Outline Homogeneous approximation Application for nonlinear tools
Homogeneity in the bi-limit as a tool for observer and feedback design Vincent Andrieu, Laurent Praly and Alessandro Astolfi
December 11, 2009
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Homogeneous approximation Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Application for nonlinear tools The domination approach Tools for a chain of integrator Tools for nonlinear systems
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Standard homogeneity DEFINITION: The function 𝜙 is homogeneous if there exists 1. a weight (r1 , . . . , rn ) with ri > 0, 2. a degree d ≥ 0, such that for all 𝜆 ≥ 0 and x : 𝜙(X ) = 𝜆d 𝜙(x)
,
X = (𝜆r1 x1 , . . . , 𝜆rn xn ) .
DEFINITION: The system x˙ = f (x) is homogeneous of degree 𝜏 if for all 𝜆 ≥ 0 and x : X˙ = 𝜆𝜏 f (X )
,
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
X = (𝜆r1 x1 , . . . , 𝜆rn xn ) .
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Illustrative example of homogeneous approximation Consider the system : x˙ 1 = x2 − x1 Note that :
x˙ 2 = −x1 − x2 + x2q
,
q
−𝜆r2 x2 + (𝜆r2 x2 )
= −𝜆r2 x2 + 𝜆q r2 x2q
If q ∕= 1 this system is not homogeneous When q > 1 and x2 ≈ 0, we have : −x2 + x2q ≈ −x2 and the linear system :
x˙ 1 = x2 − x1
,
x˙ 2 ≈ −x1 − x2
is a local homogeneous approximation.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Homogeneous approximation: local case DEFINITION: The function 𝜙(x) is homogeneous in the 0-limit if there exist 1. a weight (r0,1 , . . . , r0,n ), with r0,i > 0, 2. a degree d0 ≥ 0, 3. an approximating homogeneous function 𝜙0 , 0) lim𝜆→0 𝜙(X − 𝜙0 (x) = 0 such that on each compact set 𝜆d0 with X0 = (𝜆r0,1 x1 , . . . , 𝜆r0,n xn ).
DEFINITION: The system x˙ = f (x) is homogeneous in the 0-limit1 with approximating homogeneous system x˙ = f0 (x) if X˙ 0 lim𝜆→0 𝜆d0 − f0 (X0 ) = 0 1
Lyapunov, Kawski, Massera, Hahn, Hermes, Rosier...
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Homogeneous approximation: at infinity DEFINITION: The function 𝜙(x) is homogeneous in the ∞-limit functions if there exist 1. a weight (r∞,1 , . . . , r∞,n ), with r∞,i > 0, 2. a degree d∞ ≥ 0, 3. an approximating homogeneous function 𝜙∞ , such that for each compact set which does not contain the origin 𝜙(X∞ ) lim𝜆→∞ 𝜆𝜏∞ − 𝜙∞ (x) = 0 with X∞ = (𝜆r∞,1 x1 , . . . , 𝜆r∞,n xn ).
DEFINITION: The system x˙ = f (x) is homogeneous in the ∞-limit with approximating homogeneous system x˙ = f∞ (x) if X˙ ∞ lim𝜆→∞ 𝜆d∞ − f∞ (X∞ ) = 0
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Homogeneity in the bi-limit DEFINITION: The function 𝜙(x) is homogeneous in the bi-limit if it is 1. homogeneous in the 0-limit 2. homogeneous in the ∞-limit
DEFINITION: The system x˙ = f (x) is homogeneous in the bi-limit if it is 1. homogeneous in the 0-limit 2. homogeneous in the ∞-limit
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Local stability THEOREM(Lyapunov, Rosier,...) 1. If the origin of x˙ = f0 (x) is GAS ⇒ The origin of x˙ = f (x) is LAS. 2. If the origin of x˙ = f0 (x) is unstable ⇒ The origin of x˙ = f (x) is unstable. EXAMPLE:
x˙ 1 = x2 − x1
x˙ 2 = −x1 − x2 + x2q
,
is homogeneous in the 0-limit with approximating system : 1. When 1 < q ⇒ x˙ 1 = x2 − x1 ⇒ the origin is LAS.
