Main Content

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:

  1. Uses H synthesis to find a controller that minimizes the closed-loop gain of the nominal system.

  2. 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).

  3. Finds a new controller to minimize the scaled H norm obtained in step 2 (the K step).

  4. 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 μ¯ 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 Δperf 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

Δ(Δperf00Δunc).

musyn computes μ¯, an upper bound on the robust H performance. To do so, musyn selects a frequency grid {ω1,…,ωN}. For T with complex uncertainty only, musyn computes at each frequency ωi

μ¯i=infDiDiT0(jωi)Di1.

The frequency-dependent matrices D, which commute with Δ, are called D scalings. μ¯ is the largest result over all frequencies in the grid,

μ¯=maxiμ¯i.

When you use musyn, you can access the results of the D step in several ways.

  • The default musyn display shows μ¯ for each iteration in the Peak MU column.

  • musyn returns μ¯ for each iteration in the PeakMU field of the info output argument.

  • musyn returns Di in the DG field of the info output argument.

  • To visualize the frequency-dependence of Di, set the 'Display' option of musynOptions to 'full'.

For additional details about the computation and interpretation of μ¯, 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 {Di}. The fit yields a quantity μF called the scaled H performance,

μFD(T0)D1.

Because the fit is not exact, μF is typically somewhat larger than μ¯.

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

  • The default musyn display shows μF for each iteration in the DG Fit column.

  • musyn returns μF for each iteration in the 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

T0 depends on the choice of controller K by the relation T0 = LFT(P0,K). Therefore, minimizing μF 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 μF. The minimized quantity is the scaled H norm. For the algorithm to make progress, the new controller must improve the robust performance obtained in the D step:

DLFT(P0,K*)D1<μ¯.

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 μ¯. The algorithm in this case is called mixed-μ synthesis.

For mixed uncertainty, musyn computes μ¯i and scalings Dr(ωi), Dc(ωi), and Gcr(ωi) such that

(T0(jωi)I)H(Dr(ωi)jGcrH(ωi)jGcr(ωi)μ¯i2Dc(ωi))(T0(jωi)I)0

at each frequency in the grid.

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

F(s)=Ψ(s)(dr(s)00dc(s))

such that

  • dr(s), dc(s), and Ψ(s) are stable with stable inverse.

  • dr(s) and dc(s) approximate the square roots of the diagonal entries of Dr(ωi) and Dc(ωi).

  • F approximately satisfies

    (Dr(ωi)jGcrH(ωi)jGcr(ωi)μ2Dc(ωi))F(jωi)HJF(jωi),J=(Ir00Ic).

Finally, the scaled H performance is defined as

μFT¯(s),

where T¯(s) is the transformed system,

T¯(s)μ¯T¯1T¯21,(T¯1(s)T¯2(s))Ψ(s)(dr(s)T0(s)dc(s)).

For an exact fit of D and G, T¯(jωi)=μ¯i. Therefore, in general, μFμ¯.

Because the transformed system T¯(s) 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 T¯.

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

| |

Related Topics