ss2zp

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

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

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

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

$$H(s)=\frac{Z(s)}{P(s)}=k\frac{(s-z_1)(s-z_2)\cdots (s-z_n)}{(s-p_1)(s-p_2)\cdots (s-p_n)}$$

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

Consider a discrete-time system defined by the transfer function

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

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)

z = 2×1

         0
   -1.5000

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