Uncertain Complex Parameters and Matrices

Uncertain Complex Parameters

The ucomplex element is the Control Design Block that represents an uncertain complex number. The value of an uncertain complex number lies in a disc, centered at NominalValue, with radius specified by the Radius property of the ucomplex element. The size of the disc can also be specified by Percentage, which means the radius is derived from the absolute value of the NominalValue. The properties of ucomplex objects are

Properties	Meaning	Class
`Name`	Internal Name	`char`
`NominalValue`	Nominal value of element	`double`
`Mode`	`'Range'\|'Percentage'`	`char`
`Radius`	Radius of disk	`double`
`Percentage`	Additive variation (percent of `Radius`)	`double`
`AutoSimplify`	`'off'\| {'basic'} \|'full'`	`char`

Create Uncertain Complex Parameter

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The simplest construction requires only a name and nominal value.

a = ucomplex('a',2-j)

Uncertain complex parameter "a" with nominal value 2-1i and radius 1.

Displaying the properties shows that the default Mode is Radius, and the default radius is 1.

get(a)

    NominalValue: 2.0000 - 1.0000i
            Mode: 'Radius'
          Radius: 1
      Percentage: 44.7214
    AutoSimplify: 'basic'
            Name: 'a'

Sample the uncertain complex parameter at 400 values and plot the samples in the complex plane. The plot shows that the samples all fall within a disc of radius 1, centered in the complex plane at the value 2 – j.

asample = usample(a,400); 
plot(asample(:),'o'); 
xlim([-0.5 4.5]); 
ylim([-3 1]);

Figure contains an axes object. The axes contains a line object which displays its values using only markers.

Uncertain Complex Matrices

The uncertain complex matrix class, ucomplexm, represents the set of matrices given by the formula

N + W_LΔW_R

where N, W_L, and W_R are known matrices, and Δ is any complex matrix with $\dot{\bar{σ}} (Δ) \leq 1$ . All properties of a ucomplexm are can be accessed with get and set. The properties are

Properties	Meaning	Class
`Name`	Internal Name	`char`
`NominalValue`	Nominal value of element	`double`
`WL`	Left weight	`double`
`WR`	Right weight	`double`
`AutoSimplify`	`'off'\| {'basic'} \|'full'`	`char`

Uncertain Complex Matrix and Weighting Matrices

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Create a 4-by-3 uncertain complex matrix (ucomplexm), and view its properties. The simplest construction requires only a name and nominal value.

m = ucomplexm('m',[1 2 3; 4 5 6; 7 8 9; 10 11 12])

Uncertain complex matrix "m" with 4 rows and 3 columns.

get(m)

    NominalValue: [4x3 double]
              WL: [4x4 double]
              WR: [3x3 double]
    AutoSimplify: 'basic'
            Name: 'm'

The nominal value is the matrix you supply to ucomplexm.

mnom = m.NominalValue

mnom = 4×3

     1     2     3
     4     5     6
     7     8     9
    10    11    12

By default, the weighting matrices are the identity. For example, examine the left weighting.

m.WL

ans = 4×4

     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

Sample the uncertain matrix, and compare to the nominal value. Note the element-by-element sizes of the difference are roughly equal, indicative of the identity weighting matrices.

msamp = usample(m);
diff = abs(msamp-mnom)

diff = 4×3

    0.3309    0.0917    0.2881
    0.2421    0.3449    0.3917
    0.2855    0.2186    0.2915
    0.3260    0.2753    0.3816

Change the left and right weighting matrices, making the uncertainty larger as you move down the rows, and across the columns.

m.WL = diag([0.2 0.4 0.8 1.6]); 
m.WR = diag([0.1 1 4]);

Sample the uncertain matrix again, and compare to the nominal value. Note the element-by-element sizes of the difference, and the general trend that the smallest differences are near the (1,1) element, and the largest differences are near the (4,3) element, consistent with the trend in the diagonal weighting matrices.

msamp = usample(m);
diff = abs(msamp-mnom)

diff = 4×3

    0.0048    0.0526    0.2735
    0.0154    0.1012    0.4898
    0.0288    0.3334    0.8555
    0.0201    0.4632    1.3783