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ss2tf() - Signal Processing

```example[b,a] =
ss2tf(A,B,C,D) converts
a state-space representation of a system into an equivalent transfer
function. ss2tf returns the Laplace-transform transfer
function for continuous-time systems and the Z-transform transfer
function for discrete-time systems.
example[b,a]
= ss2tf(A,B,C,D,ni) returns
the transfer function that results when the nith
input of a system with multiple inputs is excited by a unit impulse.```

Syntax

```[b,a] =
ss2tf(A,B,C,D) example[b,a]
= ss2tf(A,B,C,D,ni) example```

Example

```Mass-Spring SystemOpen This Example
A one-dimensional discrete-time oscillating system consists of a unit mass,
, attached to a wall by a spring of unit elastic constant. A sensor samples the acceleration,
, of the mass at
Hz.

Generate 50 time samples. Define the sampling interval
.Fs = 5;
dt = 1/Fs;
N = 50;
t = dt*(0:N-1);
The oscillator can be described by the state-space equations
where
is the state vector,
and
are respectively the position and velocity of the mass, and the matrices
A = [cos(dt) sin(dt);-sin(dt) cos(dt)];
B = [1-cos(dt);sin(dt)];
C = [-1 0];
D = 1;
The system is excited with a unit impulse in the positive direction. Use the state-space model to compute the time evolution of the system starting from an all-zero initial state.u = [1 zeros(1,N-1)];

x = [0;0];
for k = 1:N
y(k) = C*x + D*u(k);
x = A*x + B*u(k);
end
Plot the acceleration of the mass as a function of time.stem(t,y,'v','filled')
xlabel('t')

Compute the time-dependent acceleration using the transfer function H(z) to filter the input. Plot the result.[b,a] = ss2tf(A,B,C,D);
yt = filter(b,a,u);
stem(t,yt,'d','filled')
xlabel('t')

The transfer function of the system has an analytic expression:
Use the expression to filter the input. 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','filled')
xlabel('t')

The result is the same in all three cases.Two-Body OscillatorOpen This Example
An ideal one-dimensional oscillating system consists of two unit masses,
and
, confined between two walls. Each mass is attached to the nearest wall by a spring of unit elastic constant. Another such spring connects the two masses. Sensors sample
and
, the accelerations of the masses, at
Hz.

Specify a total measurement time of 16 s. Define the sampling interval
.Fs = 16;
dt = 1/Fs;
N = 257;
t = dt*(0:N-1);
The system can be described by the state-space model
where
is the state vector and
and
are respectively the location and the velocity of the
-th mass. The input vector
and the output vector
. The state-space matrices are
the continuous-time state-space matrices are
and
denotes an identity matrix of the appropriate size.Ac = [0 1 0 0;-2 0 1 0;0 0 0 1;1 0 -2 0];
A = expm(Ac*dt);
Bc = [0 0;1 0;0 0;0 1];
B = Ac\(A-eye(4))*Bc;
C = [-2 0 1 0;1 0 -2 0];
D = eye(2);
The first mass,
, receives a unit impulse in the positive direction.ux = [1 zeros(1,N-1)];
u0 = zeros(1,N);
u = [ux;u0];
Use the model to compute the time evolution of the system starting from an all-zero initial state.x = [0;0;0;0];
for k = 1:N
y(:,k) = C*x + D*u(:,k);
x = A*x + B*u(:,k);
end
Plot the accelerations of the two masses as functions of time.stem(t,y','.')
xlabel('t')
legend('a_1','a_2')
title('Mass 1 Excited')
grid

Convert the system to its transfer function representation. Find the response of the system to a positive unit impulse excitation on the first mass.[b1,a1] = ss2tf(A,B,C,D,1);
y1u1 = filter(b1(1,:),a1,ux);
y1u2 = filter(b1(2,:),a1,ux);
Plot the result. The transfer function gives the same response as the state-space model.stem(t,[y1u1;y1u2]','.')
xlabel('t')
legend('a_1','a_2')
title('Mass 1 Excited')
grid

The system is reset to its initial configuration. Now the other mass,
, receives a unit impulse in the positive direction. Compute the time evolution of the system.u = [u0;ux];

x = [0;0;0;0];
for k = 1:N
y(:,k) = C*x + D*u(:,k);
x = A*x + B*u(:,k);
end
Plot the accelerations. The responses of the individual masses are switched.stem(t,y','.')
xlabel('t')
legend('a_1','a_2')
title('Mass 2 Excited')
grid

Find the response of the system to a positive unit impulse excitation on the second mass.[b2,a2] = ss2tf(A,B,C,D,2);
y2u1 = filter(b2(1,:),a2,ux);
y2u2 = filter(b2(2,:),a2,ux);
Plot the result. The transfer function gives the same response as the state-space model.stem(t,[y2u1;y2u2]','.')
xlabel('t')
legend('a_1','a_2')
title('Mass 2 Excited')
grid```