findABCD

Arguments

S

integer, the number of block rows in the block-Hankel matrices

N

integer, the system order

L

integer, the number of output

R

matrix, relevant part of the R factor of the concatenated block-Hankel matrices computed by a call to findr.

METH

integer, an option for the method to use

= 1

MOESP method with past inputs and outputs;

= 2

N4SID method;

= 3

combined method: A and C via MOESP, B and D via N4SID.

Default: METH = 3.

NSMPL

integer, the total number of samples used for calculating the covariance matrices and the Kalman predictor gain. This parameter is not needed if the covariance matrices and/or the Kalman predictor gain matrix are not desired. If NSMPL = 0, then K, Q, Ry, and S are not computed. Default: NSMPL = 0.

TOL

the tolerance used for estimating the rank of matrices. If TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition number. Default: prod(size(matrix))*epsilon_machine where epsilon_machine is the relative machine precision.

PRINTW

integer, switch for printing the warning messages.

PRINTW

= 1: print warning messages;

PRINTW

= 0: do not print warning messages.

Default: PRINTW = 0.

SYS

computes a state-space realization SYS = (A,B,C,D) (an syslin object)

K

the Kalman predictor gain K (if NSMPL > 0)

Q

state covariance

Ry

output covariance

S

state-output cross-covariance

RCND

vector, reciprocal condition numbers of the matrices involved in rank decisions, least squares or Riccati equation solutions

Description

Finds the system matrices and the Kalman gain of a discrete-time system, given the system order and the relevant part of the R factor of the concatenated block-Hankel matrices, using subspace identification techniques (MOESP and/or N4SID).

• [SYS,K] = findABCD(S,N,L,R,METH,NSMPL,TOL,PRINTW) computes a state- space realization SYS = (A,B,C,D) (an ss object), and the Kalman predictor gain K (if NSMPL > 0). The model structure is:

  x(k+1) = Ax(k) + Bu(k) + Ke(k),   k >= 1,
  y(k)   = Cx(k) + Du(k) + e(k),

  where x(k) and y(k) are vectors of length N and L, respectively.

• [SYS,K,Q,Ry,S,RCND] = findABCD(S,N,L,R,METH,NSMPL,TOL,PRINTW) also returns the state, output, and state-output (cross-)covariance matrices Q, Ry, and S (used for computing the Kalman gain), as well as the vector RCND of length lr containing the reciprocal condition numbers of the matrices involved in rank decisions, least squares or Riccati equation solutions, where

lr = 4,  if Kalman gain matrix K is not required, and
lr = 12, if Kalman gain matrix K is required.

Matrix R, computed by findR, should be determined with suitable arguments METH and JOBD. METH = 1 and JOBD = 1 must be used in findR, for METH = 1 in findABCD; METH = 1 must be used in findR, for METH = 3 in findABCD.

Examples

//generate data from a given linear system
A = [ 0.5, 0.1,-0.1, 0.2;
      0.1, 0,  -0.1,-0.1;
     -0.4,-0.6,-0.7,-0.1;
      0.8, 0,  -0.6,-0.6];
B = [0.8;0.1;1;-1];
C = [1 2 -1 0];
SYS=syslin(0.1,A,B,C);
nsmp=100;
U=prbs_a(nsmp,nsmp/5);
Y=(flts(U,SYS)+0.3*rand(1,nsmp,'normal'));

// Compute R
S=15;
[R,N1,SVAL] = findR(S,Y',U');
N=3;
SYS1 = findABCD(S,N,1,R) ;SYS1.dt=0.1;

SYS1.X0 = inistate(SYS1,Y',U');

Y1=flts(U,SYS1);
clf();plot2d((1:nsmp)',[Y',Y1'])

See also

• findAC — discrete-time system subspace identification
• findBD — initial state and system matrices B and D of a discrete-time system
• findBDK — Kalman gain and B D system matrices of a discrete-time system
• findR — Preprocessor for estimating the matrices of a linear time-invariant dynamical system
• sorder — computing the order of a discrete-time system
• sident — discrete-time state-space realization and Kalman gain