Questions and Answers of Control System Analysis And Design

The pole-zero map and the Nyquist plot of the loop transfer function GH(s) of a feedback system are shown below. For this(a) both open loop and closed loop systems are stable.(b) open loop system is
Consider the following open loop frequency response of a unity feedback system.The gain and phase margin of the system are respectively(a) \(0.00 \mathrm{~dB},-180^{\circ}\)(b) \(3.86
An underdamped second order system having a transfer function of the form \(\mathrm{M}(s)\) \(=\frac{K \omega_{n}^{2}}{s^{2}+2 \xi \omega_{n} s+\omega_{n}^{2}}\) has frequency response plot as shown
The open loop transfer function of a unity feedback control system is given as \(\mathrm{G}(s)=\frac{a s+1}{s^{2}}\). The value of ' \(a\) ' to give a phase margin of \(45^{\circ}\) is equal to(a)
A system has poles at \(0.01 \mathrm{~Hz}, 1 \mathrm{~Hz}\) and \(80 \mathrm{~Hz}\); zeros at \(5 \mathrm{~Hz}, 100 \mathrm{~Hz}\) and \(200 \mathrm{~Hz}\). The approximate phase of the system
Consider the following Nyquist plot of a feedback system having open loop transfer function \(\mathrm{GH}(s)=(s+1) /\left[s^{2}(s-2)ight]\) as shown in the diagram given below. What is the number of
Consider the following statements for a counter clockwise Nyquist path.1. For a stable closed loop system, the Nyquist plot of \(\mathrm{G}(s) \mathrm{H}(s)\) should encircle \((-1, j 0)\) point as
The gain margin of a unity feedback control system with the open loop transfer function \(\mathrm{G}(s)=\frac{s+1}{s^{2}}\) is(a) 0(b) \(1 / \sqrt{2}\)(c) \(\sqrt{2}\)(d) \(\infty\)
In the \(\mathrm{GH}(s)\) plane, the Nyquist plot of the loop transfer function \(\mathrm{G}(s) \mathrm{H}(s)=\frac{\pi e^{-0.25 s}}{s}\) passes through the negative real axis at the point(a)
If the compensated system shown in figure has a phase margin of \(60^{\circ}\) at the crossover frequency of \(1 \mathrm{rad} / \mathrm{sec}\), the value of the gain \(\mathrm{K}\) is(a) 0.366(b)
The polar diagram of a conditionally stable system for open loop gain \(K=1\) is shown in figure. The open loop transfer function of the system is known to be stable. The closed loop system is stable
The radius of constant \(\mathrm{N}\) circle of \(\mathrm{N}=1\), is(a) 2(b) \(\sqrt{2}\)(c) 1(d) \(1 / \sqrt{2}\).
A constant \(\mathrm{M}\) circle is described by equation\[x^{2}+2.25 x+y^{2}=-11.25\]where \(x=\operatorname{Re}[\mathrm{G}(j \omega)]\) and \(y=\operatorname{Im}[\mathrm{G}(j \omega)]\). The value
A constant \(\mathrm{N}\) circle has centre at \(-\frac{1}{2}+j 0\) in \(\mathrm{G}(j \omega)\) plane. It represents phase angle equal to(a) \(180^{\circ}\)(b) \(90^{\circ}\)(c) \(45^{\circ}\)(d)
The \(\mathrm{N}\) loci is described by equation\[x^{2}+x+y^{2}=0 \text { where } x=\operatorname{Re}[\mathrm{G}(j \omega)] \text { and } y=\operatorname{Im}[\mathrm{G}(j \omega)] \text {. }\]The
The root locus of a unity feedback system is shown in figure below. For design value of gain \(\mathrm{K}=8\), the root locations are shown by small square. The gain margin of system is(a) 2(b) 4(c)
Consider the following plots.The plots which represent marginally stable systems, would include(a) 1 and 2(b) 3 and 4(c) 1 and 3(d) 2 and 4. 1. |G| 0 dB ZG -180 (0 ge pe co (log scale) 2. |G| 0 dB ZG
The Bode plots of an open loop transfer function of a control system are shown in figure below. The gain margin of the system is(a) \(\mathrm{K}\)(b) \(-\mathrm{K}\)(c) \(1 / \mathrm{K}\)(d) \(-1 /
