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.

Q2 Consider a conical receiver shown in Figure 2. The inlet and outlet liquid volumetric flow rates are Fl and F2, respectiveily. ccorresponding radius in r) Figure 2 Conicul tank Develop the model equation with necessary assumption(s) with respect to the liquid height h. ii. What type of mathematical model is this? 1 R %3D Hint: Model: = F, – Fz, where the volume V=r h=nh, since = substitute V and Fz expressions and get the final form. %3D %3D %3D dt H.

The pressure rise, Ap across a centrifugal pump from a given manufacture can be expected to depend on the angular velocity of the impeller w, the Diameter D, of the impeller, the volume flow-rate Q, and the density of the fluid, p. By using the method of repeating variables show that Др ρω- D wD³ A model pump having an impeller diameter of 0.200 m is tested in the laboratory using water. The pressure rise when tested at an angular velocity of 407 rad/sec is shown on the graph. What would be the pressure rise for a geometrically similar pump with an impeller diameter of 0.30 m used to pump water operating at an angular velocity of 807 rad/sec and at a flow rate of 0.070 m³/s? Sol: 40 30 (kPa) Apm 10 0 € 0.02 Q Model data (₁ = 40 rad/s Dm = 20 cm 0.04 0.06 Qm (cubic meters per sec) 0.08

Manufacturing process with two variables x1,x2 described by the empirical model:   y=bo +b1 x1 + b2 x2 + b12 x1 x2 + b3 (x1)^2 +b4  (x2)^2    please refere to the image attached

Can you help me solve the following problem in MATLAB?

The Gilles & Retzbach model of a distillation column, the system model includes the dynamics of a boiler, is driven by the inputs of steam flow and the flow rate of the vapour side stream, and the measurements are the temperature changes at two different locations along the column. The state space model is given by: x = 0 00 -30.3 0.00012 -6.02 0 0 0 -3.77 00 0 -2.80 0 0 Is the system?: a. unstable b. C. not unstable x+ 6.15 0 0 0 0 3.04 0 0.052 not asymptotically stable d. asymptotically stable -1 u y = 0 0 0 0 -7.3 0 0 -25.0 X

An object attached to a spring undergoes simple harmonic motion modeled by the differential equation d²x = 0 where x (t) is the displacement of the mass (relative to equilibrium) at time t, m is the mass of the object, and k is the spring constant. A mass of 3 kilograms stretches the spring 0.2 meters. dt² Use this information to find the spring constant. (Use g = 9.8 meters/second²) m k = + kx The previous mass is detached from the spring and a mass of 17 kilograms is attached. This mass is displaced 0.45 meters below equilibrium and then launched with an initial velocity of 2 meters/second. Write the equation of motion in the form x(t) = c₁ cos(wt) + c₂ sin(wt). Do not leave unknown constants in your equation. x(t) = Rewrite the equation of motion in the form ä(t) = A sin(wt + ), where 0 ≤ ¢ < 2π. Do not leave unknown constants in your equation. x(t) =

1.)

Free stream 8 (x) Free stream • 8,(x) Thermal Velocity boundary layer boundary layer (a) (b) A differential energy analysis of this flow under common assumptions reduces the energy equation to 2 a²T + v. dy Cp where T = T(r, y) is the temperature of the fluid in the boundary layer, u is the velocity parallel to the plate, v is the velocity perpendicular to the plate, a is the thermal diffusivity of the fluid, v is the kinematic viscosity, and c, is the specific heat of the fluid at constant pressure. a) Find the base dimensions, in [M LT0], where 0 is temperature, of a and c, using the given equation. Recall that v has SI units of [m?s-'). Do not simply use the internet, textbooks, or your prior knowledge to determine the dimensions of these variables. b) It is common to "shift" both the fluid temperature in the boundary layer, T, and the far-field fluid temperature, To, by considering these temperatures relative to the surface temperature of the plate, T,. This effectively reduces…

For the following water liquid level system, 9mi(t) is the mass flow rate input deviation to the system and the restrictions can be considered linear resistances about the equilibrium: 9mi h2 Ro R1 a) Using basic fluids principles, develop the Dynamic Equations for the deviation of the two liquid level heights from their equilibrium height as a function of the input mass flow rate, Imi-