Questions and Answers of Control System Analysis And Design

Sketch the root loci for the system shown in Fig. P5.4 (a) and show that system is stable for all \(\mathrm{K}>0\). R(s) K (s + 0.4) s(s+3.6) (a) Y(s) -3.6 -4 K = 0 60 -2 Fig. P5.4: (a) System (b)
The Bode plots for the loop transmittance \(\mathrm{G}(s)\) of unity feedback control system are experimentally obtained as shown in Fig. P6.11 (a) and (b) with the loop gain set at nominal value.(a)
The straight line approximated torque-speed curve of AC servo motor for rated control voltage \(115 \mathrm{~V}, 50 \mathrm{~Hz}\) is shown in Fig. P9.4. The moment of inertia of motor is \(10^{-5}
Which of the following rotors are used in a two phase AC servo motor?1. Solid iron2. Squirrel cage3. Drag cup.Select the correct answer using the codes given below:Codes:(a) 1, 2 and 3(b) 1 and
Obtain overall transfer function for diagram shown in Fig. P3.2. R(s) G(s) + G(s) H, (s) + G(s) H(s) + C(s)
Reduce the diagram of Fig. P 3.3 to a single block. R(s) G(s) H(s) + G(s) H(s) G (s) H (s) Y(s)
Find closed loop transfer function of systems shown in Fig. P3.4. (a), (b) and (c). R(S) G(s) H(s) G(s) H(s) (a) + G,(s) C(s)
If all the system initial conditions are zero, find the Laplace transform of the output for given system inputs in Fig. P3.5.\[\begin{aligned}& r_{1}(t)=3 e^{-t} \\& r_{2}(t)=4 u(t) ; u(t) \text { is
Find \(\frac{\mathrm{C}_{1}(s)}{\mathrm{R}_{1}(s)}\) and \(\frac{\mathrm{C}_{2}(s)}{\mathrm{R}_{2}(s)}\) for the system shown in Fig. P 3.6 where \(\alpha_{1}, \alpha_{2}\) and \(\alpha_{3}\) are
Determine \(\frac{\mathrm{Y}(s)}{\mathrm{X}(s)}\) for Fig. P3.7. X(s). G(s) H,(s) G, (s) G(s) H(s) Y(s)
Find gains (a) \(\frac{Y_{5}}{Y_{1}}\) (b) \(\frac{Y_{2}}{Y_{1}}\) and (c) \(\frac{Y_{5}}{Y_{2}}\) for signal flow graph shown in Fig. P3.8 Y0- Y G4 G X3 - H Y6 G - H Y4 G5 G - H - H 5
Determine following transfer functions for signal flow graph shown in Fig. P 3.9.(a) \(\mathrm{T}_{11}(s)=\frac{\mathrm{Y}_{1}(s)}{\mathrm{R}_{1}(s)}\)(b)
Convert the block diagram shown in Fig. P3.11 into signal flow graph and obtain following system functions.\(\begin{array}{llll}\text { (a) }
Construct signal flow graph and find system function \(\frac{\mathrm{X}(\mathrm{S})}{\mathrm{U}(\mathrm{S})}\) for the system characterised by following equations:\[\begin{aligned}x & =x_{1}+b_{3} u
The block diagram of a Feedback Control System is shown in Fig. P 3.13. Draw the equivalent signal flow graph and determine C/R. R G pL3 H G G4 G H C
Find the transfer function \(C(s) / R(s)\) for a system whose signal flow graph is shown in Fig. P3.14. RQ 1 . S. -1 in -2 C
Find \(\mathrm{Y}(s)\) in signal flow graph shown in Fig. P3.15. R(s) O G H U(s) 1 G W(S) -H 2 1 -o Y(s)
Construct signal flow graph for electrical networks shown in Fig. P3.16(a) and (b). R mnoooo L 1 R ww I V Fig. P3.16 (a) R L vooo KI R www 2 1 ww R R
Use block diagram reduction technique and obtain the following transfer functions for the system shown in Fig. D3.3:(i) \(\frac{\mathrm{C}(s)}{\mathrm{R}(s)}\)(ii)
A block diagram of a linear feedback system is shown in Fig. D3.4. Obtain a signal flow graph for the system and hence calculate the overall gain \(\mathrm{C}(\mathrm{s}) / \mathrm{R}(\mathrm{s})\)
