Uncertain matrices (class umat
)
are built from doubles and uncertain elements, using traditional MATLAB^{®} matrix
building syntax. Uncertain matrices can be added, subtracted, multiplied,
inverted, transposed, etc., resulting in uncertain matrices. The rows
and columns of an uncertain matrix are referenced in the same manner
that MATLAB references rows and columns of an array, using parenthesis,
and integer indices. The NominalValue
of a uncertain
matrix is the result obtained when all uncertain elements are replaced
with their own NominalValue
. The uncertain elements
making up a umat
are accessible
through the Uncertainty
gateway, and the properties
of each element within a umat
can
be changed directly. The properties are:
Properties  Meaning  Class 

 Nominal value of element 

 Uncertain blocks in the matrix, stored as a structure
whose fields are named after the uncertain blocks, and contain the
uncertain elements, such as 

 Sampling grid, for 




Using usubs
, specific
values may be substituted for any of the uncertain elements within
a umat
. The command usample
generates
a random sample of the uncertain matrix, substituting random samples
(within their ranges) for each of the uncertain elements.
The command wcnorm
computes
tight bounds on the worstcase (maximum over the uncertain elements'
ranges) norm of the uncertain matrix.
Standard MATLAB numerical matrices (i.e., double
)
naturally can be viewed as uncertain matrices without any uncertainty.
You create uncertain matrices (umat
objects) by creating uncertain parameters and using them to build matrices. You can then use uncertain matrices to build uncertain statespace models. This example shows how to create an uncertain matrix, access and change its uncertain parameters, extract elements, and perform matrix arithmetic.
For example, create two uncertain real parameters, and use them to create a 3by2 uncertain matrix.
a = ureal('a',3); b = ureal('b',10,'Percentage',20); M = [a, 1/b; b, a+1/b; 1, 3]
M = Uncertain matrix with 3 rows and 2 columns. The uncertainty consists of the following blocks: a: Uncertain real, nominal = 3, variability = [1,1], 2 occurrences b: Uncertain real, nominal = 10, variability = [20,20]%, 3 occurrences Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
Examine and Modify umat
Properties
M
is a umat
object. Examine its properties using get
.
get(M)
NominalValue: [3×2 double] Uncertainty: [1×1 struct] SamplingGrid: [1×1 struct] Name: ''
The nominal value of M
is the matrix obtained by replacing all the uncertain elements with their nominal values.
M.NominalValue
ans = 3.0000 0.1000 10.0000 3.1000 1.0000 3.0000
The Uncertainty
property is a structure containing the uncertain elements (the Control Design Blocks (Control System Toolbox)) of M
.
M.Uncertainty
ans = struct with fields: a: [1×1 ureal] b: [1×1 ureal]
M.Uncertainty.a
ans = Uncertain real parameter "a" with nominal value 3 and variability [1,1].
Use the Uncertainty
property for direct access to the uncertain elements. For example, check the Range
of the uncertain element a
within M
.
M.Uncertainty.a.Range
ans = 2 4
The range is [2,4]
because you created the ureal
parameter a
with a nominal value 3 and the default uncertainty of +/ 1. Change the range to [2.5,5]
.
M.Uncertainty.a.Range = [2.5,5]
M = Uncertain matrix with 3 rows and 2 columns. The uncertainty consists of the following blocks: a: Uncertain real, nominal = 3, variability = [0.5,2], 2 occurrences b: Uncertain real, nominal = 10, variability = [20,20]%, 3 occurrences Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
This change to a
only takes place within M
. Verify that the variable a
in the MATLAB workspace still has the original range.
a.Range
ans = 2 4
You cannot combine elements that have a common internal name, but different properties. So, for example, entering M.Uncertainty.a  a
would generate an error, because the realp
parameter a
in the workspace has different properties from the element a
in M
.
