zp2ss

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

Syntax

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

Description

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

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

such that it is equivalent to a system in factored transfer function form

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

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.

Examples

State-Space Representation of Mass-Spring System

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

$$\underset{}{\overset{¨}{w}}+0.01\underset{}{\overset{\dot{}}{w}}+w=u(t).$$

The measurable quantity is the acceleration, $$y=\underset{}{\overset{¨}{w}}$$, and $$u(t)$$ is the driving force. In Laplace space, the system is represented by

$$Y(s)=\frac{s^{2}U(s)}{s^2+0.01s+1}.$$

The system has unit gain, a double zero at $$s=0$$, and two complex-conjugate poles.

z = [0 0];
p = roots([1 0.01 1])

p = 2×1 complex

  -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 = 2×2

   -0.0100   -1.0000
    1.0000         0

B = 2×1

     1
     0

C = 1×2

   -0.0100   -1.0000

D = 
1

Input Arguments

z — Zeros

vector

Zeros of the system, specified as a vector. The zeros must be real or come in complex conjugate pairs.

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

Example: [1 (1+1j)/2 (1-1j)/2]'

Data Types: double
Complex Number Support: Yes

p — Poles

vector

Poles of the system, specified as a vector. The poles must be real or come in complex conjugate pairs.

Example: [1 (1+1j)/2 (1-1j)/2]'

Data Types: double
Complex Number Support: Yes

k — Scalar gain

scalar

Scalar gain of the system, specified as a scalar.

Data Types: double

Output Arguments

A — State matrix

matrix

State matrix, returned as a matrix. If the system 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, returned as a matrix. If the system is described by n state variables, then B is n-by-1.

Data Types: single | double

C — State-to-output matrix

matrix

State-to-output matrix, returned as a matrix. If the system has 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, returned as a matrix. If the system has q outputs, then D is q-by-1.

Data Types: single | double

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.

Extended Capabilities

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

Version History

Introduced before R2006a