In control engineering and system identification, a statespace representation is a mathematical model of a physical system specified as a set of input, output, and variables related by firstorder 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 state variable values and may also depend on the input variable values.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector, the state vector.
If the dynamical system is linear, timeinvariant, and finitedimensional, then the differential and algebraic equations may be written in matrix form.^{[1]}^{[2]} The statespace method is characterized by the algebraization of general system theory, which makes it possible to use Kronecker vectormatrix structures. The capacity of these structures can be efficiently applied to research systems with or without modulation.^{[3]} The statespace representation (also known as the "timedomain 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 statespace representation is not limited to systems with linear components and zero initial conditions.
The statespace model can be applied in subjects such as economics,^{[4]} statistics,^{[5]} computer science and electrical engineering,^{[6]} and neuroscience.^{[7]} In econometrics, for example, statespace models can be used to decompose a time series into trend and cycle, compose individual indicators into a composite index,^{[8]} identify turning points of the business cycle, and estimate GDP using latent and unobserved time series.^{[9]}^{[10]} Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations.^{[11]}^{[12]}
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.^{[13]} 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 statespace 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 statespace 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.
The most general statespace representation of a linear system with inputs, outputs and state variables is written in the following form:^{[14]}
where:
In this general formulation, all matrices are allowed to be timevariant (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 statespace model representation can assume the following forms:
System type  Statespace model 
Continuous timeinvariant  
Continuous timevariant  
Explicit discrete timeinvariant  
Explicit discrete timevariant  
Laplace domain of continuous timeinvariant 

Zdomain of discrete timeinvariant 

Stability and natural response characteristics of a continuoustime LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix . The stability of a timeinvariant statespace model can be determined by looking at the system's transfer function in factored form. It will then look something like this:
The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of ,
The roots of this polynomial (the eigenvalues) are the system transfer function's poles (i.e., the singularities where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is asymptotically stable or marginally stable. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability.
The zeros found in the numerator of can similarly be used to determine whether the system is minimum phase.
The system may still be input–output stable (see BIBO stable) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are removable).
The state controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous timeinvariant linear statespace model is controllable if and only if
where rank is the number of linearly independent rows in a matrix, and where n is the number of state variables.
Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).
A continuous timeinvariant linear statespace model is observable if and only if
The "transfer function" of a continuous timeinvariant linear statespace model can be derived in the following way:
First, taking the Laplace transform of
yields
Next, we simplify for , giving
and thus
Substituting for in the output equation
giving
Assuming zero initial conditions and a singleinput singleoutput (SISO) system, the transfer function is defined as the ratio of output and input . For a multipleinput multipleoutput (MIMO) system, however, this ratio is not defined. Therefore, assuming zero initial conditions, the transfer function matrix is derived from
using the method of equating the coefficients which yields
Consequently, is a matrix with the dimension which contains transfer functions for each input output combination. Due to the simplicity of this matrix notation, the statespace representation is commonly used for multipleinput, multipleoutput systems. The Rosenbrock system matrix provides a bridge between the statespace representation and its transfer function.
Any given transfer function which is strictly proper can easily be transferred into statespace by the following approach (this example is for a 4dimensional, singleinput, singleoutput system):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
The coefficients can now be inserted directly into the statespace model by the following approach:
This statespace realization is called controllable canonical form because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
The transfer function coefficients can also be used to construct another type of canonical form
This statespace realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
Transfer functions which are only proper (and not strictly proper) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.
The strictly proper transfer function can then be transformed into a canonical statespace realization using techniques shown above. The statespace realization of the constant is trivially . Together we then get a statespace realization with matrices A, B and C determined by the strictly proper part, and matrix D determined by the constant.
Here is an example to clear things up a bit:
which yields the following controllable realization
Notice how the output also depends directly on the input. This is due to the constant in the transfer function.
A common method for feedback is to multiply the output by a matrix K and setting this as the input to the system: . Since the values of K are unrestricted the values can easily be negated for negative feedback. The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results.
becomes
solving the output equation for and substituting in the state equation results in
The advantage of this is that the eigenvalues of A can be controlled by setting K appropriately through eigendecomposition of . This assumes that the closedloop system is controllable or that the unstable eigenvalues of A can be made stable through appropriate choice of K.
For a strictly proper system D equals zero. Another fairly common situation is when all states are outputs, i.e. y = x, which yields C = I, the Identity matrix. This would then result in the simpler equations
This reduces the necessary eigendecomposition to just .
In addition to feedback, an input, , can be added such that .
becomes
solving the output equation for and substituting in the state equation results in
One fairly common simplification to this system is removing D, which reduces the equations to
A classical linear system is that of onedimensional movement of an object (e.g., a cart). Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring:
where
The state equation would then become
where
The controllability test is then
which has full rank for all and . This means, that if initial state of the system is known ( , , ), and if the and are constants, then there is a force that could move the cart into any other position in the system.
The observability test is then
which also has full rank. Therefore, this system is both controllable and observable.
The more general form of a statespace model can be written as two functions.
The first is the state equation and the latter is the output equation. If the function is a linear combination of states and inputs then the equations can be written in matrix notation like above. The argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).
A classic nonlinear system is a simple unforced pendulum
where
The state equations are then
where
Instead, the state equation can be written in the general form
The equilibrium/stationary points of a system are when and so the equilibrium points of a pendulum are those that satisfy
for integers n.