Instructions:  On a separate sheet of paper, answer each of…

Instructions:  On a separate sheet of paper, answer each of the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (15 pts)  Determine if the following discrete-time system, x=x+u=Ax+bu,{“version”:”1.1″,”math”:”\begin{eqnarray*} x &=&\left x+\leftu\\ &=& A x+ b u, \end{eqnarray*}”} is reachable, controllable or neither. Carefully justify your answer. Problem 2. (15 pts)  For the discrete-time dynamical system, x=Ax=xy=cx=x,{“version”:”1.1″,”math”:” \begin{eqnarray*} x &=& A x = \left x \\ y &=& c x =\left x, \end{eqnarray*}”} find the initial state x{“version”:”1.1″,”math”:”\( x \)”} such that y=2{“version”:”1.1″,”math”:”\( y=2 \) “} and y=6{“version”:”1.1″,”math”:”\( y=6 \)”}. Problem 3. (15 pts)  Let x˙=Ax+bu=x+u,y=cx+du=x−3u.{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{ x} &=& A x + b u =\left x + \leftu,\\ y &=& c x + du= \left x – 3u. \end{eqnarray*}”} (5 pts) Find the state transformation that transforms the pair (A,b){“version”:”1.1″,”math”:”\( ( A, b) \)”} into the controller form. (10 pts) Find the representation of the dynamical system model in the new coordinates. Problem 4. (20 pts)  Construct a state-space model for the transfer function G(s)=.{“version”:”1.1″,”math”:”G(s)=\left. “} Problem 5. (20 pts) 1. (10 pts) For what range of the parameter γ{“version”:”1.1″,”math”:”\( \gamma \)”} the quadratic form x⊤x{“version”:”1.1″,”math”:”x^{\top}\left x “}  is negative semi-definite? 2. (10 pts) For what range of the parameter γ{“version”:”1.1″,”math”:”\( \gamma \)”} this quadratic form is positive semi-definite? Problem 6. (15 pts) For the system modeled by x˙=Ax+bu=x+u,{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{ x}&=& A x+ b u\\ &=&\left x+\leftu, \end{eqnarray*}”} (10 pts) construct a state-feedback control law, u=−kx+r{“version”:”1.1″,”math”:”\( u=- k x+r \)”}, such that the closed-loop system poles are located at −1±j2{“version”:”1.1″,”math”:”\(-1 \pm j2 \)”}; (5 pts) Let y=cx+du=x−3u.{“version”:”1.1″,”math”:”\( y = c x+du=\left x -3 u. \)”} Find the transfer function,Y(s)R(s){“version”:”1.1″,”math”:”\( \frac{Y(s)}{R(s)} \)”} , of the closed-loop system. Problem 7. (20 pts) Find the transfer function of the system described by the following state-space equations,  x˙(t)=x(t)+u(t)y(t)=x(t)+u(t).{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{x}(t) &=&\leftx(t) + \leftu(t)\\ y(t) &=& \left x(t) + \leftu(t). \end{eqnarray*}”} Problem 8. (15 pts)  For the following linear time-varying (LTV) system model, x˙(t)=x(t);{“version”:”1.1″,”math”:”\dot{ x}(t)=\left x(t); “} (10 pts) Find the state transition matrix Φ(t,τ){“version”:”1.1″,”math”:”\( \Phi(t,\tau)\)”}; (5 pts) Let x(2)=⊤{“version”:”1.1″,”math”:”\( x(2)=\left^{\top}\)”}. Find x(1){“version”:”1.1″,”math”:”\( x(1) \)”}. Problem 9. (15 pts)  Solve the equation x˙(t)=(cos⁡t)x(t),x(0)=5,{“version”:”1.1″,”math”:”\”} then find x(π){“version”:”1.1″,”math”:”\( x(\pi)\)”}. Problem 10. (15 pts)  Recall that an equilibrium state of a dynamical system is a state of rest, that is, the system starting from that state stays there thereafter. Determine the equilibrium state of the following continuous-time (CT) system, x˙(t)=x(t)−.{“version”:”1.1″,”math”:”\dot{x}(t)=\leftx(t)-\left. “} Problem 11. (15 pts)  For the system described by the following state-space equations, x˙(t)=x(t)+u(t)],{“version”:”1.1″,”math”:”\dot{x}(t) =\left x(t) + \leftu(t)],”}   construct a state-feedback controller u=−kx{“version”:”1.1″,”math”:”\(u=- k x\) “} and determine the parameter a{“version”:”1.1″,”math”:”\(a \) “} such that the closed-loop system poles are all at −2{“version”:”1.1″,”math”:”\(-2\)”}. Problem 12. (20 pts)  Consider the following model of a discrete-time (DT)dynamical system: x=Ax+bu=x+uy=cx+du=x−3u.{“version”:”1.1″,”math”:”\begin{eqnarray*} x &=& A x + bu=\left x + \leftu\\ y &=& c x+du=\left x -3 u. \end{eqnarray*}”} (10 pts) Design an asymptotic state observer for the above system with the observer poles located at −3{“version”:”1.1″,”math”:”\( -3 \)”} and −4{“version”:”1.1″,”math”:”\( -4\)”}. Write the equations of the observer dynamics. (10 pts) Denote the observer state by x~{“version”:”1.1″,”math”:”\(\tilde{x}\)”}. Let u=−kx~+r{“version”:”1.1″,”math”:”\( u=- k \tilde{x} + r\)”}. Determine the feedback gain k{“version”:”1.1″,”math”:”\( k \)”} such that the controller’s poles are at −1{“version”:”1.1″,”math”:”\( -1 \)”} and −2{“version”:”1.1″,”math”:”\( -2 \)”}, that is, det=(z+1)(z+2).{“version”:”1.1″,”math”:”\=(z+1)(z+2).\]”}Find the transfer function, YR{“version”:”1.1″,”math”:”\( \frac{Y}{R} \)”},  of the closed-loop system driven by the combined observer controller compensator. *** Congratulations, you are almost done with the Final Exam.  DO NOT end the Examity session until you have submitted your work to Gradescope.  When you have answered all questions:  Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible.  Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.).  Click this link to go to Gradescope: Final Exam Submit your exam to the assignment Final Exam.  Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam.  End the Examity session.   
