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Actuator saturation means the applied control input reaches a device limit even if the controller requests a larger or smaller command. In the heater interpretation, the applied command is bounded between $0$ and $100\%$. If integral action continues accumulating a persistent tracking error beyond that limit, its stored integral can grow without producing the requested physical correction, delaying recovery when the goal changes. The unconstrained steady-state tracking error formulas require that the necessary command can actually be applied. ^[ref]

A control input taking only extreme admissible values except possibly on a switching or singular set. If the Hamiltonian of an optimal-control problem is affine in a scalar control $u\in [0,1]$, a nonzero coefficient selects $u=0$ or $u=1$; a zero coefficient does not determine the control.

A finite-dimensional closed quantum control system with fixed drift and control Hermitian matrices, real control inputs, and dynamics linear in the state at fixed controls and affine in the controls at fixed state. Its Dynamical Lie algebra is generated over the reals by $i{H}_0,i{H}_1,\dots$. Reachability depends on the admissible controls, allowed duration and phase convention.

Feedback control selects a control input using observations of the actual controlled plant. In a tracking loop, compare the reference signal $r$ with the measured output $y_m$ to form the tracking error $e=r-y_m$, then let the controller change $u$. The control input changes the plant output, which is measured again: that return path closes the loop. A controller can use earlier observations as well as the latest one. ^[ref]

For the ideal controlled plant, proportional control $u=2(1-x)$ initially commands $u(0)=2$. As $x$ approaches $1$, the command shrinks. When a positive control disturbance pushes $x$ above $1$, the tracking error becomes negative and the control input pushes back. The result is a bounded offset rather than the continuing drift of open-loop control. Integral action can remove that offset by remembering past tracking errors.

A control selected from the observed current state, rather than specified only as a function of the initial state and time. For $x_{t+1}=x_t+u_t$ and stage cost $x_t^2+2x_t{u}_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3{)}^t{x}_0$ and total cost $2x_0^2$. The finite-horizon Bellman coefficients satisfy ${\Pi }_0=0$, ${\Pi }_s=6(1+{\Pi }_{s-1})/(7+{\Pi }_{s-1})$ and increase to two, proving optimality.

A control disturbance is an influence on the controlled plant that is not the chosen control input. In $\dot{x}=u+w$, a constant $w=0.2$ increases $x$ by $0.2$ per unit $t$ whenever $u=0$.

Controllability asks whether the available control inputs can move a dynamical system between every pair of states in its declared state space. Fixing a duration $T>0$ gives the precise condition ${\mathcal{R}}_T(x_0)=X$ for every $x_0\in X$, where $X$ is the state space and ${\mathcal{R}}_T(x_0)$ its reachable set. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible control inputs, deadlines and any constraints along the path. ^[ref]

The surprising part is that one control input can control several state variables by acting at different times. For the double integrator, acceleration changes velocity directly, while the accumulated velocity changes position. Starting from rest, accelerate during the first half of the journey and brake during the second: the final velocity can vanish although the final position has changed. The figure is an analytic mathematical example in dimensionless quantities, with no experimental measurements.

Controllability is useful before trying to design a controller: a requested endpoint outside the reachable set cannot be attained by clever choice of feedback control. The Kalman rank criterion makes this an exact linear algebra calculation for the stated linear time-invariant system model. It concerns whole states, so attaining one plant output is not enough to establish controllability. In quantum control, the declared state space may be an orbit constrained by conservation laws; see quantum controllability and unitary orbit of a density operator.

For the linear time-invariant system $\dot{x}=Ax+Bu$, with $A\in {\mathbb{R}}^{n\times n}$ and $B\in {\mathbb{R}}^{n\times m}$, the controllability matrix has $n$ rows and $nm$ columns. The columns of $B$ are directions influenced directly by the control inputs; $AB,A^{2}B,\dots$ account for how the dynamics carry their influence into further directions. The column space of the controllability matrix is the zero-state reachable set for any $T>0$ when inputs are unrestricted. ^[ref]

Control theory studies how to choose control inputs for a dynamical system so that its plant output follows a reference signal, despite control disturbances and imperfect knowledge of the controlled plant. A controller turns the available information into a control signal. The design must consider both the resulting response and stability theory: a rule that corrects a small mismatch can still produce growing deviations if the interconnection is badly chosen. ^[ref]

The central distinction is informational. Open-loop control executes planned commands; feedback control uses the measured output of the actual controlled plant to update those commands. A reference signal can describe a destination, while control disturbances describe influences that the controller did not command. A model can predict where the controlled plant should be; measurement tells the controller what actually happened. The authors provide a free full-book PDF through ^[ref].

