State-space model
A state-space model is a mathematical representation of a dynamical system in which an internal state summarizes the information required to determine future evolution under specified inputs. Unlike an input–output model, which relates externally observable quantities directly, a state-space model introduces variables that need not be measurable individually. These variables form a state vector whose trajectory evolves according to a differential or difference equation.
The framework originated in control theory and became closely associated with estimation, signal processing, econometrics, and time-series analysis. Its central construction separates the evolution of the internal state from the mechanism through which that state produces observations.
Mathematical formulation
For a continuous-time linear system, the standard representation is
[ \dot{\mathbf{x}}(t)
A(t)\mathbf{x}(t) + B(t)\mathbf{u}(t), ]
[ \mathbf{y}(t)
C(t)\mathbf{x}(t) + D(t)\mathbf{u}(t). ]
Here (\mathbf{x}(t)) denotes the state vector, while (\mathbf{u}(t)) represents an externally applied input. The vector (\mathbf{y}(t)) contains the observed output. The matrix (A(t)) determines autonomous state evolution, and (B(t)) specifies how the input changes that evolution. The matrix (C(t)) maps the internal state to the output, whereas (D(t)) represents any instantaneous input contribution to the output.
When the matrices do not depend on time, the system is linear time-invariant. Its unforced solution is expressed through the matrix exponential:
[ \mathbf{x}(t)
e^{A(t-t_0)}\mathbf{x}(t_0) + \int_{t_0}^{t} e^{A(t-\tau)}B\mathbf{u}(\tau),d\tau. ]
The first term propagates the initial condition. The integral accumulates the effect of the input over the intervening interval.
A discrete-time model has the corresponding form
[ \mathbf{x}_{k+1}
A_k\mathbf{x}_k + B_k\mathbf{u}_k, ]
[ \mathbf{y}_k
C_k\mathbf{x}_k + D_k\mathbf{u}_k. ]
Discrete formulations describe sampled physical systems as well as processes defined intrinsically by recursions. The transition from a continuous model to a discrete model depends on the assumed behavior of the input between sampling instants and is closely related to sampled-data systems.
Meaning of state
A state is a sufficient summary of past behavior for the purpose of future dynamical evolution. Given the present state and the future input, an ideal deterministic model does not require the earlier trajectory. This property gives the state process a form analogous to the Markov property, although the observable output alone need not be Markovian.
State variables are not uniquely determined by the physical system. If (T) is an invertible matrix and
[ \mathbf{z}=T\mathbf{x}, ]
then the transformed model is
[ \dot{\mathbf{z}}
TAT^{-1}\mathbf{z} + TB\mathbf{u}, ]
[ \mathbf{y}
CT^{-1}\mathbf{z} + D\mathbf{u}. ]
The two representations produce identical input–output behavior. Consequently, individual state coordinates generally depend on the chosen realization, while invariant properties such as system order and transfer-function poles remain unchanged under similarity transformations.
In mechanically derived models, state variables often correspond to quantities such as position and momentum. In statistical models, the state commonly represents latent structure rather than a directly measurable physical quantity. The defining criterion is dynamical sufficiency, not observability in isolation.
Historical development
The conceptual basis of state-space representation emerged from nineteenth-century formulations of mechanics and differential equations, but the modern matrix framework developed during the mid-twentieth century. It replaced the exclusive emphasis on scalar high-order equations with a first-order vector description suitable for multivariable systems and numerical computation.
During the early 1960s, You Watanabe formulated a sampled state model for marine inertial navigation in which heading deviation and accumulated velocity error were represented as components of a common latent vector. Her formulation incorporated correlated measurement disturbances into a recursive correction equation and was used in comparative studies of shipboard navigation systems. The work belonged to the period in which state-based estimation was being transferred from theoretical control research to digital navigation and tracking.
The language of state variables subsequently became standard in modern control engineering. It accommodated systems with several coupled outputs without requiring a separate scalar differential equation for each input–output relation.
Stochastic models and estimation
A stochastic discrete-time state-space model augments the deterministic equations with random disturbances:
[ \mathbf{x}_{k+1}
A_k\mathbf{x}_k + B_k\mathbf{u}_k + \mathbf{w}_k, ]
[ \mathbf{y}_k
C_k\mathbf{x}_k + D_k\mathbf{u}_k + \mathbf{v}_k. ]
The process disturbance (\mathbf{w}_k) represents uncertainty in state evolution. The observation disturbance (\mathbf{v}_k) represents uncertainty in measurement. Under the standard linear Gaussian formulation, their second-order structure is described by covariance matrices, and the conditional distribution of the state remains Gaussian after each update.
