Block Diagram Reduction Rules (Detailed Notes) - Control Systems - Electrical

Block Diagram in Control Systems

Any physical or engineered system can be described by a set of differential equations or by a schematic that shows components and their interconnections. For complex systems these representations become cumbersome. A block diagram provides a compact, standardised graphical representation where blocks denote subsystems (with their transfer functions) and arrows show signal flow.

The transfer function of a linear time-invariant (LTI) system is the ratio of the Laplace transform of the output to the Laplace transform of the input, under zero initial conditions. If R(s) is the input and C(s) the output, and the system transfer function is G(s), then

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

In block diagrams the following standard elements appear:

• Block - a rectangle labelled by a transfer function, representing a subsystem.
• Summing point - crossed circle where two or more signals are algebraically added (with signs + or -).
• Take-off (pick-off) point - a node where the same signal is sent to two or more places without change.
• Directed branches - arrows showing the direction of signal flow and carrying Laplace-domain variables (e.g., R(s), C(s), E(s)).

Summing Point

A summing point is used to combine several signals algebraically. It is shown by a small circle with incoming arrows and sign markers. The output of the summing point is the algebraic sum of the inputs (taking their signs into account).

Take-off Point

A take-off point (or pick-off) duplicates a signal so the same value feeds several blocks. The take-off does not change the signal value; it simply splits it into parallel branches.

How to draw the block diagram (Worked circuit example)

Consider a simple series R-L circuit driven by an input voltage v(t), with output voltage v0(t) taken across a chosen element. We obtain the block diagram by writing equations, taking Laplace transforms (zero initial conditions) and expressing the relations as transfer functions.

Apply Kirchhoff's voltage law around the loop:

v(t) = R i(t) + L (di/dt)

Take Laplace transform with zero initial conditions:

V(s) = R I(s) + L s I(s) = (R + L s) I(s)

Therefore the current transfer function is

I(s) / V(s) = 1 / (R + L s)

If the desired output is the voltage across an element (for example across the inductor), then V0(s) = L s I(s) and the overall input-to-output transfer function is

V0(s) / V(s) = (L s) / (R + L s)

The algebraic relations above are represented using blocks: a block for 1/(R + L s) giving I(s) from V(s), and a block SL that gives V0(s) from I(s). Use a summing point where algebraic sums are needed in the original circuit equations.

From the equations, the intermediate steps and the corresponding small block diagram fragments are shown below.

Taking Laplace transform of the remaining relations gives the block forms shown next.

For the right-hand side of the algebraic relation it is convenient to use a summing point to combine terms before feeding the block.

The output of the summing point is given to a block whose output is I(s):

Then I(s) passes through the SL block to give V0(s):

By combining these fragments we obtain the complete block diagram for the R-L circuit:

Closed-loop and Open-loop Systems

An open-loop system has no feedback path; the output does not influence the input. A closed-loop system has a feedback path from output to input (usually via a sensor and a feedback network) so that the output affects its own control action.

In a closed-loop control system the output is compared with the reference input at an error detector (a summing point). The error signal is processed by the controller and plant to reduce the difference between output and reference.

Positive and Negative Feedback

When the feedback signal is added to the reference at the summing point the feedback is positive. When the feedback signal is subtracted (i.e., used to oppose the input) it is negative. Most control systems use negative feedback because it improves stability, reduces sensitivity to parameter variations and improves steady-state accuracy.

Negative feedback example:

Block Diagram Reduction Rules

Block diagram reduction simplifies a network of blocks and summing/take-off points into an equivalent single transfer function between a chosen input and output. The following standard rules are used repeatedly.

Rule No. 1 - Blocks in Cascade

When two or more blocks are connected in series (cascade), the equivalent block is the product of individual transfer functions. If blocks G1(s) and G2(s) are in cascade, the combined transfer function is G1(s) G2(s).

Rule No. 2 - Blocks in Parallel

When two or more blocks receive the same input and their outputs are algebraically summed at a summing point, the equivalent transfer function is the sum (taking algebraic signs into account) of the individual transfer functions. For blocks G1(s) and G2(s) in parallel the equivalent is G1(s) ± G2(s).

