Time Delay Inverse Response
Example Situation for Time Delay
Suppose an application of an input to a system is performed a length L away from the system, there will be a time delay.
The above system yields a transport delay as such:
Proof that Time Delay Systems are Infinite Order
Now consider a more general pure time-delay process whose transfer function is:
Note the only affect a has on an input is a delay. In order to find a rational version of this equation, we will rewrite it as follows:
Now we can see that a pure time delay system is an infinite order system. A Taylor Series expansion gives a numerical approximation to g(s).
Dynamic Behavior of Time Delay Systems
From Ogunnaike and Ray, consider Fig. 8.4 (from section 8.3)
In this situation we consider a to be
The relationship between the pipe inlet concentration co and the pipe outlet concentration is found to be as follows
We can write the deviation variables as
We now take the Laplace transform and obtain,
A block diagram representation is now shown as such. Note that the dynamic behavior of this system is actually that of two processes. The first is the long pipe, and the second is the reactor.
Because we know that the reactor is a first order system, we designate the deviation of the reactor outlet concentration cA(t) from the steady state concentration cAss as y(t). The transfer function relationship between the input, w(s), and y(s) is given by:
Now we consider the overall transfer function between the SYSTEM input, u(s), and output y(s).
This equation is the transfer function of a first order plus time delay system. When we take the inverse Laplace transform, we must consider e-as. In the text, table C1 (in Appendix C) is consulted to find a transfer function of the form,
The inverse Laplace Transform of the above transfer function is given as such:
In this case b=a can be seen to be the time lag. The inverse Laplace Transform in this example is then given as follows:
The catch to this is that for any value of ta the function has to be equal to zero. This makes sense, because we know nothing is happening before the response.
Consider the following simple graph. The value of a from the graph is easy to see.
The horizontal part of this graph represents a and is the time delay.
Problem
Now try a time delay problem.
Consider the dynamic behavior of a process, represented as a first order system. The steady state gain is 0.5, the time constant is 10 minutes, and the time delay is 4 minutes. What is the transfer function representation of this system? What is the unit step response of this system.
Problem Solution
From the problem statement we know that the transfer function is as follows:
After taking the inverse Laplace Transform from Appendix C, we obtain the unit step response for this system.
INVERSE RESPONSE TO BE POSTED SOON!!
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