Restructuring State Models, part 3

There is one more important kind of transition matrix structure that has both practical and theoretical importance, if not the best numerical properties.

For example, suppose that you have a model that does close to what you need, but not quite close enough. Since linear systems are in general additive, it would be helpful if you could devise a new model, starting with the model you currently have, that "adds on" some new internal state variables to provide the additional behaviors. But to do this, you need to have a structure with states that act with a high degree of independence.

 

Eigenvalue Transformation

Here is a quick review of non-symmetric matrix eigenvalue theory, emphasizing the case of a square, real-valued, non-symmetric matrix. Feel free to skip this entire installment if you are completely up-to-speed on this topic.

For a given real-valued matrix A, an eigenvector or characteristic vector e of that matrix is a special vector such that result of multiplying the matrix times this vector is the original e vector scaled by some constant λ, called an eigenvalue or characteristic value.

   A e  =  e λ  

All matrices have these (complex-valued matrices too). For our case of real-valued square matrices, some of the eigenvalues can be complex numbers, and the corresponding eigenvectors are also complex. It is easy to verify that if a complex number is an eigenvalue, then its complex conjugate is an eigenvalue also, and it has the complex conjugate eigenvector associated.

The real and imaginary parts of a complex-valued eigenvector can be separated and managed in a real valued form. When either of these special vectors is multiplied by the A matrix, the results is a mix of the two real-valued parts rather than a scaled version of the original vector.

α      Real part of the eigenvalue
β      Imaginary part of the eigenvalue

er     Real part of the eigenvector
er     Imaginary part of the eigenvector

$$\left[e_r{e}_i\right]\left[\begin{array}{c}\alpha \beta \\ -\beta \alpha \end{array}\right]$$

To make the eigenvectors unique, the convention is to scale them so that they have unit magnitude — the sum of squared terms in the normalized vector is 1.0. (For a complex eigenvector, the sum of products of the conjugate terms is 1.0 — real and imaginary parts cannot be scaled separately.)

If a vector is an eigenvector of a matrix, and a scalar scaling factor is applied to every term of this matrix, the vector remains an eigenvector, but the eigenvalue changes by the same scaling factor applied to the matrix.

The number of linearly independent eigenvalues you can find for any matrix equals the rank of the matrix. Degenerate eigenvectors with associated eigenvalues 0.0 can be artificially constructed for the case of matrices that are not full rank. This is not a concern for our models, which will be constructed to be full rank.

The eigenvectors (or special pairs) can be collected as columns of matrix E. Eigenvalues (or for the complex pair cases, eigenvalue parts) are collected to form a matrix Λ with the eigenvalues along the main diagonal in the same order as the eigenvectors. Once organized in this fashion, the relationship between the matrix Λ and the matrix E can then be summarized by the following.

      A E  =  E Λ 

The E matrix, being full rank, has an inverse. The following two relationships can then be shown to apply.

      E-1 A  E  =  Λ 
      E Λ  E-1  =  A
 

 

A little more detail

Let's follow through by applying this. Start with the original state transition model equations.

$$x^{i+1}=A{x}^i+B{u}^i$$ $$y^i=C{x}^i$$

Perform an eigenvalue analysis and from this construct the E and Λ matrices. The critical calculations are provided by the Octave eig function, but an alternative is the realeig ^[2] function that uses the eig function and then constructs the real-valued form as previously discussed. This gives you the E matrix, Λ matrix, and E-1 matrix.

Substitute the decomposed matrix form for the A matrix in the original state transition equations, and then premultiply on both sides by the E-1 matrix.

$$x^{i+1}=E\Lambda E^{-1}{x}^i+B{u}^i$$ $$E^{-1}{x}^{i+1}=E^{-1}E\Lambda E^{-1}{x}^i+E^{-1}B{u}^i$$

Now introduce a notation for the new variables generated by the eigenvector mapping. We can now see that the transformed state variable system has the transition matrix in almost-diagonal form.

$$z=E^{-1}xx=Ez$$ $$z^{i+1}=\Lambda z^i+E^{-1}B{u}^i$$ $$y^{i+1}=CE{z}^{i+1}$$

 

Let's test it...

We fire up Octave and enter the data for an almost arbitrarily selected state transition model.

Amat = [ ...
   1.03,  0.14,  0.0,    0.0 ;  ...
  -0.10,  0.92, -0.09,   0.0 ;  ...
   0.0,   0.20,  0.88,  -0.40;  ...
   0.0,   0.0,   0.00,   0.78;   ];
Cmat = [0.86, 0.42, -0.48, -0.60 ];
Bmat = [0, 0, 0, 1.00 ]';

Now apply the realeig function to perform a normal complex-valued eigenvector-eigenvalue decomposition of the matrix, and then reorganize it in real-valued form as discussed. Apply this to the Amat state transition matrix.

% Eigenvector decomposition
[reig, lam, reiginv] = realeig(Amat)

We get the following results for the E, Λ, and E-1 matrices.

reig =
   0.72095   0.28328  -0.31933  -0.20947
  -0.29264   0.16670   0.56193   0.37405
  -0.62816   0.68864   0.00000   0.81460
   0.00000   0.00000   0.00000   0.39068

lam =
   0.97317   0.00000   0.00000   0.00000
   0.00000   0.92841   0.16320   0.00000
   0.00000  -0.16320   0.92841   0.00000
   0.00000   0.00000   0.00000   0.78000

reiginv =
   1.11177   0.63179  -0.61027   1.26368
   1.01414   0.57630   0.89546  -1.87516
   0.27814   1.93763  -0.58345  -0.48949
   0.00000   0.00000   0.00000   2.55966

The Λ matrix has the expected block-diagonal form, indicating two real-valued eigenvalues and one pair of complex conjugate eigenvalues. Let's apply the decomposition formula to try restoring the original matrix.

Amat
testA = reig * lam * reig^-1

Amat =
   1.03000   0.14000  -0.00000   0.00000
  -0.10000   0.92000  -0.09000  -0.00000
  -0.00000   0.20000   0.88000  -0.40000
   0.00000   0.00000   0.00000   0.78000

testA =
   1.03000   0.14000  -0.00000   0.00000
  -0.10000   0.92000  -0.09000  -0.00000
  -0.00000   0.20000   0.88000  -0.40000
   0.00000   0.00000   0.00000   0.78000

These match. The original matrix is successfully reconstructed from the near-diagonal form. The transformation is verified. So let us now use it to construct the transformed state transition equations.

% Diagonalized state equations
Ad = lam
Bd = reiginv * Bmat
Cd = Cmat * reig

Ad =
   0.97317   0.00000   0.00000   0.00000
   0.00000   0.92841   0.16320   0.00000
   0.00000  -0.16320   0.92841   0.00000
   0.00000   0.00000   0.00000   0.78000

Bd =
   1.26368
  -1.87516
  -0.48949
   2.55966

Cd =
   0.798629  -0.016916  -0.038611  -0.648458

 

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