Lecture 15: Nonlinear adaptive systems. • Adaptive feedback linearization. •
Adaptive backstepping design. • Adaptive forwarding design c Leonid Freidovich.
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. 1/1
Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1 d x dt 2
c Leonid Freidovich.
May 21, 2008.
= x2 + f (x1 ) = u
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 2/1
Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=u
Introduce the nonlinear change of coordinates ξ 1 = x1 ξ2 = x2 + f (x1 )
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 2/1
Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
d x dt 2
= x2 + f (x1 ),
=u
Introduce the change of coordinates as ξ 1 = x1 ,
ξ2 = x2 + f (x1 )
Then d ξ dt 1 d ξ dt 2
c Leonid Freidovich.
= ξ2 =
May 21, 2008.
d x dt 2
+
d f (x1 ) dt
= u + ξ2 f ′(ξ1 )
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 2/1
Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=u
Introduce the nonlinear change of coordinates ξ 1 = x1 ,
ξ2 = x2 + f (x1 )
Then d ξ dt 1
= ξ2 ,
d ξ dt 2
= u + ξ2 f ′(ξ1 )
Making the feedback transform (u → v) u = −a2 ξ1 − a1 ξ2 − ξ2 f ′(ξ1 ) + v
we bring the system dynamics into the linear form d ξ dt 1 c Leonid Freidovich.
= ξ2 ,
May 21, 2008.
d ξ dt 2
= −a2 ξ1 − a1 ξ2 + v
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 2/1
Adaptive Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1 d x dt 2
c Leonid Freidovich.
May 21, 2008.
= x2 + θ0 · f (x1 ) = u
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 3/1
Adaptive Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ0 · f (x1 ),
d x dt 2
=u
Introduce the nonlinear change of coordinates ξ 1 = x1 ξ2 = x2 + θ · f (x1 )
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 3/1
Adaptive Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
d x dt 2
= x2 + θ0 · f (x1 ),
=u
Introduce the nonlinear change of coordinates ξ 1 = x1 ,
ξ2 = x2 + θ · f (x1 )
Then d ξ dt 1
= x2 + θ0 · f (x1 ) = ξ2 + (θ0 − θ)f (ξ1 )
d ξ dt 2
=
d x dt 2
= u+
c Leonid Freidovich.
+ d dt
d dt
[θ · f (x1 )] = u +
d dt
[θ] · f (x1 ) + θ ·
d f (x1 ) dt
[θ] · f (x1 ) + θ · f ′(x1 ) [x2 + θ0 · f (x1 )]
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 3/1
Adaptive Feedback Linearization Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ0 · f (x1 ),
d x dt 2
=u
Introduce the nonlinear change of coordinates ξ 1 = x1 ,
ξ2 = x2 + θ · f (x1 )
Then d ξ dt 1
= ξ2 + (θ0 − θ)f (ξ1 )
d ξ dt 2
= u+
d dt
[θ] · f (x1 ) + θ · f ′(x1 ) [x2 + θ0 · f (x1 )]
Introduce the feedback transform (u → v) d [θ]·f (x1 )−θ·f ′(x1 ) [x2 + θ · f (x1 )]+v u = −a2 ξ1 −a1 ξ2 − dt c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 3/1
Adaptive Feedback Linearization The system dynamics before feedback transform d ξ dt 1
= ξ2 + (θ0 − θ)f (ξ1 )
d ξ dt 2
= u+
d dt
[θ] · f (x1 ) + θ · f ′(x1 ) [x2 + θ0 · f (x1 )]
Introduce the feedback transform (u → v) d u = −a2 ξ1 −a1 ξ2 − dt [θ]·f (x1 )−θ·f ′(x1 ) [x2 + θ · f (x1 )]+v
The system dynamics after the feedback transform is # " #" # " " # " # d ξ1 f (ξ1 ) ξ1 0 1 0 + = v (θ0 −θ)+ ′ ξ2 −a2 −a1 1 θf (x1 )f (x1 ) dt ξ2
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 3/1
Adaptive Feedback Linearization (Cont’d) Choose the target dynamics for # " #" # " # " " # d ξ1 f (ξ1 ) ξ1 0 1 0 + = (θ0 −θ)+ v ′ ξ2 −a2 −a1 θf (x1 )f (x1 ) 1 dt ξ2 as d dt
"
xm1 x2m
c Leonid Freidovich.
#
=
May 21, 2008.
