Mathematical modeling of control systems. Laith Batarseh. Mathematical modeling of control systems

Transcription

1 Chapter two Laith Batareh Mathematical modeling The dynamic of many ytem, whether they are mechanical, electrical, thermal, economic, biological, and o on, may be decribed in term of differential equation Linear Sytem. A ytem i called linear if the principle of uperpoition applie. The principle of uperpoition tate that the repone produced by the imultaneou application of two different forcing function i the um of the two individual repone

2 TRANSFER FUNCTION Tranfer Function. The tranfer function of a linear, time-invariant, differential equation ytem i defined a the ratio of the Laplace tranform of the output (repone function) to the Laplace tranform of the input (driving function) under the aumption that all initial condition are zero. Conider the linear time-invariant ytem defined by the following differential equation: Comment on Tranfer Function. The tranfer function of a ytem i a mathematical model in that it i an operational method of expreing the differential equation that relate the output variable to the input variable.. The tranfer function i a property of a ytem itelf, independent of the magnitude and nature of the input or driving function. 3. The tranfer function include the unit neceary to relate the input to the output; however, it doe not provide any information concerning the phyical tructure of the ytem. (The tranfer function of many phyically different ytem can be identical.)

3 Comment on Tranfer Function 4. If the tranfer function of a ytem i known, the output or repone can be tudied for variou form of input with a view toward undertanding the nature of the ytem. 5. If the tranfer function of a ytem i unknown, it may be etablihed experimentally by introducing known input and tudying the output of the ytem. Once etablihed, a tranfer function give a full decription of the dynamic characteritic of the ytem, a ditinct from it phyical decription Block diagram Block Diagram:- i a pictorial repreentation of the ytem governing equation. In general, it i a box with two arrow. One i the input and the other i the output. Block Diagram element Block Diagram of a Cloed-Loop Sytem 3

4 AUTOMATIC CONTROL SYSTEMS Ued to convert the form of the output ignal to that of the input ignal AUTOMATIC CONTROL SYSTEMS Open-Loop Tranfer Function and Feed forward Tranfer Function Cloed-Loop Tranfer Function 4

5 Drawing block diagram Example :- draw a ingle block diagram that repreent the following ytem Drawing block diagram 5

6 Drawing block diagram Block diagram reduction Conecutive block X A U A X U 6

7 7 Block diagram reduction Node and comparator U X U U A A X U A A U A A U A 3 3 It i preferred in many cae to reduce the complex block diagram to a ingle block diagram relate the excitation with the repone

8 EXAMPLE Conider the ytem hown in Figure 3(a). Simplify thi diagram. 8

9 EXAMPLE By moving the umming point of the negative feedback loop containing H outide the poitive feedback loop containing H, we obtain Figure 3(b). EXAMPLE Eliminating the poitive feedback loop, we have Figure 3(c). 9

10 EXAMPLE The elimination of the loop containing H/ give Figure 3(d). Finally, eliminating the feedback loop reult in Figure 3(e). A. Simplify the block diagram hown in Figure 0

11 EXAMPLE A.. Firt, move the branch point of the path involving H outide the loop involving H a hown in Figure (a). Then eliminating two loop reult in Figure (b). 3. Combining two block into one give Figure (c). A. Simplify the block diagram hown in Figure 9. Obtain the tranfer function relating C() and R().

12 A. Redrawn for more identification Example.7 Dorf (008) Reduce the following block diagram to ingle block diagram ytem

13 . Move the node between 3 and 4 to after 4 :. Reduce the feedback ytem 3, 4 and H : Example [] 3. Reduce the feedback ytem The ytem become Reduce the final feed back ytem 3

14 EXAMPLE 3--5 olnaraghi (00) Reduce the following block diagram to ingle block diagram ytem. move the branch point at Y to the left of block. combining the block, 3, and 4 4

15 3. eliminating the two feedback loop Obtaining ytem function from block diagram Conider the ame ytem from example 3--5 olnaraghi (00), find the ytem equation in -domain E R Y Y Y Y Y 3 E HY Y 3 Y 3 4 Y 5