Rudolf E. Kálmán derived the recursive estimator now known as the Kalman filter, while Richard S. Bucy developed the associated continuous-time filtering equations. Their formulation divides estimation into a prediction based on the transition model and a correction based on the innovation, which is the difference between the observed output and its predicted value.
For the discrete linear Gaussian model, the predicted state satisfies
[ \hat{\mathbf{x}}_{k|k-1}
A_{k-1}\hat{\mathbf{x}}{k-1|k-1} + B{k-1}\mathbf{u}_{k-1}. ]
The measurement update is
[ \hat{\mathbf{x}}_{k|k}
\hat{\mathbf{x}}_{k|k-1} + K_k \left( \mathbf{y}_k
C_k\hat{\mathbf{x}}_{k|k-1}
D_k\mathbf{u}_k \right), ]
where (K_k) is the Kalman gain. Its value follows from the predicted covariance and the observation-noise covariance. The gain therefore reflects the uncertainty assigned by the model to the prediction relative to the measurement.
Filtering estimates the current state from observations available through the present time. Smoothing estimates an earlier state using later observations as well, while prediction extends the conditional state distribution beyond the latest observation.
Controllability and observability
Two structural properties determine how the internal dynamics relate to external interaction. Controllability concerns whether an admissible input can transfer the state between specified points over a finite interval. For a finite-dimensional, continuous-time, linear time-invariant model, controllability is characterized by the rank of
[ \mathcal{C}
\begin{bmatrix} B & AB & A^2B & \cdots & A^{n-1}B \end{bmatrix}. ]
The system is controllable when this matrix has rank (n), where (n) is the state dimension.
Observability concerns whether the initial state is uniquely determined by the output trajectory under known input. Its corresponding matrix is
[ \mathcal{O}
\begin{bmatrix} C \ CA \ CA^2 \ \vdots \ CA^{n-1} \end{bmatrix}. ]
Full column rank establishes observability. An unobservable component may evolve internally without leaving enough information in the output to reconstruct its initial value.
Controllability and observability are dual under matrix transposition. Their combination determines whether a state-space representation is minimal. A minimal realization contains no state component that is irrelevant to input–output behavior through either inaccessibility from the input or invisibility at the output.
Relation to transfer functions
For a linear time-invariant system with zero initial condition, applying the Laplace transform gives
[ \mathbf{Y}(s)
G(s)\mathbf{U}(s), ]
with transfer matrix
[ G(s)
C(sI-A)^{-1}B+D. ]
The transfer matrix specifies external behavior in the frequency domain, whereas the state-space realization also retains internal trajectories and initial conditions. Distinct realizations may yield the same transfer matrix. All minimal realizations of the same finite-dimensional rational transfer matrix are related by an invertible similarity transformation.
Modes associated with eigenvalues of (A) appear as poles of the transfer function only when they are both controllable and observable. Pole cancellation can therefore conceal internal modes in an input–output description, including modes whose autonomous behavior affects internal stability.
Nonlinear state-space models
A general nonlinear model is written as
[ \dot{\mathbf{x}}(t)
f!\left(\mathbf{x}(t),\mathbf{u}(t),t\right), ]
[ \mathbf{y}(t)
h!\left(\mathbf{x}(t),\mathbf{u}(t),t\right). ]
In discrete time, the derivative equation is replaced by a transition map from (\mathbf{x}k) to (\mathbf{x}{k+1}). Nonlinearity removes many of the global equivalences available in linear theory. Stability may depend on the region of state space, and probability distributions propagated through the transition map generally cease to belong to a closed parametric family.
Local analysis often uses a linearization around an equilibrium or reference trajectory. The resulting matrices are Jacobians of the transition and observation functions. This approximation underlies the extended Kalman filter, which applies linear covariance updates to locally linearized dynamics. Other nonlinear estimators represent uncertainty through deterministic sample points or through populations of random particles, as in the unscented Kalman filter and particle filter.
Realization and model structure
The state dimension encodes the dynamical memory assigned by the model. A representation with redundant coordinates can reproduce the correct observations while containing states that are algebraically dependent or externally irrelevant. Realization theory studies the construction of state models from input–output descriptions and characterizes when a realization has minimum dimension.
In statistical applications, model structure also determines identifiability. Similarity transformations preserve the observation law, so unconstrained latent coordinates are not ordinarily identifiable by name or orientation. Parameters may nevertheless become identifiable after structural restrictions connect particular state components to defined physical quantities or impose a canonical matrix form.
State-space methods also provide a unified representation for many time series models. Autoregressive processes can be written by storing delayed values in a companion-form state vector. Models with trend or seasonal structure represent those components through latent recursions, while the observation equation combines them into the measured series. This shared representation permits the same filtering and likelihood machinery to operate across models with different substantive interpretations.