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This example shows how to write ODE files for nonlinear grey-box models as MATLAB and C MEX files.

Grey box modeling is conceptually different to black box modeling in that it involves a more comprehensive modeling step. For IDNLGREY (the nonlinear grey-box model object; the nonlinear counterpart of IDGREY), this step consists of creating an ODE file, also called a "model file". The ODE file specifies the right-hand sides of the state and the output equations typically arrived at through physical first principle modeling. In this example we will concentrate on general aspects of implementing it as a MATLAB file or a C MEX file.

IDNLGREY supports estimation of parameters and initial states in nonlinear model structures written on the following explicit state-space form (so-called output-error, OE, form, named so as the noise e(t) only affects the output of the model structure in an additive manner):

` xn(t) = F(t, x(t), u(t), p1, ..., pNpo); x(0) = X0;`

` y(t) = H(t, x(t), u(t), p1, ..., pNpo) + e(t)`

For discrete-time structures, xn(t) = x(T+Ts) with Ts being the sample time, and for continuous-time structures xn(t) = d/dt x(t). In addition, F(.) and H(.) are arbitrary linear or nonlinear functions with Nx (number of states) and Ny (number of outputs) components, respectively. Any of the model parameters p1, ..., pNpo as well as the initial state vector X(0) can be estimated. Worth stressing is that

time-series modeling, i.e., modeling without an exogenous input signal u(t), and

static modeling, i.e., modeling without any states x(t)

are two special cases that are supported by IDNLGREY. (See the tutorials idnlgreydemo3 and idnlgreydemo5 for examples of these two modeling categories).

The first IDNLGREY modeling step to perform is always to implement a MATLAB or C MEX model file specifying how to update the states and compute the outputs. More to the point, the user must write a model file, MODFILENAME.m or MODFILENAME.c, defined with the following input and output arguments (notice that this form is required for both MATLAB and C MEX type of model files)

[dx, y] = MODFILENAME(t, x, u, p1, p2, ..., pNpo, FileArgument)

MODFILENAME can here be any user chosen file name of a MATLAB or C MEX-file, e.g., see twotanks_m.m, pendulum_c.c etc. This file should be defined to return two outputs:

dx: the right-hand side(s) of the state-space equation(s) (a column vector with Nx real entries; [] for static models)

y: the right-hand side(s) of the output equation(s) (a column vector with Ny real entries)

and it should take 3+Npo(+1) input arguments specified as follows:

t: the current time

x: the state vector at time t ([] for static models)

u: the input vector at time t ([] for time-series models)

p1, p2, ..., pNpo: the individual parameters (which can be real scalars, column vectors or 2-dimensional matrices); Npo is here the number of parameter objects, which for models with scalar parameters coincide with the number of parameters Np

FileArgument: optional inputs to the model file

In the onward discussion we will focus on writing model using either MATLAB language or using C-MEX files. However, IDNLGREY also supports P-files (protected MATLAB files obtained using the MATLAB command "pcode") and function handles. In fact, it is not only possible to use C MEX model files but also Fortran MEX files. Consult the MATLAB documentation on External Interfaces for more information about the latter.

What kind of model file should be implemented? The answer to this question really depends on the use of the model.

Implementation using MATLAB language (resulting in a *.m file) has some distinct advantages. Firstly, one can avoid time-consuming, low-level programming and concentrate more on the modeling aspects. Secondly, any function available within MATLAB and its toolboxes can be used directly in the model files. Thirdly, such files will be smaller and, without any modifications, all built-in MATLAB error checking will automatically be enforced. In addition, this is obtained without any code compilation.

C MEX modeling is much more involved and requires basic knowledge about the C programming language. The main advantage with C MEX model files is the improved execution speed. Our general advice is to pursue C MEX modeling when the model is going to be used many times, when large data sets are employed, and/or when the model structure contains a lot of computations. It is often worthwhile to start with using a MATLAB file and later on turn to the C MEX counterpart.

With this said, let us next move on to MATLAB file modeling and use a nonlinear second order model structure, describing a two tank system, as an example. See idnlgreydemo2 for the modeling details. The contents of twotanks_m.m are as follows.

