 Suggested languages for you:

Europe

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 $$u(t)$$.
See the step by step solution

## Step 1: Identify the system matrices and initial state

Firstly, we identify the system matrices A, B, and C, and the initial state x(0) as given in the problem: A = $$\begin{pmatrix} 0 & 1 \\ -\lambda^2 & 0 \end{pmatrix}$$ B = $$\begin{pmatrix} 0 \\ -1 \end{pmatrix}$$ C = $$\begin{pmatrix} 1 & 0 \end{pmatrix}$$ x(0) = $$\begin{pmatrix} 0 \\ \lambda \end{pmatrix}$$

## Step 2: Obtain the state transition matrix, Phi(t)

Next, we need to find the state transition matrix, Phi(t), which describes the evolution of the system's state over time. The state transition matrix for a linear time-invariant system with matrix A is given by $$e^{At}$$. Given that exponentiation of the matrix A is laborious, in this case, we take advantage of the given facts: $$x(t) = \begin{pmatrix} \sin\lambda t \\ \lambda\cos\lambda t \end{pmatrix}$$ The state transition matrix, Phi(t), can be directly derived from x(t) as follows: $$Phi(t) = \begin{pmatrix} \sin\lambda t & \frac{\cos\lambda t}{\lambda} \\ \lambda\cos\lambda t & -\sin\lambda t \end{pmatrix}$$

## Step 3: Compute the first-order response

The first-order response is given by the integral: $$x_1(t) = \int_0^t \Phi(t-\tau)Bu(\tau) d\tau$$ Since B is a column matrix with only the second element non-zero, this integral simplifies to: $$x_1(t) = \int_0^t \Phi(t-\tau)\begin{pmatrix} 0 \\ -u(\tau) \end{pmatrix} d\tau$$ The first-order response y_1(t) is given by: $$y_1(t) = Cx_1(t)$$

## Step 4: Compute the second-order response

The second-order response is given by the integral: $$x_2(t) = \int_0^t \Phi(t-\tau)B\left(\int_0^\tau\Phi(\tau-\sigma)Bu(\sigma) d\sigma\right) d\tau$$ This integral may be analytically computed with a reasonable amount of work. The second-order response y_2(t) is given by: $$y_2(t) = Cx_2(t)$$

## Step 5: Compute the third-order response

Similarly, the third-order response is given by the integral: $$x_3(t) = \int_0^t \Phi(t-\tau)B\left(\int_0^\tau\Phi(\tau-\sigma)B\left(\int_0^\sigma\Phi(\sigma-\rho)Bu(\rho) d\rho\right) d\sigma\right) d\tau$$ This integral is computationally intensive and may not be analytically solvable. The third-order response y_3(t) is given by: $$y_3(t) = Cx_3(t)$$ To find the first three terms of the Volterra representation of y(t), compute y_1(t), y_2(t), and y_3(t). To simplify, it is more appropriate to find a numerical solution or use approximations based on the input function u(t).

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features ## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner 