ECE509 Analysis of Linear Systems Project #3 Version 3 (100 pts.) 1. (40 pts.) Consider the SISO system from: -3 −1 0 0 21 7 1 -2 x(t) = -1702 -354 -824 x(t) + -120 209 -168 -58 100 y(t)= [135 45 8 -15] x(t) 102 u(t) Verify that this system is controllable and observable in MATLAB. In Simulink, design a state feedback controller such that the system meets or exceeds the following specifications when tracking a square wave alternating between -2 and 1 at 1.5 Hz for seconds: (a) Maximum settling time for 98% criterion: 0.15s (b) 0.95 ≤ y(t) ≤ 1.05 for 2.5s ≤ t < 2.66s (c) -2.05 y(t) ≤ -1.95 for 3.5s ≤ t < 3.66s Plot the reference signal r(t) along with the output y(t) for 5 seconds. In a separate figure, plot the control signal u(t). Also state what your final poles and k vector are. A state feedback flow diagram is given below. You can implement the model however you like in Simulink as long as the implementation can accommodate a system of arbitrary order N. u(t) x(t) x(t) r(t) KF + b S C y(t) k A 2. (40 pts.) Now consider the system from (1) and implement feedback via a state observer: Here you no longer have access to the state vector ☑(t), only the output y(t). Design a system using a state observer that is able to track a square wave alternating between 0 and 3 with a frequency of your choice following: - but make sure the frequency is chosen so that steady-state tracking can take place. Do the (a) Describe your overall control design solution. (b) State the frequency of the reference signal. (c) Simulate the system with your control solution. Demonstrate your design by plotting, in one figure, the real state and its estimate, and in another, the real and estimated output together with the reference to be tracked. Show two full periods of your reference. Make sure the plant and observer have different initial conditions (of your choice), so that the effect of the observer can be clearly seen. The figure below gives a flow diagram for state feedback control using an observer. You may implement the system in Simulink however you wish as long as the implementation can accommodate a system of arbitrary order N. u(t) x(t) x(t) r(t)-kF + b S C y(t) k + x(t) 2 A L S A - Lc 3. (20 pts.) A More Realistic Example: Mechanical Press A simple mechanical press can be modeled as two rotating masses joined by a torsional spring, as depicted in Figure 1. 01 and wi ė₁ are, respectively, the angle and angular speed of the press motor spindle. = 01, W1 J₁ 02,W2 k1, C1 и J2 Similarly, W2 = Figure 1: Schematic diagram of a mechanical press. №2. In a typical press application, it is of interest to control the motor angle, 0₁. The parameters of the system are as follows: • J₁ = 0.1722kg • m²: inertia of rotating mass 1; • J₂ = 0.1392kg • m²: inertia of rotating mass 2; . • kı 20491N m/rad: torsional spring constant; and • C₁ = 4.7521N s/m: damping coefficient. By choosing the states x(t) = [01, W1, 02, W2]T, the press can be modeled by the state equation x(t) = = 0 1 0 0 0 -k1/J1 -C1/J1 k1/J1 C1/J1 J₁ x(t) + u(t) 0 0 0 1 k1/J2 C1/J2 -k1/J2-C1/J2] 0 y(t) = [1000] x(t), and the output y(t) is in radians. It is desired to track a reference r(t) that can be produced by using the following code: % Set time vector t = [0:0.001:4]; % Set reference signal r = (t<=0.6).*t*12/0.6 + (t>0.6 & t<=1)*12 +... (t>1 & t<=2).* (18-6*t) +... (t>2 & t<=2.6)*6 + (t>2.6 & t<=3)*12 +... (t>3 & t<=4)*9; figure (1) plot(t, r 'LineWidth', 2) grid , xlabel('Time (s)') ylabel('Reference (rad)') title('Desired reference position') Set the initial conditions as follows: ✗(0) = [7, −3, 2, 1]T, and (0) = 0. Let the tracking error be defined as e(t) = r(t) − y(t). Design a state feedback controller together with a state observer that can track the reference given in the MATLAB code above while satisfying the following design specifications: (a) e(t)|≤7 rad for 0 ≤ t ≤ 0.25s. (b) e(t) ≤ 0.3 rad for t = 0.6s. (c) The settling time (98% criterion) for the step at t = 2.6s should be less than 0.05s. (d) The e(t) ≤6 rad for 2.5s < t <= 4s. A good way to solve such a problem is to first design the state feedback, temporarily assuming that the state vector x(t) is available, and meet the requirements. Then, use x(t) from the observer instead of x(t) and design the observer to still meet the requirements. Provide enough support via plots to show that your design meets all the given specifications. General notes and hints: • Your report should be well written and presented. If you need assistance with writing, contact The Write Place. Be clear and concise. A report not satisfying these requirements will lose 15% credit. • Only include your result plots and any pertinent analysis in the main portion of your report. Submit MATLAB code and Simulink files separately, NOT in the report. • All figures must be carefully labeled. The x and y axes should be labeled, a title used, and a legend should be included when more than one signal is plotted in the same figure. • Submit your report as a PDF along with MATLAB code and Simulink files in electronic form in a ZIP file via Isidore. A report submitted in any format other than PDF will automatically lose 10% credit. • Last, but not least, ensure the work you turn in is your own, from the code to the report. It is OK to talk with other students about the project. It is NOT OK to claim a design or technical explanation very similar to anyone else's as your own. Any evidence of exchanging code, models, model parameters, text, figures, etc., between students past or present will result in an automatic 0 for the project. If you have concerns regarding your design, implementation, report, etc., contact Dr. Kaufman more than 24 hours prior to the due date. 4

