On the General Theory of Control Systems

Kalman, R. E. (1960). On the General Theory of Control Systems. Proceedings of the First International Congress of the International Federation of Automatic Control (IFAC), 1(1), 481-492.

Type

Journal article

Intellectual base

Primary research

Year

1960

Volume

1

Issue

1

Pages

481-492

ISSN

1474-6670

DOI

10.1016/s1474-6670(17)70094-8

DOI registrant

Elsevier

Link

https://doi.org/10.1016/S1474-6670(17)70094-8

Cited from

engineering_design, mathematics, neuroscience, statistics_experimental_design

Cited by

8 citations across 8 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Controllability
  • Controllability is the structural property that determines whether an agent's available inputs can steer a system's state into any desired region such that: (1) a system is controllable when, for any initial state \(x_0\) and any target state \(x_1\), there exists an admissible input trajectory \(u(t)\) that drives the system from \(x_0\) to \(x_1\) in finite time — formally, for a linear time-invariant system \(\dot x = Ax + Bu\), controllability reduces to the rank condition on the controllability matrix \(\mathcal{C} = [B, AB, A^2 B, \ldots, A^{n-1} B]\) (full rank \(\Leftrightarrow\) controllable); for nonlinear systems, the analogous condition uses Lie bracket algebra (Chow's theorem, Sussmann's controllability criterion); practical operational definitions in management contexts test whether "interventions of specified type-and-magnitude can move this system's key variables through the desired range"

    This sourceIntroduces controllability and observability and the state-space formulation, including the controllability rank condition and the controllability/observability duality. Load-bearing for markers 106 (controllability definition / rank condition), 107 (Kalman controllability, pole placement, LQR), and 119 (rank-based characterization for linear systems).

• Linearity
  • Engineering and control uses linearity for LTI (linear time-invariant) systems, transfer functions, state-space models
• Measurement Uncertainty and Observational Noise
  • Kalman (1960) addressed precisely this composite problem by deriving an optimal recursive estimator that combines a noisy measurement with a model of the system to produce the minimum-variance estimate of the true state.
• Observability
  • (2) observability is the information-theoretic dual of controllability (see #391) — controllability asks "can inputs steer state?"; observability asks "do outputs reveal state?"; Kalman's 1960 seminal work established this duality via the formal correspondence \((A, B) \text{ controllable} \Leftrightarrow (A^T, B^T) \text{ observable}\), making observability and controllability reciprocal structural properties of the same state-space model; (3) observability delivers the prerequisite for monitoring, diagnosis, state-feedback control, and learning — without observability, the system's internal state is partly or fully hidden; diagnostic reasoning, state estimation (Kalman filter, Luenberger observer), and closed-loop feedback become impossible or degraded; software systems without observability incur long incident-resolution times and repeated unknown-cause outages; biological systems without observable markers resist treatment; organizations without observable KPIs cannot self-correct; (4) the concept generalizes across domains — control engineering (Kalman observability, observer design, state estimation under noise, fault detection and identification)
• Predictive Coding
  • Control and estimation: The Kalman filter advances a state prediction and corrects it by the innovation — measurement minus predicted measurement — which is the exact same residual loop with an optimal, uncertainty-weighted gain, as Kalman (1960) formalized for linear-Gaussian systems.
• Surjectivity
  • In systems and control it is controllability and reachability of a state set — every target state must be the image of some control input — and full state coverage is the surjectivity of the reachability map.

Mechanisms

• Independent-Axis Design Review
  • The bar the review is really enforcing is controllability — that the thrusters' direction vectors span the space of motions the vehicle must command.

    This sourceDefines controllability as the ability of available control inputs to span or reach the required system state space.

• Stochastic State-Space Model
  • Observability is the hard limit

    This sourceEstablishes observability as the condition under which latent state can be uniquely reconstructed from measured outputs.

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