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, dx = 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: 01 A=02x2, B=02x2, G -69 (3) where σp E R represents the degree of the uncertainty, and let us take σ₁ = 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 t = [0, 1]: (a): Consider the increments of w between each time interval t€ [tk,tk+1). Derive the analytical expression of Awk using w~N(02, 12), where Awk = w(tk+1) - w(tk). (b): Generate and plot the time history of Awk, Vk, with Atk = tk+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. (c): Derive the approximate continuous-time EoM from Eq. (1) by assuming that the same noise is applied to the system over the interval t = [tk, tk+1) while satisfying the increment Awk derived in (a). (d): 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: dx = (Ax+ Bu)dt + Gdw, dx = 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: 01 A=02x2, B=02x2, G -69 (3) where σp E R represents the degree of the uncertainty, and let us take σ₁ = 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 t = [0, 1]: (a): Consider the increments of w between each time interval t€ [tk,tk+1). Derive the analytical expression of Awk using w~N(02, 12), where Awk = w(tk+1) - w(tk). (b): Generate and plot the time history of Awk, Vk, with Atk = tk+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. (c): Derive the approximate continuous-time EoM from Eq. (1) by assuming that the same noise is applied to the system over the interval t = [tk, tk+1) while satisfying the increment Awk derived in (a). (d): 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.
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Kreith, Frank; Manglik, Raj M.
Chapter5: Analysis Of Convection Heat Transfer
Section: Chapter Questions
Problem 5.12P
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Can you use MATLAB to help me solve part (d)
![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,
dx = 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:
01
A=02x2, B=02x2, G
-69
(3)
where σp E R represents the degree of the uncertainty, and let us take σ₁ = 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
t = [0, 1]:
(a): Consider the increments of w between each time interval t€ [tk,tk+1). Derive the analytical expression
of Awk using w~N(02, 12), where Awk = w(tk+1) - w(tk).
(b): Generate and plot the time history of Awk, Vk, with Atk = tk+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.
(c): Derive the approximate continuous-time EoM from Eq. (1) by assuming that the same noise is applied
to the system over the interval t = [tk, tk+1) while satisfying the increment Awk derived in (a).
(d): 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fdec12682-e91a-4c9a-a50d-9d070c9829ac%2Fiw1z5m_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,
dx = 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:
01
A=02x2, B=02x2, G
-69
(3)
where σp E R represents the degree of the uncertainty, and let us take σ₁ = 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
t = [0, 1]:
(a): Consider the increments of w between each time interval t€ [tk,tk+1). Derive the analytical expression
of Awk using w~N(02, 12), where Awk = w(tk+1) - w(tk).
(b): Generate and plot the time history of Awk, Vk, with Atk = tk+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.
(c): Derive the approximate continuous-time EoM from Eq. (1) by assuming that the same noise is applied
to the system over the interval t = [tk, tk+1) while satisfying the increment Awk derived in (a).
(d): 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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