python-control/control/sisotool.py at 4afda82ae752df3fa3204731dd1ff201eec3fdd9 · python-control/python-control

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__all__ = ['sisotool', 'rootlocus_pid_designer']

from control.exception import ControlMIMONotImplemented

from .freqplot import bode_plot

from .timeresp import step_response

from .iosys import common_timebase, isctime, isdtime

from .xferfcn import tf

from .statesp import ss, summing_junction

from .bdalg import append, connect

from .nlsys import interconnect

from control.statesp import _convert_to_statespace

from . import config

import numpy as np

import matplotlib.pyplot as plt

import warnings

_sisotool_defaults = {

'sisotool.initial_gain': 1

}

def sisotool(sys, initial_gain=None, xlim_rlocus=None, ylim_rlocus=None,

plotstr_rlocus='C0', rlocus_grid=False, omega=None, dB=None,

Hz=None, deg=None, omega_limits=None, omega_num=None,

margins_bode=True, tvect=None, kvect=None):

"""Sisotool style collection of plots inspired by MATLAB's sisotool.

The left two plots contain the bode magnitude and phase diagrams.

The top right plot is a clickable root locus plot, clicking on the

root locus will change the gain of the system. The bottom left plot

shows a closed loop time response.

Parameters

----------

sys : LTI object

Linear input/output systems. If sys is SISO, use the same system for

the root locus and step response. If it is desired to see a different

step response than feedback(K*sys,1), such as a disturbance response,

sys can be provided as a two-input, two-output system (e.g. by using

:func:`bdgalg.connect' or :func:`iosys.interconnect`). For two-input,

two-output system, sisotool inserts the negative of the selected gain

K between the first output and first input and uses the second input

and output for computing the step response. To see the disturbance

response, configure your plant to have as its second input the

disturbance input. To view the step response with a feedforward

controller, give your plant two identical inputs, and sum your

feedback controller and your feedforward controller and multiply them

into your plant's second input. It is also possible to accomodate a

system with a gain in the feedback.

initial_gain : float, optional

Initial gain to use for plotting root locus. Defaults to 1

(config.defaults['sisotool.initial_gain']).

xlim_rlocus : tuple or list, optional

Control of x-axis range, normally with tuple

(see :doc:`matplotlib:api/axes_api`).

ylim_rlocus : tuple or list, optional

control of y-axis range

plotstr_rlocus : :func:`matplotlib.pyplot.plot` format string, optional

Plotting style for the root locus plot(color, linestyle, etc).

rlocus_grid : boolean (default = False)

If True plot s- or z-plane grid.

omega : array_like

List of frequencies in rad/sec to be used for bode plot.

dB : boolean

If True, plot result in dB for the bode plot.

Hz : boolean

If True, plot frequency in Hz for the bode plot (omega must be

provided in rad/sec).

deg : boolean

If True, plot phase in degrees for the bode plot (else radians).

omega_limits : array_like of two values

Limits of the to generate frequency vector. If Hz=True the limits

are in Hz otherwise in rad/s. Ignored if omega is provided, and

auto-generated if omitted.

omega_num : int

Number of samples to plot. Defaults to

config.defaults['freqplot.number_of_samples'].

margins_bode : boolean

If True, plot gain and phase margin in the bode plot.

tvect : list or ndarray, optional

List of timesteps to use for closed loop step response.

Examples

--------

>>> G = ct.tf([1000], [1, 25, 100, 0])

>>> ct.sisotool(G) # doctest: +SKIP

"""

from .rlocus import root_locus

# sys as loop transfer function if SISO

if not sys.issiso():

if not (sys.ninputs == 2 and sys.noutputs == 2):

raise ControlMIMONotImplemented(

'sys must be SISO or 2-input, 2-output')

# Setup sisotool figure or superimpose if one is already present

fig = plt.gcf()

if fig.canvas.manager.get_window_title() != 'Sisotool':

plt.close(fig)

fig,axes = plt.subplots(2, 2)

fig.canvas.manager.set_window_title('Sisotool')