`type twotanks_m.m`

function [dx, y] = twotanks_m(t, x, u, A1, k, a1, g, A2, a2, varargin) %TWOTANKS_M A two tank system. % Copyright 2005-2006 The MathWorks, Inc. % Output equation. y = x(2); % Water level, lower tank. % State equations. dx = [1/A1*(k*u(1)-a1*sqrt(2*g*x(1))); ... % Water level, upper tank. 1/A2*(a1*sqrt(2*g*x(1))-a2*sqrt(2*g*x(2))) ... % Water level, lower tank. ];

In the function header, we here find the required t, x, and u input arguments followed by the six scalar model parameters, A1, k, a1, g, A2 and a2. In the MATLAB file case, the last input argument should always be varargin to support the passing of an optional model file input argument, FileArgument. In an IDNLGREY model object, FileArgument is stored as a cell array that might hold any kind of data. The first element of FileArgument is here accessed through varargin{1}{1}.

The variables and parameters are referred in the standard MATLAB way. The first state is x(1) and the second x(2), the input is u(1) (or just u in case it is scalar), and the scalar parameters are simply accessed through their names (A1, k, a1, g, A2 and a2). Individual elements of vector and matrix parameters are accessed as P(i) (element i of a vector parameter named P) and as P(i, j) (element at row i and column j of a matrix parameter named P), respectively.

Writing a C MEX model file is more involved than writing a MATLAB model file. To simplify this step, it is recommended that the available IDNLGREY C MEX model template is copied to MODFILENAME.c. This template contains skeleton source code as well as detailed instructions on how to customize the code for a particular application. The location of the template file is found by typing the following at the MATLAB command prompt.

` fullfile(matlabroot, 'toolbox', 'ident', 'nlident', 'IDNLGREY_MODEL_TEMPLATE.c')`

For the two tank example, this template was copied to twotanks_c.c. After some initial modifications and configurations (described below) the state and output equations were entered, thereby resulting in the following C MEX source code.

`type twotanks_c.c`

/* Copyright 2005-2015 The MathWorks, Inc. */ /* Written by Peter Lindskog. */ /* Include libraries. */ #include "mex.h" #include <math.h> /* Specify the number of outputs here. */ #define NY 1 /* State equations. */ void compute_dx(double *dx, double t, double *x, double *u, double **p, const mxArray *auxvar) { /* Retrieve model parameters. */ double *A1, *k, *a1, *g, *A2, *a2; A1 = p[0]; /* Upper tank area. */ k = p[1]; /* Pump constant. */ a1 = p[2]; /* Upper tank outlet area. */ g = p[3]; /* Gravity constant. */ A2 = p[4]; /* Lower tank area. */ a2 = p[5]; /* Lower tank outlet area. */ /* x[0]: Water level, upper tank. */ /* x[1]: Water level, lower tank. */ dx[0] = 1/A1[0]*(k[0]*u[0]-a1[0]*sqrt(2*g[0]*x[0])); dx[1] = 1/A2[0]*(a1[0]*sqrt(2*g[0]*x[0])-a2[0]*sqrt(2*g[0]*x[1])); } /* Output equation. */ void compute_y(double *y, double t, double *x, double *u, double **p, const mxArray *auxvar) { /* y[0]: Water level, lower tank. */ y[0] = x[1]; } /*----------------------------------------------------------------------- * DO NOT MODIFY THE CODE BELOW UNLESS YOU NEED TO PASS ADDITIONAL INFORMATION TO COMPUTE_DX AND COMPUTE_Y To add extra arguments to compute_dx and compute_y (e.g., size information), modify the definitions above and calls below. *-----------------------------------------------------------------------*/ void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) { /* Declaration of input and output arguments. */ double *x, *u, **p, *dx, *y, *t; int i, np; size_t nu, nx; const mxArray *auxvar = NULL; /* Cell array of additional data. */ if (nrhs < 3) { mexErrMsgIdAndTxt("IDNLGREY:ODE_FILE:InvalidSyntax", "At least 3 inputs expected (t, u, x)."); } /* Determine if auxiliary variables were passed as last input. */ if ((nrhs > 3) && (mxIsCell(prhs[nrhs-1]))) { /* Auxiliary variables were passed as input. */ auxvar = prhs[nrhs-1]; np = nrhs - 4; /* Number of parameters (could be 0). */ } else { /* Auxiliary variables were not passed. */ np = nrhs - 3; /* Number of parameters. */ } /* Determine number of inputs and states. */ nx = mxGetNumberOfElements(prhs[1]); /* Number of states. */ nu = mxGetNumberOfElements(prhs[2]); /* Number of inputs. */ /* Obtain double data pointers from mxArrays. */ t = mxGetPr(prhs[0]); /* Current time value (scalar). */ x = mxGetPr(prhs[1]); /* States at time t. */ u = mxGetPr(prhs[2]); /* Inputs at time t. */ p = mxCalloc(np, sizeof(double*)); for (i = 0; i < np; i++) { p[i] = mxGetPr(prhs[3+i]); /* Parameter arrays. */ } /* Create matrix for the return arguments. */ plhs[0] = mxCreateDoubleMatrix(nx, 1, mxREAL); plhs[1] = mxCreateDoubleMatrix(NY, 1, mxREAL); dx = mxGetPr(plhs[0]); /* State derivative values. */ y = mxGetPr(plhs[1]); /* Output values. */ /* Call the state and output update functions. Note: You may also pass other inputs that you might need, such as number of states (nx) and number of parameters (np). You may also omit unused inputs (such as auxvar). For example, you may want to use orders nx and nu, but not time (t) or auxiliary data (auxvar). You may write these functions as: compute_dx(dx, nx, nu, x, u, p); compute_y(y, nx, nu, x, u, p); */ /* Call function for state derivative update. */ compute_dx(dx, t[0], x, u, p, auxvar); /* Call function for output update. */ compute_y(y, t[0], x, u, p, auxvar); /* Clean up. */ mxFree(p); }

