1. Process dynamic and control Tutorial 2 MO, Shengyong 02/10/2007

2. Example 1 Exercise 3.9 in textbook  Given the function X(s), can you say about x(0) and x( ) without solving for x(t)?

3. Tools  Initial Value Theorem  Final Value Theorem

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7. Note!  Before using final value theorem, we must check whether it can be used. x(t) should be bounded for. That means limits of sX(s) exist for all Re(s)>=0, where Re(s) denotes the real part of complex variables s. A simple checking method  If all roots of characteristic equations of sX(s) have Re(s) 0, sX(s) becomes infinite at these points, and limits does not exist.

8. Characteristic equation  If sX(s)=N(s)/D(s), D(s)=0 is called characteristic equation of sX(s). For, the roots of characteristic equation are s1=s2=-4, s3=-5, so final value theorem can be used. For, the roots of characteristic equation are s1=-2, s2=3+2j, s3=3-2j, so x(t) is unbounded.

9. Example 2 Calculate the Laplace transform of this graphical input signals 9 246 7 2 4 0t f(t)

10. 9 2 4 6 70t 2 4 1 2 3 4 5 6 1 2 3 4 5 6 4S(t) -(t-2)S(t-2) (t-4)S(t-4) 2(t-6)S(t-6) -2(t-7)S(t-7) -4S(t-9)

12. Example 3 For such a constant-volume stirred-tank heater system, consider the heat loss to the ambient. Assume that w, T i and T a (the ambient temperature) are constant. The area available for heat loss is A and appropriate overall heat transfer coefficient is v.  Obtain a transfer function that relates liquid outlet temperature T to q which is the heating rate of the element  Does the inclusion of ambient losses increase or decrease the speed of system response to changes in q, consider both cases where T a >T, and T a <T

13. heater q w, Ti w, T Ta