Linear Time Invariant (LTI) systems in Matlab

This page will discuss only single input-single output (SISO) systems, though Matlab can handle systems with multiple inputs and multiple outputs (MIMO).   If you are interested in seeing how to apply the commands listed below to MIMO systems, use Matlabs "help" command.

Though Matlab has 3 ways to define systems (Transfer functions, State Variables, and Pole-Zero), this document on deals with Transfer Function and State Variable Representations.

Defining a system with a Transfer Function

• tf
  » Sys1=tf(num,den)
  Create a new system by defining the transfer function  numerator and denominator polynomial.

Consider the transfer function:

To define a system Sys1 in Matlab you could type:

» Sys1=tf([2 2],[1 3 2])
Transfer function:
2 s + 2
-------------
s^2 + 3 s + 2

This defines an "system object" which has certain properties.  To list the properties

» get(Sys1)
num = {[0 2 2]}
den = {[1 3 2]}
Variable = 's'
Ts = 0
Td = 0
InputName = {''}
OutputName = {''}
Notes = {}
UserData = []

Note that the numerator and denominator polynomials are in braces "{}".  This is to handle the MIMO case.  If you want to get the system numerator, type:

» n=Sys1.num{1}
n =
0 2 2

and likewise for the denominator.    Another way to do this is with the get function (which will return a property):

» n1=Sys1.num{1};
» n=n1{1};

You can also use the command "tfdata" to get both numerator and denominator:

» [n,d]=tfdata(Sys1,'v');

The argument 'v' is important to get the actual numerator and denominator for a SISO system.  The only other object properties that you might want to use are the InputName and OutputName.   If you assign these properties, the input name and output name will be printed on all graphs, and whenever you display the transfer function.  To name the input and   output and then display the transfer function:

» Sys1.InputName='MyInput';
» Sys1.OutputName='MyOutput';
» Sys1
Transfer function from input "MyInput" to output "MyOutput":
2 s + 2
-------------
s^2 + 3 s + 2

Defining a system with State Variables

• ss
  » Sys2=ss(a,b,c,d)
  Create a new system by defining the state space representation.

Likewise a system can be defined using the state-variable representation:

» a=[0 1; -2 -3]; b=[0 1]'; c=[1 0]; d=0;
» Sys2=ss(a,b,c,d);

To view the object properties:

» get(Sys2)
a = [2x2 double]
b = [2x1 double]
c = [1 0]
d = 0
e = []
StateName = {2x1 cell}
Ts = 0
Td = 0
InputName = {''}
OutputName = {''}
Notes = {}
UserData = []

The state variables and input and output can be named:

» Sys2.InputName='S2 In';
» Sys2.OutputName='S2 Out';
» Sys2.StateName={'S2 State1' 'S2 State2'};

These names appear when viewing the object:

» Sys2
a = 
             S2 State1      S2 State2
S2 State1            0              1
S2 State2           -2             -3

b = 
                  S2 In
S2 State1             0
S2 State2             1

c = 
             S2 State1      S2 State2
S2 Out               1              0
  
d = 
             S2 In
S2 Out           0

Continuous-time system.

As with the systems defined in terms of the transfer function it is possible to get the state space properties.  To get the amatrix

» amatrix=Sys2.a
amatrix =
0  1
-2 -3

or to get all matrices at one time

» [amatrix,bmatrix,cmatrix,dmatrix]=ssdata(Sys2);

Converting Between LTI Representations

Recall, from above, that Sys1 was defined in terms of the transfer function and Sys2 was defined in terms of its state-space representation:

• Transfer function to state space.
  » Sys3=ss(Sys1);
  Creates a new state-space system, Sys3,  that is equivalent to the transfer function system Sys1.
  See also "tf2ss" and "ssdata".

• State Space to transfer function.
  » Sys4=tf(Sys2);
  Creates a new transfer function system, Sys4,  that is equivalent to the state space system Sys2.
  See also "ss2tf" and "tfdata".

• State Space to State Space.
  » Sys5=ss2ss(Sys2,T);
  Matlab performs a similarity transform to go from one state space representation to another, using the matrix "T".  Note that the transform is different from the one described in most textbooks, such that T^-1=P.
  

• Zero-pole representation
  If you want to use the zero-pole representation use the "zpk" function (and "zpk", "zp2tf", "tf2zp", "zp2tf" and "tf2zp").
  

System Characterization and Responses
    In the following, "sys" is an LTI SISO system.

• Impulse Response
  » impulse(sys);
  Plots the impulse response of "sys".  If you have given names to the inputs and outputs, these are displayed on the graph.
  » impulse(sys,t);
  Calculates the impulse response at the times specified by the vector "t".
  » [y,t,x]=impulse(sys);
  Does not plot the impulse response, but returns with the values of the output, the time vector, and the state variables

• .Step Response
  » step(sys);
  Plots the step response of "sys".  If you have given names to the inputs and outputs, these are displayed on the graph.
  » step(sys,t);
  Calculates the step response at the times specified by the vector "t".
  » [y,t,x]=step(sys);
  Does not plot the step response, but returns with the values of the output, the time vector, and the state variables.

• Response to Arbitrary Inputs
  » lsim(sys,u,t);
  Plots the response of "sys" to the input u defined at times t.
  » [y,t,x]=lsim(sys,u,t);
  Does not plot the response, but returns with the values of the output, the time vector, and the state variables.  If the system is in state variable form, it is possible to include a set of initial conditions.»
  
  y=conv(h,u);
  Convolves the impulse response h, with the input u.  Note, this does a summation, not the convolution integral and does not account for the timestep used in the data.  Note also that length(y)=length(h)+length(u)-1.

  

• Response to Initial Conditions
  » initial(sys,x0);
  Plots the response of "sys" to the initial conditions defined by x0. The system must be in state variable form.
  » [y,t,x]=initial(sys,x0);
  Does not plot the response, but returns with the values of the output, the time vector, and the state variables.

• Pole Zero Plot
  » pzmap(sys);
  Plots the pole-zero map of "sys". 
  » [p,z]=pzmap(sys);
  Does not plot the map, but returns the poles and zeros of the system.

• Bode Plot
  » bode(sys);
  Plots the bode response of "sys".  If you have given names to the inputs and outputs, these are displayed on the graphs.
  » bode(sys,w);
  Calculates the step response at the frequencies specified by the vector "w".  To specify logarithmically spaced w (link).
  » [m,p,w]=bode(sys);
  Does not plot, but returns with the magnitude and phase of the output,and the frequency vector.

• LTIViewer
  » LTIView;
  Brings up the LTI viewer which is useful for comparing several different systems.   Will generate any of the plots discussed above.

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