Row and Column Referencing
You can use standard rowcolumn referencing to extract elements from a umat
. For example, extract a 2by2 selection from M
consisting of its second and third rows.
Msub = M(2:3,:)
Msub = Uncertain matrix with 2 rows and 2 columns. The uncertainty consists of the following blocks: a: Uncertain real, nominal = 3, variability = [0.5,2], 1 occurrences b: Uncertain real, nominal = 10, variability = [20,20]%, 2 occurrences Type "Msub.NominalValue" to see the nominal value, "get(Msub)" to see all properties, and "Msub.Uncertainty" to interact with the uncertain elements.
You can use single indexing only if the umat
is a single column or row. Make a singlecolumn selection from M
and use singleindex references to access elements of it.
Msing = M([2 1 2 3],2); Msing(2)
ans = Uncertain matrix with 1 rows and 1 columns. The uncertainty consists of the following blocks: b: Uncertain real, nominal = 10, variability = [20,20]%, 1 occurrences Type "ans.NominalValue" to see the nominal value, "get(ans)" to see all properties, and "ans.Uncertainty" to interact with the uncertain elements.
You can use indexing to change the value of any element of a umat
. For example, set the (3,2) entry of M
to an uncertain parameter c
.
c = ureal('c',3,'Percentage',40); M(3,2) = c
M = Uncertain matrix with 3 rows and 2 columns. The uncertainty consists of the following blocks: a: Uncertain real, nominal = 3, variability = [0.5,2], 2 occurrences b: Uncertain real, nominal = 10, variability = [20,20]%, 2 occurrences c: Uncertain real, nominal = 3, variability = [40,40]%, 1 occurrences Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
M now has three uncertain blocks.
Matrix Operations on umat
Objects
You can perform many matrix operations on a umat
object, such as matrixmultiply, transpose, and inverse. You can also combaine uncertain matrices with numeric matrices that do not have uncertainty.
For example, premultiply M
by a 1by3
numeric matrix, resulting in a 1by2 umat
.
M1 = [2 3 1]*M;
Verify that the first entry of M1
is as expected, 2*a + 3*b + 1
.
d = M1(1)  (2*M.Uncertainty.a + 3*M.Uncertainty.b + 1)
d = Uncertain matrix with 1 rows, 1 columns, and no uncertain blocks. Type "d.NominalValue" to see the nominal value, "get(d)" to see all properties, and "d.Uncertainty" to interact with the uncertain elements.
Transpose M
, form a product, and invert it. As expected, the product of a matrix and its inverse is the identity matrix. You can verify this by sampling the result.
H = M.'*M; K = inv(H); usample(K*H,3)
ans(:,:,1) = 1.0000 0.0000 0.0000 1.0000 ans(:,:,2) = 1.0000 0.0000 0.0000 1.0000 ans(:,:,3) = 1.0000 0.0000 0.0000 1.0000
Lifting a Double Matrix to umat
You can convert a numeric matrix to a umat
object with no uncertain elements. Use the umat
command to lift a double matrix to the umat
class. For example:
Md = [1 2 3;4 5 6]; M = umat(Md)
M = Uncertain matrix with 2 rows, 3 columns, and no uncertain blocks. Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
You can also convert higherdimension numeric matrices to umat
. When you do so, the software interprets the third dimension and beyond as array dimensions. For example, convert a random threedimensional numeric array to umat
.
Md = randn(4,5,6); M = umat(Md)
M = 6x1 array of uncertain matrices with 4 rows, 5 columns, and no uncertain blocks. Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
The result is a onedimensional array of uncertain matrices, rather than a threedimensional uncertain array. Similarly, a fourdimensional numeric array converts to a twodimensional array of umat
objects.
Md = randn(4,5,6,7); M = umat(Md)
M = 6x7 array of uncertain matrices with 4 rows, 5 columns, and no uncertain blocks. Type "M.NominalValue" to see the nominal value, "get(M)" to see all properties, and "M.Uncertainty" to interact with the uncertain elements.
See Array Management for Uncertain Objects for more information about multidimensional arrays of uncertain objects.