lftdata

Decompose uncertain objects into fixed certain and normalized uncertain parts

Syntax

[M,Delta] = lftdata(A)

[M,Delta,BlockStructure] = lftdata(A)

[M,Delta,BlockStructure,normunc] = lftdata(A)

[___] = lftdata(A,List)

Description

[M,Delta] = lftdata(A) separates the uncertain object A into a certain object M and a normalized uncertain matrix Delta such that A is equal to lft(Delta,M), as shown in the following illustration.

[M,Delta,BlockStructure] = lftdata(A) returns a structure array BlockStructure, where BlockStructure(i) describes the ith normalized uncertain element. You can use this structure to specify the uncertainty structure for singular value analysis with mussv.

[M,Delta,BlockStructure,normunc] = lftdata(A) returns the cell array normunc of normalized uncertain elements. Each normalized element has 'Normalized' appended to its original name. lft(blkdiag(normunc{:}),M) is equivalent to A. The normalizations for each type of uncertain element are described in Decomposing Uncertain Objects.

[___] = lftdata(A,List) separates A into M in feedback with Delta, such that Delta that includes only the uncertain elements of A specified in List. All other uncertainty in A remains in M. lftdata(A,fieldnames(A.Uncertainty)) is the same as lftdata(A). This kind of separation is called partial decomposition. You can use this syntax with any of the output-argument combinations.

Examples

Decompose Uncertain System

Create an uncertain matrix A with 2 uncertain real parameters v1 and v2. The create an uncertain system G using A as the dynamic matrix, and numeric matrices for the input and output.

A = [ureal('p1',-3,'perc',40) 1;1 ureal('p2',-2)]; 
sys = ss(A,[1;0],[0 1],0);
sys.InputGroup.ActualIn = 1; 
sys.OutputGroup.ActualOut = 1; 

You can decompose G into a certain system, Msys, and a normalized uncertain matrix, Delta. You can see from Msys that it is certain, and that the input and output groups have been adjusted.

[Msys,Delta] = lftdata(sys); 
Msys 

Msys =
 
  A = 
       x1  x2
   x1  -3   1
   x2   1  -2
 
  B = 
          u1     u2     u3
   x1  1.095      0      1
   x2      0      1      0
 
  C = 
          x1     x2
   y1  1.095      0
   y2      0      1
   y3      0      1
 
  D = 
       u1  u2  u3
   y1   0   0   0
   y2   0   0   0
   y3   0   0   0
 
Input groups:           
      Name      Channels
    ActualIn       3    
                        
Output groups:           
      Name       Channels
    ActualOut       3    
                         
Continuous-time state-space model.

You can compute the norm on samples of the difference between the original uncertain matrix and the result formed by combining Msys and Delta.

norm(usample(sys-lft(Delta,Msys),'p1',4,'p2',3),'inf') 

ans = 4×3

     0     0     0
     0     0     0
     0     0     0
     0     0     0

Decompose Uncertain Matrix

Create an uncertain matrix A with uncertain parameters p1, and p2. Decompose A into its certain part, M, and normalized uncertain part, Delta.

p1 = ureal('p1',-3,'perc',40); 
p2 = ucomplex('p2',2); 
A = [p1 p1+p2;1 p2]; 
[M,Delta] = lftdata(A); 

You can inspect the difference between the original uncertain matrix, A, and the result formed by combining the two results from the decomposition.

simplify(A-lft(Delta,M)) 

ans = 2×2

     0     0
     0     0

M

M = 4×4

         0         0    1.0954    1.0954
         0         0         0    1.0000
    1.0954    1.0000   -3.0000   -1.0000
         0    1.0000    1.0000    2.0000

You can check the worst-case norm of the uncertain part using wcnorm. Compare samples of the uncertain part A with the uncertain matrix A.

wcn = wcnorm(Delta) 

wcn = struct with fields:
    LowerBound: 0
    UpperBound: 1.0012

usample(Delta,3) 

ans = 
ans(:,:,1) =

   0.6294 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.8776 - 0.4413i


ans(:,:,2) =

   0.8116 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.2704 + 0.2907i


ans(:,:,3) =

  -0.7460 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -0.3382 + 0.9253i

Partial Decomposition

Create an uncertain matrix A and derive an uncertain matrix B using an implicit-to-explicit conversion, imp2exp. Note that B has 2 uncertain parameters R and K. You can decompose B into certain part, M, and normalized uncertain part, Delta.

