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lftdata

Decompose uncertain objects into fixed certain and normalized uncertain parts

Syntax

[M,Delta] = lftdata(A);
[M,Delta] = lftdata(A,List);
[M,Delta,Blkstruct] = lftdata(A);
[M,Delta,Blkstruct,Normunc] = lftdata(A);

Description

lftdata decomposes an uncertain object into a fixed certain part and a normalized uncertain part. lftdata can also partially decompose an uncertain object into an uncertain part and a normalized uncertain part. Uncertain objects (umat, ufrd, uss) are represented as certain (i.e., not-uncertain) objects in feedback with block-diagonal concatenations of uncertain elements.

[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 below.

If A is a umat, then M will be double; if A is a uss, then M will be ss; if A is a ufrd, then M will be frd. In all cases, Delta is a umat.

[M,Delta] = lftdata(A,List) separates the uncertain object A into an uncertain object M, in feedback with a normalized uncertain matrix Delta. List is a cell (or char) array of names of uncertain elements of A that make up Delta. All other uncertainty in A remains in M.

lftdata(A,fieldnames(A.Uncertainty)) is the same as lftdata(A).

[M,DELTA,BLKSTRUCT] = lftdata(A) returns an N-by-1 structure array BLKSTRUCT, where BLKSTRUCT(i) describes the i-th normalized uncertain element. This uncertainty description can be passed directly to the low-level structured singular value analysis function mussv.

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

Examples

collapse all

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

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

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 
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, "get(MK)" to see all properties, and "MK.Uncertainty" to interact with the uncertain elements.
[MR,DeltaK] = lftdata(B,'K'); 
MR 
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, "get(MR)" to see all properties, 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

Introduced before R2006a