# Extract bode plot parameters

bode_plot_params = {

'omega': omega,

'dB': dB,

'Hz': Hz,

'deg': deg,

'omega_limits': omega_limits,

'omega_num' : omega_num,

'sisotool': True,

'fig': fig,

'margins': margins_bode

}

# Check to see if legacy 'PrintGain' keyword was used

if kvect is not None:

warnings.warn("'kvect' keyword is deprecated in sisotool; "

"use 'initial_gain' instead", FutureWarning)

initial_gain = np.atleast_1d(kvect)[0]

initial_gain = config._get_param('sisotool', 'initial_gain',

initial_gain, _sisotool_defaults)

# First time call to setup the bode and step response plots

_SisotoolUpdate(sys, fig, initial_gain, bode_plot_params)

# Setup the root-locus plot window

root_locus(sys, initial_gain=initial_gain, xlim=xlim_rlocus,

ylim=ylim_rlocus, plotstr=plotstr_rlocus, grid=rlocus_grid,

fig=fig, bode_plot_params=bode_plot_params, tvect=tvect, sisotool=True)

def _SisotoolUpdate(sys, fig, K, bode_plot_params, tvect=None):

title_font_size = 10

label_font_size = 8

# Get the subaxes and clear them

ax_mag, ax_rlocus, ax_phase, ax_step = \

fig.axes[0], fig.axes[1], fig.axes[2], fig.axes[3]

# Catch matplotlib 2.1.x and higher userwarnings when clearing a log axis

with warnings.catch_warnings():

warnings.simplefilter("ignore")

ax_step.clear(), ax_mag.clear(), ax_phase.clear()

sys_loop = sys if sys.issiso() else sys[0,0]

# Update the bodeplot

bode_plot_params['syslist'] = sys_loop*K.real

bode_plot(**bode_plot_params)

# Set the titles and labels

ax_mag.set_title('Bode magnitude',fontsize = title_font_size)

ax_mag.set_ylabel(ax_mag.get_ylabel(), fontsize=label_font_size)

ax_mag.tick_params(axis='both', which='major', labelsize=label_font_size)

ax_phase.set_title('Bode phase',fontsize=title_font_size)

ax_phase.set_xlabel(ax_phase.get_xlabel(),fontsize=label_font_size)

ax_phase.set_ylabel(ax_phase.get_ylabel(),fontsize=label_font_size)

ax_phase.get_xaxis().set_label_coords(0.5, -0.15)

ax_phase.tick_params(axis='both', which='major', labelsize=label_font_size)

if not ax_phase.get_shared_x_axes().joined(ax_phase, ax_mag):

ax_phase.sharex(ax_mag)

ax_step.set_title('Step response',fontsize = title_font_size)

ax_step.set_xlabel('Time (seconds)',fontsize=label_font_size)

ax_step.set_ylabel('Output',fontsize=label_font_size)

ax_step.get_xaxis().set_label_coords(0.5, -0.15)

ax_step.get_yaxis().set_label_coords(-0.15, 0.5)

ax_step.tick_params(axis='both', which='major', labelsize=label_font_size)

ax_rlocus.set_title('Root locus',fontsize = title_font_size)

ax_rlocus.set_ylabel('Imag', fontsize=label_font_size)

ax_rlocus.set_xlabel('Real', fontsize=label_font_size)

ax_rlocus.get_xaxis().set_label_coords(0.5, -0.15)

ax_rlocus.get_yaxis().set_label_coords(-0.15, 0.5)

ax_rlocus.tick_params(axis='both', which='major',labelsize=label_font_size)

# Generate the step response and plot it

if sys.issiso():

sys_closed = (K*sys).feedback(1)

else:

sys_closed = append(sys, -K)

connects = [[1, 3],

[3, 1]]

sys_closed = connect(sys_closed, connects, 2, 2)

if tvect is None:

tvect, yout = step_response(sys_closed, T_num=100)

else:

tvect, yout = step_response(sys_closed, tvect)

if isdtime(sys_closed, strict=True):

ax_step.plot(tvect, yout, '.')

