# 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*}

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## Publication details:

 Title Adaptive control of nonlinear multivariable systems using neural networks Publication Type Conference Paper Authors Narendra, K.S., and Mukhopadhyay S.

# 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 $m_t/m_{bnt}$

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## Publication details:

 Title Design, modeling and control of a hopping robot Publication Type Conference Paper Authors Rad, H., Gregorio P., and Buehler M.

# 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 $m_t/m_{bnt}$

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## Publication details:

 Title Design, modeling and control of a hopping robot Publication Type Conference Paper Authors Rad, H., Gregorio P., and Buehler M.

# Generic nonlinear system

## Model description:

\begin{align*} \dot{x}_1 &= x_1 = x_1x_4 + \theta_1x_1^2 \theta_2x_3 + x_2u_1 + u_2 + u_3 \\ \dot{x}_2 &= x_2\cos{x_3} + \theta_3x_4 + u_1 + u_4 \\ \dot{x}_3 &= \sin{x_2} - x_3 \\ \dot{x}_4 &= x_3 - x_4 + x_1x_4 + \theta_1x_1^2 + (1 + x_2)u_1 + u_2 + u_3 \\ y_1 &= x_1 \\ y_2 &= x_2, \end{align*}

where $\theta_1 = 0.5$, $\theta_2 = 2$, and $\theta_3 = 1$ are unknown constant parameters.

## Publication details:

 Title Virtual Grouping based adaptive actuator failure compensation for MIMO nonlinear systems Publication Type Journal Article Authors Tang, Xidong

# Electric drive with a four-pole squirrel-cage induction motor

## Model description:

The motor dynamics are mapped by aset of five highly coupled nonlinear differential equations as given by

\begin{align*} \dfrac{{\mathrm d}i_{qs}^r}{{\mathrm d}t} &= \dfrac{1}{L_{\Sigma}} \left[-\dfrac{L_{RM}^2r_s+M^2r_r}{L_{RM}}i_{qs}^r + \dfrac{r_rM}{L_{RM}}\Psi_{qr}^r - (L_\Sigma i_{ds}^r + M \Psi_{dr}^r) \omega_r + L_{RM}u_{qs}^r\right] \\ \dfrac{{\mathrm d}i_{ds}^r}{{\mathrm d}t} &= \dfrac{1}{L_{\Sigma}} \left[-\dfrac{L_{RM}^2r_s+M^2r_r}{L_{RM}}i_{ds}^r + \dfrac{r_rM}{L_{RM}}\Psi_{dr}^r + (L_\Sigma i_{qs}^r + M \Psi_{qr}^r) \omega_r + L_{RM}u_{ds}^r\right] \\ \dfrac{{\mathrm d}\Psi_{qr}^r}{{\mathrm d}t} &= \dfrac{r_r M}{L_{RM}}i_{qs}^r - \dfrac{r_r}{L_{RM}}\Psi_{qr}^r \\ \dfrac{{\mathrm d}\Psi_{dr}^r}{{\mathrm d}t} &= \dfrac{r_r M}{L_{RM}}i_{ds}^r - \dfrac{r_r}{L_{RM}}\Psi_{dr}^r \\ \dfrac{{\mathrm d}\omega_r}{{\mathrm d}t} &= -\dfrac{B_m}{J}\omega_r + \dfrac{P}{2J}\left[\dfrac{P}{2}\dfrac{M}{L_{RM}}(i_{qs}^r\Psi_{dr}^r - i_{ds}^r-\Psi_{qr})-T_L\right], \end{align*}

where $L_{\Sigma} = L_{SM}L_{RM}-M^2$. For the load torque we assume the expression $T_L=c_2\omega_r^2 + c_3\omega_r^3$.

The state vector is given by

$x(t) = [i_{qs}^r, i_{ds}^r, \Psi_{qr}^r, \Psi_{dr}^r, \omega_r]^{\mathrm T} = [x_1, x_2, x_3, x_4, x_5]^{\mathrm T}.$

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## Publication details:

 Title Nonlinear identification of induction motor parameters Publication Type Conference Paper Authors Pappano, V., Lyshevski S.E., and Friedland B.