Nonautonomous

Non-autonomous dynamic systems.

Chaotic nonlinear ODEs

• https://en.wikipedia.org/wiki/List_of_chaotic_maps

class nonautonomous.CSTR(nsim=1001, ninit=0, ts=0.1, seed=59)

CSTR model Original code obtained from APMonitor:

• http://apmonitor.com/do/index.php/Main/NonlinearControl

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

class nonautonomous.HindmarshRose(nsim=1001, ninit=0, ts=0.1, seed=59)

Hindmarsh–Rose model of neuronal activity

• https://en.wikipedia.org/wiki/Hindmarsh%E2%80%93Rose_model

• https://demonstrations.wolfram.com/HindmarshRoseNeuronModel/

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

class nonautonomous.InvPendulum(nsim=1001, ninit=0, ts=0.1, seed=59)

Inverted Pendulum dynamics states: x = [ heta dot{ heta}]; heta is angle from upright equilibrium input: u = input torque

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

class nonautonomous.Iver_dyn(nsim=1001, ninit=0, ts=0.1, seed=3)

Dynamic model of Unmanned Underwater Vehicle (Fossen) – Excludes hydrostatic/dynamic terms and ocean current

Cross(x)

• Input: 3d array x

• Output: cross product matrix (skew symmetric)

equations(x, t, u)

• States (6): [u, v, w, p, q, r]

• Inputs (6): [tau_X, tau_Y, tau_Z, tau_K, tau_M, tau_N]

get_U(nsim)

class nonautonomous.Iver_dyn_reduced(nsim=1001, ninit=0, ts=0.1, seed=3)

Dynamic model of Unmanned Underwater Vehicle (Yan et al) – Excludes rolling, includes hydrostate/dynamic terms, no currents

equations(x, t, u)

• States (10): [n, e, d, theta, psi, u, v, w, p, q, r]

• Inputs (5): [tau_X, tau_Y, tau_Z, tau_M, tau_N]

get_U(nsim)

class nonautonomous.Iver_dyn_simplified(nsim=1001, ninit=0, ts=0.1, seed=3)

Dynamic model of Unmanned Underwater Vehicle (modified from Stankiewicz et al) – Excludes rolling, sway, currents, Includes: hydrostate/dynamic terms, control surface deflections/propeller thrust, and actuator dynamics

equations(x, t, u)

• States (12): [px, py, pz, theta, psi, uu, w, q, r, delta_u, delta_q, delta_r]

• Inputs (3): [delta_uc, delta_qc, delta_rc] (thrust speed/deflections, normalized)

get_U(nsim)

class nonautonomous.Iver_dyn_simplified_output(nsim=1001, ninit=0, ts=0.1, seed=59)

Dynamic model of Unmanned Underwater Vehicle (modified from Stankiewicz et al) – Excludes rolling, sway, currents, Includes: hydrostate/dynamic terms, control surface deflections/propeller thrust, and actuator dynamics with non-kinematic output

equations(x, t, u)

• States (9): [theta, psi, uu, w, q, r, delta_u, delta_q, delta_r]

• Inputs (3): [delta_uc, delta_qc, delta_rc] (thrust speed/deflections, normalized)

get_U(nsim)

parameters()

class nonautonomous.Iver_kin(nsim=1001, ninit=0, ts=0.1, seed=3)

Kinetic model of Unmanned Underwater Vehicle (Fossen) – Full UAV kinematic model

equations(x, t, u)

• States (5): [n, e, d, phi, theta, psi]

• Inputs (5): [u, v, w, p, q, r]

get_U(nsim)

class nonautonomous.Iver_kin_reduced(nsim=1001, ninit=0, ts=0.1, seed=3)

Kinetic model of Unmanned Underwater Vehicle (Yan et al 2020) – UAV kinematic model with no roll

equations(x, t, u)

• States (5): [xi, eta, zeta, theta, psi]

