What Is the Continuous Lyapunov Matrix Equation Stability Calculator?
The Lyapunov matrix equation establishes a quadratic energy function V(x) = xᵀPx whose negative definite derivative proves global asymptotic stability.
How Does the Continuous Lyapunov Matrix Equation Stability Calculator Work?
Sets Q = I (identity matrix) and solves the linear system of equations for unique symmetric matrix P; tests if all leading principal minors of P are positive.
Continuous Lyapunov Matrix Equation Stability Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
A^T P + P A = -Q, quad V(x) = x^T P x > 0, quad dot{V}(x) = -x^T Q x < 0
Continuous Lyapunov matrix equation for linear time-invariant system stability.
How to Use the Continuous Lyapunov Matrix Equation Stability Calculator
- Input the 4 elements of 2x2 continuous dynamic matrix A.
Step-by-Step Example Calculation
Damped Second-Order LTI System
Input Values:
a11:-2
a12:1
a21:-1
a22:-3
Worked Steps: P = [0.286, 0.071; 0.071, 0.190], Det(P) = 0.049 > 0 -> Asymptotically Stable.
Understanding Your Result
Outputs matrix P elements (p11, p12, p22) and classifies whether the system is Asymptotically Stable.
Factors That Affect the Result
- If any eigenvalue of A has a non-negative real part, matrix P fails positive definiteness, indicating instability.
When Should You Use This Calculator?
- Adaptive control synthesis, robust H-infinity control, certifying stability of linearized systems, and LQR tuning.
Assumptions & Limitations
- Applies to continuous linear time-invariant 2nd-order dynamical systems.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Exact algebraic Cramer rule solution of the matrix equation.