Description

Problem 1: Represent the inequalities
P < ATPA + Q−ATPB(R + BTPB)−1BTPA
P > 0
where R = RT > 0, as a single linear matrix inequality (in terms of the variable P).

Problem 2: Consider the unforced system x˙ = Ax, where the system matrix A can be either (i) or (ii) below

Write code for an LMI feasibility problem to determine if each of the systems with the given A matrix has eigenvalues to the left of the vertical line s = −2 in the complex plane. Then, confirm your results by finding the eigenvalues of A for both cases in MATLAB (or Python).
Problem 3: Consider the following LTI systems

Determine if the two systems above can be stabilized by a static state-feedback control law u = Kxp. For the systems that are stabilizable, determine such a stabilizing control law, i.e., matrix gain K.
NOTE: For Problems 2 and 3, please attach your MATLAB (or Python) files and outputs.
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