## D-K Iteration Process

You can use the `musyn`

command to design a robust controller for an
uncertain plant, as described in Robust Controller Design Using Mu Synthesis. The algorithm used by
`musyn`

is an iterative process called *D-K
iteration*. In this process, the function:

Uses

*H*_{∞}synthesis to find a controller that minimizes the closed-loop gain of the nominal system.Performs a robustness analysis to estimate the robust

*H*_{∞}performance of the closed-loop system. This quantity is expressed as a scaled*H*_{∞}norm involving dynamic scalings called the*D*and*G*scalings (the*D*step).Finds a new controller to minimize the scaled

*H*_{∞}norm obtained in step 2 (the*K*step).Repeats steps 2 and 3 until the robust performance stops improving.

Both the *D* step and *K* step are mathematically
intensive computations. Details of the algorithm follow.

### D Step

In the *D* step, `musyn`

computes the upper bound $$\overline{\mu}$$ of the robust *H*_{∞} performance for
the current controller *K*. The *D* step begins with a
robust performance analysis for the closed-loop uncertain system *T* =
LFT(*P*,*K*), as in the following diagram.

Introducing a performance block Δ_{perf}
transforms the robust-performance analysis of *T* to a robust-stability
analysis of the feedback loop in the following diagram.

Here, Δ is the augmented uncertainty structure

$$\Delta \triangleq \left(\begin{array}{cc}{\Delta}_{perf}& 0\\ 0& {\Delta}_{unc}\end{array}\right).$$

`musyn`

computes $$\overline{\mu}$$, an upper bound on the robust *H*_{∞}
performance. To do so, `musyn`

selects a frequency grid {*ω*_{1},…,*ω _{N}*}. For

*T*with complex uncertainty only,

`musyn`

computes at each frequency
*ω*

_{i}$${\overline{\mu}}_{i}=\underset{{D}_{i}}{\mathrm{inf}}\Vert {D}_{i}{T}_{0}\left(j{\omega}_{i}\right){D}_{i}{}^{-1}\Vert .$$

The frequency-dependent matrices *D*, which commute with Δ, are called
*D* scalings. $$\overline{\mu}$$ is the largest result over all frequencies in the grid,

$$\overline{\mu}=\underset{i}{\mathrm{max}}{\overline{\mu}}_{i}.$$

When you use `musyn`

, you can access the results of the
*D* step in several ways.

The default

`musyn`

display shows $$\overline{\mu}$$ for each iteration in the`Peak MU`

column.`musyn`

returns $$\overline{\mu}$$ for each iteration in the`PeakMU`

field of the`info`

output argument.`musyn`

returns*D*in the_{i}`DG`

field of the`info`

output argument.To visualize the frequency-dependence of

*D*, set the_{i}`'Display'`

option of`musynOptions`

to`'full'`

.

For additional details about the computation and interpretation of $$\overline{\mu}$$, see Robust Performance Measure for Mu Synthesis.

### D-Fitting and Scaled *H*_{∞} Performance

`musyn`

fits a rational function
*D*(*s*) to the sequence of scalings
{*D _{i}*}. The fit yields a quantity

*μ*called the

_{F}*scaled*,

*H*_{∞}performance$${\mu}_{F}\triangleq {\Vert D\left({T}_{0}\right){D}^{-1}\Vert}_{\infty}.$$

Because the fit is not exact, *μ _{F}* is typically
somewhat larger than $$\overline{\mu}$$.

You can access the results of the fit in several ways.

The default

`musyn`

display shows*μ*for each iteration in the_{F}`DG Fit`

column.`musyn`

returns*μ*for each iteration in the_{F}`PeakMUFit`

field of the`info`

output argument.`musyn`

returns the fitting functions in the`dr`

and`dc`

fields of the`info`

output argument.To visualize the frequency dependence of the fitting functions, set the

`'Display'`

option of`musynOptions`

to`'full'`

.

### K Step

*T*_{0} depends on the choice of controller
*K* by the relation *T*_{0} =
LFT(*P*_{0},*K*). Therefore, minimizing *μ _{F}* with
respect to

*K*is a scaled

*H*

_{∞}synthesis problem. Thus, in the

*K*step,

`musyn`

uses
`hinfsyn`

or `hinfstruct`

to compute a controller
*K*

^{*}that minimizes

*μ*. The minimized quantity is the scaled

_{F}*H*

_{∞}norm. For the algorithm to make progress, the new controller must improve the robust performance obtained in the

*D*step:

$${\Vert D\text{\hspace{0.17em}}\text{LFT}\left({P}_{0},{K}^{*}\right)\text{\hspace{0.17em}}{D}^{-1}\Vert}_{\infty}<\overline{\mu}.$$

Otherwise, the progress is not sufficient for compensate for fitting errors. Thus
`musyn`

terminates D-K iteration process when
*K*^{*} does not improve the robust performance
within the tolerance specified by the `'TolPerf'`

option of
`musynOptions`

.

