Stability in Discrete Optimal Control Systems

Introduction
In discrete optimal control systems, stability is a crucial property that ensures the system's response remains bounded and converges to a desired state. The stability of a system is determined by the location of its poles in the z-plane. The poles represent the roots of the system's characteristic equation and have a significant impact on the system's behavior.

Explanation
The correct answer is option 'B', which states that the poles must lie within the unit circle. Let's understand why this is true and why the other options are incorrect.

a) Poles must lie outside the unit circle:
If the poles of a discrete system lie outside the unit circle, the system becomes unstable. The system's response will exhibit exponential growth, leading to instability and unpredictable behavior. Therefore, this option is incorrect.

b) Poles must lie within the unit circle:
For a discrete system to be stable, it is essential that all the poles lie within the unit circle in the z-plane. This condition ensures that the system's response remains bounded and converges to a steady state. When the poles are within the unit circle, the system is said to be marginally stable or stable. Hence, this option is correct.

c) Poles must be on the unit circle:
If the poles of a discrete system lie exactly on the unit circle, the system becomes marginally stable. In this case, the system's response oscillates without growing or decaying. However, for stability, it is not necessary for the poles to be precisely on the unit circle. So, this option is incorrect.

d) Pole must be in infinity:
A pole at infinity implies that the system has an integrator. Such a system is not stable as it can exhibit unbounded growth in response to certain inputs. Therefore, this option is incorrect.

Conclusion
In summary, for a stable system in discrete optimal control systems, the poles must lie within the unit circle in the z-plane (option 'B'). This condition ensures that the system's response remains bounded and converges to a desired state. The other options are incorrect as they either lead to instability or do not guarantee stability.