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State variable

In control theory and dynamical systems, a state variable is one of a minimal set of variables that fully describe the internal condition of a system at a given time, enabling the prediction of its future behavior under specified inputs and dynamics.^[1] These variables are typically chosen to correspond to the system's energy storage elements, such as capacitor voltages or inductor currents in electrical circuits, or positions and velocities in mechanical systems.^[2] The collection of state variables forms a state vector, which evolves according to first-order ordinary differential equations in continuous-time models or difference equations in discrete-time models.^[1] The concept of state variables underpins the state-space representation, a framework introduced by Rudolf E. Kálmán in the late 1950s and 1960s, which shifted control system analysis from input-output transfer functions to internal system descriptions.^[3] In this approach, the system's dynamics are expressed as \dot{x}(t) = Ax(t) + Bu(t) for linear time-invariant systems, where x(t) is the state vector, u(t) is the input vector, and A and B are system matrices; the output is then y(t) = Cx(t) + Du(t).^[1] This formulation facilitates advanced techniques like controllability analysis, observability, optimal control, and state estimation via Kalman filters, making it essential for modeling complex systems in engineering, physics, and economics.^[3] The minimal number of state variables required equals the order of the system, ensuring a compact yet complete description without redundancy.^[1] State variables extend beyond linear systems to nonlinear and time-varying cases, where their selection may involve phase space coordinates or other coordinates that capture the system's memory and trajectory. Applications span diverse fields, including aerospace for flight control, robotics for motion planning, and signal processing for filtering noisy measurements, highlighting their role in bridging theoretical modeling with practical implementation.^[3]

Fundamentals

Definition

In dynamical systems theory, a state variable is one of a minimal set of variables that completely describe the internal condition of a system at any instant in time, enabling the prediction of its future evolution given the system's governing equations and external inputs. This set captures all essential information about the system's "memory" or past history, such that the values of these variables at an initial time t_0 suffice to determine the system's trajectory for all subsequent times t \geq t_0.^[4] The concept, formalized in modern control theory, emphasizes that the state provides a unique snapshot from which all future behavior can be computed without recourse to earlier history.^[5] The requirement for minimality ensures that the state variables form the smallest possible set without redundancy, with the number of such variables equal to the order of the system—corresponding, for instance, to the number of independent energy-storing elements in physical realizations like mechanical or electrical systems. This minimal representation avoids superfluous variables while fully encoding the dynamics, making it foundational for analysis and design in fields such as control engineering. Initial conditions, specified by assigning values to these state variables at t_0, uniquely fix the system's response path under given inputs, highlighting their role in distinguishing different possible evolutions from the same model.^[4] State variables interact with the system's inputs and outputs in a structured manner: they evolve dynamically in response to inputs according to the system's equations, while outputs are derived as functions of the current state, the inputs, and possibly explicit time dependence. This relationship underpins the state-space framework, where inputs drive changes in the state, and the state in turn governs observable outputs, facilitating comprehensive modeling of complex interactions without loss of information.^[5]

Historical Development

The concept of state variables traces its early roots to the 19th century, particularly through James Clerk Maxwell's analysis of centrifugal governors in steam engines. In his 1868 paper "On Governors," Maxwell provided the first mathematical treatment of feedback control systems, examining stability and transient behavior in terms that implicitly captured internal system states, such as velocity and displacement, to regulate engine speed.^[6] This work laid foundational groundwork for dynamical systems theory, influencing later developments in control engineering without explicitly formalizing state-space representations.^[7] The formal emergence of state-space methods occurred in the mid-20th century amid the transition from classical frequency-domain techniques, like Laplace transforms and Bode plots, to time-domain approaches. This shift accelerated in the 1950s and 1960s during the Space Race, as aerospace applications demanded multivariable and nonlinear system modeling beyond the limitations of single-input single-output transfer functions. The Apollo program's guidance and navigation systems, for instance, relied on state-space formulations for real-time trajectory prediction and correction.^[8]^[9] A pivotal milestone came in 1960 with Rudolf E. Kalman's contributions, which established state-space as the cornerstone of modern control theory. In "A New Approach to Linear Filtering and Prediction Problems," Kalman introduced state-variable models for optimal estimation in noisy environments, while his concurrent work on controllability and observability provided criteria to assess system realizability and feedback design.^[10] These ideas, building on earlier vector space concepts, enabled the analysis of linear systems via first-order differential equations in state vectors.^[11] Following the 1960s, state-space methods integrated with optimal control frameworks, notably through the linear quadratic regulator (LQR), which minimized quadratic cost functions over state trajectories and became widely adopted for stabilizing complex systems.^[12] By the 1980s and 1990s, advancements in computational tools, such as numerical solvers in software libraries like SLICOT and MATLAB's Control System Toolbox, facilitated the simulation and design of high-dimensional state-space models, broadening their application in engineering practice.^[13]

