Block Diagrams• A line is a signal• A block is a gain• A circle is a sum• Due to h.f. noise,use proper blocks: num deg ≤ den deg• Try to use just horizontal or vertical lines– Use additional “ ” to helpe.g.Σxs++-+yzGxyy = Gxzyxs-++s = x + z - yΣBlock Diagram Algebra•Series:• Parallel:G1xyG2G1 G2xyG1xyG2++G1+ G2xy• Feedback:•Proof:xGGGy2111+=G1xyG2-+bexy2111 GGG+xGGexeGGeGGxexGGGyeGyyGbbxe2121122111211)1(1,,+=⇒=+−=+=⇒==−=G1G2++2111 GGG−-+11dn22dn212121ddnndn+>> s=tf('s')Transfer function:s>> G1=(s+1)/(s+2)Transfer function:s + 1-----s + 2>> G1=(s+1)/(s+2)Transfer function:s + 1-----s + 2>> G2=5/(s+5)Transfer function:5-----s + 5>> G=G1*G2Transfer function:5 s + 5--------------s^2 + 7 s + 10>> H=G1+G2Transfer function:s^2 + 11 s + 15---------------s^2 + 7 s + 10>> HF=feedback(G1, G2)Transfer function:s^2 + 6 s + 5---------------s^2 + 12 s + 15>> delay1=tf(1,1,'inputdelay',0.05)Transfer function:exp(-0.05*s) * 1>> H2=HF*delay1Transfer function:s^2 + 6 s + 5exp(-0.05*s) * ---------------s^2 + 12 s + 15>> stepresp=H2*1/sTransfer function:s^2 + 6 s + 5exp(-0.05*s) * -------------------s^3 + 12 s^2 + 15 s>> step(H2)0 0.5 1 1.5 2 2.500.10.20.30.40.50.60.70.80.9Step ResponseTime ( sec)AmplitudeQuarter car suspensionkbs+m1s1R(s)y+-s1Series2mskbs+R(s) +-yFeedbackkbsmskbs+++2R(s)ykbsmskbssHTF+++==2)(>> b=sym('b');>> m=sym('m');>> k=sym('k');>> s=sym('s');>> G1=b*s+kG1 =b*s+k>> G2=1/m*1/s*1/sG2 =1/m/s^2>> G=G1*G2G =(b*s+k)/m/s^2>> Gcl=G/(1+G)Gcl =(b*s+k)/m/s^2/(1+(b*s+k)/m/s^2)>> simplify(Gcl)ans =(b*s+k)/(m*s^2+b*s+k)• Move a block across a into all touching lines:– If arrow direction changes, invert– If arrow direction same, no change in block•e.g.pick-up pointsummationG1xyG2G3zalong arrowalong arrowalong arrowalong arrowno changeno changeG1xyG2G3zG1111RsL +Cs1221RsL +Uy+-2R+-VcI2I1111RsL +Cs1221RsL +Uy+-2R+-VcI211RsL+)(111RsLCs +221RsL +Uy+-2R+-11RsL+1)(111++ RsLCs221RsL +11RsL+Uy+-2R222211))((1RsLRsLRsLCs ++++11RsL+y+-2RU11222211))((1RsLRsLRsLRsLCs ++++++y2RU112222112))((..RsLRsLRsLRsLCsRFT++++++=• nodes : variables• branches : gainse.g. y = a · xe.g. y = 3x + 5z – 0.1ySignal Flow Graphxyaxzy53-0.1e.g.G1ryG2-+ uryuG11-G2e.g. (3.6)Ry++-G1G2G3H1-x+z++NRyzG31-H1xG1G2-1N1Note: One node is introduced after each summationMason’s Rule• A forward path: a path from input to output• Forward path gain Mx: total product of gains along the path• A loop gain Li: total product of gains along a loop• Loop i and loop j are non-touching if they do not share any nodes or branches• The determinant ∆:∑∑∑∑−⋅⋅⋅+⋅⋅−⋅+−=∆−loopstnallmkjiloopstnallkjiloopsofpairstouchingnonalljiialliLLLLLLLLLL4..3.....1• ∆x: The determinant of the S.F.G. after removing the k-th forward path• Mason’s Rule:∑∆∆⋅==pathsforwardallxxiMyyFTOI0..e.g. (3.6)Ry++-G1G2G3H1-x+z++NRyzG31-H1xG1G2-1N13.6: Get T.F. from N to y1 f.p.: N yM = 12 loops: L1= -H1G3L2= -G2G3∆ = same∆1: remove N, y, N y∆1= 1321311111GGHGMMNykk++=∆=∆∆=∆∆=∑3.6 (cont.): Get T.F. from R to y2 f.p.: R x z y : M1=G2G3Rz y : M2=G1G32 loops: L1= -G3H1L2= -G2G33213..11 GGHGLLLTNjialli++=⋅+−=∆∑∑HRNyNy+∆=⇒∆=1103.6 (cont.)∆1: remove M1and compute ∆∆1= 1∆2: remove M2and compute ∆∆2= 1321331321 GGHGGGGGMMRyHkkk+++=∆=∆∆==∑∑Figure 3-16 (p. 57)41321242321214123211:loops Five:paths forward TwoGGGGGHGHGGHGGGGMGGGM==4132124232121413212141321242321211R(s)Y(s):gain Total1loops. no 2,path forward removingAfter 1loops. no 1,path forward removingAfter 1:tDeterminanGGGGGHGHGGHGGGGGGGGGGGGHGHGGHGG++++++==∆∴=∆∴+++++=∆s1s1xys1b3b2b1-a1-a2-a3x2x1ex3ΣΣx3eyxx2x1b1b3b2-a3-a2-a111s1s1sExample:s1s1xys1b3b2b1-a1-a2-a3x2x1ex3ΣΣ• Forward paths:sbMsbMsbM13222331===• Loops:33322211saLsaLsaL−=−=−=Determinant:∆1: If M1is taken out, all loops are broken.therefore ∆1 = 1∆2: If M2is taken out, all loops are broken.therefore ∆2 = 1∆3: Similarly, ∆3 = 13322111sasasaLialli+++=−=∆∑33221332213211..sasasasbsbsbMMMMFTii+++++=∆=∆∆=∴∑H4H1H2H3H5H6H7Forward path:M1= H1 H2 H3M2= H4 Loops:L1= H1 H5L2= H2 H6L3= H3 H7L4= H4 H7 H6 H5L1and L3are non-touching∆1: If M1is taken out, all loops are broken.therefore ∆1 = 1∆2: If M2is taken out, the loop in the middle (L2) is still there.therefore ∆2 = 1 – L2= 1 – H2H6Total T.F.:5674736251624432162211)1(HHHHHHHHHHHHHHHHHHHMMMHkk−−−−−+=∆−+=∆∆=∑H4H1H2H3H5H6H7Forward path:M1= H1 H2 H3M2= H4 Loops:L1= H1 H5L2= H2 H6L3= H3 H7L4= H4 H7 H6 H5L1and L3are non-touching∆1: If M1is taken out, all loops are broken.therefore ∆1 = 1∆2: If M2is taken out, the loop in the middle (L2) is still there.therefore ∆2 = 1 – L2= 1 – H2H6Total