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Uncertain Complex Parameters and Matrices

Uncertain Complex Parameters

The ucomplex element is the Control Design Block (Control System Toolbox) 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. Displaying the properties shows that 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)
    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 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 σ¯˙(Δ)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

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])
m =

  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 = 

     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 = 

     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 = 

    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 = 

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

See Also

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