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systems analysis and design
Questions and Answers of
Systems Analysis And Design
Find the inverse of the following Matrices: $\begin{array}{lll}3 & 2 & 1\end{array}$ $\begin{array}{lll}1 & 6 & 3\end{array}$ (a) $\boldsymbol{A}=\begin{array}{lll}-1 & 5 & 4\end{array}$ (b)
Find the inverse of the following matrices using MATLAB. $\begin{array}{lll}3 & 2 & 0\end{array}$ $\begin{array}{lll}-4 & 2 & 5\end{array}$ $\begin{array}{lll}-1 & 2 & -5\end{array}$ (a) 24 (b)
Determine the eigenvalues and eigenvectors of the following matrices using MATLAB. 1-2 A = B = 1 5 1 5 -27
Use MATLAB to determine the following:(a) the three eigenvalues of $\boldsymbol{A}$(b) the eigenvectors of $\boldsymbol{A}$(c) Show that $\boldsymbol{A Q}=\boldsymbol{Q} \boldsymbol{d}$, where
Determine eigenvalues and eigenvector of $\mathbf{A}$ using MATLAB. (a) $\boldsymbol{A}=\begin{array}{rr}0.5 & -0.8 \\ 0. 75 & 1. 0\end{array}$ (b) $\boldsymbol{A}=\begin{array}{rr}8 & 3 \\ -3 &
Determine the eigenvalues and eigenvectors of the following matrices using MATLAB. (a) $\boldsymbol{A}=\begin{array}{rr}1 & -2 \\ 1 & 3\end{array}$ (b) $\boldsymbol{A}=\begin{array}{rr}1 & 5 \\ -2
Determine the eigenvalues and eigenvectors of $A * B$ using MATLAB.\[\mathrm{A}=\begin{array}{rrrrr}3 & -1 & 2 & 1 & \mathrm{~B}= \\1 & 2 & 7 & 4 \\7 & -1 & 8 & 6\end{array} \quad
Determine the eigenvalues and eigenvectors of the following matrices using MATLAB. (a) $\boldsymbol{A}=\begin{array}{rr}1 & -2 \\ 1 & 3\end{array}$ (b) $\boldsymbol{A}=\begin{array}{rr}1 & 5 \\ -2 &
Determine the eigenvalues and eigenvectors of A and B using MATLAB $\begin{array}{lll}4 & 5 & -3\end{array}$ $\begin{array}{lll}1 & 2 & 3\end{array}$ (a) $\boldsymbol{A}=\begin{array}{lll}-1 & 2 &
Determine the eigenvalues and eigenvectors of $A=a^{*} b$ using MATLAB. 0 1 2 3 6 -3 4 1 4 56 -1 0 4 2 6 a = b = = 1 5 4 2 1 3 8 5 2 -36 7 2 2 1 4
Determine the values of $x, y$, and $\mathrm{z}$ for the following set of linear algebraic equations:\[\begin{aligned}x_{2}-3 x_{3} & =-7 \\2 x_{1}+3 x_{2}-x_{3} & =9 \\4 x_{1}+5 x_{2}-2 x_{3} &
Determine the values of $x, y$, and $z$ for the following set of linear algebraic equations:(a) $2 x+y-3 z=11$$4 x-2 y+3 z=8$$-2 x+2 y-z=-6$(b) $2 x-y=10$$-x+2 y-z=0$$-y+z=-50$
Solve the following set of equations using MATLAB.(a) $2 x_{1}+x_{2}+x_{3}-x_{4}=12$$x_{1}+5 x_{2}-5 x_{3}+6 x_{4}=35$$-7 x_{1}+3 x_{2}-7 x_{3}-5 x_{4}=7$$x_{1}-5 x_{2}+2 x_{3}+7 x_{4}=21$(b)
Solve the following set of equations using MATLAB.(a) $2 x_{1}+x_{2}+x_{3}-x_{4}=10$$x_{1}+5 x_{2}-5 x_{3}+6 x_{4}=25$$-7 x_{1}+3 x_{2}-7 x_{3}-5 x_{4}=5$$x_{1}-5 x_{2}+2 x_{3}+7 x_{4}=11$(b)
Solve the following set of equations using MATLAB.(a) $x_{1}+2 x_{2}+3 x_{3}+5 x_{4}=21$\[\begin{aligned}& -2 x_{1}+5 x_{2}+7 x_{3}-9 x_{4}=17 \\& 5 x_{1}+7 x_{2}+2 x_{3}-5 x_{4}=23 \\& -x_{1}-3
Generate a plot of\[y(x)=e^{-0.7 x} \sin \omega x\]where $\omega=15 \mathrm{rad} / \mathrm{s}$, and $0 \leq x \leq 15$. Use the colon notation to generate the $x$ vector in increments of 0. 1 .