You can access the results of the *K* step in several ways.

The default

`musyn`

display shows the scaled*H*_{∞}norm for each iteration in the`K Step`

column.`musyn`

returns the new controller in the`K`

of the`info`

output argument, and the corresponding scaled*H*_{∞}norm for each iteration in the`gamma`

field.

### Mixed Real and Complex Uncertainty

When the system has both real and complex uncertainty and you set the
`'MixedMU'`

option of `musynOptions`

to
`'on'`

, `musyn`

uses an additional
*G*-scaling to improve the computation of $$\overline{\mu}$$. The algorithm in this case is called *mixed-μ
synthesis*.

For mixed uncertainty, `musyn`

computes $${\overline{\mu}}_{i}$$ and scalings
*D _{r}*(

*ω*),

_{i}*D*(

_{c}*ω*), and

_{i}*G*(

_{cr}*ω*) such that

_{i}$${\left(\begin{array}{c}{T}_{0}\left(j{\omega}_{i}\right)\\ I\end{array}\right)}^{H}\left(\begin{array}{cc}{D}_{r}\left({\omega}_{i}\right)& -j{G}_{cr}^{H}\left({\omega}_{i}\right)\\ j{G}_{cr}\left({\omega}_{i}\right)& -{\overline{\mu}}_{i}^{2}{D}_{c}\left({\omega}_{i}\right)\end{array}\right)\left(\begin{array}{c}{T}_{0}\left(j{\omega}_{i}\right)\\ I\end{array}\right)\le 0$$

at each frequency in the grid.

`musyn`

fits the *D* and *G*
scaling data by constructing a rational function

$$F\left(s\right)=\Psi \left(s\right)\left(\begin{array}{cc}{d}_{r}\left(s\right)& 0\\ 0& {d}_{c}\left(s\right)\end{array}\right)$$

such that

*d*(_{r}*s*),*d*(_{c}*s*), and Ψ(*s*) are stable with stable inverse.*d*(_{r}*s*) and*d*(_{c}*s*) approximate the square roots of the diagonal entries of*D*(_{r}*ω*) and_{i}*D*(_{c}*ω*)._{i}*F*approximately satisfies$$\begin{array}{c}\left(\begin{array}{cc}{D}_{r}\left({\omega}_{i}\right)& -j{G}_{cr}^{H}\left({\omega}_{i}\right)\\ j{G}_{cr}\left({\omega}_{i}\right)& -{\mu}^{2}{D}_{c}\left({\omega}_{i}\right)\end{array}\right)\approx F{\left(j{\omega}_{i}\right)}^{H}J\text{\hspace{0.17em}}F\left(j{\omega}_{i}\right),\\ J=\left(\begin{array}{cc}{I}_{r}& 0\\ 0& -I{}_{c}\end{array}\right).\end{array}$$

Finally, the scaled *H*_{∞} performance is defined
as

$${\mu}_{F}\triangleq {\Vert \overline{T}\left(s\right)\Vert}_{\infty},$$

where $$\overline{T}\left(s\right)$$ is the transformed system,

$$\overline{T}\left(s\right)\triangleq \overline{\mu}{\overline{T}}_{1}{\overline{T}}_{2}^{-1},\text{\hspace{1em}}\left(\begin{array}{c}{\overline{T}}_{1}\left(s\right)\\ {\overline{T}}_{2}\left(s\right)\end{array}\right)\triangleq \Psi \left(s\right)\left(\begin{array}{c}{d}_{r}\left(s\right){T}_{0}\left(s\right)\\ {d}_{c}\left(s\right)\end{array}\right).$$

For an exact fit of *D* and *G*, $$\Vert \overline{T}\left(j{\omega}_{i}\right)\Vert ={\overline{\mu}}_{i}$$. Therefore, in general, $${\mu}_{F}\ge \overline{\mu}.$$

Because the transformed system $$\overline{T}\left(s\right)$$ is still a linear fractional function of the controller
*K*, the *K* step for the mixed-*μ*
case proceeds by computing a controller *K**** that
minimizes $${\Vert \overline{T}\Vert}_{\infty}$$.

When using `musyn`

, you can access the *D* and
*G* scalings in several ways.

`musyn`

returns the*D*and*G*scaling data in the`DG`

field of the`info`

output argument.`musyn`

returns the fitting functions in the`dr`

,`dc`

, and`PSI`

fields of the`info`

output argument.To visualize the frequency dependence of the scaling data and fitting functions, set the

`'Display'`

option of`musynOptions`

to`'full'`

.

## See Also

`musyn`

| `musynOptions`

| `musynperf`