Chapter 3 Block Diagrams and Signal Flow Graphs Automatic Control Systems, 9th Edition Farid Golnaraghi, Simon Fraser University Benjamin C. Kuo, University of Illinois 1
Introduction In this chapter, we discuss graphical techniques for modeling control systems and their underlying mathematics. We also utilize the block diagram reduction techniques and the Mason s gain formula to find the transfer function of the overall control system. Later on in Chapters 4 and 5, we use the material presented in this chapter and Chapter 2 to fully model and study the performance of various control systems. 2
Objectives of this Chapter 1. To study block diagrams, their components, and their underlying mathematics. 2. To obtain transfer function of systems through block diagram manipulation and reduction. 3. To introduce the signal flow graphs. 4. To establish a parallel l between block diagrams and signalflow graphs. 5. To use Mason s s gain formula for finding transfer function of systems. 6. To introduce state diagrams. 7. To demonstrate the MATLAB tools using case studies. 3
3 1 BLOCK DIAGRAMS Block diagrams provide a better understanding of the composition and interconnection of the components of a system. It can be used, together with transfer functions, to describe the cause-and-effect relationships throughout the system. Figure 3 1 A simplified block diagram representation of a heating system. 4
3 1 1 Typical Elements of Block Diagrams in Control Systems The common elements in block diagrams of most control systems include: Comparators Blocks ocsepese representing individual dvdu component poe transfer sefunctions, ucos, including: cud Reference sensor (or input sensor) Output sensor Actuator Controller Plant (the component whose variables are to be controlled) Input or reference signals Output signals Disturbance signal Feedback loops 5 Figure 3 3 Block diagram representation of a general control system.
Figure 3 4 Block diagram elements of typical sensing devices of control systems. (a) Subtraction. (b) Addition. (c) Addition and subtraction. 6
7 Figure 3 5 Time and Laplace domain block diagrams.
EXAMPLE 3 1 1 Figure 3 6 Block diagrams G1(s) and G2(s) connected in series. 8
EXAMPLE 3 1 2 Figure 3 7 Block diagrams G1(s) and G2(s) connected in parallel. 9
Basic block diagram of a feedback control system Figure 3 8 Basic block diagram of a feedback control system. 10
Feedback Control System R(s) (): 기준입력(reference input), 입력(input), p 또는 command Y(s) : 출력(output, controlled variable), 또는 응답(response) B(s) : 궤환 신호(feedback signal) E(s) : 오차신호(error signal) 또는 actuating signal G(s) : 순방향경로 전달함수(forward-path transfer function) H(s) : 궤환 전달함수(feedback transfer function, feedback gain) G(s)H(s) : 루프전달함수(loop transfer function), 개루프전달함수(open-loop transfer function) M(s) = Y(s)/R(s) : 폐루프전달함수(closed-loop transfer function, system transfer function) B(s) = H(s)Y(s) ) E(s) = R(s) B(s) 11 Y(s) () = G(s)E(s) ()() = G(s)R(s) () () G(s)B(s) () () M(s) = Y(s) / R(s) = G(s) / (1 + G(s)H(s))
3 1 2 Relation between Mathematical Equations and Block Diagrams Figure 3 9 Graphical representation of Eq. (3 16) using a comparator. 12
13 Figure 3 12 (a) Factorization of 1/s term in the internal feedback loop of Fig.3 11. (b) Final block diagram representation of Eq.(3 17) in Laplace domain.
14 Figure 3 13 Block diagram of Eq.(3 17) in Laplace domain with V(s) represented as the output.
15 2 n Figure 3 14 (a) Factorization of. (b) Alternative diagram representation of Eq.(3 17) in Laplace domain.
16 Figure 3 15 A block diagram representation of Eq.(3 19) in Laplace domain.
3 1 3 Block Diagram Reduction: Branch point relocation 17 Figure 3 16 (a) Branch point relocation from point P to (b) point Q.
3 1 3 Block Diagram Reduction: Comparator relocation 18 Figure 3 17 (a) Comparator relocation from the right hand side of block G2(s) to (b) the left hand side of block G2(s).
