Robust Controller Design Using Mu Synthesis

The technique of μ synthesis extends the methods of H_∞ synthesis to design a robust controller for an uncertain plant. You can perform μ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the musyn command.

musyn seeks a controller that minimizes the robust H_∞ performance of the closed-loop system. The robust H_∞ performance, also called μ, quantifies how modeled uncertainty affects the performance of a feedback loop. For details about μ and how it is computed, see Robust Performance Measure for Mu Synthesis.

Basic μ Synthesis Workflow

You can use musyn to:

Synthesize "black box" unstructured robust controllers.
Robustly tune a fixed-order or fixed-structure controller made up of tunable components such as PID controllers, state-space models, and static gains.

μ Synthesis of Unstructured Controllers

μ synthesis of unstructured controllers is analogous to controller synthesis with hinfsyn, except that the plant includes uncertainty. As with hinfsyn, you set up your problem as the feedback system CL = lft(P,K), where P is the plant and K is the controller to design.

Plant P in LFT feedback connection to controller K. P has inputs w and u, and outputs z and y. K has output u and input y.

In the diagram:

w represents the disturbance inputs.
u represents the control inputs.
z represents the error outputs to be kept small.
y represents the measurement outputs provided to the controller.

You construct the uncertain plant P by building a state-space model with uncertain coefficients (ureal or ucomplex) blocks, uncertain dynamics (ultidyn blocks), or both. Construct the plant such that the measurement outputs y are the last outputs, and the control inputs u are the last inputs. As with hinfsyn, you can optionally augment the plant inputs and outputs with weighting functions (loop-shaping filters) that represent control objectives.

You then pass this plant to musyn, which seeks a controller K that minimizes the robust H_∞ performance. The controller is returned as a state-space model. For a simple example, see Unstructured Robust Controller Synthesis on the musyn reference page.

μ Synthesis of Fixed-Structure Controllers

Instead of obtaining a controller that is a free-form state-space model, you can specify a fixed controller structure with tunable parameters. musyn then adjusts those parameters to minimize the robust H_∞ performance of the system. μ synthesis of fixed-structure controllers is analogous to controller tuning with hinfstruct, except that the plant includes uncertainty.

To set up your problem for fixed-structure μ synthesis, you construct a generalized state-space (genss) model of the uncertain closed-loop system with tunable controller elements. To do so, you create and interconnect:

Numeric LTI models representing the fixed components of the control system
Uncertain control design blocks such as ureal and ultidyn blocks representing the uncertain components of the plant
Optional LTI weighting functions (loop-shaping filters) that represent control objectives
Tunable control design blocks such as tunablePID, tunableSS, and tunableGain to represent tunable components of the system

For an example that shows how to build such a model, see Build Tunable Control System Model with Uncertain Parameters.

You pass the tunable, uncertain closed-loop model to musyn, which seeks values of the tunable parameters that optimize the robust H_∞ performance from the model inputs to its outputs. For a simple example, see Robust Tuning of Fixed-Structure Controller on the musyn reference page.

If you have a Simulink^® model of your control system, you can use slTuner to linearize the model with specified uncertain parameters and tunable blocks. You then use getIOTransfer to extract a genss model for controller design with musyn. For an example, see Model Uncertainty in Simulink for Robust Tuning.

Interpret the Results of μ Synthesis

musyn returns a robust controller K (for unstructured controller tuning) or a tuned version of the control system CL (for fixed-structure controller tuning). It also returns the best achieved robust H_∞ performance as the CLperf output argument. This value tells you that with the controller returned by musyn, the peak gain of the closed-loop system remains below CLperf for uncertainty up to 1/CLperf in normalized units. For example:

CLperf = 0.5 means that the closed-loop gain remains below 0.5 for uncertainty up to twice the uncertainty specified in the input model. The worst-case gain for the specified uncertainty is typically smaller.
CLperf = 2 means that the closed-loop gain remains below 2 for uncertainty up to half the uncertainty specified in CL. For this value, the worst-case gain for the full specified uncertainty can be much larger. It can even be infinite, meaning that the system does not remain stable over the full range of the specified uncertainty.

For more detailed information about this quantity and how it is computed, see Robust Performance Measure for Mu Synthesis.

To find K, musyn uses an iterative process called D-K iteration. This process solves a sequence of scaled H_∞ problems. The frequency-dependent scalings, called D and G scalings, take advantage of the uncertainty structure. To perform D-K iteration, musyn:

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 the D and G scalings (the D step).
Finds a new controller to minimize the H_∞ norm obtained in step 2 (the K step).
Repeats steps 2 and 3 until the robust performance stops improving.