else:

ax_step.plot(tvect, yout)

ax_step.axhline(1.,linestyle=':',color='k',zorder=-20)

# Manually adjust the spacing and draw the canvas

fig.subplots_adjust(top=0.9,wspace = 0.3,hspace=0.35)

fig.canvas.draw()

# contributed by Sawyer Fuller, minster@uw.edu 2021.11.02, based on

# an implementation in Matlab by Martin Berg.

def rootlocus_pid_designer(plant, gain='P', sign=+1, input_signal='r',

Kp0=0, Ki0=0, Kd0=0, deltaK=0.001, tau=0.01,

C_ff=0, derivative_in_feedback_path=False,

plot=True):

"""Manual PID controller design based on root locus using Sisotool

Uses `sisotool` to investigate the effect of adding or subtracting an

amount `deltaK` to the proportional, integral, or derivative (PID) gains of

a controller. One of the PID gains, `Kp`, `Ki`, or `Kd`, respectively, can

be modified at a time. `Sisotool` plots the step response, frequency

response, and root locus of the closed-loop system controlling the

dynamical system specified by `plant`. Can be used with either non-

interactive plots (e.g. in a Jupyter Notebook), or interactive plots.

To use non-interactively, choose starting-point PID gains `Kp0`, `Ki0`,

and `Kd0` (you might want to start with all zeros to begin with), select

which gain you would like to vary (e.g. gain=`'P'`, `'I'`, or `'D'`), and

choose a value of `deltaK` (default 0.001) to specify by how much you

would like to change that gain. Repeatedly run `rootlocus_pid_designer`

with different values of `deltaK` until you are satisfied with the

performance for that gain. Then, to tune a different gain, e.g. `'I'`,

make sure to add your chosen `deltaK` to the previous gain you you were

tuning.

Example: to examine the effect of varying `Kp` starting from an intial

value of 10, use the arguments `gain='P', Kp0=10` and try varying values

of `deltaK`. Suppose a `deltaK` of 5 gives satisfactory performance. Then,

to tune the derivative gain, add your selected `deltaK` to `Kp0` in the

next call using the arguments `gain='D', Kp0=15`, to see how adding

different values of `deltaK` to your derivative gain affects performance.

To use with interactive plots, you will need to enable interactive mode

if you are in a Jupyter Notebook, e.g. using `%matplotlib`. See

`Interactive Plots <https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ion.html>`_

for more information. Click on a branch of the root locus plot to try

different values of `deltaK`. Each click updates plots and prints the

corresponding `deltaK`. It may be helpful to zoom in using the magnifying

glass on the plot to get more locations to click. Just make sure to

deactivate magnification mode when you are done by clicking the magnifying

glass. Otherwise you will not be able to be able to choose a gain on the

root locus plot. When you are done, `%matplotlib inline` returns to inline,

non-interactive ploting.

By default, all three PID terms are in the forward path C_f in the diagram

shown below, that is,

C_f = Kp + Ki/s + Kd*s/(tau*s + 1).

------> C_ff ------ d

| | |

r | e V V u y

------->O---> C_f --->O--->O---> plant --->

^- ^- |

| | |

| ----- C_b <-------|

---------------------------------

If `plant` is a discrete-time system, then the proportional, integral, and

derivative terms are given instead by Kp, Ki*dt/2*(z+1)/(z-1), and

Kd/dt*(z-1)/z, respectively.

It is also possible to move the derivative term into the feedback path

`C_b` using `derivative_in_feedback_path=True`. This may be desired to

avoid that the plant is subject to an impulse function when the reference

`r` is a step input. `C_b` is otherwise set to zero.

If `plant` is a 2-input system, the disturbance `d` is fed directly into

its second input rather than being added to `u`.

