State-space representation is a compact way to model dynamic engineering systems using vectors and matrices. Instead of describing a system with one high-order differential equation, it tracks a set of internal state variables that summarize the system at any instant. This form is especially important in controls, robotics, circuits, aerospace, and mechanical vibration because real systems often have many inputs and many outputs.
It gives engineers a clear structure for simulation, analysis, and controller design.
A continuous-time state-space model is usually written as x_dot = Ax + Bu and y = Cx + Du. The matrix A describes how the current state affects its own rate of change, B describes how inputs drive the states, C maps states to measured outputs, and D maps inputs directly to outputs. Because the equations use vectors, the same format works naturally for multi-input multi-output systems.
The model also connects directly to stability, controllability, observability, feedback control, and computer simulation.
Key Facts
- Continuous-time state equation: x_dot = Ax + Bu
- Output equation: y = Cx + Du
- x is the state vector, u is the input vector, and y is the output vector.
- If x has n states, u has m inputs, and y has p outputs, then A is n by n, B is n by m, C is p by n, and D is p by m.
- For zero input, system stability depends on the eigenvalues of A.
- State feedback often uses u = -Kx + r to change the closed-loop dynamics to x_dot = (A - BK)x + Br.
Vocabulary
- State vector
- A column vector containing the minimum set of variables needed to describe the system's current condition and future motion.
- Input vector
- A column vector containing external signals or commands applied to the system.
- Output vector
- A column vector containing the measured or desired quantities produced by the system.
- System matrix A
- The matrix that describes how the state variables interact with each other in the absence of inputs.
- Direct transmission matrix D
- The matrix that describes any immediate effect of the input on the output without passing through the state dynamics.
Common Mistakes to Avoid
- Mixing up x and y is wrong because the state vector contains internal variables while the output vector contains measured or reported quantities.
- Using incompatible matrix dimensions is wrong because Ax, Bu, Cx, and Du must produce vectors of the correct sizes for addition.
- Assuming D is always zero is wrong because some systems have direct feedthrough from input to output, especially in algebraic or sensor models.
- Treating the state variables as unique is wrong because different valid choices of states can describe the same physical system.
Practice Questions
- 1 A system has 3 states, 2 inputs, and 4 outputs. What are the dimensions of A, B, C, and D in the model x_dot = Ax + Bu and y = Cx + Du?
- 2 Given A = [[0, 1], [-6, -5]], B = [[0], [1]], x = [[2], [-1]], and u = [[3]], compute x_dot = Ax + Bu.
- 3 A mass-spring-damper system can be modeled using position and velocity as states. Explain why state-space form is more useful than a single transfer function when the system has multiple actuators and multiple sensors.