Plot simulated time response of dynamic system to arbitrary inputs; simulated response data
collapse all in page
Syntax
lsim(sys,u,t)
lsim(sys,u,t,x0)
lsim(sys,u,t,x0,p)
lsim(sys,u,t,x0,method)
lsim(sys1,sys2,...,sysN,u,t,___)
lsim(sys1,LineSpec1,...,sysN,LineSpecN,___)
y = lsim(sys,u,t)
y = lsim(sys,u,t,x0)
y = lsim(sys,u,t,x0,p)
y = lsim(sys,u,t,x0,method)
[y,tOut,x] = lsim(___)
[y,tOut,x,pOut] = lsim(sys,u,t,x0,p)
lsim(sys)
Description
Response Plots
example
lsim(sys,u,t)
plots the simulated time response of the dynamic system model sys
to the input history (t
,u
). The vector t
specifies the time samples for the simulation. For single-input systems, the input signal u
is a vector of the same length as t
. For multi-input systems, u
is an array with as many rows as there are time samples (length(t)
) and as many columns as there are inputs to sys
.
example
lsim(sys,u,t,x0)
further specifies a vector x0
of initial state values, when sys
is a state-space model.
example
lsim(sys,u,t,x0,p)
specifies the parameter trajectory p
for LPV models.
lsim(sys,u,t,x0,method)
specifies how lsim
interpolates the input values between samples, when sys
is a continuous-time model.
example
lsim(sys1,sys2,...,sysN,u,t,___)
simulates the responses of several dynamic system models to the same input history and plots these responses on a single figure. All systems must have the same number of inputs and outputs. You can also use the x0 and method input arguments when computing the responses of multiple models.
example
lsim(sys1,LineSpec1,...,sysN,LineSpecN,___)
specifies a color, line style, and marker for each system in the plot. When you need additional plot customization options, use lsimplot instead.
Response Data
example
y = lsim(sys,u,t)
returns the system response y
, sampled at the same times t
as the input. For single-output systems, y
is a vector of the same length as t
. For multi-output systems, y
is an array having as many rows as there are time samples (length(t)
) and as many columns as there are outputs in sys
. This syntax does not generate a plot.
y = lsim(sys,u,t,x0)
further specifies a vector x0
of initial state values, when sys
is a state-space model.
example
y = lsim(sys,u,t,x0,p)
specifies the parameter trajectory p
for LPV models.
y = lsim(sys,u,t,x0,method)
specifies how lsim
interpolates the input values between samples, when sys
is a continuous-time model.
example
[y,tOut,x] = lsim(___)
returns the state trajectories x
, when sys is a state-space model. x
is an array with as many rows as there are time samples and as many columns as there are states in sys
. This syntax also returns the time samples used for the simulation in tOut
.
[y,tOut,x,pOut] = lsim(sys,u,t,x0,p)
also returns the parameter trajectories pOut
, when sys
is an LPV model.
Linear Simulation Tool
lsim(sys)
opens the Linear Simulation Tool. For more information about using this tool for linear analysis, see Working with the Linear Simulation Tool (Control System Toolbox).
Examples
collapse all
Simulated Response to Arbitrary Input Signal
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
Consider the following transfer function.
sys = tf(3,[1 2 3])
sys = 3 ------------- s^2 + 2 s + 3 Continuous-time transfer function.
To compute the response of this system to an arbitrary input signal, provide lsim
with a vector of the times t
at which you want to compute the response and a vector u
containing the corresponding signal values. For instance, plot the system response to a ramping step signal that starts at 0 at time t = 0
, ramps from 0 at t = 1
to 1 at t = 2
, and then holds steady at 1. Define t
and compute the values of u
.
t = 0:0.04:8; % 201 pointsu = max(0,min(t-1,1));
Use lsim
without an output argument to plot the system response to the signal.
lsim(sys,u,t)grid on
The plot shows the applied input (u,t)
in gray and the system response in blue.
Use lsim
with an output argument to obtain the simulated response data.
y = lsim(sys,u,t);size(y)
ans = 1×2 201 1
The vector y
contains the simulated response at the corresponding times in t
.
Response to Periodic Signal
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
Use gensig (Control System Toolbox) to create periodic input signals such as sine waves and square waves for use with lsim
. Simulate the response to a square wave of the following SISO state-space model.
A = [-3 -1.5; 5 0];B = [1; 0];C = [0.5 1.5];D = 0;sys = ss(A,B,C,D);
For this example, create a square wave with a period of 10 s and a duration of 20 s.
