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systems analysis design
Questions and Answers of
Systems Analysis Design
The open-loop transfer function of a control system is as follows:Determine the value of \(k\) so that the closed-loop system has an underdamped response to a unit step input. Moreover, determine the
In the control system, shown in Fig. 5.4 , determine the value of " \(k_{1}\) " and " \(k_{2}\)," so that the damping ratio and the settling time ( \(5 \%\) criterion) of the closed-loop system are
In the control system, shown in Fig. 5.5 , determine the value of " \(k_{1}\) " and " \(k_{2}\)," so that the settling time ( \(2 \%\) criterion) and the peak time of the closed-loop system are
The differential equation of a control system with the zero-primary condition is as follows:Determine the time that the second peak in the system response occurs if the input is \(r(t)=4 u(t)\).1)
A unit step function \((f(t)=u(t))\) is applied on the mechanical system shown in Fig. 5.6 .1. The output is the horizontal position of the mass which is illustrated in Fig. 5.6 .2. Determine the
Figure 5.7 illustrates the unit step response of a second-order control system. Determine its approximate closed-loop transfer function.1) \(\frac{240}{s^{2}+136 s+240}\)2) \(\frac{240^{2}}{s^{2}+136
Consider the system shown in Fig. 7.1 . Determine the type of the system and the steady-state error of the closed-loop control system to a unit ramp function. Assume that the closed-loop system is
In the closed-loop control system shown in Fig. 7.2 , calculate the steady-state error to a unit step function.1) \(\infty\)2) 5 3) \(\frac{1}{6}\)4) 0 Figure 7.2 E(s) 10 1 R(s) Y(s) (s+1)(s+2)
Determine the static error constant to a unit ramp function if the transfer function of the closed-loop control system, shown in Fig. 7.3 , is as follows:1) \(k_{v}=\frac{3}{5}\)2)
Determine the steady-state error to a unit step function if the closed-loop control system includes a negative unity feedback and the open-loop transfer function is as follows:1) 0 2) 40 3)
Determine the steady-state error to a unit ramp function if the closed-loop control system includes a negative unity feedback and the open-loop transfer function is as follows:1) 0 2) \(\infty\)3)
The open-loop transfer function of a control system with a negative unity feedback is as follows:Determine its minimum steady-state error to a unit step function.1) 0.1 2) 0 3) 1 4) \(\infty\)
Consider the control system shown in Fig. 7.4 . Determine the steady-state error resulted from the input of \(R(s)\) and the noise of \(N(s)\) that all are unit step functions.1) Zero, zero 2)
Consider the control system shown in Fig. 7.5 . Determine the total steady-state error resulted from the input of \(R(s)\), which is a unit ramp function, and the disturbance of \(D(s)\), which is a
Figure 9.1 illustrates the root locus of a control system with a negative unity feedback (for k > 0). Determine the maximum value of loop gain so that the closed-loop system is stable.1) 1 2)
Consider the problem of 9.1 and assume that the system is in the oscillating status. Determine the angular frequency of the oscillations.1) \(1 \mathrm{rad} / \mathrm{sec}\)2) \(\sqrt{2} \mathrm{rad}
Figure 9.3 shows the root locus \((k>0)\) of the control system shown in Fig. 9.2 . Determine the value of loop gain where the root locus crosses \(j \omega-\) axis.1) 16 2) 160 3) 1.6 4) 0
If the root locus of the control system of Fig. 9.4 passes from the points of \(-1 \pm j\), determine the value of the parameters \(a\) and \(b\).1) 5,4 2) 5,3 3) 3,4 4) 3,3 Figure 9.4 R(s)
In a control system with a negative unity feedback and the open-loop transfer function below, if the parameter of \(\tau\) increases about \(10 \%\), which one of the following choices is correct?1)
Which one of the following choices shows the root locus of the control system of Fig. 9.5 for \(k>0\) ?Figure 9.5 R(s)- k C(s)
Which one of the following choices illustrates the root locus of the control system shown in Fig. \(9.7(k>0)\) ?Figure 9.7 R(s) C k x|S +C(s)
The root locus of a control system is shown in Fig. 9.9 . Determine the stability status of the system for \(k=40\).1) Stable.2) Unstable.3) Marginally stable.4) Its stability depends on the other
