Uncertain LTI Dynamics Elements

Uncertain linear, time-invariant objects, ultidyn, are used to represent unknown linear, time-invariant dynamic objects, whose only known attributes are bounds on their frequency response. 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', σ¯˙[Δ(ω)]γ 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 can be accessed 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

SampleStateDim

State-space dimension of random samples of this uncertain element

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.

You can create a 2-by-3 gain-bounded uncertain linear dynamics element. Verify its size, and check the properties.

f = ultidyn('f',[2 3]); 
size(f) 
ans = 
     2     3 
get(f) 
              Name: 'f' 
      NominalValue: [2x3 double] 
              Type: 'GainBounded' 
             Bound: 1 
    SampleStateDim: 1 
      AutoSimplify: 'basic' 

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. To see this, create a ultidyn, and view the sample time of several random samples of the element.

h = ultidyn('h',[1 1]); 
get(usample(h),'Ts') 
ans = 
     0 
get(usample(h),'Ts') 
ans = 
     0 
get(usample(h),'Ts') 
ans = 
     0 

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. This is explained and demonstrated in Interpreting Uncertainty in Discrete Time.

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