Kinematic Car Model

Continuously stirred tank reactor system

Model description: 

A schematic of the CSTR plant is shown in the attached image. The process dynamics are described by

$$\begin{align*} \dot{C}_{a} &=\frac{q}{V}(C_{a0}-c_{a})-a_{0}C_{a}e^{-\frac{E}{RT_{a}}} \\ \dot{T}_{a} &=\frac{q}{V}(T_{f}-T_{a})+a_{1}C_{a}e^{-\frac{E}{RT_{a}}}+a_{3}q_{c}\left(1-e^{\frac{a_{2}}{q_{c}}}\right)(T_{cf}-T_{a}), \end{align*}$$

where the variables $C_a$ and $T_a$ are the concentration and temperature of the tank, respectively; the coolant flow rate $q_c$ is the control input and the parameters of the plant are defined in the attached table. Within the tank reactor, two chemicals are mixed and react to produce a product compound $A$ at a concentration $C_a(t)$ with the temperature of the mixture being $T(t)$. The reaction is both irreversible and exothermic.

In the paper, authors assumed that plant parameters $q, C_{a0}, T_f$ and $V$ are at the nominal values given in the attached table. The activation energy $E/R = 1 \times 10^4K$ is assumed to be known. The state variables the input and the output are defined as $x=[x_1,x_2]^{\mathrm T}=[C_a,T_a]^{\mathrm T},u=q_c,y=C_a$. Using this notation, the CSTR plant can be re-expressed as

$$\begin{align*} \dot{x}_{1} &=1-x_{1}-a_{0}x_{1}e^{-\frac{10^4}{{\rm a}_2}} \\ \dot{x}_{2} &=T_{f}-x_{2}+a_{1}x_{1}e^{-\frac{10^4}{{\rm a}_2}}+a_{3}u\left(1-e^{-\frac{a_2}{u}}\right)(T_{cf}-x_{2}) \\ y &= x_{1}, \end{align*}$$

where the unknown constant parameters are $a_0, a_1, a_2$ and $a_3$.

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Publication details: 

TitleAdaptive nonlinear control of continuously stirred tank reactor systems
Publication TypeConference Paper
AuthorsZhang, T., and Guay M.

SISO NLTI plant

Model description: 

Consider the following unknown discrete nonlinear dynamic system:

$$\begin{align*} y(k+1)&=p[{\bf q}(k), u(k)]=0.2\cos[0.8(y(k)+y(k-1))] \\ & +0.4\sin[0.8(y(k)+y(k-1))+2u(k)+u(k-1)] \\ &+0.1[9+y(k)+y(k-1)]+\left[{2(u(k)+u(k-1))\over 1+\cos(y(k))}\right] \end{align*}$$

for $k=0,1,2,\ldots$ with $y(k)=0,u(k)=0$, for $k=0,-1,-2,\ldots$, $\Delta t := t(k+1)-t(k)=0.02sec$, for $k=0,1,2,\ldots$.

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2

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TitleRobust nonlinear adaptive control using neural networks
Publication TypeConference Paper
AuthorsAdetona, O., Sathananthan S., and Keel L.H.

2-input 2-output nonlinear system

Model description: 

The suggested tracker scheme is tested with a 2-input 2-output nonlinear system given by:

$$\begin{align*} y_{1} (k) & = 0.21y_{1} (k-1)-0.12y_{2} (k-2) \\ & + 0.3y_{1} (k-1)u_{2} (k-1)-1.6u_{2} (k-1) \\ & + 1.2u_{1} (k-1), \\ y_{2} (k) & = 0.25y_{2} (k-1)-0.1y_{1} (k-2) \\ &- 0.2 y_{2} (k-1)u_{1} (k-1)-2.6u_{1} (k-1) \\ &-1.2u_{2} (k-1). \end{align*}$$

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TitleU-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems
Publication TypeConference Paper
AuthorsAzhar, A.S.S., Al-Sunni F.M., and Shafiq M.

Kinematic Car Model

Model description: 

The attached image depicts the kinematic car in the horizontal plane. Let us suppose that the Ackermann steering assumptions hold true, hence all wheels turn around the same point (denoted by P) which lies on the line of the rear axle. It follows that the kinematics of the car can be fully described by the kinematics of a bicycle fitted in the middle of the car (see attached image. The coordinates of the rear axle midpoint are given by $x$ and $y$. The orientation of the car with respect to the axis of $x$ is denoted by 9. The angle of the front wheel of the bicycle with respect to the longitudinal symmetry axis of the car is denoted by $φ$ . One may consider $φ$ or its time derivative $u_2=\dot{φ}$ as input. The longitudinal velocity of the rear axle midpoint is denoted by $u_1$ if it is a control input (two input case) and by $v_{car}$ if not (one input case). All lengths involved in the kinematic calculations, and in particular $l$, equal to one.

$$\begin{align*} \dot{x} &= u_1 \cos{\theta},\\ \dot{y} &= u_1 \sin{\theta},\\ \dot{\theta} &= u_1 \tan{\varphi},\\ \dot{\varphi} &= u_2. \end{align*}$$

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4

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TitleOn-line time-scaling control of a kinematic car with one input
Publication TypeConference Paper
AuthorsKiss, B., and Szadeczky-Kardoss E.

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