State space (controls)

In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form. The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With <math>n<math> inputs and <math>m<math> outputs, we would otherwise have to write down <math>m n<math> Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space.

State Variables

The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. State variables must be linearly independent; a state variable cannot be a linear combination of other state variables. The minimum number of state variables required to represent a given system, <math>p<math>, is usually equal to the order of the system's defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the transfer function's denominator after it has been reduced to a proper fraction. In electronic systems, the number of state variables is the same as the number of energy storage elements in the circuit (capacitors and inductors).

Mathematical representation

The state space representation of a system with <math>n<math> inputs, <math>m<math> outputs and <math>p<math> state variables is written in the following form:

<math>\dot{\mathbf{x}}(t)

<math>\mathbf{y}(t)

where

<math>\operatorname{dim}[A] = p \times p<math>

<math>\operatorname{dim}[B] = p \times n<math>

<math>\operatorname{dim}[C] = m \times p<math>

<math>\operatorname{dim}[D] = m \times n<math>

<math>\dot{\mathbf{x}}(t)

x is the "state vector", y is the "output vector", u is the "input (or control) vector", A is the "state matrix", B is the "input matrix", C is the "output matrix", and D is the "feedthrough (or feedforward) matrix". For simplicity, <math>D<math> is often chosen to be the zero matrix, i.e. the system is chosen not to have direct feedthrough.

The same representation after application of the Laplace transform is:

<math>s X(s) = A X(s) + B U(s)<math>

<math>Y(s) = C X(s) + D U(s)<math>

The characteristic polynomial of the state space representation is:

<math>a(s) = \operatorname{det}(sI - A)<math>

References

• Nise, Norman S. 2004. Control Systems Engineering, John Wiley & Sons, Inc.

See also

• Control engineering
• Control theory