# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

Quadratic stability of polytopic or affine parameter-dependent systems

## Syntax

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

## Description

For affine parameter-dependent systems

E(p) = A(p)x, p(t) = (p1(t), . . ., pn(t))

or polytopic systems

E(t) = A(t)x, (A, E) ∊ Co{(A1, E1), . . ., (An, En)},

`quadstab` seeks a fixed Lyapunov function V(x) = xTPx with P > 0 that establishes quadratic stability. The affine or polytopic model is described by `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 problem

Minimize τ over Q = QT such that

ATQE + EQAT < τI 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 pi varies in the interval

`${p}_{i}\in \left[{p}_{i}{}_{0}-{\delta }_{i},{p}_{i0}+{\delta }_{i}\right],$`

`quadstab` computes the largest Θ > 0 such that quadratic stability holds over the parameter box

`${p}_{i}\in \left[{p}_{i0}-\Theta {\delta }_{i},{p}_{i0}+\Theta {\delta }_{i}\right]$`

This “quadratic stability margin” is returned in `tau` and `ps` is quadratically stable if `tau` ≥ 1.

Given the solution Qopt 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 109.