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) x˙ = A(t)x + B(t)u

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

  whose SYSTEM matrix takes values in a fixed polytope:

  $$\underset{S(t)}{\underbrace{\left[\begin{array}{cc}A(t)+jE(t)& B(t)\\ C(t)& D(t)\end{array}\right]}}\in \text{Co}\left\{\underset{S_1}{\underbrace{\left[\begin{array}{cc}{A}_1+j{E}_1& B_1\\ C_1& D_1\end{array}\right]}},\dots ,\underset{S_k}{\underbrace{\left[\begin{array}{cc}Ak+j{E}_k& B_k\\ C_k& D_k\end{array}\right]}}\right\}$$

  where S₁, . . ., S_k are given “vertex” systems and

  $$\text{Co}\{S_1,\mathrm{...},S_k\}=\left\{{\displaystyle \sum _{i=1}^k{\alpha }_i{S}_i:{\alpha }_i\ge 0,{\displaystyle \sum _{i=1}^k{\alpha }_i=1}}\right\}$$

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

• Affine parameter-dependent systems

  E(p)x˙ = 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, . . ., p_n) of real parameters, i.e.,

  $$\begin{array}{l}\underset{S(p)}{\underbrace{\left[\begin{array}{cc}A(p)+jE(p)& B(p)\\ C(p)& D(p)\end{array}\right]}}=\\ \underset{S_0}{\underbrace{\left[\begin{array}{cc}{A}_0+j{E}_0& B_0\\ C_0& D_0\end{array}\right]}}+p1\underset{S_1}{\underbrace{\left[\begin{array}{cc}{A}_1+j{E}_1& B_1\\ C_1& D_1\end{array}\right]}}+\dots +p_n\underset{S_n}{\underbrace{\left[\begin{array}{cc}{A}_n+j{E}_n& B_n\\ C_n& D_n\end{array}\right]}}\end{array}$$

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

The argument syslist lists the SYSTEM matrices S_i 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.

Version History

Introduced before R2006a