ReadMeFirst

PURPOSE ^

Read me first file: use of the routines for parametric plant modeling of MIMO systems with nu inputs and ny outputs.

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

 Read me first file: use of the routines for parametric plant modeling of MIMO systems with nu inputs and ny outputs. 
 These functions use the "MatrixArrayFunctions" toolbox (MIMO_ML, MIMO_WTLS, MIMO_WGTLS, MIMO_IQML, MIMO_BTLS, MIMO_ML_CRbound), 
 and the "CovRootsResidues" toolbox (MIMO_ML_CRbound).
 Examples illustrating the use of the functions can be found in the folder "examples". To run these examples one
 also needs the "LocalPolynomialMethod" and "DesignMultiSineExcitations" toolboxes.

%   1. (SAMPLE) MAXIMUM LIKELIHOOD ESTIMATE PLANT MODEL PARAMETERS

           [Theta, Cost, smax, smin, wscale] = MIMO_ML(data, Sel, Theta0, ModelVar, IterVar);

       Starting from noisy input and noisy output DFT spectra (= the sample means) and their sample covariance based on dof
       degrees of freedom, the function estimates the model parameters Theta of an ny x nu common denominator plant transfer 
       function of order nb over na. Discrete-time (z-domain), continuous-time (s-domain), as well as diffusion (sqrt(s)-domain)
       systems can be identified. The sample means and sample covariances are the output of the following functions:

           1. ArbLocalPolyAnal:    1 MIMO experiment with random inputs (can be the concatenation of several data records)
           2. FastLocalPolyAnal:    1 MIMO experiment with periodic inputs 
                                   (random phase multisines, P > 1 periods) 
           3. RobustLocalPolyAnal:    nu MIMO experiments with periodic inputs 
                                   (random phase multisines, M >=1 realisations, P > 1 periods) 

       Note: the MIMO_ML function can handle nexp >= 1 experiments with periodic signals.
       In the absence of modeling errors the expected value and the variance of the cost function are given by 

           E{Cost} = dof/(dof-ny)*ny*(F-ntheta/2)      and       var_min    <= var{Cost} <= 3*var_min 

       where

           var_min = dof^3/((dof-ny)^2*(dof-ny-1))*ny*(F-ntheta/2)

       The upper bound of the variance is reached for 1 MIMO experiment with random or periodic excitations. 
       For nu MIMO experiments with periodic excitations (or 1 MIMO experiment with periodic excitation and averaging
       over the periods only) the variance is in between the two bounds. 

       The structures Sel (definition free parameters), Theta0 (defintion parameters), ModelVar (definition model type), and 
       IterVar (definition iteration parameters), as needed by the function MIMO_ML are easily generated using the function: 

           [Sel, Theta0, ModelVar, IterVar] = MIMO_ML_DefaultValues(na, nb, nu, ny, PlantPlane, ModelStruct, Recip, Transient) 

       The transfer function matrix G(x, Theta) corresponding to the model parameters Theta and the generalised frequency 
       variable x (= j*omega for continuous-time; exp(-j*omega*Ts) for discrete-time, sqrt(j*omega) for diffusion) is calculated
       using the funtion:

           PolyTrans = MIMO_ML_CalcPolyTrans(Theta, x);

       To perform the whitness test on the residuals one should take into account the correlation � CL over the frequency 
       of the sample mean and sample covariance: the autocorrelation should be performed on residuals in steps of CL+1. 
       over the EXCITED frequencies. This is automatically done in the function: 

           [Auto_Corr, Lags, Conf_Bound, Fraction] = WhitenessTestResiduals(G, CvecG, Gest, CL, dof, PlotFig); 

       References:

           Pintelon, R., J. Schoukens, G. Vandersteen, and K. Barb�(2010). Estimation of nonparametric noise and FRF models 
           for multivariable systems - Part II: extensions, applications, Mechanical Systems and Signal Processing, vol. 24, 
           no. 3, pp. 596-616.