[u,t] = gensig("square",10,20);
gensig
returns the vector t
of time steps and the vector u
containing the corresponding values of the input signal. (If you do not specify a sample time for t
, then gensig
generates 64 samples per period.) Use these with lsim
and plot the system response.
lsim(sys,u,t)grid on
The plot shows the applied square wave in gray and the system response in blue. Call lsim
with an output argument to obtain the response values at each point in t
.
[y,~] = lsim(sys,u,t);
Response of Discrete-Time System
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
When you simulate the response of a discrete-time system, the time vector t
must be of the form Ti:dT:Tf
, where dT
is the sample time of the model. Simulate the response of the following discrete-time transfer function to a ramp step input.
sys = tf([0.06 0.05],[1 -1.56 0.67],0.05);
This transfer function has a sample time of 0.05 s. Use the same sample time to generate the time vector t
and a ramped step signal u
.
t = 0:0.05:4; u = max(0,min(t-1,1));
Plot the system response.
lsim(sys,u,t)
To simulate the response of a discrete-time system to a periodic input signal, use the same sample time with gensig
to generate the input. For instance, simulate the system response to a sine wave with period of 1 s and a duration of 4 s.
[u,t] = gensig("sine",1,4,0.05);
Plot the system response.
lsim(sys,u,t)
Plot Response of Multiple Systems to Same Input
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
lsim
allows you to plot the simulated responses of multiple dynamic systems on the same axis. For instance, compare the closed-loop response of a system with a PI controller and a PID controller. Create a transfer function of the system and tune the controllers.
H = tf(4,[1 10 25]);C1 = pidtune(H,'PI');C2 = pidtune(H,'PID');
Form the closed-loop systems.
sys1 = feedback(H*C1,1);sys2 = feedback(H*C2,1);
Plot the responses of both systems to a square wave with a period of 4 s.
[u,t] = gensig("square",4,12);lsim(sys1,sys2,u,t)grid onlegend("PI","PID")
By default, lsim
chooses distinct colors for each system that you plot. You can specify colors and line styles using the LineSpec
input argument.
lsim(sys1,"r--",sys2,"b",u,t) grid on legend("PI","PID")
The first LineSpec
"r--"
specifies a dashed red line for the response with the PI controller. The second LineSpec
"b"
specifies a solid blue line for the response with the PID controller. The legend reflects the specified colors and line styles. For more plot customization options, use lsimplot
.
Plot Simulated Response of MIMO System
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
In a MIMO system, at each time step t
, the input u(t)
is a vector whose length is the number of inputs. To use lsim
, you specify u
as a matrix with dimensions Nt
-by-Nu
, where Nu
is the number of system inputs and Nt
is the length of t
. In other words, each column of u
is the input signal applied to the corresponding system input. For instance, to simulate a system with four inputs for 201 time steps, provide u
as a matrix of four columns and 201 rows, where each row u(i,:)
is the vector of input values at the i
th time step; each column u(:,j)
is the signal applied at the j
th input.
Similarly, the output y(t)
computed by lsim
is a matrix whose columns represent the signal at each system output. When you use lsim
to plot the simulated response, lsim
provides separate axes for each output, representing the system response in each output channel to the input u(t)
applied at all inputs.
Consider the two-input, three-output state-space model with the following state-space matrices.
A = [-1.5 -0.2 1.0; -0.2 -1.7 0.6; 1.0 0.6 -1.4]; B = [ 1.5 0.6; -1.8 1.0; 0 0 ];C = [ 0 -0.5 -0.1; 0.35 -0.1 -0.15 0.65 0 0.6]; D = [ 0.5 0; 0.05 0.75 0 0];sys = ss(A,B,C,D);
Plot the response of sys
to a square wave of period 4 s, applied to the first input sys
and a pulse applied to the second input every 3 s. To do so, create column vectors representing the square wave and the pulsed signal using gensig
. Then stack the columns into an input matrix. To ensure the two signals have the same number of samples, specify the same end time and sample time.
Tf = 10;Ts = 0.1;[uSq,t] = gensig("square",4,Tf,Ts);[uP,~] = gensig("pulse",3,Tf,Ts);u = [uSq uP];lsim(sys,u,t)
Each axis shows the response of one of the three system outputs to the signals u
applied at all inputs. Each plot also shows all input signals in gray.
Plot System Evolution from Initial Condition
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
By default, lsim
simulates the model assuming all states are zero at the start of the simulation. When simulating the response of a state-space model, use the optional x0
input argument to specify nonzero initial state values. Consider the following two-state SISO state-space model.