Which one of the following options shows the root locus of a closed-loop control system (for \(k>0\) ) with a negative unity feedback and the open-loop transfer function below? G(s) = s-k s(s+1)
Consider a closed-loop control system with a negative unity feedback and the open-loop transfer function below \((k>0)\). If the settling time of the system for the large \(k\) is about eight
Consider a closed-loop control system with a negative unity feedback and the open-loop transfer function below \((k>0)\). Which one of the following statements is correct and complete?1) The
Consider a closed-loop control system with a negative unity feedback and the open-loop transfer function below \((k>0)\). Determine the range of \(p\), so that the transient response of the
Which one of the following choices shows the root locus of the control system \((kFigure 9.11 R(s) k 4 s+2 -13 S C(s)
In a control system with a negative unity feedback, the open-loop transfer function is as follows:Determine the sensitivity of the maximum overshoot percentage of the closed-loop system's response to
Consider a closed-loop control system with a negative unity feedback and the open-loop transfer function below \((k>0)\). Determine the characteristic equation of the closed-loop system if its
The state-transition matrix of a closed-loop control system with a negative unity feedback is as follows:Herein, \(k\) is the forward gain of the system. Determine the break-away/break-in point and
The open-loop transfer function of a control system with a negative unity feedback and a proportional-derivative (PD) controller is as follows:Determine the parameters of the controller, so that the
The open-loop transfer function of the control system, shown in Fig. 11.1 , is as follows:Design a proportional controller, in the form of \(G_{c}(s)=k_{p}\), so that the damping ratio of the
The open-loop transfer function of a control system that includes a negative unity feedback is as follows:Determine the proportional controller gain \(\left(k_{P}ight)\) for a
The open-loop transfer function of a control system is as follows:Determine the integral time constant \(\left(T_{I}ight.\) ) for a proportional-integral-derivative (PID) controller in
The open-loop transfer function of a control system is as follows:Design a controller \(\left(G_{c}(s)ight)\) in the feedback structure, so that \(-1 \pm j 2\) are the closed-loop poles of the
Which type of the controllers below must be used in a closed-loop control system, with the following open-loop transfer function and a negative unity feedback, to set the undamped natural frequency
The open-loop transfer function of a control system that includes a negative unity feedback is as follows:The uncontrolled closed-loop system response has the overshoot and settling time of \(16
The open-loop transfer function of the control system, which is shown in Fig. 11.2 , is as follows:Figure 11.2Design a proportional-integrate (PI) controller, in the form of
Determine the characteristic equation of a control system with the block-diagram shown in Fig. 1.1 .1) \(1+G_{2} H_{2}-G_{1} G_{2} G_{3} H_{1} H_{2}\)2) \(1+G_{1} G_{2} H_{2}-G_{1} G_{2} G_{3} H_{1}
Figure 1.2 illustrates the signal flow graph (SFG) of a control system. Determine its transfer function.1) \(\frac{G_{1}+G_{2}}{1-G_{2} H+G_{1} G_{2}}\)2) \(\frac{G_{1}+G_{2}}{1+G_{2} H-G_{1}
In the block-diagram, shown in Fig. 1.3 , determine the transfer function of \(\frac{Y(s)}{X(s)}\).1) \(\frac{5}{s(s+2)}\)2) \(\frac{5}{s^{2}+22 s+5}\)3) \(\frac{4 s+1}{s^{2}+22 s+5}\)4)
The state equations of a LTI control system, which is in zero-state, are as follows. Determine the steady-state value of its output.1) \(\left[\begin{array}{r}-\frac{1}{8} \\
In the block-diagram shown in Fig. 1.4 , determine the value of \(k_{1}, k_{2}, k_{3}\), so that the transfer function is as follows:1) \(k_{1}=1, k_{2}=\frac{2}{3}, k_{3}=6\)2) \(k_{1}=\frac{2}{3},
The differential equation of a control system is as follows:Determine the state and output equations of the system in the matrices form.1) \(\frac{d}{d t}\left[\begin{array}{l}x_{1} \\ x_{2} \\
