(SEM V) THEORY EXAMINATION 2023-24 CONTROL SYSTEM

SECTION A – Short Answers (2 × 10 = 20 Marks)

(a) Properties of a Signal Flow Graph

A signal flow graph (SFG) represents a set of linear algebraic equations.

Nodes represent system variables; directed branches represent functional relationships.

The gain along any branch is unidirectional.

There can be forward paths, feedback loops, and non-touching loops.

It is based on Mason’s Gain Formula for finding transfer functions.

(b) Define Transfer Function

The transfer function is the ratio of Laplace transform of the output to the input under zero initial conditions:

T(s)=C(s)R(s)T(s) = \frac{C(s)}{R(s)}T(s)=R(s)C(s)​

It defines the dynamic behavior of a linear time-invariant (LTI) system.

(c) Effect of Damping Ratio (ξ) on Second-Order System for Unit Step Input

(d) Standard Test Signals

Used to analyze system response:

Step input → steady-state error        Ramp input → velocity error    Parabolic input → acceleration error   Impulse input → natural response

(e) BIBO Stability (Bounded Input–Bounded Output)

                                                   A system is stable if a bounded input produces a bounded output.

Unstable if a bounded input produces an unbounded output.
                                                   Mathematically: All poles must lie in left half of s-plane.

(f) Disadvantages of Routh-Hurwitz Criterion

Does not give exact location of roots.                          Applicable only to polynomial equations.

Cannot handle time delays or nonlinear systems.       Requires all coefficients to be real and finite.

(g) Resonant Peak (Mr) and Resonant Frequency (ωr)

Resonant Peak (Mr): Maximum value of magnitude of frequency response.

Resonant Frequency (ωr): Frequency at which Mr occurs.

Mr=12ξ1−ξ2,ωr=ωn1−2ξ2M_r = \frac{1}{2ξ\sqrt{1-ξ^2}}, \quad ω_r = ω_n\sqrt{1-2ξ^2}Mr​=2ξ1−ξ2​1​,ωr​=ωn​1−2ξ2​ 

(h) Minimum and Non-Minimum Phase System

Minimum Phase: All poles and zeros lie in left half-plane (LHP).

Non-Minimum Phase: Has one or more zeros in right half-plane (RHP).
→ Non-minimum phase systems are less stable and have slower transient response.

(i) Advantages of State Variable Approach

Can analyze multi-input multi-output (MIMO) systems.   Handles time-varying and nonlinear systems.                                                                                  Provides internal state information.

Suitable for computer implementation and modern control design.

(j) Define State Space and State Trajectory

State Space: n-dimensional space whose coordinates are the state variables of a system.

State Trajectory: The path traced by state variables over time in the state space.

SECTION B – Descriptive Questions (Any 3 × 10 = 30 Marks)

(a) Mason’s Gain Formula (for Block Diagram)

T=∑k=1nPkΔkΔT = \frac{\sum_{k=1}^n P_k Δ_k}{Δ}T=Δ∑k=1n​Pk​Δk​​

where:
PkP_kPk​ = gain of k-th forward path,
Δ=1−(sum of individual loop gains)+(sum of product of two non-touching loops)−…Δ = 1 - (\text{sum of individual loop gains}) + (\text{sum of product of two non-touching loops}) - \dotsΔ=1−(sum of individual loop gains)+(sum of product of two non-touching loops)−…,
ΔkΔ_kΔk​ = Δ excluding loops touching k-th path.

(b) PD and PI Controllers

PD (Proportional-Derivative): Improves transient response and stability.

• Gc(s)=Kp(1+Tds)G_c(s) = K_p(1 + T_d s)Gc​(s)=Kp​(1+Td​s)

PI (Proportional-Integral): Eliminates steady-state error.