• Inputs (5): [u, v, w, q, r]

get_U(nsim)

class nonautonomous.LorenzControl(nsim=1001, ninit=0, ts=0.1, seed=59)

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

u_fun(t)

class nonautonomous.ODE_NonAutonomous(nsim=1001, ninit=0, ts=0.1, seed=59)

base class autonomous ODE

simulate(U=None, ninit=None, nsim=None, ts=None, x0=None)

Parameters:

• nsim – (int) Number of steps for open loop response

• ninit – (float) initial simulation time

• ts – (float) step size, sampling time

• x0 – (float) state initial conditions

Returns:

X, Y, U, D

class nonautonomous.SEIR_population(nsim=1001, ninit=0, ts=0.1, seed=59)

Susceptible, Exposed, Infected, and Recovered (SEIR) population population model. Used to model COVID-19 spread. Source of the model:

• https://apmonitor.com/do/index.php/Main/COVID-19Response

states:

• Susceptible (s): population fraction that is susceptible to the virus

• Exposed (e): population fraction is infected with the virus but does not transmit to others

• Infectious (i): population fraction that is infected and can infect others

• Recovered (r): population fraction recovered from infection and is immune from further infection

equations(x, t, u)

• Inputs (1): social distancing (u=0 (none), u=1 (total isolation))

• States (4):

• Susceptible (s): population fraction that is susceptible to the virus

• Exposed (e): population fraction is infected with the virus but does not transmit to others

• Infectious (i): population fraction that is infected and can infect others

• Recovered (r): population fraction recovered from infection and is immune from further infection

get_U(nsim)

class nonautonomous.SwingEquation(nsim=1001, ninit=0, ts=0.1, seed=3)

Power Grid Swing Equation.

• https://en.wikipedia.org/wiki/Swing_equation

• The second-order swing equation is converted to

• two first-order ODEs

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

class nonautonomous.Tank(nsim=1001, ninit=0, ts=0.1, seed=59)

Single Tank model Original code obtained from APMonitor:

• https://apmonitor.com/pdc/index.php/Main/TankLevel

equations(x, t, u)

• States (1): level in the tanks

• Inputs u(1): c - valve coefficient (kg/s / %open)

• Inputs u(2): valve in % [0-100]

get_U(nsim)

class nonautonomous.TwoTank(nsim=1001, ninit=0, ts=0.1, seed=59)

Two Tank model Original code obtained from APMonitor:

• https://apmonitor.com/do/index.php/Main/LevelControl

equations(x, t, u)

Define equations defining the dynamical system

get_U(nsim)

class nonautonomous.UAV2D_kin(nsim=1001, ninit=0, ts=0.1, seed=59)

Dubins 2D model – UAV kinematic model with no wind

equations(x, t, u)

• States (3): [x, y, z, psi]

• Inputs (1): [phi]

get_U(nsim)

class nonautonomous.UAV3D_dyn(nsim=1001, ninit=0, ts=0.1, seed=59)

UAV dynamic guidance model with no wind

equations(x, t, U)

• States (6): [x, y, z, psi, gamma, V]

• Inputs (3): [T, phi, load]

• load = Lift force / Weight

get_U(nsim)

class nonautonomous.UAV3D_kin(nsim=1001, ninit=0, ts=0.1, seed=59)

Dubins 3D model – UAV kinematic model with no wind

equations(x, t, u)

• States (4): [x, y, z]

• Inputs (3): [V, phi, gamma]

• Transformed Inputs (3): [U1, U2, U3]

get_U(nsim)

class nonautonomous.UAV3D_reduced(nsim=1001, ninit=0, ts=0.1, seed=59)

Reduced Dubins 3D model – UAV kinematic model with transformed inputs

equations(x, t, u)

• States (4): [x, y, z]

• Inputs (3): [V, phi, gamma]

• Transformed Inputs (3): [U1, U2, U3]

get_U(nsim)