           Pintelon, R., G. Vandersteen, J. Schoukens, and Y. Rolain (2011). Improved (non-)parametric identification of dynamic 
           systems excited by periodic signals - The multivariate case, Mechanical Systems and Signal Processing, vol. 25, no. 8, 
           pp. 2892-2922.

           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition, 
           IEEE Press-Wiley, Piscataway (USA). 


%   2. CRAMER-RAO LOWER BOUND - ASYMPTOTIC COVARIANCE MATRIX

           [CRbound, G, Theta, CovThetan, Thetan, Seln, wscale, TheCond] = MIMO_ML_CRbound(data, Sel, Theta, ModelVar); 

       Calculates the Cram�-Rao lower bound of the physical model parameters (numerator and denominator coefficients, poles, 
       residue matrices) and the estimated transfer function model for the following cases: 

           1. 1 MIMO experiment with random excitations (the input or the power spectrum of the input is provided); 
              or the concatenation of several data records (see the ArbLocalPolyAnal function).  
           2. 1 MIMO experiment with periodic excitations
           3. nexp >=1 MIMO experiments with periodic excitations

       The covariance matrix of the SML estimate Theta_SML(Zm) using the sample mean Zm and sample covariance CZm (result of
       the functions ArbLocalPolyAnal, or FastLocalPolyAnal) is related to the covariance matrix of the ML estimate Theta_ML(Z) 
       using the original data set Z and the known covariance CZ0 as 

           Cov(Theta_SML(Zm)) = dof*(dof-ny)/((dof-ny+1)*(dof-ny-1)) * Cov(Theta_ML(Z)) 

       where Cov(Theta_ML(Z)) = CRbound.Theta(Z, CZ0) (= CRbound.Theta with (Z, CZ0) as input data). For the result of 
       RobustLocalPolyAnal (or FastLocalPolyAnal with averaging over the periods only) Cov(Theta_ML(Z)) in the formula above is 
       replaced by CRbound.Theta(Zm, CZm0) with CZm0 the true covariance of Zm. Since in practise the true covariance CZ0 (or 
       CZm0) is unknown, Cov(Theta_ML(Z)) can only be calculated in an estimate CZ (sample cov.) of the true covariance CZ0. 
       The formula above is then replaced by 

           Cov(Theta_SML(Zm)) = dof^2/((dof-ny+1)*(dof-ny-1)) * CRbound(Z, CZ) 

       where CRbound(Z, CZ) is replaced by CRbound(Zm, CZm) for RobustLocalPolyAnal  (or FastLocalPolyAnal with averaging 
       over the periods only) 

       References:

           Pintelon, R., P. Guillaume, and J. Schoukens (2007). Uncertainty calculation in (operational) modal analysis, 
           Mechanical Systems and Signal Processing, vol. 21, no. 6, pp. 2359-2373.

           Pintelon, R., J. Schoukens, G. Vandersteen, and K. Barb�(2010). Estimation of nonparametric noise and FRF models 
           for multivariable systems - Part II: extensions, applications, Mechanical Systems and Signal Processing, vol. 24, 
           no. 3, pp. 596-616.

           Pintelon, R., G. Vandersteen, J. Schoukens, and Y. Rolain (2011). Improved (non-)parametric identification of dynamic 
           systems excited by periodic signals - The multivariate case, Mechanical Systems and Signal Processing, vol. 25, no. 8, 
           pp. 2892-2922.

           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition, 
           IEEE Press-Wiley, Piscataway (USA). 


%  STARTING VALUES: SAMPLE WEIGHTED (GENERALIZED) TOTAL LEAST SQUARES, BOOTSTRAPPED TOTAL LEAST SQUARES 

           [Theta, smax, smin, wscale] = MIMO_WTLS(data, Sel, ModelVar)

       Weighted total least squares (WTLS) estimate of the common denominator plant model parameters. The routine does not
       require any noise covariance information. In general the estimates are inconsistent. 

           [Theta, smax, smin, wscale] = MIMO_WGTLS(data, Sel, ModelVar)

       Weighted generalized total least squares (WGTLS) esimate of the common denominator plant model parameters. The routine requires
       the (sample) noise covariance information. The estimates are consistent.