Plot the following functions using MATLAB. (a) $r^{2}=5 \cos 3 t$ $0 \leq t \leq 2 \pi$ (b) $r^{2}=5 \cos 3 t$ $0 \leq t \leq 2 \pi$ $x=r \cos t, y=r \sin t$ (c) $y_{1}=e^{-2 x} \cos x \quad 0 \leq x
Use MATLAB for plotting 3. D data for the following functions:(a) $z=\cos x \cos y e^{-\sqrt{\frac{x^{2}+y^{2}}{5}}}|x| \leq 7,|y| \leq 7$(b) Discrete data plots with stems$x=t, \quad y=t \cos
Obtain the plot of the points for $0 \leq t \leq 6 \pi$ when the coordinates $x, y$, and $z$ are given as a function of the parameter $t$ as follows:\[\begin{aligned}& x=\sqrt{t} \sin (3 t) \\&
Obtain the mesh and surface plots for the function $z=\frac{2 x y^{2}}{x^{2}+y^{2}}$ over the domain $-2 \leq x \leq 6$ and $2 \leq y \leq 8$.
Plot the function $z=2^{-1.5 \sqrt{x^{2}+y^{2}}} \sin (x) \cos (0.5 y)$ over the domain $-4 \leq x \leq 4$ and $-4 \leq y \leq 4$.(a) Mesh plot(b) Surface plot(c) Mesh curtain plot(d) Mesh and
Plot the function $y=|x| \cos (x)$ for $-200 \leq x \leq 200$.
Plot the following functions on the same plot for $0 \leq x \leq 2 \pi$ using the plot function:(a) $\sin ^{2}(x)$(b) $\cos ^{2} x$(c) $\cos (x)$
(a) Generate an overlay plot for plotting three lines\[\begin{aligned}& y_{1}=\sin t \\& y_{2}=t \\& y_{3}=t-\frac{t^{3}}{3 !}+\frac{t^{5}}{5 !}+\frac{t^{7}}{7 !} \quad 0 \leq t \leq 2
(a) Plot the parametric space curve of\[\begin{aligned}& x(t)=t \\& y(t)=t^{2} \\& z(t)=t^{3}\end{aligned}\](b) $z=\frac{-7}{1+x^{2}+y^{2}}|x| \leq 10,|y| \leq 10$
(a) Plot the parametric space curve of\[ \begin{aligned} & x(t)=t \\ & y(t)=t^{2} \\ & z(t)=t^{3} \quad 0 \leq t \leq 3. 0\end{aligned}\](b) $z=-7 /\left(1+x^{2}+y^{2}\right) \quad|x| \leq 10,|y|
Perform the following symbolic operations using MATLAB. Consider the given symbolic expressions have been defined.\[\begin{aligned}& S 1=' 2 /(x-5)^{\prime} ; \\& S 2=x^{\wedge} x^{\wedge} 5+9^{*}
Solve the following equations using symbolic mathematics.(a) $x^{2}+9=0$(b) $x^{2}+5 x-8=0$(c) $x^{3}+11 x^{2}-7 x+8=0$(d) $x^{4}+11 x^{3}+7 x^{2}-19 x+28=0$(e) $x^{7}-8 x^{5}+7 x^{4}+5 x^{3}-8 x+9=0$
Determine the values of $\mathrm{x}, \mathrm{y}$, and $\mathrm{z}$ for the following set of linear algebraic equations:\[\begin{aligned}2 x+y-3 z & =11 \\4 x-2 y+3 z & =8 \\-2 x+2 y-z &
Determine the values of $x, y$, and $z$ for the following set of linear algebraic equations:\[\begin{aligned}& 2 x-y=10 \\& -x+2 y-z=0 \\& -y+z=-50\end{aligned}\]
Determine the solutions of the following first-order ordinary differential equations using MATLAB's symbolic mathematics.(a) $y^{\prime}=8 x^{2}+5$ with initial condition $y(2)=0.5$.(b) $y^{\prime}=5
