tf2ss

Convert transfer function filter parameters to state-space form

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Syntax

[A,B,C,D] = tf2ss(b,a)

Description

tf2ss converts the parameters of a transfer function representation of a given system to those of an equivalent state-space representation.

[A,B,C,D] = tf2ss(b,a) returns the A, B, C, and D matrices of a state space representation for the single-input transfer function

H(s)=B(s)A(s)=b1sn−1+⋯+bn−1s+bna1sm−1+⋯+am−1s+am=C(sI−A)−1B+D

in controller canonical form

x˙=Ax+Buy=Cx+Du

The input vector a contains the denominator coefficients in descending powers of s. The rows of the matrix b contain the vectors of numerator coefficients (each row corresponds to an output). In the discrete-time case, you must supply b and a to correspond to the numerator and denominator polynomials with coefficients in descending powers of z.

For discrete-time systems you must make b have the same number of columns as the length of a. You can do this by padding each numerator represented in b (and possibly the denominator represented in the vector a) with trailing zeros. You can use the function eqtflength to accomplish this if b and a are vectors of unequal lengths.

The tf2ss function is part of the standard MATLAB® language.

    Note   There is disagreement in the literature on naming conventions for the canonical forms. It is easy, however, to generate similarity transformations that convert these results to other forms.

Examples

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Open Script

Consider the system described by the transfer function

$$
H(s)=\frac{\left[\begin{array}{c}2s+3\\s^2+2s+1\end{array}\right]}
 {s^2+0.4s+1}.
$$

Convert it to state-space form using tf2ss.

b = [0 2 3; 1 2 1];
a = [1 0.4 1];
[A,B,C,D] = tf2ss(b,a)

A =

   -0.4000   -1.0000
    1.0000         0


B =

     1
     0


C =

    2.0000    3.0000
    1.6000         0


D =

     0
     1
Open Script

A one-dimensional discrete-time oscillating system consists of a unit mass, $m$, attached to a wall by a spring of unit elastic constant. A sensor samples the acceleration, $a$, of the mass at $F_s=5$ Hz.

Generate 50 time samples. Define the sampling interval $\Delta t=1/F_s$.

Fs = 5;
dt = 1/Fs;
N = 50;
t = dt*(0:N-1);
u = [1 zeros(1,N-1)];

The transfer function of the system has an analytic expression:

$$H(z)={{1-z^{-1}(1+\cos\Delta t)+z^{-2}\cos\Delta
t}\over{1-2z^{-1}\cos\Delta t+z^{-2}}}.$$

The system is excited with a unit impulse in the positive direction. Compute the time evolution of the system using the transfer function. Plot the response.

bf = [1 -(1+cos(dt)) cos(dt)];
af = [1 -2*cos(dt) 1];
yf = filter(bf,af,u);

stem(t,yf,'o')
xlabel('t')

Find the state-space representation of the system. Compute the time evolution starting from an all-zero initial state. Compare it to the transfer-function prediction.

[A,B,C,D] = tf2ss(bf,af);

x = [0;0];
for k = 1:N
    y(k) = C*x + D*u(k);
    x = A*x + B*u(k);
end

hold on
stem(t,y,'*')
hold off
legend('tf','ss')

See Also

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Introduced before R2006a

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