Parameters

----------

plant : :class:`LTI` (:class:`TransferFunction` or :class:`StateSpace` system)

The dynamical system to be controlled.

gain : string (optional)

Which gain to vary by `deltaK`. Must be one of `'P'`, `'I'`, or `'D'`

(proportional, integral, or derative).

sign : int (optional)

The sign of deltaK gain perturbation.

input : string (optional)

The input used for the step response; must be `'r'` (reference) or

`'d'` (disturbance) (see figure above).

Kp0, Ki0, Kd0 : float (optional)

Initial values for proportional, integral, and derivative gains,

respectively.

deltaK : float (optional)

Perturbation value for gain specified by the `gain` keywoard.

tau : float (optional)

The time constant associated with the pole in the continuous-time

derivative term. This is required to make the derivative transfer

function proper.

C_ff : float or :class:`LTI` system (optional)

Feedforward controller. If :class:`LTI`, must have timebase that is

compatible with plant.

derivative_in_feedback_path : bool (optional)

Whether to place the derivative term in feedback transfer function

`C_b` instead of the forward transfer function `C_f`.

plot : bool (optional)

Whether to create Sisotool interactive plot.

Returns

-------

closedloop : class:`StateSpace` system

The closed-loop system using initial gains.

Notes

-----

When running using iPython or Jupyter, use `%matplotlib` to configure

the session for interactive support.

"""

if plant.ninputs == 1:

plant = ss(plant, inputs='u', outputs='y')

elif plant.ninputs == 2:

plant = ss(plant, inputs=['u', 'd'], outputs='y')

else:

raise ValueError("plant must have one or two inputs")

C_ff = ss(_convert_to_statespace(C_ff), inputs='r', outputs='uff')

dt = common_timebase(plant, C_ff)

# create systems used for interconnections

e_summer = summing_junction(['r', '-y'], 'e')

if plant.ninputs == 2:

u_summer = summing_junction(['ufb', 'uff'], 'u')

else:

u_summer = summing_junction(['ufb', 'uff', 'd'], 'u')

if isctime(plant):

prop = tf(1, 1, inputs='e', outputs='prop_e')

integ = tf(1, [1, 0], inputs='e', outputs='int_e')

deriv = tf([1, 0], [tau, 1], inputs='y', outputs='deriv')

else: # discrete-time

prop = tf(1, 1, dt, inputs='e', outputs='prop_e')

integ = tf([dt/2, dt/2], [1, -1], dt, inputs='e', outputs='int_e')

deriv = tf([1, -1], [dt, 0], dt, inputs='y', outputs='deriv')

if derivative_in_feedback_path:

deriv = -deriv

# create gain blocks

Kpgain = tf(Kp0, 1, inputs='prop_e', outputs='ufb')

Kigain = tf(Ki0, 1, inputs='int_e', outputs='ufb')

Kdgain = tf(Kd0, 1, inputs='deriv', outputs='ufb')

# for the gain that is varied, replace gain block with a special block

# that has an 'input' and an 'output' that creates loop transfer function

if gain in ('P', 'p'):

Kpgain = ss([],[],[],[[0, 1], [-sign, Kp0]],

inputs=['input', 'prop_e'], outputs=['output', 'ufb'])

elif gain in ('I', 'i'):

Kigain = ss([],[],[],[[0, 1], [-sign, Ki0]],

inputs=['input', 'int_e'], outputs=['output', 'ufb'])

elif gain in ('D', 'd'):

Kdgain = ss([],[],[],[[0, 1], [-sign, Kd0]],

inputs=['input', 'deriv'], outputs=['output', 'ufb'])

else:

raise ValueError(gain + ' gain not recognized.')

# the second input and output are used by sisotool to plot step response

loop = interconnect((plant, Kpgain, Kigain, Kdgain, prop, integ, deriv,

C_ff, e_summer, u_summer),

inplist=['input', input_signal],

outlist=['output', 'y'], check_unused=False)

if plot:

sisotool(loop, initial_gain=deltaK)

cl = loop[1, 1] # closed loop transfer function with initial gains

return ss(cl.A, cl.B, cl.C, cl.D, cl.dt)

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