# Documentation

## 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'`, $\stackrel{˙}{\overline{\sigma }}\left[\Delta \left(\omega \right)\right]\le \gamma$ 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 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.