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computer science
systems analysis and design
Questions and Answers of
Systems Analysis And Design
Repeat Problem 1 for the following open-loop transfer functions.a. \(G(s)=\frac{1}{s^{2}+4 s+100}\)b. \(G(s)=\frac{s+0.5}{s^{2}+s+25}\)c. \(G(s)=\frac{s^{2}+0.1 s+25}{s^{2}+0.24 s+144}\)d.
For each of the following open-loop transfer functions, construct a Bode plot for \(K=1\) using the MATLAB command bode. Estimate the GM, PM, and their associated crossover frequencies from the plot.
The Bode plot of a dynamic system is shown in Figure 10.78, in which the asymptotes are also given. Following the rules of sketching Bode plots, find the transfer function of the system. FIGURE 10.78
Figure 10.79 shows the Bode plot for an open-loop transfer function \(K G(s)\) with \(K=500\).a. Determine the stability of the closed-loop system with \(K=500\).b. Determine the value of \(K\) that
The Bode plot for an open-loop transfer function \(K G(s)\) is shown in Figure 10.80.a. Determine the stability of the closed-loop system.b. Assume that the proportional control gain \(K\) is
Consider the unity negative feedback system shown in Figure 10.81.a. Use MATLAB to obtain the Bode plot \(K G(s)\) for \(K=5\).b. Determine the stability of the closed-loop system when \(K=5\) using
For the system shown in Figure 10.83, derive the state-space equations using the state variables indicated. Make sure to give the A, B, C, and D matrices. Also determine the poles of the system.
Repeat Problem 1 for the system shown in Figure 10.84. FIGURE 10.84 Problem 2. + -8 x1 1 x2 U(s) -Y(s) S+4 S+2 X3 2+ $ + 5
Determine the controllability and observability for each of the following systems.a. \(\left\{\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3}\end{array}\right\}=\left[\begin{array}{ccc}-5 &
Consider the two-degree-of-freedom mass-spring system as shown in Figure 10.85, in which two masses are to be controlled by two equal and opposite forces \(f\). The equations of motion of the system
Using the approach in Section 4.4, find the controllable canonical form for the plant transfer function and then design a full-state feedback controller that places the closed-loop poles at the same
A regulation system has a plant with the transfer functionG(s)=Y(s)U(s)=5s3+3s2+4s+6G(s)=Y(s)U(s)=5s3+3s2+4s+6a. Transform the plant transfer function into the state-space form with the state vector
Consider the system\[\left\{\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}\right\}=\left[\begin{array}{cc}0 & 1 \\0 & -1\end{array}\right]\left\{\begin{array}{l}x_{1}
Consider the systemG(s)=Y(s)U(s)=1(s+1)(s+2)(s+3)G(s)=Y(s)U(s)=1(s+1)(s+2)(s+3)a. Design a state-feedback controller, so that the closed-loop response has an overshoot of less than 5%5% and a rise
Consider the feedback control system as shown in Figure 10.91. Determine the range of \(K\) for closed-loop stability. FIGURE 10.91 Problem 1. 5+2 1 R(s) K Y(s) s+10 (s2-1)(s+5)
Consider the feedback control system as shown in Figure 10.93.a. Assuming \(C(s)=k_{\mathrm{p}}\), determine the value of the proportional gain that makes the closed-loop system marginally stable.
Consider a unity negative feedback system with the open-loop transfer function\[K G(s)=\frac{K}{s(s+3)(s+6)}\]a. Use MATLAB to draw the Bode plots for \(K=1\). Determine the range of \(K\) for which
The Bode plot of a dynamic system is shown in Figure 10.95, in which the asymptotes are also given. Following the rules of sketching Bode plots, find the transfer function of the system. FIGURE 10.95
Consider the system\[\left\{\begin{array}{l}\dot{x}_{1} \\\dot{x}_{2}\end{array}\right\}=\left[\begin{array}{cc}0 & 1 \\0 & -10\end{array}\right]\left\{\begin{array}{l}x_{1}
Consider the two-degree-of-freedom quarter-car model shown in Figure 5.34, in which the force \(f\), applied between the car body and the wheel-tire-axle assembly, is controlled by feedback and
A Consider the DC motor-driven wheeled mobile robot shown in Figure 6.82, in which the voltage applied to the DC motor is computed by a controller. Assume that \(v_{\mathrm{a}}=2.56 i-0.37 x+4.61
Consider the cart-inverted-pendulum system shown in Figure 5.79. Assume that the mass of the cart is \(0.8 \mathrm{~kg}\), the mass of the pendulum is \(0.2 \mathrm{~kg}\), and the length of the
Determine the damping ratio associated with a second-order system in the standard form of Equation 8.32 that corresponds to a maximum (peak) logarithmic magnitude of \(15.22 \mathrm{~dB}\).
