# State Space Control Theory Controllability/Observability

• ccmuggs13
In summary, the conversation discusses a practice exam question about a system's controllability and observability based on its state space representation. It is determined that the system as a whole is neither controllable nor observable due to zeros in the B and C matrices. However, it is possible for parts of the system to be controllable or observable, which can be computed using the controllability and observability matrices.
ccmuggs13
I'm studying for an upcoming final and this is a question from a practice exam we were given. It looks simple but for some reason I just can't wrap my brain around it. The question is attached and gives a state space representation of a system (A, B, C matrices) and asks what part of the system is controllable, observable, both and neither. It is clear from the zeros in the B and C matrices that the system as a whole is neither controllable nor observable, but I am not entirely sure what it means as far as the "parts" of the system that are controllable or observable.

#### Attachments

• State Space.PNG
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ccmuggs13 said:
It is clear from the zeros in the B and C matrices that the system as a whole is neither controllable nor observable, but I am not entirely sure what it means as far as the "parts" of the system that are controllable or observable.
A system can be 100% controllable with some of the B matrix being zero.
the same can be said for a system with some of the C matrix being zero.

why don't you start by computing the controlability and observability matrices.

## What is state space control theory?

State space control theory is a mathematical approach to modeling and controlling dynamic systems. It involves representing a system in terms of its state variables and using mathematical equations to describe how these variables change over time.

## What is controllability in state space control theory?

Controllability refers to the ability to control a system's state variables to reach a desired state. A system is considered controllable if there exists a control input that can drive the system from any initial state to any desired final state in a finite amount of time.

## What is observability in state space control theory?

Observability refers to the ability to measure or observe the state variables of a system based on its output. A system is considered observable if the entire state of the system can be determined from its input and output data.

## What is the role of controllability and observability in state space control theory?

Controllability and observability are important concepts in state space control theory as they determine the feasibility and effectiveness of controlling a system. A system that is both controllable and observable can be fully controlled and its behavior can be predicted and analyzed accurately.

## What are some real-world applications of state space control theory?

State space control theory has various applications in fields such as engineering, economics, and biology. Some examples include the design of aircraft control systems, economic modeling and prediction, and the analysis of biological systems such as the spread of diseases.

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