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Course Name: Linear Algebra and Linear Control Systems Instructions: All questions are compulsory. The exam contains 5 questions. Submit the answer script with pdf format by July 10, 2021.

Q.1) Consider an 1000 × 1000 sparse random matrix A by using MATLAB command sprand and a vector b ∈ R1000×1 now solve a linear system Ax = b for x using: ? Matrix inversion. ? MATLAB Backslash i.e., x = A \ b ? The lu decomposition of A with row and column Permutation matrices. 20 marks Q.2) Consider a state space system: (A, B, C, D) = ? ? ? ? ?1 2 3 0 ?6 1 0 0 ?12 ? ? , ? ? 0 1 1 ? ? , 1 0 0 , 0 ? ? . Compute the controllability and the observability Gramians of the system and hence comment whether the system is balanced or not. If the system is not balanced compute balancing transformations and find a balanced system. 20 marks Q.3) Write some advantages and disadvantages of balanced truncation and iterapolatory projection techniques via IRKA. Apply both the methods to ISS for finding same dimensional (e.g., 50 or 30) ROM. Now discuss the results to compare the efficiency of the methods. 20 marks Q.4) Compute different dimensional reduced models e.g., 30, 50, and 70 applying the IRKA to the CD player (CDP) model. Using MATLAB R function hsvplot plot the singular values of the original and reduced models on a single graph and discuss the results. 20 marks Q.5) Using MATLAB R function lyap find the controllability and observability Gramians of the CDP. Find the rank of the Gramians. If the Gramians have rank deficiency approximate them by their low-ranks factors. Then find the system Hankel singular values using the Gramian factors. Compare them with the original system Hankel singular values. 20 marks

COM S 331: Theory of Computing, Summer 2021 Homework Assignment 8 Due at 11:59PM, Wednesday, July 7, on Gradescope.

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