2.153 Adaptive Control Fall 2019 Lecture 11: Neural ...

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2.153 Adaptive Control Fall 2019 Lecture 11: Neural Networks and Adaptive Control Anuradha Annaswamy [email protected] October 9, 2019 ( [email protected]) October 9, 2019 1 / 15

Transcript of 2.153 Adaptive Control Fall 2019 Lecture 11: Neural ...

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2.153 Adaptive ControlFall 2019

Lecture 11: Neural Networks and Adaptive Control

Anuradha Annaswamy

[email protected]

October 9, 2019

( [email protected]) October 9, 2019 1 / 15

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Problem Statement

Problem: minxf(x)

f(x) unknown

ଵ ଶ

Identifying f(x) at all x is impossibleWe convert f(x) into basis functions

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Approximation

f(x) =

N∑i=1

ai sin(ωix)

𝑁 = 3

Fourier Series says that

|f(x)−N∑i=1

ai sin(ωix)| ≤ ε

Taylor Series says that

|f(x)−N∑i=1

aixi| ≤ ε, for x ∈ [x1, x2]

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Neural Networks∑ai sin(ωix),

∑aix

i : linearly parameterized

Departure: Neural Networks

+

y =

N∑i=1

αiσ(wTi x+ θi

)x =

[x1

]Let σ

(wTi x+ θi

)= σ

(wTi x)

= σi (x); Σ = [σ1(x), σ2(x), . . . , σN (x)]T

y = αTΣ(x)

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Universal Approximation

Theorem

σ is any continuous sigmoidal function. Let

N (x) =

N∑i=1

αiσ(wTi x).

Then for any f that is continuous and ε > 0, there exists N such that

|N (x)− f(x)| < ε, ∀x ∈ X, (1)

where X: compact set

Instead of (1), one can have∫X|N (x)− f(x)|dx < ε

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Stone-Weierstrass Theorem

x

How do we adjust

wi in Σ(x)?αi in αT ?

Use the back propagation algorithm:

J = e2

α ∝ −∂J∂α

wi ∝ − ∂J

∂wi

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Back Propagation Algorithm

e = y − y =

N∑i=1

αiσ(wTi x)− y

∂e

∂αi

∣∣∣∣αi

= σ(wTi x)

; ˙αi = −γe ∂e∂αi

∣∣∣∣αi

∂e

∂wi

∣∣∣∣wi

= αiσ′∣∣∣∣wT

i x

· x; ˙wi = −γe ∂e∂wi

∣∣∣∣wi

We did not discuss stability properties

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Back Propagation Algorithm

e = y − y =

N∑i=1

αiσ(wTi x)− y

∂e

∂αi

∣∣∣∣αi

= σ(wTi x)

; ˙αi = −γe ∂e∂αi

∣∣∣∣αi

∂e

∂wi

∣∣∣∣wi

= αiσ′∣∣∣∣wT

i x

· x; ˙wi = −γe ∂e∂wi

∣∣∣∣wi

We did not discuss stability properties

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider a very simple example:

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider a very simple example:

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider a very simple example:

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

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Control of a Simple Nonlinear Plant

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

= −axp + (ασ(wxp)− ασ(wxp) + ασ(wxp)− ασ(wxp)) + r

= −axp + αε− ασ(wxp) + r

Reference Model:xm = −axm + r

Error model:e = −ae− ασ(wxp) + αε

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Control of a Simple Nonlinear Plant

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

= −axp + (ασ(wxp)− ασ(wxp) + ασ(wxp)− ασ(wxp)) + r

= −axp + αε− ασ(wxp) + r

Reference Model:xm = −axm + r

Error model:e = −ae− ασ(wxp) + αε

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Control of a Simple Nonlinear Plant

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

= −axp + (ασ(wxp)− ασ(wxp) + ασ(wxp)− ασ(wxp)) + r

= −axp + αε− ασ(wxp) + r

Reference Model:xm = −axm + r

Error model:e = −ae− ασ(wxp) + αε

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Control of a Simple Nonlinear Plant