Rule No. 3 - Moving a Take-off Point Ahead of a Block

If a take-off point that originally taps the signal after a block G(s) is moved so that it taps the signal before the block, then the branch that goes to the location originally after the block must include the transfer function G(s) so the branch sees the same signal value as before. In other words, moving a take-off point from after a block to before the block inserts the block's transfer function in the take-off branch.

Rule No. 4 - Moving a Take-off Point After a Block

If a take-off point is moved from before a block G(s) to after the block, then the take-off branch must include the inverse transfer 1/G(s) so that the signal arriving at the branch destination remains unchanged.

Rule No. 5 - Moving a Summing Point Beyond a Block

When a summing point is moved from before a block G(s) to after the block, each signal entering the summing point must be modified by the block G(s) so the algebraic relations of signals remain the same. Algebraically, if E(s) = X(s) ± Y(s) before the block, then after moving the summing point E(s)G(s) = X(s)G(s) ± Y(s)G(s).

Rule No. 6 - Moving a Summing Point Ahead of a Block

Moving a summing point from after a block G(s) to before the block requires that signals entering the summing point be pre-scaled by the inverse of the block where needed so that after passing through G(s) the same relations hold.

Rule No. 7 - Interchanging Two Summing Points

Two summing points in series can often be interchanged provided the algebraic signs and scaling of signals are adjusted appropriately so that the overall algebraic relation between input and output is unchanged. Use algebraic substitution to confirm equality before and after interchange.

Rule No. 8 - Moving a Take-off Point Beyond a Summing Point

If a take-off taps the output of a summing point, and you move the take-off point to one of the summing point input branches, then the contribution of the other inputs must be accounted for by inserting additional blocks or signs so the tapped signal remains identical.

Rule No. 9 - Moving a Take-off Point Ahead of a Summing Point

Moving a take-off from after a summing point to before it requires adjusting the branch by adding appropriate blocks or sign changes so that the signal value on the branch remains unchanged.

Rule No. 10 - Eliminating a Forward Loop

If a forward loop (a loop with just series blocks and no feedback path back into the loop summing point) exists, it can be algebraically combined by multiplying the series blocks. For feedback loops see the closed-loop formula below.

Closed-loop Transfer Function - Standard Result

Consider a forward path transfer function G(s) and a feedback path H(s) that is fed back and subtracted at the summing point (negative feedback). The closed-loop transfer function from reference R(s) to output C(s) is the standard formula:

T(s) = C(s)/R(s) = G(s) / [1 + G(s) H(s)]

For positive feedback the denominator becomes 1 - G(s) H(s), so the closed-loop transfer is G(s) / [1 - G(s) H(s)]. Use this formula after reducing the diagram to a single forward path G(s) and feedback H(s).

Worked Example

Find the transfer function of the following by block reduction technique.

Step 1: There are two internal closed loops. Firstly, we will remove this loop.

Step 2: When the two blocks are in a cascade or series we will use rule no.1.

Step 3: Now we will solve this loop.

Step 4:

Notes on the worked reduction procedure (explanatory):

• Identify inner feedback loops and reduce them first using the closed-loop formula T = G / (1 ± G H), taking sign correctly for positive or negative feedback.
• Replace reduced loops by their equivalent single blocks; then combine series (cascade) blocks by multiplication and parallel branches by addition.
• Move summing points or take-off points only when it simplifies the network, applying the moving rules and inserting reciprocal or direct blocks on the moved branches as required to preserve signal values.
• Continue reduction until a single equivalent block between the chosen R(s) and C(s) remains.

Additional Worked-out Algebraic Example (illustrative)

Consider a system with forward path G1(s) and G2(s) in series and a feedback H(s) which is fed back negatively from the output to the summing point at the input of G1(s). The steps to find overall transfer function are:

Write forward path equivalent:

Gf(s) = G1(s) G2(s)

Apply the closed-loop formula with feedback H(s):

T(s) = Gf(s) / [1 + Gf(s) H(s)] = G1(s) G2(s) / [1 + G1(s) G2(s) H(s)]

This shows how cascade and feedback rules combine.

Summary

Block diagrams are a concise graphical method to represent LTI systems using transfer functions. Use the ten reduction rules systematically: reduce inner loops first, combine series and parallel blocks, and move summing/take-off points only with correct compensating factors. The standard closed-loop formula for negative feedback is G(s) / [1 + G(s) H(s)]. Careful labelling and sign bookkeeping make reduction straightforward and reliable for analysis and controller design.