Math 466 Spring 2024 Assignment 9

1.   Consider the linear input output system:

where ∈ Rn,  ∈ Rp , and  ∈ Rm. It is desired to input signals of the form  = −K +  where the

feedback term −K serves to stabilize the system so that resonance can be averted.  The term  represents an external disturbance or forcing function. Unfortunately, however, we are unable to measure the entire    state, , and can only observe  as given by the output equation  = C .    To remedy this, we design an

observer system the state of which approximates  .  We then input a signal of the form

and attempt to design the feedback gain Kin such away that we obtain the desired result.  The observer system takes the form

where the matrix L is an appropriately chosen gain.

a.   What are the dimensions of the design matrices Kand L?

b.   Let  =  − , and use (1)-(3), (5),(6) to show that  satisfies the homogeneous linear system given by

c.   Give a design condition on the gain L that will guarantee that lim (t) = 0 (and therefore that t→∞

t(l)(t) − (t)) = 0.

d.   Now substitute the input given by (4) with (t) = 0(⃗⃗) into (1) and (5) and use (1)-(8) to find a 2n

dimensional first order homogeneous linear system describing the evolution of the 2n

dimensional  vector function given by

e.   Give a design condition on the gain K, that when used in conjunction with the design criteria

given in part (c) will guarantee that lim (t) = 0 and therefore the design objective has been met. t→∞

(Hint: For a block upper triangular matrix of the form T  =  [ 0(u)    v(w)] , it follows that

det T = detudetv .   If you do not see how this hint is helpful, then you may have made an error in parts ((a) –(d)).

f.    In the case of the undamped simple harmonic oscillator with both mass and stiffness set equal to one (i.e. m = k = 1) and the ability to measure only displacement and not velocity, provide

explicit conditions on the elements in the gains L and kthat will ensure that the conditions you set out in parts (c) and (e) are satisfied.

g.   Use Simulink to demonstrate that your design in part (f) works by choosing values for the gains L and kthat satisfy the conditions you set out in part (f) for the cases (i) v(t) = 0, and (ii) v(t) =    sint.

2. A United States Navy Contractor is designing the next generation of sono-buoy.  A sono-buoy

is a device used to track enemy submarines.  It is dropped into the ocean from low flying aircraft  and sends out low frequency sound waves which travel through the water and bounce off of large metallic objects such as a submarine.  The new generation of sono-buoy is cylindrical in shape

with circular bases and floats on the surface of the ocean with the circular bases at the top and

bottom.  See the figure on the next page. The radius of the circular bases is rand the height of the cylinder ish.  The average mass density of the sono-buoy isp and the mass density of seawater   is pw.

a.   What is the mass m of the sono-buoy in terms of p,T, and ℎ.

b. The buoyancy (which is a force!) of an object is equal in magnitude to the weight of water being displaced. In terms of l, at equilibrium with gravity, what is the buoyancy?

c.   Based on part (b), what must lbe equal to?

d.   In order to be able to accurately process the data it receives from the sono-buoy, the Navy requires a mathematical model for the vertical motion of the device as it bobs up and

down on the surface of the ocean. Let u(t) denote any external forces acting on the

device at time t ≥ 0, and use Newton’s second law to derive a differential equation for the quantity measured in reference to the coordinate system shown in the figure.  Write your model as a first order linear input/output system.

e.   Using either Laplace transforms or the matrix exponential and the variation of constant formula, provide an expression for the output, y, in terms of the input u.

f.    If incoming ocean waves aremodeled in the form of a cosine function of the form,

u(t) = R cos幼t

where 幼 is never larger than some frequency 幼max,  in designing the sono-buoy, what values of the sono-buoy dimensions,h, r and p will the Navy engineers want to avoid and why?

g.   If in their design, the Navy decides to include a damping device that can be modeled via a term that is proportional to the vertical velocity of the sono-buoy, with a positive constant of proportionality b, where the value of b depends on the design of the damper, how

should the damper be designed (i.e. how should the gain b be chosen) so as to achieve critical damping?

h.   Choose reasonable values for p,T, and ℎ, lookup pw, and simulate your model using

Simulink for different values of R, b and 幼.  In your simulations, be certain to include    examples where you choose values for 幼 that cause the system response to exhibit beats and resonance.  Be certain to pay careful attention to units and make conversions where  necessary.