Thyrоxine (T4) is а hоrmоne thаt certаin animals use to regulate the activity of cells. The production of T4 depends on the concentrations of two other hormones: thyrotropin releasing hormone (TRH) and thyroid stimulating hormone (TSH). TRH comes from a part of the brain called the hypothalamus. This hormone can bind to receptors in another part of the brain, called the pituitary gland. Upon binding to a receptor, TRH activates a signaling pathway that causes a cell to release TSH. The figure shows the signaling pathway that controls the production of T4. TSH binds to receptors in the thyroid gland, causing cells of this gland to produce T4. The production of T4 depends on an enzyme called thyroid peroxidase (TPO), which creates T4 by adding an iodide ion (I-) to a molecule. A nonpolar structure enables T4 to diffuse through the phospholipid bilayer of any cell, including cells of the hypothalamus and the pituitary gland, where T4 inhibits the release of TRH and TSH, respectively. About 1 in 200 people have a genetic condition called Graves’ disease. A person with this condition produces an excess of proteins that bind to TSH receptors, activating more of these receptors than usual. The next figure shows four hypothetical relationships between the number of TSH receptors activated by proteins and the concentration of T4 in a cell, labeled A through D. Which relationship accurately describes the expected relationship in a patient with Graves’ disease?
Prоblem 1. (15 pts) Cоnsider the system depicted in the аbоve block diаgrаm, with the input (u) and output (y). Note that the three blocks marked with "(s^{-1})" are the integrators (i.e., "(int)"). Identify a set of state variables and derive the state space model of the system. Problem 2. (30 pts) Let (A=begin{bmatrix}1& 2\-3&-4end{bmatrix}). (a) (5 pts) Find the eigenvalues of (A). (Hint: they are two integers.) (b) (10 pts) Express (A^{100}) as a proper linear combination of (A) and the identity matrix (I). (c) (10 pts) Use your favorite method to find the analytic expression of (e^{At}). (d) (5 pts) Write the solution (x(t)) starting from x(0)={"version":"1.1","math":"x(0)="}1-1{"version":"1.1","math":"1-1"} as a linear combination of the modes. Problem 3. (24 pts) Consider the matrix defined below ((T) is nonsingular): A=v1v2v3v4⏟T{"version":"1.1","math":"A=v1v2v3v4⏟T"}-0.5-110-10⏟J{"version":"1.1","math":"-0.5-110-10⏟J"}w1Tw2Tw3Tw4T⏟T-1{"version":"1.1","math":"w1Tw2Tw3Tw4T⏟T-1"} First consider the continuous-time LTI system (dot x=Ax). (a) (2 pts) Is the continuous-time system stable, marginally stable, or unstable? Explain why. (b) (4 pts) Find all the modes of the system (dot x=Ax). (c) (6 pts) For each of the following properties, determine if there exists some (x(0)neq 0) so that the resulting solution (x(t)) satisfies the property. If yes, find one such (x(0) ); If no, explain why. (i) (x(t)to 0) (ii) (x(t)) is unbounded (iii) (x(t)) is bounded and does not converge to (0) Next consider the discrete-time LTI system (x[k+1]=Ax[k] ). (d) (2 pts) Is the discrete-time system stable, marginally stable, or unstable? Explain why. (e) (4 pts) Find all the modes of the system (x[k+1]=Ax[k] ). (f) (6 pts) For each of the following properties, determine if there exists some (x[0]neq 0) so that the resulting solution (x[k]) satisfies the property. If yes, find one such (x[0] ); If no, explain why. (i) (x[k]to 0) (ii) (x[k]) is unbounded (iii) (x[k]) is bounded and does not converge to (0) Problem 4. (15 pts) Find the fundamental matrix (Phi(t)) of the LTV system (dot x(t) = begin{bmatrix} -frac{1}{t+2} & e^{t} \ 0 & -1 end{bmatrix} x(t)). Problem 5. (16 pts) Consider a system (dot x=Ax) with (xinmathbb R^3) and (Ainmathbb R^{3times 3}). Suppose the system has a solution x(t)={"version":"1.1","math":"x(t)="}e-t+11+te-t(1+t)e-t{"version":"1.1","math":"e-t+11+te-t(1+t)e-t"}. (a) (5 pts) Find the eigenvalues of (A) and their Jordan block sizes. (b) (5 pts) Write the given (x(t)) as a linear combination of the modes of the system. (c) (6 pts) Find (A) and (x(0)) that results in the solution (x(t)). Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Honorlock 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 to submit your work: Midterm Exam 1 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.