The \(\mathrm{dB}\) (Bode plot) of transfer function \(\mathrm{G}(\mathrm{s})\) is shown in figure below.Now, consider the following statements.I. \(\mathrm{G}(s)\) has corner frequencies at
The function \(\mathrm{A}(f)\) corresponding to Bode plot of shown below is(a) \(\mathrm{A}(f)=j f / f\)(b) \(\mathrm{A}(f)=1 /\left(1-j f_{1} / fight)\)(c) \(\mathrm{A}(f)=1 /\left(1+j f /
The \(\mathrm{dB}\) magnitude-phase angle plot for a typical open loop transfer function is shown below. Gain margin and phase margin respectively are(a) \(4 \mathrm{~dB}, 20^{\circ}\)(b) \(-4
The Bode \(\mathrm{dB}\) plot is shown below. The corresponding transfer function model is(a) \(\frac{10^{4}(1+j \omega)}{(10+j \omega)(100+j \omega)^{2}}\)(b) \(\frac{10^{-1}(1+j \omega)}{(10+j
If \(x=\operatorname{Re}[G(j \omega)]\) and \(y=\operatorname{Im}[\mathrm{G}(j \omega)]\), then for \(\omega ightarrow 0^{+}\), the Nyquist plot for \(\mathrm{G}(s)=1 / \mathrm{s}(s+1)(s+2)\), is(a)
The input-output transfer function of a plant \(\mathrm{H}(s)=\frac{100}{s(s+10)^{2}}\). The plant is placed in a unity negative feedback configuration as shown in the figure below.The gain margin of
The input-output transfer function of a plant \(\mathrm{H}(s)=\frac{100}{s(s+10)^{2}}\). The plant is placed in a unity negative feedback configuration as shown in the figure below.The signal flow
Construct simulation diagram in phase variable form for systems with following transfer functions and develop state space model in matrix form.(a)
Construct simulation diagram in dual phase variable form for systems with following transfer functions and develop state space model in matrix form.(a)
Develop state space model for the system with signal flow graph shown in Fig. P7.3. R(s) o R(s) o 1 - 64 4/(s + 2) 10 2+4 1 1/(s + 1) 1 -OY, (s) -0 Y(s)
Find diagonal state equations for a system with transfer function\[\mathrm{T}(s)=\frac{\mathrm{Y}(s)}{\mathrm{R}(s)}=\frac{s^{3}+18 s^{2}+50 s+50}{(s+3)(s+4)(s+5)}\]
The SISO system with transfer functioninvolves complex characteristic roots. Obtain the diagonal form of state space model. Also obtain an alternative block diagonal model which does not involve
The SISO system with repeated characteristic roots is described by transfer function\[\mathrm{T}(s)=\frac{\mathrm{Y}(s)}{\mathrm{U}(s)}=\frac{7 s^{3}}{(s+2)^{2}(s+6)^{2}}\]Find state space model in
Construct state space model for the mechanical system shown in Fig. P7.7. N oooooo 000000 4 Control force U 5 3.5 Y Ty 2.5
Find characteristic equation for each of the following systems. Then for each, determine if they are stable.(a) \(\left[\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\
The following transfer functions do not share a common denominator polynomial, but they may be made to do so by multiplying their numerators and denominators by appropriate factors. Obtain simulation
Find decoupled state equations for the system described as\[\begin{aligned}{\left[\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}ight] } & =\left[\begin{array}{rr}0 & 1 \\-6 &
Obtain controllability and observability matrices and investigate whether or not the following systems are completely controllable and/or completely observable.(a) \(\left[\begin{array}{l}\dot{x}_{1}
Find the state response and system response for the systems described as follows:(a) \(\left[\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2}\end{array}ight]=\left[\begin{array}{ll}-4 & 1 \\ -3 &
A system modelled as˙x(t)=Ax(t)x˙(t)=Ax(t)generates state response for initial vector Missing \end{array} and for Missing \end{array}. Find the system matrix A and state transition matrix
