System Identification Toolbox

Initial Parameter Values

When only orders and delays are specified, the functions armax, bj, oe, and pem use a startup procedure to produce initial values. The startup procedure goes through two to four least squares and instrumental variables steps. It is reasonably efficient in that it usually saves several iterations in the minimization phase. Sometimes, however, it might pay to use other initial conditions. For example, you can use an iv4 estimate computed earlier as an initial condition for estimating an output-error model of the same structure.

• m1 = iv4(Data,[na nb nk]);
  set(m1,'a',1,'f',m1.a)
  m2= oe(Data,m1);
  

Another example is when you want to try a model with one more delay (for example, three instead of two) because the leading b-coefficient is quite small.

• m1 = armax(Data,[3 3 2 2]);
  m1.b(3) = 0
  m2 = armax(Data,m1); 
  

If you decrease the number of delays, remember that leading zeros in the B-polynomial are treated as delays. Suppose you go from three to two delays in the above example:

• m1 = armax(z,[3 3 2 3]);
  m1.b(3) = 0.00001;
  m2 = armax(Data,m1);
  

Note that when you construct homemade initial conditions, the conditions must correspond to a stable predictor (C and F being Hurwitz polynomials), and they should not contain any exact pole-zero cancellations.

For user-defined structured state-space and multioutput models, you must provide the initial parameter values (initial model) when defining the structure in idss or idgrey. The basic approach is to use physical insight to choose initial values of the parameters with physical significance, and try some different (randomized) initial values for the others. The routine init can be used for that.

For "free" state-space parameterizations, it can sometimes be difficult to reach the global minimum. If you see that the minimization routine seems to get stuck (turn trace on and check the improvements per iteration), it may be a good idea to transform state-space matrices to other realizations, as in

• m = pem(Data,5,'trace','on')
  m.ss = 'can';
  m = pem(Data,m);
  m = balreal(m); % If you have the Control System Toolbox
  m = pem(Data,m);
  

Local Minima

Initial State