Consider the model of the eye shown below (we will consider rotations in the horizontal plane only). The elements K3 and B3 represent the restoring forces of tendons and other structures attached to the eye (these are rotational elements, the torque they generate is proportional to angular position and velocity). The eye's moment of inertia is J (all other elements are massless). To the left and the right of the eye are two muscles, the medial rectus (moves the eye toward the body's centerline) and the lateral rectus (moves the eye away from the centerline). The muscles (and associated connective tissue) are modeled by a parallel force generator and friction, which are in series with a spring -- these elements are translational. Rotational elements are in capital letters, and blue; translational elements are in lowercase and green.

Develop a state space model with two inputs (f_{1} and f_{2})
and one output (θ). Choose as state variables, x_{1}, x_{2},
θ and ω. The equations of motion are given by:

Consider the circuit shown below:

a) Derive a state space model using energy storage variables (i.e., inductor currents and capacitor voltages) as the state variables.

b) Use impedances to derive the transfer function H(s)=E_{out}(s)/E_{in}(s).

c) Use the transfer function to derive a second state space model.

a)

b)

c) One possible result is given below. Other answers are possible.

Consider a state space system given by:

a) Find the state transition matrix, **Φ**(s).

b) Find the transfer function H(s)=Y(s)/U(s)

Recall that the inverse of a 2x2 matrix is given by:

a) Use Matlab

>> syms s >> A=[-5 -3; 2 0;]; B=[1 0]'; C=[1 0.5]; D=0; >> Phi=inv(s*eye(2)-A) Phi = [ s/(s^2 + 5*s + 6), -3/(s^2 + 5*s + 6)] [ 2/(s^2 + 5*s + 6), (s + 5)/(s^2 + 5*s + 6)]

b)

>> H=C*Phi*B+D H = s/(s^2 + 5*s + 6) + 1/(s^2 + 5*s + 6)

The principles of linear constant coefficient differential equations can be applied to many physiological systems. One straightforward application is compartmental analysis.

The main assumption of this technique is that the body is made up of many compartments. Drugs, or other substances, move out of source compartments into one or more destination compartments at a rate proportional to the concentration of drugs in the source compartment. Each destination compartment has its own rate constant. We will consider a very simple case with only two compartments, and drug flow only in one direction (in the general case, flow can be bi-directional with a different rate constant in each direction).

Consider the metabolism of alcohol, and assume a two-compartment model. Alcohol is ingested into the first compartment, the gastro-intestinal (or GI -- this consists of your stomach, intestines…) tract. Alcohol moves out of the GI to the second compartment, the blood, at a rate that is dependent on the concentration of blood in the GI tract. Alcohol leaving the GI tract enters the blood, and alcohol leaves the blood (through metabolism and excretion) at a rate proportional to the concentration of alcohol in the blood.

The quantities in the system are given by:

- i(t) is alcohol consumption
- g(t) is the concentration of alcohol in the GI tract.
- c(t) is the concentration of alcohol in the blood./li>
- kk0 is the rate constant for alcohol leaving the GI tract and entering the blood.
- k1 is the rate constant for alcohol leaving the blood.

A block diagram is shown below:

We can write differential equations for the GI concentration. g(t) and the blood concentration c(t):

For this system it has been experimentally determined that k0=0.12
mn^{-1}, k1=0.006 mn^{-1}. (mn^{-1}=inverse minutes)
Note: keep the unit of time as minutes instead of seconds.

Write a state space model with i(t) as input and c(t) as output. Explicity identify the values of **A**, **B**, **C** and **D**.

The system shown below is from an article entitled "Effects of Muscle
Fatigue on the Ground Reaction Force and Soft-Tissue Vibrations During
Running: A Model Study," by Nikooyan and Zadpoor in *IEEE Transactions on
Biomedical Engineering*, Vol 59, No. 3, page 797, March 2012. The
model is that of the human body as it comes in contact with the ground.
The paper examines fatigue as indicated by the amount of shaking of m_{2} (leg muscles and tissue) and
m_{4 }(upper body tissues, organs and whatnot).

The paper describes the model as follows "Two of these masses represent
the upper body rigid (m_{3}) and wobbling (m_{4}) masses,
and the other two represent the lower body rigid (m_{1}) and wobbling (m_{2})
masses. The springs and dampers represent the stiffness and damping
properties of the human body hard and soft tissues." As the foot
hits the ground, there is an opposing force determined by the "ground
reaction model" that depends in a complex way on the kind of turf and the
kind of shoes being worn, and is too complex to consider here.

We will simplify the model by assuming that m_{1} is in direct contact with
the ground (so x_{1}(t)=0). The simplified model is
redrawn below.

a) Derive equations of motion.

b) Derive a state space model with three outputs, x_{2}, x_{3}
and x_{4}. Note there is no input, so you don't need a **
B** or **D** matrix.

a) the two spring attached to mass 4 act as one larger spring so let k_{4}+k_{5}=k_{45}=28000
N/m.

b) Choose positions and velocity as state variables.