Develop state space model for each of the electrical networks shown below. Investigate if each one of them is completely controllable and/or completely observable. Substantiate the result with
A system using state feedback is governed by the following set of equations:\[\begin{aligned}{\left[\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2} \\\dot{x}_{3}\end{array}ight] } &
A system with state feedback is depicted below in Fig. P7.16. Find the values of \(K_{1}, K_{2}\) and \(\mathrm{K}_{3}\) so that the system satisfies the following performance
For the system given below, an observer is to be designed to estimate the state variables. Select the observer gain and write the equations describing the observer dynamics. Also develop the block
Find diagonal state equations for a system with transfer function\[\mathrm{T}(s)=\frac{\mathrm{Y}(s)}{\mathrm{R}(s)}=\frac{2 s^{2}+3 s-7}{(s+2)\left(s^{2}+13 s+40ight)}\]
Construct phase variable form simulation diagram for the following transfer functions and develop state space model in matrix form(a) \(\mathrm{T}(s)=\frac{\mathrm{Y}(s)}{\mathrm{U}(s)}=\frac{10
Determine the transfer functions for the system modelled as[˙x1˙x2]=[01−3−7][x1x2]+[024−8][u1u2]y=[1−4][x1x2][˙x1˙x2]=[01−3−7][x1x2]+[024−8][u1u2]y=[1−4][x1x2]
Construct dual phase variable form simulation diagram for the following transfer functions and develop state space model in matrix form.(a) \(\frac{\mathrm{Y}(s)}{\mathrm{R}(s)}=\mathrm{T}(s)=\frac{2
Given a system\[\begin{aligned}{\left[\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2} \\\dot{x}_{3}\end{array}ight] } & =\left[\begin{array}{rrr}0 & -2 & 3 \\0 & -4 & -1 \\0 & 1 &
Consider the following SISO system with complex characteristic roots. Develop state space model in matrix form with diagonal state equations. Also find block diagonal model involving real
Obtain controllability and observability matrices and investigate whether or not the following system is completely controllable and/or completely observable.\[\begin{aligned}&
The systems together with inputs and initial conditions are given below. Determine system response in each.(a) \(\quad \dot{x}=-2 x+3 r(t)\)\[y=4 x\]\[\begin{aligned}x(0) & =10 \\r(t) & =5
A system is described by the following signal flow graph. Write state and output equations in matrix form: U(s) c (s + 3) 4/s - 2 5s 2 S +6 -o Y(s)
A system using state feedback control is governed by the following set of equations. Determine feedback gains so as to place the closed loop system poles at \(s=-4\) and \(-4 \pm j
A system is described as follows:whereWhat restrictions should be imposed on α1,α2,β1α1,α2,β1 and β2β2 so that the system is completely controllable and observable. x y = Ax + Bu = Cx
The block diagram of a system together with state variable assignment as labelled therein, is shown in Fig. P7.12. Test controllability and observability. Comment on test result. U 3 1 s+2 1 S X +
For the system with matricesDesign an observer such that the observer eigen values are placed at ( −20,−20−20,−20 ). Develop observer equations and signal flow graph showing interconnection
The state variable description of a single input single output linear system is given bywhereThe system is(a) Controllable and observable(b) Controllable but unobservable(c) Uncontrollable but
Which of the following properties are associated with the state transition matrix \(\phi(t)\) ?1. \(\phi\left(t_{1} / t_{2}ight)=\phi\left(t_{1}ight) \cdot \phi^{-1}\left(t_{2}ight)\)2.
