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idpoly - Linear polynomial input-output model

Syntax

m = idpoly(A,B)
m = idpoly(A,B,C,D,F,NoiseVariance,Ts)
m = idpoly(A,B,C,D,F,NoiseVariance,Ts,'Property1',Value1,... 
        'PropertyN',ValueN)
m = idpoly(mi)

Description

idpoly creates a model object containing parameters that describe the general multiple-input single-output model structure.

A, B, C, D, and F specify the polynomial coefficients.

For single-input systems, these are all row vectors.

A = [1 a1 a2 ... ana]

consequently describes

A, C, D, and F all start with 1, while B contains leading zeros to indicate the delays. See What Are Black-Box Polynomial Models?.

For multiple-input systems, B and F are matrices with one row for each input.

For time series, B and F are entered as empty matrices.

B = [];  F = [];

NoiseVariance is the variance of the white noise sequence , while Ts is the sampling interval.

Trailing arguments C, D, F, NoiseVariance, and Ts can be omitted, in which case they are taken as 1. (If B = [], then F is taken as [].) The property name/property value pairs can start directly after B.

Ts = 0 means that the model is a continuous-time one. Then the interpretation of the arguments is that

A = [1 2 3 4]

corresponds to the polynomial in the Laplace variable s, and so on. For continuous-time systems, NoiseVariance indicates the level of the spectral density of the innovations. A sampled version of the model has the innovations variance NoiseVariance/Ts, where Ts is the sampling interval. The continuous-time model must have a white noise component in its disturbance description. See Spectrum Normalization.

For discrete-time models (Ts > 0), note the following: idpoly strips any trailing zeros from the polynomials when determining the orders. It also strips leading zeros from the B polynomial to determine the delays. Keep this in mind when you use idpoly and polydata to modify earlier estimates to serve as initial conditions for estimating new structures.

idpoly can also take any single-output idmodel or LTI object mi as an input argument. If an LTI system has an input group with name 'Noise', these inputs are interpreted as white noise with unit variance, and the noise model of the idpoly model is computed accordingly.

Properties

na, nb, nc, nd, nf, nk: The orders and delays of the polynomials. Integers or row vectors of integers.
a, b, c, d, f: The polynomials, described by row vectors and matrices as detailed above.
da, db, dc, dd, df: The estimated standard deviation of the polynomials. Cannot be set.
'InitialState': How to deal with the initial conditions that are required to compute the prediction of the output. Possible values are:
- 'Estimate': The necessary initial states are estimated from data as extra parameters.
- 'Backcast': The necessary initial states are estimated by a backcasting (backward filtering) process, described in Knudsen (1994).
- 'Zero': All initial states are taken as zero.
- 'Auto': An automatic choice among the above is made, guided by the data.

In addition, any idpoly object also has all the properties of idmodel. See idmodel properties and Algorithm Properties.

Note that you can set or retrieve all properties either with the set and get commands or by subscripts. Autofill applies to all properties and values, and these are case insensitive.

m.a=[1 -1.5 0.7];
set(m,'ini','b')
p = roots(m.a)

For a complete list of property values, use get(m). To see possible value assignments, use set(m).

Examples

To create a system of ARMAX, type

A = [1 -1.5 0.7];
B = [0 1 0.5];
C = [1 -1 0.2];
m0 = idpoly(A,B,C);

This gives a system with one delay (nk = 1).

Create the continuous-time model

Sample it with T = 0.1 and then simulate it without noise.

B=[0 1;1 3];
F=[1 1 0;1 2 4]
m = idpoly(1,B,1,1,F,1,0)
md = c2d(m,0.1)
y = sim(md,[u1 u2])

Note that the continuous-time model is automatically sampled to the sampling interval of the data, when simulated, so the above is also achieved by

u = iddata([],[u1 u2],0.1)
y = sim(m,u)

idpoly Definition of States

The states of an idpoly model are defined as those corresponding to the model obtained by converting them to the state-space format using the idss command. For example, if you have an idpoly model defined by m1 = idpoly([1 2 1],[2 2]), then the initial states of this model correspond to those of m2 = idss(m1). The concept of states is useful for functions such as sim, predict, compare and findstates.

References

Ljung (1999) Section 4.2 for the model structure family.

Knudsen, T., (1994), "New method for estimating ARMAX models," In Proc. 10th IFAC Symposium on System Identification, pp. 611-617, Copenhagen, Denmark, for the backcast method.