EXAMPLE 3 1 5 Find the input output transfer function of the system Figure 3 18 (a) Original block diagram. (b) Moving the branch point at Y1 to the left of block G2. (c) Combining the blocks G1, G2, and G3. (d) Eliminating the inner feedback loop. 19
20 Figure 3 18 (Continued)
3 1 4 Block Diagram of Multi Input Systems Special Case: Systems with a Disturbance Figure 3 19 Block diagram of a system undergoing disturbance. 21
Figure 3 20 Block diagram of the system in Fig. 3 19 when D(s) = 0. Figure 3 21 Block diagram of the system in Fig. 3 19 when R(s) = 0. 22
Figure 3 22 Block diagram representations of a multivariable system. Figure 3 22 Block diagram representations of a multivariable feedback control system. 23
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3 2 SIGNAL FLOW GRAPHS (SFGs) 25
Signal Flow Graphs(SFG, 신호흐름선도) 신호흐름도는 신호의 입 출력관계를 cause and effect 원리에 따라 대수적으로 나타낸 흐름도 로서 절점(node)과 가지(branch)로 구성되며, 아래그림과 같이 각 node는 변수(variable)를나 타내고 branch는 전달되는 변수의 이득(gain)과 방향을 나타낸다. [x j =a ij x i 를 나타낸 node와 branch] output = gain x input Y j (s) = G kj (s) Y k (s) 즉, j th output = (gain from k to j) x (kth cause) SFG Terms의 정의 입력 노드(Input node, Source) 나가는 방향의 branch만 연결되어 있는 node 예] 위의 그림에서 x 1 출력 노드(Output node, Sink) 들어오는 방향의 branch만 연결되어 있는 node 예] 위의 그림에서 x 4 26
이득(Gain) branch로 연결되어 있는 변수간의 비율 예) x 1 과 x 2 를 연결하는 branch의 이득은 a 21 이며, x 2 = a 21 x 1 +(다른 입력에 의한 항들)의 관계를 나타냄. (주의 : x 2 /x 1 = a 21 이라는 것은 아님) 경로(Path) 지정된 방향으로 연결된 branch의 집합으로 어떤 한 변수에서 출발하여, 지정된 어떤 변수 에 이르는 경로를 이룬다. 단, 경로가 되기 위한 조건으로, 경로를 따라 신호가 전달될 때 어 떤 경우에도 같은 node를 두 번 지나서는 안된다. 예) x 1 에서 x 3 로가는path는 다음과 같이 두 개의 경로가 있다. 전방향 경로(Forward path) 입력 node에서 출력 node에 전방향으로 전달하는 path 예) x 1 x 4 의 forward path는 는 아래와 같이 2개의 경로가 가있다. 27
궤환 경로(Feedback path) 입출력 node간을 역방향으로 되돌아 진행하는 path. Loop, Self loop 경로 중에서 출발 노드와 도착 노드가 동일한 경로를 루프(loop)라고 하고, 그 경로내부에 다른 node가가 없으면 (또는는 한개의 branch로 구성된 loop) self loop 라고 함. 예) Nontouching loops Loop중에서 공통인 node가없는loop 경로 이득(Path gain) 정해진 path를 이루는 각 branch gain의 의 곱. 예) path : 에대한path gain은 a 21 a 42 (주의 : 이 예에서 path gain이 a 21 a 42 라고해서 x 4 /x 1 =a 21 a 42 라는 뜻은 아님) Loop gain 지정된 loop을 형성하는 각 branch gain의곱(loop의 path gain) 28 예) loop: loop gain은 a 23 a 32
3 2 4 SFG Algebra Figure 3 29~31 Signal flow graph. 29
3 2 7 Gain Formula for SFG 30
Gain Formula for SFG (Mason's gain rule) M : The gain between input node y in and output node y out M = y out / y in = M k k /, k = 1,, N 여기서, N : Total number of forward path M k : k번째 forward path의 gain : signal flow graph determinant 또는 characteristic function = 1 L i1 + L j2 L k3 +.. L th mr = r nontouching loops 의 m possible combination의 gain product ( 1 r L ) = 1 (모든 각각의 loop 이득의 합) + (2개의 비접 loop의 가능한 모든 조합의 이득곱의 합) (3개의 비접 loop의 가능한 모든 조합의 이득곱의 합) +.. L = loops의수 k : k th forward path와 nontouching하는 part k = k번째의 전향경로와 접하지 않는 graph의 부분에 대한 의값 k번째 경로의 모든 branch를 제거한 신호흐름도에서 구한 31
32 Figure 3 32 Signal flow graph of the feedback control system shown in Fig. 3 8.
Figure 3 33 33 Signal flow graph for Example 3 2 3. 3. 33
34 Figure 3 33 Signal flow graph for Example 3 2 4.
Ex. 3 2 2 M 1 = G(s) L 11 = G(s)H(s) 1 = 1 = 1 + G(s)H(s) Closed loop transfer function M = Y(s) / R(s) = M 1 1 / = G(s) / (1 + G(s)H(s)) Ex. 3 2 4 y 2 / y 1 = y 4 / y 1 = * 는 chosen output에 관계없이 same Noninput node와 와 output tnode사이의 gain y out / y 2 = (y out / y in ) / (y 2 / y in ) = ( M k k from y in to y out / ) / ( M k k from y in to y 2 / ) = ( M k k from y in to y out ) / Ex. 3 2 5 & 3 2 6 ( M k k from y in to y 2 ) 35
3 2 9 Application of the Gain Formula to Block Diagrams EXAMPLE 3 2 6 36 Figure 3 34 (a) Block diagram of a control system. (b) Equivalent signal flow graph.
37 3 2 10 Simplified Gain Formula