A linear system is described by the state equations\[\begin{aligned}{\left[\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}ight] } & =\left[\begin{array}{ll}1 & 0 \\1 &
Consider the following properties attributed to state model of a system.1. State model is unique.2. State model can be derived from the system transfer function.3. State model can be derived for time
A system is described by the state equation\[\left[\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}ight]=\left[\begin{array}{ll}2 & 0 \\0 & 2\end{array}ight]\left[\begin{array}{l}x_{1}
The state and output equations of a system areThe systems is(a) Neither state controllable nor output controllable(b) State controllable but not output controllable(c) Output controllable but not
The state equation of a linear system is given by \(\dot{x}=\mathrm{A} x+\mathrm{B} u\), where;\[A=\left[\begin{array}{rr}0 & 2 \\-2 & 0\end{array}ight] \text { and } B=\left[\begin{array}{r}0
The state variable description of a linear autonomous system is ˙x=Axx˙=Ax where xx is a state vector andThe poles of the system are located at(a) -2 and +2(b) -2 and -2(c) −2j−2j and +2j+2j(d)
Consider the Laplace transform of state transition matrix;\[\phi(s)=\left[\begin{array}{cc}\frac{s+6}{s^{2}+6 s+5} & \frac{1}{s^{2}+6 s+5} \\\frac{-5}{s^{2}+6 s+5} & \frac{s}{s^{2}+6
Which one of the following is NOT a correct statement about the state-space model of a physical system ?(a) State-space model can be obtained only for a linear system(b) Eigen values of the system
A linear second-order continuous time system is described by the following set of differential equations.\[\begin{aligned}\dot{x}_{1}(t) & =-2 x_{1}(t)+4 x_{2}(t) \\\dot{x}_{2}(t) & =-2
For the system described by the state equation\[\dot{x}=\left[\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\0.5 & 1 & 2\end{array}ight] x+\left[\begin{array}{l}0 \\0 \\1\end{array}ight] u\]if the control
The matrix of any state-space equations for the transfer function \(C(s) / R(s)\) of the system, shown below in figure is(a) \(\left[\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}ight]\)(b)
Given the homogeneous state-space equation \(\dot{x}=\left[\begin{array}{rr}-3 & 1 \\ 0 & -2\end{array}ight] x\). The steady state value of \(x_{s s}=\lim _{t ightarrow \infty} x(t)\), given the
The zero-input response of a system given by(a)(b)(c)(d) 1 0 N-HN = and 1 1 [x(0)] x2 (0) = 0 is:
The state-space representation in phase-variable form for the transfer function\[G(s)=\frac{2 s+1}{s^{2}+7 s+9} \quad \text { is }\](a) \(\dot{x}=\left[\begin{array}{rr}0 & 1 \\ -9 &
Letwhere bb is an unknown constant.This system is(a) observable for all values of bb(b) unobservable for all values of bb(c) observable for all non-zero values of bb(d) unobservable for all non-zero
Consider the following statements with respect to a system represented by its statespace model\[\dot{x}=\mathrm{A} x+\mathrm{B} u \text { and } y=\mathrm{C} x\]1. The state vector \(x\) of the system
A state variable system \(\dot{x}(t)=\left[\begin{array}{rr}0 & 1 \\ 0 & -3\end{array}ight] x(t)+\left[\begin{array}{l}1 \\ 0\end{array}ight] u(t)\), with the initial condition
A state variable system \(\dot{x}(t)=\left[\begin{array}{rr}0 & 1 \\ 0 & -3\end{array}ight] x(t)+\left[\begin{array}{l}1 \\ 0\end{array}ight] u(t)\), with the initial condition
Consider the system \(s_{1}\) modelled as\[\dot{x}=\left[\begin{array}{rr}2 & 0 \\0 & -1\end{array}ight] x+\left[\begin{array}{l}1 \\0\end{array}ight] u\]and system \(s_{2}\) modelled
Consider system \(s_{1}\) modelled as\[\begin{aligned}& \dot{x}=\left[\begin{array}{rr}2 & 0 \\0 & -1\end{array}ight] x+\left[\begin{array}{l}1 \\0\end{array}ight] u \\& y=\left[\begin{array}{ll}1 &
The signal flow graph together with state variable assignment is shown below:The condition for complete state controllability and complete observability is:(a) \(d eq 0\) and \(a, b, c\) can be
A system is described by state transition matrix \(\phi(t)=\left[\begin{array}{cc}e^{-t} & 0 \\ 0 & e^{-2 t}\end{array}ight]\) and has initial conditions \(\left[\begin{array}{l}x_{1}(0) \\
