Description

Problem 1: Consider the following linear uncertain system
x˙ = Ax + KΦ (1)
Ψ = Mx + HΦ (2)
with the uncertainty interconnection Φ = ∆Ψ. Use the LMI-based representation of the Small Gain Theorem (SGT) to answer the questions below.
1. Consider the following uncertain system

and write the system model in the form of (1)-(2) with ∆ = δ.
2. Use the LMI representation of SGT to compute the maximum bound for δ that guarantees stability of the uncertain system.
3. Plot the root locus of the system as a function of δ (use MATLAB for this). What is the maximum value of |δ| such that the system has eigenvalues with negative real part? Is this consistent with the result in part 2?
4. Repeat parts 2 and 3 for the following uncertain system

5. Simulate this system when δ(t) = cos(2t). Is the system stable or unstable for the given δ(t)? Is your answer consistent with the result from the SGT analysis? Why?
6. Comment on the stability of the system to time-invariant and time-varying perturbations. Can eigenvalue conditions guarantee stability to time-varying perturbations? What about the SGT condition? Problem 2: Consider the following linear plant model

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and the following controller
x˙c = −4xc + 2z
u = xc − 2z.
Using MATLAB commands, determine the closed-loop system equations
x˙cl = Aclxcl + Bclw
y = Cclxcl + Dclw.
Then, examine the system stability (whether the closed-loop system is stable or not), and calculate the H∞ norm of the closed-loop system.
Problem 3: Consider the following linear plant models

For each of the above systems, determine if the system 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: Please attach your MATLAB (or Python) files and outputs.
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