lecture 15: nonlinear adaptive systems - umu.se · lecture 15: nonlinear adaptive systems •...

Lecture 15: Nonlinear adaptive systems

• Adaptive feedback linearization

• Adaptive backstepping design

• Adaptive forwarding design

c©Leonid Freidovich. May 21, 2008. Elements of Iterative Learning and Adaptive Control: Lecture 15 – p. 1/12

Feedback Linearization

Consider the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + f(x1)

ddt

x2 = u




ddt

x1 = x2 + f(x1),ddt

x2 = u

Introduce the nonlinear change of coordinates

ξ1 = x1

ξ2 = x2 + f(x1)




ddt

x1 = x2 + f(x1),ddt

x2 = u

Introduce the change of coordinates as

ξ1 = x1, ξ2 = x2 + f(x1)

Then

ddt

ξ1 = ξ2

ddt

ξ2 = ddt

x2 + ddt

f(x1) = u + ξ2f ′(ξ1)


Feedback LinearizationConsider the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + f(x1),ddt

x2 = u


ξ1 = x1, ξ2 = x2 + f(x1)

Thenddt

ξ1 = ξ2, ddt

ξ2 = u + ξ2f ′(ξ1)

Making the feedback transform (u → v)

u = −a2ξ1 − a1ξ2 − ξ2f ′(ξ1) + v

we bring the system dynamics into the linear form

ddt

ξ1 = ξ2, ddt

ξ2 = −a2ξ1 − a1ξ2 + v


Adaptive Feedback Linearization


ddt

x1 = x2 + θ0 · f(x1)

ddt

x2 = u




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


ξ1 = x1

ξ2 = x2 + θ · f(x1)




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


ξ1 = x1, ξ2 = x2 + θ · f(x1)

Then

ddt

ξ1 = x2 + θ0 · f(x1) = ξ2 + (θ0 − θ)f(ξ1)

ddt

ξ2 = ddt

x2 + ddt

[θ · f(x1)] = u + ddt

[θ] · f(x1) + θ · ddt

f(x1)

= u + ddt

[θ] · f(x1) + θ · f ′(x1) [x2 + θ0 · f(x1)]




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


ξ1 = x1, ξ2 = x2 + θ · f(x1)

Then

ddt

ξ1 = ξ2 + (θ0 − θ)f(ξ1)

ddt

ξ2 = u + ddt

[θ] · f(x1) + θ · f ′(x1) [x2 + θ0 · f(x1)]

Introduce the feedback transform (u → v)

u = −a2ξ1−a1ξ2− ddt

[θ]·f(x1)−θ·f ′(x1) [x2 + θ · f(x1)]+v



The system dynamics before feedback transform

ddt

ξ1 = ξ2 + (θ0 − θ)f(ξ1)

ddt

ξ2 = u + ddt

[θ] · f(x1) + θ · f ′(x1) [x2 + θ0 · f(x1)]

Introduce the feedback transform (u → v)

u = −a2ξ1−a1ξ2− ddt

[θ]·f(x1)−θ·f ′(x1) [x2 + θ · f(x1)]+v

The system dynamics after the feedback transform is

d

dt

[

ξ1

ξ2

]

=

[

0 1

−a2 −a1

] [

ξ1

ξ2

]

+

[

f(ξ1)

θf ′(x1)f(x1)

]

(θ0−θ)+

[

0

1

]

v


Adaptive Feedback Linearization (Cont’d)

Choose the target dynamics for

d

dt

[

ξ1

ξ2

]

=

[

0 1

−a2 −a1

] [

ξ1

ξ2

]

+

[

f(ξ1)

θf ′(x1)f(x1)

]

(θ0−θ)+

[

0

1

]

v

as

d

dt

[

xm1

x2m

]

=

[

0 1

−a2 −a1

] [

xm1

x2m

]

+

[

0

a2

]

vm



Choose the target dynamics for

d

dt

[

ξ1

ξ2

]

=

[

0 1

−a2 −a1

] [

ξ1

ξ2

]

+

[

f(ξ1)

θf ′(x1)f(x1)

]

(θ0−θ)+

[

0

1

]

v

as

d

dt

[

xm1

x2m

]

=

[

0 1

−a2 −a1

] [

xm1

x2m

]

+

[

0

a2

]

vm

Thenv = a2vm



Introducing the error signal

e(t) = ξ(t) − xm(t)

we get the error dynamics

d

dt

[

e1

e2

]

=

[

0 1

−a2 −a1

]

︸︷︷︸

A

[

e1

e2

]

+

[

f(ξ1)

θf ′(x1)f(x1)

]

︸︷︷︸

B

(θ0 − θ)

It is almost as we have before, but B is dependent on states!

