# Documentation

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.