GATE E&C Engineering Control Systems Syllabus

What is a regenerative (positive) feedback and one of its effects?

Regenerative feedback adds the feedback signal to the input ($E = R + B$). It increases gain and bandwidth but tends to drive the system toward instability/oscillation; it is the basis of oscillators.

Why is negative feedback generally preferred over positive feedback in control systems?

Negative feedback reduces error, reduces sensitivity to parameter changes and disturbances, and improves stability and accuracy, whereas positive feedback increases gain but degrades stability and increases sensitivity.

What is a block diagram in control systems?

A block diagram is a pictorial representation of a control system that shows the functional relationships among components using blocks (transfer functions) connected by directed lines (signals) along with summing points and take-off points.

Name the four basic elements of a block diagram.

1) Blocks (each containing a transfer function), 2) signal/arrow lines indicating signal flow direction, 3) summing points (where signals are added/subtracted), and 4) take-off (pickoff) points where a signal branches off.

What does a block represent in a block diagram?

A block represents the transfer function of a component; the output of the block equals the input multiplied by the block's transfer function, i.e. $C(s) = G(s)R(s)$.

What is a summing point in a block diagram?

A summing point is a symbol (circle) where two or more signals are algebraically added or subtracted, with plus/minus signs indicating the sign of each incoming signal; it produces a single output equal to their algebraic sum.

What is a take-off (branch/pickoff) point in a block diagram?

A take-off point is a point from which a signal is tapped and sent to more than one block or summing point without changing the value of the signal; the same signal continues along the main path.

What is the block-diagram reduction rule for two blocks $G_1$ and $G_2$ in cascade (series)?

Cascaded blocks multiply: the equivalent transfer function is $G(s) = G_1(s)\,G_2(s)$ (valid when there is no loading between them).

What is the block-diagram reduction rule for two blocks $G_1$ and $G_2$ in parallel?

Parallel blocks (with a common input and a summing point) add algebraically: $G(s) = G_1(s) \pm G_2(s)$.

State the block-diagram reduction rule for a feedback loop.

A forward block $G$ with feedback block $H$ reduces to $$\frac{G}{1 \pm GH}$$ where the $+$ sign is used for negative feedback and the $-$ sign for positive feedback.

How do you move a summing point from after a block $G$ to before that block?

When moving a summing point ahead of (before) a block $G$, the signal entering at that summing point must be multiplied by $\frac{1}{G}$ to keep the diagram equivalent.

How do you move a summing point from before a block $G$ to after that block?

When moving a summing point beyond (after) a block $G$, the signal entering at that summing point must be multiplied by $G$ to maintain equivalence.

How do you move a take-off point from after a block $G$ to before that block?

When shifting a take-off point ahead of (before) a block $G$, the branched signal must be multiplied by $G$ to preserve its original value.

How do you move a take-off point from before a block $G$ to after that block?

When shifting a take-off point beyond (after) a block $G$, the branched signal must be multiplied by $\frac{1}{G}$ to preserve its original value.

Can two adjacent summing points be interchanged in a block diagram?

Yes. The order of two adjacent (associative) summing points can be interchanged without affecting the output, since algebraic addition is associative and commutative.

What is the standard canonical (single-loop) form of a feedback control block diagram?

It consists of: reference input $R(s)$ entering a summing point that subtracts feedback $B(s)$, producing error $E(s)$; $E(s)$ feeding forward block $G(s)$ to give output $C(s)$; and $C(s)$ feeding feedback block $H(s)$ to produce $B(s)=H(s)C(s)$.

In the canonical feedback form, write the expressions for the error ratio and feedback ratio.

Error ratio: $\dfrac{E(s)}{R(s)} = \dfrac{1}{1+G(s)H(s)}$. Feedback ratio: $\dfrac{B(s)}{R(s)} = \dfrac{G(s)H(s)}{1+G(s)H(s)}$.

What limitation must be remembered when cascading two blocks and multiplying their transfer functions?

Multiplying cascaded transfer functions ($G_1G_2$) is valid only when there is no loading effect between the blocks (the second block does not draw power/current that alters the first block's output).

What is a transfer function?

A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input of a linear time-invariant system, with all initial conditions assumed zero: $G(s)=\dfrac{C(s)}{R(s)}$.

List the main advantages of block diagram representation.

It gives a clear functional picture of the system, shows signal flow and the contribution of each component, allows easy evaluation of the overall transfer function by reduction, and simplifies analysis of complex systems.

State one important limitation of block diagrams.

A block diagram of a given system is not unique (many diagrams can represent the same system), and it shows functional, not physical, construction — it hides the source of energy and internal physical details.

What is the purpose of block diagram reduction?

Block diagram reduction systematically simplifies a multi-loop, multi-block diagram into a single equivalent block, yielding the overall closed-loop transfer function relating output to input.

In a block diagram, what does the arrowhead on a connecting line indicate?

The arrowhead indicates the direction of signal flow; signals travel only in the direction of the arrow, representing a unidirectional transmission of information.

What is a multi-input multi-loop system handled by in block diagram analysis?

By the superposition principle: for a linear system, each input (including disturbances) is considered acting alone (others set to zero), the output for each is found, and the results are summed to get the total output.

For unity negative feedback, express the output $C(s)$ and error $E(s)$ in terms of $R(s)$ and $G(s)$.

$$C(s) = \frac{G(s)}{1+G(s)}R(s), \qquad E(s) = \frac{1}{1+G(s)}R(s)$$

What is the effect of feedback on noise generated within the forward path elements?

Negative feedback can reduce the effect of noise/disturbances introduced in the forward path by the loop-gain factor $1+GH$, improving the signal-to-noise behavior provided the noise enters after a high-gain stage.

Why must the feedback element $H(s)$ be highly stable and accurate in a high-gain feedback system?

Because for large loop gain ($GH \gg 1$), $T = \frac{G}{1+GH} \approx \frac{1}{H}$, so the closed-loop response is determined almost entirely by $H$; any error in $H$ directly affects the output.

What does the term 'actuating signal' refer to in a feedback control system?

The actuating (error) signal is the output of the summing point, $E(s)=R(s)-B(s)$, which actuates the controller/plant to reduce the difference between desired and actual output.

Convert a non-unity feedback system $T=\frac{G}{1+GH}$ to an equivalent unity-feedback form. What is the equivalent forward transfer function?

The equivalent unity-feedback forward transfer function is $$G_{eq}(s) = \frac{G(s)H(s)}{1 + G(s)H(s) - G(s)H(s)}$$ — more simply, define $G' = GH$ giving closed-loop $\frac{G'}{1+G'}$, then divide by $H$: $G_{eq}=\dfrac{G}{1+GH-G}$ so that $\dfrac{G_{eq}}{1+G_{eq}}=\dfrac{G}{1+GH}$.

What is meant by 'loading effect' in cascaded block diagrams, and why does it matter?

Loading effect occurs when a following stage draws current/power from the preceding stage, altering its output. It matters because the simple product rule $G_1G_2$ for cascaded blocks holds only if loading is negligible (e.g., a buffer isolates the stages).