2.153 Adaptive Control Fall 2019 Lecture 13: Adaptive PI...

2.153 Adaptive ControlFall 2019

Lecture 13: Adaptive PI Control

Anuradha Annaswamy

[email protected]

October 22, 2019

( [email protected]) October 22, 2019 1 / 13

Adaptive Control of a Second-order PlantGc(s)

1s(Js+B)

+ e τ

−

Plant: Jω + Bω = τ J > 0

PI Control: Gc(s) = kp +ki

sτ = kpe(t) + ki

∫e(τ)dτ

Adaptive PI Control: τ = kp(t)e(t) + ki(t)∫

e(τ)dτ

PID Control: Gc(s) = kp + kds + kis

τ = kpe(t) + ki∫

e(τ)dτ + kddedt

Adaptive PID Control: τ = kp(t)e(t) + ki(t)∫

e(τ)dτ + kd(t)e(t)

J and B are unknown. Adjust kp(t), ki(t) and kd(t) so that theclosed-loop system is stable and limt→∞ e(t) = 0.



1s(Js+B)

+ e τ

−



s

τ = kpe(t) + ki∫

e(τ)dτ


e(τ)dτ


τ = kpe(t) + ki∫

e(τ)dτ + kddedt






1s(Js+B)

+ e τ

−



sτ = kpe(t) + ki

∫e(τ)dτ


e(τ)dτ


τ = kpe(t) + ki∫

e(τ)dτ + kddedt





Adaptive Control of a Second-order Plant

AdaptiveController

1s(Js+B)

+ e τ

−



sτ = kpe(t) + ki

∫e(τ)dτ


e(τ)dτ


τ = kpe(t) + ki∫

e(τ)dτ + kddedt





PI -Control: Algebraic Part

NominalController

1Js+B

r + er τ x

−

Gc(s) = kp +ki

s

Parameterize kp = K > 0, ki = Kλ > 0

Closed-loop transfer function:K(s + λ)

s(Js + B) + K(s + λ)

=K(s + λ)

Js2 + s(B + K) + Kλ

Stable if K > |B|. Design the controller so that x→ xd


Anu

Typewritten Text

(Angular velocity)


NominalController

1Js+B

r + er τ x

−

Gc(s) = kp +ki

sParameterize kp = K > 0, ki = Kλ > 0


s(Js + B) + K(s + λ)

=K(s + λ)

Js2 + s(B + K) + Kλ




NominalController

1Js+B

r + er τ x

−

Gc(s) = kp +ki



s(Js + B) + K(s + λ)

=K(s + λ)

Js2 + s(B + K) + Kλ

Stable if K > |B|.

Design the controller so that x→ xd



NominalController

1Js+B

r + er τ x

−

Gc(s) = kp +ki



s(Js + B) + K(s + λ)

=K(s + λ)

Js2 + s(B + K) + Kλ



PI Control - Algebraic Part: Tracking

Gc(s) 1Js+B

r + er τ x

−

Wcl(s)

Wcl(s) =Gc(s)

Js + B + Gc(s)

W−1cl (s) = 1 + (Js + B)G−1

cl (s)

r = W−1cl (s)[xd]

= xd +((Js + B)G−1

cl (s))[xd]

= xd + (Js + B)[ωd] = xd + Bωd + Jωd



W−1cl (s) Gc(s) 1

Js+Bτxd r + er x

−

Wcl(s)

Wcl(s) =Gc(s)

Js + B + Gc(s)

W−1cl (s) = 1 + (Js + B)G−1

cl (s)

r = W−1cl (s)[xd]

= xd +((Js + B)G−1

cl (s))[xd]




W−1cl (s) Gc(s) 1

Js+Bτxd r + er x

−

Wcl(s)

Wcl(s) =Gc(s)

Js + B + Gc(s)

W−1cl (s) = 1 + (Js + B)G−1

cl (s)

r = W−1cl (s)[xd]

= xd +((Js + B)G−1

cl (s))[xd]

= xd + (Js + B)[ωd]

