Contributions to the Theory of Optimal Control (Kalman, 1960)
Kalman's primary paper formulates the continuous-time quadratic regulator rigorously and connects controllability, observability, Hamilton-Jacobi reasoning, stability, and the matrix Riccati equation.
This 18-page paper develops the regulator problem for a linear plant with quadratic cost and exact state knowledge. Its lasting contribution is not merely the feedback formula: it supplies conditions under which the regulator exists and stabilizes the plant, and it makes controllability and observability central to those claims.
Mechanism #
For a linear plant and a quadratic value function, the Hamilton-Jacobi equation closes within quadratic forms. Matching coefficients produces a matrix Riccati differential equation; minimizing the Hamiltonian produces linear state feedback. In the time-invariant infinite-horizon limit, the stabilizing Riccati solution gives a constant feedback law and a Lyapunov function for the closed loop.
The paper assumes the state is known. It explicitly separates state reconstruction from regulation; GTK therefore does not smuggle estimator dynamics into the LQR derivation and links observers separately.
Boundaries #
- Positive-definite or semidefinite weight assumptions matter. An arbitrary symmetric matrix does not define a sensible nonnegative performance index.
- Stability depends on system properties, not on the mere existence of a symbolic Riccati equation.
- The unconstrained solution does not enforce actuator saturation, rate limits, or state constraints.
Visual audit #
All 18 pages were read and rendered. The source is equation- and proof-dense and contains no figure needed by GTK. The article uses an independent backward-Riccati/forward-state diagram.