Function reference

The Python Control Systems Library (control) provides common functions for analyzing and designing feedback control systems.

System creation

ss(*args) Create a state space system.
tf(*args) Create a transfer function system.

Frequency domain plotting

bode(syslist[, omega, dB, Hz, deg, Plot]) Bode plot for a system
bode_plot(syslist[, omega, dB, Hz, deg, Plot]) Bode plot for a system
nyquist(syslist[, omega, Plot, color, labelFreq]) Nyquist plot for a system
nyquist_plot(syslist[, omega, Plot, color, ...]) Nyquist plot for a system
gangof4(P, C[, omega]) Plot the “Gang of 4” transfer functions for a system
gangof4_plot(P, C[, omega]) Plot the “Gang of 4” transfer functions for a system
nichols(syslist[, omega, grid]) Nichols plot for a system
nichols_plot(syslist[, omega, grid]) Nichols plot for a system

Time domain simulation

forced_response(sys[, T, U, X0, transpose]) Simulate the output of a linear system.
impulse_response(sys[, T, X0, input, ...]) Impulse response of a linear system
initial_response(sys[, T, X0, input, ...]) Initial condition response of a linear system
step_response(sys[, T, X0, input, output, ...]) Step response of a linear system
phase_plot(odefun[, X, Y, scale, X0, T, ...]) Phase plot for 2D dynamical systems

Convention for Time Series

This is a convention for function arguments and return values that represent time series: sequences of values that change over time. It is used throughout the library, for example in the functions forced_response(), step_response(), impulse_response(), and initial_response().


This convention is different from the convention used in the library scipy.signal. In Scipy’s convention the meaning of rows and columns is interchanged. Thus, all 2D values must be transposed when they are used with functions from scipy.signal.


  • Arguments can be arrays, matrices, or nested lists.
  • Return values are arrays (not matrices).

The time vector is either 1D, or 2D with shape (1, n):

T = [[t1,     t2,     t3,     ..., tn    ]]

Input, state, and output all follow the same convention. Columns are different points in time, rows are different components. When there is only one row, a 1D object is accepted or returned, which adds convenience for SISO systems:

U = [[u1(t1), u1(t2), u1(t3), ..., u1(tn)]
     [u2(t1), u2(t2), u2(t3), ..., u2(tn)]
     [ui(t1), ui(t2), ui(t3), ..., ui(tn)]]

Same for X, Y

So, U[:,2] is the system’s input at the third point in time; and U[1] or U[1,:] is the sequence of values for the system’s second input.

The initial conditions are either 1D, or 2D with shape (j, 1):

X0 = [[x1]

As all simulation functions return arrays, plotting is convenient:

t, y = step(sys)
plot(t, y)

The output of a MIMO system can be plotted like this:

t, y, x = lsim(sys, u, t)
plot(t, y[0], label='y_0')
plot(t, y[1], label='y_1')

The convention also works well with the state space form of linear systems. If D is the feedthrough matrix of a linear system, and U is its input (matrix or array), then the feedthrough part of the system’s response, can be computed like this:

ft = D * U

Block diagram algebra

series(sys1, sys2) Return the series connection sys2 * sys1 for –> sys1 –> sys2 –>.
parallel(sys1, sys2) Return the parallel connection sys1 + sys2.
feedback(sys1[, sys2, sign]) Feedback interconnection between two I/O systems.
negate(sys) Return the negative of a system.

Control system analysis

dcgain(*args) Compute the gain of the system in steady state.
evalfr(sys, x) Evaluate the transfer function of an LTI system for a single complex number x.
freqresp(sys, omega) Frequency response of an LTI system at multiple angular frequencies.
margin(*args) Calculate gain and phase margins and associated crossover frequencies
stability_margins(sysdata[, returnall, epsw]) Calculate stability margins and associated crossover frequencies.
phase_crossover_frequencies(sys) Compute frequencies and gains at intersections with real axis in Nyquist plot.
pole(sys) Compute system poles.
zero(sys) Compute system zeros.
pzmap(sys[, Plot, title]) Plot a pole/zero map for a linear system.
root_locus(sys[, kvect, xlim, ylim, ...]) Calculate the root locus by finding the roots of 1+k*TF(s) where TF is self.num(s)/self.den(s) and each k is an element of kvect.

Matrix computations

care(A, B, Q[, R, S, E]) (X,L,G) = care(A,B,Q,R=None) solves the continuous-time algebraic Riccati
dare(A, B, Q, R[, S, E]) (X,L,G) = dare(A,B,Q,R) solves the discrete-time algebraic Riccati
lyap(A, Q[, C, E]) X = lyap(A,Q) solves the continuous-time Lyapunov equation
dlyap(A, Q[, C, E]) dlyap(A,Q) solves the discrete-time Lyapunov equation
ctrb(A, B) Controllabilty matrix
obsv(A, C) Observability matrix
gram(sys, type) Gramian (controllability or observability)

Control system synthesis

acker(A, B, poles) Pole placement using Ackermann method
lqr(*args, **keywords) Linear quadratic regulator design
place(A, B, p) Place closed loop eigenvalues

Model simplification tools

minreal(sys[, tol, verbose]) Eliminates uncontrollable or unobservable states in state-space models or cancelling pole-zero pairs in transfer functions.
balred(sys, orders[, method]) Balanced reduced order model of sys of a given order.
hsvd(sys) Calculate the Hankel singular values.
modred(sys, ELIM[, method]) Model reduction of sys by eliminating the states in ELIM using a given method.
era(YY, m, n, nin, nout, r) Calculate an ERA model of order r based on the impulse-response data YY.
markov(Y, U, M) Calculate the first M Markov parameters [D CB CAB ...] from input U, output Y.

Utility functions and conversions

unwrap(angle[, period]) Unwrap a phase angle to give a continuous curve
db2mag(db) Convert a gain in decibels (dB) to a magnitude
mag2db(mag) Convert a magnitude to decibels (dB)
drss([states, outputs, inputs]) Create a stable discrete random state space object.
isctime(sys[, strict]) Check to see if a system is a continuous-time system
isdtime(sys[, strict]) Check to see if a system is a discrete time system
issys(obj) Return True if an object is a system, otherwise False
pade(T[, n]) Create a linear system that approximates a delay.
sample_system(sysc, Ts[, method, alpha]) Convert a continuous time system to discrete time
canonical_form(xsys[, form]) Convert a system into canonical form
reachable_form(xsys) Convert a system into reachable canonical form
ss2tf(*args) Transform a state space system to a transfer function.
ssdata(sys) Return state space data objects for a system
tf2ss(*args) Transform a transfer function to a state space system.
tfdata(sys) Return transfer function data objects for a system
timebase(sys[, strict]) Return the timebase for an LTI system
timebaseEqual(sys1, sys2) Check to see if two systems have the same timebase