# 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 $k_N=54.26$

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

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

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

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

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

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

## Publication details:

 Title Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks Publication Type Journal Article Year of Publication 1994 Authors Puskorius, G.V., and Feldkamp L.A. Journal IEEE Transactions on Neural Networks Volume 5 Issue 2 Start Page 279 Pagination 279-297 Date Published 1994 ISSN 1045-9227 Accession Number 4685633 Keywords filtering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets Abstract Although 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. DOI 10.1109/72.279191

# 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.

2

## Publication details:

 Title Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks Publication Type Journal Article Year of Publication 1994 Authors Puskorius, G.V., and Feldkamp L.A. Journal IEEE Transactions on Neural Networks Volume 5 Issue 2 Start Page 279 Pagination 279-297 Date Published 1994 ISSN 1045-9227 Accession Number 4685633 Keywords filtering and prediction theory, Kalman filters, nonlinear control systems, Nonlinear dynamical systems, recurrent neural nets Abstract Although 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. DOI 10.1109/72.279191

# Generic nonlinear system 2

## Model description:

\begin{align*} x_1(k+1) &= 0.9x_1(k)\sin{[x_2(k)]} + \left(2 + 1.5 \dfrac{x_1(k)u_1(k)}{1+x_1^2(k)u_1^2(k)}\right)u_1(k) + \left(x_1(k) + \dfrac{2x_1(k)}{1+x_1^2(k)}\right)u_1(k)\\ x_2(k+1) &= x_3(k)(1+\sin{[4x_3(k)]}+ \dfrac{x_3(k)}{1+x_3^2(k)}\\ x_3(k+1) &= (3 + \sin{[2x_1(k)]})u_2(k)\\ y_1(k)&=x_1(k)\\ y_2(k)&=x_2(k) \end{align*}

3

## Publication details:

 Title Adaptive control of nonlinear multivariable systems using neural networks Publication Type Conference Paper Year of Publication 1993 Authors Narendra, K.S., and Mukhopadhyay S. Conference Name Proceedings of the 32nd IEEE Conference on Decision and Control, 1993. Date Published 12/1993 Publisher IEEE Conference Location San Antonio, TX ISBN Number 0-7803-1298-8 Accession Number 4772091 Keywords adaptive control, multivariable systems, neural nets, nonlinear systems Abstract In this paper we examine the problem of control of multivariable systems using neural networks. The problem is discussed assuming different amounts of prior information concerning the plant and hence different levels of complexity. In the first stage it is assumed that the state equations describing the plant are known and that the state of the system is accessible. Following this the same problem is considered when the state equations are unknown. In the last stage the adaptive control of the multivariable system using only input-output data is discussed in detail. The objective of the paper is to demonstrate that results from nonlinear control theory and linear adaptive control theory can be used to design practically viable controllers for unknown nonlinear multivariable systems using neural networks. The different assumptions that have to be made, the choice of identifier and controller architectures and the generation of adaptive laws for the adjustment of the parameters of the neural networks form the core of the paper DOI 10.1109/CDC.1993.325299

# Hopping robot - Flight

## Model description:

$$\dot{x}=A_Fx+b_F\tau+e_F(x)$$

with

$A_S = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & -K \dfrac{m_n}{\beta_F} & -C \dfrac{m_n}{\beta_F} \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -K \dfrac{m_{bnt}}{\beta_F} & -C \dfrac{m_{bnt}}{\beta_F} \\ \end{bmatrix},\\ x = \begin{bmatrix} z\\ \dot{z}\\ p\\ \dot{p} \end{bmatrix}, b_F = \dfrac{\eta}{\beta_{F}r} \begin{bmatrix} 0\\ m_n\\ 0\\ m_{bn} \end{bmatrix},\\ e_F(x)=\dfrac{1}{\beta_F} \begin{bmatrix} 0\\ \alpha(m_{bnt}g + f_{fF} + m_n(m_ng - k(s_0 - l_0) - f_p))\\ 0\\ -m_nf_{fF} - m_{bnt}(k(s_0-l_0) + f_a)\\ \end{bmatrix}.$