Use block diagram reduction rules to find C(s) for the system shown in Fig. D3.5. R(s) G (s) R(s) + H(s) + + R4(s) R(s) G (s) H(s) C(s)
Find six transfer functions of the system shown in Fig. D3.6. R,(s) o Y(s) - Ra(s) O 2 1 3 4 R(s) 0 Y (8)
Compute Z/Y of system shown in Fig. D3.7. Xo- Y G H G H 3 G5 G - H G - H OZ
Construct signal flow graph for following set of system equations and hence find \(x_{4} / x_{1}\).\[\begin{aligned}& x_{2}=k_{1} x_{1}+k_{3} x_{3} \\& x_{3}=k_{4} x_{1}+k_{2} x_{2}+k_{5} x_{3} \\&
Find \(\mathrm{C}(s) / \mathrm{R}(s)\) for systems shown in Fig. D3.9 (a), (b), (c) and (d). R(s) o R(s) o ! G 5 H 10. G - 1 (a) Go G3 (b) 2 G G GA 10 -o C(s) -OC(s)
Draw signal flow graph and obtain C(s)/R(s)C(s)/R(s) for the systems shown in Fig. D3.10 (a), (b) and (c)(c). R(s). 3 s+4 (a) 10 s(s+1) 0.5s N(s) = 0 C(s)
Draw signal flow graph and hence obtain transfer function of network shown in Fig. D3.11. V,(s) R R R www wwww R4 V (s)
In the signal flow graph of a closed loop system shown below, \(T_{D}\) represents the disturbance in the forward path.The effect of the disturbance can be reduced by(a) increasing
A system is represented by the block diagram shown below.Which one of the following represents the input-output relationship of the above diagram?(a) \(\mathrm{R}(s) \longrightarrow \mathrm{G}_{1}
In the feedback system shown below, the noise component of output is given by (assume high loop gain at frequencies of interest):(a) \(\frac{-\mathrm{N}(s)}{\mathrm{H}_{1}(s)}\)(b)
In the system shown below, to eliminate the effect of disturbance D(s)D(s) on C(s)C(s), the transfer function Gd(s)Gd(s) should be(a) (s+10)10(s+10)10(b) s(s+10)10s(s+10)10(c) 10(s+10)10(s+10)(d)
The closed-loop system shown below is subjected to a disturbance \(\mathrm{N}(\mathrm{s})\). The transfer function \(\mathrm{C}(s) / \mathrm{N}(s)\) is given by(a) \(\frac{\mathrm{G}_{1}(s)
The transfer function of the system shown below is(a) \(\frac{\mathrm{Q}}{\mathrm{R}}=\frac{\mathrm{ABC}}{1+\mathrm{ABC}}\)(b)
For block diagram shown below, \(\mathrm{C}(s) / \mathrm{R}(s)\) is given by(a) \(\frac{\mathrm{G}_{1} \mathrm{G}_{2} \mathrm{G}_{3}}{1+\mathrm{H}_{2} \mathrm{G}_{2} \mathrm{G}_{3}+\mathrm{H}_{1}
The sum of the gains of the feedback paths in the above signal flow graph is(a) \(a f+b e+c d+a b e f+b c d e+a b c d e f\)(b) \(a f+b e+c d+a b e f+b c d e\)(c) \(a f+b e+c e+a b e f+a b c d e
Which one of four signal flow graphs shown in \((a),(b),(c)\) and \((d)\) represents the diagram shown below? R- G G + + G H, C
For the signal flow graph shown below the value of \(x_{2}\) is(a) \(a_{12} x_{1}\)(b) \(a_{12} x_{1}+a_{32} x_{3}\)(c) \(a_{12} x_{1}-a_{23} x_{3}+a_{32} x_{3}\)(d) \(-a_{23} x_{3}-a_{34} x_{4}\)
Consider the following signal flow graphs.The value of gain is two for(a) 1(b) 2(c) 2 and 3(d) 1,2 and 3 1. Input 1 1 Output
The signal flow graph of system is shown below.In this graph, the number of three non-touching loops is(a) zero(b) 1(c) 2(d) 3. R(s) -O C(s)
The sum of gain products of all possible combinations of two non-touching loops in the following signal flow graph is:(a) \(t_{23} t_{32} t_{44}\)(b) \(t_{23} t_{32}+t_{34} t_{43}\)(c) \(t_{23}
