Example of State-Space Diagonalization

For the example of Eq.(G.7), we obtain the following results:

>> % Initial state space filter from example above:
>> A = [-1/2, -1/3; 1, 0]; % state transition matrix
>> B = [1; 0];
>> C = [2-1/2, 3-1/3];
>> D = 1;
>>
>> eig(A) % find eigenvalues of state transition matrix A

ans =
  -0.2500 + 0.5204i
  -0.2500 - 0.5204i

>> roots(den) % find poles of transfer function H(z)

ans =
  -0.2500 + 0.5204i
  -0.2500 - 0.5204i

>> abs(roots(den)) % check stability while we're here

ans =
    0.5774
    0.5774

% The system is stable since each pole has magnitude < 1.

Our second-order example is already in real form, because it is only second order. However, to illustrate the computations, let's obtain the eigenvectors and compute the complex modal representation:

>> [E,L] = eig(A)  % [Evects,Evals] = eig(A)

E =

  -0.4507 - 0.2165i  -0.4507 + 0.2165i
        0 + 0.8660i        0 - 0.8660i


L =

  -0.2500 + 0.5204i        0          
        0            -0.2500 - 0.5204i

>> A * E - E * L   % should be zero (A * evect = eval * evect)

ans =
  1.0e-016 *
        0 + 0.2776i        0 - 0.2776i
        0                  0          

% Now form the complete diagonalized state-space model (complex):

>> Ei = inv(E); % matrix inverse
>> Ab = Ei*A*E % new state transition matrix (diagonal)

Ab =
  -0.2500 + 0.5204i   0.0000 + 0.0000i
  -0.0000            -0.2500 - 0.5204i

>> Bb = Ei*B   % vector routing input signal to internal modes

Bb =
   -1.1094
   -1.1094

>> Cb = C*E    % vector taking mode linear combination to output

Cb =
  -0.6760 + 1.9846i  -0.6760 - 1.9846i

>> Db = D      % feed-through term unchanged

Db =
     1

% Verify that we still have the same transfer function:

>> [numb,denb] = ss2tf(Ab,Bb,Cb,Db)

numb =
   1.0000             2.0000 + 0.0000i   3.0000 + 0.0000i

denb =
   1.0000             0.5000 - 0.0000i   0.3333          

>> num = [1, 2, 3]; % original numerator
>> norm(num-numb)

ans =
  1.5543e-015

>> den = [1, 1/2, 1/3]; % original denominator
>> norm(den-denb)

ans =
  1.3597e-016

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.