Third-order nonlinear discrete-time system #1

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
Year of Publication1994
AuthorsPuskorius, G.V., and Feldkamp L.A.
JournalIEEE Transactions on Neural Networks
Volume5
Issue2
Start Page279
Pagination279-297
Date Published1994
ISSN1045-9227
Accession Number4685633
Keywordsfiltering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets
AbstractAlthough the potential of the powerful mapping and representational capabilities of recurrent network architectures is generally recognized by the neural network research community, recurrent neural networks have not been widely used for the control of nonlinear dynamical systems, possibly due to the relative ineffectiveness of simple gradient descent training algorithms. Developments in the use of parameter-based extended Kalman filter algorithms for training recurrent networks may provide a mechanism by which these architectures will prove to be of practical value. This paper presents a decoupled extended Kalman filter (DEKF) algorithm for training of recurrent networks with special emphasis on application to control problems. We demonstrate in simulation the application of the DEKF algorithm to a series of example control problems ranging from the well-known cart-pole and bioreactor benchmark problems to an automotive subsystem, engine idle speed control. These simulations suggest that recurrent controller networks trained by Kalman filter methods can combine the traditional features of state-space controllers and observers in a homogeneous architecture for nonlinear dynamical systems, while simultaneously exhibiting less sensitivity than do purely feedforward controller networks to changes in plant parameters and measurement noise.
DOI10.1109/72.279191

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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Publication details: 

TitleNeurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks
Publication TypeJournal Article
Year of Publication1994
AuthorsPuskorius, G.V., and Feldkamp L.A.
JournalIEEE Transactions on Neural Networks
Volume5
Issue2
Start Page279
Pagination279-297
Date Published1994
ISSN1045-9227
Accession Number4685633
Keywordsfiltering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets
AbstractAlthough the potential of the powerful mapping and representational capabilities of recurrent network architectures is generally recognized by the neural network research community, recurrent neural networks have not been widely used for the control of nonlinear dynamical systems, possibly due to the relative ineffectiveness of simple gradient descent training algorithms. Developments in the use of parameter-based extended Kalman filter algorithms for training recurrent networks may provide a mechanism by which these architectures will prove to be of practical value. This paper presents a decoupled extended Kalman filter (DEKF) algorithm for training of recurrent networks with special emphasis on application to control problems. We demonstrate in simulation the application of the DEKF algorithm to a series of example control problems ranging from the well-known cart-pole and bioreactor benchmark problems to an automotive subsystem, engine idle speed control. These simulations suggest that recurrent controller networks trained by Kalman filter methods can combine the traditional features of state-space controllers and observers in a homogeneous architecture for nonlinear dynamical systems, while simultaneously exhibiting less sensitivity than do purely feedforward controller networks to changes in plant parameters and measurement noise.
DOI10.1109/72.279191

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
Year of Publication2008
AuthorsTar, J.K., and Rudas I.J.
Conference NameInternational Conference on Intelligent Engineering Systems, 2008. INES 2008.
Date Published02/2008
PublisherIEEE
Conference LocationMiami, FL
ISBN Number978-1-4244-2082-7
Accession Number9965899
Keywordsadaptive control, asymptotic stability, initial value problems, iterative methods, MIMO systems, nonlinear control systems, robots, singular value decomposition
AbstractIn 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.
DOI10.1109/INES.2008.4481264

Third-order nonlinear discrete-time system #2

Model description: 

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

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

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3

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TitleNonlinear system identification using genetic algorithms with application to feedforward control design
Publication TypeConference Paper
Year of Publication1998
AuthorsLuh, Guan-Chun, and Rizzoni G.
Conference NameProceedings of the 1998 American Control Conference, 1998.
Date Published06/1998
PublisherIEEE
Conference LocationPhiladelphia, PA
ISBN Number0-7803-4530-4
Accession Number6076036
Keywordsautoregressive processes, continuous time systems, discrete time systems, feedforward, genetic algorithms, identification, inverse problems, nonlinear systems
AbstractA 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
DOI10.1109/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.

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

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3

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Publication details: 

TitleNonlinear system identification using genetic algorithms with application to feedforward control design
Publication TypeConference Paper
Year of Publication1998
AuthorsLuh, Guan-Chun, and Rizzoni G.
Conference NameProceedings of the 1998 American Control Conference, 1998.
Date Published06/1998
PublisherIEEE
Conference LocationPhiladelphia, PA
ISBN Number0-7803-4530-4
Accession Number6076036
Keywordsautoregressive processes, continuous time systems, discrete time systems, feedforward, genetic algorithms, identification, inverse problems, nonlinear systems
AbstractA 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
DOI10.1109/ACC.1998.703056

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