09 MRAS Gradient Method 0th 1st 2nd Order Systems

09 MRAS Gradient Method 0th 1st 2nd Order Systems - SYS635...

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SYS635 Adaptive Control Systems Ka C. Cheok MRAS Gradient Method 0th 1st 2nd Order Systems 22 Oct ‘05 1 MODEL REFERENCE ADAPTIVE SYSTEM (MRAS) MRAS Objective: Based on information y m , y, u and u c , devise a controller that automatically adjusts itself so that the behavior of the closed-loop control plant output (y) closely follows that (y m ) of the reference model. In other words, make y mimic y m . Illustration: Can you think of an example for such a system? Dance choreographer? Coach? 5.2 The MIT Rule (read Chap 5.2) Example of a cost function 2 1 ( ) ( ) where is a function of 2 J θ εθ ε = MIT Rule says that the time rate of change of is proportional to negative gradient of J w.r.t. . That is dd Jd dt d d γγ =− Integrating yields the continuous-time equation for updating : 00 ( ) () () tt J d t t d t dt d θθ γ =+ ∫∫ Since we can d td t , this leads to the Delta Rule that says the increment (delta) in is approximately dJ t d θγ ∆≈ −∆ and the discrete-time equation for updating is new old Water surface He/she is looking for the deepest spot in the river, should he/she go forward or backward? What logic did he used? This may be a vector although drawn as a scalar 0 J < 0 J > J y u u c y m Reference Model (in computer/or a physical system) Controller Physical Plant Adjustment Mechanism
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SYS635 Adaptive Control Systems Ka C. Cheok MRAS Gradient Method 0th 1st 2nd Order Systems 22 Oct ‘05 2 () ( ) 1 2 2 1 Sigmoid function = 1 1 1 y 1 Derivative ( 1) 1 1 1 11 1 1 1 h h h h hh h h h h y e e e e ee h e e yy e e −−   +  ∂+ + == = + = ∂∂ + = ++ + + Application of MIT Rule to a single neuron model Let’s suppose we are shown a input-output pattern or phenomenon in the form of y m versus x curve. In the following example, it so happens that the pattern behaves as shown in the figure. We introduce a neuron model whose math description tends to produce an I/O relationship similar to the given pattern. That single neuron math model is described by a sigmoidal funtion 1 1 h yh w x b e + + where w is called a weight and b is a bias. The error between the pattern and the neuron model is given by 1 1 mm m h wx b y y e e ε −+ =− = − = + + We would like to find w and b so that y follows m y . A way to do this is to find w and b so that they minimize the error cost function 2 1 , 2 Jwb = The gradients of J w.r.t. w and b are: ( ) 2 2 1 1( 1 ) ( h m w xb JJ y h e yy x x wy h w y h w εε ∂− + ∂∂∂∂ + = = 2 21 1 ( 1)1 ( 1) h m w y h e by h b y h b + + = = An important property of sigmoid function Given pattern x y m y m
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This note was uploaded on 04/17/2011 for the course SYS 635 taught by Professor Re during the Spring '11 term at Albany College of Pharmacy and Health Sciences.

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09 MRAS Gradient Method 0th 1st 2nd Order Systems - SYS635...

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