State-space representation

In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector.

For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial conditions. This approach turns systems theory into an algebraic framework, making it possible to use Kronecker structures for efficient analysis.

State-space models are applied in fields such as economics, statistics, computer science, 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.

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, ${\displaystyle n}$, 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.

The most general state-space representation of a linear system with ${\displaystyle p}$ inputs, ${\displaystyle q}$ outputs and ${\displaystyle n}$ state variables is written in the following form: $${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)}$$ $${\displaystyle \mathbf {y} (t)=\mathbf {C} (t)\mathbf {x} (t)+\mathbf {D} (t)\mathbf {u} (t)}$$

where:

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 ${\displaystyle t}$ can be continuous (e.g. ${\displaystyle t\in \mathbb {R} }$) or discrete (e.g. ${\displaystyle t\in \mathbb {Z} }$). In the latter case, the time variable ${\displaystyle k}$ is usually used instead of ${\displaystyle t}$. 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:

Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix ${\displaystyle \mathbf {A} }$. The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form. It will then look something like this: