Control Systems MCQ – 33 previous year questions with solutions

Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response. Key relation: Characteristic form: s²+2ζω_n s+ω_n²=0.

Exam focus: ζ1 overdamped. Common trap: Overdamped responses do not sustain oscillations.

Percent overshoot fixes the damping ratio through M_p=e^{−πζ/√(1−ζ²)}. For G(s)=K/[s(sT+1)], the closed-loop damping ratio varies inversely with √K. Converting 75% and 25% overshoot to ζ values and squaring their ratio gives the required gain reduction factor, matching 20.

For unity feedback, first form T(s)=G(s)/(1+G(s)), then multiply by the step input 1/s and perform partial fractions. For G(s)=4/[s(s+5)], the closed-loop denominator is s²+5s+4=(s+1)(s+4), giving the response 1−(1/3)(4e⁻ᵗ−e⁻⁴ᵗ).

A standard first-order system has one real pole and an exponential step response controlled by a time constant τ. For τ=2 s, y(t)=1−e^(−t/2). At t=2 s, y≈0.632. Key relation: G(s)=1/(τs+1); y(t)=1−e^(−t/τ).

Exam focus: 63% at τ, ~95% at 3τ, ~99% at 5τ. Common trap: Do not confuse one time constant with 95% settling.

Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. With ζ=0.5 and ωn=4 rad/s, ωd=ωn√(1−ζ²)≈3.46 rad/s, so the response is underdamped and has nonzero overshoot. Key relation: Characteristic form: s²+2ζω_n s+ω_n²=0.

Exam focus: ζ1 overdamped. Common trap: Overdamped responses do not sustain oscillations.

The response 1−e⁻⁵ᵗ−5te⁻⁵ᵗ is the standard unit-step response of a critically damped second-order system with repeated pole at −5. Hence ζ=1, and differentiating the step response gives impulse response h(t)=25te⁻⁵ᵗ. Statement 1 is false because there is no underdamped natural frequency.

A PI controller adds an integrator pole at the origin and a finite zero, increasing system type and reducing/eliminating steady-state error for appropriate inputs. The assertion is therefore true. The reason is false because a PI controller's added pole is specifically at the origin; the finite negative-real-axis point is the controller zero, not an arbitrary added pole.

A PI controller introduces an integrator 1/s, so a first-order plant normally produces a second-order closed-loop characteristic equation: Statement 1 is true. For a standard plant K/(τs+1) with fixed K_p, ζ = (1 + KK_p)/(2√(τKK_i)); increasing K_i reduces ζ, so Statement 2 is false.

Statement 1 is false because negative feedback does not always guarantee stable closed-loop poles. Statement 2 is true because negative feedback can reduce parameter sensitivity; for a standard loop, S = 1/[1 + G(s)H(s)]. Therefore option C is correct.

Proportional action generally speeds response but leaves finite error; integral action raises low-frequency gain and can eliminate step steady-state error; derivative action adds predictive damping and can reduce overshoot. None of these by itself guarantees stability for every plant/tuning. Key relation: u(t)=K_p e + K_i∫e dt + K_d de/dt.

Exam focus: I fixes offset; D improves damping; P sets overall response strength. Common trap: PID is not an unconditional stability guarantee.

A controller processes the error/control information and modifies the control signal so the plant responds according to desired performance requirements. It may amplify, integrate, differentiate, shape or otherwise compensate the signal.

Gain margin and phase margin quantify distance from the critical feedback condition and are widely used as relative-stability measures; pole and zero locations shape the frequency response. At gain crossover, phase=−150°, so PM=180−150=30°. Gain margin must be evaluated at phase crossover, so it is not automatically zero here.

Exam focus: Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions. Common trap: Frequency response is not independent of poles and zeros.

In the standard negative-feedback frequency-response stability test used in this question, both margins must be positive: GM > 0 dB and PM > 0°. Positive margins mean the loop has not yet reached the critical instability boundary.

Each first-order pole contributes −20 dB/decade and each first-order zero contributes +20 dB/decade after its corner frequency. With 4 poles and 2 zeros, the high-frequency slope is 20(2−4)=−40 dB/decade.

Frequency response describes how a linear system's gain and phase vary with sinusoidal frequency. Gain margin and phase margin quantify distance from the critical feedback condition and are widely used as relative-stability measures; pole and zero locations shape the frequency response. Key relation: At gain crossover ω_gc: PM=180°+∠G(jω_gc).

Exam focus: Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions. Common trap: Frequency response is not independent of poles and zeros.

Linear gain margin is the reciprocal of the open-loop magnitude at the phase-crossover frequency: GM = 1/|L(jω_pc)|. Doubling the loop gain doubles that magnitude, so GM_new = GM_old/2. Thus it decreases by a factor of 2.

