Getting Started

Constructing SISO Models

Once you have a set of differential equations that describe your plant, you can construct SISO models using simple commands in the Control System Toolbox. The following sections discuss:

• Constructing a state-space model of the DC motor
• Converting between model representations
• Creating transfer function and zero/pole/gain models

Constructing a State-Space Model of the DC Motor

Listed below are nominal values for the various parameters of a DC motor.

• R= 2.0 % Ohms
  L= 0.5 % Henrys
  Km = .015 % Torque constant
  Kb = .015 % emf constant
  Kf = 0.2 % Nms
  J= 0.02 % kg.m^2/s^2
  

Given these values, you can construct the numerical state-space representation using the ss function.

• A = [-R/L -Kb/L; Km/J -Kf/J]
  B = [1/L; 0];
  C = [0 1];
  D = [0];
  sys_dc = ss(A,B,C,D)
  

This is the output of the last command.

• a = 
                          x1           x2
             x1           -4        -0.03
             x2         0.75          -10
   
   
  b = 
                          u1
             x1            2
             x2            0
   
   
  c = 
                          x1           x2
             y1            0            1
   
   
  d = 
                          u1
             y1            0
  
  

Converting Between Model Representations

Now that you have a state-space representation of the DC motor, you can convert to other model representations, including transfer function (TF) and zero/pole/gain (ZPK) models.

Transfer Function Representation.   You can use tf to convert from the state-space representation to the transfer function. For example, use this code to convert to the transfer function representation of the DC motor.

• sys_tf = tf(sys_dc)
   
  Transfer function:
         1.5
  ------------------
  s^2 + 14 s + 40.02
  

Zero/Pole/Gain Representation.   Similarly, the zpk function converts from state-space or transfer function representations to the zero/pole/gain format. Use this code to convert from the state-space representation to the zero/pole/gain form for the DC motor.

• sys_zpk = zpk(sys_dc)
   
  Zero/pole/gain:
          1.5
  -------------------
  (s+4.004) (s+9.996)
  

Note    The state-space representation is best suited for numerical computations. For highest accuracy, convert to state space prior to combining models and avoid the transfer function and zero/pole/gain representations, except for model specification and inspection. See Reliable Computations in the online Control System Toolbox documentation for more information on numerical issues.

Constructing Transfer Function and Zero/Pole/Gain Models

In the DC motor example, the state-space approach produced a set of matrices that represents the model. If you choose a different approach, you can construct the corresponding models using tf, zpk, ss, or frd.

• sys = tf(num,den)               % Transfer function
  sys = zpk(z,p,k)                % Zero/pole/gain
  sys = ss(a,b,c,d)               % State-space
  sys = frd(response,frequencies) % Frequency response data
  

For example, if you want to create the transfer function of the DC motor directly, use these commands.

• s = tf('s');
  sys_tf = 1.5/(s^2+14*s+40.02)
  

The Control System Toolbox builds this transfer function.

• Transfer function:
          1.5
  --------------------
  s^2 + 14 s + 40.02
  

Alternatively, you can create the transfer function by specifying the numerator and denominator with this code.

• sys_tf = tf(1.5,[1 14 40.02])
   
  Transfer function:
         1.5
  ------------------
  s^2 + 14 s + 40.02
  

To build the zero/pole/gain model, use this command.

• sys_zpk = zpk([],[-9.996 -4.004], 1.5)
  

This is the resulting zero/pole/gain representation.

• Zero/pole/gain:
          1.5
  -------------------
  (s+9.996) (s+4.004)
  

SISO Example: the DC Motor

Discrete Time Systems