Determine the impulse response of system - determine the

Question

Problem 01

A set of floating magnets on a frictionless rod, with every other magnet having its poles reversed, makes an interesting demonstration of magnetic fields. The equations for nine such magnets are given by

x¨_m = -1 + 1/(x_m - x_m-1)² - 1/(x_m+1 - x_m)²,

with x₀ ≡ 0 and x₁₀ ≡ 10, for 1 ≤ m ≤ 9. The initial conditions are x_m(t) = m and x^.m(t) = 0 for all t ≤ 0.

It is assumed that there is no friction. Plot all nine magnet positions vs time (units in seconds) on the same plot using the Halijak double integrator, which approximates a double integrater as

1/s² = H(z) = T²z (z²- 2z + 1) .

Please do not use operational substitution!

Problem 02

A "pulse echoer" has the continuous-time transfer function

G_c(s) = (1 - e^-s + 1/2e^-4s - 1/2e^-5s)/s

(A) Analytically determine the impulse response of this system.

(B) Using Pade approximation, obtain a rational transfer function reasonably well. Determine an impulse response for G^{^}_c(s).

(C) Determine a discrete-time approximation G_d(z) for G_c(s). Determine an impulse response for G_d(z).

(D) Compare the three impulse responses.

Problem 03

An approximation to the fractional-order integrator G_c(s) = 1/s^3/8 is given by

       G_oust(s) = (303.8s⁴ + 4.85 × 104s³ + 5.655 × 104s² + 485s + 0.03038)/(s⁵ + 3440s⁴ + 8.641 × 104s³ + 1.597 × 104s² + 21.7s + 0.0002154)

1. Determine the stiffness constant (stiffness ratio)of G_oust(s).

2. Plot the Bode magnitude plots of G_c(s) and G_oust(s) and determine the bandwith (range of frequencies ω over which the approximation is good. Also estimate the allowable error in dB when the approximation was designed. MATLAB commands that might be useful include logspace, semilogx, and log₁₀.

3. Design a third-order accurate integrater that would allow accurate simulation of G_oust(s) for a time-step T = 0.01. Use either the stability region placement technique or the matrix stability region placement technique.

4. Use the designed integrater to obtain a step response for G_oust(s).

Problem 04

A continuous stirred tank reactor, commonly abbreviated CSTR, is a tank in which the non- isothermal, first-order irreversible reaction a → B is taking place. The dynamic model for a continuous stirred tank reactor is

d/dt.CA = F/V (C_A0 - C_A) - k₀C_Aexp(-E/RT)

d/dt.T = F/V (T₀ - T) - (ΔH_rk₀/ρC_p).C_Aexp(-E/RT) - Q/VρC_p

Q = aF_c^b+1/(Fc + (aF_cB/2ρC_pc)).(T - T_cin),

where C_A the concentration of reagent A and T the reactor temperature are the state variables. The steady-state input flow of A, denoted by F and the input flow rate of coolant, denoted by F_c are the inputs. The inlet concentration of A, denoted C_A0, the inlet temperature of A, denoted T₀, and the inlet temperature of the coolant T_cin are considered disturbances. All other variables in the model are parameters. Table 1 identifies the disturbances, inputs, and parameters in the model, and gives typical values for most of these quantities.

It is suggested that with a = 516 × 10³, T_cin = 343K, and T₀ = 310K, that the system will have three possible steady-states (equilibrium points). One of the steady states (equilibrium points) is unstable at C_A = 1.3718 mol/m³ and T = 349.905K. The other two steady states (equilibrium points) are stable. Simulate this system using the method of your choosing for three initial conditions, each initial condition resulting in each of the three steady states. You must find the initial conditions that lead to each of the three equilibrium points.

Problem 05

The equations that we used for Chua's system in homework were actually a variation on Chua's system that were proposed by Hartley. The Hartley-Chua system is given by

x? = α(y + (x - 2x³)/7)

y? = x - y + z

z? = - 100/7.y,

Chua's original system was described by

x? = α (y - g(x))

y? = x - y + z

z? = -100/7 y,

Quantity	Description	Units	Classification	Typical Value
b	-	-	Parameter	0.5
CA	Reactor Concentration of reagent A	kmol m3	State Variable	-
CA0	Inlet Concentration of reagent A	kmol m3	Disturbance	2.0
Cp	Specific Heat Capacity of reagent A	cal g0 C	Parameter	1.0
Cpc	Specific Heat Capacity of coolant	cal g0 C	Parameter	1.0
E R	-	K	Paremteter	8330
F	Steady State Flow of Reagent  A	m3 min	Parameter	1.0
Fc	Steady State Flow of  Coolant	m3 min	Parameter	15.0
?Hr	Heat of Reaction	cal kmol	Parameter	-130 × 106
k0	First Order Reaction Rate Constant	min-1	Parameter	1010
T	Reactor  Temperature	K	State Variable	-
Tcin	Temperature of Coolant	K	Disturbance	-
T0	Inlet Temperature of Reagent A	K	Disturbance	-
V	Volume  of the CSTR	m3	Parameter	1.0
ρ	Density of Reagent A	g m3	Parameter	106
ρc	Density of Coolant	g m3	Parameter	106

Table 1: Typical Parameter Values for Continuous Stirred Tank Reactor.

where g(x) is a piecewice-linear function given by

            bx + (a + b), x ≤ -1
g(x) =  ax, -1 < x < 1 .
           bx - (a + b), x ≥ 1

Hartley's version of this system is to replace g(x) with gˆ(x) = ax + bx³, specifically using a = -1/7 and b = 2/7. For α = 10, and initial conditions x(0) = √2/10 , y(0) = 0, and z(0) = -√2/10, use the standard fourth-order Runge-Kutta method to simulate both the Chua system and the Hartley-Chua system. Create phase-plane plots (which start after the transient has died) and compare the behavior of the simulations. Additionally, create a time plots of the difference between the state variable values obtained via the Chua system simulation and the state variable values obtained via the Hartley-Chua system simulation.

Problem 06

A dynamical model of a zombie outbreak was formulated by Munz et al. This paper concluded that the only situation where both humans and zombies could coexist is when there was a treatment for "zombie-ism." This model for an outbreak consists of four groups, the suceptable group S of humans that could contract zombie-ism, the infected group I, that has contracted zombie-ism and will become zombies if the illness is left to take its course, the group of zombies Z, and finally, the individuals that have been removed, either humans who have died non-zombie deaths or zombies that have been eliminated. This model allows zombies that have been killed to be reanimated as zombies again. An additional group K of humans or zombies that are unable to be reanimated has been added. The modified model is given by

S? = ΠS - βZS - δS +cZ
I? = βZS - ρI - δI
Z? = ρI + ζR - αSZ -cZ
R? = δS - ζR,
K? = δI + αSZ

where the parameters and possible values are given in Table 2. Note that state variables are in terms of thousands, and the timescale of t is in days. For example, S(6) = 25, indicates that the susceptable population is 25, 000 people at day six of the outbreak. Simulate, using any simulation technique that produces reasonable results, at least 180 days worth of data (more time would be needed of the simulation does not reach a steady state) for the four cities shown in Table . Create six plots: one plot for each city including the S, Z, I, and R populations, and a plot that includes the S populations for each city normalized to the largest value of S, and a plot that shows the total Z populations for each city. For your reference, a simulation has been performed for Middletown, OH (population 97,500) and is shown in Figure 1