In the signal flow graph shown below, the value of the \(C / R\) ratio is(a) \(\frac{28}{57}\)(b) \(\frac{40}{57}\)(c) \(\frac{40}{81}\)(d) \(\frac{28}{81}\) Ro 5 2 -OC
The forward paths and feedback loops in the signal flow graph shown below are respectively(a) 4,4(b) 4,3(c) 3,4(d) 3,3 X O D k b h d m n e g
In the signal flow graph shown below, the gain \(c / r\) will be(a) \(\frac{11}{9}\)(b) \(\frac{22}{15}\)(c) \(\frac{24}{23}\)(d) \(\frac{44}{23}\) 5 3 CO
For signal flow graph shown below, the transmittance between \(x_{2}\) and \(x_{1}\) is(a) \(\frac{r s u}{1-s t}+\frac{e f h}{1-f g}\)(b) \(\frac{r s u}{1-f g}+\frac{e f h}{1-s t}\)(c) \(\frac{e f
Consider the system I and system II shown below. The system I can be reduced to the form as shown in system II with(a) \(\mathrm{X}=c_{0} s+c_{1}\), Y \(=\frac{1}{s^{2}+a_{0} s+a_{1}},
For the following polynomials, determine how many roots are in RHP, how many are in LHP and how many are on the imaginary axis. Also comment on stability of systems characterised by the
Are the systems shown in Fig P 4.2 (a), (b) and (c) stable or unstable? R(s). 4 2 S + 4 3 s +3 (a) -Y(s)
For what range(s) of the adjustable parameter \(\mathrm{K}\), do the following polynomials have all roots in the LHP ?(a) \(s^{3}+(2+K) s^{2}+(8+K) s+6\)(b) \(s^{4}+(10+K) s^{3}+3 s^{2}+9 s+11\)
Find the range(s) of adjustable parameter \(\mathrm{K}>0\) for which the systems of Fig. P 4.4 (a) and (b) are stable. R(s) R(s) o K s + K K (s + 1) S 3 (a) 1 s+ 0.1 (b) 3 s+3 -2 3 S -o Y(s) -C(s)
The open loop transfer function of a unity feedback control system is given by\[\mathrm{G}(s) \mathrm{H}(s)=\frac{\mathrm{K}}{s(s+1)(1+2 s)(1+3 s)}\]Determine the value of \(\mathrm{K}\)(i) for which
A linear feedback control system has an open-loop transfer function\[A(s)=\frac{\mu(s+\alpha)^{2}}{\alpha^{2}(1+s) s^{3}}\]where \(\mu\) and \(\alpha\) are adjustable parameters. Find the relation
The loop transfer function of a feedback control system is given by:\[\mathrm{G}(s) \mathrm{H}(s)=\frac{\mathrm{K}(s+1)}{s(1+s \mathrm{~T})(1+2 s)}, \mathrm{K}>0\]Use Routh Hurwitz criterion to
Consider the closed-loop feedback control system shown in Fig. P4.8Use Routh-Hurwitz criterion to determine the range of \(\mathrm{K}\) for which the system is stable. Find also the number of roots
For the system shown in Fig. P 4.9, establish relation between ' \(k\) ' and ' \(a\) ' so that system is stable and show stable region on a plane with ' \(k\) ' on \(y\) axis and ' \(a\) ' on \(x\)
A unity feedback system has open loop transfer functionDetermine stability of closed-loop system as a function of \(\mathrm{K}\). Determine value of \(\mathrm{K}\) that will cause sustained
Determine the values of \(\mathrm{K}\) and \(\alpha\), so that system of Fig. \(\mathrm{P} 4.11\) oscillates at a frequency of \(2 \mathrm{rad} / \mathrm{sec}\). K (s + 1) 2 +as +2s +1 3 S
A unity feedback system has open-loop transfer function\[\mathrm{G}(s)=\frac{\mathrm{K} e^{-s}}{s\left(s^{2}+5 s+9ight)}\]Determine maximum value of \(\mathrm{K}\) for closed-loop system to be stable.
Show that the system with closed-loop transfer function\[\mathrm{T}(s)=\frac{20}{(s+2)^{2}\left(s^{2}+5 s+12ight)}\]has relative stability of at least 2 units.
