Final answer:
Two scenarios are provided with different numbers of inputs and outputs, and the matrices are determined based on the system equations for each scenario.
Step-by-step explanation:
The student is asking for the state-variable models to be represented in terms of matrices A, B, C, and D. For the first scenario, where the outputs are x1 and x2 and the input is u, the system equations are x1' = -7x1 + 4x2 and x2' = -3x2 + 8u. The matrices can be obtained as:
- A = [ [-7, 4], [0, -3] ]
- B = [ [0], [8] ]
- C = [ [1, 0], [0, 1] ]
- D = [ [0], [0] ]
For the second scenario, where the output is x1 and the inputs are u1 and u2, the system equations are x1' = -7x1 + 5x2 + 3u1 and x2' = -9x2 + 2u2. The matrix representations would be:
- A = [ [-7, 5], [0, -9] ]
- B = [ [3], [2] ]
- C = [ [1, 0] ]
- D = [ [0, 0] ]
In both cases, matrix A represents the system dynamics, matrix B links the input to the state derivatives, matrix C maps the states to the outputs, and matrix D represents direct feedthrough from inputs to outputs.