 $z$ Body Height $p$ Actuator Length $\tau$ Motor Torque $\theta$ Motor angle, $\theta = p/r$ $s$ Spring Length $m_b$ 9.5kg Upper Leg Mass $m_n$ 0.25kg Ball Nut Mass $m_t$ 0.5kg Toe Mass $k$ 400 N/m Spring Constant $F_p$ 5.0N Leg Dry Friction $F_z$ 1.5N Planarized Dry Friction $F_a$ 0N Ball Screw Dry Friction $c$ 5.5Ns/m Spring Viscous Friction $\hat{\tau}$ 1.78Nm Stall Torque $\hat{\omega}$ 2800RPM Max Speed $\eta$ 0.95 Ball Screw Efficiency $s_0$ 0.608m Spring Rest Length $l_0$ 0.595m Maximum Leg Length $J$ 2.7$\times$10$^{-4}$kgm$^2$ Motor Inertia $\alpha$ 0.34kgm $J/r^2+m_n$ $\mu$ 0.05 $m_t/m_{bnt}$

4

## Publication details:

 Title Design, modeling and control of a hopping robot Publication Type Conference Paper Year of Publication 1993 Authors Rad, H., Gregorio P., and Buehler M. Conference Name Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, IROS '93. Date Published 06/1993 Publisher IEEE Conference Location Yokohama ISBN Number 0-7803-0823-9 Accession Number 5050001 Keywords legged locomotion Abstract The authors report progress towards model based, dynamically stable legged locomotion with energy efficient, electrically actuated robots. The present the mechanical design of a prismatic robot leg which is optimized for electrical actuation. A dynamical model of the robot and the actuator as well as the interaction with ground is derived and validated by demonstrating close correspondence between simulations and experiments. A new continuous, and exactly implementable open loop torque control algorithm is introduced which stabilizes a limit cycle of the underlying fourth order intermittent robot/actuator/environment dynamics DOI 10.1109/IROS.1993.583877

# Hopping robot - Stance

## Model description:

$$\dot{x}=A_Sx+b_S\tau+e_S(x)$$

with

$A_S = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -K\dfrac{J}{\beta_Sr^2} & -C\dfrac{J}{\beta_Sr^2} & K\dfrac{J}{\beta_Sr^2} & C\dfrac{J}{\beta_Sr^2} \\ 0 & 0 & 0 & 1 \\ K\dfrac{m_b}{\beta_s} & C\dfrac{m_b}{\beta_s} & -K\dfrac{m_b}{\beta_s} & -C\dfrac{m_b}{\beta_s} \\ \end{bmatrix},\\ x = \begin{bmatrix} z\\ \dot{z}\\ p\\ \dot{p} \end{bmatrix}, b_S = \dfrac{\eta}{\beta_{S}r} \begin{bmatrix} 0\\ m_n\\ 0\\ m_{bn} \end{bmatrix},\\ e_S(x)=\dfrac{1}{\beta_s} \begin{bmatrix} 0\\ \alpha(ks_0 - m_{bn}g- f_{fS} + m_n(m_ng - ks_0 - f_a)\\ 0\\ 0\\ \end{bmatrix}.$

 $z$ Body Height $p$ Actuator Length $\tau$ Motor Torque $\theta$ Motor angle, $\theta = p/r$ $s$ Spring Length $m_b$ 9.5kg Upper Leg Mass $m_n$ 0.25kg Ball Nut Mass $m_t$ 0.5kg Toe Mass $k$ 400 N/m Spring Constant $F_p$ 5.0N Leg Dry Friction $F_z$ 1.5N Planarized Dry Friction $F_a$ 0N Ball Screw Dry Friction $c$ 5.5Ns/m Spring Viscous Friction $\hat{\tau}$ 1.78Nm Stall Torque $\hat{\omega}$ 2800RPM Max Speed $\eta$ 0.95 Ball Screw Efficiency $s_0$ 0.608m Spring Rest Length $l_0$ 0.595m Maximum Leg Length $J$ 2.7$\times$10$^{-4}$kgm$^2$ Motor Inertia $\alpha$ 0.34kgm $J/r^2+m_n$ $\mu$ 0.05 $m_t/m_{bnt}$

4

## Publication details:

 Title Design, modeling and control of a hopping robot Publication Type Conference Paper Year of Publication 1993 Authors Rad, H., Gregorio P., and Buehler M. Conference Name Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems '93, IROS '93. Date Published 06/1993 Publisher IEEE Conference Location Yokohama ISBN Number 0-7803-0823-9 Accession Number 5050001 Keywords legged locomotion Abstract The authors report progress towards model based, dynamically stable legged locomotion with energy efficient, electrically actuated robots. The present the mechanical design of a prismatic robot leg which is optimized for electrical actuation. A dynamical model of the robot and the actuator as well as the interaction with ground is derived and validated by demonstrating close correspondence between simulations and experiments. A new continuous, and exactly implementable open loop torque control algorithm is introduced which stabilizes a limit cycle of the underlying fourth order intermittent robot/actuator/environment dynamics DOI 10.1109/IROS.1993.583877