Block Diagrams & Signal Flow Graphs Lectures 5 & 6 5-6.pdf · Rotational Mechanical Systems Block...

Rotational Mechanical Systems Block Diagrams Signal Flow Graph Method

Block Diagrams & Signal Flow GraphsLectures 5 & 6

M.R. Azimi, Professor

Department of Electrical and Computer EngineeringColorado State University

Fall 2016

M.R. Azimi Control Systems


Rotational Mechanical Systems

Similar to translational system, but using torques and angular distances ratherthan forces and translational displacements.

Example:

Free Body Diagram for J1:

J1d2θ1(t)

dt2+D1

dθ1(t)

dt+K1(θ1(t)− θ2(t)) +D2

d(θ1(t)− θ2(t))

dt= 0

[J1s2 + (D1 +D2)s+K1]Θ1(s)− (D2s+K1)Θ2(s) = 0 (1)



Rotational Mechanical Systems-Cont.

Free Body Diagram for J2:

J2d2θ2(t)

dt2+K2θ2(t) +K1(θ2(t)− θ1(t)) +D2

d(θ2(t)− θ1(t))

dt= T (t)

[J2s2 +D2s+ (K1 +K2)]Θ2(s)− (D2s+K1)Θ1(s) = T (s) (2)

Gear Trains:

Then

r1

r2=N1

N2=θ1

θ2=T1

T2=⇒ T1θ1 = T2θ2

where N : Number of teeth on the gear; T1(t): Applied torque, T2(t): Deliveredtorque; and θ(t): angular displacement



Gear Trains-Cont.

Example:

J d2θ2(t)dt2 +D dθ2(t)

dt +Kθ2 = T2(t) =⇒ (Js2 +Ds+K)Θ2(s) = T2(s)

Solve for Θ2(s) and then find Θ1(s) using the gear train equation.



Block Diagrams and Transfer Functions

Block Diagram: Used to represent composition and interconnection of a system.

Transfer Function: Used to capture cause-effect relationships in s-domain.

Basic Blocks:

1 Adder/subtractor

2 LTI-Subsystem (Linear Time Invariant)

3 Multiplier



Transfer Function: Different Configurations

(a) Series Combination / Cascaded Systems

Y1(s) = H1(s)U(s)Y2(s) = H2(s)Y1(s) = H1(s)H2(s)U(s)

...YN (s) = HN (s)YN−1(s) = HN (s) . . . H1(s)U(s)

Overall transfer function G(s) = YN (s)U(s) = ΠN

i=1Hi(s)

(b) Parallel combination




Y (s) = Y1(s) + · · ·+ YN (s)Yi(s) = Hi(s)R(s), i = 1, .., N

Y (s) =(∑N

i=1Hi(s))R(s)

Overall transfer function G(s) =∑Ni=1Hi(s)

(c) Closed-loop Systems (SISO)SISO - single input, single output.

Plant: C(s) = G(s)E(s)Feedback: F (s) = H(s)C(s)Error: E(s) = R(s)− F (s) = R(s)−H(s)C(s)C(s) = G(s)[R(s)−H(s)C(s)]

Closed-loop transfer function M(s) = C(s)R(s) = G(s)

1+G(s)H(s)




(d) Open-loop System (MIMO)MIMO - Multi-Input, Multi-Output.

Using superpositionC1(s) = G11(s)R1(s) +G12(s)R2(s) + · · ·+G1M (s)RM (s)C2(s) = G21(s)R1(s) +G22(s)R2(s) + · · ·+G2M (s)RM (s)

...CN (s) = GN1(s)R1(s) +GN2(s)R2(s) + · · ·+GNM (s)RM (s)To find Gij(s) set all inputs except Rj to zero and look at ith output i.e.

Gij(s) = Ci(s)Rj(s) when all other Rk(s) = 0, k 6= j.



Open-Loop MIMO System

Input vector:

R(s) =

R1(s)...

RM (s)

Output vector:

C(s) =

C1(s)...

CN (s)

Matrix transfer function (N ×M) with element Gij(s):

G(s) =[Gij(s)

]Then

C(s) = G(s)R(s)

Notation: An underbar represents a vector and hat indicates a matrix.



Closed-Loop MIMO System

Note: Thick lines represent multiple parallel inputs/outputs.Overall transfer function matrix:

M(s) =(I + G(s)H(s)

)−1

G(s)

where I is an N ×N identity matrix, M(s) is an N ×M matrix, G(s) is anN ×M matrix, and H(s) is an M ×N matrix.



