lec14 - Outline Self Tuning Control MRAC with full state...

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1 Outline Self Tuning Control stems MRAC with full state feedback Extensions to Nonlinear Systems Examples ME6402, Nonlinear Control Sys Motivation ± The parameter estimation routines can be used to adjust controller parameters ± STC tunes control parameters based on the identified model ± MRAC identifies controller parameters directly
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2 Self-Tuning Regulation r stems plant controller estimator ME6402, Nonlinear Control Sys Controller : linear model based control such as PID, LQR, State feedback, etc . Estimator : Identifies the plant model Comments on Self-Tuning Controller ± An identification routine is used to estimate the system parameters ± A feedback/feedforward controller such as PID, state feedback, etc. is used to control the system It is assumed that the identification ± It is assumed that the identification speed is faster than parameter variations
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3 Estimator Type ± Either gradient or RLS estimation may be used stems ± Gradient method can handle varying parameter but has poor convergence ± RLS method cannot handle time-varying parameters since its gain P(t) 0! recalling ME6402, Nonlinear Control Sys () τ τ τ = t T d t 0 1 ) ( ) ( w w P Modifications of the RLS Algorithm Covariant Resetting: P is reset to a ± Covariant Resetting: P is reset to a predetermined positive definite value when λ min (P) gets too small. ± Adding a forgetting factor to covariant updating Parameter Projection: Confining the ± Parameter Projection: Confining the parameters to a finite region
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4 RLS Algorithm with Forgetting Factor ± Replace by fo λ >0 prevents P -1 from becoming too ) ( w ) ( w P 1 τ τ = T dt d ) ( w ) ( w P 1 1 τ τ + λ = T P dt d stems for 1 from becoming too small! ± RLS with Forgetting Factor : 0 0 ) ) ( ) ( ) ( ) ( > λ = θ P P w w P P P w P t e t t t T & & ME6402, Nonlinear Control Sys ± Addition of the forgetting factor keeps the updating ‘alive’ for all time provided w is PE. ) ( ), ( ) ( ) ( ) ( ) ( ) ( = t t t t t t Stability Proof ± Assume y= w T (t) θ * for some θ * R n . T T ± Then e=w (t) θ -y=w (t) θ ± Assume w is PE and bounded so that α I P ≤ β I for some finite α , β >0 ± System: ) ( ~ ) ( ) ( ) ( ) ( ~ t t t t t w w P θ = θ & ± Choose Lyapunov candidate ) ( ) ( ) ( ) ( 1 1 t t t t dt d T w w P P + λ = ) ( ~ ) ( ) ( ~ 1 t t t V θ θ = P V V λ = &
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5 Illustrative STC Example Consider the d.c. servo system stems ± Identify the parameters using both gradient descent and RLS algorithms using u and v. ± Design PI controller for this system to track a ku bv v m = + & ME6402, Nonlinear Control Sys desired speed vd ± Use the identification routine to update the PI controller parameters Identification Model ku bv v m = + & Physical model:
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6 PI Tracking Controller u b y a y 0 0 = + & Model : 1 stems Control law: ( ) + = edt K e K y a y b u i p d d 0 0 & Closed loop system: ME6402, Nonlinear Control Sys Simulink Block Diagram r To Workspace3 tc To Workspace1 Clock x' = Ax+Bu In1 In2 In3 Out1 controller y To Workspace2 Sine Wave1 Sine Wave In1 Out1 Plant y = Cx+Du y-filter lam s+lam u-filter th To Workspace rlssys RLS
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7 Simulation Results: RLS with
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lec14 - Outline Self Tuning Control MRAC with full state...

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