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 Problem 24. Plot the response \(x_{1}\) by simulating the Simulink model of the system by using the state-space block.Data From Problem 24:In the mechanical system in Figure 8.20, the
Reconsider Problem 24 but assume \(f_{1}=0\) and \(f_{2}\) is a unit-impulsive force. All parameter values remain unchanged. Assuming zero initial conditions, plot the response \(x_{2}\).Data From
The mathematical model of a dynamic system is derived asa. If \(f(t)\) is the input and \(x_{1}\) and \(\dot{x}_{1}\) are the outputs, obtain the state-space form.b. \(A\) Determine if the system is
Decide whether the system in Problem 1 is stable. A linear dynamic system is stable if the homogeneous solution of its mathematical model, subjected to the prescribed initial conditions, decays. More
Decide whether the system in Problem 5 is stable. A linear dynamic system is stable if the homogeneous solution of its mathematical model, subjected to the prescribed initial conditions, decays. More
Decide whether the system in Problem 4 is stable. A linear dynamic system is stable if the homogeneous solution of its mathematical model, subjected to the prescribed initial conditions, decays. More
Perform the indicated operations, if defined, for the following vectors and matrices.\(\mathbf{w}^{T} \mathbf{A}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\left(\mathbf{A B}^{T}\right) \mathbf{v}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\left(\mathbf{B}-\mathbf{v} \mathbf{w}^{T}\right) \mathbf{A}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\mathbf{A w}-\mathbf{B}^{T} \mathbf{v}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\mathbf{A}\left(\mathbf{B A}+\mathbf{v} \mathbf{w}^{T}\right)\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\mathbf{v}^{T} \mathbf{B} \mathbf{w}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Perform the indicated operations, if defined, for the following vectors and matrices.\(\left(\mathbf{A}^{2}+\mathbf{B}^{T} \mathbf{B}\right) \mathbf{W}\) -2 1 -3 1 1 A = 1 -3 2 1 32 B = V= W = 0 0 4 5
Build a Simscape model for an RL circuit ( R=5Ω,L=0.5HR=5Ω,L=0.5H ), driven by an applied voltage represented by a pulse (amplitude 0.6 for 1<t<31<t<3 ). Run the simulation and generate
Express \(z=-1-2 j\) in polar form.
Simplify \((-1-2 j)^{4}\).
a. Express \(z_{1} / z_{2}\) in rectangular form.b. Verify that \(\left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right|\).\(\frac{-2 j}{3-4 j}\)
a. Express \(z_{1} / z_{2}\) in rectangular form.b. Verify that \(\left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right|\).\(\frac{1-j}{2-\frac{1}{2} j}\)
a. Express \(z_{1} / z_{2}\) in rectangular form.b. Verify that \(\left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right|\).\(\frac{\frac{1}{3}}{-2+j}\)
a. Express \(z_{1} / z_{2}\) in rectangular form.b. Verify that \(\left|z_{1} / z_{2}\right|=\left|z_{1}\right| /\left|z_{2}\right|\).\(\frac{\frac{2}{3}+2 j}{-3 j}\)
Express each complex number in its polar form.\(\sqrt{3}-2 j\)
Express each complex number in its polar form.\(-\frac{2}{3}+j\)
Express each complex number in its polar form.\(\sqrt{3}+j\)
Express each complex number in its polar form.\(-\frac{1}{2} j\)
Express each complex number in its polar form.\(\frac{1+j}{-1+2 j}\)
Express each complex number in its polar form.\(\frac{1+j \sqrt{3}}{1-j \sqrt{3}}\)
Perform the operations by using the polar form and express the result in rectangular form.\(\frac{\frac{2}{3} j}{1+j \sqrt{3}}\)
Perform the operations by using the polar form and express the result in rectangular form.\(\frac{1-\frac{1}{3} j}{\frac{1}{3}+j}\)
Perform the operations by using the polar form and express the result in rectangular form.\(\frac{4+j}{(4+3 j)^{3}}\)
Perform the operations by using the polar form and express the result in rectangular form.\((0.9239+0.3827 j)^{12}\)
Perform the operations by using the polar form and express the result in rectangular form.\(\left(\frac{j}{1+4 j}\right)^{3}\)
Find all possible values for each expression.\((-1+2 j)^{1 / 3}\)
Find all possible values for each expression.\((-1)^{1 / 4}\)
Find all possible values for each expression.\((\sqrt{3}-j)^{1 / 4}\)
Find all possible values for each expression.\(\sqrt{1+j \sqrt{2}}\)
Find all possible values for each expression.\(\left(-\frac{2}{3}+j\right)^{2 / 3}\)
Solve the linear, first-order IVP.\(\dot{x}+\frac{1}{3} x=\cos 2 t, \quad x(0)=1\)
Solve the linear, first-order IVP.\(\frac{1}{2} \dot{x}+t x=\frac{1}{2} t, x(0)=\frac{1}{3}\)
