Source-linked AI summary
Feedback Stabilization for Sampled Linear Systems with Control-linear Noise
Cynthia Cheng, Xudong Chen
TL;DR
The paper asks how control-noise intensity and sampling intervals affect mean-square feedback stabilization in continuous stochastic systems with sampled measurements. It connects stabilization to a Riccati-type fixed-point condition and infinite-horizon optimal control, and reports that stabilizability is preserved as noise intensity or sampling interval decreases.
Problem
The paper asks how noise intensity σ² and sampling interval τ jointly constrain feedback stabilizability for continuous dynamics with sampled measurements.
Source: §1 · pp. 1–2Method
It relates mean-square stabilization to a fixed point of a Riccati-type map and an infinite-horizon optimal-control solution.
Source: §2.1 · p. 3, §3.2.2 · p. 10Results
Stabilizability is preserved when σ² and τ are reduced; a unique fixed point of f also implies mean-square feedback stabilizability.
Source: §1 · p. 2, §2.1 · p. 3Takeaways & Limitations
For fixed (A, B), reduced noise intensity and shorter sampling intervals preserve stabilizability.
Source: §1 · p. 2The measurement model uses exact state samples, y_{kτ}=x_{kτ}.
Source: §1 · p. 1
Abstract
from arXiv · showhide
In this paper, we consider linear stochastic systems with control-linear noise and periodically sampled measurements. We address the problem of feedback stabilization in the mean-square sense. The main contribution of the paper is to provide a necessary and sufficient condition for feedback stabilization. In particular, we relate feedback stabilizability of the stochastic system to the existence of a fixed point of a Riccati-type algebraic equation and, further, to the existence of a solution to an infinite-horizon optimal control problem.
1 Introduction
The paper studies mean-square feedback stabilization for continuous stochastic dynamics with control-linear noise and periodically sampled measurements. It asks how noise intensity and sampling interval constrain stabilizability, in a hybrid setting not previously addressed for this problem.
1 Introduction
- The model combines continuous-time linear stochastic dynamics with control-linear noise and measurements sampled at intervals τ.The measurements satisfy y_{kτ}=x_{kτ}, and the paper addresses stabilization in the mean-square sense.Source: §1 · p. 1
- Earlier stabilization studies related stochastic-system stabilizability to infinite-horizon optimal control and Riccati equations, assuming perfect full-state measurements at every time instant or step.This paper instead considers continuous dynamics with discrete measurements.Source: §1 · pp. 1–2
- Although earlier work studied finite-horizon optimal control in this hybrid setting, feedback stabilization for the paper’s system had not been addressed, to the authors’ knowledge.The setting is motivated in part by spacecraft guidance and cyber-physical systems with scheduled or sporadic sensing and communications.Source: §1 · p. 2
- For fixed (A, B), stabilizability is preserved when σ² or τ is reduced, so greater noise intensity and longer sampling intervals are less favorable.The system is parameterized by (A, B, σ, τ).Source: §1 · p. 2
- The central question is to determine the largest sampling interval or noise intensity that still permits feedback stabilization for fixed (A, B).The paper frames necessary and sufficient conditions as a step toward finding these limits.Source: §1 · p. 2
2 Main Results
The section gives a necessary-and-sufficient fixed-point criterion for mean-square feedback stabilization and links it to solvability of an infinite-horizon optimal-control problem. It also describes a numerical search over noise-variance thresholds and sampling intervals.
2 Main Results
- Mean-square feedback stabilizability requires an L2 feedback gain for which the expected squared state norm tends to zero for every initial condition.The paper defines any gain satisfying this condition as a stabilizing feedback gain.Source: §2 · p. 3
2.1 Necessary and sufficient condition
- A fixed point of f exists if and only if system (1) is mean-square feedback stabilizable; when it exists, it is unique and yields a stabilizing gain.The condition is stated for positive definite L and R.Source: §2.1 · p. 3
2.2 Connections with stochastic optimal control
- An infinite-horizon optimal-control solution exists for every initial state if and only if f has a fixed point; when it does, the minimizer is unique and uses the stabilizing gain.The converse also holds: a solution for every initial state implies a unique fixed point.Source: §2.2 · pp. 4–5
2.3 Numerical study
- At τ increments of 0.2, the study binary-searches maximum σ² with f_N(0) convergence for five A_i cases under B=I and B=1.The two sets of resulting (τ, σ²) curves are shown in Figures 1 and 2.Source: §2.3 · p. 5, Figure 1 · p. 6
3 Analysis and Proofs
The proofs proceed from finite-horizon optimal control to the fixed-point results. They establish sufficiency and necessity for stabilization and optimal-control solvability, then prove uniqueness of the fixed point.
