This paper deals with the parameter estimation problem for multivariable nonlinear systems described by MIMO state-space Wiener models. Recursive parameters and state estimation algorithms are presented using the least squares technique, the adjustable model, and the Kalman filter theory. The basic idea is to estimate jointly the parameters, the state vector, and the internal variables of MIMO Wiener models based on a specific decomposition technique to extract the internal vector and avoid problems related to invertibility assumption. The effectiveness of the proposed algorithms is shown by an illustrative simulation example.

Over the last years, modeling, identification, and parameter estimation theories have received much attention by various research teams [

Recently, much attention has been paid to blocks-oriented state-space systems which have been successfully used for control algorithms, identification schemes, and signal filtering [

The main difficulty in the identification of Wiener models is that the internal variables, acting between linear and nonlinear blocks, are almost unavailable and the input-output available data are not enough to provide all information on these unknown variables. To overcome this difficulty, most published works, addressing the identification of Wiener systems, assume one of these assumptions: the invertibility of the unknown nonlinear element [

This paper introduces a recursive identification method for MIMO Wiener model. This model is characterised by a linear dynamic block as an observer state-space model and a nonlinear block as combined and arbitrary (reversible or irreversible) nonlinearities. A recursive algorithm which combines the least squares technique, the adjustable model, and the Kalman filter principle is developed to resolve the parameters and state estimation problem with less computational effort and a fast convergence rate. Indeed, in the proposed method, the parameters of the linear part and nonlinear part of the MIMO Wiener model are estimated separately in order to decrease the dimension of the unknown parameters matrices and reduce the parameter redundancy. Moreover, a modified Kalman filter and a specific decomposition technique are developed to extract and estimate the unknown internal vector without any research of the inverse nonlinear functions.

The remainder of this paper is organized as follows. Section

Consider the MIMO discrete-time Wiener model Figure

The MIMO Wiener model.

Assume that the degrees

In the rest of this paper, we propose to rewrite the system output vector

The first submodel is given by the following equation:

However, the second submodel is based on a decomposition technique of nonlinear functions

Using (

From (

In order to simplify the formulation of the parametric and state estimation problem, this section is divided into two subsections. The basic idea is to estimate recursively the dynamic linear part parameters and the static nonlinear part parameters of the considered MIMO Wiener model and then to estimate the state vector

If the state vector

To avoid the parametric redundancy problem and construct the estimate parameters of the static nonlinear part (

In reality, the regression matrix

In the area of state estimation algorithms, the extended Kalman filter (EKF) is the most widely used nonlinear estimation method [

This part is based essentially on the Kalman filter principle, which is defined by the following.

Define

In a first step, applying this theorem to the linear state-space equation (

However, the

Combining (

Thus, we can replace

Combining (

The procedure for computing the parameter, the state, and the internal vector estimates using the RPSE algorithm is listed as follows:

To initialize, let

Collect the input-output vectors

Form

Compute the covariance matrix

Compute the state gains

Increase

The flowchart of the recursive learning algorithms which are used for the parameters estimation of nonlinear MIMO Wiener models is shown in Figure

Flowchart for recursive estimation of MIMO Wiener models.

Consider the following state-space model:

