Quadratic stability of polytopic or affine parameter-dependent systems

[tau,P] = quadstab(ps,options)

For affine parameter-dependent systems

*E*(*p*)*x˙* = *A*(*p*)*x*,
*p*(*t*) = (*p*_{1}(*t*),
. . ., *p _{n}*(

or polytopic systems

*E*(*t*)*x˙* = *A*(*t*)*x*,
(*A*, *E*) ∊ Co{(*A*_{1}, *E*_{1}),
. . ., (*A _{n}*,

`quadstab`

seeks a fixed
Lyapunov function V(*x*) = *x ^{T}*

`ps`

(see `psys`

).The task performed by `quadstab`

is selected
by `options(1)`

:

if

`options(1)=0`

(default),`quadstab`

assesses quadratic stability by solving the LMI problemMinimize τ over

*Q*=*Q*such that^{T}*A*^{T}*QE*+*EQA*< τ^{T}*I*for all admissible values of (*A*,*E*)*Q*>*I*The global minimum of this problem is returned in

`tau`

and the system is quadratically stable if`tau`

< 0.if

`options(1)=1`

,`quadstab`

computes the largest portion of the specified parameter range where quadratic stability holds (only available for affine models). Specifically, if each parameter*p*varies in the interval_{i}$${p}_{i}\in [{p}_{i}{}_{0}-{\delta}_{i},{p}_{i0}+{\delta}_{i}],$$

`quadstab`

computes the largest Θ > 0 such that quadratic stability holds over the parameter box$${p}_{i}\in [{p}_{i0}-\Theta {\delta}_{i},{p}_{i0}+\Theta {\delta}_{i}]$$

This "quadratic stability margin" is returned in

`tau`

and`ps`

is quadratically stable if`tau`

≥ 1.

Given the solution *Q*_{opt} of
the LMI optimization, the Lyapunov matrix *P* is
given by *P* = $${Q}_{\text{opt}}^{-1}$$.
This matrix is returned in `P`

.

Other control parameters can be accessed through `options(2)`

and `options(3)`

:

if

`options(2)=0`

(default),`quadstab`

runs in fast mode, using the least expensive sufficient conditions. Set`options(2)=1`

to use the least conservative conditions`options(3)`

is a bound on the condition number of the Lyapunov matrix*P*. The default is 10^{9}.

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