"
0 1 −a2 −a1
#"
xm1 x2m
#
+
"
0 a2
Elements of Iterative Learning and Adaptive Control:
#
vm
Lecture 15
– p. 4/1
Adaptive Feedback Linearization (Cont’d) Choose the target dynamics for # " #" # " # " " # d ξ1 f (ξ1 ) ξ1 0 1 0 + = (θ0 −θ)+ v ′ ξ2 −a2 −a1 θf (x1 )f (x1 ) 1 dt ξ2 as d dt
"
xm1 x2m
#
=
"
0 1 −a2 −a1
#"
xm1 x2m
#
+
"
0 a2
#
vm
Then v = a2 vm
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 4/1
Adaptive Feedback Linearization (Cont’d) Introducing the error signal e(t) = ξ(t) − xm(t)
we get the error dynamics d dt
"
e1 e2
#
#"
"
0 1 = −a2 −a1 {z } | A
e1 e2
#
"
#
f (ξ1 ) + (θ0 − θ) ′ θf (x1 )f (x1 ) {z } | B
It is almost as we have before, but B is dependent on states! The possible choice for θ -dynamics is d dt
θ = γB(·)T P e
where γ > 0 and P is a solution for: AT P + P A < 0 c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 5/1
Backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1 d x dt 2
c Leonid Freidovich.
May 21, 2008.
= x2 + f (x1 ) = u
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 6/1
Backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + f (x1 ),
c Leonid Freidovich.
May 21, 2008.
x2 = v1
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 6/1
Backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + f (x1 ),
x2 = v1
Use Lyapunov-based design with V1 (x1 ) =
1 2
x21
⇒
V˙ 1 = x1 (v1 + f (x1 )) < 0
so that we can take with c1 > 0 v1 = −f (x1 ) − c1 x1
c Leonid Freidovich.
May 21, 2008.
⇒
V˙ 1 = −c1 x21 < 0
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 6/1
Backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + f (x1 ),
x2 = v1
Use Lyapunov-based design with V1 (x1 ) =
1 2
x21
⇒
V˙ 1 = x1 (v1 + f (x1 )) < 0
so that we can take with c1 > 0 v1 = −f (x1 ) − c1 x1
⇒
V˙ 1 = −c1 x21 < 0
Define α1 (x1 ) = v1 to be the desired law of change for x2 . c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 6/1
Backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
d x dt 2
= x2 + f (x1 ),
=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: d x dt 1
= f (x1 ) + α1 (x1 ) + (x2 − α1 (x1 )), | {z } | {z } −c1 x1
c Leonid Freidovich.
May 21, 2008.
d x dt 2
=u
z2
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 7/1
Backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 ): d x dt 1
= −c1 x1 + z2 ,
c Leonid Freidovich.
May 21, 2008.
d z dt 2
d = u − α′1 (x1 )( dt x1 )
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 7/1
Backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 ): d x dt 1
= −c1 x1 + z2 ,
c Leonid Freidovich.
May 21, 2008.
d z dt 2
= u − α′1 (x1 )(−c1 x1 + z2 )
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 7/1
Backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 ): d x dt 1
= −c1 x1 + z2 ,
d z dt 2
= u − α′1 (x1 )(−c1 x1 + z2 )
Use Lyapunov-based design with V2 = V1 (x1 )+ 12 z22
⇒
d z2 ) < 0 V˙ 2 = −c1 x21 +x1 z2 +z2 ( dt
so that we can take with c2 > 0 u = α2 (x1 , z2 ) ⇒ V˙ 2 = −c1 x21 − c2 z22 < 0 where α2 (x1 , z2 ) = α′1 (x1 )(−c1 x1 + z2 ) − x1 − c2 z2 . c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 7/1
Adaptive backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1 d x dt 2
c Leonid Freidovich.
May 21, 2008.
= x2 + θ 0 · f (x1 ) = u
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 8/1
Adaptive backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + θ 0 · f (x1 ),
c Leonid Freidovich.
May 21, 2008.
x2 = v1
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 8/1
Adaptive backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + θ 0 · f (x1 ),
x2 = v1
Use Lyapunov-based design with θ˜ = θ − θ 0 ˜ = V1 (x1 , θ)
1 2
(x21 + γ1 θ˜2 ) ⇒ V˙ 1 = x1 (v1 +θ 0 ·f (x1 ))+ γ1 θ˜θ˙ ≤ 0
so that we can take with c1 > 0 and γ > 0 v1 = −θ·f (x1 )−c1 x1 ,
c Leonid Freidovich.
May 21, 2008.
θ˙ = γ x1 f (x1 )
⇒
V˙ 1 = −c1 x21 ≤ 0
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 8/1
Adaptive backstepping design Consider the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
Treat x2 as a virtual control signal v1 for x1 in the 1st equation x˙ 1 = v1 + f (x1 ),
x2 = v1
Use Lyapunov-based design with θ˜ = θ − θ 0 ˜ = V1 (x1 , θ)
1 2
(x21 + γ1 θ˜2 ) ⇒ V˙ 1 = x1 (v1 +θ 0 ·f (x1 ))+ γ1 θ˜θ˙ ≤ 0
so that we can take with c1 > 0 and γ > 0 v1 = −θ·f (x1 )−c1 x1 ,
θ˙ = γ x1 f (x1 )
⇒
V˙ 1 = −c1 x21 ≤ 0
Define α1 (x1 , θ) = v1 to be the desired law of change for x2 . c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 8/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=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: d x dt 1
= θ 0 · f (x1 ) + α1 (x1 , θ) + (x2 − α1 (x1 , θ)), {z } | {z } | ˜ f(x1 ) −c1 x1 −θ
c Leonid Freidovich.
