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Problem 1

The transmission of FM signals is based on the modulation of frequency based on an input voltage. A simple circuit generating such modulated signals is the one satisfying the differential equation $$ \ddot{y}(t)+\left[\lambda^{2}+u(t)\right] y(t)=0, $$which corresponds to a harmonic oscillator with frequency controlled about the nominal value \(\lambda\). We model this as a time-invariant bilinear system $$ \begin{aligned} \dot{x} &=\left(\begin{array}{cc} 0 & 1 \\ -\lambda^{2} & 0 \end{array}\right) x+u\left(\begin{array}{cc} 0 & 0 \\ -1 & 0 \end{array}\right) x \\ y &=\left(\begin{array}{ll} 1 & 0 \end{array}\right) x \end{aligned} $$ with \(x=\mathbb{R}^{2}, u=\mathbb{R}\), and \(y=\mathbb{R}\). (The " \(B\) " matrix is identically zero.) When \(u \equiv 0\) one has a standard harmonic oscillator, and $$ x(t)=\left(\begin{array}{c} \sin \lambda t \\ \lambda \cos \lambda t \end{array}\right) $$ is a solution, with $$ x(0)=\left(\begin{array}{l} 0 \\ \lambda \end{array}\right) $$ Find the first three terms of the Volterra representation of \(y(t)\) for \((2.68)\), with this initial state.

Expert verified

In summary, the first three terms of the Volterra representation of \(y(t)\) are obtained by computing the first-order, second-order, and third-order responses, \(y_1(t)\), \(y_2(t)\), and \(y_3(t)\), respectively. The state transition matrix, \(\Phi(t)\), is given by
\(\Phi(t) = \begin{pmatrix} \sin\lambda t & \frac{\cos\lambda t}{\lambda} \\ \lambda\cos\lambda t & -\sin\lambda t \end{pmatrix}\)
To obtain the first three Volterra terms, we have to compute the iterated integrals \(x_1(t), x_2(t),\) and \(x_3(t)\) using the system matrices and the state transition matrix. These integrals can be computationally intensive, and it is more appropriate to find a numerical solution or use approximations based on the input function \(u(t)\).

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Chapter 2

Let \(m, n\) be nonnegative integers, $x:=\mathbb{K}^{n}, \mathcal{U}:=\mathbb{K}^{m} .$ Assume that $$ f(t, x, u)=A(t) x+B(t) u $$ where \(A(t)\) is an \(n \times n\) matrix and \(B(t)\) is an \(n \times m\) matrix, each of whose entries is a locally essentially bounded (measurable) function \(\mathbb{R} \rightarrow \mathbb{K}\). Equivalently, $$ f: \mathbb{R} \times \mathcal{X} \times \mathcal{U} \rightarrow \mathcal{X} $$ is linear in \((x, u)\) for each fixed \(t\), and is locally bounded in \(t\) for each fixed \((x, u)\). Conclude that \(f\) is a rhs.

Chapter 2

If \(\Sigma\) is linear and \(\Lambda\) is an integral behavior, then $$ \Lambda_{\Sigma, 0}=\Lambda $$ if and only if $$ \widetilde{K}(t, \tau)=C(t) \Phi(t, \tau) B(\tau) $$ for almost all \((t, \tau)\). Thus, the integral behavior \(\Lambda\) is realizable by some linear system if and only if there exist time-varying matrices $$ (A(t), B(t), C(t)) $$ such that the above conditions hold.

Chapter 2

Exercise 2.7.15 simplifying assumptions, and choosing appropriate units and time scales, we have an equation $$ \ddot{\theta}=\sin \theta-u \cos \theta $$

Chapter 2

Assume that \(K\) in formula (2.27) is continuous. Pick any fixed $i \in\\{1, \ldots, m\\}\( and any \)T>0\(. For each \)\varepsilon>0\(, let \)\omega_{e}$ be the control defined on the interval \([0, T]\) by $$ \omega_{\varepsilon}(t):= \begin{cases}\frac{1}{\varepsilon} e_{i} & \text { if } 0 \leq t<\varepsilon \\ 0 & \text { otherwise }\end{cases} $$ where \(e_{i}=(0, \ldots, 0,1,0, \ldots, 0)^{\prime}\) is the \(i\) th canonical basis vector. Prove that $\lim _{\varepsilon \rightarrow 0} y_{\varepsilon}(T)=K_{i}(T)$,

Chapter 2

Prove that, for any linear discrete-time system and any pair of integers \(\sigma<\tau, \phi(\tau, \sigma, \cdot, \cdot)\) is linear.

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