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.
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:
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.
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.
The concept of the attainable set proves valuable in various applications related to controllability:
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.
Instructions: Choose the best answer for each question.
1. What does the attainable set, K(t₀, t₁), represent?
a) The collection of all possible states a system can reach at time t₁ starting from zero initial conditions at time t₀. b) The set of all possible input sequences that can be applied to the system. c) The set of all possible initial states the system can start from. d) The set of all possible output signals the system can produce.
a) The collection of all possible states a system can reach at time t₁ starting from zero initial conditions at time t₀.
2. Which of the following is NOT a factor in determining the attainable set?
a) The initial state of the system. b) The input matrix at each time step. c) The state transition matrix at each time step. d) The output matrix at each time step.
d) The output matrix at each time step.
3. A discrete system is considered controllable in the interval [t₀, t₁] if:
a) Its attainable set is empty. b) Its attainable set spans the entire state space. c) Its attainable set is a single point. d) Its attainable set is a line in the state space.
b) Its attainable set spans the entire state space.
4. What is the practical significance of the attainable set concept?
a) It helps determine the stability of a system. b) It helps design controllers that achieve desired states. c) It helps understand the system's response to different inputs. d) All of the above.
d) All of the above.
5. Which of the following is NOT a potential application of the attainable set concept?
a) Analyzing the controllability of a system. b) Designing optimal control strategies. c) Predicting the future behavior of a system. d) Determining the stability of a system.
d) Determining the stability of a system.
Problem: Consider a discrete-time system with the following state-space representation:
Task: Determine the attainable set K(0, 2) for this system.
The attainable set K(0, 2) is the set of all possible states the system can reach at time t = 2, starting from zero initial conditions at time t = 0.
We can calculate the attainable set using the formula:
K(0, 2) = {x ∈ ℝ² | x = Σ_(j=0)^(1) F(2, j+1)B(j)u(j)}
For t = 2, j = 0 and j = 1.
So, we have:
x = F(2, 1)B(0)u(0) + F(2, 2)B(1)u(1)
F(2, 1) = F * F = [[1 1], [0 1]] * [[1 1], [0 1]] = [[1 2], [0 1]]
F(2, 2) = F = [[1 1], [0 1]]
Therefore,
x = [[1 2], [0 1]] * [[1], [0]] * u(0) + [[1 1], [0 1]] * [[1], [0]] * u(1)
x = [[1], [0]] * u(0) + [[1], [0]] * u(1)
x = [[u(0) + u(1)], [0]]
Thus, the attainable set K(0, 2) is the set of all states of the form [u(0) + u(1), 0], where u(0) and u(1) are arbitrary inputs.
This means that the system can reach any state on the x-axis, but cannot reach any state with a non-zero y-coordinate. Therefore, the system is not controllable in the interval [0, 2].
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