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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

The simplest construction requires only a name and nominal value. The default Mode is Radius, and the default radius is 1.

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

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

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

Sample the uncertain complex parameter at 400 values, and plot in the complex plane. Clearly, the samples appear to be from 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]);

Uncertain Complex Matrices

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

N + WLΔWR

where N, WL, and WR are known matrices, and Δ is any complex matrix with . 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

The simplest construction requires only a name and nominal value. The default left and right weight matrices are identity.

You can create a 4-by-3 ucomplexm element, and view its properties.

m = ucomplexm('m',[1 2 3;4 5 6;7 8 9;10 11 12]) 
Uncertain Complex Matrix: Name m, 4x3 
get(m) 
            Name: 'm' 
    NominalValue: [4x3 double] 
              WL: [4x4 double] 
              WR: [3x3 double] 
    AutoSimplify: 'basic' 
m.NominalValue 
ans = 
     1     2     3 
     4     5     6 
     7     8     9 
    10    11    12 
m.WL 
ans = 
     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 generally equal, indicative of the default (identity) weighting matrices that are in place.

abs(usample(m)-m.NominalValue) 
ans = 
    0.2948    0.1001    0.2867 
    0.3028    0.2384    0.2508 
    0.3376    0.1260    0.2506 
    0.2200    0.3472    0.1657 

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, 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, which is completely expected by choice of the diagonal weighting matrices.

abs(usample(m)-m.NominalValue) 
ans = 
    0.0091    0.0860    0.2753 
    0.0057    0.1717    0.6413 
    0.0304    0.2756    1.4012 
    0.0527    0.4099    1.8335 
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