,
x˙ 2 = −x1 − x2
2. When 1 > q ⇒ x˙ 1 = x2 ⇒ the origin is unstable.
,
x˙ 2 = x2q
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
THEOREM If the origin of x˙ = f∞ (x) is GAS ⇒ There exists a compact set C which is GAS for x˙ = f (x).
EXAMPLE:The system : 1
x˙ 1 = x2 − x12−p
,
p
x˙ 2 = −x12−p − x2p + x2q
p>q
is Homogeneous in the ∞-limit with approximating systems p
1
x˙ 1 = x2 − x12−p , x˙ 2 = −x12−p − x2p ⇒ There exists a GAS compact subset.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Finite-time convergence THEOREM If the origins of : x˙ = f (x) , x˙ = f0 (x) , x˙ = f∞ (x) are GAS, then there exists a homogeneous in the bi-limit Lyapunov function V such that : ( ) dV +𝜏0 dV +𝜏∞ 0 ∞ ∂V (x)f (x) < −c V (x) dV0 + V (x) dV∞ ∂x FINITE TIME CONVERGENCE If 𝜏0 < 0 and 𝜏∞ > 0 then we get convergence to the origin in finite time and uniformly in the initial condition. dV +𝜏∞ ∞ V˙ < V dV∞ ⇒ Finite time convergence to V = 1 V˙ < V
dV +𝜏0 0 dV 0
⇒ Finite time convergence from V = 1 to V = 0
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
Robustness COROLLARY If the origins of : x˙ = f (x)
,
x˙ = f0 (x)
,
x˙ = f∞ (x)
are GAS, then there exists a positive real number cr such that the origin of the system : x˙ = f (x) + 𝔇(t) is GAS where 𝔇(t) = (𝛿1 (t), . . . , 𝛿n (t)) and { { }} 𝜏0 +r0,i 𝜏∞ +r∞,i ∣𝛿i (t)∣ ≤ cr max 𝛽(∣𝔇(0)∣, t), sup ∣x(𝜅)∣ r0 + ∣x(𝜅)∣ r∞ 0≤𝜅≤t
where 𝛽 is a class 𝒦ℒ function. ⇒ Robustness to polynomial disturbances !
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
Standard homogeneity Homogeneous Approximation Homogeneous approximation and system analysis
EXAMPLE : When 0 < p < r < q < 2, the system : 1 1 ) ( r y˙ = x − y 2−p − y 2−q 2−r + ∣x∣r p q , 𝛿 = c ∣y ∣ x˙ = −y 2−p − y 2−q − x p − x q +𝛿 is Homogeneous in the bi-limit, with approximating systems : 1
1. In the 0-limit :
y˙ = x − y 2−p
2. In the ∞-limit :
y˙ = x − y 2−q
1
p
,
x˙ = −y 2−p − x p ⇒ The origin is GAS.
,
x˙ = −y 2−q − x q ⇒ The origin is GAS.
q
Also, the origin of the nominal system without 𝛿 is GAS ⇒ Using the Corollary, if c is small enough the origin of the nominal system is GAS. ⇒ if p < 1 and q > 1 we get uniforme finite time convergence.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
The domination approach Construct a feedback or an observer on a chain of integrator and handle nonlinear terms by domination Consider the system ⎧ x˙ 1 ⎨ x˙ i ⎩ x˙ n
: = x2 .. .
+
𝛿1 (x)
= xi +1 .. .
