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**Q1:**4. Given the system equations a. Using only amplifiers and integrators draw a signal-flow graph representation of the system where U(s) is the input and X2(s) is the output. You may assume zero initial conditions. b. Find the transfer function X2(s)/U(s) using Mason's Gain formula. Check your result using an algebraic approach. \frac{d x_{1}}{d t}=x_{1}+5 x_{2} \frac{d x_{2}}{d t}=2 x_{1}+uSee Answer**Q2:**3. Given the signal flow graph below determine the transfer matrix A where Aij = Yi/Xj. Note that Aij = Yi/Xj given that all other inputs equal to zero.are \left.\left[\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right]=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \begin{array}{l} X_{1} \\ X_{2} \end{array}\right] See Answer**Q3:**2. Given the block diagram shown below а.Determine its signal flow graph realization. b. Using Mason's gain formula determine Y (s)/X(s).See Answer**Q4:**Problem 1 Fourier transform analysis of previous homework problem. Problem 2 from Homework 3 is revisited. Consider the low-pass system y = -y +2u with input u(t): = -et t<0 t>0 (1) Explain why a Fourier transform approach can be used to solve this problem. Use it to find y(t) and compare the result to the convolution approach from Homework 3.See Answer**Q5:**Problem 2 Fourier transform analysis using Library of transforms. This is another Fourier transform analysis problem. Consider a one degree-of-freedom damped spring-mass system governed by the differential equation ÿ+2y+26y = 26u, where y is the position of the mass relative to it's equilibrium position and u is a force that is applied to the mass. The force input u is the same as the previous problem, i.e. (1). Solve for y on the time interval (-∞, ∞). Graph y on the interval [-3,3] second. Hint: Once ŷ is determined, use a partial fraction expansion and the "Library" from Homework 6 to reverse-engineer the time functions associated with the terms in the partial fraction expansion.See Answer**Q6:**Problem 4 Periodic input to multi-flywheel system. Return to the three flywheel system from Home- work 6 bi Use the same parameter values as the analysis in Homework 6. The motor moment, s, is the periodic signal shown below: 2 bs Lbs Here are some general facts established from prior homework (see Problem 3, Homework 3): ● Because u is bounded and the system is asymptotically stable, fly, fly and fly are bounded. Because u is periodic all dependent variables are periodic/nThese points provide a priori justification for using Fourier series to analyze the dependent vari ables. 1. Compute the Fourier series coefficients for u. 2. Write the Fourier series representations for f₁, fly and fly. Hint: use the Fourier series repre- sentation of the input and also the transfer functions/frequency response functions determined in Homework 6. 3. Graph, in a single figure, the periodic responses off, f and fly and the input u by computing a partial sum of the Fourier series expression over the index range &-0,1,2,...,1000. Use the time grid t-[-2:0.0001:2]; 4. Note how 1₂ appears "smoother than fh, and fly is smoother than f₂. How can this be rigorously explained? Hint: Graph the frequency response magnitudes associated with fh, fly and fly on the interval 0.01 Hz to 100 Hz and note how the magnitudes decrease at the frequencies represented in the Fourier series.See Answer**Q7:**Problem 3 Fourier series analysis of moving average filter with periodic input. Revisit Problem 4, Homework 3. Consider a moving average filter with the following impulse response A (the impulse response is zero outside of the displayed time interval): The input to this filter is the following periodio cignal se (T- T The dashed line shows one isolated period. Answer the following, 1. Determine the Fourier series for the periodic inputs. In particular, derive an expression for the Fourier series coefficients. 2. Determine the frequency response of the moving average filter. Hint: compute the Fourier transform of the impulse response. 