A = [-1.5 -3; 3 -1];B = [1.3; 0];C = [1.15 2.3];D = 0; sys = ss(A,B,C,D);
Suppose that you want to allow the system to evolve from a known set of initial states with no input for 2 s, and then apply a unit step change. Specify the vector x0
of initial state values, and create the input vector.
x0 = [-0.2 0.3];t = 0:0.05:8;u = zeros(length(t),1);u(t>=2) = 1;lsim(sys,u,t,x0)grid on
The first half of the plot shows the free evolution of the system from the initial state values [-0.2 0.3]
. At t = 2
there is a step change to the input, and the plot shows the system response to this new signal beginning from the state values at that time.
Extract Simulated Response Data
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
When you use lsim
with output arguments, it returns the simulated response data in an array. For a SISO system, the response data is returned as a column vector of the same length as t
. For instance, extract the response of a SISO system to a square wave. Create the square wave using gensig
.
sys = tf([2 5 1],[1 2 3]);[u,t] = gensig("square",4,10,0.05);[y,t] = lsim(sys,u,t);size(y)
ans = 1×2 201 1
The vector y
contains the simulated response at each time step in t
. (lsim
returns the time vector t
as a convenience.)
For a MIMO system, the response data is returned in an array of dimensions N-by-Ny-by-Nu, where Ny and Nu are the number of outputs and inputs of the dynamic system. For instance, consider the following state-space model, representing a three-state system with two inputs and three outputs.
A = [-1.5 -0.2 1.0; -0.2 -1.7 0.6; 1.0 0.6 -1.4]; B = [ 1.5 0.6; -1.8 1.0; 0 0 ];C = [ 0 -0.1 -0.2; 0.7 -0.2 -0.3 -0.65 0 -0.6]; D = [ 0.1 0; 0.1 1.5 0 0];sys = ss(A,B,C,D);
Extract the responses of the three output channels to the square wave applied at both inputs.
uM = [u u];[y,t] = lsim(sys,uM,t);size(y)
ans = 1×2 201 3
y(:,j)
is a column vector containing response at the jth output to the square wave applied to both inputs. That is, y(i,:)
is a vector of three values, the output values at the ith time step.
Because sys
is a state-space model, you can extract the time evolution of the state values in response to the input signal.
[y,t,x] = lsim(sys,uM,t);size(x)
ans = 1×2 201 3
Each row of x
contains the state values [x1,x2,x3]
at the corresponding time in t
. In other words, x(i,:)
is the state vector at the ith time step. Plot the state values.
plot(t,x)
Response of Systems in Model Array
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
The example Plot Response of Multiple Systems to Same Input shows how to plot responses of several individual systems on a single axis. When you have multiple dynamic systems arranged in a model array, lsim
plots all their responses at once.
Create a model array. For this example, use a one-dimensional array of second-order transfer functions having different natural frequencies. First, preallocate memory for the model array. The following command creates a 1-by-5 row of zero-gain SISO transfer functions. The first two dimensions represent the model outputs and inputs. The remaining dimensions are the array dimensions. (For more information about model arrays and how to create them, see Model Arrays (Control System Toolbox).)
sys = tf(zeros(1,1,1,5));
Populate the array.
w0 = 1.5:1:5.5; % natural frequencieszeta = 0.5; % damping constantfor i = 1:length(w0) sys(:,:,1,i) = tf(w0(i)^2,[1 2*zeta*w0(i) w0(i)^2]);end
Plot the responses of all models in the array to a square wave input.
[u,t] = gensig("square",5,15);lsim(sys,u,t)
lsim
uses the same line style for the responses of all entries in the array. One way to distinguish among entries is to use the SamplingGrid
property of dynamic system models to associate each entry in the array with the corresponding w0
value.
sys.SamplingGrid = struct('frequency',w0);
Now, when you plot the responses in a MATLAB figure window, you can click a trace to see which frequency value it corresponds to.
Simulate Response of Identified Model
Open Live Script
Load estimation data to estimate a model.
load dcmotordataz = iddata(y,u,0.1,'Name','DC-motor');
z
is an iddata
object that stores the one-input two-output estimation data with a sample time of 0.1 s.
Estimate a state-space model of order 4 using estimation data z
.
[sys,x0] = n4sid(z,4);
sys
is the estimated model and x0
is the estimated initial states.
Simulate the response of sys
using the same input data as the one used for estimation and the initial states returned by the estimation command.
[y,t,x] = lsim(sys,z.InputData,[],x0);
Here, y
is the system response, t
is the time vector used for simulation, and x
is the state trajectory.
