Pendulum system with Coulomb friction

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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.

Publication details:

 Title Analysis of the Fixed Point Transformation Based Adapive Robot Control Publication Type Conference Paper Year of Publication 2008 Authors Tar, J.K., and Rudas I.J. Conference Name International Conference on Intelligent Engineering Systems, 2008. INES 2008. Date Published 02/2008 Publisher IEEE Conference Location Miami, FL ISBN Number 978-1-4244-2082-7 Accession Number 9965899 Keywords adaptive control, asymptotic stability, initial value problems, iterative methods, MIMO systems, nonlinear control systems, robots, singular value decomposition Abstract In this paper the properties of a novel adaptive nonlinear control recently developed at Budapest Tech for "Multiple Input-Multiple Output (MIMO) Systems" is compared with that of the sophisticated "Adaptive Control by Slotine & Li" widely used in robot control literature. While this latter traditional method utilizes very subtle details of the structurally and formally exact analytical model of the robot in each step of the control cycle in which only the exact values of the parameters are unknown, the novel approach is based on simple geometric considerations concerning the method of the "Singular Value Decomposition (SVD)". Furthermore, while the proof of the asymptotic stability and convergence to an exact trajectory tracking of Slotine's & Li's control is based on "Lyapunov's 2nd Method", in the new approach the control task is formulated as a Fixed Point Problem for the solution of which a Contractive Mapping is created that generates an Iterative Cauchy Sequence. Consequently it converges to the fixed point that is the solution of the control task. Besides the use of very subtle analytical details the main drawback of the Slotine & Li method is that it assumes that the generalized forces acting on the controlled system are exactly known and are equal with that exerted by the controlled drives. So unknown external perturbations can disturb the operation of this sophisticated method. In contrast to that, in the novel method the computationally relatively costly SVD operation on the formally almost exact model need not to be done within each control cycle: it has to be done only one times before the control action is initiated. In the control cycle the inertia matrix is modeled only by a simple scalar. In a more general case the SVD of some approximate model can be done only in a few typical points of the state space of a Classical Mechanical System. To illustrate the usability of the proposed method adaptive control of a Classical M- echanical paradigm, a cart plus crane plus hamper system is considered and discussed by the use of simulation results. DOI 10.1109/INES.2008.4481264

Third-order nonlinear discrete-time system #2

Model description:

Image below shows the block diagram of a discrete-time system.

\begin{align*} H_1(z) &=\dfrac{0.2z^{-1}}{z^{-1}-0.21z^{-2}} \\ H_2(z) &=\dfrac{0.1z^{-1}}{1-1.1z^{-1}+0.3z^{-2}} \\ H_3(z) &=\dfrac{0.3z^{-1}}{1-0.4z^{-1}} \end{align*}

3

Publication details:

 Title Nonlinear system identification using genetic algorithms with application to feedforward control design Publication Type Conference Paper Year of Publication 1998 Authors Luh, Guan-Chun, and Rizzoni G. Conference Name Proceedings of the 1998 American Control Conference, 1998. Date Published 06/1998 Publisher IEEE Conference Location Philadelphia, PA ISBN Number 0-7803-4530-4 Accession Number 6076036 Keywords autoregressive processes, continuous time systems, discrete time systems, feedforward, genetic algorithms, identification, inverse problems, nonlinear systems Abstract A GAMAS-based system identification scheme is developed to construct NARX model of nonlinear systems. Several simulated examples demonstrate that it can be applied to identify both nonlinear continuous-time systems and discrete-time systems with acceptable accuracy. Inverting the identified NARX model, a feedforward controller may be derived to track desired time varying signal of nonlinear systems. Sufficient conditions of the invertibility of NARX model are proposed to investigate the existence of the inverse model. Simulation results depict the effectiveness of the feedforward controller with the aid of simple feedback controller designed for regulation purpose DOI 10.1109/ACC.1998.703056

Third-order nonlinear discrete-time system #1

Model description:

The block diagram of a third-order nonlinear discrete time system adopted by Fakhouri for identification evaluation is shown below.

