State-Space Calculator

• •Use the State-Space Calculator when solving homework or exam problems that require quick numerical verification of your hand calculations — instant feedback helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The State-Space Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Analyze state-space systems: compute eigenvalues of A, controllability rank, observability rank, and stability from A, B, C, D matrices All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Inputs support both metric (SI) and imperial unit systems, with unit conversion handled automatically — simply select your preferred unit from the dropdown next to each field. Results are computed instantly in the browser without sending data to a server, ensuring both speed and privacy. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

State-space representation of a linear time-invariant system uses the first-order differential equation form dx/dt = A·x + B·u, y = C·x + D·u, where x is the state vector (n × 1 for an n-th order system), u is the input vector, y is the output vector, and A, B, C, D are constant matrices. This modern representation handles multi-input multi-output (MIMO) systems naturally and is the foundation of modern control theory. The eigenvalues of A are the poles of the system and determine stability (all must have negative real parts). Controllability is the ability to drive the state from any initial condition to any final condition in finite time, checked by the rank of the controllability matrix [B, AB, A²B, ..., A^(n-1)B]. Observability is the ability to determine the initial state from output measurements, checked by the rank of the observability matrix. If both controllability and observability hold, state-space analysis and design (pole placement, LQR optimal control, Kalman filtering) can be applied. State-space systems are solved in closed form using the matrix exponential: x(t) = e^(At)·x(0) + ∫e^(A(t−τ))·B·u(τ)dτ. Numerical simulation uses matrix-based ODE integrators. State-space is the standard formulation in modern control textbooks and is required for problems with multiple inputs and outputs, time-varying systems, and nonlinear systems linearized at operating points.

• •Multi-variable system design: design controllers for systems with multiple inputs and outputs (e.g., aircraft flight control, chemical plants with multiple manipulated variables).
• •Optimal control design: LQR (Linear Quadratic Regulator) and MPC (Model Predictive Control) use state-space formulations to compute optimal control laws.
• •Kalman filter design: state estimation from noisy sensor measurements requires state-space models and Kalman filtering for optimal estimates.
• •Modern robotics control: robot manipulator dynamics are inherently multi-variable and best represented in state-space form.
• •Aerospace guidance and control: missile, aircraft, and spacecraft dynamics use state-space for 6-DOF rigid body motion plus linearized aerodynamics.