Uncertain LTI Dynamics Elements

Uncertain linear, time-invariant objects, ultidyn, are used to represent unknown linear, time-invariant dynamics, whose only known attributes are bounds on their frequency response.

Create Uncertain LTI Dynamics

You can create a 1-by-1 (scalar) positive-real uncertain linear dynamics element, whose frequency response always has real part greater than -0.5. Set the SampleStateDimension property to 5. Plot a Nyquist plot of 30 instances of the element.

g = ultidyn('g',[1 1],'Type','Positivereal','Bound',-0.5); 
g.SampleStateDimension = 5;

nyquist(usample(g,30)) 
xlim([-2 10]) 
ylim([-6 6]);

Properties of ultidyn Elements

Uncertain linear, time-invariant objects have an internal name (the Name property), and are created by specifying their size (number of outputs and number of inputs).

The property Type specifies whether the known attributes about the frequency response are related to gain or phase. The property Type may be 'GainBounded' or 'PositiveReal'. The default value is 'GainBounded'.

The property Bound is a single number, which along with Type, completely specifies what is known about the uncertain frequency response. Specifically, if Δ is an ultidyn element, and if γ denotes the value of the Bound property, then the element represents the set of all stable, linear, time-invariant systems whose frequency response satisfies certain conditions:

If Type is 'GainBounded', $$\dot{\overline{\sigma }}\left[\Delta \left(\omega \right)\right]\le \gamma$$ for all frequencies. When Type is 'GainBounded', the default value for Bound (i.e., γ) is 1. The NominalValue of Δ is always the 0-matrix.

If Type is 'PositiveReal', Δ(ω) + Δ^*(ω) ≥ 2γ· for all frequencies. When Type is 'PositiveReal', the default value for Bound (i.e., γ) is 0. The NominalValue is always (γ + 1 +2|γ|)I.

All properties of a ultidyn are accessible with get and set (although the NominalValue is determined from Type and Bound, and not accessible with set). The properties are

Properties	Meaning	Class
Name	Internal Name	char
NominalValue	Nominal value of element	See above
Type	'GainBounded' |'PositiveReal'	char
Bound	Norm bound or minimum real	scalar double
SampleStateDimension	State-space dimension of random samples of this uncertain element	scalar double
SampleMaxFrequency	Maximum natural frequency for random sampling	scalar double
AutoSimplify	'off' | {'basic'} |'full'	char

The SampleStateDim property specifies the state dimension of random samples of the element when using usample. The default value is 1. The AutoSimplify property serves the same function as in the uncertain real parameter.

Time Domain of ultidyn Elements

On its own, every ultidyn element is interpreted as a continuous-time, system with uncertain behavior, quantified by bounds (gain or real-part) on its frequency response. However, when a ultidyn element is an uncertain element of an uncertain state space model (uss), then the time-domain characteristic of the element is determined from the time-domain characteristic of the system. The bounds (gain-bounded or positivity) apply to the frequency-response of the element.

Interpreting Uncertainty in Discrete Time

The interpretation of a ultidyn element as a continuous-time or discrete-time system depends on the nature of the uncertain system (uss) within which it is an uncertain element.

For example, create a scalar ultidyn object. Then, create two 1-input, 1-output uss objects using the ultidyn object as their “D” matrix. In one case, create without specifying sample-time, which indicates continuous time. In the second case, force discrete-time, with a sample time of 0.42.

delta = ultidyn('delta',[1 1]); 
sys1 = uss([],[],[],delta) 
USS: 0 States, 1 Output, 1 Input, Continuous System 
  delta: 1x1 LTI, max. gain = 1, 1 occurrence 
sys2 = uss([],[],[],delta,0.42) 
USS: 0 States, 1 Output, 1 Input, Discrete System, Ts = 0.42 
  delta: 1x1 LTI, max. gain = 1, 1 occurrence 

Next, get a random sample of each system. When obtaining random samples using usample, the values of the elements used in the sample are returned in the 2nd argument from usample as a structure.

[sys1s,d1v] = usample(sys1); 
[sys2s,d2v] = usample(sys2); 

Look at d1v.delta.Ts and d2v.delta.Ts. In the first case, since sys1 is continuous-time, the system d1v.delta is continuous-time. In the second case, since sys2 is discrete-time, with sample time 0.42, the system d2v.delta is discrete-time, with sample time 0.42.

d1v.delta.Ts 
ans = 
     0 
d2v.delta.Ts 
ans = 
    0.4200 

Finally, in the case of a discrete-time uss object, it is not the case that ultidyn objects are interpreted as continuous-time uncertainty in feedback with sampled-data systems. This very interesting hybrid theory is beyond the scope of the toolbox.