Chapter 4 Interconnection of LTI Systems

Transcription

1 Chapter 4 Interconnection of LTI Sytem 4. INTRODUCTION Block diagram and ignal flow graph are commonly ued to decribe a large feedback control ytem. Each block in the ytem i repreented by a tranfer function, which indicate a proportional relationhip between two Laplace-tranformed ignal known a tranmittance. The tranmittance relate the incoming and outgoing ignal a indicated within each block of the ytem. MATLAB ha numerou function that can be ued to interconnect two or more block to form a larger feedback control ytem. With thee function, you will be able to deign a large feedback control ytem without olving for the cloed-loop tranfer function. 4. THE TRANSFER MATRI The tranfer matrix i a matrix whoe entrie are the tranfer function relating an input to an output, and ha the form G n G G G G G G. (4.) Gm Gm Gmn n The matrix equation realization of a MIMO ytem with m output and n input i Y G n Y G G G Y G G Gn Y G G G m m m mn n (4.) 40

2 where, Y T Y Y Y and ( ) ( ) ( ) ( ) m n are the output and input vector, repectively. If the tranfer matrix i quare and diagonal, then the ubytem in the matrix are aid to be independent from each other, i.e., having n imultaneou tranfer function independent from each other. A diagonal tranfer matrix can be generated in MATLAB uing the function ytappend(y,y,,yn), where y, y,,yn are the ubytem to be appended. Liting 4. how an example of a cript that generate a 3 3tranfer matrix of the ytem model G 0 0 Y + Y 0 G Y3 3 ( ) G33 Liting 4. >> ytf([ 0],[ ]); >> ytf(,[ 3 -]); >> y3tf([ ],[ -5 5]); >> ytappend(y,y,y3) Tranfer function from input to output... #: #: 0 #3: 0 Tranfer function from input to output... #: 0 #: ^ #3: 0 Tranfer function from input 3 to output... #: 0 #: 0 + #3: ^ T 4

3 For a ingle-input-multiple-output ytem, the function tf can alo be ued. The firt tep i to define all numerator and denominator vector for each ubytem. Then ue tf to generate the SIMO ytem defined by your ubytem. If the tranfer matrix contain 3 ubytem y, y, and y3, then the tf function can be ued a follow yt tf({num;num;num3},{den,den,den3}). Alternatively, the SIMO ytem can be generated by defining a column vector of individual tranfer function. yt [y;y;y3] Liting 4. how an example of a cript that generate a ingle-input-three-output ytem of the ytem model Y G G G Liting 4. >> % Uing the ubytem previouly developed. >> yt[y;y;y3] Tranfer function from input to output... #: #: ^ #3: ^ The function tf can alo be ued to generate a MIMO ytem. The firt tep i to define all the ubytem and plug it into the tf function a to make a tranfer matrix. Liting 4.3 how an example of a cript that generate a MIMO ytem of the model G G G. Y Y G G G

4 Liting 4.3 >> y(tf(,[ ])) Tranfer function: >> y(tf(,[ 0-4])) Tranfer function: ^ - 4 >> y3(tf([ 0],[ 5])) Tranfer function: >> y(tf(,[ 0])) Tranfer function: - >> y(tf([ ],[ ])) Tranfer function: >> y3(tf([ 3],[ 3 4])) Tranfer function: ^ >> yt[y y y3; y y y3] Tranfer function from input to output... #: #: - 43

5 Tranfer function from input to output... #: ^ #: Tranfer function from input 3 to output... #: #: ^ SERIES INTERCONNECTION OF SYSTEMS y y yn Fig. 4. Sytem connected in erie. Fig. 4. how an example of ytem connected in erie. If all the ytem are ingleinput-ingle-output, then the total ytem tranfer function can be expreed a T G G G (4.3) n where, T() i the total tranfer function, and G G G,,, n are the individual SISO tranfer function. However, if two MIMO ytem are interconnected a hown in Fig 4., then the multiplication of tranfer function i not anymore applicable. y W W U U y Fig. 4. Two ytem connected in cacade. Y Y 44

6 The MATLAB function yterie(y,y,output,input) i ued to interconnect two ytem in erie of any configuration. y and y are the two ytem to be connected in erie, output i a vector coniting of output of y that will be connected to the input of y defined by the vector input. Liting 4.4 how an example of a cript that interconnect two ytem with the following matrix equation: Sytem : Y G G,,, + 4, Y,, G, 0 Sytem : Y, + 3 G The two ytem are interconnected a hown in Fig. 4.3., 3+,, G, + 6,,, y Y, Y,, y Y, Fig 4.3 An example of two ytem connected in cacade. Liting 4.4 >> y tf(,[ 0 0]) Tranfer function: --- ^ >> y tf(,[ 4]) Tranfer function: >> y tf([ 0],[ 0 3]) 45

7 Tranfer function: ^ + 3 >> yt [y y ; y 0] Tranfer function from input to output... #: --- ^ #: ^ + 3 Tranfer function from input to output... #: #: 0 >> y tf(,[ -3 ]) Tranfer function: ^ >> y tf([ 0],[ 0 6]) Tranfer function: ^ + 6 >> yt [y y] Tranfer function from input to output: ^ Tranfer function from input to output: ^ + 6 >> ytotal erie(yt,yt,,) 46

