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.

Can you use MATLAB?

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)dt + Gdw, dx = f(x,u,t)dt+Gdw, (1) (2) where x 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 = [000] (3) where σp Є R represents the degree of the uncertainty, and let us take σ₁ = 2 and σ2 = 3. Assume that the initial state is deterministic and e(t = 0) = [0,0]. Take the following steps to simulate the given SDE for 1 € [0, 1]: (a): Consider the increments of w between each time interval [+1). Derive the analytical expression of Aw using w~N(02, 12), where Aw, w(tk+1) — w(tk). (c): Derive the approximate continuous-time EoM from…

Can you help me by providing the MATLAB code?