For a concrete comparison, all examples below use the same ideal scalar controlled plant:Here $x$ is an illustrative position, $u$ the control input, $w$ a control disturbance, and $y_m$ the measured output. Both $x$ and $t$ are dimensionless quantities; the numbers are chosen to expose the mechanism, with no empirical interpretation. The ordinary differential equation says that the present control input and control disturbance accumulate into $x$. The measured output equals the entire state only in this ideal example; a general plant output need not reveal the entire state.

Such a quantum state remains an eigenstate of $H_0+f(t)H_1$ for every control input. Its projector commutes with all available generators, so its population is conserved. This obstructs full-space pure-state controllability even when the orthogonal bright subspace can be controlled. An instantaneous dark state of a slowly varying Quantum Hamiltonian instead supports adiabatic quantum control if a spectral gap and slowness condition are maintained.

Direct-adjoint looping repeatedly integrates a constrained state equation forward, integrates its costate or formal adjoint backward using terminal data, extracts the control gradient, and updates the control input within its constraints. It gives a practical route to local optimization but does not by itself prove a global minimum.

The double integrator is the linear time-invariant system in which one control input is integrated into velocity, and velocity into position. It is a mathematical idealization of one-dimensional motion under commanded acceleration; for a constant mass $m$ and net force $F$, Newton's second law gives $u=F/m$. The worked examples here use normalized dimensionless quantities and assume no control disturbances, rather than describing a laboratory record. ^[ref]

Writing $x=(p,v{)}^T$ givesThe Kalman rank criterion therefore gives controllability in every $T>0$ for unrestricted real control inputs. More concretely, choose constants $a,b$ on the first and second halves. Starting at $x_0=(p_0,v_0{)}^T$, direct integration givesThe endpoint map has determinant $T^3/8\ne 0$, so these two amplitudes can supply any desired endpoint displacement. For a rest-to-rest move of distance $d$, take $a=4d/T^2$, $b=-4d/T^2$. At $T=2$ and $d=1$ these are the two commands in the first figure. No impulse is needed; shorter deadlines require larger finite commands.

A feedback gain sets the strength of a controller's response to its observations. For proportional control, $u=ke$ means that a tracking error $e=0.1$ produces a control input $u=0.2$ when $k=2$.

Hamiltonian engineering designs accessible interactions or their effective time average to implement desired quantum states or unitary operators. The available control inputs, Quantum Hamiltonian generators and laboratory constraints determine which transformations can actually be achieved. Model-based open-loop control computes a waveform before running it; adaptive experiments instead evaluate its performance on the physical system.

A plant input is a signal supplied to the controlled plant. The chosen plant input is its control input; other influences can enter as control disturbances. This use concerns a dynamical system, rather than the input of a computation.

Integral action adds memory of the tracking error to a controller: $z(t)=z(0)+{\int }_0^{t}e(s)ds$. A persistent tracking error keeps changing $z$, so an equilibrium point with constant $z$ must have $e=0$. The accumulated value can remain nonzero after the tracking error has vanished, maintaining the control input needed to cancel a constant control disturbance. Convergence still requires a stable feedback control design. ^[ref]

The Kalman filter alternates a Kalman prediction step with a Kalman measurement update. It carries a state estimate forward using a state-space model, then adjusts it by the part of the new observation that was not predicted: the innovation. The Kalman gain determines the adjustment from the uncertainty of the prediction and the measurement noise. A noisy measurement should have less influence than an equally surprising precise one.