R = ureal('R',1,'Percentage',[-10 40]); 
K = ureal('K',2e-3,'Percentage',[-30 30]); 
A = [1 -R 0 -K;0 -K 1 0]; 
Yidx = [1 3]; 
Uidx = [4 2]; 
B = imp2exp(A,Yidx,Uidx); 
[M,Delta] = lftdata(B); 

The same operation can be performed by defining the uncertain parameters, K and R, to be extracted.

[MK,DeltaR] = lftdata(B,'R'); 
MK 

Uncertain matrix with 3 rows and 3 columns.
The uncertainty consists of the following blocks:
  K: Uncertain real, nominal = 0.002, variability = [-30,30]%, 2 occurrences

Type "MK.NominalValue" to see the nominal value and "MK.Uncertainty" to interact with the uncertain elements.

[MR,DeltaK] = lftdata(B,'K'); 
MR 

Uncertain matrix with 4 rows and 4 columns.
The uncertainty consists of the following blocks:
  R: Uncertain real, nominal = 1, variability = [-10,40]%, 1 occurrences

Type "MR.NominalValue" to see the nominal value and "MR.Uncertainty" to interact with the uncertain elements.

simplify(B-lft(Delta,M)) 

ans = 2×2

     0     0
     0     0

simplify(B-lft(DeltaR,MK)) 

ans = 2×2

     0     0
     0     0

simplify(B-lft(DeltaK,MR)) 

ans = 2×2

     0     0
     0     0

Examine the result formed by combining the two results from the decomposition.

[Mall,Deltaall] = lftdata(B,{'K';'R'}); 
Mall-M

ans = 5×5

     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0
     0     0     0     0     0

Input Arguments

A — Uncertain system or matrix

uss | ufrd | umat

Uncertain system or matrix to decompose, specified as a uss model, a ufrd model, or a umat object.

List — Uncertain elements

string array | cell array of character vectors

Uncertain elements to extract, specified as a string array or cell array of character vectors. The entries in List specify the names of the uncertain elements of A that lftdata separates into Delta. Other elements are returned in M.

Example: {'p1','p3'}

Example: ["p1","p3"]

Output Arguments

M — Portion in feedback with extracted uncertainty

ss | frd | matrix | uss | ufrd | umat

Portion of decomposed A in feedback with the extracted uncertainty Delta, returned as follows:

• If List is omitted or specifies all uncertain elements of A, then:

  • If A is a uss model, then M is an ss model.

  • If A is a ufrd model, then M is an frd model.

  • If A is a umat, then M is a numeric matrix.

• If List specifies a subset of the uncertain elements of A, then M is of the same type as A.

M is such that A = lft(M,Delta)

Delta — Extracted uncertainty

umat

Extracted uncertainty, returned as a umat uncertain matrix. The uncertain elements of Delta are the same as the uncertain elements of A (or the subset specified in List). For more information about the structure of Delta see Decomposing Uncertain Objects.

BlockStructure — Description of uncertain elements

structure array

Description of uncertain elements in Delta, returned as an N-by-1 structure array, where N is the number of uncertain elements in Delta. The structure BlockStructure(i) contains information about the ith uncertain element of Delta. The fields of the structure are:

• Name — Name of uncertain element, returned as a character vector.

• Size — Dimensions of element, returned as a vector.

• Type — Element type, returned as a character vector, such as 'ureal', umat, or ucomplex.

• Occurrences — Number of instances of the element in Delta, returned as an integer.

• Simplify — Autosimplify setting of the element, returned as an integer that encodes the value of the AutoSimplify property of the element as follows:

  • 2 — basic

  • 3 — full

  • 0 — off

normunc — Normalized uncertainty

cell array

Normalized uncertainty, returned as a cell array in which each entry is the normalized version of the corresponding block of Delta. Normalization transforms the uncertain blocks to standard nominal values and uncertainty ranges as follows:

• ureal objects have nominal value of 0, and range from –1 to 1.

• ultidyn objects are norm bounded, with norm bound of 1.

• umargin objects are converted to norm-bounded ultidyn objects with norm bound of 1.

• ucomplex objects have nominal value of 0, and radius 1.

• ucomplexm objects have nominal value of 0, and identity matrices for each of the WL and WR weights.

The normalized uncertainties are such that lft(blkdiag(normunc{:}),M) is equivalent to A. Each normalized element in normunc has 'Normalized' appended to its original name to avoid confusion. For instance, if Delta contains an uncertain element with name 'p1', the corresponding entry in normunc has name 'p1Normalized'.

Version History

Introduced before R2006a