Pendulum system with Coulomb friction

Engine model operating under idle

Model description: 

The dynamic engine model, with parameters for a 1.6 liter, 4-cylinder fuel injected engine, is a two-input/two-output system, given by the following differential equations: $\dot{P}=k_P(\dot{m}_{ai}-\dot{m}_{ao}),$ where $k_p=42.40$

$\dot{N}=k_N(T_i-T_L),$ where $k_N=54.26$

$\dot{m}_{ai}=(1+0.907\theta+0.0998\theta^2)g(P)$

$g{P}= \begin{cases} 1, & P<50.6625 \\ 0.0197(101.325P - P^2)^{\frac{1}{2}}, &P \geq 50.6625 \end{cases}$

$\dot{m}_{ao} = -0.0005968N-0.1336P+0.0005341NP+0.000001757NP^2$

$m_{ao} = \dot{m}_{ao}(t-\tau)/(120N), \ \tau=45/N$

$T_i = -39.22+325024m_{ao}-0.0112\delta^2+0.635\delta+(0.0216+0.000675\delta)N(2\pi/60)-0.000102N^2(2\pi/60)^2$

$T_L = (N/263.17)^2+T_d$.

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TitleNeurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks
Publication TypeJournal Article
AuthorsPuskorius, G.V., and Feldkamp L.A.

Continuous flow stirred tank reactor

Model description: 

Coupled nonlinear differential equations describing a process involving a continuous flow stirred tank reactor are given by

$$\begin{align*} \dot{C}_1 &= -C_1u + C_1(1-C_2)e^{C_2/\Gamma} \\ \dot{C}_2 &= -C_2u + C_1(1-C_2)e^{C_2/\Gamma}\dfrac{1+\beta}{1+\beta-C_2}. \end{align*}$$

In these equations, the state variables $C_1$ and $C_2$ represent dimensionless forms of cell mass and amount of nutrients in a constant volume tank, bounded between zero and unity. The control $u$ is the flow rate of nutrients into the tank, and is the same rate at which contents are removed from the tank. The constant parameters $\Gamma$ and $\beta$ determine the rates of cell formation and nutrient consumption; these parameters are set to $\Gamma$= 0.48 and $\beta$ = 0.02 for the nominal benchmark specification.

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2

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TitleNeurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks
Publication TypeJournal Article
AuthorsPuskorius, G.V., and Feldkamp L.A.

Cart plus crane plus hammer

Model description: 

The Euler-Lagrange equations of motion of the system are given as follows

$$\begin{pmatrix} (M+m) & mL\cos{q_1} & 0\\ mL\cos{q_1} & mL^2+\Theta & \dfrac{\Theta}{2}\\ 0 & \dfrac{\Theta}{2} & \Theta \end{pmatrix} \begin{pmatrix} \ddot{x}\\ \ddot{q_1}\\ \ddot{q_2} \end{pmatrix} + \begin{pmatrix} -mL\sin{q_1\dot{q_1}^2}\\ -mLg_g\sin{q_1}\\ 0 \end{pmatrix} = \begin{pmatrix} Q_x\\ Q_1\\ Q_2 \end{pmatrix},$$

where $Q_x (N)$ is the generalized force pushing the cart in the horizontal “$x$” direction, $Q_1$ and $Q_2$ are torques in $(N · m)$ rotating the beam of the crane around a horizontal axis orthogonal to “$x$” and counter-rotating the hamper at the free end of the beam to avoid turning out the worker from the hamper. $L (m)$ denotes the lenght of the crane’s beam, $g_g$ ($m/s^2$) is the gravitational acceleration, $m$ ($kg$) and $\Theta$ $(kg · m^2)$ denote the momentum (with respect to its own center of mass that was supposed to be on the rotational axle) and the mass of the hamper.

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TitleAnalysis of the Fixed Point Transformation Based Adapive Robot Control
Publication TypeConference Paper
AuthorsTar, J.K., and Rudas I.J.

Coupled electric drives

Model description: 

This particular laboratory-scale process simulates the actual industrial problems in tension and speed controls as they occur in magnetic tape drives, textile machines, paper mills, strip metal production plants, etc. To simulate these problems. the coupled electric drives consists of two similar servo-motors which drive a jockey pulley via a continuous flexible belt (see attached image).The jockey pulley assembly constitutes a simulated work station. The basic control problem is to regulate the belt speed and tension by varying the two servo-motor torque.

The structure of the transfer function matrix model (numerator and denominator orders of each transfer function) have been determined from the theoretical modelling of the coupled electric drive system which has given

$$\begin{bmatrix} Y_1(s)\\ Y_2(s) \end{bmatrix} =G(s) \begin{bmatrix} U_1(s)\\ U_2(s) \end{bmatrix}$$

with

$ G(s) = \begin{bmatrix} \dfrac{b_{1,1,0}}{s^2+a_{11}s+a_{12}} & \dfrac{b_{1,2,0}}{s^2+a_{11}s+a_{12}}\\ \dfrac{b_{2,1,0}s+b_{2,1,1}}{s^3+a_{21}s^2+a_{22}s+a_{23}} & \dfrac{b_{2,2,0}}{s^3+a_{21}s^2+a_{22}s+a_{23}} \end{bmatrix} $

The inputs $U(s)$ to the system are the drive voltages to the servo-motor power amplifiers. The outputs $Y_1(s)$ and $Y_2(s)$ are the jockey pulley velocity and the belt tension respectively.

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TitleA new bias-compensating least-squares method for continuous-time MIMO system identification applied to a laboratory-scale process
Publication TypeConference Paper
AuthorsGarnier, H., Sibille P., and Nguyen H.L.

Pendulum system with Coulomb friction

Model description: 

Consider a pendulum system with Coulomb friction and external perturbation

$$ \ddot {\theta} = \frac{1}{J}u - \frac{g}{L}\sin \theta - \frac{V_s}{J}\dot{\theta } - \frac{P_s}{J}\mathrm{sgn}(\dot{\theta}) + \upsilon, $$

where parameters have the following values $M=1.1$, $L=0.9$, $J=ML^2=0.891$, $V_s=0.18$, $P_s=0.18$, $P_s=0.45$, $g=9.815$, and $v$ is an uncertain external perturbation $|\upsilon| \leq 1$.

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TitleA Simple Nonlinear Observer for a Class of Uncertain Mechanical Systems
Publication TypeJournal Article
AuthorsSu, Yuxin, Müller P.C., and Zheng Chunhong

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