Consider a two-link rigid robot manipulator moving a horizontal plane. The dynamic equations of this MIMO system are

$$\left[\matrix{ \ddot{q}_{1}\cr \ddot{q}_{2} }\right]=\left[\matrix{ M_{11} & M_{12}\cr M_{21} & M_{22} }\right]^{-1} \left\{\left[\matrix{ u_{1}\cr u_{2} }\right]-\left[\matrix{ -h\dot{q}_{2} & -h(\dot{q}_{1}+\dot{q}_{2})\cr h\dot{q}_{1} & 0 }\right]\left[\matrix{ \dot{q}_{1}\cr \dot{q}_{2} }\right]\right\},$$

where

$\begin{align*} M_{11}&=a_{1}+2a_{3}\cos(q_{2})+2a_{4} \sin (q_{2}),\ M_{22}=a_{2} \\ M_{12}&=M_{21}=a_{2}+\alpha_{3}\cos(q_{2})+a_{4}\sin(q_{2}) \\ h&=a_{3}\sin(q_{2})-a_{4}\cos(q_{2}) \end{align*}$

with

$\begin{align*} a_{1}&=I_{1}+m_{1}l_{c1}^{2}+I_{e}+m_{e}l_{ce}^{2}+m_{e}l_{1}^{2} \\ a_{2}&=I_{e}+m_{e}l_{ce}^{2} \\ a_{3}&=m_{e}l_{1}l_{ce}\cos(\delta_{e}) \\ a_{4}&=m_{e}l_{1}l_{ce}\sin(\delta_{e}). \end{align*}$

The state model of the knee-quadriceps can be expressed as

$$\begin{cases} \begin{align*} \dot{x}_1 &= \left[ s_0 \alpha K_m + s_v q\dfrac{s_0\alpha F_mx_1 - s_ux_2x_1}{1 + px_1 - s_vqx_2}\right] u_{ch} - s_ux_1u_{ch} - \dfrac{s_v ax_1 r_p x_4}{L_0 (1+px_1-s_vqx_2)}\\ \dot{x}_2 &= \left[ \dfrac{s_0\alpha F_m - s_ux_2}{1 + px_1 - s_vqx_2} \right]u_{ch} + \dfrac{bx_1r_px_4 - s_vax_2r_px_4}{L_0(1+px_1-s_vqx_2)}\\ \dot{x}_3 &= x_4\\ \dot{x}_4 &= \dfrac{1}{I}[x_2r_p - \lambda x_3 - \mu x_4 - mgl_c \cos{x_3}], \end{align*} \end{cases}$$

where $\textbf{x}=[x_1, \ldots, x_4]^{\mathrm T} = [K_c, F_c, \theta, \dot{\theta}]^{\mathrm T}$ is the state vector and $\textbf{u}=[u_{ch},\alpha ]^{\mathrm T}$ the control vector. The variable $\theta$ represents the knee joint angle and the variables $K_c, F_c, u_{ch}, \alpha$ represent the state variables of the quadriceps muscle model.

In the operation of a power plant superheater, exacting demands are made on the steam temperature maintenance at the outlet. For temperature control at the outlet of a superheater, the relevant system state is the temperature pattern along the superheater tube. This is described by a distributed-parameter system, which involves an infinite number of state variables. To derive a simplified model for control purposes, the superheater is divided into segments, and a lumped model is derived, which represents a finite number of intermediate temperatures.

Assuming that the pressure inside the tube is constant, the enthalpy of the steam satisfies the relation $dH = C_pdT(kcal/kg)$ , where $C_p(kcal/kg^{\circ}C)$ is the constant-pressure specific heat. Hence, we conclude that the heat supplied to the following fluid(steam) only increases its enthalpy, $dH = dQ$ , where $Q$ denotes the heat. In the above equations, it is assumed that convection is the exclusive heat transfer mode for the superheater. Hence the heat transfer from to metal $Q_{ms}(kcal/s)$ and from gas to metal $Q_{gm}(kcal/s)$ are expressed in terms of the heat transfer rates from gas to metal $\alpha_{gm}(kcal/m^2s^{\circ}C)$ and from metal to steam $\alpha_{ms}(kcal/m^2s^{\circ}C)$ and heating surface $S(m^2)$ :

$$\begin{align*} \alpha_{ms}S_1(T(l,t)-T(l,t)) &=Q_{ms} \\ \alpha_{gm}S_2(T_m(l,t)-T(l,t)) &=Q_{gm}. \end{align*}$$

It is also assumed that the heat transfer rates $\alpha_{gm}$ and $\alpha_{ms}$ are constants.

Now, to simulate the profile of superheated steam precisely, it is necessary to divide the superheater into $n$ segments as shown in the attached image.

In the first segment, the desuperheater is included and system is modified as follows:

$$\begin{align*} V_s\rho C_p\frac{{\mathrm d} x_1}{{\mathrm d}t} &={C_{p}}{T_{i}}{w_{i}}-{C_{p}}({w_{i}}+{w_{d}}){x_{1}} +{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}})+{C_{pd}}{T_{d}}{w_{d}}\\ M_mC_m \frac{{\mathrm d}z_1}{{\mathrm d}t} &={\alpha_{gm}}{S_{2}}(T{g_{1}}-{z_{1}})-{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}}), \end{align*}$$

where $x=[x_1,x_2,\ldots,x_n]^{\mathrm T}=[T_1,T_2,\ldots,T_n]^{\mathrm T}$, $z=[z_1,z_2,\ldots,z_n]^{\mathrm T}=[T_{m1},T_{m2},\ldots,t_{mn}]^{\mathrm T}$, and $T_{mi}(^{\circ}C)$ are metal temperature, $T_i(^{\circ}C)$ are steam temperature, $i=1,\ldots,n$.

Consider a linear system represented by the transfer function

$$G(s)=\dfrac{c}{s(s+a)}$$

where $a$ and $c>0$ are unknowns constants, and the reference model

$$G_m(s)=\dfrac{\omega^2}{s^2 + 2\zeta\omega s + \omega^2}.$$

A single link manipulator with flexible joints and negligible damping can be represented by

$$\begin{align*} I\ddot{q}_1 + MgL\sin{q_1} + k(q_1 - q_2) &= 0 \\ J\ddot{q}_2-k(q_1-q_2) &=u, \end{align*}$$

where $q_1$ and $q_2$ are the angular positions, and $u$ is a torque input. The physical parameters $g, I, J, k, L,$ and $M$ are all positive. Taking $y=q_1$ as the output, it can be verified that $y$ satisfies the fourth-order differential equation

$$y^{(4)}=\dfrac{gLM}{I}(\dot{y}^2\sin{y}-\ddot{y}\cos{y})- \left(\dfrac{k}{I}+\dfrac{k}{J}\right)\ddot{y}-\dfrac{gkLM}{IJ}\sin{y}+\dfrac{k}{IJ}u.$$