           [Theta, Cost, smax, smin, wscale] = MIMO_IQML(data, Sel, Theta0, ModelVar, IterVar) 

       Iterative quadratic maximum likelihood (IQML) estimate of the common denominator plant model parameters. The routine  
       requires starting values and the (sample) noise covariance information. In general the estimates are inconsistent. 

           [Theta, Cost, smax, smin, wscale] = MIMO_BTLS(data, Sel, Theta0, ModelVar, IterVar) 

       Bootstrapped total least squares (BTLS) estimate of the common denominator plant model parameters. The routine requires 
       starting values and the (sample) noise covariance information. The estimates are consistent. 

       References:

           Pintelon R., P. Guillaume, G. Vandersteen and Y. Rolain (1998). Analyses, development and applications of TLS 
           algorithms in frequency-Domain System Identification, SIAM J. Matrix Anal. Appl., vol. 19, no. 4, pp. 983-1004.

           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition, 
           IEEE Press-Wiley, Piscataway (USA). 


%

 Copyright (c) Rik Pintelon, Vrije Universiteit Brussel - dept. ELEC, 5 November 2009
 All rights reserved.
 Software can be used freely for non-commercial applications only.
 version 24 October 2011

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 % Read me first file: use of the routines for parametric plant modeling of MIMO systems with nu inputs and ny outputs.
0002 % These functions use the "MatrixArrayFunctions" toolbox (MIMO_ML, MIMO_WTLS, MIMO_WGTLS, MIMO_IQML, MIMO_BTLS, MIMO_ML_CRbound),
0003 % and the "CovRootsResidues" toolbox (MIMO_ML_CRbound).
0004 % Examples illustrating the use of the functions can be found in the folder "examples". To run these examples one
0005 % also needs the "LocalPolynomialMethod" and "DesignMultiSineExcitations" toolboxes.
0006 %
0007 %%   1. (SAMPLE) MAXIMUM LIKELIHOOD ESTIMATE PLANT MODEL PARAMETERS
0008 %
0009 %           [Theta, Cost, smax, smin, wscale] = MIMO_ML(data, Sel, Theta0, ModelVar, IterVar);
0010 %
0011 %       Starting from noisy input and noisy output DFT spectra (= the sample means) and their sample covariance based on dof
0012 %       degrees of freedom, the function estimates the model parameters Theta of an ny x nu common denominator plant transfer
0013 %       function of order nb over na. Discrete-time (z-domain), continuous-time (s-domain), as well as diffusion (sqrt(s)-domain)
0014 %       systems can be identified. The sample means and sample covariances are the output of the following functions:
0015 %
0016 %           1. ArbLocalPolyAnal:    1 MIMO experiment with random inputs (can be the concatenation of several data records)
0017 %           2. FastLocalPolyAnal:    1 MIMO experiment with periodic inputs
0018 %                                   (random phase multisines, P > 1 periods)
0019 %           3. RobustLocalPolyAnal:    nu MIMO experiments with periodic inputs
0020 %                                   (random phase multisines, M >=1 realisations, P > 1 periods)
0021 %
0022 %       Note: the MIMO_ML function can handle nexp >= 1 experiments with periodic signals.
0023 %       In the absence of modeling errors the expected value and the variance of the cost function are given by
0024 %
0025 %           E{Cost} = dof/(dof-ny)*ny*(F-ntheta/2)      and       var_min    <= var{Cost} <= 3*var_min
0026 %
0027 %       where
0028 %
0029 %           var_min = dof^3/((dof-ny)^2*(dof-ny-1))*ny*(F-ntheta/2)
0030 %
0031 %       The upper bound of the variance is reached for 1 MIMO experiment with random or periodic excitations.
0032 %       For nu MIMO experiments with periodic excitations (or 1 MIMO experiment with periodic excitation and averaging
