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# zpk

Zero pole gain system representation

### Syntax

```S = zpk(Z,P,K,dt)
S = zpk(z,p,k,dt)
S = zpk(sys)```

### Arguments

Z

a m by n cell of real or complex vectors, Z{i,j} is the transmission zeros of the transfer from the the jth intput to the ith output.

P

a m by n cell of real or complex vectors, P{i,j} is the poles of the transfer from the the jth intput to the ith output.

K

a m by n matrix of real numbers, K(i,j) is the gain of the transfer from the the jth intput to the ith output.

z

a real or complex vector, the transmission zeros of the siso transfer function.

p

a real or complex vector, the poles of the siso transfer function.

k

a real scalar, the gain of the siso transfer function.

dt

a character string with possible values "c" or "d", [] or a real positive scalar, the system time domain (see syslin).

sys

A linear dynamical system in transfer function or state spece representation (see syslin).

S

a mlist with the fields Z , P, K and dt.

Z

a m by n cell array of real or complex vectors, S.Z{i,j} contains the zeros of the transfer from the the jth intput to the ith output

P

a m by n cell array of real or complex vectors, S.P{i,j} contains the poles of the transfer from the the jth intput to the ith output

K

a m by n matrix of real numbers, S.K(i,j) is the gain of the transfer from the the jth intput to the ith output. output.

dt

a positive scalar or "c" or "d" the time domain

### Description

`S=zpk(Z,P,K,dt)` forms the multi-input, multi-output zero pole gain system representation given the cell arrays of the transmission zeros,poles and gain.

`S=zpk(z,p,k,dt)` forms the single-input, single output zero pole gain system representation given the vecteors of the transmission zeros and poles and the scalar gain.

`S=zpk(sys)` converts the system representation into a zero-pole-gain representation.

The poles and zeros of each transfer function are sorted in decreasing order of the real part.

Most functions and operations than can act on state-space or rational transfer function representations can be also applied to zero-pole-gain representations.

### Examples

```//Form system from zeros, poles and gain
//SISO case
z11=[1 -0.5];p11=[-3+2*%i -3-2*%i  -2];k11=1;
S11=zpk(z11,p11,k11,"c")

//MIMO case
z21=0.3;p21=[-3+2*%i -3-2*%i];k21=1.5;
S21=zpk(z21,p21,k21,"c")
S=zpk({z11 [];z21 1},{p11,0;p21 -3},[k11 1;k21 1],"c")

//system representation conversion
h=syslin("c",5*(%s^2+2*%s+1)/(%s^2-4))
sh=zpk(h)

//operations with zpk representations
S(1,:)

S'

S(1,1)=sh

sh*S11

sh./S11```

### See Also

• tf2zp — SIMO transfer function to zero pole gain representation
• zpk2tf — Zero pole gain to transfer function
• zpk2ss — Zero pole gain to state space

### History

 Version Description 6.0 Function added.

### Comments

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