Signal Flow Graph- S. J. Mason 1953

Though the block diagram approach is commonly used for simple systems, itquickly gets complicated when there are multiple loops and subsystems or inMIMO cases. Thus, we need a more streamlined and systematic approach forsuch systems.

Signal Flow Graph (SFG): Pictorial representation of a system of equations, inwhich:variables → nodes of SFGrelationship between variables → branches of SFGcoefficients → gains of branches in SFG.Example: F = Ma



Signal Flow Graph-Cont.

Key Definitions:

1 Input Node: Node with only outgoing branches;

2 Output Node: Node with incoming branches.Note: Any non-input node can be made an output node by adding abranch with gain= 1.

3 Path: Collection of branches linked together in same direction.

4 Forward Path: Path from input node to output node where node is visitedmore than once.

5 Gain of Forward Path: Product of all gains of branches in the forward path.

6 Loop: Path that originates and terminates at the same node. No othernode is visited more than once.

7 Loop Gain: Product of branch gains in a loop.

8 Non-Touching: Two parts of a SFG are non-touching if they do not shareat least one node.




Example:

Input Node: x1

Output Node: All nodes besides x1.Forward Path: Assume x5 as output node, then Path 1 = x1, x2, x3, x4, x5;Path 2 = x1, x2, x4, x5

Gain of Forward Path: Path 1: M1 = abcd; Path 2: M2 = afdLoop: x3 → x4 → x5

Loop Gain: P1 = −ce




Algebra of SFG:

1 Output variable of a node = weighted sum (by the gains of branches) of allincoming branches.x2 = ax1, x3 = bx2 − ex4, x4 = cx3 + fx2, x5 = dx4

2 Parallel branches

Note: all branches must be in same direction (otherwise they form a loop).

3 Series branches

Note: No intermediate incoming or outgoing branches between x1 and x4.




Mason Gain Formula:

Let yin : Input variable (s-domain)yout : Output variable (s-domain)

Then the gain, M , is:

M =youtyin

=

∑Nk=1Mk∆k

∆

Mk: Gain of kth forward path between yout and yin.N : Number of forward paths.∆: 1−

∑Pm1 +

∑Pm2 −

∑Pm3 . . .

Pm1: Loop gain of mth loopPm2: Product of mth combination of pairs of non-touching loop gainsPm3: Product of mth combination of triplets of non-touching loop gains∆k: value of ∆ for part of SFG which is non-touching with kth forward path.




Steps:

1 Arrange SFG (from Block Diagram) and identify input and output nodes.

2 List all forward paths and gains Mks.

3 List all loops and gains, Pm1, Pm2, . . . and form ∆.

4 Determine state of path k with loops and form ∆ks.

5 Apply Mason Gain Formula.




Example 1:

Consider a standard closed-loop system as shown.

R - inputC - outputM1 = GP11 = −GH∆ = 1− P11 = 1 +GH∆1 = 1 since path 1 is touching with the loop.

M(s) = M1∆1

∆ = G(s)1+G(s)H(s)




Example 2:

Find Gij(s), i, j = 1, 2 i.e.

Gij(s) =Ci(s)

Rj(s)

∣∣∣∣Rk(s)=0

, k 6= j




Step 1:Input: R1, output: C1

G11(s) = C1(s)R1(s)

∣∣∣R2(s)=0

Step 2:M1 = G1

Step 3:Loop gains: P11 = −G1G2G3G4

Thus ∆ = 1− P11 = 1 +G1G2G3G4

Step 4:∆1 = 1 since forward path is touching with the loop.

Step 5:G11(s) = M1∆1

∆ = G1

1+G1G2G3G4




Step 1:Input: R2, output: C1

G12(s) = C1(s)R2(s)

∣∣∣R1=0

Step 2:M12 = G1G3G4

Step 3:∆ does not change.

Step 4:∆1 = 1

Step 5:G12(s) = M1∆1

∆ = G1G3G4

1+G1G2G3G4




Input: R1, output: C2

G21(s) = C2(s)R1(s)

∣∣∣R2(s)=0

= −G1G2G3

1+G1G2G3G4

Input: R2, output: C2

G22(s) = C2(s)R2(s)

∣∣∣R1(s)=0

= G3

1+G1G2G3G4

Now write equation for whole system:[C1(s)C2(s)

]=

[G11(s) G12(s)G21(s) G22(s)

] [R1(s)R2(s)

]


Block Diagrams & Signal Flow Graphs Lectures 5 & 6 5-6.pdf · Rotational Mechanical Systems Block...

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