Simulation of First Order System (Matlab & Simulink)

Question

Yesterday, I tried to find constants $k_1$ and $k_2$ for the following system:

$$\frac{\mathrm d RH}{\mathrm dt}=k_1+k_2(RH_{ext}-RH).$$

Now, I have some first computations for $k_1$ and $k_2$ and would like to test how good a simulation with these parameters works. I think I could also write this as state space model:

$$\dot{x}=-k_2x+\left[\begin{matrix} k_1 & k_2\end{matrix}\right]\left[ \begin{matrix} u_1 \\ u_2\end{matrix} \right],$$

$$y = x,$$

where $x = RH$, $u_1=1$ ($const.$) and $u_2=RH_{ext}$. I have measurement data for $RH_{ext}$ in Matlab (vector), a time vector $t$ with the same length and would like to do a simulation for $RH$.

Any suggestions how I could proceed? Simulink is possible, but just a script-based solution would be even better.

Answer 1

Yes, if you have values for k1 and k2, data for RH_ext and t, your suggestion would work although I think you have your matrices wrong. You probably need to define x as RH - k1/k2 to get rid of the constant term:

A = -k2;
B = k2; % x_dot = -k2*x + k2*u
C = 1;
D = 0; % y = x
sys = ss(A,B,C,D);
y = lsim(sys,RH_ext,t);
RH_sim = y + k1/k2;