(a) Given the differential equation\[\frac{d^{2} x}{d t^{2}}+7 \frac{d x}{d t}+5 x=8 u(t) \quad t \geq 0\]Using MATLAB program, find(i) $x(t)$ when all the initial conditions are zero(ii) $x(t)$ when
For the following differential equations, use MATLAB to find $x(t)$ when $(a)$ all the initial conditions are zero, $(b) x(t)$ when $x(0)=1$ and $\dot{x}(0)=-1$.(a) $\frac{d^{2} x}{d t^{2}}+10
Obtain the first and second derivatives of the following functions using MATLAB's symbolic mathematics.(a) $F(x)=x^{5}-8 x^{4}+5 x^{3}-7 x^{2}+11 x-9$(b) $F(x)=\left(x^{3}+3
Determine the values of the following integrals using MATLAB's symbolic functions.(a) $\int 5 x^{7}-l x^{5}+3 x^{3}-8 x^{2}+7$(b) $\int \sqrt{x} \cos x$(c) $\int x^{2 / 3} \sin ^{2} 2 x$(d)
Given the differential equation\[\frac{d^{2} x}{d t^{2}}+3 \frac{d x}{d t}+x=98 \quad t \geq 0\]Using MATLAB program, find(a) $x(t)$ when all the initial conditions are zero(b) $x(t)$ when $x(0)=0$
Determine the inverse of the following matrix using MATLAB.\[A=\begin{array}{rrr}3 s & 2 & 0 \\7 s & -s & -5 \\3 & 0 & -3 s\end{array}\]
Expand the following function $F(s)$ into partial fractions with MATLAB:\[F(s)=\frac{5 s^{3}+7 s^{2}+8 s+30}{s^{4}+15 s^{3}+62 s^{2}+85 s+25}\]
Determine the Laplace transform of the following time functions using MATLAB.(a) $f(t)=u(t+9)$(b) $f(t)=e^{5 t}$(c) $f(t)=(5 t+7)$(d) $f(t)=5 u(t)+8 e^{7 t}-12 e^{-8 t}$(e) $f(t)=e^{-t}+9 t^{3}-7
Determine the inverse Laplace transform of the following rotational function using MATLAB.\[F(s)=\frac{7}{s^{2}+5 s+6}=\frac{7}{(s+2)(s+3)}\]
Determine the inverse transform of the following function having complex poles\[F(s)=\frac{15}{\left(s^{3}+5 s^{2}+11 s+10\right)}\]
Determine the inverse Laplace transform of the following functions using MATLAB: (a) $F(s)=\frac{s}{s(s+2)(s+3)(s+5)}$ (b) $F(s)=\frac{1}{s^{2}(s+7)}$ (c) $F(s)=\frac{5 s+9}{\left(s^{3}+8
Reduce the system shown in Fig. P 3. 1 to a single transfer function, $T(s)=C(s) / R(s)$ using MATLAB. The transfer functions are given as\[\begin{aligned}& G_{1}(s)=1 /(s+3) \\& G_{2}(s)=1
Obtain the unit-step response plot for the unity-feedback control system whose open loop transfer function is\[G(s)=\frac{8}{s(s+1)(s+3)}\]using MATLAB. Determine also the rise time, peak time,
Obtain the unit-acceleration response curve of the unity-feedback control system whose open loop transfer function is given by\[G(s)=\frac{8(s+1)}{s^{2}(s+3)}\]using MATLAB. The unit-acceleration
The feed forward transfer function $G(s)$ of a unity-feedback system is given by\[G(s)=\frac{k(s+3)^{2}}{\left(s^{2}+5\right)(s+4)^{2}}\]Plot the root loci for the system using MATLAB.
