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ME-GY 6703 Linear Control Theory and Design, Spring 2024

Homework Assignment 1

1. Given the matrix A below, compute  such that its entry on the first row and first column ()11 is equal to 4.

2. Given the matrix A below, compute eA via diagonalization.

3. If A(t) is a continuously-differentiable N × N matrix function that is invertible at each t, show that

4. Compute the spectral norm of the following matrices:

5. Consider the nonlinear state equation, with a, b, c, d, f nonzero parameters,

Show that the origin is an equilibrium and derive a linearized model to describe perturbations in the vicinity of the origin.

6. Linearize the nonlinear state equation

about the nominal trajectory arising from  (0) = 0,  (0) = 1, and  (t) = 1 for all t ≥ 0. Is the linear system time-varying?