Compare the simulated response y
to the measured response z.OutputData
for both outputs.
subplot(211), plot(t,z.OutputData(:,1),'k',t,y(:,1),'r')legend('Measured','Simulated')subplot(212), plot(t,z.OutputData(:,2),'k',t,y(:,2),'r')legend('Measured','Simulated')
Time Response of Linear-Parameter Varying State-Space Model
This example uses:
- Control System ToolboxControl System Toolbox
Open Live Script
For this example, fcnMaglev.m
defines the matrices and offsets of a magnetic levitation system. The magnetic levitation controls the height of a levitating ball using a coil current that creates a magnetic force on the ball. This example simulates the model in open loop.
Create an LPV model.
lpvSys = lpvss('h',@fcnMaglev)
Continuous-time state-space LPV model with 1 outputs, 1 inputs, 2 states, and 1 parameters.Model Properties
You can set additional properties using the dot notation.
lpvSys.StateName = {'h','hdot'};lpvSys.InputName = 'current';lpvSys.InputName = 'height';
Simulate the response of this model to an arbitrary sinusoidal input current.
h0 = 1;[~,~,~,~,~,~,x0,u0,~] = fcnMaglev([],h0);t = 0:1e-2:1;u = u0*(1+0.1*sin(10*t));y = lsim(lpvSys,u,t,x0,@(t,x,u) x(1));
**iter = 20, residual: 3.32e-01**iter = 7, residual: 4.33e-10
Plot the response.
plot(t,y,t,u/u0)legend('height','current')
The ball is attracted to the magnet when the current first increases (h
decreases). The subsequent decrease in current is not enough to bring it back.
h
= 0 is a singularity for this model, that is, the ball hits the magnet. The LPV model ceases to be valid at this point.
Effect of Sample Time on Simulation
The choice of sample time can drastically affect simulation results. To illustrate why, consider the following second-order model.
Simulate the response of this model to a square wave with period 1 s, using a sample time of 0.1 s.
w2 = 62.83^2;sys = tf(w2,[1 2 w2]);tau = 1;Tf = 5;Ts = 0.1;[u,t] = gensig("square",tau,Tf,Ts);lsim(sys,u,t)
lsim
simulates the model using the specified input signal, but it issues a warning that the input signal is undersampled. lsim
recommends a sample time that generates at least 64 samples per period of the input u
. To see why this recommendation matters, simulate sys
again using a sample time smaller than the recommended maximum.
figureTs2 = 0.01;[u2,t2] = gensig("square",tau,Tf,Ts2);lsim(sys,u2,t2)
This response exhibits strong oscillatory behavior that is hidden in the undersampled version.
Input Arguments
collapse all
sys
— Dynamic system
dynamic system model | model array
Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems whose responses you can simulate include:
Continuous-time or discrete-time numeric LTI models, such as
tf
,zpk
, orss
models.Generalized or uncertain LTI models such as
genss
oruss
models. (Using uncertain models requires Robust Control Toolbox™ software.)For tunable control design blocks, the function evaluates the model at its current value for both plotting and returning response data.
For uncertain control design blocks, the function plots the nominal value and random samples of the model. When you use output arguments, the function returns response data for the nominal model only.
Sparse state-space models such as
sparss
andmechss
models.Identified LTI models, such as
idtf
,idss
, oridproc
models. For identified models, you can also use the sim command, which can compute the standard deviation of the simulated response and state trajectories.sim
can also simulate all types of models with nonzero initial conditions, and can simulate nonlinear identified models.Linear time-varying (ltvss (Control System Toolbox)) and linear parameter-varying (lpvss (Control System Toolbox)) models.
lsim
does not support frequency-response data models such as frd
, genfrd
, or idfrd
models.
If sys
is an array of models, the function plots the responses of all models in the array on the same axes. See Response of Systems in Model Array (Control System Toolbox).
u
— Input signal
vector | array
Input signal for simulation, specified as a vector for single-input systems, and an array for multi-input systems.
For single-input systems,
u
is a vector of the same length as t.For multi-input systems,
u
is an array with as many rows as there are time samples (length(t)
) and as many columns as there are inputs to sys. In other words, each rowu(i,:)
represents the values applied at the inputs ofsys
at timet(i)
. Each columnu(:,j)
is the signal applied to thej
th input ofsys
.
t
— Time samples
vector
Time samples at which to compute the response, specified as a vector of the form T0:dT:Tf
. The lsim
command interprets t
as having the units specified in the TimeUnit
property of the model sys.