Determine matrix \(\mathbf{A}\) in the state equations \((\dot{\mathbf{X}}=\mathbf{A X}+\mathbf{B} u)\) for the block-diagram of Fig. 1.5 if \(\mathbf{X}=\left[\begin{array}{l}x_{1}(t) \\
Determine the transfer function of a control system with the following state equations:1) \(\frac{m}{s^{2}+b s+k}\)2) \(\frac{k}{b s^{2}+m s+k}\)3) \(\frac{b}{m s^{2}+b s+k}\)4) \(\frac{1}{m s^{2}+b
The state equations of a control system are as follows. Determine the state-transition matrix of the system \((\boldsymbol{\varphi}(t))\).1) \(\left[\begin{array}{cc}(1+t) e^{-2 t} & t e^{-2 t}
Consider the LTI control system below.1) \(e^{-t}+e^{-2 t}\)2) \(e^{-t}+2 e^{-2 t}\)3) \(e^{-t}+1.5 e^{-2 t}\)4) \(1.5 e^{-t}+e^{-2 t}\) X = AX, y=CX Determine the output of the system based on the
Determine the state equations of the control system shown in Fig. 1.6.1) \(\frac{d}{d t}\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}ight]=\left[\begin{array}{ccc}-1 & -1 & 0 \\ 0
In the rotational mechanical system shown in Fig. 1.7, determine the transfer function of \(\frac{\theta_{2}(s)}{T(s)}\).1) \(\frac{J_{1} s^{2}+k}{s^{2}\left(J_{1} J_{2}
The equation below shows the characteristic equation of a closed-loop control system. Determine its stability status.1) The system is stable.2) The system has one unstable root.3) The system has
Which one of the transfer functions below has a non-zero primary time response?1) \(\frac{1}{s^{2}+2 s+2}\)2) \(\frac{s}{s^{2}+2 s+2}\)3) \(\frac{s+1}{s^{2}+2 s+2}\)4) \(\frac{s^{2}+2 s+1}{s^{2}+2
Which one of the following choices is correct about a closed-loop control system with the characteristic equation of \(4 s^{3}+2 s^{2}+k s+1=0\) ?1) For \(k=2\), it oscillates with the angular
The open-loop transfer function of a control system with a negative unity feedback is as follows:For what value of \(k\), does the closed-loop system response oscillate?1) -15 2) 15 3) 34 4) 64
Determine the period of oscillations of the closed-loop control system's response illustrated in Fig. 3.1 .1) \(11.2 \mathrm{sec}\)2) \(6.5 \mathrm{sec}\)3) \(2.2 \mathrm{sec}\)4) \(2 \sqrt{10}
The differential equations of a control system are as follows:For what value of " \(a\) " and " \(b\) ", the system is stable?1) \(a b \geq 0\)2) \(a>0, b3) \(a>0, b=0\)4) \(a (x(t) +x(t) 2u(t)
The state equations of a control system are as follows. For what value of " \(k\) ", the system is stable?1) \(k>-2\)2) \(k>-1\)3) \(-24) \((-2,-1.5) \cup(-1, \infty)\) (t) = -k-2 k+1
In the control system shown in Fig. 3.2 , determine the range of " \(p\) ", so that the system is stable.1) \(p>0\)2) \(p>-1\)3) \(-34) \(-3Figure 3.2 1 R(s)- Y(s). s+1 s+p
For a control system with the signal flow graph (SFG), shown in Fig. 3.3 , and the transfer function of \(\frac{C(s)}{R(s)}\), which one of the following choices is correct?1) The system is always
For the control system, shown in Fig. 3.4 , determine the hidden modes of the system.1) \(1, \pm j\)2) 0 3) \(-1,0\)4) 1,0 Figure 3.4 X(s) 2+1 (s+2) Y(s)
For what range of \(k\), the control system, shown in Fig. 3.5 , is stable?1) \(-\frac{4}{3}2) \(-\frac{1}{3}3) \(-\frac{5}{3}4) \(-\frac{2}{3}Figure 3.5 X(s)- k Y(s) S s-1 s2+2s+1
The equation below shows the characteristic equation of a control system. How many unstable poles does it have?1) 1 2) 2 3) 3 4) 0 +s+53 +5s + 12s+ 100
The differential equations of a control system are as follows:For what value of " \(a\) " the system is stable?1) \(a2) \(a>-2\)3) \(-24) \(1 xi(t) = axi(t) +xz(t)+u(t) x2(t) = -2x1(t) +x2(t)
Determine the transfer function of \(\frac{C(s)}{R(s)}\) for the control system, shown in Fig. 3.6 . Is this system internally stable or unstable?1) \(\frac{3}{(s+1)(s+2)}\), stable 2)
In the control system, shown in Fig. 3.7 , the controller is in the form of \(G_{c}(s)=k_{P}+\frac{k_{l}}{s}\). Which one of the choices, illustrated in Fig. 3.8 , graphically shows the stability
For a control system with a negative unity feedback and the following open-loop transfer function, which one of the choices, shown in Fig. 3.9 , graphically shows the stability area of both
Use the defining integral to find the Fourier transform of \(f(t)=A[u(t+1)-u(t-2)]\).