The possible choice for θ-dynamics is

d

dtθ = γB(·)T Pe

where γ > 0 and P is a solution for: AT P + P A < 0


Backstepping design


ddt

x1 = x2 + f(x1)

ddt

x2 = u


Backstepping design


ddt

x1 = x2 + f(x1),ddt

x2 = u

Treat x2 as a virtual control signal v1 for x1 in the 1st equation

x1 = v1 + f(x1), x2 = v1


Backstepping design


ddt

x1 = x2 + f(x1),ddt

x2 = u


x1 = v1 + f(x1), x2 = v1

Use Lyapunov-based design with

V1(x1) = 12

x21 ⇒ V1 = x1(v1 + f(x1)) < 0

so that we can take with c1 > 0

v1 = −f(x1) − c1 x1 ⇒ V1 = −c1 x21 < 0


Backstepping designConsider the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + f(x1),ddt

x2 = u


x1 = v1 + f(x1), x2 = v1


V1(x1) = 12

x21 ⇒ V1 = x1(v1 + f(x1)) < 0


v1 = −f(x1) − c1 x1 ⇒ V1 = −c1 x21 < 0

Define α1(x1) = v1 to be the desired law of change for x2.


Backstepping design (Cont’d)

For the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + f(x1),ddt

x2 = u

Define α1(x1) to be the desired law of change of the variable x2

α1(x1) = −f(x1) − c1 x1

and introduce the new variable z2 to measure the difference:

ddt

x1 = f(x1) + α1(x1)︸︷︷︸

−c1 x1

+ (x2 − α1(x1))︸︷︷︸

z2

, ddt

x2 = u




ddt

x1 = x2 + f(x1),ddt

x2 = u


α1(x1) = −f(x1) − c1 x1

and introduce the new variable z2 = x2 − α1(x1):

ddt

x1 = −c1 x1 + z2, ddt

z2 = u − α′

1(x1)(ddt

x1)




ddt

x1 = x2 + f(x1),ddt

x2 = u


α1(x1) = −f(x1) − c1 x1


ddt

x1 = −c1 x1 + z2, ddt

z2 = u − α′

1(x1)(−c1 x1 + z2)




ddt

x1 = x2 + f(x1),ddt

x2 = u


α1(x1) = −f(x1) − c1 x1


ddt

x1 = −c1 x1 + z2, ddt

z2 = u − α′

1(x1)(−c1 x1 + z2)


V2 = V1(x1)+12

z22 ⇒ V2 = −c1 x2

1+x1 z2+z2 ( ddt

z2) < 0


u = α2(x1, z2) ⇒ V2 = −c1 x21 − c2 z2

2 < 0

where α2(x1, z2) = α′

1(x1)(−c1 x1 + z2) − x1 − c2 z2.


Adaptive backstepping design


ddt

x1 = x2 + θ0 · f(x1)

ddt

x2 = u




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


x1 = v1 + θ0 · f(x1), x2 = v1




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


x1 = v1 + θ0 · f(x1), x2 = v1

Use Lyapunov-based design with θ = θ − θ0

V1(x1, θ) = 12(x2

1+1γθ2) ⇒ V1 = x1(v1+θ0·f(x1))+

1γθθ ≤ 0

so that we can take with c1 > 0 and γ > 0

v1 = −θ·f(x1)−c1 x1, θ = γ x1 f(x1) ⇒ V1 = −c1 x21 ≤ 0


Adaptive backstepping designConsider the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


x1 = v1 + f(x1), x2 = v1

Use Lyapunov-based design with θ = θ − θ0

V1(x1, θ) = 12(x2

1+1γθ2) ⇒ V1 = x1(v1+θ0·f(x1))+

1γθθ ≤ 0

so that we can take with c1 > 0 and γ > 0

v1 = −θ·f(x1)−c1 x1, θ = γ x1 f(x1) ⇒ V1 = −c1 x21 ≤ 0

Define α1(x1, θ) = v1 to be the desired law of change for x2.


