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# ss2ss

State coordinate transformation for state-space model

## Syntax

sysT = ss2ss(sys,T)

## Description

Given a state-space model sys with equations

$\begin{array}{l}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}$

or the innovations form used by the identified state-space (IDSS) models:

$\begin{array}{c}\frac{dx}{dt}=Ax+Bu+Ke\\ y=Cx+Du+e\end{array}$

(or their discrete-time counterpart), ss2ss performs the similarity transformation $\overline{x}=Tx$ on the state vector x and produces the equivalent state-space model sysT with equations.

$\begin{array}{l}\stackrel{˙}{\overline{x}}=TA{T}^{-1}\overline{x}+TBu\\ y=C{T}^{-1}\overline{x}+Du\end{array}$

or, in the case of an IDSS model:

$\begin{array}{l}\stackrel{˙}{\overline{x}}=TA{T}^{-}{}^{1}\overline{x}+TBu+TKe\\ y=C{T}^{-}{}^{1}\overline{x}+Du+e\end{array}$

(IDSS models require System Identification Toolbox™ software.)

sysT = ss2ss(sys,T) returns the transformed state-space model sysT given sys and the state coordinate transformation T. The model sys must be in state-space form and the matrix T must be invertible. ss2ss is applicable to both continuous- and discrete-time models.

## Examples

Perform a similarity transform to improve the conditioning of the A matrix.

```T = balance(sys.a)
sysb = ss2ss(sys,inv(T))
```