Mathematical Representation

Continuous-Time Systems

In continuous-time systems, the state-space representation provides a framework for modeling the dynamics of linear time-invariant (LTI) systems using first-order differential equations. The core state equation is given by

\dot{x}(t) = A x(t) + B u(t),

where x(t) \in \mathbb{R}^n is the state vector capturing the system's internal conditions at time t, A \in \mathbb{R}^{n \times n} is the system matrix describing the natural evolution of the states, B \in \mathbb{R}^{n \times m} is the input matrix relating the input vector u(t) \in \mathbb{R}^m to the state dynamics, and \dot{x}(t) denotes the time derivative of the state vector.^[14]^[15] This formulation, introduced by Kalman, unifies the description of multi-input multi-output systems by focusing on the state as a minimal set of variables sufficient to predict future behavior given the inputs.^[14] The output of the system is related to the states and inputs through the equation

y(t) = C x(t) + D u(t),

where y(t) \in \mathbb{R}^p is the output vector, C \in \mathbb{R}^{p \times n} is the output matrix mapping states to outputs, and D \in \mathbb{R}^{p \times m} is the feedthrough matrix accounting for direct transmission from inputs to outputs without state involvement.^[15] For LTI systems, the matrices A, B, C, and D are constant, ensuring the system's response is linear in the states and inputs and invariant to time shifts.^[14] The general solution to the state equation, assuming an initial state x(t_0) at time t_0, is obtained by solving the linear ordinary differential equation:

x(t) = e^{A(t - t_0)} x(t_0) + \int_{t_0}^t e^{A(t - \tau)} B u(\tau) \, d\tau,

where e^{A(t - t_0)} is the matrix exponential serving as the state transition matrix that propagates the homogeneous solution forward in time.^[14] This integral form incorporates the forced response due to the input, leveraging the superposition principle inherent to linear systems.^[15] While the state-space model primarily addresses LTI systems under linearity and time-invariance assumptions, it extends to nonlinear continuous-time systems by generalizing the equations to \dot{x}(t) = f(x(t), u(t)) and y(t) = h(x(t), u(t)), where f and h are nonlinear functions.^[16]

Discrete-Time Systems

In discrete-time systems, the state-space model describes the evolution of the state vector at successive time steps, typically indexed by the integer k. The state update equation is given by

x(k+1) = A x(k) + B u(k),

where x(k) \in \mathbb{R}^n is the state vector, u(k) \in \mathbb{R}^m is the input vector, A \in \mathbb{R}^{n \times n} is the state transition matrix, and B \in \mathbb{R}^{n \times m} is the input matrix. This formulation, introduced by Rudolf E. Kalman in his foundational work on linear filtering, captures the dynamics of systems such as digital controllers and sampled-data processes, where the state at each step depends linearly on the previous state and the current input. The output equation complements the state update, relating the observed output y(k) \in \mathbb{R}^p to the current state and input via

y(k) = C x(k) + D u(k),

with C \in \mathbb{R}^{p \times n} as the output matrix and D \in \mathbb{R}^{p \times m} as the feedthrough matrix. This pair of equations forms the canonical discrete-time state-space representation, enabling analysis of systems like signal processing algorithms and discrete event simulations. The model assumes linearity and time-invariance unless specified otherwise, distinguishing it from continuous-time counterparts by using difference equations rather than differentials.^[17] The solution to the state update equation can be expressed iteratively, starting from an initial state x(k_0) at time k_0:

x(k) = A^{k - k_0} x(k_0) + \sum_{m = k_0}^{k-1} A^{k-1-m} B u(m),

for k \geq k_0. This closed-form expression highlights the Markovian property of the state, where future states depend only on the current state and inputs, facilitating computations in recursive algorithms like the Kalman filter. It arises directly from repeated substitution of the state equation, underscoring the model's suitability for prediction in time-series data.^[17] Discrete-time models are often derived from continuous-time systems through sampling with a zero-order hold (ZOH), which maintains constant input over each sampling interval T. The discretized matrices are

