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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: da (Ax + Bu)dl + Gdw, d = 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. I. Problem Set 9 Linear Stochastic Process In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]T (no control), and dw€ R². The matrices A, B, and G are given as follows: A-02x2, B-02x2, G= (3) where σp E IR represents the degree of the uncertainty, and let us take o₁ = 2 and 2 = 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]: Perform Monte Carlo simulation (again M = 20) by propagating the linear SDE with the approx- imated Brownian motion, and show the time history of each element of a over time; include 3-0 bounds (i.e., ±30) in the plot and discuss the consistency.

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: da (Ax + Bu)dl + Gdw, d = 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. I. Problem Set 9 Linear Stochastic Process In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]T (no control), and dw€ R². The matrices A, B, and G are given as follows: A-02x2, B-02x2, G= (3) where σp E IR represents the degree of the uncertainty, and let us take o₁ = 2 and 2 = 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]: Perform Monte Carlo simulation (again M = 20) by propagating the linear SDE with the approx- imated Brownian motion, and show the time history of each element of a over time; include 3-0 bounds (i.e., ±30) in the plot and discuss the consistency.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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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: da (Ax + Bu)dl + Gdw, d = 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. I. Problem Set 9 Linear Stochastic Process In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]T (no control), and dw€ R². The matrices A, B, and G are given as follows: A-02x2, B-02x2, G= (3) where σp E IR represents the degree of the uncertainty, and let us take o₁ = 2 and 2 = 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]: Perform Monte Carlo simulation (again M = 20) by propagating the linear SDE with the approx- imated Brownian motion, and show the time history of each element of a over time; include 3-0 bounds (i.e., ±30) in the plot and discuss the consistency.
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: da (Ax + Bu)dl + Gdw, d = 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. I. Problem Set 9 Linear Stochastic Process In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0]T (no control), and dw€ R². The matrices A, B, and G are given as follows: A-02x2, B-02x2, G= (3) where σp E IR represents the degree of the uncertainty, and let us take o₁ = 2 and 2 = 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]: Perform Monte Carlo simulation (again M = 20) by propagating the linear SDE with the approx- imated Brownian motion, and show the time history of each element of a over time; include 3-0 bounds (i.e., ±30) in the plot and discuss the consistency.
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