3. Determine the Fourier series coefficients for the periodic output p 4. Use Matlab to compute a partial sum of the Fourier series for y with -1000,-900,1,0,1,999, 1000 on the time grid t = [-3:0.001:3]. Compare this result to the graphical convolution ap- proach from Homework 3.See Answer**Q8:**Problem 5 Pulse width modulation analysis of flywheel system. This problem extends the flywheel analysis of Problem 4. Pulse width modulation (PWM) is a common technique for specifying the moment applied by an electric motor. In PWM, the motor current is either "on" or "off" with the switching occurring with regularity (often called the repetition rate). This is done to maximize the efficiency of the amplifier. It is assumed that the moment created by the motor also follows the 3 current switching. The rapid switching of the moment is filtered by the inertia of the motor load (the flywheels in this problem) so that the angular velocities of the flywheels are a "smoothed" version of the moment. This problem explores these ideas in a quantitative manner. 1. The average value of a periodic signal 9 is defined (d, T- signal period Prove the following general result: if an asymptotically stable system is subjected to a periodic input, then Have (0) Ma (2) where (0) is the frequency response evaluated at -0, u is the periodic input and y is the periodic output. Refer to the graphs of fly, fly and fly from Problem 4 to see that they agree with (2). Also refer to Problem 3 to show its agreement with (2). Hist: Apply the "averaging operator" to the Fourier series representations of the input and output and note that t=0 for all 0 (recall -20/T). 2. Although the average value of f, can be specified according to (2) there can be considerable variation in the angular velocity about the mean value. Some applications require that the angular velocity not deviate too much from the mean value. A conservative upper limit on the period I will be derived that guarantees that a specified deviation from the mean value will not be exceeded. Consider these steps: (a) Consider the rectangular wave with period 7>0 and duty cycle a € [0, 1]. One period for t€ 0,7] is defined below: (1 teBar) (t)= 10 t€aTT) The duty cycle determines the duration that the input is "on" or "off" in one period: a-0 means - 0 for all t; a-1 means -1 for all t; a-0.5 means -1 for half the period and zero for the remaining half. Find the Fourier series coefficients, denoted (b) Determine an upper bound for ea, &0, where are the Fourier series coefficients of . This upper bound will have the index k. (e) Let Hy be the frequency response function associated with f. Find an approximation for H₁ for sufficiently large . "Sufficiently large" means the dominant terms in the numerator and denominator are the are the highest powers of ./nnumerator and denominator are the are the highest powers of us. (d) The Fourier series for fly can be written as - - Σκουλαρίκια 12(1) - +2ΣRe [8 (kuva) - +2 4 What is the relation between fave and a? Thus, conclude that over a certain range simply by selecting a. can be specified (e) Despite the relation between a and Shaw, the "ripple" in the angular velocity cannot be too large. Thus, consider the following sequence of bounds ||$21 (1) — £21,000| — | |-|(2Σe [ (int) Re 52 Rei ام و لا (س) 2 > (س) - Substitute the upper bound for a/T and the "high frequency" approximation of |H₂|- Then, do the sum to find an upper bound for (1) -- The period I should be a parameter. Hint: 1² (f) Based on the bound, determine the largest value of T so that f(t) — £₁,| ≤0.01. In other words, fly(t) is very close to its mean value. (g) Graph four periods of fly with the selected value of T and a(0.25, 0.5, 0.75). Use -1000,0,1000) in the partial sum of the Fourier series. The time grid can need to be customized. Select the time spacing in the grid, t, so that there are 1000 points per period. Show that the deviation of f(t) from he doe not exceed C all cases of a.See Answer**Q9:**Problem 1 Solve an IVP using the unilateral Laplace transform. The IVP problem from the recitation on Friday, June 2, 2023, will be solved using a Laplace transform approach. Here is the problem restatement: y + 4y = 4u y(0) = 1.3 u(t) = [1 0 uk (t) Note that u is periodic for t ≥ 0, in other words, u(t +T) T = = 1 second. Answer the following: 1. First, compute the unilateral Laplace transform of a single rectangular pulse, denoted uo, 1 t € [0, 0.5] t> 0.5 so it can be argued that t = [0, 0.5] t≤ (0.5, 1) = (one period), t≥0 uo(t) Argue that ROCuo is the entire complex plane. 2. Now compute the unilateral Laplace transform of a delayed rectangular pulse (delayed by k seconds, where k is a positive integer), = 0 t = [0, k) te [k, k +0.5], k = 1,2,3,... 0 t>k+0.5 In fact, show ûk = ûoe-ks. In other words, the Laplace transform of a time-shifted function has a simple relation with the transform of the unshifted function. 