We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form of a set of stochastic differential equation (SDE) as follows: dx=(Ax+ Bu)dt + Gdw, da f(x, u, t)dt + Gdw, (1) (2) where is the state, u is the control, and dw is a differential increment of standard Brownian motion, i.e., E[dw] = 0 and E [dw(t)dw (t)] = dt-1. In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]" (no control), and dw R². The matrices A, B, and G are given as follows: A 02x2, B-02x2, G= 02 (3) where σ, E R represents the degree of the uncertainty, and let us take o₁ = 2 and 02 = 3. Assume that the initial state is deterministic and (t = 0) = [0,0]. Take the following steps to simulate the given SDE for tЄ [0, 1]: Consider the increments of w between each time interval te [tktk+1). Derive the analytical expression of Awk using w~N(02, 12), where Awkw(tk+1) - w(tk). Generate and plot the time history of Awk, Vk, with Atktk+1-tk=10-3, Vk, for sample number M = 20; include 3-0 bounds in the plot and discuss the consistency with the Monte Carlo result. Hint: an increment of standard Brownian motion is white Gaussian noise.
We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form of a set of stochastic differential equation (SDE) as follows: dx=(Ax+ Bu)dt + Gdw, da f(x, u, t)dt + Gdw, (1) (2) where is the state, u is the control, and dw is a differential increment of standard Brownian motion, i.e., E[dw] = 0 and E [dw(t)dw (t)] = dt-1. In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]" (no control), and dw R². The matrices A, B, and G are given as follows: A 02x2, B-02x2, G= 02 (3) where σ, E R represents the degree of the uncertainty, and let us take o₁ = 2 and 02 = 3. Assume that the initial state is deterministic and (t = 0) = [0,0]. Take the following steps to simulate the given SDE for tЄ [0, 1]: Consider the increments of w between each time interval te [tktk+1). Derive the analytical expression of Awk using w~N(02, 12), where Awkw(tk+1) - w(tk). Generate and plot the time history of Awk, Vk, with Atktk+1-tk=10-3, Vk, for sample number M = 20; include 3-0 bounds in the plot and discuss the consistency with the Monte Carlo result. Hint: an increment of standard Brownian motion is white Gaussian noise.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form
of a set of stochastic differential equation (SDE) as follows:
dx=(Ax+ Bu)dt + Gdw,
da f(x, u, t)dt + Gdw,
(1)
(2)
where is the state, u is the control, and dw is a differential increment of standard Brownian motion, i.e.,
E[dw] = 0 and E [dw(t)dw (t)] = dt-1.
In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]"
(no control), and dw R². The matrices A, B, and G are given as follows:
A 02x2, B-02x2, G=
02
(3)
where σ, E R represents the degree of the uncertainty, and let us take o₁ = 2 and 02 = 3. Assume that the
initial state is deterministic and (t = 0) = [0,0]. Take the following steps to simulate the given SDE for
tЄ [0, 1]:
Consider the increments of w between each time interval te [tktk+1). Derive the analytical expression
of Awk using w~N(02, 12), where Awkw(tk+1) - w(tk).
Generate and plot the time history of Awk, Vk, with Atktk+1-tk=10-3, Vk, for sample number
M = 20; include 3-0 bounds in the plot and discuss the consistency with the Monte Carlo result.
Hint: an increment of standard Brownian motion is white Gaussian noise.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fc6e0f6cc-d8ce-4b90-8492-c04ef0230260%2Fl6was4t_processed.png&w=3840&q=75)
Transcribed Image Text:We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form
of a set of stochastic differential equation (SDE) as follows:
dx=(Ax+ Bu)dt + Gdw,
da f(x, u, t)dt + Gdw,
(1)
(2)
where is the state, u is the control, and dw is a differential increment of standard Brownian motion, i.e.,
E[dw] = 0 and E [dw(t)dw (t)] = dt-1.
In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]"
(no control), and dw R². The matrices A, B, and G are given as follows:
A 02x2, B-02x2, G=
02
(3)
where σ, E R represents the degree of the uncertainty, and let us take o₁ = 2 and 02 = 3. Assume that the
initial state is deterministic and (t = 0) = [0,0]. Take the following steps to simulate the given SDE for
tЄ [0, 1]:
Consider the increments of w between each time interval te [tktk+1). Derive the analytical expression
of Awk using w~N(02, 12), where Awkw(tk+1) - w(tk).
Generate and plot the time history of Awk, Vk, with Atktk+1-tk=10-3, Vk, for sample number
M = 20; include 3-0 bounds in the plot and discuss the consistency with the Monte Carlo result.
Hint: an increment of standard Brownian motion is white Gaussian noise.
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