Characteristic Equation

Definition

The characteristic equation is a polynomial equation derived from the system's differential equations or transfer function, used to determine the system's dynamic behavior and stability. It connects directly to the eigenvalues of a system, indicating how the system responds to inputs, which is crucial for analyzing stability and designing control strategies.

5 Must Know Facts For Your Next Test

1. The characteristic equation is formed by setting the determinant of the system's matrix minus lambda times the identity matrix equal to zero.
2. The roots of the characteristic equation are known as eigenvalues, which are critical for understanding system stability.
3. A characteristic equation with all negative roots indicates that the system is stable, while positive or complex roots can suggest instability.
4. In state-space representation, the characteristic equation can be derived from the system's state matrix.
5. The Routh-Hurwitz criterion uses the coefficients of the characteristic equation to determine stability without explicitly calculating the roots.

Review Questions

• How does the characteristic equation relate to a system's eigenvalues and stability?
  • The characteristic equation provides a polynomial whose roots are the eigenvalues of the system. These eigenvalues are crucial for assessing stability; if all eigenvalues have negative real parts, the system is stable. Conversely, if any eigenvalue has a positive real part or has complex components with positive real parts, it indicates that the system may become unstable over time. Thus, analyzing the characteristic equation is vital for predicting how a system behaves under different conditions.
• Describe how to derive the characteristic equation from a state-space representation and what information it reveals about the system.
  • To derive the characteristic equation from a state-space representation, you start with the state matrix A. The characteristic polynomial is obtained by calculating the determinant of (sI - A), where I is the identity matrix and s is a complex variable. Setting this determinant equal to zero gives you the characteristic equation. This equation reveals important information about the system's dynamic behavior and stability by indicating where its poles are located in the complex plane.
• Evaluate how different forms of a characteristic equation can affect control design strategies such as pole placement and feedback control.
  • Different forms of a characteristic equation can significantly influence control design strategies like pole placement and feedback control. When designing controllers, engineers often aim to modify the characteristic equation's roots through feedback mechanisms to achieve desired dynamics. If a controller can successfully shift all poles to locations that ensure stability (e.g., left-half plane for continuous systems), then it enhances performance metrics such as transient response and steady-state error. Thus, understanding how to manipulate the characteristic equation is essential for creating effective control solutions tailored to specific performance criteria.

Related terms

Eigenvalues: Scalar values that indicate the behavior of a linear transformation; in control systems, they are used to analyze stability and response characteristics.

Transfer Function: A mathematical representation of the relationship between the input and output of a linear time-invariant system, often expressed in terms of Laplace transforms.

Stability: The property of a system that determines whether its output will converge to a steady state or diverge over time, influenced by the roots of the characteristic equation.