Disk-based stability margins of feedback loops

`[DM,MM] = diskmargin(L)`

`MMIO = diskmargin(P,C)`

`___ = diskmargin(___,E)`

`[`

computes the disk-based stability margins for the SISO or MIMO negative feedback loop
`DM`

,`MM`

] = diskmargin(`L`

)`feedback(L,eye(N))`

, where `N`

is the number of inputs
and outputs in `L`

.

The `diskmargin`

command returns loop-at-a-time stability margins in
`DM`

and multiloop margins in `MM`

. Disk-based
margin analysis provides a stronger guarantee of stability than the classical gain and phase
margins. For general information about disk margins, see
Stability Analysis Using Disk Margins.

`diskmargin`

assumes negative feedback. To compute the disk margins of a positive feedback system, use`diskmargin(-L)`

or`diskmargin(P,-C)`

.To compute disk margins for a system modeled in Simulink

^{®}, first linearize the model to obtain the open-loop response at a particular operating point. Then, use`diskmargin`

to compute stability margins for the linearized system. For more information, see Stability Margins of a Simulink Model.To compute classical gain and phase margins, use

`allmargin`

.

For SISO *L*, the uncertainty model for disk-margin analysis incorporates
a multiplicative complex uncertainty Δ into the loop transfer function as follows:

$$L\to {L}_{\Delta}=L\frac{1+\Delta \left(1-E\right)/2}{1-\Delta \left(1+E\right)/2}=L\left(1+{\delta}_{L}\right),\text{\hspace{1em}}\left|\Delta \right|<\alpha .$$

For Δ = 0, the multiplicative factor is 1, corresponding to the nominal
*L*. As Δ varies in the ball |Δ| < *α*, the gain and
phase of the multiplicative factor are a model for gain and phase variation in
*L*. The eccentricity parameter *E* varies the shape of
the applied uncertainty in the complex plane. The *disk margin* is the
smallest radius *α* at which the closed-loop system becomes
unstable[1]. From the disk margin
*α*, `diskmargin`

derives the minimum gain and phase
margins.

For MIMO systems, `diskmargin`

applies an analogous uncertainty model
that allows the uncertainty to vary independently in each channel.

For further details about the computation and interpretation of disk margins, see Stability Analysis Using Disk Margins.

[1] Blight, J.D., R.L. Dailey, and D.
Gangsaas. "Practical Control Law Design for Aircraft Using Multivariable Techniques."
*International Journal of Control*. Vol. 59, Number 1, 1994, pp.
93–137.