Solution Found!
Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and
Chapter , Problem 18E(choose chapter or problem)
Problem 18E
Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and let represent a sequence of input vectors in
a. Set from equation (1), and write a formula for x4 involving the controllability matrix M appearing in equation (2). (Note: The matrix M is constructed as a partitioned matrix. Its overall size here is 4 × 8.)
b. Suppose (A, B) is controllable and v is any vector in . Explain why there exists a control sequence in such that x4 = v.
Questions & Answers
QUESTION:
Problem 18E
Suppose A is a 4 × 4 matrix and B is a 4 × 2 matrix, and let represent a sequence of input vectors in
a. Set from equation (1), and write a formula for x4 involving the controllability matrix M appearing in equation (2). (Note: The matrix M is constructed as a partitioned matrix. Its overall size here is 4 × 8.)
b. Suppose (A, B) is controllable and v is any vector in . Explain why there exists a control sequence in such that x4 = v.
ANSWER:
Solution 18E
Step 1 of 2
Consider A be the matrix of order and B be the matrix of order , and the
vectors represents a sequence of input vectors in .
(a)
The objective is to compute the vectors by setting and write a formula for involving the controllability matrix M.
To compute the vectors, set and use the following equation:
For,
For,
For,
For,
Here,
and
Since the order of the matrix B is , the matrix M has 4 rows then only the product holds. The matrix M have 8 columns because the matrix B and each of the matrices have 2 columns.
Therefore, the vector is in.