3. The input to the system u can be expressed as a superposition of these shifted pulses, u(t) for all t ≥ 0. The period is ∞ u(t) = Σ uk(t), t> 0 k=0 1 ∞ û(s) = Σûk(s) (1) k=0 What is the ROC associated with u, though? Hint: it is no longer the entire complex plane. 4. If s € ROCu, show that the geometric series formula can be applied to (1). What is the sum in this case? 5. Apply the unilateral Laplace transform to the ODE to write a complete expression for the unilateral Laplace transform of the IVP solution that is valid for t € [0, k]. In other words, compute ŷ. 6. The inverse Laplace transform is Since s = o + jw, where o is in the ROC of ŷ, then ds = y(t) = lim 1 y(t) = lim R→∞ j2π o-jR = = eat wo 2π = Wo 2π = eat wo kwo, k This integral can be approximated by a Riemann sum by discretizing w as w = . . .‚ —2, —1, 0, 1, 2,… The frequency “step" is wo, i.e. dw wo. The Riemann sum shares many similarities with the Fourier series synthesis formula, y(t) ~ Σ ŷ(o + jkwo)e(o+jkwo)t k=-∞ y(t) ~ R→∞ 2π po jR [ +1² (8)est ds. R #Lio R 2π ∞ Σ ŷ (o + jkwo) ej kwot k=-∞ Wo 2 (163- 2π jdw and integral can be written as + jw)e(o+jw)t 20000 ŷ(o) + 2 Σ Re ŷ(o k=1 (20 ŷ(0) +2ΣRe [ŷ(0 + jkwo)ej kwot] k=1 = Use Matlab to numerically approximate y on the interval t = [0,3] using wo (confirm this is in the ROC of y), and the following limits on k, 2 t dw. = = 1 1) +jkwo) ej kwot Use a time grid so that the time step is ts 200000 This will ensure at least four time steps fall within one period of the highest frequency sinusoid in the sum. Graph the numerical approximations of u and y for t = [0,3] seconds. Note that Gibbs phenomenon is present in u. = 0.1 rad/s, o = 1See Answer**Q10:**Problem 2 Fill out the table of unilateral Laplace transforms below (all signals are defined on the interval t € [0, 0)). Also state the region of convergence (ROC). Signal, g ROC ελέμ(t), λεσ e(t-thi) (t-tshift), AEC, fshift > 0 text μ(t), XE C eat cos(wot)μ(t), o ER et sin(wot)u(t), o ER Σμ( -- kT), T > 0, k=0 Laplace transform, ĝ Remark: The last signal is a "staircase" function in which a "step" occurs every T seconds. It is useful to sketch it to see what it looks like. A closed-form expression for its Laplace transform can be determined using the geometric series summation formula when 8 € ROCg. 3See Answer**Q11:**Problem 3 Unilateral Laplace transform analysis of transfer function zeros in a flywheel system. Return to the system of three flywheels: 221 23 b3 +0₂ byl As in Homework 6, J₁ = J2 = J3 = 1, b₁ = b2 = b3 = b = b5 = 1. Answer the following: 1. The three coupled first-order ODEs governing this system where derived to be U ₁ = 29₁ +₂+u 2₂=₁-302 +03 03-0₂-203 Apply the unilateral Laplace transform to each ODE and solve for ₁, ₂, and f, in terms of {1(0), 2(0), 3(0)} and û. This involves a bit of algebra but yields the complete IVP solu- tion for each dependent variable, albeit in the Laplace domain. Split up the final expressions for 21, 22, and 3 in terms of the zero-input response (the part with the initial conditions) plus the zero-state response (the part with the transfer function times û). 2. The transfer function associated with ₁ (denoted H₁ in prior homework) is H₁ = s²+58 +5 (8+1)(8+2)(8+4) This transfer function has two zeros. Label them z₁ and 22 such that 22 <₁ <0. Let u(t) = e²¹¹µ(t), t≥0. Find initial conditions {₁(0), ₂(0¯), N₂(0¯)} such that ₁ = 0, t≥0. Hint: ₁ = 0 if and only if f₁ = 0. 3. Consider the ICs and input from the previous part, i.e. u(t) = e²¹tu(t), t≥ 0. Show that N₂(t) = N₂(0-)eit Na(t) = N₂(0)eit t20™ Hint: the easiest way is to show (2) satisfies the ODEs for ₂ and 3, 0₂=91 35₂ +93 123 = 22-203 (2) 4. Now consider the input u(t) = etu(t), t≥ 0. Find new initial conditions (2₁(0¯), N₂(0¯), N3(0)} such that ₁ = 0, t≥ 0. Also show N₂(t) = N₂(0¯)et and N3(t) = N3(0¯)e²²ª, t≥ 0¯. 4See Answer**Q12:**Problem 4 Unilateral Laplace transform analysis of another IVP. Consider the system from Problem 3, Homework 5: torsional → spring k = 1 U disks rub with friction O c=24 J = 2 J = 1 The dependent variables are the left flywheel angle, 01, and right flywheel angular velocity, 2. 1. Apply the unilateral Laplace transform to the second order ODE for 0₁ and the first order ODE for ₂. The natural initial conditions are {0₁(0), 6(0), 2₂(0)}. Determine 0₁ and 2₂ in terms of the ICs and û. There is no need to find the corresponding time-domain signals -the point is to show how the unilateral Laplace transform yields both the zero-input response and zero-state response parts of the IVP solution. 