0033 %       over the periods only) the variance is in between the two bounds.
0034 %
0035 %       The structures Sel (definition free parameters), Theta0 (defintion parameters), ModelVar (definition model type), and
0036 %       IterVar (definition iteration parameters), as needed by the function MIMO_ML are easily generated using the function:
0037 %
0038 %           [Sel, Theta0, ModelVar, IterVar] = MIMO_ML_DefaultValues(na, nb, nu, ny, PlantPlane, ModelStruct, Recip, Transient)
0039 %
0040 %       The transfer function matrix G(x, Theta) corresponding to the model parameters Theta and the generalised frequency
0041 %       variable x (= j*omega for continuous-time; exp(-j*omega*Ts) for discrete-time, sqrt(j*omega) for diffusion) is calculated
0042 %       using the funtion:
0043 %
0044 %           PolyTrans = MIMO_ML_CalcPolyTrans(Theta, x);
0045 %
0046 %       To perform the whitness test on the residuals one should take into account the correlation � CL over the frequency
0047 %       of the sample mean and sample covariance: the autocorrelation should be performed on residuals in steps of CL+1.
0048 %       over the EXCITED frequencies. This is automatically done in the function:
0049 %
0050 %           [Auto_Corr, Lags, Conf_Bound, Fraction] = WhitenessTestResiduals(G, CvecG, Gest, CL, dof, PlotFig);
0051 %
0052 %       References:
0053 %
0054 %           Pintelon, R., J. Schoukens, G. Vandersteen, and K. Barb�(2010). Estimation of nonparametric noise and FRF models
0055 %           for multivariable systems - Part II: extensions, applications, Mechanical Systems and Signal Processing, vol. 24,
0056 %           no. 3, pp. 596-616.
0057 %
0058 %           Pintelon, R., G. Vandersteen, J. Schoukens, and Y. Rolain (2011). Improved (non-)parametric identification of dynamic
0059 %           systems excited by periodic signals - The multivariate case, Mechanical Systems and Signal Processing, vol. 25, no. 8,
0060 %           pp. 2892-2922.
0061 %
0062 %           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition,
0063 %           IEEE Press-Wiley, Piscataway (USA).
0064 %
0065 %
0066 %%   2. CRAMER-RAO LOWER BOUND - ASYMPTOTIC COVARIANCE MATRIX
0067 %
0068 %           [CRbound, G, Theta, CovThetan, Thetan, Seln, wscale, TheCond] = MIMO_ML_CRbound(data, Sel, Theta, ModelVar);
0069 %
0070 %       Calculates the Cram�-Rao lower bound of the physical model parameters (numerator and denominator coefficients, poles,
0071 %       residue matrices) and the estimated transfer function model for the following cases:
0072 %
0073 %           1. 1 MIMO experiment with random excitations (the input or the power spectrum of the input is provided);
0074 %              or the concatenation of several data records (see the ArbLocalPolyAnal function).
0075 %           2. 1 MIMO experiment with periodic excitations
0076 %           3. nexp >=1 MIMO experiments with periodic excitations
0077 %
0078 %       The covariance matrix of the SML estimate Theta_SML(Zm) using the sample mean Zm and sample covariance CZm (result of
0079 %       the functions ArbLocalPolyAnal, or FastLocalPolyAnal) is related to the covariance matrix of the ML estimate Theta_ML(Z)
0080 %       using the original data set Z and the known covariance CZ0 as
0081 %
0082 %           Cov(Theta_SML(Zm)) = dof*(dof-ny)/((dof-ny+1)*(dof-ny-1)) * Cov(Theta_ML(Z))
0083 %
0084 %       where Cov(Theta_ML(Z)) = CRbound.Theta(Z, CZ0) (= CRbound.Theta with (Z, CZ0) as input data). For the result of
0085 %       RobustLocalPolyAnal (or FastLocalPolyAnal with averaging over the periods only) Cov(Theta_ML(Z)) in the formula above is
0086 %       replaced by CRbound.Theta(Zm, CZm0) with CZm0 the true covariance of Zm. Since in practise the true covariance CZ0 (or