= xd + Bωd + Jωd



W−1cl (s) Gc(s) 1

Js+Bτxd r + er x

−

Wcl(s)

Wcl(s) =Gc(s)

Js + B + Gc(s)

W−1cl (s) = 1 + (Js + B)G−1

cl (s)

r = W−1cl (s)[xd]

= xd +((Js + B)G−1

cl (s))[xd]




Using r = Jωd + Bωd + xd the block diagram can be represented as

J

B Gc(s) 1Js+B

ωd

ωd

xd

+

++ r + er τ x

−

which can then be simplified to

Gc(s)

B

J

1Js+B

Plantxd +

xd

xd

+e + τ x

−


PI Control - Algebraic Part: Tracking - Revised Design

J: reparametrize Gc(s):Gc(s) = kp +

ki

s= K +

Kλs

=Ks + Kλ

s

Change to Gc(s) =(K + Jλ)s + Kλ

s

Closed-loop transfer function:(K + Jλ)s + Kλ)

s(Js + B) + (K + Jλ)s + Kλ)

Move B from feedforward - to feedback

Closed-loop transfer function:(K + Jλ)s + Kλ

Js2 + (K + Jλ)s + Kλ



Gc(s)

B

J

1Js+B

Plantxd +

xd

xd

+e + τ x

−

Gc(s) =(K + Jλ)s + Kλ

s



J

Gc(s) 1Js+B

B

xd +

xd

+e + τ x

+−


s



(K+Jλ)s+Kλs

1Js+B

B

r + er + τ x

+−

Wcl(s) :(K + Jλ)s + Kλ


ωn =

√KλJ

ζ =K + Jλ

2J·√

JKλ

=K + Jλ2√

JKλ

Less sensitive to uncertainties in JLess sensitive to uncertainties in B


PI Control - Algebraic Part: Tracking - CompleteDesign

J

Gc(s)1

Js+B

B

xd +

xd

+e + τ x

+−


s,

Wcl(s) =(K + Jλ)s + Kλ


r = W−1cl (s)[xd]

= xd +((Js)G−1

cl (s))[xd]

= xd + Jωd

τ = Bx + Jxd + Gc(s)[e]

= Bx + Jxd + (K + Jλ)e + Kλ∫

e(τ)dτ

= J (xd + λe) + Bx + K(

e + λ

∫e(τ)dτ

)= θ∗Tφ(t)



J

Gc(s)1

Js+B

B

xd +

xd

+e + τ x

+−


s,Wcl(s) =

(K + Jλ)s + KλJs2 + (K + Jλ)s + Kλ

r = W−1cl (s)[xd]

= xd +((Js)G−1

cl (s))[xd]

= xd + Jωd


= Bx + Jxd + (K + Jλ)e + Kλ∫

e(τ)dτ

= J (xd + λe) + Bx + K(

e + λ

∫e(τ)dτ

)= θ∗Tφ(t)



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


= Je1(t) + Bx(t) + Ke2(t) = θ∗Tφ(t)

e1 = (xd + λe) , e2 =

(e + λ

∫e(τ)dτ

)

φ = [ e1 x e2 ]> , θ∗ = [ J B K ]>

Adaptive PI control:

τ = J(t)e1 + B(t)x + Ke2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


= Je1(t) + Bx(t) + Ke2(t) = θ∗Tφ(t)

e1 = (xd + λe) ,

e2 =

(e + λ

∫e(τ)dτ

)

φ = [ e1 x e2 ]> , θ∗ = [ J B K ]>


τ = J(t)e1 + B(t)x + Ke2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


= Je1(t) + Bx(t) + Ke2(t) = θ∗Tφ(t)

e1 = (xd + λe) , e2 =

(e + λ

∫e(τ)dτ

)

φ = [ e1 x e2 ]> , θ∗ = [ J B K ]>


τ = J(t)e1 + B(t)x + Ke2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


= Je1(t) + Bx(t) + Ke2(t) = θ∗Tφ(t)

e1 = (xd + λe) , e2 =

(e + λ

∫e(τ)dτ

)