Let us go through the contents of this file. As a first observation, we can divide the work of writing a C MEX model file into four separate sub-steps, the last one being optional:

Inclusion of C-libraries and definitions of the number of outputs.

Writing the function computing the right-hand side(s) of the state equation(s), compute_dx.

Writing the function computing the right-hand side(s) of the output equation(s), compute_y.

Optionally updating the main interface function which includes basic error checking functionality, code for creating and handling input and output arguments, and calls to compute_dx and compute_y.

Before we address these sub-steps in more detail, let us briefly comment upon a couple of general features of the C programming language.

High-precision variables (all inputs, states, outputs and parameters of an IDNLGREY object) should be defined to be of the data type "double".

The unary * operator placed just in front of the variable or parameter names is a so-called dereferencing operator. The C-declaration "double *A1;" specifies that A1 is a pointer to a double variable. The pointer construct is a concept within C that is not always that easy to comprehend. Fortunately, if the declarations of the output/input variables of compute_y and compute_dx are not changed and all unpacked model parameters are internally declared with a *, then there is no need to know more about pointers from an IDNLGREY modeling point of view.

Both compute_y and compute_dx are first declared and implemented, where after they are called in the main interface function. In the declaration, the keyword "void" states explicitly that no value is to be returned.

For further details of the C programming language we refer to the book

`B.W. Kernighan and D. Ritchie. The C Programming Language, 2nd`

`edition, Prentice Hall, 1988.`

In the first sub-step we first include the C-libraries "mex.h" (required) and "math.h" (required for more advanced mathematics). The number of outputs is also declared per modeling file using a standard C-define:

` /* Include libraries. */`

` #include "mex.h"`

` #include "math.h"`

` /* Specify the number of outputs here. */`

` #define NY 1`

If desired, one may also include more C-libraries than the ones above.

The "math.h" library must be included whenever any state or output equation contains more advanced mathematics, like trigonometric and square root functions. Below is a selected list of functions included in "math.h" and the counterpart found within MATLAB:

` C-function MATLAB function`

` ========================================`

` sin, cos, tan sin, cos, tan`

` asin, acos, atan asin, acos, atan `

` sinh, cosh, tanh sinh, cosh, tanh`

` exp, log, log10 exp, log, log10`

` pow(x, y) x^y`

` sqrt sqrt`

` fabs abs`

Notice that the MATLAB functions are more versatile than the corresponding C-functions, e.g., the former handle complex numbers, while the latter do not.

