Compute step-response characteristics such as rise time, settling time, and overshoot for a dynamic system model. For this example, use the continuous-time transfer function:

Create the transfer function and examine its step response.

sys = tf([1 5 5],[1 1.65 5 6.5 2]);
step(sys)

The plot shows that the response rises in a few seconds, and then rings down to a steady-state value of about 2.5. Compute the characteristics of this response using stepinfo.

S = stepinfo(sys)

S = struct with fields:
        RiseTime: 3.8456
    SettlingTime: 27.9762
     SettlingMin: 2.0689
     SettlingMax: 2.6873
       Overshoot: 7.4915
      Undershoot: 0
            Peak: 2.6873
        PeakTime: 8.0530

By default, the settling time is the time it takes for $$y(t)-y_{final}$$ to fall below 2% of its peak value, where $$y(t)$$is the system response at time t and $$y_{final}$$ is the steady-state response. The result S.SettlingTime shows that for sys, this condition occurs after about 28 seconds. The default definition of rise time is the time it takes for the response to go from 10% of its steady-state value to 90% of that value. S.RiseTime shows that for sys, this rise occurs in less than 4 seconds. The maximum overshoot is returned in S.Overshoot. For this system, the peak value S.Peak, which occurs at the time S.PeakTime, overshoots the steady-state value by about 7.5% of the steady-state value.

For a MIMO system, stepinfo returns a structure array in which each entry contains the response characteristics of the corresponding I/O channel of the system. For this example, use a two-output, two-input discrete-time system. Compute the step-response characteristics.

A = [0.68 -0.34; 0.34 0.68];
B = [0.18 -0.05; 0.04 0.11];
C = [0 -1.53; -1.12 -1.10];
D = [0 0; 0.06 -0.37];
sys = ss(A,B,C,D,0.2);

S = stepinfo(sys)

S=2×2 struct array with fields:
    RiseTime
    SettlingTime
    SettlingMin
    SettlingMax
    Overshoot
    Undershoot
    Peak
    PeakTime

Access the response characteristics for a particular I/0 channel by indexing into S. For instance, examine the response characteristics for the response from the first input to the second output of sys, corresponding to S(2,1).

S(2,1)

ans = struct with fields:
        RiseTime: 0.4000
    SettlingTime: 2.8000
     SettlingMin: -0.6724
     SettlingMax: -0.5188
       Overshoot: 24.6476
      Undershoot: 11.1224
            Peak: 0.6724
        PeakTime: 1

To access a particular value, use dot notation. For instance, extract the rise time of the (2,1) channel.

rt21 = S(2,1).RiseTime

rt21 = 0.4000

By default, stepinfo defines settling time as the time it takes for the error $\left|\mathit{y}\left(\mathit{t}\right)-{\mathit{y}}_{\mathit{final}}\right|$ between the response $\mathit{y}\left(\mathit{t}\right)$and the steady-state response ${\mathit{y}}_{\mathit{final}}$ to come within 2% of ${\mathit{y}}_{\mathit{final}}$. Also, stepinfo defines the rise time as the time it takes for the response to rise from 10% of ${\mathit{y}}_{\mathit{final}}$ to 90% of ${\mathit{y}}_{\mathit{final}}$. You can change these definitions using SettlingTimeThreshold and RiseTimeThreshold. For this example, use the system given by:

Create the transfer function.

sys = tf([1 5 5],[1 1.65 5 6.5 2]);

Compute the time it takes for the error in the response of sys to to reach 0.5% of the steady-state response. To do so, set SettlingTimeThreshold to 0.5%, or 0.005.

S1 = stepinfo(sys,'SettlingTimeThreshold',0.005);
st1 = S1.SettlingTime

st1 = 46.1325

Compute the time it takes the response of sys to rise from 5% to 95% of the steady-state value. To do so, set RiseTimeThreshold to a vector containing those bounds.

S2 = stepinfo(sys,'RiseTimeThreshold',[0.05 0.95]);
rt2 = S2.RiseTime

rt2 = 4.1690

You can define both settling time and rise time in the same computation.

S3 = stepinfo(sys,'SettlingTimeThreshold',0.005,'RiseTimeThreshold',[0.05 0.95])

S3 = struct with fields:
        RiseTime: 4.1690
    SettlingTime: 46.1325
     SettlingMin: 2.0689
     SettlingMax: 2.6873
       Overshoot: 7.4915
      Undershoot: 0
            Peak: 2.6873
        PeakTime: 8.0530

You can extract step-response characteristics from step-response data even if you do not have a model of your system. For instance, suppose you have measured the response of your system to a step input, and saved the resulting response data in a vector y of response values at the times stored in another vector, t. Load the response data and examine it.

load StepInfoData t y
plot(t,y)

Compute step-response characteristics from this response data using stepinfo. If you do not specify the steady-state response value yfinal, then stepinfo assumes that the last value in the response vector y is the steady-state response.Because there is some noise in the data, the last value in y is likely not the true steady-state response value. When you know what the steady-state value should be, you can provide it to stepinfo. For this example, suppose that the steady-state response is 2.4.

S1 = stepinfo(y,t,2.4)

S1 = struct with fields:
        RiseTime: 1.2713
    SettlingTime: 19.6478
     SettlingMin: 2.0219
     SettlingMax: 3.3302
       Overshoot: 38.7575
      Undershoot: 0
            Peak: 3.3302
        PeakTime: 3.4000

Because of the noise in the data, the default definition of the settling time is too stringent, resulting in an arbitrary value of almost 20 seconds. To allow for the noise, increase the settling-time threshold from the default 2% to 5%.

S2 = stepinfo(y,t,2.4,'SettlingTimeThreshold',0.05)

S2 = struct with fields:
        RiseTime: 1.2713
    SettlingTime: 10.4201
     SettlingMin: 2.0219
     SettlingMax: 3.3302
       Overshoot: 38.7575
      Undershoot: 0
            Peak: 3.3302
        PeakTime: 3.4000