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]

4 Find and sketch the Fourier transforms for the following signals. u(t)=(1-|t|) I_{[-1,1]}(t) v(t)=\operatorname{sinc}(2 t) \operatorname{sinc}(4 t) \text {. } \text { (c) } s(t)=v(t) \cos (200 \pi t) \text {. } (d) Classify cach of the signals in (a) (c) as baseband or passband.

A signal x(t) has a Fourier Transform given by X(\omega)=\frac{5(1+j \omega)}{8-\omega^{2}+6 j \omega} Without finding x(t), find the Fourier Transform of the following: а. х(t-3) b. x(4t) C. ei1.e12x(t) d. x(-2t)

1.Consider the coupled acoustic-mechanical system. The velocity of the masses are given by u and applied force by the variable f. The variables k represent the mechanical stiffness, M the mass and b the damping coefficient. The closed open pipe is filled with a fluid having mass density Po, sound speed c,cross sectional area A , length L. а. Using mobility analogy where the velocity as the "across" variable, determine the an equivalent circuit for the system. b. Determine the equations of motion in the Laplace-domain. c. Determine the equations of motion in the time-domain. d. Find the transfer function U2(s)/Uo(s).

. Obtain the Fourier Transform of the following functions \text { a. } x(t)=\left\{\begin{array}{l} e^{a t}, t<0 \\ e^{-a t}, t>0 \end{array}\right\} b. Signal shown below:

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}+u

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.

2. Let s be a periodic signal with period To = 2 and s(t)=\left\{\begin{array}{ll} -t(t-1) & 0 \leq t<1 \\ (t-1)(t-2) & 1 \leq t<2 \end{array}\right. ) Find the first, second, and third derivatives r = Ds, u= D²s, and v = D³s. s) Find the Fourier coefficients of each of the four signals: §, î, û, and û. :) (12 pts) For each of the four signals, compute the power with a time domain calculation and compute the power in frequencies ±1/2 (the positive and negative fundamental frequencies) with a frequency domain calculation. What fraction of the power is in the positive and negative fundamental frequencies? Express all answers both symbolically and with an approximate decimal representation.

1. (30 pts) For each of the following systems, determine whether it is linear and whether it is time-invariant. Justify your answers. If it is LTI, find the impulse response function h(t). Each system is specified by the output y that is produced from an input r. \text { (a) } y(t)=x(t+7) \text { (b) } y(t)=x(3 t) \text { (c) } y(t)=|x(10)| y(t)=\int_{-\infty}^{\infty} I_{[0,+\infty)}(t-\tau) \exp (\tau-t) x(\tau) d \tau y(t)=\int_{-\infty}^{\infty} \frac{1}{1+\tau^{2}} x(\tau-t) d \tau y(t)=\int_{-1}^{0}(\tau-1) x(t+\tau) d \tau y(t)=\min (1, \max (-1, x(t-4))) n) Let (a1,. , ak) be a vector of k nonnegative reals and let (T1,.., Tk) E R*. y(t)=\underset{x \in \mathbb{R}}{\operatorname{argmin}} \sum_{i=1}^{k} a_{j}\left(z-x\left(t-\tau_{i}\right)\right)^{2} The argmin, is the value of z (the argument) that minimizes the expression.

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.