ss2zp

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

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Syntax

[z,p,k] = ss2zp(A,B,C,D)

[z,p,k] = ss2zp(A,B,C,D,ni)

Description

[z,p,k] = ss2zp(A,B,C,D) converts a state-space representation

$\begin{array}{l} \dot{x} = A x + B u \\ y = C x + D u \end{array}$

of a given continuous-time or discrete-time system to an equivalent zero-pole-gain representation

$H (s) = \frac{Z (s)}{P (s)} = k \frac{(s - z_{1}) (s - z_{2}) \dots (s - z_{n})}{(s - p_{1}) (s - p_{2}) \dots (s - p_{n})}$

whose zeros, poles, and gains represent the transfer function in factored form.

[z,p,k] = ss2zp(A,B,C,D,ni) indicates that the system has multiple inputs and that the nith input has been excited by a unit impulse.

Examples

Zeros, Poles, and Gain of a Discrete-Time System

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Consider a discrete-time system defined by the transfer function

$H (z) = \frac{2 + 3 z^{- 1}}{1 + 0.4 z^{- 1} + z^{- 2}} .$

Determine its zeros, poles, and gain directly from the transfer function. Pad the numerator with zeros so it has the same length as the denominator.

b = [2 3 0];
a = [1 0.4 1];
[z,p,k] = tf2zp(b,a)

p = 2×1 complex

  -0.2000 + 0.9798i
  -0.2000 - 0.9798i

k = 
2

Express the system in state-space form and determine the zeros, poles, and gain using ss2zp.

[A,B,C,D] = tf2ss(b,a);
[z,p,k] = ss2zp(A,B,C,D,1)

z = 2×1

   -1.5000
   -0.0000

p = 2×1 complex

  -0.2000 + 0.9798i
  -0.2000 - 0.9798i

k = 
2

Input Arguments

`A` — State matrix
matrix

State matrix. If the system has r inputs and q outputs and is described by n state variables, then A is n-by-n.

Data Types: single | double

`B` — Input-to-state matrix
matrix

Input-to-state matrix. If the system has r inputs and q outputs and is described by n state variables, then B is n-by-r.

Data Types: single | double

`C` — State-to-output matrix
matrix

Input-to-state matrix. If the system has r inputs and q outputs and is described by n state variables, then C is q-by-n.

Data Types: single | double

`D` — Feedthrough matrix
matrix

Feedthrough matrix. If the system has r inputs and q outputs and is described by n state variables, then D is q-by-r.

Data Types: single | double

`ni` — Input index
`1` (default) | integer scalar

Input index, specified as an integer scalar. If the system has r inputs, use ss2zp with a trailing argument ni = 1, …, r to compute the response to a unit impulse applied to the nith input. Specifying this argument causes ss2zp to use the nith columns of B and D.

Data Types: single | double

Output Arguments

`z` — Zeros
matrix

Zeros of the system, returned as a matrix. z contains the numerator zeros in its columns. z has as many columns as there are outputs (rows in C).

`p` — Poles
column vector

Poles of the system, returned as a column vector. p contains the pole locations of the denominator coefficients of the transfer function.

`k` — Gains
column vector

Gains of the system, returned as a column vector. k contains the gains for each numerator transfer function.

Algorithms

ss2zp finds the poles from the eigenvalues of the A array. The zeros are the finite solutions to a generalized eigenvalue problem:

z = eig([A B;C D],diag([ones(1,n) 0]);

In many situations, this algorithm produces spurious large, but finite, zeros. ss2zp interprets these large zeros as infinite.

ss2zp finds the gains by solving for the first nonzero Markov parameters.

References

[1] Laub, A. J., and B. C. Moore. "Calculation of Transmission Zeros Using QZ Techniques." Automatica. Vol. 14, 1978, p. 557.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Outputs z and p are always complex.
The order of outputs, z and p, might be different in MATLAB^® and the generated code.

Version History

Introduced before R2006a

See Also

sos2zp | ss2sos | ss2tf | tf2zp | tf2zpk | zp2ss