Multicriteria Analysis of Visual Servos through Rational Systems, Biquadratic Lyapunov Functions, and LMIs - Visual Servoing via Advanced Numerical Methods - page 170

m

σ ×

n

matrix function Λ σ ( .,. ) ∈ R

σ can be exhibited such that

Λ σ ( x , χ ) σ ( x , χ )= 0

˜

X×X χ , then the following is a less conservative sufficient condition

is true on

to (10.14) :

˜

n σ × m σ such that

∃ L ∈ R ∀

( x

, χ ∈V X×X χ ) , Σ τ l , P ,..., L ( x , χ ≤ ,

) ( ) 0 (10.15)

Λ σ ( x L . with Σ τ l , P ,..., L ( x , χ ) Σ τ l , P ,... ( x , χ )+ LΛ σ ( x , χ )+ , χ )

Each constraint Σ τ l , P ,..., L ( x , χ ) ≤ 0isanLMIon τ l , P ,..., L computed at a vertex

˜

( x , χ ) of X×X χ . The matrix function Λ σ ( .,. ) is said a linear annihilator of σ ( .,. ).

10.4 Multicriteria Analysis through Biquadratic Lyapunov

Functions

Constructive results to the multicriteria analysis of visual servos by biquadratic and
piecewise-biquadratic Lyapunov functions are hereafter detailed.

10.4.1 Mathematical Background

Equivalent representations of the closed-loop system and of the constraints are first
introduced. Then, the classes of Lyapunov functions candidates are presented.

Definition 10.4 (Differential Algebraic Representations [24, 8]). A differential
algebraic representation (DAR) of an uncertain nonlinear rational system is de-
fined by

x =

A 1 x + A 2 π

(10.16)

0 = Ω 1 ( x , χ ) x + Ω 2 ( x , χ ) π ,

n

where π = π ( x , χ ) ∈ R is a nonlinear vector function of ( x , χ ) , A 1 , A 2 are con-

π

stant matrices and Ω 1 ( .,. ) , Ω 2 ( .,. ) are affine matrix functions. The above repre-

˜

sentation is well-posed if X×X χ .The

rational closed loop system (10.4) can be turned into (10.16) by gathering nonlin-
ear terms in

Ω 2 ( .,. ) is column full rank for all ( x , χ ) ∈

π and augmenting this vector so that its entries can be united to ( x , χ )

through some affine matrix functions Ω 1 ( .,. ) and Ω 2 ( .,. ) . Importantly, a DAR is not

unique.
Similarly, the additional variables ζ j = Z j ( x , χ ) x defined in (10.5) write as [12]

ζ j = K 1j x + K 2j ρ
0 =

(10.17)

Υ 1j ( x , χ ) x + Υ 2j ( x , χ ) ρ ,

with ρ = ρ ( x , χ ) a nonlinear vector function of ( x , χ ) , K 1j , K 2j constant vectors,

˜

Υ 1j ( .,. ) , Υ 2j ( .,. ) affine matrix functions, and Υ 2j ( .,. ) column full rank on X×X χ .

The considered classes of Lyapunov function candidates are as follows.