Analysis of Systems with Time Delays
For example, consider the following control loop, where the plant is modeled as first-order plus dead time:
You can model the closed-loop system from
the following commands:
s = tf('s'); P = 5*exp(-3.4*s)/(s+1); C = 0.1 * (1 + 1/(5*s)); T = feedback(P*C,1);
T is a state-space model with an internal
delay. For more information about models with internal delays, see Closing Feedback Loops with Time Delays.
Plot the step response of
For more complicated interconnections, you can name the input
and output signals of each block and use
automatically take care of the wiring. Suppose, for example, that
you want to add feedforward to the control loop of the previous model.
You can derive the corresponding closed-loop model
F = 0.3/(s+4); P.InputName = 'u'; P.OutputName = 'y'; C.InputName = 'e'; C.OutputName = 'uc'; F.InputName = 'r'; F.OutputName = 'uf'; Sum1 = sumblk('e','r','y','+-'); % e = r-y Sum2 = sumblk('u','uf','uc','++'); % u = uf+uc Tff = connect(P,C,F,Sum1,Sum2,'r','y');
and compare its response with the feedback only design.
stepplot(T,Tff) legend('No feedforward','Feedforward')
The state-space representation keeps track of the internal delays in both models.
Considerations to Keep in Mind when Analyzing Systems with Internal Time Delays
The time and frequency responses of delay systems can look odd and suspicious to those only familiar with delay-free LTI analysis. Time responses can behave chaotically, Bode plots can exhibit gain oscillations, etc. These are not software or numerical quirks but real features of such systems. Below are a few illustrations of these phenomena.
s = tf('s'); G = exp(-5*s)/(s+1); T = feedback(G,.5); bodemag(T)
G = 1 + 0.5 * exp(-3*s); bodemag(G)
Jagged step response:
G = exp(-s) * (0.8*s^2+s+2)/(s^2+s); T = feedback(G,1); stepplot(T)
Note the rearrivals (echoes) of the initial step function.
G = 1/(s+1) + exp(-4*s); T = feedback(1,G); stepplot(T,150)
You can use Control System Toolbox™ tools to model and analyze these and other strange-appearing artifacts of internal delays.