8 Tranfer function from input to output: ^4 + 6 ^ Tranfer function from input to output: ^3 + 4 ^ Interconnecting the two ytem with the configuration hown in Fig. 4.3 will reult to two tranfer function: T T,, Y, and Y, , Thu, the overall ytem model become Y PARALLEL INTERCONNECTION OF SYSTEMS y y y3 Fig. 4.4 Sytem connected in parallel. 47

9 Fig. 4.4 how an example of ytem connected in parallel. If all the ytem are ingleinput-ingle-output, then the total ytem tranfer function can be expreed a T G G G (4.4) n where, T() i the total tranfer function, and G G G,,, n are the individual SISO tranfer function. However, if two MIMO ytem are interconnected a hown in Fig 4.5, then the addition of tranfer function i not anymore applicable. y Y U( ) W y Y Fig. 4.5 Two ytem connected in parallel. The MATLAB function ytparallel(y,y,in,in,out,out) i ued to interconnect two ytem in parallel of any configuration. y and y are the two ytem to be connected in parallel, in and in are the input to be tied together, and out and out are the output to be ummed together. Liting 4.5 how an example of a cript that interconnect two ytem with the following matrix equation given in the previou ection. The two ytem are interconnected a hown in Fig. 4.6., Y, U ( ) W y, Y, U W U3,, y Y, Fig. 4.6 An example of two ytem connected in parallel. 48

10 Liting 4.5 >> %Ue the previou ubytem generated in the previou ection. >> y tf(,[ 0 0]) Tranfer function: --- ^ >> y tf(,[ 4]) Tranfer function: >> y tf([ 0],[ 0 3]) Tranfer function: ^ + 3 >> yt [y y ; y 0] Tranfer function from input to output... #: --- ^ #: ^ + 3 Tranfer function from input to output... #: #: 0 >> y tf(,[ -3 ]) Tranfer function: ^

11 >> y tf([ 0],[ 0 6]) Tranfer function: ^ + 6 >> yt [y y] Tranfer function from input to output: ^ Tranfer function from input to output: ^ + 6 >> ytotal parallel(yt,yt,,,,) Tranfer function from input to output... #: --- ^ #: ^ + 3 Tranfer function from input to output... #: #: ^ Tranfer function from input 3 to output... #: 0 #: ^ + 6 Interconnecting the two ytem with the configuration hown in Fig. 4.6 will reult to ix tranfer function ince the overall ytem ha three input and two output. The tranfer function are: T W U 50

12 T T T T 3 W 3 4 U + W 0 U W 4 U + 3 W 0 + U T Thu, the overall ytem model become W +. 6 U Y + 4 Y FEEDBACK INTERCONNECTION OF SYSTEMS y y Fig. 4.7 Sytem connected a a feedback pair. Fig. 4.7 how an example of two ytem connected a a feedback pair. If all the ytem are ingle-input-ingle-output, then the total ytem tranfer function can be expreed a T G (4.4) + G G auming negative feedback, where, T() i the total tranfer function, and G and G are the individual SISO tranfer function. However, if two MIMO ytem are interconnected a hown in Fig 4.8, then the feedback formula i not anymore applicable. 5

13 , ( ) Y, U ( ) W y, Y, U W Y, y, Fig. 4.8 An example of two ytem connected a a feedback pair. The MATLAB function ytfeedback(y,y,feedin,feedout,ign) i ued to interconnect two ytem in feedback of any configuration. y and y are the two ytem to be connected in feedback, feedin i a vector containing indice into the input vector of y and pecifie which input are involved in the feedback loop, and feedout pecifie which output of y are ued for feedback. If ign, then the function ue negative feedback, otherewie (ign0), poitive feedback i ued. Liting 4.6 how an example of a cript that interconnect two ytem with the following matrix equation given in the previou ection. The two ytem are interconnected a hown in Fig. 4.8 with the following ytem model: Sytem : Y,,, G G + 4, Y Sytem : Y + 5,, G, 0 Liting 4.6 >> % Uing y in the previou example. >> yt Tranfer function from input to output... #: --- ^ 5

14 #: ^ + 3 Tranfer function from input to output... #: #: 0 >> yttf([ ],[ 5]) Tranfer function: >> feedback(yt,yt,,) Tranfer function from input to output... 8 ^3 + 3 ^ #: ^6 + 9 ^5 + 3 ^4 + 7 ^ ^ #: ^ + 3 Tranfer function from input to output... #: #: 0 The overall ytem reulted to a two-input-two output model Y Y

15 4.5 EERCISES Given the ytem model Sytem Sytem Sytem 3 G G H H + 3 G ( ) + G G G G G H + G H + H3 ( ) 0 H H H H H H H3 H3 W y3 + + Y y + y Y Y 3 y3 Fig. 4.9 A complex feedback control ytem for No.. 54

16 . Generate the three ytem.. Connect the three ytem a hown in Fig. 4.9 and determine all the tranfer function in the overall ytem. 3. Determine the tranfer matrix. 4. What i the difference between a tranfer function and a tranfer matrix? 5. If a MIMO ytem ha 3 input and 4 output, how many combination of tranfer function can be derived? 55