Determine whether the largest time constant of characteristic equation given below is greater than, less than or equal to 1 sec.\[s^{3}+4 s^{2}+6 s+4=0\]
For each of following polynomials, how many roots are in the LHP, how many are in the RHP, and how many are on the imaginary axis ?(a) \(s^{4}+3 s^{2}+4\)(b) \(s^{5}+2 s^{4}+3 s^{3}+6 s^{2}+2
For what range(s), if any, of the adjustable constant \(\mathrm{K}\), are all the roots of following polynomials in the left half of the complex plane ?(a) \(s^{4}+s^{3}+3 s^{2}+2 s+4+\mathrm{K}\)(b)
Find the range(s) of positive constant \(\mathrm{K}\), if any, for which the systems shown in Fig. D4.3 (a) and \((b)\), are stable. R(s) o R(s) 1/s 1/s (a) 4 3 s +3 (b) R,(s) 10 s + ks + 10 K 1/s K
Show that the characteristic polynomial\[s^{4}+14 s^{3}+73 s^{2}+168 s+144\]has relative stability of at least 2 units.
The unity feedback system given by\[\mathrm{G}(s)=\frac{4}{s\left(s^{2}+a s+2 bight)} ; a \text { and } b \text { are positive constants. }\]is limitedly stable and oscillates with frequency \(4
Investigate stability of unity feedback system whose open-loop transfer function isAns. Stable for \(\mathrm{T}
The characteristic equation of a system is\[s^{3}+10 s^{2}+(50+\mathrm{A}) s+\mathrm{K}=0\]Determine the regions on A-K plane (A on \(\mathrm{X}\) axis and \(\mathrm{K}\) on \(\mathrm{Y}\) axis) in
A plot such as the example sketch of Fig. D4.8 shows the range of values of two parameters \(K_{1}\) and \(K_{2}\) for which a system is stable. It is called a stability boundary diagram. Draw such a
For the closed loop system shown in Fig. D4.9.(a) For what values at \(\mathrm{K}\) is the system stable?(b) For what value of \(\mathrm{K}\) is the system marginally stable?(c) For the value of
The closed-loop transfer function of a control system is\[T(s)=\frac{K}{s^{4}+6 s^{3}+30 s^{2}+60 s+K}\]What should be the upper limit on \(\mathrm{K}\) if all the poles of \(\mathrm{T}(s)\) are
The open-loop transfer functions with unity feedback are given below for different systems.1. \(\mathrm{G}(s)=\frac{2}{s+2}\)2. \(G(s)=\frac{2}{s(s+2)}\)3. \(G(s)=\frac{2}{s^{2}(s+2)}\)4.
The open-loop transfer function of a control system is given by\[\frac{\mathrm{K}(s+10)}{s(s+2)(s+a)}\]The smallest possible value of ' \(a\) ' for which this system is stable in the closed-loop for
The open-loop transfer function of a unity negative feedback control system is given by\[\mathrm{G}(s)=\frac{\mathrm{K}(s+2)}{[(s+1)(s-7)]}\]For \(\mathrm{K}>6\), the stability characteristics of the
For the block diagram shown in the Figure below, the limiting values of KKKK for stability of inner loop is found to be XKYXKYXKYXKY. The overall system will be stable if and only if(a) \(4
The open-loop transfer function of a unity feedback control system is\[\mathrm{G}(s) \mathrm{H}(s)=\frac{30}{s(s+1)(s+\mathrm{T})}\]where \(\mathrm{T}\) is a variable parameter. The closed-loop
While forming Routh's array, the situation of a row of zeros indicates that the system(a) has symmetrically located roots(b) is not sensitive to variations in gain(c) is stable(d) unstable.
