attainable set for discrete system

Understanding the Attainable Set: A Key Concept for Controllability in Discrete Systems

In the realm of control theory, understanding how a system behaves under various inputs is crucial for achieving desired outcomes. For discrete-time systems, a fundamental concept in this endeavor is the attainable set. This article delves into the concept of the attainable set, highlighting its significance in analyzing the controllability of discrete systems.

Definition and Interpretation

The attainable set, denoted by K(t₀, t₁), represents the collection of all possible states a discrete system can reach at time t₁ starting from zero initial conditions at time t₀. In other words, it encapsulates the "reachable space" of the system within the specified time interval.

Mathematically, the attainable set is defined as:

K(t₀, t₁) = {x ∈ ℝⁿ | x = Σ_(j=t₀)^(t₁-1) F(t₁, j+1)B(j)u(j)}

where:

• x is the state vector at time t₁
• F(t₁, j+1) is the state transition matrix from time j+1 to time t₁
• B(j) is the input matrix at time j
• u(j) is the input vector at time j
• ℝⁿ represents the n-dimensional real space

This definition emphasizes that the attainable set is constructed by applying all possible input sequences u(j) over the interval [t₀, t₁] and observing the resulting state vectors x.

Controllability and the Attainable Set

The concept of the attainable set is closely intertwined with the notion of controllability. A discrete system is said to be controllable in the interval [t₀, t₁] if any state x in the state space can be reached from the initial state x(t₀) using an appropriate input sequence.

Importantly, the controllability of a discrete system within a given time interval is directly related to its attainable set. The system is controllable in [t₀, t₁] if and only if its attainable set K(t₀, t₁) spans the entire state space ℝⁿ.

Example: Consider a system with a 2-dimensional state space. If the attainable set K(t₀, t₁) is a line in this space, the system is not controllable because it cannot reach states outside this line. However, if K(t₀, t₁) encompasses the entire 2-dimensional space, the system is controllable.

Applications of the Attainable Set

The concept of the attainable set proves valuable in various applications related to controllability:

• Determining controllability: By analyzing the structure and properties of the attainable set, one can determine whether a system is controllable within a given time interval.
• Optimal control: The attainable set can provide valuable insights for designing optimal control strategies that achieve desired states in the shortest possible time or with minimal energy consumption.
• Robust control: Understanding the attainable set allows for designing controllers that are robust to uncertainties and disturbances in the system dynamics.
• Reachability analysis: The attainable set forms the basis for reachability analysis, which involves determining the set of states that can be reached from a given initial state under certain constraints.

Conclusion

The attainable set is a fundamental concept in the analysis of discrete-time systems. It provides a powerful tool for understanding controllability, designing optimal controllers, and performing reachability analysis. By leveraging the insights gained from the attainable set, researchers and engineers can gain deeper understanding of system behavior and develop effective control strategies for a wide range of applications.