The recursive parameter estimates and errors with

| 100 | 200 | 500 | 1000 |
---|---|---|---|---|

| −0.0166 | −0.0008 | −0.0002 | −0.0001 |

| 0.4769 | 0.4989 | 0.5000 | 0.5001 |

| −0.3965 | −0.3998 | −0.4000 | −0.4000 |

| 0.0048 | 0.0002 | 0.0001 | 0.0000 |

| 0.001 | 0.0022 | 0.0001 | 0.0000 |

| 0.8976 | 0.8991 | 0.9108 | 0.9219 |

| −0.2654 | −0.5653 | −0.7731 | −0.7744 |

| −0.1022 | 0.0101 | 0.0105 | −0.0002 |

| 1.0012 | 1.0000 | 1.0000 | 1.0000 |

| 0.0019 | 0.0000 | 0.0000 | 0.0000 |

| −0.0007 | −0.0000 | −0.0000 | −0.0000 |

| 0.9989 | 1.0000 | 1.0000 | 1.0000 |

| 0.9573 | 0.9902 | 1.0027 | 1.000 |

| 0.1502 | 0.1853 | 0.2889 | 0.3001 |

| 0.0167 | −0.0027 | 0.0016 | 0.0010 |

| 0.1202 | 0.1153 | 0.1108 | 0.0979 |

| 0.0027 | 0.00620 | 0.0015 | 0.0001 |

| −0.0793 | −0.0046 | 0.0003 | 0.0000 |

| 0.0036 | 0.0043 | 0.0014 | 0.0002 |

| 0.2306 | 0.2490 | 0.2913 | 0.3001 |

| 0.3797 | 0.5472 | 0.9894 | 0.9989 |

| 0.9941 | 0.9526 | 1.0008 | 1.0001 |

| 31.6697 | 19.9767 | 4.2708 | 3.6902 |

The recursive parameter estimates and errors with

| 100 | 200 | 500 | 1000 |
---|---|---|---|---|

| −0.0166 | −0.0008 | 0.0000 | 0.0000 |

| 0.4769 | 0.4989 | 0.5002 | 0.5000 |

| −0.3965 | −0.3998 | −0.4000 | −0.4000 |

| 0.0048 | 0.0002 | 0.0000 | 0.0001 |

| 0.0012 | 0.0011 | 0.0010 | 0.0010 |

| 0.8974 | 0.8996 | 0.8999 | 0.9105 |

| −0.2654 | −0.5651 | −0.7659 | −0.7699 |

| −0.1022 | −0.0101 | −0.0101 | −0.0091 |

| 1.0012 | 1.0000 | 1.0000 | 1.0000 |

| 0.0019 | 0.0000 | 0.0000 | 0.0000 |

| −0.0007 | 0.0000 | 0.0000 | 0.0000 |

| 0.9989 | 0.9995 | 1.0000 | 1.0000 |

| 0.9071 | 0.9567 | 1.0082 | 0.9976 |

| 0.1710 | 0.1773 | 0.2705 | 0.2995 |

| 0.0123 | −0.0052 | 0.0013 | 0.001 |

| 0.1410 | 0.0473 | 0.1205 | 0.0992 |

| −0.0597 | 0.00731 | −0.00384 | −0.0025 |

| −0.0016 | −0.0075 | −0.0010 | 0.0013 |

| 0.0044 | 0.0044 | 0.0014 | 0.0001 |

| 0.4817 | 0.3529 | 0.2921 | 0.2997 |

| 0.4528 | 0.5773 | 0.9541 | 0.9903 |

| 0.9679 | 0.9486 | 1.0023 | 1.0010 |

| 31.4375 | 18.7664 | 5.1010 | 3.8043 |

The internal variables

The estimation errors

The output variances

From the simulation results in Tables

The estimated parameters converge to real ones and the parameter estimation errors given by the RPSE algorithm become smaller when

The estimated internal outputs

The output variances

The estimation quality is better when the parametric gains

The variances of the parameters estimates, using the Monte Carlo simulation, are small which improves the effectiveness of the RPSE algorithm; see Table

The proposed algorithm can achieve a satisfactory estimation quality through appropriately choosing the parametric gains and the innovation length.

The Monte Carlo estimates and variances with

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

500 | −0.0002 ± 0.0001 | 0.5000 ± 0.0045 | −0.4000 ± 0.0135 | 0.0001 ± 0.0006 | 0.0001 ± 0.0004 | 0.9108 ± 0.0089 | −0.7731 ± 0.0077 | |

1000 | −0.0001 ± 0.00005 | 0.5001 ± 0.0026 | −0.4000 ± 0.00256 | 0.0000 ± 0.0002 | 0.0000 ± 0.00001 | 0.9219 ± 0.006897 | −0.7744 ± 0.00755 | |

True values | 0.0000 | 0.5000 | −0.4000 | 0.0000 | 0.0000 | 0.9220 | −0.7746 | |

| ||||||||

| | | | | | | | |

| ||||||||

500 | 0.0105 ± 0.0006 | 1.0000 ± 0.0090 | 0.0000 ± 0.00018 | −0.0000 ± 0.0004 | 1.0000 ± 0.009814 | 1.0027 ± 0.02173 | 0.2889 ± 0.03178 | |

1000 | −0.0002 ± 0.0001 | 1.0000 ± 0.0090 | 0.0000 ± 0.0001 | −0.0000 ± 0.00008 | 1.0000 ± 0.00968 | 1.0000 ± 0.0211 | 0.3001 ± 0.0233 | |

True values | 0.0000 | 1.0000 | 0.0000 | 0.0000 | 1.0000 | 1.0000 | 0.3000 | |

| ||||||||

| | | | | | | | |

| ||||||||

500 | 0.0016 ± 0.0008 | 0.1108 ± 0.0086 | 0.0015 ± 0.0160 | 0.0003 ± 0.0039 | 0.0014 ± 0.0005 | 0.2913 ± 0.0253 | 0.9894 ± 0.099 | 1.0008 ± 0.0195 |

1000 | 0.0010 ± 0.0002 | 0.0979 ± 0.0081 | 0.0001 ± 0.0100 | 0.0000 ± 0.0033 | 0.0002 ± 0.0002 | 0.3001 ± 0.0230 | 0.9989 ± 0.017 | 1.0001 ± 0.0130 |

True values | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.3000 | 1.0000 | 1.0000 |

The state estimates

This paper presents a recursive parameter and state estimation algorithm by combining the least square technique, the adjustable model, and the Kalman filter principal for estimating jointly the parameters, the state vector, and the internal variables of MIMO Wiener state-space models. By estimating the parameters of the linear and nonlinear parts separately and using a specific decomposition technique, we can remove the redundant parameters and avoid problems related to computing the inverse nonlinear functions. The proposed algorithm can be combined with adaptive control schemes and extended to other blocks-oriented models.

The authors declare that there are no competing interests regarding the publication of this paper.

This work was supported by the ministry of higher education and scientific research of Tunisia.