## Robustness and Worst-Case Analysis

In robust control design, performance is expressed and measured in terms of the peak
gain (the *H*_{∞} norm or peak singular
value) of a system. The smaller this gain is, the better the system performance. The
performance of a nominally stable uncertain system generally degrades as the amount of
uncertainty increases. Use robustness analysis and worst-case analysis to examine how
the amount of uncertainty in your system affects the stability and peak gain of the
system.

### Robustness Analysis

Robustness analysis is about finding the maximum amount of uncertainty compatible with stability or with a given performance level. The following illustration shows a typical tradeoff curve between performance and robustness. Here, the peak gain (peak magnitude on a Bode plot or singular-value plot) characterizes the system performance.

The *x*-axis quantifies the normalized amount of uncertainty. The
value *x* = 1 corresponds to the uncertainty ranges specified in
the model. *x* = 2 represents the system with twice as much
uncertainty. *x* = 0 corresponds to the nominal system. (See
`actual2normalized`

for more details
about normalized uncertainty ranges.) The *y*-axis is performance,
measured as the peak gain of some closed-loop transfer function. For instance, if
the closed-loop transfer function measures the sensitivity of an error signal to
some disturbance, then higher peak gain corresponds to poorer disturbance
rejection.

When all uncertain elements are set to their nominal values (*x*
= 0), the gain of the system is its nominal value. In the figure, the nominal system
gain is about 1. As the range of values that the uncertain elements can take
increases, the peak gain over the uncertainty range increases. The heavy blue line
represents the peak gain, and is called the *system performance
degradation curve*. It increases monotonically as a function of the
uncertainty amount.

#### Robust Stability Margin

The system performance degradation curve typically has a vertical asymptote
corresponding to the *robust stability margin*. This margin
is the maximum amount of uncertainty that the system can tolerate while
remaining stable. For the system of the previous illustration, the peak gain
becomes infinite at around *x* = 2.3. In other words, the
system becomes unstable when the uncertainty range is 2.3 times that specified
in the model (in normalized units). Therefore, the robust stability margin is
2.3. To compute the robust stability margin for an uncertain system model, use
the `robstab`

function.

#### Robust Performance Margin

The *robust performance margin* for a given gain,
*γ*, is the maximum amount of uncertainty the system can
tolerate while having a peak gain less than *γ*. For example,
in the following illustration, suppose that you want to keep the peak
closed-loop gain below 1.8. For that peak gain, the robust performance margin is
about 1.7. This value means that the peak gain of the system remains below 1.8
as long as the uncertainty remains within 1.7 times the specified uncertainty
(in normalized units).

To compute the robust performance margin for an uncertain system model, use
the `robgain`

function.

### Worst-Case Gain Measure

The *worst-case gain* is the largest value that the peak gain
can take over a specific uncertainty range. This value is the counterpart of the
robust performance margin. While the robust performance margin measures the maximum
amount of uncertainty compatible with a particular peak gain level, the worst-case
gain measures the maximum gain associated with a particular uncertainty amount. For
instance, in the following illustration, the worst-case gain for the uncertainty
amount specified in the model is about 1.20. If that uncertainty amount is doubled,
the worst-case gain increases to 2.5.

To compute the worst-case gain for an uncertain system model, use the `wcgain`

function. The
`ULevel`

option of the `wcOptions`

command lets you compute
the worst-case gain for different amounts of uncertainty.