Adaptive backstepping design (Cont’d)


ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u

Define α1(x1, θ) to be the desired law of change of x2

α1(x1, θ) = −θ f(x1) − c1 x1

and introduce the new variable z2 to measure the difference:

ddt

x1 = θ0 · f(x1) + α1(x1, θ)︸︷︷︸

−c1 x1−θ f(x1)

+ (x2 − α1(x1, θ))︸︷︷︸

z2

, ddt

x2 = u




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


α1(x1, θ) = −θ f(x1) − c1 x1

and introduce the new variable z2 = x2 − α1(x1, θ):

ddt

x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = u−∂α1

∂x1

( ddt

x1)−∂α1

∂θθ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


α1(x1, θ) = −θ f(x1) − c1 x1


ddt

x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = u−∂α1

∂x1

( ddt

x1)+f(x1) θ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


α1(x1, θ) = −θ f(x1) − c1 x1


ddt

x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = u−∂α1

∂x1

( ddt

x1)+f(x1) θ

We will use Lyapunov-based design with

V2(x1, z2, θ) = V1(x1, θ) + 12

z22 = 1

2

(

x21 + 1

γθ2 + z2

2

)

achieving with c2 > 0

V2 = −c1 x21 − c2 z2

2 ≤ 0

with u = α2(x1, z2, θ).




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u

introduce new state variables z2 and θ

z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1

Dynamics can be rewritten asddt

x1 = −c1 x1 − θ f(x1) + z2,

ddt

z2 = u+(θ f ′(x1)+ c1) (−c1 x1 − θ f(x1)+ z2)+ f(x1) θ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1


x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = v−(θ f ′(x1)+c1) θ f(x1)

v = u + (θ f ′(x1) + c1) (−c1 x1 + z2) + f(x1) θ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1


x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = v−(θ f ′(x1)+c1) θ f(x1)

v = u + (θ f ′(x1) + c1) (−c1 x1 + z2) + f(x1) θ

Derivative of the Lyapunov function

V2(x1, z2, θ) =1

2

(

x21 + 1

γθ2+ z2

2

)

is given byV2 = x1

ddt

x1 + z2ddt

z2 + 1γθθ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1


x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = v−(θ f ′(x1)+c1) θ f(x1)

v = u + (θ f ′(x1) + c1) (−c1 x1 + z2) + f(x1) θ


V2(x1, z2, θ) =1

2

(

x21 + 1

γθ2+ z2

2

)

is given by

V2 = x1(−c1 x1−θ f(x1)+z2)+z2(v−(θ f ′(x1)+c1) θ f(x1))+1γθθ




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u


z2 = x2 − α1(x1, θ) = x2 + θ f(x1) + c1 x1


x1 = −c1 x1−θ f(x1)+z2, ddt

z2 = v−(θ f ′(x1)+c1) θ f(x1)

v = u + (θ f ′(x1) + c1) (−c1 x1 + z2) + f(x1) θ


V2(x1, z2, θ) =1

2

(

x21 + 1

γθ2+ z2

2

)

is given by

V2 = −c1 x21 − c2 z2

2 + z2 (v + x1 + c2 z2) + θ(· · · + 1γθ)



Finally, for the nonlinear system with f(·) ∈ C1

ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u

andz2 = x2 + θ f(x1) + c1 x1




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u

andz2 = x2 + θ f(x1) + c1 x1

We have

V2(x1, z2, θ) =1

2

(

x21 + 1

γθ2+ z2

2

)

, V2 = −c1 x21−c2 z2

2




ddt

x1 = x2 + θ0 · f(x1),ddt

x2 = u

andz2 = x2 + θ f(x1) + c1 x1

We have

V2(x1, z2, θ) =1

2

(

x21 + 1

γθ2+ z2

2

)

, V2 = −c1 x21−c2 z2

2

provided

−x1 − c2 z2 = u + (θ f ′(x1) + c1) (−c1 x1 + z2) + f(x1) θ

and

θ = γ(

x1 + z2 (θ f ′(x1) + c1))

f(x1)


Next Lecture / Assignments:

Last meetings: May 23, 13:00-15:00, in A206Tekn – Recitations;June 9, 10:00-12:00, in A206Tekn: – Project reports / EXAMHomework problem: Consider the nonlinear system

x1 = x2 + a x21, x2 = b x1 x2 + u

where a and b are unknown parameters.• Design an adaptive state feedback controller using the

backstepping method.• Argue why x1(t) → 0 and x2(t) → 0.• Simulate the closed-loop system with a = b = 1.• How to modify the design to ensure that the output y = x1

follows the modelGm(s) =

1

s2 + 1.4 s + 1

driven by a unit step signal uc(t)?


lecture 15: nonlinear adaptive systems - umu.se · lecture 15: nonlinear adaptive systems •...

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