This example deals with a model describing the control of insulin on glucose utilization in humans. The model is shown in Fig. 2. The experiment consists of an impulse input of glucose labeled with a tracer and of the measurement in plasma of glucose, labeled glucose and insulin concentrations. The measured insulin concentration acts as model input $u$, while the model output $y$ is the measured tracer glucose concentration. The control by insulin on the glucose system is exerted by insulin in a remote compartment $(x_3)$. The glucose system is described by two compartments which represent, respectively, glucose in rapidly $(x_1)$ and slowly equilibrating tissues $(x_2)$ which include the muscle tissues. Insulin control is exerted on glucose utilization in compartment 3 (insulin-dependent tissues) while glucose utilization in compartment 1 refers to insulin-independent tissues.

The system-experiment model is

$$\begin{align*} \dot{x}_1(t) &= - \left(k_p + \frac{F_{01}/V_1}{g(t)} + k_{21}\right)x_1(t) + k_{12}x_2(t)\\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + x_3(t) + k_{12})x_2(t) \\ \dot{x}_3(t) &= -k_bx_3(t) + k_au(t)\\ y_1(t) &= x_1(t)/V \\ \end{align*}$$

The initial conditions are $x_1(0)=i_1,x_2(0)$, and $x_3(0)=0$. System parameters are presented in the table below.

The two compartment model describes the kinetics of a drug in the human body. The drug is injected into the blood (compartment 1) where it exchanges linearly with the tissues (compartment 2); the drug is irreversibly removed with a nonlinear saturative characteristic from compartment 1 and with a linear one from compartment 2. The I/O experiment takes place in compartment 1.

The system-experiment model is

$$\begin{align*} \dot{x}_1(t) &= - \left(k_{21} + \frac{V_M}{K_m + x_1}\right)x_1(t) + k_{12}x_2(t) + b_1u(t) \\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + k_{12})x_2(t) \\ y(t) &= c_1x_1(t) \\ \end{align*}$$

The initial conditions are $x_1(0) = 0$ and $x_2(0) = 0$. System parameters are presented in the table below.

The ball and plate system is a system, where a metal ball stays on a rigid square plate with each side length of 1m. The slope of the plate can be manipulated by two perpendicularly installed step motors, so that the tilting of the plate will make the ball moving.

$$\begin{align*} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2\\ \dot{x}_3\\ \dot{x}_4\\ \dot{x}_5\\ \dot{x}_6\\ \dot{x}_7\\ \dot{x}_8\\ \end{bmatrix} &= \begin{bmatrix} x_2 \\ B(x_1x_4^2 + x_4x_5x_8 - g \sin x_3)\\ x_4\\ 0\\ x_6\\ B(x_5x_8^2 + x_1x_4x_8 - g \sin x_7)\\ x_8\\ 0\\ \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0\\ 1 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} u_x\\ u_y\\ \end{bmatrix}, \\ Y &= h(X) = (x_1,x_5)^{\mathrm T}, \end{align*}$$

where $B= m/(m + J/R^2)$ and $X = (x_1; x_2; x_3; x_4; x_5; x_6; x_7; x_8)^{\mathrm T} = (x; \dot{x}; \theta_x; \dot{\theta}_x; y; \dot{y}; \theta_y; \dot{\theta}_y)^{\mathrm T}$

Parameters are presented in the table below.

$$\begin{align*} \dot x_1 &= -k_1x_1 - k_3x_1^2 + u(c - x_1) \\ \dot x_2 &= k_1x_1 - k_2x_2 - ux_2 \\ y &= x_2, \end{align*}$$

where $c$ and $x_1$ are the concentrations of the input and reactant substance, $x_2$ is the concentration of the desired output product, $u$ is the normalized input flow rate of the reactant substance. The output $y$ here reflects the grade of the final product. The parameters $k_1, k_2, k_3$ and $c$ are positive constants under the isothermal considered conditions.