tf2ss

Convert transfer function filter parameters to state-space form

Syntax

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

Description

[A,B,C,D] = tf2ss(b,a) converts a continuous-time or discrete-time single-input transfer function into an equivalent state-space representation.

Examples

Convert Transfer Function to State-Space Form

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 = 2×2

   -0.4000   -1.0000
    1.0000         0

B = 2×1

     1
     0

C = 2×2

    2.0000    3.0000
    1.6000         0

D = 2×1

     0
     1

Mass-Spring System

A one-dimensional discrete-time oscillating system consists of a unit mass, $\mathit{m}$, attached to a wall by a spring of unit elastic constant. A sensor samples the acceleration, $\mathit{a}$, of the mass at ${\mathit{F}}_{\mathrm{s}}=5$ Hz.

Generate 50 time samples. Define the sampling interval $\Delta \mathit{t}=1/{\mathit{F}}_{\mathrm{s}}$.

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:

$\mathit{H}\left(\mathit{z}\right)=\frac{1-{\mathit{z}}^{-1}\left(1+\cos \Delta \mathit{t}\right)+{\mathit{z}}^{-2}\cos \Delta \mathit{t}}{1-2{\mathit{z}}^{-1}\cos \Delta \mathit{t}+{\mathit{z}}^{-2}}$.

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')

Input Arguments

b — Transfer function numerator coefficients

vector | matrix

Transfer function numerator coefficients, specified as a vector or matrix. If b is a matrix, then each row of b corresponds to an output of the system.

• For discrete-time systems, b contains the coefficients in descending powers of z.

• For continuous-time systems, b contains the coefficients in descending powers of s.

For discrete-time systems, b must have a number of columns equal to the length of a. If the numbers differ, make them equal by padding zeros. You can use the function eqtflength to accomplish this.

a — Transfer function denominator coefficients

vector

Transfer function denominator coefficients, specified as a vector.

• For discrete-time systems, a contains the coefficients in descending powers of z.

• For continuous-time systems, a contains the coefficients in descending powers of s.

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

More About

Transfer Function

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

• For discrete-time systems, the state-space matrices relate the state vector x, the input u, and the output y:

  $$\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right).\end{array}$$

  The transfer function is the Z-transform of the system’s impulse response. It can be expressed in terms of the state-space matrices as

  $$H\left(z\right)=C{\left(zI-A\right)}^{-1}B+D.$$

• For continuous-time systems, the state-space matrices relate the state vector x, the input u, and the output y:

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

  The transfer function is the Laplace transform of the system’s impulse response. It can be expressed in terms of the state-space matrices as

  $$H(s)=\frac{B(s)}{A(s)}=\frac{b_1{s}^{n-1}+\cdots +b_{n-1}s+b_n}{a_1{s}^{m-1}+\cdots +a_{m-1}s+a_m}=C{\left(sI-A\right)}^{-1}B+D.$$

Extended Capabilities

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