Q: Q.4. A unity feedback system has open loop transfer function G(s) = 200/[s(s+…

For G(s)=200/[s(s+1)(s+10)], converting to Bode (time-constant) form gives G(jω)=20/[jω(1+jω)(1+jω/10)]; the Bode plot shows corner frequencies at ω=1 and ω=10 rad/s; calculation gives gain crossover frequency ≈4.23 rad/s, phase crossover frequency ≈3.16 rad/s, gain margin ≈-5.19 dB, and phase margin ≈-9.66°, both negative, indicating the closed-loop system is unstable.

Q: Q.4 OR (a) Draw the Bode plot (magnitude only) for the following transfer fun…

For G(s)=10/[s(1+5s)(1+0.25s)], the Bode magnitude plot has corner frequencies at ω=0.2 rad/s and ω=4 rad/s, with the gain crossover frequency calculated as approximately 1.368 rad/s; phase crossover frequency is the frequency at which the open-loop phase equals -180°, gain crossover frequency is where the open-loop magnitude equals 0dB (unity), phase margin is 180° plus the phase at gain crossover, and gain margin is the reciprocal (in dB, the negative) of the magnitude at phase crossover.

Q: Q.5 (a) Obtain the time response of following system: [ẋ1;ẋ2] = [[1,0],[1,1]]…

For the given state equation ẋ1=x1+u, ẋ2=x1+x2+u with x1(0)=1, x2(0)=0 and unit-step u(t), solving each first-order equation sequentially gives x1(t)=2e^t-1 and x2(t)=2t·e^t; for the given SISO state-space system with A=[[0,1,0],[0,-3,-3],[1,0,-1]], B=[0,3,0]ᵀ, C=[1,0,0], the transfer function is derived as G(s)=(3s+3)/(s³+4s²+3s+3).

Q: Q.5 OR Write short note on: (a) Lead-lag compensation network [8] (b) PID con…

A lead-lag compensation network combines a lead network (providing phase lead/advance to improve transient response and stability margin, typically implemented as a zero closer to the origin than its associated pole) with a lag network (providing improved steady-state accuracy via increased low-frequency gain, typically implemented as a pole closer to the origin than its associated zero), together achieving both improved transient and steady-state performance simultaneously; a PID controller combines Proportional, Integral, and Derivative control actions, generating a control signal u(t)=Kp·e(t)+Ki∫e(t)dt+Kd·de(t)/dt, providing fast response (P), zero steady-state error (I), and improved damping/stability (D).

Question 1

Q.1. (a) What is control system? Explain difference between open loop & close…

Accepted Answer

A control system is an arrangement of components that manages, commands, directs or regulates the behavior of other systems to achieve a desired output; open-loop systems have no feedback (output does not influence the control action), while closed-loop systems continuously feed the output back for comparison with the reference, correcting for disturbances and modeling errors; for the given mechanical system with dashpots f1, f2 and spring k, the transfer function is Y(s)/X(s) = f1·s / [(f1+f2)s + k].

Question 2

Q.1 OR (a) Obtain the signal flow graph representation for a system whose blo…

Accepted Answer

A signal flow graph represents a system of linear algebraic/differential equations using directed branches (with gains) between nodes (representing variables), from which forward paths (unbroken node-to-node paths from input to output) and loops (closed paths returning to the starting node) can be identified and combined using Mason's gain formula; for a multi-loop block diagram, the overall transfer function is found by systematically applying block-diagram reduction rules (combining series/parallel/feedback blocks) or, equivalently, by converting to a signal flow graph and applying Mason's gain formula directly.

Question 3

Q.2 (a) Derive the expression for time response of second order control syste…

Accepted Answer

The time response of a standard second-order system to a unit step input is derived by taking the inverse Laplace transform of C(s) = ωn²/[s(s²+2ζωn·s+ωn²)], yielding c(t) = 1 - [e^(-ζωnt)/√(1-ζ²)]·sin(ωd·t + φ) for the underdamped case; applying this to G(s)=1/(s²+10s+15) with unity feedback gives a closed-loop system with ζ=1.25 (overdamped), so peak time and overshoot do not exist in the classical sense, with rise time (10-90%) ≈ 1.16s and 2% settling time (dominant pole) ≈ 2s.

