Problem 1. (14 pts)  Suppose (a) Find constants \(\alpha,\beta,\gamma\in\mathbb R\) such that \(A^{10}=\alpha A^2 +\beta A+\gamma I\). (b) Find all the modes for the continuous-time system \(\dot x=Ax\). Is the system stable, marginally stable, or unstable? (c) Find all the modes for the discrete-time system \(x[k+1]=Ax[k]\). Is the system stable, marginally stable, or unstable? Problem 2. (10 pts)  Consider the system \(\dot x=Ax+Bu\), \(y=Cx\), with A=-201-1-11-100{“version”:”1.1″,”math”:”A=-201-1-11-100″} the same matrix as given in Problem 1, B=100{“version”:”1.1″,”math”:”B=100″}, and C=011{“version”:”1.1″,”math”:”C=011″}.  Suppose x(0)={“version”:”1.1″,”math”:”x(0)=”} 111{“version”:”1.1″,”math”:”111″} and the input \(u(t)\equiv 1\) is the unit step signal.  Find the output \(y(t)\), \(t\ge 0\). Problem 3. (10 pts)  Suppose A=-200001000{“version”:”1.1″,”math”:”A=-200001000″} Consider the LTV system \(\dot x(t)=A(t) x(t)\) with \(A(t)=tA\).  Find the fundamental matrix \(\Phi(t)\) of the system. Problem 4. (10 pts)  Consider the system x·=1-3-1-1x+11u, y=10x{“version”:”1.1″,”math”:”x·=1-3-1-1x+11u, y=10x”}.  Find the transfer function \(\frac{Y(s)}{U(s)}\). Is the system BIBO stable? Problem 5. (10 pts)  Consider the nonlinear system\begin{align*}  \dot x_1 &= 2x_1-x_2-x_1^2 \\  \dot x_2 &= x_1-2x_2+x_2^2\end{align*}which is known to have two equilibrium points xe1=00T{“version”:”1.1″,”math”:”xe1=00T”} and xe2=11T{“version”:”1.1″,”math”:”xe2=11T”}. For each of these two equilibrium points, determine its local stability, if possible. Problem 6. (10 pts) Find a quadratic Lyapunov function \(V(x)=x^T Px\) for some \(P\succ 0\) for the discrete-time LTI system xk+1={“version”:”1.1″,”math”:”xk+1=”}-0.5010.5{“version”:”1.1″,”math”:”-0.5010.5″}x[k]{“version”:”1.1″,”math”:”x[k]”}.  Problem 7. (14 pts)  Consider the following discrete-time LTI system: x[k+1]=Ax[k]+Bu[k]={“version”:”1.1″,”math”:”x[k+1]=Ax[k]+Bu[k]=”}-1210{“version”:”1.1″,”math”:”-1210″}x[k]+{“version”:”1.1″,”math”:”x[k]+”}1-1{“version”:”1.1″,”math”:”1-1″}u[k],    k=0, 1, ….{“version”:”1.1″,”math”:”u[k],    k=0, 1, ….”} (a) Find a state coordinate transform \(x=T\tilde x\) and the transformed system \((\tilde A, \tilde B)\) where the controllable and uncontrollable parts are separated. (b) Describe the set of all possible values of the eigenvalues of \(A-BK\) for arbitrary \(K\in\mathbb R^{1\times 2}\).  (c) Can the poles of the closed-loop system \(A-BK\) be placed at \(\{-2,-1\}\)? If so, design one such gain \(K\); if not, explain why. Problem 8. (12 pts)  Consider the following discrete-time LTI system x[k+1]= Ax=-1210x[k],     y[k]=Cx[k]=1-1x[k] (a) Given some noisy measurements of the output \(\hat y[0]=1\), \(\hat y[1]=0\), \(\hat y[2]=1\), find the estimate of \(x[0]\) resulting in the least squared error between the predicted and the measured outputs. (b) Design a gain matrix \(L\) such that the poles of \(A-LC\) are placed at \(\{0,0\}\).  (c) Plot the block diagram of the system with the state observer designed in (b). Problem 9 (10 pts) Find a state-space realization of the transfer function  H(s)={“version”:”1.1″,”math”:”H(s)=”}1s+21s(s+2)s+1s(s+2)s-1s{“version”:”1.1″,”math”:”1s+21s(s+2)s+1s(s+2)s-1s”}. Congratulations, you are almost done with the Final Exam.  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 submit your work 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 Honorlock session.   

Part II (10 points) – Structural and Behavioral Modeling Canvas is a course management system. A system administrator uses the Canvas system to create course templates and to enroll students. A professor can use the system to add teaching materials including lecture notes and assignments, and to enter student grades. A student can access his/her enrolled courses to read and download teaching materials, to submit assignments, and to check grades. Canvas ensures that only an authorized person can access the relevant information, i.e. only the professor of a class can enter/change the grades of the students taking that class, and only a student can check his/her own grades. (1) (5 points) Develop a Structural Model (class diagram) based on the above system description (show essential entity classes, a few boundary and controller classes, and their correct relationships): (2) (5 points) Develop a UML Behavioral Model (sequence diagram) for the scenario that a student successfully submits a homework assignment including the authentication step (show instances of all relevant classes – boundary, controller, and entity classes; and all important messages – operations/methods with necessary parameters).

In the final paragraph, what does the speaker say he lacks on earth?    

What epithet does the Rood use to describe Jesus?  

In the sentence begins “Of old,” what does the Rood say about itself?  

Just based on the correlation matrix, Jeff believes that sugars is the most important attribute customers consider in ratings. Which of the following is correct regarding Jeff’s conjecture?

Making food the world loves Jeff Harmening, the CEO of General Mills, strives to achieve their slogan of “making food the world loves.” He first collected data on his dependent variable of consumer ratings from 0 (low) to 100 (high) for all cereal brands. He then considered several independent (x) variables. He created an indicator variable generalmills to compare cereals produced by General Mills to cereals produced by competitors. He also collected data on the nutrition of each, including the amount of calories, fiber, and sugars. Below is the correlation matrix of the variables:

The operating expenses of a distributor were 38% of gross sales. If the operating expenses were $4,250.00, what were the gross sales? 

Which of the following is the null and alternative hypothesis used to calculate the p-value for the density variable in the regression output?

Which of the following R code would give the p-value for fiber that is in the last column of the regression output? Hint: Think carefully about whether you are looking at a one-sided (left-/right-tail) or a two sided alternative hypothesis.