For the unity feedback shown in Fig. P 3. 5, where\[G(s)=\frac{K}{s(s+3)(s+4)(s+5)}\]Obtain the following:(a) display a root locus and pause(b) draw a close-up of the root locus where the axes go
For the system shown in Fig. P 3. 6, determine the following using MATLAB(a) display a root locus and phase(b) display a close-up of the root locus where the axes go from -2 to 2 on the real axis and
Write a program in MATLAB to obtain a Bode plot for the transfer function\[G(s)=\frac{\left(5 s^{3}+51 s^{2}+20 s+400\right)}{\left(s^{4}+12 s^{3}+60 s^{2}+300 s+250\right)}\]
Write a program in MATLAB for the unity feedback system with $G(s)=K /[s(s+7)$ $(\mathrm{s}+15)]$ so that the value of gain $K$ can be input. Display the Bode plots of $t$ a system for the input
Write a program in MATLAB for the system shown in Fig. P 3. 9 so that the value of $K$ can be input $(K=40)$.Fig. P 3. 9(a) Display the closed-loop magnitude and phase frequency response for unity
Write a program in MATLAB for a unity feedback system with the forward-path transfer function given by\[G(s)=\frac{7(s+3)}{s\left(s^{2}+4 s+12\right)}\](a) Draw a Nichols plot of an open-loop
For the system shown in Fig. P 3. 11, write a program in MATLAB that will use an open-loop transfer function $G(s)$.System 1System 2Fig. P 3. 11(a) Obtain a Bode plot(b) Estimate the percent
Write a program in MATLAB for a unity-feedback system with\[G(s)=\frac{K(s+3)}{\left(s^{2}+5 s+80\right)\left(s^{2}+4 s+20\right)}\](a) plot the Nyquist diagram(b) display the real-axis crossing
Write a program in MATLAB to obtain the Nyquist and Nichols plots for the following transfer function for $k=30$.\[G(s)=\frac{k(s+1)(s+2+5 i)(s+2-5 i)}{(s+2)(s+5)(s+7)(s+2+7 i)(s+2-7 i)}\]
For a unit feedback system with the forward-path transfer function.\[G(s)=\frac{K}{s(s+3)(s+10)}\]and a delay of 0. 5 second, estimate the percent overshoot for $K=40$ using a secondorder
For the control system shown in Fig. 3. 16:(a) plot the root loci of the system(b) find the value of gain $K$ such that the damping ratio $\xi$ of the dominant closed-loop poles is 0. 5(c) obtain all
Fig. P 3. 17 shows a position control system with velocity feedback. What is the response $c(t)$ to the unit step input?Fig. P 3. 17 R(s) + + 80 s(s +3) 0.15 C(s) 1/s
The open-loop transfer function $G(s) H(s)$ of a control system is\[G(s) H(s)=\frac{K}{s(s+0.5)\left(s^{2}+0.5 s+8\right)}=\frac{K}{s^{4}+s^{3}+8.25 s^{2}+4 s}\]Plot the root loci for the system
Design a compensator for the system shown in Fig. P 3. 19 such that the dominant closed-loop poles are located at $s=-1 \pm j \sqrt{3}$.Fig. P 3. 19 1 Ge(s) 2 S
For the control system shown in Fig.3.20:(a) design a PID control $G_{c}(s)$ such that the dominant closed-loop poles located at $s=-1$ $\pm j 1$.(b) select $a=0.6$ for the PID controller and find
Draw a Bode diagram of the open-loop transfer function $G(s)$ of the closed-loop system shown in Fig. P 3. 21 and obtain the phase margin and gain margin.Fig. P 3. 21 R(s) 18(s +1) s(s +3)(s2 +2s
A block diagram of a process control system is shown in Fig. P 3. 22. Find the range of gain for stability.Fig. P 3. 22 Ke S S+1
For the control system shown in Fig. P 3. 23:(a) draw a Bode diagram of the open-loop transfer function(b) find the value of the gain $K$ such that the phase margin is $50^{\circ}$(c) find the gain
Obtain the unit-step response and unit-ramp response of the following system using MATLAB.\[ \begin{aligned} & \begin{array}{llllll} \dot{x}_{1} & -5 & -25 & -5 & x_{1} & 1 \end{array} \\ &