None of the poles of a linear control system lie in the right half of s-plane. For a bounded input the output of this system(a) is always bounded(b) could be unbounded(c) always tends to zero(d) none
For what range of \(K\) is the following system asymptotically stable ? Assume \(K \geq 0\)(a) \(0(b) \(0(c) \(0(d) \(1 S-5 s+4 K
The system represented by the transfer function \(G(s)=\frac{s^{2}+10 s+24}{s^{4}+6 s^{3}-39 s^{2}+18 s+84}\) has(a) 2 poles in the right half \(s\)-plane(b) 4 poles in the left half s-plane(c) 3
An electromechanical closed-loop control system has the following characteristic equation\[s^{3}+6 \mathrm{~K} s^{2}+(\mathrm{K}+2) s+8=0\]where \(\mathrm{K}\) is the forward gain of the system. The
The feedback control system shown in Figure below is stable.(a) for all \(\mathrm{K} \geq 0\)(b) only if \(\mathrm{K} \geq 1\)(c) only if \(0 \leq \mathrm{K}(d) only if \(0 \leq \mathrm{K} \leq 1\)
The first two rows of Routh's array of a fourth-order system areThe number of roots of the system lying on the right half of \(s\)-plane is(a) zero(b) 2(c) 3(d) 4 4 S 3 S 1 2 10 20 5
The first element of each of the rows of a Routh-Hurwitz stability test showed the sign as followsThe number of roots of the system lying in the right half of \(s\)-plane is(a) 2(b) 3(c) 4(d) 5 Rows
The first two rows of Routh array of a third order system areselect the correct answer from the following:(a) system has one root in the RHP.(b) system has two roots on \(j \omega\) axis at \(s= \pm
Consider the system of order three with characteristics equation\[s^{3}+\mathrm{Ts}^{2}+(\mathrm{K}+2) s+(1+\mathrm{K})=0\]The values of \(\mathrm{K}\) and \(\mathrm{T}\) such that the system has two
Consider the following statements.I. A continuous system generates output of form \(y=t\) when excited by a step function. This system is unstable.II. The dynamics of an integrator is given by \(d y
Consider the system shown in figure below. Which one of the following statements is true?(a) Both open loop and closed loop system are stable.(b) Both open loop and closed loop system are
A continuous system with time delay, is described by characteristic equation\[s^{2}+s+e^{-s \mathrm{~T}}=0\]Now, consider the following statements.I. Strictly speaking, this equation has an infinite
The system shown in figure below is designed to be an oscillator. What is corresponding frequency of oscillation in \(\mathrm{rad} / \mathrm{sec}\).?(a) 1(b) 2(c) 3(d) 4 K s(s+ 3)
Consider the following statements in relation with an active network with transfer function modelwhere \(\mathrm{K}\) is gain and \(\tau=\mathrm{RC}\).I. The network is unstable for all values of
The design goals for a unity feedback control system having an open loop transfer function\[\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+1)(s+2)} \text { are }\]1. velocity error coefficient \(K_{V} \geq 10
An open loop transmittance is \(G(s)=\frac{K(1-s)}{(1+s)}\). This is unity feedback configuration, will be stable for(a) \(|\mathrm{K}|>1\)(b) \(\mathrm{K}>1\)(c) \(\mathrm{K}
A negative feedback system has loop transfer function \(K(s+3) /(s+8)^{2}\) where \(K\) is be adjusted that the system exhibits sustained oscillation. The corresponding frequency of oscillation,
For a system with open-loop transfer function\[\mathrm{G}(s) \mathrm{H}(s)=\frac{\mathrm{K}}{s(s+3)\left(s^{2}+6 s+64ight)}\]Sketch the root locus showing all details thereon. Comment on the
Consider the system shown in Fig. P 5.2 (a).(a) Sketch root loci as \(\mathrm{K}\) varies from 0 to \(\infty\).(b) Find the range of \(\mathrm{K}\) for which system exhibits underdamped and
A simplified model of an auto pilot of an aircraft is given by\[\mathrm{G}(s) \mathrm{H}(\mathrm{s})=\frac{\mathrm{K}(s+\alpha)}{s(s-\beta)\left(s^{2}+2 \xi \omega_{n} s+\omega_{n}^{2}ight)}\]Choose
Sketch the root loci for the control system shown in Fig. P5.5(a). Determine the range of parameter \(\mathrm{K}\) for stability. Apply the angle criterion to show that root locus branches consist of
The \(\mathrm{G}(s) \mathrm{H}(\mathrm{s})\) product of a feedback system is given by\[\mathrm{G}(s) \mathrm{H}(\mathrm{s})=\frac{\mathrm{K}}{s(s+1)(s+9)}\]Determine if the points given below, lie on
Sketch the root locus for the product\[\mathrm{G}(s) \mathrm{H}(s)=\frac{\mathrm{K}}{s(s+3)\left(s^{2}+3 s+11.25ight)}\]and comment on stability.
Sketch the root locus for a system with product\[\mathrm{G}(s) \mathrm{H}(s)=\frac{\mathrm{K}}{s(s+2)(s+4)(s+8)}\]and determine the location of dominant characteristic roots for damping ratio of
Consider the system shown in Fig. P 5.9(a).(a) Sketch the root locus as \(\mathrm{z}\) varies from 0 to \(\infty\).(b) Determine the value of \(\mathrm{z}\) so that damping ratio of dominant

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