(a) Use the defining integral to find the Fourier transform of the following waveform:\[f(t)=A[u(t+10)-u(t-10)]\](b) Use the MATLAB function fourier to find the same transform.
Use MATLAB and the defining integral to find the Fourier transform of the following waveform:\[f(t)=5 \pi \cos (\pi t / 4)[u(t)-u(t-8)]\]
Use the inversion integral to find the inverse transform of the following function:\[F(\omega)=10 \pi[u(\omega+10)-u(\omega-10)]\]
Use MATLAB and the inversion integral to find the inverse transform of the following function:\[F(\omega)=\cos (\pi \omega / 4)[u(\omega+4)-u(\omega-4)]\]
First find the transforms of the following functions. Then determine what type of characteristics they possess.(a) \(f_{1}(t)=100 e^{-100 t} u(t)\)(b) \(f_{2}(t)=1.25\left[e^{-100 t}-e^{-500 t}ight]
Find the inverse transforms of the following functions:(a) \(F_{1}(\omega)=\frac{10,000}{j \omega(j \omega+100)(j \omega+1000)}\)(b) \(F_{2}(\omega)=\frac{-10 \omega^{2}}{j \omega(j \omega+20)(j
Use MATLAB's fourier function to find the Fourier transforms of the following waveforms:(a) \(f_{1}(t)=3 u(-t)-3\)(b) \(f_{2}(t)=-\operatorname{sgn}(t)-u(-t)\)(c) \(f_{3}(t)=\operatorname{sgn}(t)+1\)
Find the Fourier transforms of the following waveforms:(a) \(f_{1}(t)=\frac{10}{j}\left(e^{i 2 t}-e^{-j 2 t}ight)+10\left(e^{j i t}+e^{-j 2 t}ight)\)(b) \(f_{2}(t)=\frac{10}{t}(\sin 5 t)\)
Find the Fourier transforms of the following waveforms:(a) \(f_{1}(t)=100 \sin [2 \pi(t-10)]\)(b) \(f_{2}(t)=5 e^{j 10 t} \operatorname{sgn}(t)\)
Find the inverse transforms of the following functions:(a) \(F_{1}(\omega)=6 \pi \delta(\omega)+6 \pi \delta(\omega-3)+6 \pi \delta(\omega-4)\)(b) \(F_{2}(\omega)=4 \pi \delta(\omega)-j 6 / \omega+4
Use the duality property to find the inverse transforms of the following functions:(a) \(F_{1}(\omega)=50 \cos (100 \omega)\)(b) \(F_{2}(\omega)=10 u(\omega)-5\)(c) \(F_{3}(\omega)=6 e^{-|2 \omega|}\)
Use the time-shifting property to find the inverse transforms of the following functions:(a) \(F_{1}(\omega)=[6 \pi \delta(\omega) j 6 / \omega] e^{-j 5 \omega}\)(b) \(F_{2}(\omega)=50 e^{-j 4
Given that the Fourier transform of \(f(t)\) is\[F(\omega)=\frac{100,000}{(j \omega+500)(j \omega+1000)}\]Use the integration property to find the waveform\[g(t)=\int_{-\infty}^{t} f(x) d x\]
Use the reversal property to show that\[\mathscr{F}\left\{A e^{-\alpha t \mid} \operatorname{sgn}(t)ight\}=\frac{-2 A j \omega}{\omega^{2}+\alpha^{2}}\]
Use the frequency shifting property to show that\[\mathscr{F}\{\cos (\beta t) u(t)\}=\frac{j \omega}{\beta^{2}-\omega^{2}}+\frac{\pi}{2}[\delta(\omega-\beta)+\delta(\omega+\beta)]\]
The input in Figure P18-17 is \(v_{1}(t)=5 e^{-|t|} \mathrm{V}\). Use Fourier transforms to find \(v_{2}(t)\). V(t) + 0.1 uF + 10 V(1)
