(SEM VI) THEORY EXAMINATION 2023-24 CONTROL SYSTEM

CONTROL SYSTEM – KEC602

Section-wise Important Questions & Ready Answers

SECTION A

(Attempt all – 2 marks each)

(a) Difference between Open Loop and Closed Loop System

An open-loop system operates without feedback, meaning the output has no effect on the control action. These systems are simple, economical, and fast but lack accuracy. A closed-loop system uses feedback to compare output with input and correct errors. Though more complex and costly, closed-loop systems provide higher accuracy and better disturbance rejection.

(b) Why disturbances are introduced in closed-loop systems?

In closed-loop control systems, disturbances are introduced because feedback allows the system to sense deviations caused by external or internal factors. The controller continuously adjusts the input to minimize the effect of disturbances, making closed-loop systems more robust compared to open-loop systems.

(c) Conditions for a System to be Controllable

A system is said to be controllable if it is possible to transfer the system from any initial state to any desired final state in finite time using a suitable control input. Mathematically, the controllability matrix must be of full rank.

(d) Advantages of State-Space Model over Transfer Function

The state-space model can represent multiple-input multiple-output (MIMO) systems and includes internal state variables. It is suitable for time-varying and nonlinear systems, whereas transfer functions are limited to linear time-invariant systems and do not provide state information.

(e) Advantage of Calculating Overshoot in Control System

Overshoot indicates how much the system exceeds the desired output. Calculating overshoot helps in assessing system stability and transient performance. It is essential in designing controllers that avoid excessive oscillations and mechanical stress.

(f) Difference between Rise Time and Fall Time

Rise time is the time taken by the system output to rise from a lower percentage to a higher percentage of its final value, typically from 10% to 90%. Fall time is the time taken for the output to fall from a higher value to a lower value, usually from 90% to 10%.

(g) Relation between Pole Location and Stability

The stability of a system depends on the location of poles in the s-plane. If all poles lie in the left half of the s-plane, the system is stable. Poles on the imaginary axis indicate marginal stability, while poles in the right half make the system unstable.

(h) Measurement of Departure Angle

The angle of departure is measured at complex poles in root locus analysis. It is calculated by subtracting the angles of all other poles and zeros from 180°, ensuring that the total phase condition of the root locus is satisfied.

(i) Significance of Gain Margin and Phase Margin

Gain margin indicates how much gain can be increased before the system becomes unstable, while phase margin shows how much additional phase lag can be tolerated. Both margins measure relative stability 
and robustness of control systems.

(j) Significance of Polar Coordinates

Polar coordinates are used in frequency response analysis to represent magnitude and phase of a system. They are especially useful in Nyquist and polar plots, helping in stability and performance assessment.

SECTION B

(Attempt any three – 10 marks each)

1. Transfer Function from Block Diagram

The transfer function is obtained by simplifying the block diagram using block diagram reduction rules such as series, parallel, and feedback combination. The final expression relates output C(s) to input R(s), representing overall system dynamics.

2. State Space Model of the Given Mechanical System

The system consists of two rotating inertias connected through springs and dampers. The state variables are chosen as angular positions and velocities. Using Newton’s law, differential equations are formed and written in matrix form as state equations, with torque as input and angular displacement as output.

3. Reduction of Peak Overshoot (Numerical)

Peak overshoot depends on damping ratio. Reducing overshoot from 75% to 25% requires increasing damping, which is achieved by reducing system gain. The required reduction factor is obtained using the overshoot-damping ratio relation.

4. Stability Using Routh-Hurwitz Criterion

The Routh array is constructed from the characteristic equation. Stability is determined by checking sign changes in the first column. If no sign change occurs, the system is stable; otherwise, it is unstable or conditionally stable.

5. Gain Margin, Phase Margin & Their Practical Use

Gain crossover frequency occurs when magnitude is unity, while phase crossover occurs when phase is −180°. These parameters help determine how close the system is to instability and guide controller design.

SECTION C

Q3(a) Signal Flow Graph & Mason’s Gain Formula

The signal flow graph is drawn by converting equations into nodes and branches. Mason’s gain formula is then applied to find the overall transfer function by considering forward paths, loops, and non-touching loops.

Q3(b) Block Diagram Reduction to Canonical Form

The diagram is reduced step by step by eliminating inner loops first. The final simplified block gives the overall transfer function C(s)/R(s).

Q4(a) Controllability and Observability (State Model)

The controllability matrix is formed using system matrices. If it has full rank, the system is controllable. Observability is checked similarly using the observability matrix. Both properties ensure effective system control and monitoring.

Q5(a) Correlation between Peak Overshoot and Resonant Peak

The correlation exists when damping ratio is less than 0.707. Under this condition, a resonant peak appears in the frequency response corresponding to peak overshoot in the time domain.

Q5(b) Transfer Function from Step Response

By comparing the given output equation with standard second-order response, parameters such as natural frequency and damping ratio are identified. These values are substituted into the standard transfer function.

Q6(a) Routh Stability (Higher Order Polynomial)

The Routh array is constructed carefully. The number of sign changes in the first column gives the number of poles in the right half plane, determining stability.

Q6(b) Root Locus Plot

Root locus shows how system poles move as gain varies. It starts at open-loop poles and ends at open-loop zeros. Stability is assessed by checking pole locations.

Q7(a) Bode Plot and Stability Comment

Magnitude and phase plots are drawn. Gain and phase margins are identified. If both margins are positive, the system is stable.

Q7(b) Bode Plot with Calculations

From the plotted curves, crossover frequencies are obtained. Gain margin and phase margin are calculated, and system stability is concluded accordingly.