Observable vs Non-Observable Systems

Introduction

In the realm of systems engineering and control theory, understanding the concept of observability is paramount. This lesson explores the differences between observable and non-observable systems, emphasizing their characteristics, implications, and practical applications.

Definitions

• Observable System: A system where the internal states can be determined by its output.
• Non-Observable System: A system where the internal states cannot be fully inferred from the output.

Observable Systems

In observable systems, every state can be inferred from the output. This is crucial in control systems where feedback is needed for stability and performance. A common example is a linear time-invariant (LTI) system represented by the state-space model:

ẋ = Ax + Bu
y = Cx + Du
                

Here, x is the state vector, y is the output, and the matrices A, B, C, and D define the system dynamics. A system is observable if the observability matrix O = [C; CA; CA²; ...; CA^(n-1)] has full rank.

Non-Observable Systems

Non-observable systems present challenges in control and monitoring. In such systems, certain internal states cannot be determined from the outputs. An example can be illustrated as:

ẋ = Ax + Bu
y = Cx
                

If the rank of the observability matrix O is less than the number of states, then the system is non-observable. This may lead to issues in system control, as not all states can be monitored or controlled directly.

Best Practices

• Conduct observability analysis during the design phase of the system.
• Ensure that the observability matrix is computed accurately.
• Utilize state observers or estimators to reconstruct unobservable states when necessary.
• Regularly review and test system performance to adapt to changes in observability.

FAQ