A state-space model represents internal dynamics using a minimal set of state variables. It naturally handles MIMO systems and incorporates initial conditions. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

A state-space model represents internal dynamics using a minimal set of state variables. Output generally depends on both state and direct input through y=Cx+Du. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

Statement 1 gives the standard meaning of complete controllability: the input can drive the state to any desired state in finite time. Statement 2 gives the observability idea: the initial state can be reconstructed from output measurements. For an n-state LTI model, the algebraic checks are rank(𝒞)=n and rank(𝒪)=n. Hence option A.

A state-space model represents internal dynamics using a minimal set of state variables. Stability of a continuous-time LTI model is tied to eigenvalues of A: strictly negative real parts imply asymptotic stability. Key relation: ẋ=Ax+Bu; y=Cx+Du.

Exam focus: Eigenvalues of A are the state-model characteristic roots. Common trap: Output is not generally independent of state variables.

Use G(s)=C(sI−A)^−1B+D. Here det(sI−A)=s^2+8. Multiplying C·adj(sI−A)·B gives 8s+1; therefore G(s)=(8s+1)/(s^2+8).

A transfer function is the Laplace-domain ratio of output to input for a linear time-invariant system under zero initial conditions. Its poles and zeros determine important input-output stability, transient and frequency-response features, but the transfer function does not uniquely reveal the internal state realization. Key relation: G(s)=Y(s)/U(s) with zero initial conditions.

Exam focus: Poles are denominator roots; zeros are numerator roots. Common trap: Transfer functions do not contain complete unique internal-state information.

For a unit-step input, U(s) = 1/s and G(s) = Y(s)/U(s) = sY(s). Transforming the given response gives Y(s) = (s + 2)/[s(s + 1)]; therefore G(s) = (s + 2)/(s + 1).

D — Characteristics of the system. The denominator gives the characteristic equation; its roots determine the poles and natural response of the represented system.

The factor e^−2s is a 2 s transport delay. Therefore y(t)=6e^−4(t−2)u(t−2), which is zero for t<2 s. Hence y(0)=0.

It determines the order of polynomial input that can be tracked with zero/finite/infinite steady-state error, provided the closed loop is stable. Static-error constants directly produce the truth set {1,2,3}. Key relation: Type 1: Kp=lim_{s→0}GH=∞ → ess,step=1/(1+Kp)=0; Kv=lim_{s→0}sGH finite → ess,ramp=1/Kv.

Exam focus: Type-1 mnemonic: step → zero error; ramp → finite error; parabolic → infinite error. Common trap: Ignoring the closed-loop stability condition when applying steady-state error formulas.

System type equals the number of pure integrators (poles at the origin) in the open-loop transfer function G(s)H(s). Here the denominator contains s², so there are two poles at the origin and the system is Type 2.

It determines which standard polynomial inputs can be tracked with zero or finite steady-state error. Integral action increases low-frequency gain and improves steady-state accuracy, but it can also affect transient stability margins. Key relation: Kp=lim G(s), Kv=lim sG(s), Ka=lim s²G(s) as s→0 for unity feedback.

Exam focus: Type-0/1/2 classification comes from poles at the origin. Common trap: Do not infer stability from system type alone.

Investigating stability by constructing a Lyapunov function V(x) and examining its derivative along system trajectories is called Lyapunov's direct (second) method. It avoids explicitly solving the nonlinear differential equations.

Nonlinear systems do not obey superposition, but they can still be stable or unstable. Linearization around an operating point uses a local first-order approximation and is useful for small-signal analysis. Key relation: Local linearization uses the Jacobian evaluated at the operating point.

Exam focus: Nonlinear does not mean unstable; stability depends on the dynamics and operating point. Common trap: Do not apply global conclusions from a local linear model without checking validity range.

Root-locus asymptote angles depend only on the excess number of open-loop poles over zeros: θ_k=(2k+1)180°/(n−m). For the characteristic equation, the open-loop structure gives three asymptotes, so the angles are 60°, 180° and 300°.

Branches start at open-loop poles and end at finite zeros or zeros at infinity; the number of branches equals the number of open-loop poles. Real-axis breakaway/break-in points can occur where multiple branches meet, and loci can enter the right-half plane. Key relation: Characteristic equation 1+KG(s)H(s)=0.

Exam focus: Start at poles, finish at zeros/infinity. Common trap: Root locus is not confined to the left-half plane.

Both statements are true: the Routh criterion converts a characteristic polynomial into an algebraic array and is especially convenient for determining the number of right-half-plane roots of higher-order systems without explicitly solving for all roots. However, the second statement is a use of the method rather than the precise reason the array formulation is convenient, so the final-key choice is B.