(a) A signal is given by

x[t] = cos(28πt) + 2cos(40πt) + 3cos(70πt)

This signal is sampled at 90 samples/s to get discrete-time signal x(n).

(i) Find the periodicity of the individual components in the signal and hence find the periodicity N₀ of t

Question

(a) A signal is given by

x[t] = cos(28πt) + 2cos(40πt) + 3cos(70πt)

This signal is sampled at 90 samples/s to get discrete-time signal x(n).

(i) Find the periodicity of the individual components in the signal and hence find the periodicity N₀ of the signal x(n).

(ii) Find the harmonic indices m (0 ≤ m < N₀) of the complex DTFS coefficient Dₘ, where Dₘ is non-zero.

(iii) By inspection, write the magnitude of the coefficients |Dₘ| for the indices found above. 20 marks

(b) For a unity feedback time delay system with open-loop transfer function

G(s) = Ke⁻ᵀˢ/s(s+2)

calculate—

(i) the maximum tolerable value of delay T, when K = 1;

(ii) phase margin when K = √5 and delay T = 0·5 second. 20 marks

(c) Given a system in state space representation as

[ẋ₁]   [0  1][x₁]   [0]
[ẋ₂] = [0 -3][x₂] + [1] u

y = [1 0][x₁]
        [x₂]

(i) Check whether the system is observable or not.

(ii) Find the state transition matrix.

(iii) Design a state feedback controller to place closed-loop poles at −1±2j. 20 marks

UPSC Answer Check · Accepted Answer

Calculate all numerical results with systematic derivations, allocating approximately 35% time to part (a) on DTFS periodicity and harmonic analysis, 30% to part (b) on time-delay stability using Nyquist/Bode criteria, and 35% to part (c) on state-space design including observability test, matrix exponential computation, and pole placement via Ackermann's formula or direct method. Present each sub-part with clear headings, show all intermediate steps, and conclude with boxed final answers.
- For (a)(i): Compute digital frequencies ω₁=28π/90, ω₂=40π/90, ω₃=70π/90; find periods N₁=45, N₂=9, N₃=9; determine fundamental period N₀=LCM(45,9,9)=45
- For (a)(ii): Identify harmonic indices m₁=14, m₂=20, m₃=35 (or equivalently m₃=-10 mod 45 = 35) where Dₘ is non-zero within 0≤m<45
- For (a)(iii): State |D₁₄|=0.5, |D₂₀|=1, |D₃₅|=1.5 with correct conjugate symmetry |D₄₅₋ₘ|=|Dₘ|
- For (b)(i): Apply Nyquist stability criterion; find phase crossover frequency ωₚc=√2 rad/s; compute maximum delay Tₘₐₓ=π/(2√2)≈1.11s for K=1
- For (b)(ii): At K=√5, find gain crossover frequency ωgc=1 rad/s; calculate phase margin without delay as 90°-arctan(0.5)=63.43°; subtract delay phase ωgc×T×180/π=28.65° to get PM≈34.8°
- For (c)(i)-(iii): Check observability rank[O]=rank[C;CA]=2 (observable); compute state transition matrix Φ(t)=[1 (1-e⁻³ᵗ)/3; 0 e⁻³ᵗ]; design state feedback K=[5 3] using desired characteristic equation s²+2s+5=0

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	12	Correctly applies Nyquist sampling theorem for (a), recognizes time-delay as non-minimum phase element for (b), and properly uses observability rank test, matrix exponential properties, and pole placement theory for (c); no conceptual confusion between continuous and discrete domains	Minor errors in identifying harmonic folding or aliasing in (a); applies Bode plot approximately for (b) with some phase margin errors; attempts observability test but with matrix dimension errors in (c)	Fundamental misunderstanding of periodicity in sampled signals, treats e⁻ᵀˢ as simple gain, or confuses controllability with observability; fails to recognize state feedback requires controllability check
Numerical accuracy	20%	12	All numerical values precise: N₀=45, correct harmonic indices, Tₘₐₓ=1.1107s, PM≈34.8° or 35°, state feedback gains K₁=5, K₂=3; uses exact fractions where appropriate (π/2√2)	Correct order of magnitude but rounding errors in final values; e.g., N₀=45 correct but harmonic indices off by 1 due to floor/ceiling confusion; PM within ±5° of correct value	Major calculation errors: wrong LCM for periodicity, incorrect phase crossover frequency, or state feedback gains that don't produce desired poles; arithmetic mistakes in matrix operations
Diagram quality	15%	9	Clear unit circle diagram showing digital frequencies and harmonic locations for (a); labeled Bode magnitude/phase sketches showing gain/phase crossover for (b); signal flow graph or block diagram for state feedback implementation in (c)	Rough sketches without proper labels or scales; missing key features like -180° line on phase plot or conjugate symmetry indication on DTFS diagram	No diagrams where clearly needed, or completely incorrect diagrams (e.g., s-plane pole-zero plot instead of z-plane for discrete signal, wrong Nyquist contour)
Step-by-step derivation	25%	15	Explicit derivation: digital frequency conversion ω=ΩTₛ, period calculation via 2π/ω=N/M reduction, complete Nyquist criterion application with phase condition -180°-ωT×180/π, full observability matrix construction, Laplace transform inversion for Φ(t), and direct comparison method or Ackermann's formula for K	Skips key steps like explicit LCM calculation or assumes Φ(t) without derivation; uses shortcuts without justification; attempts pole placement but with algebraic errors in characteristic equation matching	Jumps to final answers without derivation; no intermediate steps shown; incorrect formula application (e.g., wrong state transition matrix formula, confused stability criteria)
Practical interpretation	20%	12	Interprets N₀=45 as 0.5s period in continuous time; explains Tₘₐₓ as stability limit for Indian telecom/networked control systems; discusses state feedback as modern control alternative to PID for MIMO systems like power plant turbine-governor control	Brief mention of practical relevance without specific context; generic statements about stability importance without linking to time-delay systems	No physical interpretation; purely mathematical answer without connecting to engineering applications like signal reconstruction, process control with transport lag, or state estimation in power systems

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