Kalman CriterionEdit
The Kalman Criterion refers to a fundamental set of rank conditions used to judge whether a linear dynamical system is observable or controllable from input-output data. Originating with the work of Rudolf E. Kalman, these criteria provide a precise, checkable standard for when a system’s internal state can be inferred from outputs or steered to any desired state through appropriate inputs. In engineering practice, these criteria underpin the reliability of state estimation and controller design, and they form a backbone for modern approaches to navigation, robotics, and process control. The ideas are closely tied to the development of the Kalman filter and its extensions, which rely on a sound understanding of observability and controllability to function correctly.
The Kalman Criterion is most often stated in two dual forms: one for observability and another for controllability. In the usual state-space formulation, a linear time-invariant system is written as ẋ = A x + B u, y = C x (+ D u in some formulations), with x representing the state vector, u the input, and y the measured output. The notions below apply to both discrete-time and continuous-time versions, with the appropriate matrices A, B, and C.
Mathematical formulation
Discrete-time observability criterion
For a system x_{k+1} = A x_k + B u_k, y_k = C x_k, the observability matrix is
O = [ C CA CA^2 ... CA^{n-1} ],
where n is the dimension of the state x. The Kalman criterion states that the system is observable if rank(O) = n.
Controllability criterion (Kalman rank condition)
Dually, for the same system form, the controllability matrix is
Q = [ B AB A^2B ... A^{n-1}B ].
The system is controllable if rank(Q) = n. This condition guarantees that an appropriate sequence of inputs can drive the state from any initial value to any final value within finite time.
Duality and decomposition
Observability and controllability are mirror concepts: the pair (A, C) characterizes observability, while (A, B) characterizes controllability. The two notions are related through a form of duality, and together they yield the Kalman decomposition, which separates the state space into observable/unobservable and controllable/uncontrollable subspaces. This decomposition clarifies which state components can be inferred from outputs and which can be influenced by inputs, guiding practical design decisions Kalman decomposition.
Continuous-time perspective
In continuous time, the same ideas apply with ẋ = A x + B u and y = C x. The observability of (A, C) is determined by the rank of an observability matrix built from C and its Lie derivatives CA, CA^2, etc., or by equivalent rank conditions derived from the system’s impulse response. Many texts express the criterion in algebraic form identical in spirit to the discrete-time case, just adapted to the differential-time setting.
Historical context and significance
Rudolf E. Kalman introduced these rank conditions in the 1960s as part of a broader theory connecting state estimation and control. His work laid the foundation for the modern Kalman filter, a recursive estimator that assumes the system is observable in the Kalman sense and thereby produces optimal state estimates under Gaussian noise assumptions. The criteria gained immediate practical traction in aerospace, navigation, and later in industrial automation, where knowing that the internal state could be reconstructed from measurements or steered through controls was essential for reliability and performance. Over time, the Kalman Criterion has become a standard reference in control theory, systems engineering, and related disciplines Rudolf E. Kalman.
Examples and intuition
A simple two-state system provides an accessible illustration. Suppose A = [[1, 1], [0, 1]] and C = 1, 0. The observability matrix O becomes O = [[1, 0], [1, 1]], which has full rank 2, so the system is observable. A different C, say C = 0, 1, yields O = [[0, 1], [1, 0]], also of full rank 2, again observable. If instead C = 1, 1, then O = [[1, 1], [2, 1]], which still has rank 2 in this case, preserving observability. These little calculations illustrate how the choice of output channels (C) interacts with the system dynamics (A) to determine observability, a core idea behind the Kalman Criterion.
The same kind of rank test applies to controllability with the matrix B. If B is poorly aligned with the dynamics encoded in A, the columns of Q = [B, AB, ..., A^{n-1}B] may fail to span the entire state space, and some directions in state space cannot be reached by input.
Applications and limitations
Practical engineering uses: The Kalman Criterion informs the design of estimators and controllers across domains such as aerospace guidance, automotive control, and industrial process control. When a system is observable and controllable, engineers can confidently implement observers like the Kalman filter and design state-feedback controllers that achieve desired performance.
Nonlinear and uncertain settings: Real-world systems are not perfectly linear or known, so the strict linear Kalman Criterion may not hold globally. In nonlinear contexts, extensions such as the Extended Kalman filter and the Unscented Kalman filter are used, though they rely on local or approximate observability concepts. In severely uncertain environments, alternative approaches like particle filtering may be preferred, but the core ideas about when state information can be extracted or controlled remain central.
Model and measurement considerations: Observability is a property of the model (A, C) and not merely of the data collected. If the model omits critical dynamics or if measurements are highly corrupted, the practical realization of observability may be compromised, even if the nominal Kalman Criterion holds for the nominal model.