\begin{align*} H_1(z) &=\dfrac{0.1z^{-1}}{1-0.5z^{-1}} \\ H_2(z) &=\dfrac{0.1z^{-1}}{1-1.3z^{-1}+0.42z^{-2}} \\ H_3(z) &=\dfrac{1.0z^{-1}}{1-0.7z^{-1}} \end{align*}

3

Publication details:

 Title Nonlinear system identification using genetic algorithms with application to feedforward control design Publication Type Conference Paper Year of Publication 1998 Authors Luh, Guan-Chun, and Rizzoni G. Conference Name Proceedings of the 1998 American Control Conference, 1998. Date Published 06/1998 Publisher IEEE Conference Location Philadelphia, PA ISBN Number 0-7803-4530-4 Accession Number 6076036 Keywords autoregressive processes, continuous time systems, discrete time systems, feedforward, genetic algorithms, identification, inverse problems, nonlinear systems Abstract A GAMAS-based system identification scheme is developed to construct NARX model of nonlinear systems. Several simulated examples demonstrate that it can be applied to identify both nonlinear continuous-time systems and discrete-time systems with acceptable accuracy. Inverting the identified NARX model, a feedforward controller may be derived to track desired time varying signal of nonlinear systems. Sufficient conditions of the invertibility of NARX model are proposed to investigate the existence of the inverse model. Simulation results depict the effectiveness of the feedforward controller with the aid of simple feedback controller designed for regulation purpose DOI 10.1109/ACC.1998.703056

Continuous stirred-tank reactor system

Model description:

The following CSTR system developed by Liu(1967). The reaction is exothermic first-order, $A \rightarrow B$, and is given by the following mass and energy balances. One should notice that the energy balance includes cooling water jacket dynamics. The following model was identified using regression techniques on the energy balance equations:

\begin{align*} y(k) &= 1.3187y(k-1) - 0.2214y(k-2) - 0.1474y(k-3) \\ &- 8.6337u(k-1) + 2.9234u(k-2) + 1.2493u(k-3) \\ &- 0.0858y(k-1)u(k-1) + 0.0050y(k-2)u(k-1) \\ &+ 0.0602y(k-2)u(k-2) + 0.0035y(k-3)u(k-1) \\ &- 0.0281y(k-3)u(k-2) + 0.0107y(k-3)u(k-3). \end{align*}

3

Publication details:

 Title Identification and Control of Bilinear Systems Publication Type Conference Paper Year of Publication 1992 Authors Bartee, James F., and Georgakis Christos Conference Name American Control Conference, 1992 Date Published 06/1992 Publisher IEEE Conference Location Chicago, Illinois ISBN Number 0-7803-0210-9 Keywords Algorithm design and analysis, Chemical processes, Continuous-stirred tank reactor, control system synthesis, Control systems, Control theory, linear systems, nonlinear control systems, nonlinear systems, process control Abstract The research presented in this paper combines the problem of identifiction and control of nonlinear processes. This is done by approximating the process with a bilinear model and designing model-based control structures (Reference System Controllers) based on the bilinear approximation. The identification of the bilinear model and the construction of the controller are described below. An example of the identification and control of an exothermic CSTR is also presented. URL http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=4792606&queryText%3DIdentification+and+Control+of+Bilinear+Systems

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 $|\upsilon| \leq 1$.

2

Publication details:

 Title A Simple Nonlinear Observer for a Class of Uncertain Mechanical Systems Publication Type Journal Article Year of Publication 2007 Authors Su, Yuxin, Müller P.C., and Zheng Chunhong Journal IEEE Transactions on Automatic Control Volume 52 Issue 7 Start Page 1340 Pagination 1340-1345 Date Published 07/2007 ISSN 0018-9286 Accession Number 9606706 Keywords asymptotic stability, MIMO systems, nonlinear control systems, observers, uncertain systems Abstract A simple nonlinear observer is proposed for a class of uncertain nonlinear multiple-input-multiple-output (MIMO) mechanical systems whose dynamics are first-order differentiable. The global asymptotic observation of the proposed observer is proved. Thus, the observer can be designed independently of the controller. Furthermore, the proposed observer is formulated without any detailed model knowledge of the system. These advantages make it easy to implement. Numerical simulations are included to illustrate the effectiveness of the proposed observer. DOI 10.1109/TAC.2007.900851