Next, in the file we find the functions for updating the states, compute_dx, and the output, compute_y. Both these functions hold argument lists, with the output to be computed (dx or y) at position 1, after which follows all variables and parameters required to compute the right-hand side(s) of the state and the output equations, respectively.

All parameters are contained in the parameter array p. The first step in compute_dx and compute_y is to unpack and name the parameters to be used in the subsequent equations. In twotanks_c.c, compute_dx declares six parameter variables whose values are determined accordingly:

` /* Retrieve model parameters. */`

` double *A1, *k, *a1, *g, *A2, *a2;`

` A1 = p[0]; /* Upper tank area. */`

` k = p[1]; /* Pump constant. */`

` a1 = p[2]; /* Upper tank outlet area. */`

` g = p[3]; /* Gravity constant. */`

` A2 = p[4]; /* Lower tank area. */`

` a2 = p[5]; /* Lower tank outlet area. */`

compute_y on the other hand does not require any parameter for computing the output, and hence no model parameter is retrieved.

As is the case in C, the first element of an array is stored at position 0. Hence, dx[0] in C corresponds to dx(1) in MATLAB (or just dx in case it is a scalar), the input u[0] corresponds to u (or u(1)), the parameter A1[0] corresponds to A1, and so on.

In the example above, we are only using scalar parameters, in which case the overall number of parameters Np equals the number of parameter objects Npo. If any vector or matrix parameter is included in the model, then Npo < Np.

The scalar parameters are referenced as P[0] (P(1) or just P in a MATLAB file) and the i:th vector element as P[i-1] (P(i) in a MATLAB file). The matrices passed to a C MEX model file are different in the sense that the columns are stacked upon each other in the obvious order. Hence, if P is a 2-by-2 matrix, then P(1, 1) is referred as P[0], P(2, 1) as P[1], P(1, 2) as P[2] and P(2, 2) as P[3]. See "Tutorials on Nonlinear Grey Box Identification: An Industrial Three Degrees of Freedom Robot : C MEX-File Modeling of MIMO System Using Vector/Matrix Parameters", idnlgreydemo8, for an example where scalar, vector and matrix parameters are used.

The state and output update functions may also include other computations than just retrieving parameters and computing right-hand side expressions. For execution speed, one might, e.g., declare and use intermediate variables, whose values are used several times in the coming expressions. The robot tutorial mentioned above, idnlgreydemo8, is a good example in this respect.

compute_dx and compute_y are also able to handle an optional FileArgument. The FileArgument data is passed to these functions in the auxvar variable, so that the first component of FileArgument (a cell array) can be obtained through

` mxArray* auxvar1 = mxGetCell(auxvar, 0);`

Here, mxArray is a MATLAB-defined data type that enables interchange of data between the C MEX-file and MATLAB. In turn, auxvar1 may contain any data. The parsing, checking and use of auxvar1 must be handled solely within these functions, where it is up to the model file designer to implement this functionality. Let us here just refer to the MATLAB documentation on External Interfaces for more information about functions that operate on mxArrays. An example of how to use optional C MEX model file arguments is provided in idnlgreydemo6, "Tutorials on Nonlinear Grey Box Identification: A Signal Transmission System : C MEX-File Modeling Using Optional Input Arguments".

The main interface function should almost always have the same content and for most applications no modification whatsoever is needed. In principle, the only part that might be considered for changes is where the calls to compute_dx and compute_y are made. For static systems, one can leave out the call to compute_dx. In other situations, it might be desired to only pass the variables and parameters referred in the state and output equations. For example, in the output equation of the two tank system, where only one state is used, one could very well shorten the input argument list to

` void compute_y(double *y, double *x)`

and call compute_y in the main interface function as

` compute_y(y, x);`

The input argument lists of compute_dx and compute_y might also be extended to include further variables inferred in the interface function. The following integer variables are computed and might therefore be passed on: nu (the number of inputs), nx (the number of states), and np (here the number of parameter objects). As an example, nx is passed to compute_y in the model investigated in the tutorial idnlgreydemo6.

