9.1 Consider the system

where u(k) ? ? ^m and y(k) ? ? ^m with m>1. We are given the sequences of input-output data pairs

such that

with e _i being the ith column of the n n identity matrix. Assume that N ? s, and show how the output data sequences for i=1, 2, , m must be stored in the matrix such that

9.2 Consider the subspace identification method for impulse input signals, described in Section 9.2.3.

1. Write a Matlab program to determine the system matrices A, B, and C up to a similarity transformation.

2. Test this program using 20 data points obtained from the following system:

   
   

   Check the eigenvalues of the estimated A matrices, and compare the outputs from the models with the real output of the system.

9.3 Consider the minimal and asymptotically stable LTI system

with x(0)=0 and u(k) equal to a step sequence given by

1. Show that the data equation (9.7) on page 296 can be written as

   
   

   where _j ? ? ^j denotes the vector with all entries equal to unity.

2. Show that

   
   

3. Show that

   
   

   and use it to prove that

   
   

4. Use the results derived above to prove that

   
   

9.4 When the input to the state-space system

is periodic, the output will also be periodic. In this case there is no need to build a block Hankel...