Description

Problem 1: Consider the following signal

where f1(t),f2(t) and f3(t) are as shown below. Determine the L1,L2 and L∞ norm of the signal f(t).

Problem 2: Consider the following state-space model of the short-period approximation of the F-16 longitudinal dynamics
,
with the state vector
,
where α is the angle of attack, q is the pitch rate and wg is the vertical wind gust acting as the disturbance. The vertical acceleration of the aircraft is given by
y = 15.87875α + 1.48113q.
(a) Compute the energy-to-peak gain (Γep) of the system.
(b) Consider a pulse disturbance wg(t) = 2 for 0 ≤ t ≤ 1 and wg(t) = 0 for t > 1. Calculate the energy of this disturbance signal ∥wg∥L2. Simulate the response of the system using MATLAB. Is the system response consistent with the system gain Γep? Why?
(c) Compute the energy-to-energy gain (Γee) (H∞ norm) of the system (by solving the LMI problem in Bounded Real Lemma). Estimate the energy of the response of the system, i.e., ∥y∥L2, to the pulse disturbance wg(t). Is the system response consistent with the system gain Γee? Why?
(d) Let G(s) be the transfer function of the above system. Plot |G(jω)| as a function of ω. Verify that the peak value of the plot provides the energy-to-energy gain of the system.
Problem 3: Consider the following linear uncertain system
q¨(t) + (5 + 2δ)q˙(t) + (4 + δ)q(t) = 0
where δ(t) is a time-varying uncertainty.
1. Write the system in the state-space form
x˙ = Ax + Kϕ ψ = Mx + Hϕ
with an uncertainty interconnection
ϕ = δ(t)ψ.
2. Discuss in detail how to find a bound γ such that the uncertain system is stable for all uncertainties δ = δ(t) with |δ(t)| < γ.
3. Discuss a solution approach for the same problem when the uncertainty δ is time invariant. Then, find an interval for δ that guarantees that the uncertain system is stable.
NOTE: For Problems 2 and 3, please attach your MATLAB (or Python) files and outputs.
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