In control engineering and system identification, a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables and may also depend on the values of the input variables.
The state space or phase space is the geometric space in which the variables on the axes are the state variables. The state of the system can be represented as a vector, the state vector, within state space.
If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general system theory, which makes it possible to use Kronecker vector-matrix structures. The capacity of these structures can be efficiently applied to research systems with modulation or without it. 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 inputs and outputs, we would otherwise have to write down 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.
The state-space model can be applied in subjects such as economics, statistics, computer science and electrical engineering, and neuroscience. In econometrics, for example, state-space models can be used to decompose a time series into trend and cycle, compose individual indicators into a composite index, identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations.
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. The minimum number of state variables required to represent a given system, , is usually equal to the order of the system's defining differential equation, but not necessarily. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. It is important to understand that converting a state-space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. In electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as capacitors and inductors. The state variables defined must be linearly independent, i.e., no state variable can be written as a linear combination of the other state variables, or the system cannot be solved.
Linear systems
The most general state-space representation of a linear system with inputs, outputs and state variables is written in the following form:
where: is called the "state vector", ; is called the "output vector", ; is called the "input (or control) vector", ; is the "state (or system) matrix", , is the "input matrix", , is the "output matrix", , is the "feedthrough (or feedforward) matrix" (in cases where the system model does not have a direct feedthrough, is the zero matrix), , .
In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common LTI case, matrices will be time invariant. The time variable can be continuous (e.g. ) or discrete (e.g. ). In the latter case, the time variable is usually used instead of . Hybrid systems allow for time domains that have both continuous and discrete parts. Depending on the assumptions made, the state-space model representation can assume the following forms: