ultidyn
Create uncertain linear time-invariant object
Syntax
H = ultidyn('Name',iosize)
H = ultidyn('Name',iosize,'Property1',Value1,'Property2',Value2,...)
Description
H = ultidyn('Name',iosize) creates an uncertain linear, time-invariant objects are used to represent unknown dynamic objects whose only known attributes are bounds on their frequency response. Uncertain linear, time-invariant objects have a name (the Name property), and an input/output size (ioSize property).
Trailing Property/Value pairs are allowed in the construction.
H = ultidyn('name',iosize,'Property1',Value1,'Property2',Value2,...)
The property Type is 'GainBounded' (default) or 'PositiveReal', and describes in what form the knowledge about the object's frequency response is specified.
If
Typeis'GainBounded', then the knowledge is an upper bound on the magnitude (i.e., absolute value), namelyabs(H)<= Boundat all frequencies. The matrix generalization of this is ∥H∥<= Bound.If
Typeis'PositiveReal'then the knowledge is a lower bound on the real part, namelyReal(H) >= Boundat all frequencies. The matrix generalization of this isH+H' >= 2*Bound
The property Bound is a real, scalar that quantifies the bound on the frequency response of the uncertain object as described above.
The property SampleStateDimension is a positive integer, defining the state
dimension of random samples of the uncertain object when sampled with
usample. The default value is 3.
The property AutoSimplify controls how expressions involving the uncertain matrix are simplified. Its default value is 'basic', which means elementary methods of simplification are applied as operations are completed. Other values for AutoSimplify are 'off', no simplification performed, and 'full' which applies model-reduction-like techniques to the uncertain object.
Use the property SampleMaxFrequency to limit the natural frequency for sampling. Randomly sampled uncertain dynamics are no faster than the specified value. The default value is Inf (no limit).
To model frequency-dependent uncertainty levels, multiply the ultidyn object by a suitable shaping filter. For example, for a ultidyn object dH, the following commands specify an uncertainty bound that increases from 0.1 at low frequencies to 10 at high frequencies.
W = tf([1 .1],[.1 1]); dH = W*dH;
Examples
MIMO Uncertain Dynamics
Create a ultidyn object with internal name 'H', norm bounded by 7, with three inputs and two outputs.
H = ultidyn('H',[2 3],'Bound',7)
H = Uncertain LTI dynamics "H" with 2 outputs, 3 inputs, and gain less than 7.
Typically, when you use uncertain dynamics, you apply a weighting function to emphasize the uncertain contribtion in a certain bandwidth. For instance, suppose that the behavior of your system is modestly uncertain (say 10%) at low frequencies, while the high-frequency behavior beyond 20 rad/s is not accurately modeled. Use makeweight to create a shaping filter that captures this behavior.
W = makeweight(.1,20,50); bodemag(W)
Apply the weighting filter at the block outputs. Examine samples of the unmodeled dynamics.
Hw = blkdiag(W,W)*H; bodemag(Hw)
Nyquist Plot of Uncertain Dynamics
Create a scalar ultidyn object with an internal name 'B', whose frequency response has a real part greater than 2.5.
B = ultidyn('B',[1 1],'Type','PositiveReal','Bound',2.5)
B = Uncertain LTI dynamics "B" with 1 outputs, 1 inputs, and positive real bound of 2.5.
Change the SampleStateDimension to 5, and plot the Nyquist plot of 30 random samples.
B.SampleStateDimension = 5; nyquist(usample(B,30))