For a precise discrete-time statement, let the state and observation be vectors in real vector spaces obeying the linear state-space modelThe matrices and control inputs are known. Assume a Gaussian random vector $x_0$ with mean $m_0$ and covariance matrix $P_0$, and mutually independent centered Gaussian noises $w_t,v_t$, independent of $x_0$, with covariance matrices $Q_t,R_t$. Suppose $R_t$ is positive-definite, so the observation innovation covariance is invertible. Then the recursions below give the exact posterior mean $m_t=\mathbb{E}[x_t\mid y_1,\dots ,y_t]$ and posterior covariance matrix $P_t$, hence the full conditional multivariate normal distribution. They also minimize mean squared error. Without the Gaussian assumptions, suitable second-order assumptions still give best affine state estimates, but do not generally give an exact posterior distribution. ^[ref].

For an exact scalar example, suppose the predicted state has normal distribution $N(10,4)$ and a measurement reports $y=14$ in the same units. With measurement noise variance $R=1$, the Kalman gain is $4/(4+1)=4/5$ and the posterior distribution isWith the same prediction and observed value but measurement noise variance $R=16$, the Kalman gain becomes $1/5$ and the posterior distribution is instead $N(10.8,3.2)$. Both updates move toward the observation; the more precise measurement moves the estimate further and reduces uncertainty more.

The useful compression is exact under the stated model: the two objects $m_t,P_t$ contain everything from the past observations needed for the next update. Their size depends on the state dimension, not on the length of the observation history. This makes causal state filtering practical when observations arrive continually. It also gives Linear-quadratic-Gaussian control an estimate of the state on which to act.

Rudolf E. Kalman's 1960 paper organized linear filtering around state transitions, error covariance and recursive computation, connecting estimation with optimal regulation. Its introduction acknowledges earlier work by Wiener, Bode, Shannon and others; the useful historical claim is this particular formulation, rather than that estimation began in 1960. ^[ref].

The Kalman prediction step advances the previous posterior mean and posterior covariance matrix through the state transition:The minus superscript marks information before observing $y_t$. The transpose and matrix multiplication propagate the old uncertainty; $Q_t$ adds the uncertainty of the new process noise. A known control input changes the mean but contributes no additional variance. If the control input itself is uncertain, that uncertainty needs its own model.

For a finite-dimensional continuous-time linear time-invariant system $\dot{x}=Ax+Bu$ on ${\mathbb{R}}^n$, with constant matrices $A,B$ over the real numbers, no state constraints, and every piecewise continuous function $u:[0,T]\to {\mathbb{R}}^m$ allowed as a control input without an amplitude bound, controllability in any fixed $T>0$ is equivalent to the controllability matrix having rank of a matrix $n$. The same equivalence holds with all square-integrable functions as inputs. A stronger useful statement iswhere $e^{AT}$ is a matrix exponential and $\mathrm{im}\mathcal{C}$ is the column space. Thus a rank of a matrix $r<n$ leaves only an $r$-dimensional affine subspace reachable at that time. ^[ref]

The Kalman rank criterion allows arbitrarily large commands as $T$ shrinks. It does not assert controllability with bounded control inputs, nor test a general nonlinear dynamical system just by counting its inputs. For the double integrator, $B=(0,1{)}^T$ and $AB=(1,0{)}^T$ give two independent directions, despite there being only one control input.

Measurement-based quantum feedback adapts a control input from a measurement record or conditional quantum state estimate. A measurement contributes backaction and generally stochastic dynamics. Exact instantaneous expectation values cannot be read from a single system without these effects. A feedback law derived for deterministic closed-system evolution therefore needs a new conditional dynamical analysis before being applied as measurement feedback.

An open-loop control is prescribed as a function of time, possibly after calculation from an initial state and a dynamical model. It is executed without using observations of the actual current state to change the input. A simulated feedback control law can generate an open-loop waveform, but the resulting laboratory control does not automatically correct errors in the actual state or model.

For the ideal controlled plant with $w=0$, plan the control inputIntegration gives $x(t)=\min (t,1)$, so the planned command reaches the reference signal $r=1$ at $t=1$. If an unplanned constant control disturbance $w=0.2$ begins at $t=2$, the same open-loop control gives $x(t)=1+0.2(t-2)$ for $t\ge 2$: $x(4)=1.4$ and $x(8)=2.2$. Knowing the nominal response did not provide a way to correct the actual plant output.