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.
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]);
Uncertain linear, time-invariant objects have an internal name
Name property), and are created by specifying
their size (number of outputs and number of inputs).
Type specifies whether the known
attributes about the frequency response are related to gain or phase.
Type may be
The default value is
Bound is a single number, which
Type, completely specifies what is known
about the uncertain frequency response. Specifically, if Δ is
ultidyn element, and if
γ denotes the value of the
then the element represents the set of all stable, linear, time-invariant
systems whose frequency response satisfies certain conditions:
all frequencies. When
the default value for
Bound (i.e., γ)
is 1. The
NominalValue of Δ is always the
'PositiveReal', Δ(ω) + Δ*(ω) ≥ 2γ·
for all frequencies. When
the default value for
Bound (i.e., γ) is
NominalValue is always (γ + 1 +2|γ|)I.
All properties of a
can be accessed with
NominalValue is determined from
and not accessible with
set). The properties are
Nominal value of element
Norm bound or minimum real
State-space dimension of random samples of this uncertain element
Maximum natural frequency for random sampling
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.
On its own, every
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 (
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 .
For example, create a scalar
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
[sys1s,d1v] = usample(sys1); [sys2s,d2v] = usample(sys2);
In the first case, since
sys1 is continuous-time,
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
it is not the case that
are interpreted as continuous-time uncertainty in feedback with sampled-data
systems. This very interesting hybrid theory is beyond the scope of