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This question values 45% of final exam.   Assume you are the…

This question values 45% of final exam.   Assume you are the contracting officer. In each of the following four scenarios, assume a government customer comes to you with a requirement (as described below).   Advise how you would structure the requirement to include:  contract type; appropriate contract incentive(s) and legitimate penalty/penalties, if any; contract structure (single requirement; options; multiple award contracts…) whether the requirement must be competed or not (and the basis for not competing); and The contracting methodology you would use to solicit offers/proposals (e.g., sealed bid, negotiated procurement, etc.).  You may propose more than one methodology.  If you choose a negotiated procurement method, you do not have to proceed further and describe the evaluation factors that should be considered.   The four scenarios are: The US Public Health Service has a responsibility to promote and advance the health and safety of the United States.   It responds to natural disasters, disease outbreaks and terrorist attacks in the United States and overseas.  It also operates a number of public health hospitals across the United States.    To better concentrate on its core mission, the department decides it needs a contractor to manage and staff its facilities.   The contractor will control access to each facility, and provide supplemental medical staffing at the hospitals, to include physicians, specialists, nurses, pharmacists, and support personnel.    The staffing requirements at each facility are not fully known, but the department needs an instrument that is flexible that will allow it to add staffing.  This requirement initially will last for five years.  The government project manager does not want to pay for any price adjustments due to inflation or other causes, but is willing to reward the contractor for quality performance and responsiveness.  Congress has directed the Army to complete an independent study of its mechanized infantry and armor needs, to include staffing and organization of armor, and for equipping and modernizing the force.  Congress has given the Army a date for completing and submitting the report;  if the report is not submitted on time, armor units will be moved into the national guard and will not be immediately available for mobilization.   The study must be completed within six weeks, to allow for its staffing and review by the Army and Defense Department leadership.  There is also not enough time to allow offerors 30 days to submit proposals if the report is to be completed on time.  The Armor School has identified three potential contractors who have the experience and capabilities to complete the study on time. Fort Bragg maintains a centralized supply office for government offices to obtain their supplies.  The supply office requires a number of contractors to provide necessary supply items.   The requirement is continuing in nature and will last for at least three years.  The supply office is willing to pay contractors for price increases on an annual basis.  The office however must meet customer demands.  It wants the contractors to pay costs for not meeting established delivery dates.  Countries that can launch ballistic missiles at the United States may include dummy warheads in the single warhead fired at the United States or its allies.  The Missile Defense Agency has been tasked to develop a multiple kill vehicle (MKV).   The MKV is designed to destroy multiple warheads (to include dummy warheads) fired from the same enemy missile.   The winning contractor must possess capabilities to detect and discriminate against multiple warheads and destroy them, as well as the ability to develop a MKV vehicle.  The government project manager advises appropriate provisions should be included to incentivize the contractor to meet technical and cost objectives. 
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