step

Plot the step response of a continuous-time system represented by the following transfer function.

For this example, create a tf model that represents the transfer function. You can similarly plot the step response of other dynamic system model types, such as zero-pole gain (zpk) or state-space (ss) models.

sys = tf(4,[1 2 10]);

Plot the step response.

step(sys)

The step plot automatically includes a dotted horizontal line indicating the steady-state response. In a MATLAB® figure window, you can right-click on the plot to view other step-response characteristics such as peak response and settling time. For more information about these characteristics, see stepinfo (Control System Toolbox).

Plot the step response of a discrete-time system. The system has a sample time of 0.2 s and is represented by the following state-space matrices.

A = [1.6 -0.7;
      1  0];
B = [0.5; 0];
C = [0.1 0.1];
D = 0;

Create the state-space model and plot its step response.

sys = ss(A,B,C,D,0.2);
step(sys)

The step response reflects the discretization of the model, showing the response computed every 0.2 seconds.

Examine the step response of the following transfer function.

sys = zpk(-1,[-0.2+3j,-0.2-3j],1) * tf([1 1],[1 0.05]) 

sys =
 
            (s+1)^2
  ----------------------------
  (s+0.05) (s^2 + 0.4s + 9.04)
 
Continuous-time zero/pole/gain model.

step(sys)

By default, step chooses an end time that shows the steady state that the response is trending toward. This system has fast transients, however, which are obscured on this time scale. To get a closer look at the transient response, limit the step plot to t = 15 s.

step(sys,15)

Alternatively, you can specify the exact times at which you want to examine the step response, provided they are separated by a constant interval. For instance, examine the response from the end of the transient until the system reaches steady state.

t = 20:0.2:120;
step(sys,t)

Even though this plot begins at t = 20, step always applies the step input at t = 0.

Consider the following second-order state-space model:

A = [-0.5572,-0.7814;0.7814,0];
B = [1,-1;0,2];
C = [1.9691,6.4493];
sys = ss(A,B,C,0);

This model has two inputs and one output, so it has two channels: from the first input to the output, and from the second input to the output. Each channel has its own step response.

When you use step, it computes the responses of all channels.

step(sys)

The left plot shows the step response of the first input channel, and the right plot shows the step response of the second input channel. Whenever you use step to plot the responses of a MIMO model, it generates an array of plots representing all the I/O channels of the model. For instance, create a random state-space model with five states, three inputs, and two outputs, and plot its step response.

sys = rss(5,2,3);
step(sys)

In a MATLAB figure window, you can restrict the plot to a subset of channels by right-clicking on the plot and selecting I/O Selector.

step allows you to plot the 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 2 10]);
C1 = pidtune(H,'PI');
C2 = pidtune(H,'PID');

Form the closed-loop systems and plot their step responses.

sys1 = feedback(H*C1,1);
sys2 = feedback(H*C2,1);
step(sys1,sys2)
legend('PI','PID','Location','SouthEast')

By default, step chooses distinct colors for each system that you plot. You can specify colors and line styles using the LineSpec input argument.

 step(sys1,'r--',sys2,'b')
 legend('PI','PID','Location','SouthEast')

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 linestyles. For more plot customization options, use stepplot.

The example Compare Responses of Multiple Systems shows how to plot responses of several individual systems on a single axis. When you have multiple dynamic systems arranged in a model array, step 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.

 sys = tf(zeros(1,1,1,5));

Populate the array.

w0 = 1.5:1:5.5;    % natural frequencies
zeta = 0.5;        % damping constant
for i = 1:length(w0)
   sys(:,:,1,i) = tf(w0(i)^2,[1 2*zeta*w0(i) w0(i)^2]);
end

(For more information about model arrays and how to create them, see Model Arrays (Control System Toolbox).) Plot the step responses of all models in the array.

step(sys)

step uses the same linestyle 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.

When you give it an output argument, step returns an array of response data. For a SISO system, the response data is returned as a column vector of length equal to the number of time points at which the response is sampled. You can provide the vector t of time points, or allow step to select time points for you based on system dynamics. For instance, extract the step response of a SISO system at 101 time points between t = 0 and t = 5 s.

sys = tf(4,[1 2 10]);
t = 0:0.05:5;
y = step(sys,t);
size(y)

ans = 1×2

   101     1

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 two-input, one-output system.

A = [-0.5572,-0.7814;0.7814,0];
B = [1,-1;0,2];
C = [1.9691,6.4493];
sys = ss(A,B,C,0);

Extract the step response of this system at 200 time points between t = 0 and t = 20 s.

t = linspace(0,20,200);
y = step(sys,t);
size(y)

ans = 1×3

   200     1     2

y(:,i,j) is a column vector containing the step response from the jth input to the ith output at the times t. For instance, extract the step response from the second input to the output.

y12 = y(:,1,2);
plot(t,y12)

Create a feedback loop with delay and plot its step response.

s = tf('s');
G = exp(-s) * (0.8*s^2+s+2)/(s^2+s);
sys = feedback(ss(G),1);
step(sys)

The system step response displayed is chaotic. The step response of systems with internal delays may exhibit odd behavior, such as recurring jumps. Such behavior is a feature of the system and not software anomalies.

