## Documentation |

The linearized model of a continuous stirred-tank reactor (CSTR) involving an exothermic (heat-generating) reaction is represented by the following differential equations:[9]

$$\frac{d{{C}^{\prime}}_{A}}{dt}={a}_{11}{{C}^{\prime}}_{A}+{a}_{12}{T}^{\prime}+{b}_{11}{{T}^{\prime}}_{c}+{b}_{12}{{C}^{\prime}}_{Ai}$$

$$\frac{d{T}^{\prime}}{dt}={a}_{21}{{C}^{\prime}}_{A}+{a}_{22}{T}^{\prime}+{b}_{21}{{T}^{\prime}}_{c}+{b}_{22}{{C}^{\prime}}_{Ai}$$

where *C _{A}* is the concentration
of a key reactant,

Measurement of reactant concentrations is often difficult, if
not impossible. Let us assume that *T* is a measured
output, *C _{A}* is an unmeasured
output,

The model fits the general state-space format

$$\frac{dx}{dt}=Ax+Bu$$

$$y=Cx+Du$$

where

$$x=\left[\begin{array}{c}{{C}^{\prime}}_{A}\\ {T}^{\prime}\end{array}\right]\text{}u=\left[\begin{array}{c}{{T}^{\prime}}_{c}\\ {{C}^{\prime}}_{Ai}\end{array}\right]\text{}y=\left[\begin{array}{c}{T}^{\prime}\\ {{C}^{\prime}}_{A}\end{array}\right],$$

$$A={\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]}_{}\text{}B=\left[\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right]\text{}C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\text{}D=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$

The following code shows how to define such a model for some
specific values of the *a _{ij}* and

A = [-0.0285 -0.0014 -0.0371 -0.1476]; B = [-0.0850 0.0238 0.0802 0.4462]; C = [0 1 1 0]; D = zeros(2,2); CSTR = ss(A,B,C,D);

The following code sets some of the `CSTR` model's
optional properties:

CSTR.InputName = {'T_c', 'C_A_i'}; CSTR.OutputName = {'T', 'C_A'}; CSTR.StateName = {'C_A', 'T'}; CSTR.InputGroup.MV = 1; CSTR.InputGroup.UD = 2; CSTR.OutputGroup.MO = 1; CSTR.OutputGroup.UO = 2;

To view the properties of `CSTR`, enter:

CSTR

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