For the PD compensated system shown in Fig. P8.1, compute \(K\) and \(\mathbf{b}\) such that the system exhibits peak overshoot of \(2.5 \%\) and settling time of 0.5 seconds. R(s)- bs + + K s(s+12)
A unity feedback system has open loop transfer function\[G(s)=\frac{K}{(s+3)(s+6)}\]and operates with peak overshoot of \(4.32 \%\). Design a PI compensator via root locus such that the system is
Design a lag-compensator for a unity feedback system with forward transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{(s+1)(s+3)(s+5)}\]to yield the following specifications:(i) peak overshoot \(\leq 10
Design a cascade PD compensator for the unity feedback system with transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{(s+2)^{2}(s+3)}\]so that the following design objectives are met:(i) \% peak
For a system in unity feedback configuration with forward transmittance\[G(s)=\frac{K}{s^{2}},\]a cascade compensator is to be designed to achieve settling time of 1.6 seconds and peak overshoot of
Design PID compensator for the system with pole-zero function\[G(s)=\frac{K}{(s+3)(s+6)}\]in unity feedback configuration so as to meet the following specifications:Peak overshoot \(\leq 1.18
A system in unity feedback configuration, has the transmittance\[G(s)=\frac{\mathrm{K}}{s(s+3)(s+9)}\](a) What value of \(\mathrm{K}\) will force the system to exhibit peak overshoot of \(20 \%\) to
A system with transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+6)^{2}}\]is placed in rate feedback control organisation as shown in Fig. P 8.8. Find \(\mathbf{K}\) and \(\mathbf{b}\) so as to meet
A feedback compensated system is shown below. Do the following:(a) Find the value of \(\mathbf{a}\) and \(\mathbf{b}\) in minor feedback loop so as to achieve the settling time of 1 second with \(5
Use frequency response methods to design a lag-compensator for a system in unity feedback configuration, with open loop transmittanceThe design goals are\[\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+1)(0.2
A unity feedback system has the loop transmittance\[\mathrm{G}(s)=\frac{1000 \mathrm{~K}}{s(s+40)(s+100)}\]Design a lead compensator so as to achieve the following specifications:(i) Peak overshoot
Use frequency response approach to design a lag-lead compensator for a unity feedback system where the loop transmittance is\[\mathrm{G}(s)=\frac{\mathrm{K}(s+8)}{s(s+4)(s+20)}\]The design goals
A system in unity feedback configuration, has the pole-zero function \(\frac{K}{s(s+50)(s+100)}\)Design the value of gain \(\mathrm{K}\) for \(15 \%\) peak overshoot in closed loop step response
Design a PI compensator for the position control system shown in Fig. P 8.15, to achieve the following design goals:(i) Steady state error for ramp input =0=0(ii) Peak overshoot for step input
A unity feedback system with loop transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+5)(s+20)}\]operates with approximately \(55 \%\) peak overshoot and 0.5 seconds peak time when the gain
A system in unity feedback configuration, has the transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{(s+3)(s+9)(s+15)}\]Use frequency response methods to determine the value of K. Such that system is
A system is placed in a feedback organisation as shown in Fig. D8.2. Design a lead compensator so as to meet the following design goals:(i) Error coefficient \(\mathrm{K}_{v}=2\)(ii) Phase margin
The open loop transfer function of a system in unity feedback configuration, is\[\mathrm{G}(s)=\frac{\mathrm{K}}{(s+1)(s+2)(s+10)}\]Use the root locus approach to do the following:(a) Determine the
Use the root locus approach to design a lag compensator for the system of drill problem D8.3 to achieve the following design goals:(i) Peak overshoot ≤57.5%≤57.5%(ii) The steady state error for a
A system in unity feedback configuration, has the loop transmittance\[\mathrm{G}(s)=\frac{\mathrm{K}}{s(s+4)(s+6)}\]Use the root locus methods and do the following:(a) Sketch root locus and find the
Design a lead compensator while using root locus for the system of drill problem D8.5, to meet the design specifications as follows:(i) Peak overshoot \(\leq 30 \%\)(ii) Settling time

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