Final answer:
Determining if each of the systems given is controllable and/or observable involves using controllability and observability matrices. For all systems, controllability is determined using the controllability matrix, while observability is checked using the observability matrix. System D cannot be determined without specific matrix values.
Step-by-step explanation:
To determine whether each of the systems below is controllable and/or observable, we delve into system theory applied within engineering, specifically control engineering. The controllability and observability of a system are fundamental properties that ensure a system can be controlled to any state and its internal state can be inferred from its outputs respectively.
For system (a), x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k), controllability can be checked using the controllability matrix [B, AB, A2B, ...]. If this matrix is full-rank, then the system is controllable. Observability is determined by the observability matrix [CT, ATCT, ...]. If this matrix is full-rank as well, the system is observable.
System (b), x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k) + Du(k), has an additional direct feedthrough term Du(k) from the input to the output. The presence of D does not affect the controllability or observability, which are still determined by the aforementioned criteria.
System (c) introduces a new observation equation, z(k) = Ex(k). This would potentially expand the observability matrix, but the criteria for controllability and observability remain the same.
Option (d), None of these, is not applicable without further specifics. The determination of controllability and observability requires computation and cannot be ascertained without the actual matrices A, B, C, D, and E.
In conclusion, for each system, the controllability and observability must be checked using the respective matrices and criteria specific to each property. The matrices A, B, C (and E for observability in system c) are crucial for this purpose.