For continuous-time sys
, the lsim
command uses the time step dT
to discretize the model. If dT
is too large relative to the system dynamics (undersampling), lsim
issues a warning recommending a faster sampling time. For further discussion of the impact of sampling time on simulation, see Effect of Sample Time on Simulation (Control System Toolbox).
For discrete-time sys
, the time step dT
must equal the sample time of sys
. Alternatively, you can omit t
or set it to []
. In that case, lsim
sets t
to a vector of the same length as u that begins at 0 with a time step equal to sys.Ts
.
x0
— Initial state values
vector of zeros (default) | vector
Initial state values for simulating a state-space model, specified as a vector having one entry for each state in sys. If you omit this argument, then lsim
sets all states to zero at t = 0
.
p
— LPV model parameter trajectory
matrix | function handle
Parameter trajectory of the LPV model, specified as a matrix or a function handle.
For exogenous or explicit trajectories, specify
p
as a matrix with dimensions N-by-Np, where N is the number of time samples and Np is the number of parameters.Thus, the row vector
p(i,:)
contains the parameter values at the ith time step.For endogenous or implicit trajectories, specify
p
as a function handle of the form p = F(t,x,u) in continuous time and p = F(k,x,u) in discrete time that gives parameters as a function of time t or time sample k, state x, and input u.This option is useful when you want to simulate quasi-LPV models.
method
— Discretization method
'zoh
| 'foh'
Discretization method for sampling continuous-time models, specified as one of the following.
'zoh'
— Zero-order hold'foh'
— First-order hold
When sys is a continuous-time model, lsim
computes the time response by discretizing the model using a sample time equal to the time step dT = t(2)-t(1)
of t. If you do not specify a discretization method, then lsim
selects the method automatically based on the smoothness of the signal u. For more information about these two discretization methods, see Continuous-Discrete Conversion Methods (Control System Toolbox).
Output Arguments
collapse all
y
— Simulated response data
array
Simulated response data, returned as an array.
For single-input systems,
y
is a column vector of the same length as t.For multi-output systems,
y
is an array with as many rows as there are time samples (length(t)
) and as many columns as there are outputs in sys. Thus, the jth column ofy
, ory(:,j)
, contains the response at the jth output to u applied at all inputs.
tOut
— Time vector
vector
Time vector used for simulation, returned as a column vector. When you specify an input time vector t of the form 0:dT:Tf
, then tOut = t
. If t
is nearly equisampled, lsim
adjusts the sample times for simulation and returns the result in tOut
. For discrete-time sys, you can omit t
or set it to []
. In that case, lsim
sets t
to a vector of the same length as u that begins at 0 with a time step equal to sys.Ts
, and returns the result in tOut
.
x
— State trajectories
array
State trajectories, returned as an array. When sys is a state-space model, x
contains the evolution of the states of sys
in response to the input. x
is an array with as many rows as there are time samples (length(t))
and as many columns as there are states in sys
.
pOut
— Parameter trajectories
array
Parameter trajectories, returned as an array. When sys is a linear-parameter varying (lpvss
) model, pOut
contains the evolution of the parameters of sys
. pOut
is an array with as many rows as there are time samples (length(t))
and as many columns as there are parameters in sys
.
Tips
When you need additional plot customization options, use lsimplot instead.
Algorithms
For a discrete-time transfer function,
lsim
filters the input based on the recursion associated with this transfer function:
For discrete-time zpk
models, lsim
filters the input through a series of first-order or second-order sections. This approach avoids forming the numerator and denominator polynomials, which can cause numerical instability for higher-order models.
For discrete-time state-space models, lsim
propagates the discrete-time state-space equations,
For continuous-time systems, lsim
first discretizes the system using c2d
, and then propagates the resulting discrete-time state-space equations. Unless you specify otherwise with the method input argument, lsim
uses the first-order-hold discretization method when the input signal is smooth, and zero-order hold when the input signal is discontinuous, such as for pulses or square waves. The sample time for discretization is the spacing dT
between the time samples you supply in t.
Version History
Introduced in R2012a
See Also
Functions
- gensig (Control System Toolbox) | impulse | initial (Control System Toolbox) | step | sim | lsiminfo | lsimplot
Apps
- Linear System Analyzer (Control System Toolbox)
MATLAB Command
You clicked a link that corresponds to this MATLAB command:
Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.
Select a Web Site
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
You can also select a web site from the following list:
Americas
- América Latina (Español)
- Canada (English)
- United States (English)
Europe
- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)
- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- Deutsch
- English
- Français
- United Kingdom (English)
Asia Pacific
- Australia (English)
- India (English)
- New Zealand (English)
- 中国
- 日本 (日本語)
- 한국 (한국어)
Contact your local office