2. The transfer function associated with ₁ has a zero at s= -1. Therefore, let u = e-¹μ(t), +20, and find initial conditions (01(0), 0(0), 2(0)}, such that 01 (t) = 0, + > 0. Hint: use the expression for 8₁ to find ICs for which ₁ = 0. 3. The transfer function associated with 2 has a zero a s = 0. Therefore, let u= u(t), t≥ 0¯, and find initial conditions {0₁(0), 6(0), ₂(0)} such that ₂(t) = 0, t≥ 0. Hint: find ICs so ₂ = 0.See Answer**Q13:**Problem 5 A system is tested on a moving platform u -moving base The position of the mass relative to an inertial frame is y. The actual measurement of the mass motion, however, is its acceleration, ÿ, which is provided by an accelerometer attached to the mass. The mass, spring rate, and damping rate are 1, 4 and 8, respectively, so the ODE is ÿ + 4y + 8y = 4ů + 8u. 5 1. Solve the unforced (u= 0) IVP for y with y(0) = -1, g(0) = 1, by computing the characteristic roots A₁ and ₂ and letting y(t) = Aet + Bet, t≥ 0. The parameters A and B are determined by enforcing the initial conditions. Once y(t) is determined (it is a real-valued signal), compute the mass acceleration by differentiation (note: there are no discontinuities in y or any of its time derivatives in a neighborhood of t = 0). 2. Now use the unilateral Laplace transform to find the acceleration due to these initial con- ditions (u= 0 still) by following these steps: 1) apply the transform to the ODE, 2) iso- late the expression for ŷ, 3) use the Derivative Theorem again for the mass acceleration, ỹ = s²ŷ - sy(0) - (0), 4) "invert" the expression for to determine ÿ(t) for, t≥ 0. Compare to the result from Part 1. Note that when differentiating a dependent variable it is always necessary to apply the Derivative Theorem to account for possible non-zero initial conditions associated with it. 3. Now consider a forced IVP in which the ICs are the same as Part 1, however, u(t) = tµ(t), t20. Use the unilateral Laplace transform to determine ÿ, t≥ 0. Note that "external" inputs are always considered "abrupt", so u(0) = 0, ú(0) = 0, etc. Hint: develop the expression for first, then apply the Derivative Theorem to determine , then reverse engineer to time-domain signals. Make sure to distinguish which signals in ÿ are abrupt, versus those that are not.See Answer**Q14:**Find the Fourier transform of the shown waveform. f(1) A 10 5 0 1 2 tSee Answer**Q15:**3. Consider the system tx' = [22]x, t> 0 with initial condition X(2) = [−12]. A Assuming solutions of the form x = where X, V are an eigenvalue/eigenvector pair of the given matrix, use techniques similar to those used to construct solutions to the constant coefficient linear homogeneous systems to solve the given initial value problem. Write your answer as a single vector.See Answer**Q16:**4. Solve the initial value problem ਬਚਤਰ ਹੈ। 이 rigk |||| 3r-z -3y-z 2y-2 Bou dt with z(0) = -5, y(0) = 13, 2(0) = -26 using eigenvalue/eigenvector techniques.See Answer**Q17:**5. Consider the system x' following values of c: (a) c = (0,00) 9 x where c is a parameter. Classify the geometry and stability properties of the system for the (b) c = 0 (c) c = (-1,0) (d) c= -1 (e) c = (-∞, -1)See Answer**Q18:**esc ZILLDIFFEQ9 10.2.010. Classify the critical point (0, 0) of the given linear system by computing the trace r and determinant A and using the figure. X'= -9x + 3y y' = 6x - 5y O center Stable node unstable node O stable spiral O unstable spiral Osaddle Ostable node Degenerate stable node O degenerate unstable node O degenerate stable node PREVIOUS ANSWERS 2 Stable spiral 80 44 Unstable spiral T²-44 <0 Center Saddle T²=4A Unstable node Degenerate unstable node ZE DII il later. $19 be downloaded. Please try again MY NOTES F12See Answer**Q19:**2. Use the following figure to construct a model for the number of pounds of salt ₁ (t), ₂(t), and z3(t) at time t in tanks A, B, and C, respectively. Write the model in matrix form and then solve it using eigenvalue/eigenvector techniques assuming that 21 (0) = 15, 22(0) = 10, and 23 (0) = 5. Will all of the tanks eventually be free of salt? Use your solution to justify your answer. pure water 4 gal/min 200 gal mixture 4 gal/min 150 gal mixture 4 gal/min امع 1000 mixture 4 gal/minSee Answer**Q20:**For t E[1,5], the upward velocity of a rocket is given by a quadratic expression v(t)=at2+bt+c. The upward velocity of the rocket is recordedSee Answer

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