A_d = e^{A T}, \quad B_d = \int_0^T e^{A \tau} \, d\tau \, B,

where A and B are the continuous-time matrices. This exact discretization preserves the system's response to piecewise-constant inputs, making it essential for implementing analog controllers on digital hardware, such as in aerospace and robotics applications. The matrix exponential e^{A T} can be computed via series expansion or eigenvalue decomposition, ensuring numerical stability for small T.^[18]

Properties and Analysis

Controllability

In control theory, controllability refers to the ability of a system to be driven from any initial state to any desired final state within a finite time interval using appropriate input signals.^[11] This property is fundamental for linear time-invariant (LTI) systems represented in state-space form, where the state vector x(t) evolves according to \dot{x}(t) = Ax(t) + Bu(t), with A as the system matrix, B as the input matrix, and u(t) as the control input.^[11] For continuous-time LTI systems, controllability is determined by the controllability matrix \mathcal{C} = [B, AB, A^2B, \dots, A^{n-1}B], where n is the dimension of the state space.^[11] The system is controllable if and only if the rank of \mathcal{C} equals n, ensuring that the columns of \mathcal{C} span the entire state space.^[11] This rank condition, known as the Kalman rank condition, provides a necessary and sufficient criterion for controllability and was established as part of the foundational framework for state-space analysis.^[11] In discrete-time LTI systems, described by x(k+1) = Ax(k) + Bu(k), the controllability matrix takes the identical form \mathcal{C} = [B, AB, A^2B, \dots, A^{n-1}B], and the system is controllable if \operatorname{[rank](/page/Rank)}(\mathcal{C}) = n.^[19] The Kalman rank condition applies similarly here, confirming that the reachable subspace from the origin covers the full state space under admissible inputs.^[19] The Kalman rank condition has critical implications for control design, particularly in techniques like pole placement, where controllability ensures that state feedback can arbitrarily assign the closed-loop eigenvalues of the system matrix to achieve desired dynamic performance.^[20] For instance, in multi-input systems, this equivalence allows for the synthesis of stabilizing controllers by transforming the system into a controllable canonical form.^[20]

Observability

In control theory, observability refers to the property of a dynamic system that allows the initial state x(t_0) to be uniquely reconstructed from the knowledge of the input u(t) and output y(t) over a finite time interval [t_0, t_0 + T] for some T > 0.^[21] This concept is fundamental for state estimation, as it ensures that all internal states influencing the system's behavior can be inferred from external measurements, distinguishing the system from unobservable cases where certain states produce identical input-output responses regardless of their values.^[22] For linear time-invariant (LTI) systems, observability is tested using the observability matrix. In continuous-time systems described by \dot{x} = Ax + Bu, y = Cx + Du, the observability matrix is defined as

\mathcal{O} = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{bmatrix},

where n is the dimension of the state vector x, A is the state matrix, and C is the output matrix. The system is observable if and only if \mathcal{O} has full rank n, meaning its rows span the entire n-dimensional state space.^[23] For discrete-time LTI systems given by x_{k+1} = Ax_k + Bu_k, y_k = Cx_k + Du_k, the observability matrix takes the analogous form

\mathcal{O} = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{bmatrix},

and the system is observable if \mathcal{O} has full rank n.^[23] This rank condition guarantees that the initial state can be uniquely determined from a finite sequence of inputs and outputs. A key application of observability is in state estimation via the Luenberger observer, which reconstructs the state vector when direct measurement is unavailable. For a continuous-time LTI system, the observer dynamics are given by

\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x}),

where \hat{x} is the estimated state, u is the input, y is the measured output, and L is the observer gain matrix chosen to ensure stability of the error dynamics \dot{e} = (A - LC)e with e = x - \hat{x}.^[24] This structure requires the pair (A, C) to be observable, allowing the estimation error to converge asymptotically to zero under appropriate gain selection.^[24]

