Time Domain Simulation

Time responses

control.forced_response(sys, T=None, U=0.0, X0=0.0, transpose=False, **keywords)

Simulate the output of a linear system.

As a convenience for parameters U, X0: Numbers (scalars) are converted to constant arrays with the correct shape. The correct shape is inferred from arguments sys and T.

For information on the shape of parameters U, T, X0 and return values T, yout, xout see: time-series-convention

Parameters:

sys: Lti (StateSpace, or TransferFunction)

LTI system to simulate

T: array-like

Time steps at which the input is defined, numbers must be (strictly monotonic) increasing.

U: array-like or number, optional

Input array giving input at each time T (default = 0).

If U is None or 0, a special algorithm is used. This special algorithm is faster than the general algorithm, which is used otherwise.

X0: array-like or number, optional

Initial condition (default = 0).

transpose: bool

If True, transpose all input and output arrays (for backward compatibility with MATLAB and scipy.signal.lsim)

**keywords:

Additional keyword arguments control the solution algorithm for the differential equations. These arguments are passed on to the function scipy.integrate.odeint(). See the documentation for scipy.integrate.odeint() for information about these arguments.

Returns:

T: array

Time values of the output.

yout: array

Response of the system.

xout: array

Time evolution of the state vector.

See also

step_response, initial_response, impulse_response

Examples

>>> T, yout, xout = forced_response(sys, T, u, X0)

control.initial_response(sys, T=None, X0=0.0, input=0, output=None, transpose=False, **keywords)

Initial condition response of a linear system

If the system has multiple inputs or outputs (MIMO), one input and one output have to be selected for the simulation. The parameters input and output do this. All other inputs are set to 0, all other outputs are ignored.

For information on the shape of parameters T, X0 and return values T, yout see: time-series-convention

Parameters:

sys: StateSpace, or TransferFunction

LTI system to simulate

T: array-like object, optional

Time vector (argument is autocomputed if not given)

X0: array-like object or number, optional

Initial condition (default = 0)

Numbers are converted to constant arrays with the correct shape.

input: int

Index of the input that will be used in this simulation.

output: int

Index of the output that will be used in this simulation. Set to None to not trim outputs

transpose: bool

If True, transpose all input and output arrays (for backward compatibility with MATLAB and scipy.signal.lsim)

**keywords:

Additional keyword arguments control the solution algorithm for the differential equations. These arguments are passed on to the function lsim(), which in turn passes them on to scipy.integrate.odeint(). See the documentation for scipy.integrate.odeint() for information about these arguments.

Returns:

T: array

Time values of the output

yout: array

Response of the system

See also

forced_response, impulse_response, step_response

Examples

>>> T, yout = initial_response(sys, T, X0)

control.step_response(sys, T=None, X0=0.0, input=0, output=None, transpose=False, **keywords)

Step response of a linear system

If the system has multiple inputs or outputs (MIMO), one input has to be selected for the simulation. Optionally, one output may be selected. The parameters input and output do this. All other inputs are set to 0, all other outputs are ignored.

For information on the shape of parameters T, X0 and return values T, yout see: time-series-convention

Parameters:

sys: StateSpace, or TransferFunction

LTI system to simulate

T: array-like object, optional

Time vector (argument is autocomputed if not given)

X0: array-like or number, optional

Initial condition (default = 0)

Numbers are converted to constant arrays with the correct shape.

input: int

Index of the input that will be used in this simulation.

output: int

Index of the output that will be used in this simulation. Set to None to not trim outputs

transpose: bool

If True, transpose all input and output arrays (for backward compatibility with MATLAB and scipy.signal.lsim)

**keywords:

Additional keyword arguments control the solution algorithm for the differential equations. These arguments are passed on to the function lsim(), which in turn passes them on to scipy.integrate.odeint(). See the documentation for scipy.integrate.odeint() for information about these arguments.

Returns:

T: array

Time values of the output

yout: array

Response of the system

See also

forced_response, initial_response, impulse_response

Examples

>>> T, yout = step_response(sys, T, X0)

Phase portraits

phaseplot.phase_plot(odefun, X=None, Y=None, scale=1, X0=None, T=None, lingrid=None, lintime=None, logtime=None, timepts=None, parms=(), verbose=True)

Phase plot for 2D dynamical systems

Produces a vector field or stream line plot for a planar system.

Call signatures:

phase_plot(func, X, Y, ...) - display vector field on meshgrid phase_plot(func, X, Y, scale, ...) - scale arrows phase_plot(func. X0=(...), T=Tmax, ...) - display stream lines phase_plot(func, X, Y, X0=[...], T=Tmax, ...) - plot both phase_plot(func, X0=[...], T=Tmax, lingrid=N, ...) - plot both phase_plot(func, X0=[...], lintime=N, ...) - stream lines with arrows

Parameters:

func : callable(x, t, ...)

Computes the time derivative of y (compatible with odeint). The function should be the same for as used for scipy.integrate. Namely, it should be a function of the form dxdt = F(x, t) that accepts a state x of dimension 2 and returns a derivative dx/dt of dimension 2.

X, Y: ndarray, optional

Two 1-D arrays representing x and y coordinates of a grid. These arguments are passed to meshgrid and generate the lists of points at which the vector field is plotted. If absent (or None), the vector field is not plotted.

scale: float, optional

Scale size of arrows; default = 1

X0: ndarray of initial conditions, optional

List of initial conditions from which streamlines are plotted. Each initial condition should be a pair of numbers.

T: array-like or number, optional

Length of time to run simulations that generate streamlines. If a single number, the same simulation time is used for all initial conditions. Otherwise, should be a list of length len(X0) that gives the simulation time for each initial condition. Default value = 50.

lingrid = N or (N, M): integer or 2-tuple of integers, optional

If X0 is given and X, Y are missing, a grid of arrows is produced using the limits of the initial conditions, with N grid points in each dimension or N grid points in x and M grid points in y.

lintime = N: integer, optional

Draw N arrows using equally space time points

logtime = (N, lambda): (integer, float), optional

Draw N arrows using exponential time constant lambda

timepts = [t1, t2, ...]: array-like, optional

Draw arrows at the given list times

parms: tuple, optional

List of parameters to pass to vector field: func(x, t, *parms)

See also

box_grid, Y

phaseplot.box_grid(xlimp, ylimp)

box_grid generate list of points on edge of box

list = box_grid([xmin xmax xnum], [ymin ymax ynum]) generates a list of points that correspond to a uniform grid at the end of the box defined by the corners [xmin ymin] and [xmax ymax].