Solve the linear, first-order IVP.\((t-1) \dot{u}+t u=2 t, u(0)=1\)
Solve the linear, first-order IVP.\(\dot{u}=(1-u) \cos t, u(0)=\frac{1}{2}\)
Solve the linear, second-order IVP.\(\ddot{x}+4 \dot{x}+4 x=e^{-t}, x(0)=1, \dot{x}(0)=1\)
Solve the linear, second-order IVP.\(\ddot{x}+4 x=t, x(0)=0, \dot{x}(0)=1\)
Solve the linear, second-order IVP.\(\ddot{x}+x=\sin t, x(0)=1, \dot{x}(0)=0\)
Solve the linear, second-order IVP.\(\ddot{x}+4 \dot{x}+3 x=2 e^{-3 t}, x(0)=0, \dot{x}(0)=-1\)
Solve the linear, second-order IVP.\(6 \ddot{u}+7 \dot{u}+2 u=65 \cos t, u(0)=0, \dot{u}(0)=5\)
Solve the linear, second-order IVP.\(4 \ddot{u}+4 \dot{u}+5 u=0, u(0)=0, \dot{u}(0)=1\)
Write the expression in the form \(D \sin (\omega t+\phi)\).\(\cos t+\frac{1}{3} \sin t\)
Write the expression in the form \(D \sin (\omega t+\phi)\).\(\frac{1}{2} \cos 3 t-\sin 3 t\)
Write the expression in the form \(D \sin (\omega t+\phi)\).\(\cos \left(2 t+\frac{1}{3} \pi\right)\)
Write the expression in the form \(D \sin (\omega t+\phi)\).\(3 \sin \omega t-\cos \omega t\)
Write the expression in the form \(D \cos (\omega t+\phi)\).\(\cos t-\frac{3}{4} \sin t\)
Write the expression in the form \(D \cos (\omega t+\phi)\).\(\sin t-\cos t\)
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(e^{a t+b}, a, b=\mathrm{const}\) TABLE 2.2 Laplace Transform Pairs No.
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(\cos (\omega t+\phi), \omega, \phi=\mathrm{const}\) TABLE 2.2 Laplace
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(\sin (\omega t-\phi), \omega, \phi=\mathrm{const}\) TABLE 2.2 Laplace
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(t^{3}-\frac{1}{2}\) TABLE 2.2 Laplace Transform Pairs No. f(t) F(s) 1 Unit
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(\sin ^{2} t\) TABLE 2.2 Laplace Transform Pairs No. f(t) F(s) 1 Unit
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(t \cosh t\) TABLE 2.2 Laplace Transform Pairs No. f(t) F(s) 1 Unit impulse
a. Find the Laplace transform of the given function. Use Table 2.2 when applicable.b. Confirm the result of (a) in MATLAB.\(t^{2} \sin \left(\frac{1}{2} t\right)\) TABLE 2.2 Laplace Transform Pairs
a. Express the signal in terms of unit-step functions.b. Find the Laplace transform of the expression in (a) by using the shift on \(t\)-axis.\(g(t)\) in Figure 2.15 FIGURE 2.15 Signal in Problem 9.
a. Express the signal in terms of unit-step functions.b. Find the Laplace transform of the expression in (a) by using the shift on \(t\)-axis.\(g(t)\) in Figure 2.16 FIGURE 2.16 Signal in Problem 10
a. Express the signal in terms of unit-step functions.b. Find the Laplace transform of the expression in (a) by using the shift on t-axis.\(g(t)= 0 if t
a. Express the signal in terms of unit-step functions.b. Find the Laplace transform of the expression in (a) by using the shift on \(t\)-axis.\(g(t)= 0 if t
Find the Laplace transform of each periodic function whose definition in one period is given.\(h(t)=\left\{\begin{array}{ccc}1 & \text { if } & 0
Find the Laplace transform of each periodic function whose definition in one period is given.\(h(t)=\left\{\begin{array}{lll}t & \text { if } & 0
Find the Laplace transform of each periodic function whose definition in one period is
Find the Laplace transform of each periodic function whose definition in one period is given.\(h(t)=2-t, 0
Find the Laplace transform of the given periodic function of \(f(t)\) in Figure 2.17 FIGURE 2.17 Periodic function in Problem 17. 1 ^(t) 0 2
Find the Laplace transform of the given periodic function of \(f(t)\) in Figure 2.18. a f(t) 0 a 2a 4a FIGURE 2.18 Periodic function in Problem 18.
a. Find the partial-fraction expansion.b. AVerify (a) by using the residue command in MATLAB.\(\frac{s}{(s+1)\left(s^{2}+2 s+2\right)}\)
Find the inverse Laplace transform by using the partialfraction expansion method.\(\frac{s+1}{s^{2}(s+3)}\)
Find the inverse Laplace transform by using the partialfraction expansion method.\(\frac{4 s+5}{s^{2}\left(s^{2}+4 s+5\right)}\)
Find the inverse Laplace transform by using the partialfraction expansion method.\(\frac{s+10}{s\left(s^{2}+2 s+5\right)}\)
Find the inverse Laplace transform by using the partialfraction expansion method.\(\frac{5 s+8}{s(s+2)^{2}}\)
Find the inverse Laplace transform by using the partialfraction expansion method.\(\frac{2 s^{2}+3 s-4}{(s+3)\left(s^{2}+2 s+2\right)}\)
a. Solve the IVP.b. Confirm the result of (a) in MATLAB.\(\frac{2}{3} \dot{x}+2 x=u(t)-u(t-1), x(0)=0\)
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