+
𝛿i (x)
=u
+
𝛿n (x)
,
y = x1 ,
Which can be rewritten : x˙ = S x + B u + | {z } Chain of integrator part
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
𝛿(x) |{z} Nonlinearities = Disturbances
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Homogeneous in the bi-limit state feedback THEOREM Consider 1. The degrees −1 < 𝜏0 ≤ 𝜏∞ <
1 n−1
2. The weights defined by : r0,n = 1 , r0,i = r0,i +1 − 𝜏0 r∞,n = 1 , r∞,i = r∞,i +1 − 𝜏∞ Then there exists a homogeneous in the bi-limit state feedback u = 𝜙(x) such that the system x˙ = S x + B 𝜙(x) is homogeneous in the bi-limit and, its origin and the origin of x˙ = Sx + B𝜙0 (x)
,
x˙ = Sx + B𝜙∞ (x)
are GAS equilibriums. Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Homogeneous in the bi-limit observer design Consider a chain of integrator : x˙ = S x + B u THEOREM Consider 1. The degrees −1 < 𝜏0 ≤ 𝜏∞ <
,
y = x1
1 n−1
2. The weights defined by : r0,n = 1 , r0,i = r0,i +1 − 𝜏0 r∞,n = 1 , r∞,i = r∞,i +1 − 𝜏∞ Then there exists a homogeneous in the bi-limit observer : xˆ˙ = S xˆ + B u + K (ˆ x1 − x1 ) such that the origin of the error system : E˙ = SE + K (e1 ) , E˙ = SE + K0 (e1 )
,
E˙ = SE + K∞ (e1 )
are GAS equilibriums. Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Output feedback for feedback systems Consider the system x˙ = S x + B u + 𝔇(x) , THEOREM If there exist −1 < 𝜏0 ≤ 𝜏∞ < ∣𝛿i (x)∣ ≤ c𝛿
i ∑
1−𝜏0 (n−i )
1 n−1
y = x1
and c𝛿 > 0 such that : 1−𝜏∞ (n−i )
∣xj ∣ 1−𝜏0 (n−j −1) + ∣xj ∣ 1−𝜏∞ (n−j −1)
j=1
⇒ System in feedback form
then the output feedback u = Ln 𝜙(ˆ x)
,
xˆ˙ = L (S xˆ + B𝜙(ˆ x ) + K (ˆ x1 − y )) ,
where L is the high-gain parameter, large enough stabilizes the origin.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Output feedback for feedfoward systems
THEOREM If there exist −1 < 𝜏0 ≤ 𝜏∞ < ∣𝛿i (x)∣ ≤ c𝛿
n ∑
1−𝜏0 (n−i )
1 n−1
and c𝛿 > 0 such that : 1−𝜏∞ (n−i )
∣xj ∣ 1−𝜏0 (n−j −1) + ∣xj ∣ 1−𝜏∞ (n−j −1)
j=i +2
⇒ System in feedforward form
then then the output feedback u = Ln 𝜙(ˆ x)
,
xˆ˙ = L (S xˆ + B𝜙(ˆ x ) + K (ˆ x1 − y )) ,
where L is the low-gain parameter, small enough stabilizes the origin.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Uniform finite time observer
THEOREM If 𝛿 is globally Lipschitz, i.e., ∃ c such that ∣𝛿i (x + x˜) − 𝛿i (x)∣ ≤ c
i ∑
∣˜ xj ∣
j=1
then for all L sufficiently large, xˆ˙ = S xˆ + Bu + 𝛿(ˆ x ) + diag(L, L2 , Ln )K (ˆ x1 − y ) with K homogeneous in the bi-limit with weight 𝜏0 < 0 < 𝜏∞ , estimates the state in finite time uniformly in the initial condition.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
Outline Homogeneous approximation Application for nonlinear tools
The domination approach Tools for a chain of integrator Tools for nonlinear systems
Conclusion
1. We have introduced the homogeneity in the bi-limit contexte. 2. Allow robustness with respect to polynomial disturbances and uniform finite time convergence. 3. In combination with the domination approach provides results for output feedback and observer design.
Vincent Andrieu, Laurent Praly and Alessandro Astolfi
Homogeneity in the bi-limit as a tool for observer and feedback