For the mechanical system shown in Fig. P 3. 25, the input and output are the displacement $x$ and $y$ respectively. The input is a step displacement of $0.4 \mathrm{~m}$. Assuming the system remains
Using MATLAB, write the state equations and the output equation for the phase variable representation for the following systems in Fig. P 3. 26.(a)(b)Fig. P 3. 26 R(s) 3s +7 s4 ts3 +2s +7s +5 C(s)
Determine the transfer function and poles of the system represented in state space as following using MATLAB.\[\begin{aligned}& \dot{x}=\begin{array}{rrrr}9 & -5 & 2 & 2 \\-4 & 1 & 0 & x+5 \\3 & 5 &
Obtain the root locus diagram of a system defined in state space using MATLAB. The system equations are\[\dot{x}=A x+B u \quad \text { and } y=C x+D u \quad \text { and } \quad u=r-y\]where $r$ is
Obtain the Bode diagram of the following system using MATLAB.\[\begin{aligned}& \begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}=\begin{array}{rrl}0 & 1 \\-30 & 7\end{array} \quad
A control system is defined byWrite a MATLAB program to obtain the following plots:(a) two Nyquist plots for the input $u_{1}$ in one diagram(b) two Nyquist plots for the input $u_{2}$ in one
Obtain the unit-ramp response of the system defined by\[\begin{aligned}& y=\left[\begin{array}{ll}1 & 0\end{array}\right] \begin{array}{l}x_{1} \\x_{2}\end{array}\end{aligned}\]where $u$ is the
Obtain the response curves $y(t)$ using MATLAB for the following system.\[\begin{aligned}& \begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}=\begin{array}{lll}-1 & 1 \\-1 & 0 & x_{1}
Plot the step response using MATLAB for the following system represented in state space, where $u(t)$ is the unit step.\[\begin{aligned}& \begin{array}{rrrr}-3 & 2 & 0 & 0 \\\dot{x}= & -7 & 1 \\0 &
Diagonalize the following system using MATLAB.\[\begin{aligned}& \dot{x}=\begin{array}{rrrr}-10 & -5 & 7 & 1 \\15 & 4 & -12 \\-8 & -3 & 6 & 2 \\& \\y=\left[\begin{array}{lll}1 & -2 &
Determine to unit-ramp response of the system defined by\[\begin{aligned}\dot{x}_{1} & =\begin{array}{rrr}0 & 2 \\-3 & -3 & x_{1} \\\dot{x}_{2}\end{array}+{ }_{2}^{0} \\y & =\left[\begin{array}{ll}1
Obtain the unit-impulse response of the following system using MATLAB\[\begin{aligned}& \begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}=\begin{array}{rrr}0 & 1 \\-1 & -2 & x_{1}
A control system is defined by\[ \begin{aligned} & \begin{array}{llllll} \dot{x}_{1} & -1 & -3 & -3 & x_{1} & 3 \end{array} \\ & \dot{x}_{2}=0 \quad-2 \quad 1 \quad x_{2}+0 u \\ &
Determine the eigenvalues of the following system using MATLAB.\[\begin{aligned}& \dot{x}=\begin{array}{rrrrr}0 & 1 & 0 & 0 \\0 & 1 & -5 & x+ & 0 \\-2 & 1 & 3 & 1\end{array} \\&
For the following path of a unity feedback system in state space representation, determine if the closed-loop system is stable using the Routh-Hurwitz criterion and MATLAB.\[\begin{aligned}&
Consider the differential equation system given by\[\ddot{y}=4 \dot{y}+3 y=0 ; \quad y(0)=0.2 ; \quad \dot{y}(0)=0.1\]Find the state-space equation for the system. Also, obtain the response $y(t)$ of
a. Plot the response \(x\) by using the step and initial commands, showing the step response, the initial response and the total response in one figure. Verify the steady-state value of \(x\).b. Plot
a. Plot the response \(x\) by using the step and initial commands, showing the step response, the initial response and the total response in one figure. Verify the steady-state value of \(x\).b. Plot
