Calculates the left null-space of the basis of similarity transformations.
U2 = simlns(A,B,C,,,)
U2 = simlns(A,B,C,K,fD,fx)
The function simlns calculates the left null-space of an LTI system's similarity map Mtheta. In the most general case, when A, B, C, D, K and x0 are part of the parameter vector, this matrix is given by :
A QR-factorization is used to obtain the left null-space.
A,B,C are the system matrices describing the LTI State Space system.
K is the (optional) Kalman gain, specify as empty matrix when not present.
fD (optional) specifies whether D is part of the parameter vector, specify as empty, 0 or 1.
fx (optional) specifies whether x0 is part of the parameter vector, specify as empty, 0 or 1.
U2 is the left null-space of the similarity map.
Specifying fx=1 only causes an n x n identify-matrix to be appended to the lower right of the left null-space matrix; in a non-linear optimization, applying the left null-space ensures that the state-basis does not change. It thus does not have to be projected.
The manifold matrix Mtheta is calculated according to . A QR-factorization is used subsequently to obtain the left null-space
 L. H. Lee and K. Poolla, "Identification of linear parameter-varying systems using nonlinear programming", Journal of Dynamic Systems, Measurement and Control, vol. 121, pp. 71-78, Mar. 1999.