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.
Understanding Engineering: State-Space Representation
A state is a quantity whose present value carries information from the past into the future. Position and velocity form two states for a moving mass on a spring. The mass cannot respond correctly if a model knows only its position.
Its velocity matters because momentum keeps it moving. In an electric circuit, capacitor voltage stores information about past charge, while inductor current stores information about past magnetic effects.
These stored quantities are often the best starting point when choosing states. A useful model uses enough states to predict later behavior once the input is known.
Engineers build the matrices by starting with physical laws. For mechanical systems, Newton's second law connects force, mass, velocity, and acceleration. For circuits, Kirchhoff's laws connect currents and voltages.
The original laws may contain second order or higher order derivatives. Each higher order equation can be split into several first order equations by defining extra states. For example, velocity is the rate of change of position.
Acceleration is the rate of change of velocity. This step may feel like extra work, but it makes complicated systems easier for computers to handle. It also makes it clear which physical quantities are stored inside the model.
Initial conditions are essential. A system can receive no new input yet still change because it begins with stored energy. A released spring moves because its initial displacement or velocity is not zero.
A charged capacitor can discharge without a battery connected. In a simulation, the initial state sets the starting point, then the model calculates small changes over time. Digital controllers work at separate sampling instants rather than continuously.
Their models use a discrete version in which the next state depends on the current state and current input. Choosing a sample time matters.
If it is too slow, fast motion or electrical changes can be missed. If it is very fast, measurements may contain more noise and computation increases.
Not every internal state can be measured directly. A drone may measure position and angular rate, but not every force affecting its body. A motor controller may measure current and shaft speed, while estimating load torque.
Observability describes whether measurements over time contain enough information to infer the hidden states. Controllability describes whether available inputs can move the states in useful directions. These ideas guide sensor placement and actuator choice before a controller is built.
When studying a model, check units carefully. Matrix entries must convert quantities consistently, such as volts into rates of current change or newtons into acceleration.
Check matrix sizes, signs, initial conditions, and whether a direct input to output path is physically realistic. Small mistakes in these details can produce simulations that look convincing but describe the wrong system.
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.