a. Plot the response \(x\) by using the step and initial commands, showing the step response, the initial response and the total response in one figure. Verify the steady-state value of \(x\).b. Plot
a. Plot the response \(x\) by using the step and initial commands, showing the step response, the initial response and the total response in one figure. Verify the steady-state value of \(x\).b. Plot
The transfer function of a dynamic system is given by\[G(s)=\frac{s+4}{s^{5}+2 s^{4}+3 s^{3}+8 s^{2}+4 s+5}\]Determine the stability of the systema. Using Routh's stability criterion without solving
Consider a second-order system whose transfer function is in standard form as in Equation 10.7. Assume that the requirements for the system unit-step response are rise time \(t_{\mathrm{r}} \leq 0.1
The unit-step response of a dynamic system is shown in Figure 10.12. Find the transfer function of the system if it can be approximated as \(a /\left(s^{2}+2 \zeta \omega_{n}
Consider an unstable plant\[G(s)=\frac{s+2}{s^{3}+4 s^{2}-5 s}\]with feedback control, as shown in Figure 10.15.a. Using Routh's stability criterion, determine the range of the control gain \(K\) for
Consider the closed-loop control system shown in Figure 10.24.a. Using MATLAB, plot the unit-step responses of the system for the following values of \(K: 5,50\), and 500 .b. Compute the steady-state
Consider a mass-damper-spring system \(G(s)=Y(s) / U(s)=1 /\left(m s^{2}+b s+k\right)\), where \(m=1 \mathrm{~kg}, b=8 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\), and \(k=40 \mathrm{~N} /
Consider the mass-damper-spring system in Example 10.9,\[G(s)=\frac{1}{s^{2}+8 s+40}\]Use Simulink to build a block diagram for proportional feedback control. Find the unit-step responses for
Use the Simulink block diagram built in Example 10.10 to find the unit-step responses for \(k_{\mathrm{I}}=50,500\), and 1550 . Set \(k_{\mathrm{p}}=250\) and \(k_{\mathrm{D}}=0\). Discuss the
Use the Simulink block diagram built in Example 10.10 to find the unit-step responses for \(k_{\mathrm{D}}=1,10\), and 50 . Set \(k_{\mathrm{p}}=250\) and \(k_{\mathrm{I}}=500\). Discuss the effects
Consider the feedback control system shown in Figure 10.35, in which the plant is the DC motor-driven cart given in Example 10.2. The input to the plant is the voltage applied to the DC motor, and
For the system in Figure 10.49, find the locus of closed-loop poles with respect to K. R(s) K C(s) FIGURE 10.49 Block diagram for the feedback control in Example 10.14. s+1 (s+3)(s + 2s+2) G(s) 1
Refer to the root locus obtained in Example 10.14. Comment on the stability and performance of the closed-loop system when \(K\) varies from 0 to \(\infty\).Data From Example 10.14:For the system in
Reconsider Example 10.7. Using the final-value theorem, verify the steady-state errors to a unit-step input for open and closed-loop control without- and with disturbance.Data From Example
Reconsider Example 10.9. Build Simulink block diagrams to simulate open and closed-loop control with parameter variations. Verify the steady-state response values \(y_{\mathrm{ss}}\) obtained in
Sketch the asymptotes of the Bode plot magnitude and phase for the following open-loop transfer functions. Make sure to give the corner frequencies, slopes of the magnitude plot, and phase angles.
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