3.1
- In Subsection 3.1, the analysis investigates finite-horizon optimal control problem (7), establishes Theorem 3, and validates Algorithm 1; part of the analysis serves as a cornerstone for proving other theorems.Source: §3 · p. 6
3.2
- The fixed-point existence proof establishes sufficiency for mean-square feedback stabilization and existence and uniqueness of an infinite-horizon optimal-control solution.This proof does not rely on uniqueness of the fixed point.Source: §3 · p. 6
3.3
- A separate proof establishes the necessity directions for the stabilization and infinite-horizon optimal-control results.Source: §3 · p. 6
3.4
- The final proof establishes fixed-point uniqueness using the connections among f, feedback stabilizability, and optimal-control solutions.Source: §3 · p. 6
3.1 Proof of Theorem 3
The proof of Theorem 3 uses dynamic programming to characterize the finite-horizon optimal control problem. Its interval-wise problem has a unique minimizer given by the stated control law.
3.1 Proof of Theorem 3
- Dynamic programming defines the value-to-go at each sampled time, with the terminal boundary condition set at the horizon.The proof suppresses the horizon argument while writing the value functions and associated gains.Source: §3.1 · p. 7
- The proof derives the conditional mean and covariance between samples from linear ordinary differential equations, then uses their endpoint solutions in the interval optimization.The conditional mean and covariance start from x and zero covariance at the sampling time.Source: §3.1 · pp. 7–8
- The interval-wise optimal control problem has a unique minimizer, given by the restriction of u*_N to that sampling interval.The proof considers the larger space of all L2-integrable controls on the interval.Source: §3.1 · p. 8
3.2 Proof of Sufficiency
A fixed point of f yields mean-square feedback stabilization and the unique optimizer of the infinite-horizon control problem. The proof also shows that the finite-horizon cost comparison identifies that optimizer uniquely.
3.2 Proof of Sufficiency
- If f has a fixed point W, system (1) is mean-square feedback stabilizable and the infinite-horizon problem has u* as its unique minimizer.The proof uses the feedback gain K(s) associated with the fixed point.Source: §3.2 · p. 9, §3.2.1 · p. 9
3.2.1 Proof of item 1 of Theorem 2
- The fixed-point argument makes the sampled quadratic quantity nonincreasing, and the proof concludes that the state’s mean-square norm tends to zero.The proof first obtains decay at sampling times, then extends the conclusion to all times using bounded matrix-valued functions.Source: §3.2.1 · p. 9
3.2.2 Proof of item 1 of Theorem 4
- For any infinite-horizon optimizer u, the cost satisfies J(x,u) ≥ x⊤Wx, with equality if and only if u = u*.The interval-wise cost term is minimized by u* and has minimum zero.Source: §3.2.2 · p. 10
3.3 Proof of Necessity
The necessity proof combines monotonicity of finite-horizon Riccati iterates with an upper bound supplied by mean-square stabilizability. Their limit is a fixed point of f.
3.3.1 Proof of item 2 of Theorem 4
- With P = 0, the finite-horizon optimal values increase with the horizon, as shown by comparing the N- and (N+1)-step problems.The monotonicity is stated for the sequence f^N(0).Source: §3.3.1 · pp. 10–11, §3.3 · p. 10
- A solution to the infinite-horizon optimal control problem also bounds the finite-horizon iterates, and their limit is a fixed point of f.The proof obtains the limit using monotone convergence.Source: §3.3.1 · p. 11
3.3.2 Proof of item 2 of Theorem 2
- The sequence f^N(0) is monotonically increasing and, under mean-square feedback stabilizability, is bounded above by a positive matrix.These properties yield a finite matrix limit that satisfies W = f(W).Source: §3.3.2 · pp. 11–12
3.4 Proof of Theorem 1
Once f has a fixed point W, its iterates converge to W from every positive semidefinite starting matrix. The proof identifies W as the limit from zero and establishes convergence for arbitrary P.
3.4 Proof of Theorem 1
- For every P ≥ 0, f^N(P) converges to the fixed point W, which is unique.The proof establishes uniqueness by applying the convergence result to any other fixed point.Source: §3.4 · p. 12
- The proof chooses W as the limit of f^N(0), after using the fixed-point sufficiency and necessity results to establish that this limit is a fixed point.Source: §3.4 · p. 12
- Under the optimal feedback, the proof uses exponential decay of the sampled state and convergence of f^N(0) to control the difference between f^N(P) and W.For sufficiently large N, the quadratic-form difference is bounded by an arbitrary positive tolerance times ∥x∥².Source: §3.4 · pp. 12–13