SYST 611 Homework 4

Issued: September 25, 2009

Due: October 1, 2009

Two problems, fifty points each.

1.       Consider the following differential equation:

.

(a)  Recast the system in state-space format using observable canonical form.

(b)  Is the system asymptotically stable, marginally stable, or unstable?

(c)  Find the Transfer Function for this system.

(d)  Discretize the system using the forward difference method (with an arbitrary step size h).

(e)  Is the discretized system in part (d) stable, marginally stable, or unstable?

(f)     Now assume that h=0.1, and use it to determine whether the discretized system is asymptotically stable, marginally stable, or unstable.  Is your answer in parts (e) and (f) the same?  Why or why not?

2.       Consider a wage dispute between labor and management.  At each state of the negotiations, labor representatives submit a wage demand to management that, in turn, presents a counter offer.  Since the wage offer will be usually less than the wage demand, further negotiations are required.  One can formulate this situation as a dynamic system, where at each period management “updates” its previous offer by the addition of some fraction a of the difference between last period’s demand and offer.  Labor also “updates” its previous demand by the subtraction of some fraction b of the difference between the demand and offer of the last period.  Let x₁ equal the management offer and x₂ equal the labor demand.  Write the dynamic state equations (in matrix form) for this situation.