The input in Figure P18-18 is \(v_{1}(t)=1 \operatorname{Osgn}(t)\) V. Use Fourier transforms to find \(v_{2}(t)\). V(t) +1 10 www 0.1 F 10 V2(t)
The input in Figure P18-19 is \(v_{1}(t)=10 e^{10 t} u(-t)\) V. Use Fourier transforms to find \(v_{2}(t)\). 1 w + V(t) + 1 F 1 V2(1)
The input in Figure P18-19 is \(v_{1}(t)=2 \operatorname{sgn}(t) \mathrm{V}\). Use Fourier transforms to find \(v_{2}(t)\). V(t) 1 ww + 1 + 1 F: 1 v2(1) 19
(a) The input in Figure P18-21 is \(v_{1}(t)=3 u(-t) \mathrm{V}\). Use Fourier transforms to find \(v_{2}(t)\).(b) Will the OP AMP saturate? 1 0.1 F ww + V(t) 10 w + V2(t) Vcc = 15 V
The input in Figure P18-22 is \(v_{1}(t)=3 e^{-2|t|} u(-t) \mathrm{V}\). Use Fourier transforms to find \(v_{2}(t)\). 0.1 F 2 10 w ww + + V(1) + V(1)
The input in Figure P18-23 is \(i_{1}(t)=10 e^{-5|t|} \mathrm{mA}\). Use Fourier transforms to find \(i_{2}(t)\). i2(t) i(t) 0.1 uF 10 mH 1
The impulse response of a linear system is \(h(t)=5 e^{-3 t}\) \(u(t)\). Find the output for an input \(x(t)=u(-t)\).
The impulse response of a linear system is \(h(t)=e\) \(-2|t|\). Find the output for an input \(x(t)=u(-t)\).
The impulse response of a linear system is \(h(t)=\delta(t\) )\(-5 e^{-2 t} u(t)\). Use MATLAB and Fourier transform techniques to find the output for an input \(x(t)=\operatorname{sgn}(-t)\).
The impulse response of a linear system is \(h(t)\) \(=A\left[\delta(t)-\alpha e^{-\alpha t} u(t)ight]\), with \(\alpha>0\). Let \(A=10\) and \(\alpha=3\) and use MATLAB to plot \(|H(\omega)|\). On
The impulse response of a linear system is \(h(t)=A\) \([\delta(t)-\sin (\beta t) / \pi t]\). Let \(A=5\) and \(\beta=2\) and use MATLAB to plot \(|H(\omega)|\). On the same axes, plot
Use MATLAB's ifourier function to find the system impulse response \(h(t)\) if the frequency response of a linear system is shown in Figure P18-29. 2 H(0) 1 -2 - 0 + +2 3
Find the \(1-\Omega\) energy carried by the signal \(F(\omega)=25 /\left(\omega^{2}ight.\) \(+625)\).
Compute the \(1-\Omega\) energy carried by the signal \(f(t)=\) \(9 e^{4.5^{t}} u(-t)\).
Find the \(1-\Omega\) energy carried by the signal\[F(\omega)=\frac{j \omega A}{\omega^{2}+\alpha^{2}}\]Then, find the percentage of the \(1-\Omega\) energy carried in the frequency band \(|\omega|
The impulse response of a filter is \(h(t)=3 e^{-200}\) \({ }^{t} u(t)\). Find the \(1-\Omega\) energy in the output signal when the input is \(x(t)=4 e^{-20 t} u(t)\). Verify your result using
The impulse response of a filter is \(h(t)=50 e^{-20 t} u(t)\). Find the \(1-\Omega\) energy in the output signal when the input is \(x(t)\) \(=u(t)\).
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