step

Step response plot of dynamic system

Syntax

step(sys)
step(sys,Tfinal)
step(sys,t)
step(sys1,sys2,...,sysN)
step(sys1,sys2,...,sysN,Tfinal)
step(sys1,sys2,...,sysN,t)
y = step(sys,t)
[y,t] = step(sys)
[y,t] = step(sys,Tfinal)
[y,t,x] = step(sys)
[y,t,x,ysd] = step(sys)
[y,...] = step(sys,...,options)

Description

step calculates the step response of a dynamic system. For the state space case, zero initial state is assumed. When it is invoked with no output arguments, this function plots the step response on the screen.

step(sys) plots the step response of an arbitrary dynamic system model sys. This model can be continuous or discrete, and SISO or MIMO. The step response of multi-input systems is the collection of step responses for each input channel. The duration of simulation is determined automatically, based on the system poles and zeros.

step(sys,Tfinal) simulates the step response from t = 0 to the final time t = Tfinal. Express Tfinal in the system time units, specified in the TimeUnit property of sys. For discrete-time systems with unspecified sampling time (Ts = -1), step interprets Tfinal as the number of sampling periods to simulate.

step(sys,t) uses the user-supplied time vector t for simulation. Express t in the system time units, specified in the TimeUnit property of sys. For discrete-time models, t should be of the form Ti:Ts:Tf, where Ts is the sample time. For continuous-time models, t should be of the form Ti:dt:Tf, where dt becomes the sample time of a discrete approximation to the continuous system (see Algorithms). The step command always applies the step input at t=0, regardless of Ti.

To plot the step response of several modelssys1,..., sysN on a single figure, use

step(sys1,sys2,...,sysN)

step(sys1,sys2,...,sysN,Tfinal)

step(sys1,sys2,...,sysN,t)

All of the systems plotted on a single plot must have the same number of inputs and outputs. You can, however, plot a mix of continuous- and discrete-time systems on a single plot. This syntax is useful to compare the step responses of multiple systems.

You can also specify a distinctive color, linestyle, marker, or all three for each system. For example,

step(sys1,'y:',sys2,'g--')

plots the step response of sys1 with a dotted yellow line and the step response of sys2 with a green dashed line.

When invoked with output arguments:

y = step(sys,t)

[y,t] = step(sys)

[y,t] = step(sys,Tfinal)

[y,t,x] = step(sys)

step returns the output response y, the time vector t used for simulation (if not supplied as an input argument), and the state trajectories x (for state-space models only). No plot generates on the screen. For single-input systems, y has as many rows as time samples (length of t), and as many columns as outputs. In the multi-input case, the step responses of each input channel are stacked up along the third dimension of y. The dimensions of y are then

(lengthoft)×(numberofoutputs)×(numberofinputs)

and y(:,:,j) gives the response to a unit step command injected in the jth input channel. Similarly, the dimensions of x are

(lengthoft)×(numberofstates)×(numberofinputs)

For identified models (see idlti and idnlmodlel) [y,t,x,ysd] = step(sys) also computes the standard deviation ysd of the response y (ysd is empty if sys does not contain parameter covariance information).

[y,...] = step(sys,...,options) specifies additional options for computing the step response, such as the step amplitude or input offset. Use stepDataOptions to create the option set options.

Examples

Step Response Plot of Dynamic System

Plot the step response of the following second-order state-space model:

$$\begin{array}{l}
\left[ {\begin{array}{*{20}{c}}
{{{\dot x}_1}}\\
{{{\dot x}_2}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{ - 0.5572}&{ - 0.7814}\\
{0.7814}&0
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_1}}\\
{{x_2}}
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
1&{ - 1}\\
0&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{u_1}}\\
{{u_2}}
\end{array}} \right]\\
y = \left[ {\begin{array}{*{20}{c}}
{1.9691}&{6.4493}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_1}}\\
{{x_2}}
\end{array}} \right]
\end{array}$$

a = [-0.5572,-0.7814;0.7814,0];
b = [1,-1;0,2];
c = [1.9691,6.4493];
sys = ss(a,b,c,0);
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.

Step Response Plot of Feedback Loop with Delay

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);
T = feedback(ss(G),1);
step(T)

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.

Step Responses of Identified Models with Confidence Regions

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

load iddata1 z1
sys1 = ssest(z1,4);

parametric model

sys2 = impulseest(z1); 

non-parametric model

[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:')

Validate Linearization of Identified Nonlinear ARX Model

Validation the linearization of a nonlinear ARX model by comparing their small amplitude step responses.

nlsys = nlarx(z2,[4 3 10],'tree','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)

Now validate the usefulness of sys by comparing its small-amplitude step response to that of nlsys. The nonlinear system nlsys is operating an equilibrium level dictated by (u0, y0). About this steady-state, we introduce a step perturbation of size 0.1. The corresponding response is computed as follows:

opt = stepDataOptions;
opt.InputOffset = u0;
opt.StepAmplitude = 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 nonlinear system's equilibrium values. The step response of the linear system is:

opt = stepDataOptions;
opt.StepAmplitude = 0.1;
yl = step(sys, t, opt);

To compare, add the steady-state offset, y0, to the response of the linear system:

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

Step Response of Identified Time-Series Model

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 iddata9
sys = ar(z9, 4);

ys 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".

step(sys)

More About

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Tips

You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots.

Algorithms

Continuous-time models without internal delays are converted to state space and discretized using zero-order hold on the inputs. The sampling period, dt, is chosen automatically based on the system dynamics, except when a time vector t = 0:dt:Tf is supplied (dt is then used as sampling period). The resulting simulation time steps t are equisampled with spacing dt.

For systems with internal delays, Control System Toolbox™ software uses variable step solvers. As a result, the time steps t are not equisampled.

References

[1] L.F. Shampine and P. Gahinet, "Delay-differential-algebraic equations in control theory," Applied Numerical Mathematics, Vol. 56, Issues 3–4, pp. 574–588.

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