Discrete time state space model: solving for input

[nweibley]

Preface: This homework assignment was due long ago. At this point I am only trying to understand the problem (or really if the posted solution follows the problem) before my exam. I have no real indication that this problem (or even one like it) will be on my final, but I feel that my inability to solve it shows a weakness in my understanding of the material that I should fix before I sit down to take the test. The PDF for the assignment is http://csc.list.ufl.edu/3105/fall08/assignment/assignment-8.pdf. So here goes...

Homework Statement

This the exact wording of the problem:

Use a discrete time state space approach to model a mortgage payment
system and find the fixed monthly payment required to amortize
a loan of $54546.00 at 9% per annum over a period of 60 months.
(Answer should be 1132.29 using effective monthly interest as .09/12)

Homework Equations

$$\mathbf{x}[n+1] = A\mathbf{x}[n] + Bu[n]$$
$$\mathbf{y}[n] = C\mathbf{x}[n] + Du[n]$$

The Attempt at a Solution

Well... first I tried to setup a difference equation of the form:
$$\-1y[n+1] + \left(1+\frac{.09}{12}\right) y[n]=u[n]$$
and set
$$x_{1}[n]=y[n]$$
$$x_{2}[n]=y[n+1]$$
and
$$x_{1}[n+1]=y[n+1]=x_{2}[n]$$However, I came to realize that this is either a) false or b) going to yield a state space representation with 1 state (A is a 1x1 matrix).

I'd be thrilled with a lead to chase on this one; I can't tell if the question as-asked makes sense insomuch as I'm going to find a system matrix of dimensions m x m (m > 1) which are the only state spaces I can remember working in class.
The posted solutions from the TA show a difference equation of the form:
$$x[n+1] = ax[n]-ku[n]$$
Which is then Z transformed to yield:
$$X(z) = -k\left(\frac{z}{\left(z-1\right)\left(z-a\right)}\right)+\frac{z}{z-a}x[0]$$
At which point I believe the TA made a superficial error (but am probably wrong about that).

So, any guidance? Is there a state space solution that makes sense for this problem? And if so, does it essentially present itself like scalars?

Many thanks for any pointers.

 

[mighty2000]

Hello,

Thank you for sharing your question and attempts at solving the problem. I can understand your frustration and desire to fully understand the material before your exam.

After reviewing the problem and your attempts at solving it, I believe the key to understanding this problem lies in properly defining the state variables. In this case, the state variables should represent the current balance of the loan and the number of monthly payments made so far. This will result in a state space representation with two states (m=2).

I suggest setting up the difference equation using these state variables and then using the Z-transform to solve for the transfer function. From there, you can solve for the fixed monthly payment required to amortize the loan.

I hope this helps and good luck on your exam! Remember to always approach problems with a clear and structured mindset, and don't hesitate to seek help if needed.