Create Uncertain Frequency Response Data Models

Uncertain frequency responses (ufrd) arise naturally when computing the frequency response of an uncertain state-space model (uss). They also arise when frequency response data in an frd model object is combined with an uncertain matrix (umat) such as by adding, multiplying, or concatenating.

To take the frequency response of an uncertain state-space model, use the ufrd command. Construct an uncertain state-space model.

p1 = ureal('p1',10,'pe',50); 
p2 = ureal('p2',3,'plusm',[-.5 1.2]); 
p3 = ureal('p3',0); 
A = [-p1 p2;0 -p1]; 
B = [-p2;p2+p3]; 
C = [1 0;1 1-p3]; 
D = [0;0]; 
sys = ss(A,B,C,D) 

Uncertain continuous-time state-space model with 2 outputs, 1 inputs, 2 states.
The model uncertainty consists of the following blocks:
  p1: Uncertain real, nominal = 10, variability = [-50,50]%, 2 occurrences
  p2: Uncertain real, nominal = 3, variability = [-0.5,1.2], 2 occurrences
  p3: Uncertain real, nominal = 0, variability = [-1,1], 2 occurrences

Type "sys.NominalValue" to see the nominal value and "sys.Uncertainty" to interact with the uncertain elements.

Compute the uncertain frequency response of the uncertain system. Use ufrd command with a frequency grid of 100 points. The result is an uncertain frequency response model object, a ufrd model.

sysg = ufrd(sys,logspace(-2,2,100))  

Uncertain continuous-time FRD model with 2 outputs, 1 inputs, 100 frequency points.
  p1: Uncertain real, nominal = 10, variability = [-50,50]%, 2 occurrences
  p2: Uncertain real, nominal = 3, variability = [-0.5,1.2], 2 occurrences
  p3: Uncertain real, nominal = 0, variability = [-1,1], 2 occurrences

Type "sysg.NominalValue" to see the nominal value and "sysg.Uncertainty" to interact with the uncertain elements.

Properties of ufrd Model Objects

View the properties of the model object.

get(sysg)

        Frequency: [100x1 double]
    FrequencyUnit: 'rad/TimeUnit'
     ResponseData: [2x1x100x1 umat]
     NominalValue: [2x1 frd]
      Uncertainty: [1x1 struct]
       InputDelay: 0
      OutputDelay: [2x1 double]
        InputName: {''}
        InputUnit: {''}
       InputGroup: [1x1 struct]
       OutputName: {2x1 cell}
       OutputUnit: {2x1 cell}
      OutputGroup: [1x1 struct]
            Notes: [0x1 string]
         UserData: []
             Name: ''
               Ts: 0
         TimeUnit: 'seconds'
     SamplingGrid: [1x1 struct]

The properties ResponseData and Frequency behave the same as the corresponding properties in Control System Toolbox™ frd objects, except that ResponseData is an uncertain matrix (umat). The properties InputName, OutputName, InputGroup, and OutputGroup behave in exactly the same manner as for all of the Control System Toolbox model objects such as ss, zpk, tf, and frd.

The NominalValue property is an frd object. Hence all functions you can use to analyze frd objects can also analyze ufrd objects. When you use analysis commands such as bode or step with an uncertain model, the command plots random samples of the response to give you a sense of the variation. For instance, plot sampled Bode responses of the system along with the nominal response, using a dot marker so that you can see the individual frequency points.

bode(sysg,'r.',sysg.NominalValue,'b.')

Just as with umat uncertain matrices and uss uncertain models, the Uncertainty property of the ufrd model is a structure containing the uncertain elements. In the model sysg, all uncertain elements are ureal parameters. Change the nominal value of the uncertain element p1 within sysg to 14, and plot the Bode response of the (new) nominal system.

sysg.Uncertainty.p1.NominalValue = 14

Uncertain continuous-time FRD model with 2 outputs, 1 inputs, 100 frequency points.
  p1: Uncertain real, nominal = 14, variability = [-50,50]%, 2 occurrences
  p2: Uncertain real, nominal = 3, variability = [-0.5,1.2], 2 occurrences
  p3: Uncertain real, nominal = 0, variability = [-1,1], 2 occurrences

Type "sysg.NominalValue" to see the nominal value and "sysg.Uncertainty" to interact with the uncertain elements.

bode(sysg.NominalValue)

Lifting an frd Model to a ufrd Model

A non-uncertain frequency response model is equivalent to an uncertain frequency response model with no uncertain elements. Use the ufrd command to “lift” an frd model to the ufrd class.

sys = rss(3,2,1); 
sysg = frd(sys,logspace(-2,2,100)); 
usysg = ufrd(sysg) 

Uncertain continuous-time FRD model with 2 outputs, 1 inputs, 100 frequency points, and no uncertain blocks.

Type "usysg.NominalValue" to see the nominal value and "usysg.Uncertainty" to interact with the uncertain elements.

You can also lift arrays of frd objects. See Array Management for Uncertain Objects for more information about how arrays of uncertain objects are handled.