0087 %       CZm0) is unknown, Cov(Theta_ML(Z)) can only be calculated in an estimate CZ (sample cov.) of the true covariance CZ0.
0088 %       The formula above is then replaced by
0089 %
0090 %           Cov(Theta_SML(Zm)) = dof^2/((dof-ny+1)*(dof-ny-1)) * CRbound(Z, CZ)
0091 %
0092 %       where CRbound(Z, CZ) is replaced by CRbound(Zm, CZm) for RobustLocalPolyAnal  (or FastLocalPolyAnal with averaging
0093 %       over the periods only)
0094 %
0095 %       References:
0096 %
0097 %           Pintelon, R., P. Guillaume, and J. Schoukens (2007). Uncertainty calculation in (operational) modal analysis,
0098 %           Mechanical Systems and Signal Processing, vol. 21, no. 6, pp. 2359-2373.
0099 %
0100 %           Pintelon, R., J. Schoukens, G. Vandersteen, and K. Barb�(2010). Estimation of nonparametric noise and FRF models
0101 %           for multivariable systems - Part II: extensions, applications, Mechanical Systems and Signal Processing, vol. 24,
0102 %           no. 3, pp. 596-616.
0103 %
0104 %           Pintelon, R., G. Vandersteen, J. Schoukens, and Y. Rolain (2011). Improved (non-)parametric identification of dynamic
0105 %           systems excited by periodic signals - The multivariate case, Mechanical Systems and Signal Processing, vol. 25, no. 8,
0106 %           pp. 2892-2922.
0107 %
0108 %           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition,
0109 %           IEEE Press-Wiley, Piscataway (USA).
0110 %
0111 %
0112 %%  STARTING VALUES: SAMPLE WEIGHTED (GENERALIZED) TOTAL LEAST SQUARES, BOOTSTRAPPED TOTAL LEAST SQUARES
0113 %
0114 %           [Theta, smax, smin, wscale] = MIMO_WTLS(data, Sel, ModelVar)
0115 %
0116 %       Weighted total least squares (WTLS) estimate of the common denominator plant model parameters. The routine does not
0117 %       require any noise covariance information. In general the estimates are inconsistent.
0118 %
0119 %           [Theta, smax, smin, wscale] = MIMO_WGTLS(data, Sel, ModelVar)
0120 %
0121 %       Weighted generalized total least squares (WGTLS) esimate of the common denominator plant model parameters. The routine requires
0122 %       the (sample) noise covariance information. The estimates are consistent.
0123 %
0124 %           [Theta, Cost, smax, smin, wscale] = MIMO_IQML(data, Sel, Theta0, ModelVar, IterVar)
0125 %
0126 %       Iterative quadratic maximum likelihood (IQML) estimate of the common denominator plant model parameters. The routine
0127 %       requires starting values and the (sample) noise covariance information. In general the estimates are inconsistent.
0128 %
0129 %           [Theta, Cost, smax, smin, wscale] = MIMO_BTLS(data, Sel, Theta0, ModelVar, IterVar)
0130 %
0131 %       Bootstrapped total least squares (BTLS) estimate of the common denominator plant model parameters. The routine requires
0132 %       starting values and the (sample) noise covariance information. The estimates are consistent.
0133 %
0134 %       References:
0135 %
0136 %           Pintelon R., P. Guillaume, G. Vandersteen and Y. Rolain (1998). Analyses, development and applications of TLS
0137 %           algorithms in frequency-Domain System Identification, SIAM J. Matrix Anal. Appl., vol. 19, no. 4, pp. 983-1004.
0138 %
0139 %           Pintelon, R., and J. Schoukens (2012). System Identification: A Frequency Domain Approach, second edition,
0140 %           IEEE Press-Wiley, Piscataway (USA).
0141 %
0142 %
0143 %%
0144 %
0145 % Copyright (c) Rik Pintelon, Vrije Universiteit Brussel - dept. ELEC, 5 November 2009
0146 % All rights reserved.
0147 % Software can be used freely for non-commercial applications only.
0148 % version 24 October 2011
0149 %
0150 
0151

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