a x b u θ Φ ( x )Lyapunov function candidateV ( e ( t ) , a ( t ) , b ( t ) , θ ( t ) ) e γ a γ b θ Γθ θ 20mmmmmmmm2E. Lavretskyaˆ , bˆ,
Robust and Adaptive Control Workshop Adaptive Control: Introduction, Overview, and Applications Lyapunov's Direct Method (Overview) Method - based on generalization of energy concepts Procedure - generate a scalar "energy-like function (Lyapunov function) for the dynamic system, and examine its variation in time, (derivative
Sybase Adaptive Server Enterprise 11.9.x-12.5. DOCUMENT ID: 39995-01-1250-01 LAST REVISED: May 2002 . Adaptive Server Enterprise, Adaptive Server Enterprise Monitor, Adaptive Server Enterprise Replication, Adaptive Server Everywhere, Adaptive Se
Adaptive Control, Self Tuning Regulator, System Identification, Neural Network, Neuro Control 1. Introduction The purpose of adaptive controllers is to adapt control law parameters of control law to the changes of the controlled system. Many types of adaptive controllers are known. In [1] the adaptive self-tuning LQ controller is described.
Adaptive Control - Landau, Lozano, M'Saad, Karimi Adaptive Control versus Conventional Feedback Control Conventional Feedback Control System Adaptive Control System Obj.: Monitoring of the "controlled" variables according to a certain IP for the case of known parameters Obj.: Monitoring of the performance (IP) of the control
Chapter Two first discusses the need for an adaptive filter. Next, it presents adap-tation laws, principles of adaptive linear FIR filters, and principles of adaptive IIR filters. Then, it conducts a survey of adaptive nonlinear filters and a survey of applica-tions of adaptive nonlinear filters. This chapter furnishes the reader with the necessary
Highlights A large thermal comfort database validated the ASHRAE 55-2017 adaptive model Adaptive comfort is driven more by exposure to indoor climate, than outdoors Air movement and clothing account for approximately 1/3 of the adaptive effect Analyses supports the applicability of adaptive standards to mixed-mode buildings Air conditioning practice should implement adaptive comfort in dynamic .
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adaptive control, nonlinear systems, system identification, uncertain systems 1 INTRODUCTION Adaptive control methods provide a technique to achieve a control objective despite uncertainties in the system model. Adaptive estimates are developed through insights from a Lyapunov-based analysis as a means to yield a desired objec-tive.
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