Math Problem Statement

Given the state-space model: ẋ = Ax + Bu, y = Cx, where A, B, C matrices are provided. Analyze the system.

Solution

The system given in the image appears to represent a state-space model for a dynamical system. Let's break it down into its components:

  1. State equation (the first equation): x˙=Ax+Bu\dot{x} = A x + B u Where:

    • x˙\dot{x} is the derivative of the state vector xx with respect to time.
    • AA is the state matrix, which defines the system's dynamics.
    • BB is the input matrix, which defines how the input uu affects the system.

    From the image:

    -4 & 5 & -6 & -4 \\ -15 & 15 & -17 & -11 \\ 4 & -4 & 7 & 4 \\ -16 & 15 & -21 & -12 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$$ Therefore, the system equation is: $$\dot{x} = A x + B u$$ where $$u$$ is a scalar input, and $$x$$ is a 4-dimensional state vector.
  2. Output equation (the second equation): y=Cxy = C x Where:

    • yy is the system output.
    • CC is the output matrix, which maps the state vector xx to the output.

    From the image: C=[1000]C = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}

    This means that the output yy is simply the first state variable x1x_1 in the state vector xx.

Summary of the system:

  • State equation: x˙=Ax+Bu\dot{x} = A x + B u where AA and BB are defined above.

  • Output equation: y=Cxy = C x where CC is defined above.

Would you like further details on how to analyze the stability, controllability, or observability of this system? Let me know if you want to explore specific aspects of this model!

Related questions:

  1. How do we analyze the stability of this state-space system?
  2. What is the system's controllability matrix, and how do we determine if the system is controllable?
  3. How do we calculate the system's observability matrix?
  4. How would we design a controller for this system using state feedback?
  5. How do we simulate the time response of this system for a given input u(t)u(t)?

Tip:

In state-space models, controllability and observability are key concepts for determining whether the system can be controlled or fully observed from its inputs and outputs.

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Math Problem Analysis

Mathematical Concepts

State-Space Representation
Linear Systems
Dynamical Systems

Formulas

State equation: ẋ = A * x + B * u
Output equation: y = C * x

Theorems

Controllability (Kalman rank condition)
Observability (Kalman rank condition)
Stability of linear systems

Suitable Grade Level

Undergraduate Level