Robust Control Toolbox™

Decomposing Uncertain Objects (for Advanced Users)

Each uncertain object (umat, uss, ufrd) is a generalized feedback connection (lft) of a not-uncertain object (e.g., double, ss, frd) with a diagonal augmentation of uncertain atoms (ureal, ultidyn, ucomplex, ucomplexm, udyn). In robust control jargon, if the uncertain elements are normalized, this decomposition is often called "the M/D form."

The purpose of the uncertain objects (ureal, ultidyn, umat, uss, etc.) is to hide this underlying decomposition, and allow the user to focus on modeling and analyzing uncertain systems, rather than the details of correctly propagating the M/D representation in manipulations. Nevertheless, advanced users may want access to the familiar M/D form. The command lftdata accomplishes this decomposition.

Since ureal, ucomplex and ucomplexm do not have their NominalValue necessarily at zero, and in the case of ureal objects, are not symmetric about the NominalValue, some details are required in describing the decomposition.

Normalizing Functions for Uncertain Atoms

Associated with each uncertain element is a normalizing function. The normalizing function maps the uncertain element into a normalized uncertain element.

If is an uncertain real parameter, with range [L R] and nominal value N, then the normalizing function F is

with the property that for all satisfying LR, it follows that -1F()1, moreover, F(L)=-1, F(N)=0, and F(R)=1. If the nominal value is centered in the range, then it is easy to conclude that

, , C=1, D=0

It is left as an algebra exercise for the user to work out the various values for A, B, C and D when the nominal value is not centered.

If E is an uncertain gain-bounded, linear, time-invariant dynamic uncertainty, with gain-bound , then the normalizing function F is

If E is an uncertain positive-real, linear, time-invariant dynamic uncertainty, with positivity bound , then the normalizing function F is

where =2||+1.

The normalizing function for an uncertain complex parameter , with nominal value C and radius is

The normalizing function for uncertain complex matrices H, with nominal value N and weights W_L and W_R is

In each case, as the uncertain atom varies over its range, the absolute value of the normalizing function (or norm, in the matrix case) varies from 0 and 1.

Properties of the Decomposition

Take an uncertain object A, dependent on uncertain real parameters ₁,...,_N, uncertain complex parameters ₁,...,_K, uncertain complex matrices H₁,...,H_B, uncertain gain-bounded linear, time-invariant dynamics E₁,...,E_D, and uncertain positive-real linear, time-invariant dynamics P₁,...,P_Q.

Write A(,,H,E,P) to indicate this dependence. Using lftdata, A can be decomposed into two separate pieces: M and (,,H,E,P) with the following properties:

• M is certain (i.e., if A is uss, then M is ss; if A is umat, then M is double; if A is ufrd, then M is frd).
• is always a umat, depending on the same uncertain elements as A, with ranges, bounds, weights, etc., unaltered.
• The form of is block diagonal, with elements made up of the normalizing functions acting on the individual uncertain elements.

• A(,,H,E,P) is given by a linear fractional transformation of M and (,,H,E,P),

The order of the normalized atoms making up A is not the simple order shown above. It is actually the same order as given by the command fieldnames(M.Uncertainty). See Advanced Syntax of lftdata for more information.

Syntax of lftdata

The decomposition is carried out by the command lftdata.

You can create a 2-by-2 umat named A using three uncertain real parameters.

• delta = ureal('delta',2); 
  eta = ureal('eta',6); 
  rho = ureal('rho',-1); 
  A = [3+delta+eta delta/eta;7+rho rho+delta*eta] 
  UMAT: 2 Rows, 2 Columns 
    delta: real, nominal = 2, variability = [-1  1], 2 occurrences 
      eta: real, nominal = 6, variability = [-1  1], 3 occurrences 
      rho: real, nominal = -1, variability = [-1  1], 1 occurrence 
  

Note that A depends on two occurrences of delta, three occurrences of eta and one occurrence of rho.

Decompose A into M and Delta. Note that M is a double, and Delta has the same uncertainty dependence as A.

• [M,Delta] = lftdata(A); 
  class(M) 
  ans = 
  double 
  Delta 
  UMAT: 6 Rows, 6 Columns 
    delta: real, nominal = 2, variability = [-1  1], 2 occurrences 
      eta: real, nominal = 6, variability = [-1  1], 3 occurrences 
      rho: real, nominal = -1, variability = [-1  1], 1 occurrence 
  

Sample Delta at 5 points. Things to note are: it is diagonal; the values range between -1 and 1; there are 3 independent values, and duplication of the entries is consistent with the dependence of Delta and A on the 3 uncertain real parameters.

