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The two dominant forms of model uncertainty are as follows:

Uncertainty in parameters of the underlying differential equation models

Frequency-domain uncertainty, which often quantifies model uncertainty by describing absolute or relative uncertainty in the process's frequency response

Using these two basic building blocks, along with conventional
system creation commands (such as `ss`

and `tf`

),
you can easily create uncertain system models.

An uncertain parameter has a name (used to identify it within an uncertain system with many uncertain parameters) and a nominal value. Being uncertain, it also has variability, described in one of the following ways:

An additive deviation from the nominal

A range about the nominal

A percentage deviation from the nominal

Create a real parameter, with name '|bw|', nominal value 5, and a percentage uncertainty of 10%.

bw = ureal('bw',5,'Percentage',10)

bw = Uncertain real parameter "bw" with nominal value 5 and variability [-10,10]%.

This command creates a `ureal`

object that stores a number of parameters in its properties. View the properties of `bw`

.

get(bw)

NominalValue: 5 Mode: 'Percentage' Range: [4.5000 5.5000] PlusMinus: [-0.5000 0.5000] Percentage: [-10 10] AutoSimplify: 'basic' Name: 'bw'

Note that the range of variation (`Range`

property) and the additive deviation from nominal (the `PlusMinus`

property) are consistent with the `Percentage`

property value.

You can create state-space and transfer function models with uncertain real coefficients using `ureal`

objects. The result is an uncertain state-space (`uss`

) object. As an example, use the uncertain real parameter `bw`

to model a first-order system whose bandwidth is between 4.5 and 5.5 rad/s.

H = tf(1,[1/bw 1])

H = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 1 states. The model uncertainty consists of the following blocks: bw: Uncertain real, nominal = 5, variability = [-10,10]%, 1 occurrences Type "H.NominalValue" to see the nominal value, "get(H)" to see all properties, and "H.Uncertainty" to interact with the uncertain elements.

Note that the result `H`

is an uncertain system, called a `uss`

model. The nominal value of `H`

is a state-space (`ss`

) model. Verify that the pole is at -5, as expected from the uncertain parameter's nominal value of 5.

pole(H.NominalValue)

ans = -5

Next, use `bodeplot`

and `stepplot`

to examine the behavior of `H`

. These commands plot the responses of the nominal system and a number of random samples of the uncertain system.

bodeplot(H,{1e-1 1e2});

stepplot(H)

While there are variations in the bandwidth and time constant of `H`

, the high-frequency rolls off at 20 dB/decade regardless of the value of `bw`

. You can capture the more complicated uncertain behavior that typically occurs at high frequencies using the `ultidyn`

uncertain element.

An informal way to describe the difference between the model of a process and the actual process behavior is in terms of bandwidth. It is common to hear “The model is good out to 8 radians/second.” The precise meaning is not clear, but it is reasonable to believe that for frequencies lower than, say, 5 rad/s, the model is accurate, and for frequencies beyond, say, 30 rad/s, the model is not necessarily representative of the process behavior. In the frequency range between 5 and 30, the guaranteed accuracy of the model degrades.

The uncertain linear, time-invariant dynamics object `ultidyn`

can
be used to model this type of knowledge. An `ultidyn`

object
represents an unknown linear system whose only known attribute is
a uniform magnitude bound on its frequency response. When coupled
with a nominal model and a frequency-shaping filter, `ultidyn`

objects can be used to capture uncertainty associated with the model dynamics.

Suppose that the behavior of the system modeled by `H`

significantly
deviates from its first-order behavior beyond 9 rad/s, for example,
about 5% potential relative error at low frequency, increasing to
1000% at high frequency where `H`

rolls off. In order
to model frequency domain uncertainty as described above using `ultidyn`

objects, follow these steps:

Create the nominal system

`Gnom`

, using`tf`

,`ss`

, or`zpk`

.`Gnom`

itself might already have parameter uncertainty. In this case`Gnom`

is`H`

, the first-order system with an uncertain time constant.Create a filter

`W`

, called the “weight,” whose magnitude represents the relative uncertainty at each frequency. The utility`makeweight`

is useful for creating first-order weights with specific low- and high-frequency gains, and specified gain crossover frequency.Create an

`ultidyn`

object`Delta`

with magnitude bound equal to 1.

The uncertain model `G`

is formed by ```
G
= Gnom*(1+W*Delta)
```

.

If the magnitude of `W`

represents an absolute
(rather than relative) uncertainty, use the formula ```
G = Gnom
+ W*Delta
```

instead.

The following commands carry out these steps:

bw = ureal('bw',5,'Percentage',10); H = tf(1,[1/bw 1]); Gnom = H; W = makeweight(.05,9,10); Delta = ultidyn('Delta',[1 1]); G = Gnom*(1+W*Delta)

G = Uncertain continuous-time state-space model with 1 outputs, 1 inputs, 2 states. The model uncertainty consists of the following blocks: Delta: Uncertain 1x1 LTI, peak gain = 1, 1 occurrences bw: Uncertain real, nominal = 5, variability = [-10,10]%, 1 occurrences Type "G.NominalValue" to see the nominal value, "get(G)" to see all properties, and "G.Uncertainty" to interact with the uncertain elements.

Note that the result `G`

is also an uncertain
system, with dependence on both `Delta`

and `bw`

.
You can use `bode`

to make a Bode plot of 20 random
samples of `G'`

s behavior over the frequency range
[0.1 100] rad/s.

bode(G,{1e-1 1e2})

In the next section, you design and compare two feedback controllers
for `G`

.

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