• Gc(s)=Kp(1+1Tis)G_c(s) = K_p\left(1 + \frac{1}{T_i s}\right)Gc​(s)=Kp​(1+Ti​s1​)

(c) Range of K for Stability (Routh Criterion)

For G(s)=K(s+1)s(s−1)(s+6)G(s) = \frac{K(s+1)}{s(s-1)(s+6)}G(s)=s(s−1)(s+6)K(s+1)​:
Construct Routh array → find K range ensuring all first-column elements positive → gives stable region.
Marginal stability → one row becomes zero → determine K at imaginary axis crossing.

(d) Mapping Theorem & Nyquist Criterion

Mapping theorem: Relates encirclements of origin in G(s)H(s)-plane to those of (-1,0) in Nyquist plot.

Nyquist Criterion:

• N=Z−PN = Z - PN=Z−P

where N = encirclements of (-1,0), Z = zeros (RHP poles of closed-loop system), P = poles (RHP poles of open-loop).
→ Used for frequency-domain stability analysis.

(e) Transfer Function from State Model

For given:

x˙=Ax+Bu,y=Cx+Du\dot{x} = Ax + Bu,\quad y = Cx + Dux˙=Ax+Bu,y=Cx+Du T(s)=C(sI−A)−1B+DT(s) = C(sI - A)^{-1}B + DT(s)=C(sI−A)−1B+D

Plug in A, B, C matrices and simplify.

SECTION C – Long Questions (Any 1 from each)

3(a) Open vs Closed Loop Systems

4(a) Peak Time & Peak Overshoot

For 2nd order system G(s)=ωn2s2+2ξωns+ωn2G(s) = \frac{ω_n^2}{s^2 + 2ξω_n s + ω_n^2}G(s)=s2+2ξωn​s+ωn2​ωn2​​:

Tp=πωn1−ξ2,Mp=e−πξ1−ξ2×100%T_p = \frac{π}{ω_n\sqrt{1-ξ^2}}, \quad M_p = e^{\frac{-πξ}{\sqrt{1-ξ^2}}} × 100\%Tp​=ωn​1−ξ2​π​,Mp​=e1−ξ2​−πξ​×100% 

5(a) Root Locus

Steps:

Identify open-loop poles & zeros.   Number of branches = poles – zeros.  Symmetry about real axis.

Asymptotes:

1. Centroid =∑p−∑zn−m\text{Centroid } = \frac{\sum p - \sum z}{n - m}Centroid =n−m∑p−∑z​ Angles=(2q+1)180°n−m\text{Angles} = \frac{(2q+1)180°}{n-m}Angles=n−m(2q+1)180°​

Breakaway & intersection points found using dK/ds = 0.

6(a) Nyquist Plot for Given Transfer Function

For G(s)H(s)=(4s+1)s2(2s+1)(s+1)G(s)H(s) = \frac{(4s+1)}{s^2(2s+1)(s+1)}G(s)H(s)=s2(2s+1)(s+1)(4s+1)​:
Plot frequency response in complex plane → analyze encirclement of (-1,0) → comment on stability using Nyquist Criterion.

7(a) Compensators

Types: Lead, Lag, Lead-Lag compensators.      Lead compensator: improves transient response.

• Gc(s)=Ks+zs+p, (z<p)G_c(s) = K\frac{s+z}{s+p},\ (z < p)Gc​(s)=Ks+ps+z​, (z<p)

Frequency of max phase lead:                         ωm=ω1ω2ω_m = \sqrt{ω_1 ω_2}ωm​=ω1​ω2​​

(Geometric mean of corner frequencies.)

7(b) Controllability and Observability

For

A=[−2−34301010],B=[001],C=[0 5 1]A = \begin{bmatrix}-2 & -3 & 4\\ 3 & 0 & 1\\ 0 & 1 & 0\end{bmatrix},\quad B = \begin{bmatrix}0\\0\\1\end{bmatrix},\quad C = [0\ 5\ 1]A=​−230​−301​410​​,B=​001​​,C=[0 5 1]

Controllability Matrix: [B AB A²B] → if rank = n ⇒ controllable.

Observability Matrix: [C;CA;CA2][C; CA; CA^2][C;CA;CA2] → if rank = n ⇒ observable.