By default, step applies an input signal that changes from 0 to 1 at t = 0. To customize the amplitude and bias, use RespConfig. For instance, compute the response of a SISO state-space model to a signal that changes from 1 to –1 to at t = 0.

A = [1.6 -0.7;
      1  0];
B = [0.5; 0];
C = [0.1 0.1];
D = 0;
sys = ss(A,B,C,D,0.2);

opt = RespConfig;
opt.Bias = 1;
opt.Amplitude = -2;

step(sys,opt)

For responses to arbitrary input signals, use lsim (Control System Toolbox).

Compare the step response of a parametric identified model to a non-parametric (empirical) model. Also view their 3 $$\sigma$$ confidence regions.

Load the data.

load iddata1 z1

Estimate a parametric model.

sys1 = ssest(z1,4);

Estimate a non-parametric model.

sys2 = impulseest(z1);

Plot the step responses for comparison.

t = (0:0.1:10)';
[y1, ~, ~, ysd1] = step(sys1,t);
[y2, ~, ~, ysd2] = step(sys2,t);
plot(t, y1, 'b', t, y1+3*ysd1, 'b:', t, y1-3*ysd1, 'b:')
hold on
plot(t, y2, 'g', t, y2+3*ysd2, 'g:', t, y2-3*ysd2, 'g:')

Compute the step response of an identified time-series model.

A time-series model, also called a signal model, is one without measured input signals. The step plot of this model uses its (unmeasured) noise channel as the input channel to which the step signal is applied.

Load the data.

load iddata9;

Estimate a time-series model.

sys = ar(z9, 4);

sys is a model of the form A y(t) = e(t) , where e(t) represents the noise channel. For computation of step response, e(t) is treated as an input channel, and is named e@y1.

Plot the step response.

step(sys)

Validate the linearization of a nonlinear ARX model by comparing the small amplitude step responses of the linear and nonlinear models.

Load the data.

load iddata2 z2;

Estimate a nonlinear ARX model.

nlsys = nlarx(z2,[4 3 10],idTreePartition,'custom',...
    {'sin(y1(t-2)*u1(t))+y1(t-2)*u1(t)+u1(t).*u1(t-13)',...
    'y1(t-5)*y1(t-5)*y1(t-1)'},'nlr',[1:5, 7 9]);

Determine an equilibrium operating point for nlsys corresponding to a steady-state input value of 1.

u0 = 1;
[X,~,r] = findop(nlsys, 'steady', 1);
y0 = r.SignalLevels.Output;

Obtain a linear approximation of nlsys at this operating point.

sys = linearize(nlsys,u0,X);

Validate the usefulness of sys by comparing its small-amplitude step response to that of nlsys.

The nonlinear system nlsys is operating at an equilibrium level dictated by (u0,y0). Introduce a step perturbation of size 0.1 about this steady-state and compute the corresponding response.

opt = RespConfig;
opt.InputOffset = u0;
opt.Amplitude = 0.1;
t = (0:0.1:10)';
ynl = step(nlsys, t, opt);

The linear system sys expresses the relationship between the perturbations in input to the corresponding perturbation in output. It is unaware of the nonlinear system's equilibrium values.

Plot the step response of the linear system.

opt = RespConfig;
opt.Amplitude = 0.1;
yl = step(sys, t, opt);

Add the steady-state offset, y0 , to the response of the linear system and plot the responses.

plot(t, ynl, t, yl+y0)
legend('Nonlinear', 'Linear with offset')

Compute and plot the step response of an LPV (lpvss (Control System Toolbox)) model. This example simulates the closed-loop step response of a levitating ball model defined in fcnMaglev.m to a disturbance $\mathit{du}$.

Create the model and discretize it.

hmin = 0.05; 
hmax = 0.25;
h0 = (hmin+hmax)/2;
Ts = 0.01;
Glpv = lpvss("h",@fcnMaglev,0,0,h0);
Glpvd = c2d(Glpv,Ts,"tustin"); 

Sample the LPV model for three height values and tune a PID controller.

hpid = linspace(hmin,hmax,3);
[Ga,Goffset] = sample(Glpvd,[],hpid);
wc = 50;
Ka = pidtune(Ga,"pidf",wc);
Ka.Tf = 0.01;

Create the gain-scheduled PID controller.

Ka.SamplingGrid = struct("h",hpid);
Koffset = struct("y",{Goffset.u});
Clpv = ssInterpolant(ss(Ka),Koffset);

Create the closed-loop model.

CL = feedback(Glpvd*[1,Clpv],1,2,1);
CL.InputName = {'du';'href'};
CL.OutputName = "h";

Get steady-state current for $\mathit{h}$ = ${\mathit{h}}_0$ to compute an appropriate size for the step disturbance at the plant input.

[~,~,~,~,~,~,~,u0] = Glpv.DataFunction(0,h0);

Compute and plot the response to the input disturbance and step change in reference. Set the baseline input signals $\mathit{du}$ = 0 and $\mathit{h}$ = ${\mathit{h}}_0$ to specify the starting steady-state condition.

t = 0:Ts:2;
pFcn = @(k,x,u) x(1);
Config = RespConfig( ...
    Bias=[0;h0], ...
    Amplitude=0.2*[u0;h0]*Ts, ...
    Delay=0.5, ...
    InitialParameter=h0);
step(CL,t,pFcn,Config)
title("Current Step Disturbance and Height Step Change")