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# zp2ss

Convert zero-pole-gain filter parameters to state-space form

## Syntax

`[A,B,C,D] = zp2ss(z,p,k)`

## Description

`zp2ss` converts a zero-pole-gain representation of a given system to an equivalent state-space representation.

`[A,B,C,D] = zp2ss(z,p,k)` finds a single input, multiple output, state-space representation

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

given a system in factored transfer function form.

`$H\left(s\right)=\frac{Z\left(s\right)}{P\left(s\right)}=k\frac{\left(s-{z}_{1}\right)\left(s-{z}_{2}\right)\cdots \left(s-{z}_{n}\right)}{\left(s-{p}_{1}\right)\left(s-{p}_{2}\right)\cdots \left(s-{p}_{n}\right)}$`

Column vector `p` specifies the pole locations, and matrix `z` the zero locations with as many columns as there are outputs. The gains for each numerator transfer function are in vector `k`. The `A`, `B`, `C`, and `D` matrices are returned in controller canonical form.

`Inf` values may be used as place holders in `z` if some columns have fewer zeros than others.

## Examples

collapse all

Generate the state-space representation of a damped mass-spring system that obeys the differential equation

``` ```

The measurable quantity is the acceleration, , and is the driving force. In Laplace space, the system is represented by

``` ```

The system has unit gain, a double zero at , and two complex-conjugate poles.

```z = [0 0]; p = roots([1 0.01 1])```
```p = -0.0050 + 1.0000i -0.0050 - 1.0000i ```
`k = 1;`

Use `zp2ss` to find the state-space matrices.

`[A,B,C,D] = zp2ss(z,p,k)`
```A = -0.0100 -1.0000 1.0000 0 ```
```B = 1 0 ```
```C = -0.0100 -1.0000 ```
```D = 1 ```

## Algorithms

`zp2ss`, for single-input systems, groups complex pairs together into two-by-two blocks down the diagonal of the `A` matrix. This requires the zeros and poles to be real or complex conjugate pairs.