CS545 Contents XVI. l Adaptive Control. l Reading Assignment for Next Class
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1 CS545 Contents XVI Adaptive Control Model Reference Adaptive Control Self-Tuning Regulators Linear Regression Recursive Least Squares Gradient Descent Feedback-Error Learning Reading Assignment for Next Class See
2 The Adaptive Control Problem Characterize the desired behavior of the closed loop system Determine a suitable control law with adjustable parameters Find a mechanism for adjusting the parameters Implement the control law
3 Model-Reference Adaptive Control (Direct Learning) Performance is given to correspond to a particular reference model m x + b x + c = u E.g. Adjustment of controller is done directly E.g., adjust controller parameter by gradient descent Model y desired Adjustment Mechanism adjustment Controller Robot x desired u y
4 Model-Reference Adaptive Control Example Consider the generic control system!x = f ( x) + g( x)u For this example, make this an even simpler system!x = f ( x) + u Assume that f is unkown and needs to be estimated by a learning process. Thus, we can formulated a control law: ( ) ( ) u =! ˆf ( x) +!x d! k x! x d =!x! ˆ" +!x d! k x! x d Where we replaced f with a simple linear function
5 Model-Reference Adaptive Control Example The goal of model-reference adaptive control is to adjust the open parameter and the control law such that the system is ALWAYS stable The system dynamics are now Define errors!x = x!! " x! ˆ! +!x d " k( x " x d ) e = x d! x!" = "! ˆ" Thus:!x = x! "! +!x d + ke 0 = x! "! +!e + ke!e = "x! "! " ke
6 Model-Reference Adaptive Control Example Define a Lyapunov function V = 1 2 e !!!" #1!! "V = e"e +!!!" #1!"! = e"e #!!!" #1 ˆ"! ( ) #!!!" #1 ˆ"! = e #x!!! # ke = #ex!! # ke 2 #!!!" #1 ˆ"! Thus, choose:!ex! "!! "!#!1 ˆ" " = 0
7 Model-Reference Adaptive Control Example Thus:!ex! "!! "!#!1 ˆ" " = 0 ( ) = 0!"!ex! #!1 ˆ" " ˆ" " =!#ex I.e., the guaranteed stable parameter adapation law is ˆ!! = "#ex
8 Nonlinear Example
9 Self-tuning Regulators (Indirect Learning) Controller is redesigned based on some estimated parameters certainty equivalence principle E.g., LQR controller is redesigned based on estimated model This corresponds to an indirect update of the controller Specifications adjustment Controller Design Process Parameters Estimation Controller Robot x desired u y
10 Example: Estimate the Robot Model From Data How to obtain data? Try random commands u, observe the state and change of state Don t destroy the robot How to estimate the model? Model is nonlinear Need nonlinear estimation techniques (e.g., neural networks) Model is linear Use linear regression or recursive least squares Essential ingredients of estimation: A cost criterion: N ( ) 2 Usually least squares J = 1! t i " y i 2 i=1 Some adjustable parameter ( a data generating mode)
11 Linear Regression for One Output The data generating model y = Least Squares Cost Function w T x + w 0 +! = w T x +! where x = x T,1 [ ] T ",w = w % # $ &', E! w 0 { } = 0 J = 1 ( 2 t! y ( )T t! y) = 1 ( 2 t! Xw ( )T t! Xw) Minimize Cost t 1 T " % " x 1 % $ t ' $ T ' 2 where : t = $ ', X = $ x 2 ' $ ' $ ' $ ' $ T ' # t n & # x n &! J!w = 0 =! J!w " # $ =!t T X + Xw 1 2 t! Xw ( )T t! Xw ( ) ( ) T X =!t T X + w T X T X thus : t T X = w T X T X or X T t = X T Xw -1 result : ( )!1 X T t w = X T X % & ' =! t! Xw ( ) T X
12 Remarks on Least Squares the Pseudo-Inverse X # = ( X T X)!1 X T pseudo inverses are a special solution to an infinite set of solutions of a non-unique inverse problem the matrix inversion above may still be ill-defined if X X is close to singular and so-called Ridge regression needs to be applied more about inverse problems in dedicated class Ridge Regression X # = ( X T X +!I) "1 X T where! << 1 Multiple Outputs: just like multiple single output regressions W = ( X T X)!1 X T Y
13 Geometrical Interpretation of Least Squares Subspace S spanned by the columns of X Vector of residual errors (othorgonal to y) [ X] 1 t t! y = t! Xw = t! "[ X] i w i i [ X] 0 y y is the optimal reconstruction of t in the range of X y is an orthogonal Projection of t on S
14 Physical Interpretation of Least Squares y all springs have the same spring constant points far away generate more force (danger of outliers) springs are vertical solution is the minimum energy solution achieved by the springs x
15 Recursive Least Squares The Sherman-Morrison-Woodbury Theorem ( A! zz T )!1 = A!1 + A!1 zz T A!1 1! z T A!1 z More General: The Matrix Inversion Theorem ( A! BC)!1 = A!1 + A!1 B( I + CA!1 B)!1 CA!1 Recursive Least Squares Update Initialize : P n = I 1! where! <<1 (note P " ( X T X) #1 For every new data point ( x,t) (note that x includes the bias) : P n +1 = 1 %' $ Pn # Pn x x T P n (* + 1 if no forgetting where $ =, & $ + x T P n x) - < 1 if forgetting W n +1 = W n + P n+1 x ( t # W nt x) T
16 Recursive Least Squares (cont d) Some amazing facts about recursive least squares Results for W are EXACTLY the same as for normal least squares update (batch update) after every data point was added once! (no iterations) NO matrix inversion necessary anymore NO learning rate necessary Guaranteed convergence to optimal W (linear regression is an optimal estimator under many conditions) Forgetting factor allows to forget data in case of changing target functions Computational load is larger than batch version of linear regression But don t get fooled: if data is singular, you still will have problems!
17 Feedback Error Learning Feedback Error Learning is a special adaptive control architecture to learn an inverse dynamics feedforward controller Specifications Inverse Model Process Parameters u ff x desired Feedback Controller u fb Robot y How would we change recursive least squares to deal with and error signal instead of an absolute teacher signal?
18 Gradient Descent (a.k.a. LMS, Delta rule, Widrow-Hoff rule, Adaline rule) Gradient descent can be used even if the model is nonlinear in the parameters Idea: Change parameters incrementally to reduce the least squares cost iteratively Stochastic Update given : J i = 1 2 t i! y i ( ) 2 and a i = w T x i "J i "w =! ( t! y i i ) " y i "w =!( t! y ) " f ( a ) i i i "w ( ) " f ( a i ) ( ) # update : =! t i! y i " a i "a i "w =! t! y i i w n+1 = w n %!$ "J i' = w n + $ t &"w( i! y i w n+1 = w n T +$ ) i f #( a i )x i where $ is the learning rate T ( ) # T f ( a i )x i T f ( a i )x i
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