May 21, 2008.
d x dt 2
=u
z2
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 9/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 , θ): d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
c Leonid Freidovich.
May 21, 2008.
d z dt 2
∂α1 ˙ d 1 θ = u− ∂α x )− ( 1 ∂x1 dt ∂θ
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 9/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 , θ): d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
c Leonid Freidovich.
May 21, 2008.
d z dt 2
d 1 = u− ∂α x )+f (x1 ) θ˙ ( ∂x1 dt 1
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 9/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=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 = x2 − α1 (x1 , θ): d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
d z dt 2
d 1 = u− ∂α x )+f (x1 ) θ˙ ( ∂x1 dt 1
We will use Lyapunov-based design with V2 (x1 , z2 , θ) = V1 (x1 , θ) +
1 2
z22 =
1 2
x21 + γ1 θ˜2 + z22
achieving with c2 > 0 V˙ 2 = −c1 x21 − c2 z22 ≤ 0 with u = α2 (x1 , z2 , θ). c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 9/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
Dynamics can be rewritten as d x dt 1 d z dt 2
= −c1 x1 − θ˜ f (x1 ) + z2 ,
= u + (θ f ′(x1 ) + c1 ) (−c1 x1 − θ˜ f (x1 ) + z2 ) + f (x1 ) θ˙
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
Dynamics can be rewritten as d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
d z dt 2
= v−(θ f ′(x1 )+c1 ) θ˜ f (x1 )
v = u + (θ f ′(x1 ) + c1 ) (−c1 x1 + z2 ) + f (x1 ) θ˙
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
Dynamics can be rewritten as d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
d z dt 2
= 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 1 2 2 V2 (x1 , z2 , θ) = x1 + γ1 θ˜ + z22 2 is given by d d x1 + z2 dt z2 + γ1 θ˜θ˙ V˙ 2 = x1 dt c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
Dynamics can be rewritten as d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
d z dt 2
= 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 1 2 2 V2 (x1 , z2 , θ) = x1 + γ1 θ˜ + z22 2 is given by V˙ 2 = x1 (−c1 x1 −θ˜ f (x1 )+z2 )+z2 (v−(θ f ′(x1 )+c1 ) θ˜ f (x1 ))+ γ1 θ˜θ˙ c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) For the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
introduce new state variables z2 and θ z2 = x2 − α1 (x1 , θ) = x2 + θ f (x1 ) + c1 x1
Dynamics can be rewritten as d x dt 1
= −c1 x1 −θ˜ f (x1 )+z2 ,
d z dt 2
= 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 1 2 2 V2 (x1 , z2 , θ) = x1 + γ1 θ˜ + z22 2 is given by ˜ · · + 1 θ) ˙ V˙ 2 = −c1 x21 − c2 z22 + z2 (v + x1 + c2 z2 ) + θ(· γ c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 10/1
Adaptive backstepping design (Cont’d) Finally, for the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
and z2 = x2 + θ f (x1 ) + c1 x1
c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 11/1
Adaptive backstepping design (Cont’d) Finally, for the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
and z2 = x2 + θ f (x1 ) + c1 x1
We have V2 (x1 , z2 , θ) =
c Leonid Freidovich.
1
2
2
x21 + γ1 θ˜ + z22 ,
May 21, 2008.
V˙ 2 = −c1 x21 −c2 z22
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 11/1
Adaptive backstepping design (Cont’d) Finally, for the nonlinear system with f (·) ∈ C 1 d x dt 1
= x2 + θ 0 · f (x1 ),
d x dt 2
=u
and z2 = x2 + θ f (x1 ) + c1 x1
We have V2 (x1 , z2 , θ) =
1
2
2
x21 + γ1 θ˜ + z22 ,
V˙ 2 = −c1 x21 −c2 z22
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 ) c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 11/1
Next Lecture / Assignments: Last meetings: May 23, 13:00-15:00, in A206Tekn – Recitations; June 9, 10:00-12:00, in A206Tekn: – Project reports / EXAM Homework problem: Consider the nonlinear system x˙ 1 = x2 + a x21 ,
x˙ 2 = 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 model Gm(s) =
1 s2 + 1.4 s + 1
driven by a unit step signal uc(t)? c Leonid Freidovich.
May 21, 2008.
Elements of Iterative Learning and Adaptive Control:
Lecture 15
– p. 12/1
