Source
Başar, Meyn & Perkins (2020), Lecture Notes on Control System Theory and Design, arXiv:2007.01367v2, LIT-051, read in full in NOTE-062. The reader recomputed the worked examples, including the Pendubot plant, its LQR poles and its compensator.
What was actually shown
The setting. Continuous-time, deterministic, linear time-invariant plant ẋ = Ax + Bu with one input, cost weights Q ≥ 0 and r > 0, and the infinite-horizon LQR gain K from the algebraic Riccati equation. The loop gain broken at the input is L(s) = K(sI − A)⁻¹B.
The identity (§10.6, eqs. 10.27–10.32). The return-difference equation gives |1 + L(jω)|² = 1 + |P(jω)|²/r ≥ 1 at every frequency, with P the transfer function from input to the weighted output. So the sensitivity S = 1/(1 + L) satisfies |S(jω)| ≤ 1. The Nyquist plot stays outside the unit disk around −1, which gives infinite gain margin, tolerance of the gain halved, and phase margin of at least 60° (p. 191). The same chapter gives the symmetric root locus and Kalman’s inequality.
The counterexample to carrying it over (Chs 9–10). The Pendubot is a two-link underactuated pendulum. With one sensor it is controllable and observable, but its transfer function has a zero at s ≈ +6.54 between two unstable poles. Any stabilising compensator must then itself be unstable, and the observer-based design has a compensator pole at s ≈ 46.4. Its gain margin is “less than 10⁻²” (p. 192). NOTE-062 found that the printed compensator, rounded to its printed precision, does not stabilise the printed plant, which makes the fragility sharper than the notes say. Full-state LQR on the same plant keeps the margins above.
What could have come out otherwise. The return difference could have dipped below 1 at some frequency for some choice of Q, which would have meant a margin that depends on the weights. It cannot, for any Q ≥ 0 and r > 0.
What this does not say
- That the margins hold for multi-input LQR as stated. The derivation is for a single input. The multi-input version needs more, for example a diagonal R, and the notes do not discuss it (NOTE-062, C5).
- That an observer-based LQG design inherits them. The Pendubot shows it does not. Controllable-and-observable says nothing about robustness; zeros and sensor choice do.
- That the infinite-horizon LQR exists in the case used. Existence, uniqueness and stability under stabilisability and detectability (Thm 10.6) are stated without proof in the notes.
- That the notes bridge control and reinforcement learning. The 2024 preface promises it; the text has no RL, no stochastic control and no discrete-time Riccati equation (NOTE-062, corrections).
- That “60%” is meant. p. 187 prints “phase margin at least 60%”; p. 191 has 60°.
Connections
- THEORY-325 holds that a non-normal linear system stable at every instant can be destabilised by time dependence. The margins here are for a fixed LTI loop and say nothing about time-varying perturbations of A.