For mathematical details about how this algorithm works, see D-K Iteration Process.

musyn gives you two ways to monitor and interpret the progress of the algorithm: the default display and the full display.

Default `musyn` Display

By default, musyn provides a brief display of algorithm progress in the MATLAB^® command window. For instance:

DG-K ITERATION SUMMARY:
-------------------------------------------------------------------
                       Robust performance                 Fit order
-------------------------------------------------------------------
  Iter         K Step       Peak MU       DG Fit           D      G
    1             100        5.747        6.394           10      4
    2           5.221        3.433        4.607           10      6
    3           2.682        2.263        2.627           10      4
    4           1.987        1.687         2.18           10      6
    5           1.287        1.192        1.377           10      8
    6           1.079        1.087         1.09           10      8
    7           1.076        1.046        1.055            8      6
    8           1.049        1.024        1.044           10      6
    9           1.045        1.022        1.039            8      6
   10            1.04        1.023        1.033            8      6

Best achieved robust performance: 1.02

The display includes information about each D-K iteration.

K Step column — For the first iteration, this value is the H_∞ performance of the closed-loop nominal system after controller synthesis. For remaining iterations, this column shows the scaled H_∞ norm after controller synthesis.
Peak MU column — Robust performance ( $\bar{μ}$ , an upper bound on μ) for the controller designed in the K Step.
DG fit column — Scaled H_∞ performance after fitting the D and G scalings with rational functions.
Fit order columns — Orders of the rational function used to fit the scalings in that iteration. If the system has only complex uncertainty, or when the 'MixedMU' option of musynOptions is set to 'off', then musyn does not apply G scaling. In that case, only the D fit order is listed.

If you see a large difference between the Peak MU and DG Fit values in a given iteration, it is a sign that musyn cannot find a good fit for the scalings. In that case, you can try increasing the maximum fit order using the 'FitOrder' option of musynOptions.

For other ways to improve the results, see Improve Results of Mu Synthesis.

Full `musyn` Display

You can obtain a more detailed view into the progress of D-K iteration by setting the 'Display' option of musynOptions to 'full'. If you turn on the full display, then musyn pauses after each D-K iteration so that you can view the detailed results of the iteration. In addition to the information described in Default musyn Display, the full display:

Shows detailed computation information for the controller synthesis (K step) of the current iteration. For unstructured controllers, see hinfsyn for information about this display. For fixed-structure controllers, see hinfstructOptions.
Shows information about the fits for the D scalings and the G scalings (if any) of the current iteration. The information includes the fit order of the scalings for each uncertain block. It also includes a goodness-of-fit score. A score less than or equal to 1 indicates adequate fit for μ synthesis.
Generates plots that let you visualize the D and G fits, the robust performance before fitting, and the scaled H_∞ performance after fitting. Examining these plots can help you determine if the maximum fit order is high enough to capture all the frequency-dependent variation in the scalings (see the FitOrder option of musynOptions for more information).

The D Fit or D,G Fit plot shows the scaling data and the corresponding rational fits.

Example full display of musyn results in a figure window with four axes. Each axis shows the magnitude of D as a function of frequency for an uncertain block in the tuned system. At the bottom of the figure window are radio buttons labeled D (diagonal), D (off-diagonal), and jG.

Use the radio buttons to select which scalings to inspect:

D (diagonal) shows the magnitude of the diagonal elements of the D scalings.
D (offdiagonal) shows the magnitude and phase of the off-diagonal elements of the D scalings. This plot is available when your system has repeated uncertain blocks. (See the 'FullDG' option of musynOptions for more information.)
jG shows the magnitude and phase of the G-scalings. G scalings are present only when there is real uncertainty and the MixedMU option of musynOptions is 'on'. See Improve Results of Mu Synthesis.

The Robust Performance plot shows the performance of the closed-loop system before and after fitting.

Sample Robust Performance plot with title Performance with D,G data vs. fit. The plot shows Performance (abs) vs Frequency on a log-log axis.

The traces on this plot are:

MU upper bound — Robust performance, the upper bound $\bar{μ}$ as a function of frequency
Scaled CL for D,G data — Scaled H_∞ performance before fitting the D and G scaling data with rational functions
Scaled CL for fitted D,G — Scaled H_∞ performance after fitting
Scaled CL for fitted D only — Shows what behavior would not be captured if G were omitted

For detailed information about the D-K iteration algorithm and the meaning of all the quantities in the full display, see D-K Iteration Process.