## Estimate Continuous-Time Grey-Box Model for Heat Diffusion

This example shows how to estimate the heat conductivity and the heat-transfer coefficient of a continuous-time grey-box model for a heated-rod system.

This system consists of a well-insulated metal rod of length *L* and a
heat-diffusion coefficient $$\kappa $$. The input to the system is the heating power *u(t)* and
the measured output *y(t)* is the temperature at the other end.

Under ideal conditions, this system is described by the heat-diffusion equation—which is a partial differential equation in space and time.

$$\frac{\partial x(t,\xi )}{\partial t}=\kappa \frac{{\partial}^{2}x(t,\xi )}{\partial {\xi}^{2}}$$

To get a continuous-time state-space model, you can represent the second-derivative using the following difference approximation:

$$\begin{array}{l}\frac{{\partial}^{2}x(t,\xi )}{\partial {\xi}^{2}}=\frac{x\left(t,\xi +\Delta L\right)-2x(t,\xi )+x\left(t,\xi -\Delta L\right)}{{\left(\Delta L\right)}^{2}}\\ \text{where}\xi =k\cdot \Delta L\end{array}$$

This transformation produces a state-space model of order $$n={\scriptscriptstyle \frac{L}{\Delta L}}$$, where the state variables $$x(t,k\cdot \Delta L)$$ are lumped representations for $$x(t,\xi )$$ for the following range of values:

$$k\cdot \Delta L\le \xi <\left(k+1\right)\Delta L$$

The dimension of *x* depends on the spatial grid size $$\Delta L$$ in the approximation.

The heat-diffusion equation is mapped to the following continuous-time state-space model structure to identify the state-space matrices:

$$\begin{array}{l}\dot{x}(t)=Fx(t)+Gu(t)+\tilde{K}w(t)\\ y(t)=Hx(t)+Du(t)+w(t)\\ x(0)=x0\end{array}$$

The state-space matrices are parameterized by the heat diffusion coefficient κ and the
heat transfer coefficient at the far end of the rod
*h _{tf}*. The expressions also depend upon the grid
size,

*Ngrid*, and the length of the rod L. The initial conditions x0 are a function of the initial room temperature, treated as a known quantity in this example.

Create a MATLAB

^{®}file.The following code describes the state-space equation for this model. The parameters are κ and

*h*while the auxiliary variables are_{tf}*Ngrid*,*L*and initial room temperature`temp`

. The grid size is supplied as an auxiliary variable so that the ODE function can be easily adapted for various grid sizes.function [A,B,C,D,K,x0] = heatd(kappa,htf,T,Ngrid,L,temp) % ODE file parameterizing the heat diffusion model % kappa (first parameter) - heat diffusion coefficient % htf (second parameter) - heat transfer coefficient % at the far end of rod % Auxiliary variables for computing state-space matrices: % Ngrid: Number of points in the space-discretization % L: Length of the rod % temp: Initial room temperature (uniform) % Compute space interval deltaL = L/Ngrid; % A matrix A = zeros(Ngrid,Ngrid); for kk = 2:Ngrid-1 A(kk,kk-1) = 1; A(kk,kk) = -2; A(kk,kk+1) = 1; end % Boundary condition on insulated end A(1,1) = -1; A(1,2) = 1; A(Ngrid,Ngrid-1) = 1; A(Ngrid,Ngrid) = -1; A = A*kappa/deltaL/deltaL; % B matrix B = zeros(Ngrid,1); B(Ngrid,1) = htf/deltaL; % C matrix C = zeros(1,Ngrid); C(1,1) = 1; % D matrix (fixed to zero) D = 0; % K matrix: fixed to zero K = zeros(Ngrid,1); % Initial states: fixed to room temperature x0 = temp*ones(Ngrid,1);

Use the following syntax to define an

`idgrey`

model object based on the`heatd`

code file:m = idgrey('heatd',{0.27 1},'c',{10,1,22});

This command specifies the auxiliary parameters as inputs to the function, include the model order (grid size)

`10`

, the rod length of 1 meter, and an initial temperature of`22`

degrees Celsius. The command also specifies the initial values for heat conductivity as`0.27`

, and for the heat transfer coefficient as`1`

.For given

`data`

, you can use`greyest`

to estimate the grey-box parameter values:me = greyest(data,m)

The following command shows how you can specify to estimate a new model with different auxiliary variables:

m.Structure.ExtraArguments = {20,1,22}; me = greyest(data,m);

This syntax uses the `ExtraArguments`

model structure attribute to
specify a finer grid using a larger value for `Ngrid`

. For more information
about linear grey-box model properties, see the `idgrey`

reference page.