Applications and Examples

In Mechanical Systems

In mechanical systems, state variables provide a framework for modeling the dynamics of physical components such as masses, springs, and dampers by capturing the energy storage mechanisms inherent to the system. A canonical example is the second-order mass-spring-damper system, which consists of a mass m attached to a spring with stiffness k and a damper with coefficient c, subjected to an external force u(t). The governing differential equation for the displacement x(t) of the mass is given by

m \ddot{x} + c \dot{x} + k x = u(t),

where \ddot{x} and \dot{x} represent acceleration and velocity, respectively.^[25]^[4] To represent this in state-space form, suitable state variables are selected as the position x_1 = x and the velocity x_2 = \dot{x}, which together fully describe the system's configuration and momentum at any instant. These choices align with the general continuous-time state-space representation, where the state vector evolves according to first-order differential equations. The resulting state equations are

\dot{x_1} = x_2, \quad \dot{x_2} = -\frac{k}{m} x_1 - \frac{c}{m} x_2 + \frac{1}{m} u.

This formulation transforms the second-order equation into a pair of coupled first-order equations, facilitating analysis and simulation.^[25]^[4] In matrix notation, the state-space model is expressed as \dot{\mathbf{x}} = A \mathbf{x} + B u with output y = C \mathbf{x} (assuming position as the measured output and no direct feedthrough), where

A = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{c}{m} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}.

The matrix A encodes the internal dynamics, including the restoring force from the spring and dissipative effects from the damper, while B describes how the input force influences the acceleration.^[25]^[4] The state variables x_1 and x_2 directly correspond to the system's energy storage: x_1 relates to potential energy stored in the spring (\frac{1}{2} k x_1^2), and x_2 to kinetic energy of the mass (\frac{1}{2} m x_2^2). This interpretation underscores how state variables encapsulate the minimal information needed to predict future behavior, accounting for both conservative and dissipative elements in mechanical systems. For instance, in vibration isolation or vehicle suspension design, these states enable precise control strategies by tracking energy distribution over time.^[25]^[4]

In Electrical Circuits

In electrical circuits, state-space representation is particularly useful for modeling dynamic systems involving energy storage elements, such as inductors and capacitors. A classic example is the series RLC circuit, consisting of a resistor (R), inductor (L), and capacitor (C) connected in series with an input voltage source u(t). The governing differential equation for the current i(t) through the circuit is derived from Kirchhoff's voltage law:

L \frac{di}{dt} + R i + \frac{1}{C} \int i \, dt = u(t).

This second-order equation captures the circuit's behavior, where the integral term represents the voltage across the capacitor.^[26]^[27] To express this in state-space form, appropriate state variables are selected based on the energy-storing components: the inductor current x_1 = i (which stores magnetic energy) and the capacitor voltage x_2 = \frac{1}{C} \int i \, dt (which stores electric energy). These choices align with the principle that states should reflect independent energy storage mechanisms in the circuit, allowing the system's future behavior to be determined from current states and inputs. Differentiating the integral term yields \dot{x_2} = \frac{x_1}{C}, while from the governing equation, \dot{x_1} = -\frac{R}{L} x_1 - \frac{1}{L} x_2 + \frac{1}{L} u. Thus, the state equations are:

\dot{x_1} = -\frac{R}{L} x_1 - \frac{1}{L} x_2 + \frac{1}{L} u, \quad \dot{x_2} = \frac{x_1}{C}.

For an output such as the capacitor voltage, the measurement equation is y = x_2.^[28]^[29]^[30] In matrix form, the state-space model is compactly represented as \dot{\mathbf{x}} = A \mathbf{x} + B u, y = C \mathbf{x}, where

A = \begin{bmatrix} -\frac{R}{L} & -\frac{1}{L} \\ \frac{1}{C} & 0 \end{bmatrix}, \quad B = \begin{bmatrix} \frac{1}{L} \\ 0 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 1 \end{bmatrix}.

This formulation facilitates analysis of the circuit's transient response, stability, and control design, emphasizing how the states encapsulate the circuit's memory through its passive elements.^[26]^[31]

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