Feedback Control-5.ppt

1、To build a typical element by using OA,To build initial element,-,+,-,+,4.7k,4.7k,10k,C,R,k,ur,uc,Ur(s),Uc(s),To build a typical 2nd order system,k,Ur(s),Uc(s),What if?,3.9 Signal Flow Graphs,As we know the system is the relationship of variables.,By using of transfer function the variable relations

2、hip of linear system is linear algebra relation.,Signal flow graph is one of best tools for linear algebra,3.9 Signal Flow Graphs,For a linear dynamic system,where the Yj(s) is variable while Gij(s) is the TF of element or subsystem.,effect,cause,3.9 Signal Flow Graphs,In signal flow graph, junction

3、 points or nodes are used to represent the variables (signal).,Branches with arrow are used to represent elements or subsystem.,3.9 Signal Flow Graphs,The gain of element normally is put above the branch.,An example of signal flow graph,a32,(a) y2=a12y1+a32y3,y3,y4,y5,y2,y1,a12,a23,a32,a43,(b) y2=a1

4、2y1+a32y3, y3=a23y2+a43y4,y3,y4,y5,y2,y1,a12,a32,(a) y2=a12y1+a32y3,y3,y4,y5,y2,y1,a12,An example of signal flow graph,(c) y2=a12y1+a32y3, y3=a23y2+a43y4, y4=a24y2+a34y3+a44y4,An example of signal flow graph,Summary of Basic rules of Signal Flow Graph （1）,A signal flow graph applies only to linear s

5、ystems. The equations based on which a signal flow graph is drawn must be algebraic equations in the form of effects as functions of causes. Nodes are used to represent variables. Normally, the nodes are arranged from left to right, following a succession of causes and effects through the system.,Si

6、gnal travel along branches only in the direction described by the arrows of branches. The branch directing from node yk to yj, represents the dependence of the variable yj, upon yk, but not the reverse. A signal yk traveling along a branch between nodes yk and yj is multiplied by the gain of the bra

7、nch, akj, so that a signal akj yk is delivered at node yj.,Summary of Basic rules of Signal Flow Graph （2）,Definition for Signal Flow Graphs(1),Output node (sink). An output node is a node which has only incoming branches.,Input node (source). An input node is a node that has only outgoing branches.

8、,Definition for Signal Flow Graphs(2),Path. A path is any collection of continuous succession of branches traversed in the same direction. The definition of a path is entirely general since it does not prevent any node to be traversed more than once.,a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,a3

9、4,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,Definition for Signal Flow Graphs(2),Path. A path is any collection of continuous succession of branches traversed in the same direction. The definition of a path is entirely general since it does not prevent any node to be traversed more than once.,Defin

10、ition for Signal Flow Graphs(2),Path. A path is any collection of continuous succession of branches traversed in the same direction. The definition of a path is entirely general since it does not prevent any node to be traversed more than once.,Definition for Signal Flow Graphs(3),Forward path. A fo

11、rward path is a path that starts at an input node and ends at an output node and along which no node is traversed more than once.,a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,Definition for Signal Flow Graphs(3),Forward path. A forward path is a path

12、that starts at an input node and ends at an output node and along which no node is traversed more than once.,a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,Definition for Signal Flow Graphs(3),Forward path. A forward path is a path that starts at an input node and ends at an output node and along wh

13、ich no node is traversed more than once.,Loop. A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once.,Definition for Signal Flow Graphs(4),a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,

14、a12,Loop. A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once.,Definition for Signal Flow Graphs(4),a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,Loop. A loop is a path that originates and terminates on the same node and along

15、 which no other node is encountered more than once.,Definition for Signal Flow Graphs(4),a34,a25,a44,y3,a24,a23,a32,a43,y4,y5,y2,y1,a12,Loop. A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once.,Definition for Signal Flow Graph

16、s(4),Definition for Signal Flow Graphs(5),Loop gain (transmittance). Loop gain is defined as the path gain of a loop.,Path gain (transmittance). The product of the path gains encountered in traversing a path is called the path pain.,Forward path gain (transmittance). Forward path gain is defined as

17、the path gain of a forward path.,Signal Flow Graph Algebra （1）,The value of the variable represented by a node is equal to the sum of all the signals entering the node.,y3=a23y2+a43y4,Signal Flow Graph Algebra （2）,Parallel branches in the same direction connected between two nodes can be replaced by

18、 a single branch with gain equal to the sum of the gains of the parallel branches.,Signal flow graph with parallel paths replaced by a single branch,a+b+c,y,x,Signal Flow Graph Algebra （3）,A series connection of unidirectional branches, can be replaced by a single branch with gain equal to the produ

19、ct of branch gains.,Signal flow graph with cascaded unidirectional paths replaced by a single branch,Signal Flow Graph Algebra （4）,Signal flow graph of a feedback system,Signal flow graph of feedback system,Mason Gain Formula(1),One of greatness of signal flow graph is that the general gain (transmi

20、ttance) or overall transfer function can be found by a formula.,Pn is the gain of each forward path between the input node and output, is the graph determinant which found from,L1 is the gain of each loop, and L1 is the sum of the gains of all loops in the signal flow graph.,L2 is the product of the

21、 gain of two non touching loops.,L2 is the sum of gains in all possible combinations of non touching loops taken two at a time.,L3 is the product of the gain of three non touching loops.,L3 is the sum of gains in all possible combinations of non touching loops taken three at a time.,Mason Gain Formu

22、la(2),n is the cofactor of Pn.,It is the determinant of the remaining sub- graph when forward path that produces Pn is removed.,n is equal to unity when the forward path touches all the loops in the signal flow graph or when the graph contains no loops.,Two parts of a signal flow graph are said to b

23、e non touching if they do not share a common node.,Mason Gain Formula(3),Mason Gain Formula(4),Please draw a signal flow graph for the system and find the overall gain between the R(s) and C(s).,e,w,x,y,C(s),R(s),G4,G3,G2,H2,G1,H1,H3,Mason Gain Formula(5),This system has four loops, whose gains are

24、G2H1, G4H2, G1G2G3G4, andG2G3G4H3, therefore,Mason Gain Formula(6),This system has only one forward path, 1G1G2G3G41.,Only two loops are non touching, therefore,Mason Gain Formula(7),there is no set of three loops that are non touching; therefore,Then the systems determinant can be obtained:,1 =1,Ma

25、son Gain Formula(8),Then the overall gain between R(s) and C(s) is,Mason Gain Formula(9),Another Example,Another Example,The system has two forward paths, 1G11G21G31 and 1G41,It has four loops, whose gains are G4H3, G1H1, G1G2G3H3, andG2G3H2, therefore,Another Example,There are two non touching loop

26、s, G2G3H2 and G4H3,Another Example,There are two forward paths, G1G2G3 and G4,For path G1G2G3,For path G4,then,Assignment,3-47,Once the TF has been determined by any of available methods, we can start to analyze the response of the system it represents.,3.3 Effect of Pole Location,When the system eq

27、uations are simultaneous ordinary differential equations (ODEs), the TF that results will be a ratio of polynomials:,So the values which make A(s)=0 are defined as systems poles,If the input signal is unit impulse,3.3 Effect of Pole Location,So, the output is,Then the inverse Laplace transform of G(s) is the unit impulse response of the system.,It also called the natural response of the system.,3.3 Effect of Pole Location,The poles and zeros to compute the corresponding time response and thus identify time histories