• usample(Delta,5) 
  ans(:,:,1) = 
     -0.7106         0         0         0         0         0 
           0   -0.7106         0         0         0         0 
           0         0    0.6374         0         0         0 
           0         0         0    0.6374         0         0 
           0         0         0         0    0.6374         0 
           0         0         0         0         0   -0.1258 
  ans(:,:,2) = 
     -0.5850         0         0         0         0         0 
           0   -0.5850         0         0         0         0 
           0         0   -0.3021         0         0         0 
           0         0         0   -0.3021         0         0 
           0         0         0         0   -0.3021         0 
           0         0         0         0         0    0.0803 
  ans(:,:,3) = 
      0.7013         0         0         0         0         0 
           0    0.7013         0         0         0         0 
           0         0   -0.6749         0         0         0 
           0         0         0   -0.6749         0         0 
           0         0         0         0   -0.6749         0 
           0         0         0         0         0    0.3967 
  ans(:,:,4) = 
      0.4262         0         0         0         0         0 
           0    0.4262         0         0         0         0 
           0         0    0.0795         0         0         0 
           0         0         0    0.0795         0         0 
           0         0         0         0    0.0795         0 
           0         0         0         0         0   -0.9959 
  ans(:,:,5) = 
     -0.8392         0         0         0         0         0 
           0   -0.8392         0         0         0         0 
           0         0    0.8467         0         0         0 
           0         0         0    0.8467         0         0 
           0         0         0         0    0.8467         0 
           0         0         0         0         0    0.6732 
  

In fact, verify that the maximum gain of Delta is indeed 1.

• maxnorm = wcnorm(Delta) 
  maxnorm = 
      LowerBound: 1.0000 
      UpperBound: 1.0004 
  

Finally, verify that lft(Delta,M) is the same as A. Subtract (and use the 'full' option in simplify)

• simplify(lft(Delta,M)-A,'full') 
  ans = 
       0     0 
       0     0 
  

Advanced Syntax of lftdata

Even for the advanced user, the variable Delta will actually not be that useful, as it is still a complex object. On the other hand, its internal structure is described completely using a 3rd (and 4th) output argument.

• [M,Delta,BlkStruct,NormUnc] = lftdata(A); 
  

The rows of BlkStruct correspond to the uncertain atoms named in fieldnames(A.Uncertainty). Note that the range/bound information about each uncertain atom is not included in BlkStruct.

The elements of BlkStruct describe the size, type and number-of-copies of the uncertain atoms in A, and implicitly delineate the exact block-diagonal structure of Delta. Note that the range/bound information about each uncertain atom is not included in BlkStruct.

• BlkStruct(1) 
  ans = 
             Name: 'delta' 
             Size: [1 1] 
             Type: 'ureal' 
      Occurrences: 2 
  BlkStruct(2) 
  ans = 
             Name: 'eta' 
             Size: [1 1] 
             Type: 'ureal' 
      Occurrences: 3 
  BlkStruct(3) 
  ans = 
             Name: 'rho' 
             Size: [1 1] 
             Type: 'ureal' 
      Occurrences: 1 
  

Together, these mean Delta is a block diagonal augmentation of the normalized version of 3 uncertain atoms.

The first atom is named 'delta'. It is 1-by-1; it is of class ureal; and there are 2 copies diagonally augmented.

The second atom is named 'eta'. It is 1-by-1; it is of class ureal; and there are 3 copies diagonally augmented.

The third atom is named 'rho'. It is 1-by-1; it is of class ureal; and there is 1 copy,

The 4th output argument contains a cell array of normalized uncertain elements. The cell array contains as many occurrences of each element as there are occurrences in the original uncertain object A.

• size(NormUnc) 
  ans = 
       6     1 
  NormUnc{1} 
  Uncertain Real Parameter: Name deltaNormalized, NominalValue 0, 
  variability = [-1  1] 
  isequal(NormUnc{2},NormUnc{1}) 
  ans = 
       1 
  NormUnc{3} 
  Uncertain Real Parameter: Name etaNormalized, NominalValue 0, 
  variability = [-1  1] 
  isequal(NormUnc{4},NormUnc{3}) 
  ans = 
       1 
  isequal(NormUnc{5},NormUnc{3}) 
  ans = 
       1 
  NormUnc{6} 
  Uncertain Real Parameter: Name rhoNormalized, NominalValue 0, 
  variability = [-1  1] 
  

Each normalized element has the string 'Normalized' appended to its original name to avoid confusion. By normalized,

• ureal objects have nominal value of 0, and range from -1 to 1.
• ultidyn objects are norm bounded, 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 possible behaviors of Delta and blkdiag(NormUnc{:}) are the same. Consequently, the possible behaviors of A and lft(blkdiag(NormUnc{:}),M) are the same.

Hence, by manipulating M, BlkStruct and NormUnc, a power-user has direct access to all of the linear fractional transformation details, and can easily work at the level of the theorems and algorithms that underlie the methods.

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Array Management for Uncertain Objects

Generalized Robustness Analysis

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