Control System Toolbox 

This example shows how to plot the time and frequency responses of SISO and MIMO linear systems.
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For illustration purposes, create the following thirdorder transfer function:
sys = tf([8 18 32],[1 6 14 24])
sys = 8 s^2 + 18 s + 32  s^3 + 6 s^2 + 14 s + 24 Continuoustime transfer function.
You can plot the step and impulse responses of this system using the step and impulse commands:
subplot(211), step(sys) subplot(212), impulse(sys)
You can also simulate the response to an arbitrary signal, for example, a sine wave, using the lsim command. The input signal appears in gray and the system's response in blue.
clf
t = 0:0.01:4;
u = sin(10*t);
lsim(sys,u,t) % u,t define the input signal
You can use the plotting commands with continuous or discrete TF, SS, or ZPK models. For statespace models, you can also plot the unforced response from some given initial state, for example:
A = [0.8 3.6 2.1;3 1.2 4.8;3 4.3 1.1];
B = [0; 1.1; 0.2];
C = [1.2 0 0.6];
D = 0.6;
G = ss(A,B,C,D);
x0 = [1;0;2]; % initial state
initial(G,x0)
grid
Frequencydomain analysis is key to understanding stability and performance properties of control systems. Bode plots, Nyquist plots, and Nichols chart are three standard ways to plot and analyze the frequency response of a linear system. You can create these plots using the bode, nichols, and nyquist commands. For example:
sys = tf([8 18 32],[1 6 14 24])
sys = 8 s^2 + 18 s + 32  s^3 + 6 s^2 + 14 s + 24 Continuoustime transfer function.
bode(sys), grid
nyquist(sys), grid
nichols(sys), grid
The poles and zeros of a system contain valuable information about its dynamics, stability, and limits of performance. For example, consider the feedback loop in Figure 1 where
Figure 1: SISO Feedback Loop.
For the gain value k = 0.7, you can plot the closedloop poles and zeros using pzmap:
s = tf('s');
G = (2*s+1)/(s^2+3*s+2);
k = 0.7;
T = feedback(G*k,1);
pzmap(T)
grid, axis([2 0 1 1])
The closedloop poles (marked by blue x's) lie in the left halfplane so the feedback loop is stable for this choice of gain k. You can read the damping ratio of the closedloop poles from this chart (see labels on the radial lines). Here the damping ratio is about 0.7, suggesting a welldamped closedloop response as confirmed by:
clf step(T)
To further understand how the loop gain k affects closedloop stability, you can plot the locus of the closedloop poles as a function of k:
rlocus(G), grid
Clicking where the locus intersects the y axis reveals that the closedloop poles become unstable for k = 1.51. So the loop gain should remain smaller than 1.5 for closedloop stability.
Rightclicking on response plots gives access to a variety of options and annotations. In particular, the Characteristics menu lets you display standard metrics such as rise time and settling time for step responses, or peak gain and stability margins for frequency response plots.
Using the example from the previous section, plot the closedloop step response:
step(T)
Now, rightclick on the plot to display the Peak Response and Settling Time Characteristics, and click on the blue dots to read the corresponding overshoot and settling time values:
All commands mentioned so far fully support multiinput multioutput (MIMO) systems. In the MIMO case, these commands produce arrays of plots. For example, the step response of the twoinput, twooutput system
sys = rss(3,2,2); sys.a = [0.5 0.3 0.2 ; 0 1.3 1.7; 0.4 1.7 1.3];
is a 2by2 array of plots where each column shows the step response of a particular input channel:
step(sys)
If desired, you can group all four responses on a single plot by rightclicking on the plot and selecting the I/O Grouping > All submenu. The resulting plot is shown below.
The following additional plots are useful for analyzing MIMO systems:
Singular value plot (sigma), which shows the principal gains of the frequency response
Pole/zero map for each I/O pair (iopzplot)
For example, plot the peak gain of sys as a function of frequency:
sigma(sys), grid
You can plot multiple systems at once using any of the response plot commands. You can assign a specific color, marker, or line style to each system for easy comparison. Using the feedback example above, plot the closedloop step response for three values of the loop gain k in three different colors:
k1 = 0.4; T1 = feedback(G*k1,1); k2 = 1; T2 = feedback(G*k2,1); step(T,'b',T1,'r',T2,'g') legend('k = 0.7','k = 0.4','k = 1')