Question 4

Q.2 OR (a) With the help of Routh Hurwitz criterion, comment upon the stabili…

Accepted Answer

Applying the Routh-Hurwitz criterion to the characteristic equation (correcting the evident duplicate-term typo as s⁶+s⁵-2s⁴-10s³-4s²-4=0) reveals 3 sign changes in the first column of the Routh array, indicating 3 roots in the right-half s-plane, so the system is unstable; stability is the property of a system to produce a bounded output for a bounded input (BIBO stability) and to return to equilibrium after a disturbance, of central importance since an unstable control system is unusable in practice, with absolute stability being a simple stable/unstable classification while relative stability quantifies how far a stable system is from the boundary of instability.

Question 5

Q.3. Sketch the root locus, if the transfer function (open loop) is G(s)H(s) …

Accepted Answer

For G(s)H(s) = K/[s(s+3)(s²+2s+2)], the root locus has 4 branches starting at poles s=0, s=-3, s=-1±j1 and going to infinity along 4 asymptotes at 45°, 135°, 225°, 315° centered at the centroid s=-1.25; applying the Routh-Hurwitz criterion to the characteristic equation shows the maximum gain for closed-loop stability is K_max = 8.16, with the root locus crossing the jω-axis at ω ≈ ±1.095 rad/s at this critical gain.

Question 6

Q.3 OR (a) What is M & N circle? Explain the physical significance of this ci…

Accepted Answer

M-circles and N-circles are families of constant-magnitude and constant-phase contours plotted on the G(jω) plane, used to graphically determine the closed-loop frequency response directly from the open-loop Nyquist/polar plot without needing to algebraically compute the closed-loop transfer function; applying the Nyquist stability criterion to G(s)H(s)=K/[s(1+sT1)(1+sT2)] (a Type-1 system with no open-loop RHP poles, P=0) shows the Nyquist plot crosses the negative real axis at magnitude KT1T2/(T1+T2), so the closed-loop system is stable provided this crossing magnitude is less than 1, i.e., K < (T1+T2)/(T1T2).

Question 7

Q.4. A unity feedback system has open loop transfer function G(s) = 200/[s(s+…

Accepted Answer

For G(s)=200/[s(s+1)(s+10)], converting to Bode (time-constant) form gives G(jω)=20/[jω(1+jω)(1+jω/10)]; the Bode plot shows corner frequencies at ω=1 and ω=10 rad/s; calculation gives gain crossover frequency ≈4.23 rad/s, phase crossover frequency ≈3.16 rad/s, gain margin ≈-5.19 dB, and phase margin ≈-9.66°, both negative, indicating the closed-loop system is unstable.

Question 8

Q.4 OR (a) Draw the Bode plot (magnitude only) for the following transfer fun…

Accepted Answer

For G(s)=10/[s(1+5s)(1+0.25s)], the Bode magnitude plot has corner frequencies at ω=0.2 rad/s and ω=4 rad/s, with the gain crossover frequency calculated as approximately 1.368 rad/s; phase crossover frequency is the frequency at which the open-loop phase equals -180°, gain crossover frequency is where the open-loop magnitude equals 0dB (unity), phase margin is 180° plus the phase at gain crossover, and gain margin is the reciprocal (in dB, the negative) of the magnitude at phase crossover.

Question 9

Q.5 (a) Obtain the time response of following system: [ẋ1;ẋ2] = [[1,0],[1,1]]…

Accepted Answer

For the given state equation ẋ1=x1+u, ẋ2=x1+x2+u with x1(0)=1, x2(0)=0 and unit-step u(t), solving each first-order equation sequentially gives x1(t)=2e^t-1 and x2(t)=2t·e^t; for the given SISO state-space system with A=[[0,1,0],[0,-3,-3],[1,0,-1]], B=[0,3,0]ᵀ, C=[1,0,0], the transfer function is derived as G(s)=(3s+3)/(s³+4s²+3s+3).

Question 10

Q.5 OR Write short note on: (a) Lead-lag compensation network [8] (b) PID con…

Accepted Answer

A lead-lag compensation network combines a lead network (providing phase lead/advance to improve transient response and stability margin, typically implemented as a zero closer to the origin than its associated pole) with a lag network (providing improved steady-state accuracy via increased low-frequency gain, typically implemented as a pole closer to the origin than its associated zero), together achieving both improved transient and steady-state performance simultaneously; a PID controller combines Proportional, Integral, and Derivative control actions, generating a control signal u(t)=Kp·e(t)+Ki∫e(t)dt+Kd·de(t)/dt, providing fast response (P), zero steady-state error (I), and improved damping/stability (D).

Control System