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


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


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


given a system in factored transfer function form.


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.


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


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


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


C = 1×2

   -0.0100   -1.0000

D = 1


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.

See Also

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Introduced before R2006a