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Fibonacci sequence. The Fibonacci sequence yo, y1, Y2₂, ... starts with yo = 0, y₁ = 1, and for t = 2, 3,..., Yt Yt-1+Yt-2, i.e., the sum of the previous two entries. (a) Express this as a time-invariant linear dynamical system with state xt = t = 1,2,.... That is, find A and C, where Xt+1 = Axt and Yt = Cxt. (yt, Yt-1) and output yt, for (b) Use your matrix expression above and MATLAB to compute and plot the Fibonacci sequence y, for t = 0, ..., 20. (c) Using MATLAB, simulate a modified Fibonacci sequence Zo, Z₁, Z2, ..., which starts with the same = 1, but for t = 2, 3,..., values Zo = 0 and Z₁ Zt Zt-1 Zt-2, i.e., the difference of the two previous values. Plot this sequence z, for t = 0, ..., 20.

Fibonacci sequence. The Fibonacci sequence yo, y1, Y2₂, ... starts with yo = 0, y₁ = 1, and for t = 2, 3,..., Yt Yt-1+Yt-2, i.e., the sum of the previous two entries. (a) Express this as a time-invariant linear dynamical system with state xt = t = 1,2,.... That is, find A and C, where Xt+1 = Axt and Yt = Cxt. (yt, Yt-1) and output yt, for (b) Use your matrix expression above and MATLAB to compute and plot the Fibonacci sequence y, for t = 0, ..., 20. (c) Using MATLAB, simulate a modified Fibonacci sequence Zo, Z₁, Z2, ..., which starts with the same = 1, but for t = 2, 3,..., values Zo = 0 and Z₁ Zt Zt-1 Zt-2, i.e., the difference of the two previous values. Plot this sequence z, for t = 0, ..., 20.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.4: Fractional Expressions
Problem 65E
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## Fibonacci Sequence The Fibonacci sequence \( y_0, y_1, y_2, \ldots \) starts with \( y_0 = 0 \), \( y_1 = 1 \), and for \( t = 2, 3, \ldots \), \[ y_t = y_{t-1} + y_{t-2}, \] i.e., the sum of the previous two entries. ### Tasks: (a) **Linear Dynamical System Representation** - Express this as a time-invariant linear dynamical system with state \( x_t = (y_t, y_{t-1}) \) and output \( y_t \), for \( t = 1, 2, \ldots \). That is, find \( A \) and \( C \), where \[ x_{t+1} = Ax_t \quad \text{and} \quad y_t = Cx_t. \] (b) **MATLAB Simulation** - Use your matrix expression above and MATLAB to compute and plot the Fibonacci sequence \( y_t \) for \( t = 0, \ldots, 20 \). (c) **Modified Fibonacci Sequence** - Using MATLAB, simulate a modified Fibonacci sequence \( z_0, z_1, z_2, \ldots \), which starts with the same values \( z_0 = 0 \) and \( z_1 = 1 \), but for \( t = 2, 3, \ldots \), \[ z_t = z_{t-1} - z_{t-2}, \] i.e., the difference of the two previous values. Plot this sequence \( z_t \) for \( t = 0, \ldots, 20 \).
Transcribed Image Text:## Fibonacci Sequence The Fibonacci sequence \( y_0, y_1, y_2, \ldots \) starts with \( y_0 = 0 \), \( y_1 = 1 \), and for \( t = 2, 3, \ldots \), \[ y_t = y_{t-1} + y_{t-2}, \] i.e., the sum of the previous two entries. ### Tasks: (a) **Linear Dynamical System Representation** - Express this as a time-invariant linear dynamical system with state \( x_t = (y_t, y_{t-1}) \) and output \( y_t \), for \( t = 1, 2, \ldots \). That is, find \( A \) and \( C \), where \[ x_{t+1} = Ax_t \quad \text{and} \quad y_t = Cx_t. \] (b) **MATLAB Simulation** - Use your matrix expression above and MATLAB to compute and plot the Fibonacci sequence \( y_t \) for \( t = 0, \ldots, 20 \). (c) **Modified Fibonacci Sequence** - Using MATLAB, simulate a modified Fibonacci sequence \( z_0, z_1, z_2, \ldots \), which starts with the same values \( z_0 = 0 \) and \( z_1 = 1 \), but for \( t = 2, 3, \ldots \), \[ z_t = z_{t-1} - z_{t-2}, \] i.e., the difference of the two previous values. Plot this sequence \( z_t \) for \( t = 0, \ldots, 20 \).
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