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Specify LMIs with the LMI Editor GUI

The LMI Editor lmiedit is a graphical user interface (GUI) to specify LMI systems in a straightforward symbolic manner. Typing

lmiedit

calls up a window with several editable text areas and various buttons.

In more detail, to specify your LMI system,

  1. Declare each matrix variable (name and structure) in the upper half of the worksheet. The structure is characterized by its type (S for symmetric block diagonal, R for unstructured, and G for other structures) and by an additional “ structure” matrix. This matrix contains specific information about the structure and corresponds to the second argument of lmivar.

    Please use one line per matrix variable in the text editing areas.

  2. Specify the LMIs as MATLAB® expressions in the lower half of the worksheet. For instance, the LMI

    (ATX+XAXBBTX−I)<0

    is entered by typing

    [a'*x+x*a x*b; b'*x -1] < 0

    if x is the name given to the matrix variable X in the upper half of the worksheet. The left- and right-hand sides of the LMIs should be valid MATLAB expressions.

Once the LMI system is fully specified, the following tasks can be performed by clicking the corresponding button:

  • Visualize the sequence of lmivar/lmiterm commands needed to describe this LMI system (view commands button). Conversely, the LMI system defined by a particular sequence of lmivar/lmiterm commands can be displayed as a MATLAB expression by clicking on the describe... buttons.

    Beginners can use this facility as a tutorial introduction to the lmivar and lmiterm commands.

  • Save the symbolic description of the LMI system (save button). This description can be reloaded later on by clicking the load button.

  • Read a sequence of lmivar/lmiterm commands from a file (read button). You can then click on describe the matrix variables or describe the LMIs to visualize the symbolic expression of the LMI system specified by these commands. The file should describe a single LMI system but may otherwise contain any sequence of MATLAB commands.

    This feature is useful for code validation and debugging.

    Write in a file the sequence of lmivar/lmiterm commands needed to describe a particular LMI system (write button).

    This is helpful to develop code and prototype MATLAB functions based on the LMI Lab.

  • Generate the internal representation of the LMI system by clicking create. The result is written in a MATLAB variable named after the LMI system (if the “name of the LMI system” is set to mylmi, the internal representation is written in the MATLAB variable mylmi). Note that all LMI-related data should be defined in the MATLAB workspace at this stage.

    The internal representation can be passed directly to the LMI solvers or any other LMI Lab function.

Keyboard Shortcuts

As with lmiterm, you can use various shortcuts when entering LMI expressions at the keyboard. For instance, zero blocks can be entered simply as 0 and need not be dimensioned. Similarly, the identity matrix can be entered as 1 without dimensioning. Finally, upper diagonal LMI blocks need not be fully specified. Rather, you can just type (*) in place of each such block.

Limitations

Though fairly general, lmiedit is not as flexible as lmiterm and the following limitations should be kept in mind:

  • Parentheses cannot be used around matrix variables. For instance, the expression

    (a*x+b)'*c + c'*(a*x+b)

    is invalid when x is a variable name. By contrast,

    (a+b)'*x + x'*(a+b)

    is perfectly valid.

  • Loops and if statements are ignored.

  • When turning lmiterm commands into a symbolic description of the LMI system, an error is issued if the first argument of lmiterm cannot be evaluated. Use the LMI and variable identifiers supplied by newlmi and lmivar to avoid such difficulties.

How lmivar and lmiterm Manage LMI Representation

Users familiar with MATLAB may wonder how lmivar and lmiterm physically update the internal representation LMISYS since LMISYS is not an argument to these functions. In fact, all updating is performed through global variables for maximum speed. These global variables are initialized by setlmis, cleared by getlmis, and are not visible in the workspace. Even though this artifact is transparent from the user's viewpoint, be sure to:

  • Invoke getlmis only once and after completely specifying the LMI system.

  • Refrain from using the command clear global before the LMI system description is ended with getlmis.

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