System Definition
State Matrix (A)
State Transition Matrix Φ(t) = eAt
System Analysis
Phase Portrait x(t) vs x(0)
How to Use the Calculator
Get your complete system analysis in three steps.
Define Your System
Set the size of your square state matrix (A) using the stepper. Then, input the coefficients of your matrix.
Set the Time Evolution
Use the slider to adjust the time variable 't'. This will evolve the state transition matrix and the phase portrait trajectory.
Analyze the Results
Instantly review the calculated state transition matrix, the system's eigenvalues, its stability classification, and the 2D phase portrait.
What is the State Transition Matrix?
The key to understanding how a system evolves over time.
For a linear time-invariant system described by the state equation ẋ = Ax, the State Transition Matrix (STM), denoted as Φ(t), maps the initial state of the system x(0) to the state at any given time 't'. The relationship is given by:
The STM is calculated by taking the matrix exponential of the state matrix A multiplied by time t: Φ(t) = eAt. It is a fundamental tool in modern control theory for analyzing the behavior and stability of dynamic systems. Our state transition matrix calculator automates this complex computation for you.
Interpreting System Stability
Stability is determined by the eigenvalues (λ) of the state matrix A.
Stable
The system is stable if all eigenvalues have negative real parts. Any initial state will return to the origin over time. This is the desired behavior for most control systems.
Marginally Stable
The system is marginally stable if at least one eigenvalue has a zero real part and is not repeated, with all others having negative real parts. The system will oscillate indefinitely without decaying.
Unstable
The system is unstable if any eigenvalue has a positive real part, or if there are repeated eigenvalues with zero real parts. The state will grow to infinity over time.
An Advanced Control Systems Lab
Features designed for complete system analysis.
Full System Analysis
This isn't just a matrix calculator. It computes the eigenvalues and automatically classifies the system's stability, providing a complete diagnostic.
Live Phase Portrait
For 2x2 systems, a live phase portrait visualizes the system's trajectory from an initial point, offering deep insight into its dynamic behavior (e.g., spirals, nodes, saddles).
Interactive Time Evolution
Use the time slider to smoothly and interactively see how the state transition matrix and the system's state evolve as 't' changes.
