Consider the nonlinear dynamical system: * (1)=x(1)+3tan.x2 (1) + u(1), *2 (1) sinx2 (1)+3x1(1), y(t)=x(1). which...

Transcribed Image Text:

Consider the nonlinear dynamical system: * (1)=x(1)+3tan.x2 (1) + u(1), *2 (1) sinx2 (1)+3x1(1), y(t)=x(1). which has two states x () and x2(); a single input u(1); and a single output y(r). (a) (5 points) Linearize this dynamical system about the zero equilibrium point, and write the linearized dynamical system in state space form. (b) (5 points) Compute the transfer function Y(s)/U(s). (c) (5 points) Show that the linearized-system is controllable, despite being unstable. (d) (5 points) Use Ackermann's formula to place the closed loop system eigenvalues at -3 and -5. What is your feedback gain matrix? VI. (10 points) Consider the two algorithms for finding the nth Fibonacci number that were discussed in class. A. The top-down divide and conquer algorithm, where the basic operation is a recursive call, has a complexity function T(n) which is (Circle the correct answer) (a) linear in n (b) quadratic in n (c) logarithmic in n (d) exponential in n B. The bottom-up dynamic programming algorithm, where addition is the basic operation, has a complexity function T(n) which is (Circle the correct answer) (a) linear in n (b) quadratic in n (c) logarithmic in n (d) exponential in n VII. (10 points) Consider the two algorithms for finding the binomial coefficients that were discussed in class. A. The top-down divide and conquer algorithm, where the basic operation is a recursive call, has a complexity function T(n, k) which is (Circle the correct answer) (a) less than (1) (b) about equal to (1) (c) greater than (2) (d) less than (k) B. The bottom-up dynamic programming algorithm, where addition is the basic operation, has a complexity function T(n, k) which is (Circle the correct answer) (a) less than nk (b) exactly nk (c) greater than nk (d) equal to (k) Consider the nonlinear dynamical system: * (1)=x(1)+3tan.x2 (1) + u(1), *2 (1) sinx2 (1)+3x1(1), y(t)=x(1). which has two states x () and x2(); a single input u(1); and a single output y(r). (a) (5 points) Linearize this dynamical system about the zero equilibrium point, and write the linearized dynamical system in state space form. (b) (5 points) Compute the transfer function Y(s)/U(s). (c) (5 points) Show that the linearized-system is controllable, despite being unstable. (d) (5 points) Use Ackermann's formula to place the closed loop system eigenvalues at -3 and -5. What is your feedback gain matrix? VI. (10 points) Consider the two algorithms for finding the nth Fibonacci number that were discussed in class. A. The top-down divide and conquer algorithm, where the basic operation is a recursive call, has a complexity function T(n) which is (Circle the correct answer) (a) linear in n (b) quadratic in n (c) logarithmic in n (d) exponential in n B. The bottom-up dynamic programming algorithm, where addition is the basic operation, has a complexity function T(n) which is (Circle the correct answer) (a) linear in n (b) quadratic in n (c) logarithmic in n (d) exponential in n VII. (10 points) Consider the two algorithms for finding the binomial coefficients that were discussed in class. A. The top-down divide and conquer algorithm, where the basic operation is a recursive call, has a complexity function T(n, k) which is (Circle the correct answer) (a) less than (1) (b) about equal to (1) (c) greater than (2) (d) less than (k) B. The bottom-up dynamic programming algorithm, where addition is the basic operation, has a complexity function T(n, k) which is (Circle the correct answer) (a) less than nk (b) exactly nk (c) greater than nk (d) equal to (k)