The completed C MEX model file must be compiled before it can be used for IDNLGREY modeling. The compilation can readily be done from the MATLAB command line as

` mex MODFILENAME.c`

Notice that the mex-command must be configured before it is used for the very first time. This is also achieved from the MATLAB command line via

` mex -setup`

With an execution ready model file, it is straightforward to create IDNLGREY model objects for which simulations, parameter estimations, and so forth can be carried out. We exemplify this by creating two different IDNLGREY model objects for describing the two tank system, one using the model file written in MATLAB and one using the C MEX file detailed above (notice here that the C MEX model file has already been compiled).

Order = [1 1 2]; % Model orders [ny nu nx]. Parameters = [0.5; 0.003; 0.019; ... 9.81; 0.25; 0.016]; % Initial parameter vector. InitialStates = [0; 0.1]; % Initial values of initial states. nlgr_m = idnlgrey('twotanks_m', Order, Parameters, InitialStates, 0)

nlgr_m = Continuous-time nonlinear grey-box model defined by 'twotanks_m' (MATLAB file): dx/dt = F(t, u(t), x(t), p1, ..., p6) y(t) = H(t, u(t), x(t), p1, ..., p6) + e(t) with 1 input(s), 2 state(s), 1 output(s), and 6 free parameter(s) (out of 6). Status: Created by direct construction or transformation. Not estimated.

`nlgr_cmex = idnlgrey('twotanks_c', Order, Parameters, InitialStates, 0)`

nlgr_cmex = Continuous-time nonlinear grey-box model defined by 'twotanks_c' (MEX-file): dx/dt = F(t, u(t), x(t), p1, ..., p6) y(t) = H(t, u(t), x(t), p1, ..., p6) + e(t) with 1 input(s), 2 state(s), 1 output(s), and 6 free parameter(s) (out of 6). Status: Created by direct construction or transformation. Not estimated.

In this tutorial we have discussed how to write IDNLGREY MATLAB and C MEX model files. We finally conclude the presentation by listing the currently available IDNLGREY model files and the tutorial/case study where they are being used. To simplify further comparisons, we list both the MATLAB (naming convention FILENAME_m.m) and the C MEX model files (naming convention FILENAME_c.c), and indicate in the tutorial column which type of modeling approach that is being employed in the tutorial or case study.

` Tutorial/Case study MATLAB file C MEX-file`

` ======================================================================`

` idnlgreydemo1 (MATLAB) dcmotor_m.m dcmotor_c.c`

` idnlgreydemo2 (C MEX) twotanks_m.m twotanks_c.c`

` idnlgreydemo3 (MATLAB) preys_m.m preys_c.c`

` (C MEX) predprey1_m.m predprey1_c.c`

` (C MEX) predprey2_m.m predprey2_c.c`

` idnlgreydemo4 (MATLAB) narendrali_m.m narendrali_c.c`

` idnlgreydemo5 (MATLAB) friction_m.m friction_c.c`

` idnlgreydemo6 (C MEX) signaltransmission_m.m signaltransmission_c.c`

` idnlgreydemo7 (C MEX) twobodies_m.m twobodies_c.c`

` idnlgreydemo8 (C MEX) robot_m.m robot_c.c`

` idnlgreydemo9 (MATLAB) cstr_m.m cstr_c.c`

` idnlgreydemo10 (MATLAB) pendulum_m.m pendulum_c.c`

` idnlgreydemo11 (C MEX) vehicle_m.m vehicle_c.c`

` idnlgreydemo12 (C MEX) aero_m.m aero_c.c`

` idnlgreydemo13 (C MEX) robotarm_m.m robotarm_c.c`

The contents of these model files can be displayed in the MATLAB command window through the command "type FILENAME_m.m" or "type FILENAME_c.c". All model files are found in the directory returned by the following MATLAB command.

` fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'examples')`