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

= −axp + (ασ(wxp)− ασ(wxp) + ασ(wxp)− ασ(wxp)) + r

= −axp + αε− ασ(wxp) + r

Reference Model:xm = −axm + r

Error model:e = −ae− ασ(wxp) + αε

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Control of a Simple Nonlinear Plant

xp = −axp + ασ(wxp) + u

u(t) = −ασ(wxp) + r

Defineε = σ(wxp)− σ(wxp) bounded

Closed-loop Equations:

xp = −axp + ασ(wxp)− ασ(wxp) + r

= −axp + (ασ(wxp)− ασ(wxp) + ασ(wxp)− ασ(wxp)) + r

= −axp + αε− ασ(wxp) + r

Reference Model:xm = −axm + r

Error model:e = −ae− ασ(wxp) + αε

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Stability

Error model:

e = −ae− ασ(wxp) + αε

˙α = −eσ(wxp))

V = 1/2(e2 + α2

)V ≤ −ae2 + αeε

Stability result:Suppose r is persistently exciting with the property

1

T

∫ t1+T

t1

|r(τ)dτ ≥ αε

a+ δ

then e(t) and α(t) are bounded for all initial conditions e(t0) and α(t0).

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Stability

Error model:

e = −ae− ασ(wxp) + αε

˙α = −eσ(wxp))

V = 1/2(e2 + α2

)V ≤ −ae2 + αeε

Stability result:Suppose r is persistently exciting with the property

1

T

∫ t1+T

t1

|r(τ)dτ ≥ αε

a+ δ

then e(t) and α(t) are bounded for all initial conditions e(t0) and α(t0).( [email protected]) October 9, 2019 10 / 15

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider the plant:

xp = −axp +

N∑j=1

αjσ(wjxp)

+ u

u(t) = −

N∑j=1

αjσ(wjxp)

+ r

ε =

N∑j=1

σ(wjxp)− σ(wjxp)

bounded

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider the plant:

xp = −axp +

N∑j=1

αjσ(wjxp)

+ u

u(t) = −

N∑j=1

αjσ(wjxp)

+ r

ε =

N∑j=1

σ(wjxp)− σ(wjxp)

bounded

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Control of a Nonlinear Dynamic System

xp = −axp + f(xp) + u

f(xp) ≈

N∑j=1

αjσ(wjxp)

Consider the plant:

xp = −axp +

N∑j=1

αjσ(wjxp)

+ u

u(t) = −

N∑j=1

αjσ(wjxp)

+ r

ε =

N∑j=1

σ(wjxp)− σ(wjxp)

bounded

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contd.

xp = −axp +

N∑j=1

αjσ(wjxp)

+ u

u(t) = −

N∑j=1

αjσ(wjxp)

+ r

ε =

N∑j=1

σ(wjxp)− σ(wjxp)

bounded

xp = −axp +

N∑j=1

[αjσ(wjxp)− αjσ(wjxp)] + r

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contd.

xp = −axp +

N∑j=1

[αjσ(wjxp)− αjσ(wjxp)] + r

= −axp +

N∑j=1

αjε− αjσ(wjxp) + r

Error model:

e = −ae−N∑j=1

αjε− αjσ(wjxp)

˙αj = eσ(wjxp))

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Stability Analysis

Error model:

e = −ae−N∑j=1

αjε− αjσ(wjxp)

˙αj = eσ(wjxp))

V = 1/2

e2 +

N∑j=1

α2j

V ≤ −ae2 + e

N∑j=1

αjε

= −ae2 + ed0; d0 : bounded

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Stability Analysis

Error model:

e = −ae−N∑j=1

αjε− αjσ(wjxp)

˙αj = eσ(wjxp))

V = 1/2

e2 +

N∑j=1

α2j

V ≤ −ae2 + e

N∑j=1

αjε

= −ae2 + ed0; d0 : bounded

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Stability Result

Error model:

e = −ae−N∑j=1

αjε− αjσ(wjxp)

˙αj = eσ(wjxp))

Suppose r is persistently exciting with the property

1

T

∫ t1+T

t1

|r(τ)dτ ≥ d0a

+ δ

then e(t) and α(t) are bounded for all initial conditions e(t0) and α(t0).

( [email protected]) October 9, 2019 15 / 15