# Documentation

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# psys

Specify polytopic or parameter-dependent linear systems

## Syntax

```pols = psys(syslist)
affs = psys(pv,syslist)
```

## Description

`psys ` specifies state-space models where the state-space matrices can be uncertain, time-varying, or parameter-dependent.

`psys` supports two types of uncertain state-space models:

• Polytopic systems

E(t) = A(t)x + B(t)u

y = C(t)x + D(t)u

whose `SYSTEM` matrix takes values in a fixed polytope:

`$\underset{S\left(t\right)}{\underbrace{\left[\begin{array}{cc}A\left(t\right)+jE\left(t\right)& B\left(t\right)\\ C\left(t\right)& D\left(t\right)\end{array}\right]}}\in \text{Co}\left\{\underset{{S}_{1}}{\underbrace{\left[\begin{array}{cc}{A}_{1}+j{E}_{1}& {B}_{1}\\ {C}_{1}& {D}_{1}\end{array}\right]}},\dots ,\underset{{S}_{k}}{\underbrace{\left[\begin{array}{cc}Ak+j{E}_{k}& {B}_{k}\\ {C}_{k}& {D}_{k}\end{array}\right]}}\right\}$`

where S1, . . ., Sk are given "vertex" systems and

`$\text{Co}\left\{{S}_{1},...,{S}_{k}\right\}=\left\{\sum _{i=1}^{k}{\alpha }_{i}{S}_{i}:{\alpha }_{i}\ge 0,\sum _{i=1}^{k}{\alpha }_{i}=1\right\}$`

denotes the convex hull of S1, . . ., Sk (polytope of matrices with vertices S1, . . ., Sk)

• Affine parameter-dependent systems

E(p) = A(p)x + B(p)u

y = C(p)x + D(p)u

where A(· ); B(· ), . . ., E(· ) are fixed affine functions of some vector p = (p1, . . ., pn) of real parameters, i.e.,

`$\begin{array}{l}\underset{S\left(p\right)}{\underbrace{\left[\begin{array}{cc}A\left(p\right)+jE\left(p\right)& B\left(p\right)\\ C\left(p\right)& D\left(p\right)\end{array}\right]}}=\\ \underset{{S}_{0}}{\underbrace{\left[\begin{array}{cc}{A}_{0}+j{E}_{0}& {B}_{0}\\ {C}_{0}& {D}_{0}\end{array}\right]}}+p1\underset{{S}_{1}}{\underbrace{\left[\begin{array}{cc}{A}_{1}+j{E}_{1}& {B}_{1}\\ {C}_{1}& {D}_{1}\end{array}\right]}}+\dots +{p}_{n}\underset{{S}_{n}}{\underbrace{\left[\begin{array}{cc}{A}_{n}+j{E}_{n}& {B}_{n}\\ {C}_{n}& {D}_{n}\end{array}\right]}}\end{array}$`

where S0, S1, . . ., Sn are given `SYSTEM` matrices. The parameters pi can be time-varying or constant but uncertain.

The argument `syslist` lists the `SYSTEM` matrices Si characterizing the polytopic value set or parameter dependence. In addition, the description `pv` of the parameter vector (range of values and rate of variation) is required for affine parameter- dependent models (see `pvec` for details). Thus, a polytopic model with vertex systems S1, . . ., S4 is created by

```pols = psys([s1,s2,s3,s4]) ```

while an affine parameter-dependent model with 4 real parameters is defined by

```affs = psys(pv,[s0,s1,s2,s3,s4]) ```

The output is a structured matrix storing all the relevant information.