φ = [ e1 x e2 ]> ,

θ∗ = [ J B K ]>


τ = J(t)e1 + B(t)x + Ke2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


= Je1(t) + Bx(t) + Ke2(t) = θ∗Tφ(t)

e1 = (xd + λe) , e2 =

(e + λ

∫e(τ)dτ

)

φ = [ e1 x e2 ]> , θ∗ = [ J B K ]>


τ = J(t)e1 + B(t)x + Ke2


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2

Plant+controller: x =1J(−Bx + τ)

=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)

e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)

e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)

= −KJ

e2 +1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)

− Error Model 3


Adaptive PI Control

τ = J(t)e1 + B(t)x + Ke2


=1J

(−Bx + J(t)e1 + B(t)x + Ke2

)e2 =

(e + λ

∫e(τ)dτ

)e2 = e + λe = xd − x + λe

= xd −1J

(Je1 + Bx + Ke2

)+ λe

=

(1− J

J

)e1 −

1J

(Bx + Ke2

)= −K

Je2 +

1J

(−Je1 − Bx

)− Error Model 3


Adaptive PI Control - Stability

J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

τ = J(t)e1 + B(t)x + Ke2

Error Equation: e2 = −KJ

e2 +1J

(−Je1 − Bx

)Adaptive Law: ˙J = γ1e2e1,

˙B = γ1e2x

Lyapunov function: V =12

(e2

2 +1J

(J2

γ1+

B2

γ2

))V = −K

Je2

2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

τ = J(t)e1 + B(t)x + Ke2


e2 +1J

(−Je1 − Bx

)

Adaptive Law: ˙J = γ1e2e1,˙B = γ1e2x


(e2

2 +1J

(J2

γ1+

B2

γ2

))V = −K

Je2

2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

τ = J(t)e1 + B(t)x + Ke2


e2 +1J

(−Je1 − Bx


˙B = γ1e2x


(e2

2 +1J

(J2

γ1+

B2

γ2

))V = −K

Je2

2



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

τ = J(t)e1 + B(t)x + Ke2


e2 +1J

(−Je1 − Bx


˙B = γ1e2x


(e2

2 +1J

(J2

γ1+

B2

γ2

))V = −K

Je2

2( [email protected]) October 22, 2019 12 / 13

Adaptive PI Control - Stability and Asymptotic Tracking

J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

e2, J, B ∈ L∞

=⇒ e2 ∈ L∞e2 ∈ L∞ =⇒ e ∈ L∞e ∈ L∞ =⇒ e1 ∈ L∞Therefore e2 ∈ L∞; e2 ∈ L2

=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

e2, J, B ∈ L∞ =⇒ e2 ∈ L∞

e2 ∈ L∞ =⇒ e ∈ L∞e ∈ L∞ =⇒ e1 ∈ L∞Therefore e2 ∈ L∞; e2 ∈ L2

=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

e2, J, B ∈ L∞ =⇒ e2 ∈ L∞e2 ∈ L∞ =⇒ e ∈ L∞

e ∈ L∞ =⇒ e1 ∈ L∞Therefore e2 ∈ L∞; e2 ∈ L2

=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

e2, J, B ∈ L∞ =⇒ e2 ∈ L∞e2 ∈ L∞ =⇒ e ∈ L∞e ∈ L∞ =⇒ e1 ∈ L∞

Therefore e2 ∈ L∞; e2 ∈ L2

=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x

e2, J, B ∈ L∞ =⇒ e2 ∈ L∞e2 ∈ L∞ =⇒ e ∈ L∞e ∈ L∞ =⇒ e1 ∈ L∞Therefore e2 ∈ L∞; e2 ∈ L2

=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


=⇒ limt→∞ e2(t) = 0

=⇒ limt→∞ e(t) = 0



J

K

B

1Js+B

xd

xd

e1

e2++

+

τ x


=⇒ limt→∞ e2(t) = 0 =⇒ limt→∞ e(t) = 0


2.153 Adaptive Control Fall 2019 Lecture 13: Adaptive PI...

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