The project will be done in groups of two.  Regard this project as a take home exam: you should not col- laborate with others (except for your partner to this project).  You are required to use Matlab and attach code and simulation results as part of your project. In your answers, provide a clear theoretical explanation to your results.

Consider the following system:

Consider the state vector as x = [vc1(t),vc2(t)]T , i.e., the voltage across capacitor 1 and the voltage across capacitor 2. The input is defined as u = [v1 (t),v2 (t)]T  and the output is defined as y = [vo1(t),vo2(t)]T .

Questions hereon refer to the system presented above.

1.  (Ch 2) Derive the state and observation equations for this problem given by

y(t) = Cx(t) + Du(t).                                                                  (1)

Show that the aforementioned matrices are:

C    =    [ 0(1)

1(1) ] ,

D = [0(0)

0(0) ] .

For the rest of the project proceed with the following values of the parameters of the system: R2  = R1 and C2   = 4C1 .   For all the analytic expressions, feel free to replace R1 C1   = T.   For any numerical simulation, use T = 1[ms].

2.  (Ch 20) Simulation:  To simulate the continuous-time system in (1), we would like to use the second approximation taught in class, in which a continuous time-domain state equation is set as a difference equation, i.e.,

y[k] = C\ x[k] + D\ u[k],                                                               (2)

where A\  = Φ((k + 1)∆,k∆) and B\  = 后 Φ((k + 1)∆,σ)B(σ)dσ , C\  = C, and D\  = D, and ∆ is the sampling period such that x[k] = x(k∆), u[k] = u(k∆), and y[k] = y(k∆). Select ∆ =  [ms].