On mean-square boundedness of stochastic linear systems with quantized observations

Technical Summary: Mean-Square Boundedness of Stochastic Linear Systems with Quantized Observations

Problem Formulation
The paper addresses the control of a discrete-time linear stochastic system evolving in $\mathbb{R}^d$, described by the equation $x_{t+1} = A x_t + B u_t + w_t$. The system is subject to a mean-zero sequence of mutually independent noise vectors $(w_t)$ with bounded fourth moments. The primary challenge is that the controller does not have access to perfect state measurements; instead, it receives quantized state measurements $q(x_t)$ generated by a quantizer $q: \mathbb{R}^d \to Q$, where $Q$ is a finite set of vectors (an alphabet).

The objective is to design a quantizer with a finite alphabet and a corresponding bounded control policy such that the closed-loop system states remain mean-square bounded (i.e., $\sup_{t} \mathbb{E}[\|x_t\|^2] < \infty$). The control inputs must also be uniformly bounded ($\|u_t\| \leq U_{max}$).

The authors focus specifically on systems where the matrix $A$ is Lyapunov stable (eigenvalues have magnitude $\leq 1$, with those on the unit circle having equal geometric and algebraic multiplicities). This distinguishes the work from the extensive literature on stabilizing systems with unstable eigenvalues (outside the unit disc) under communication constraints. The authors note that for stochastic systems with unbounded noise and at least one eigenvalue with magnitude greater than 1, local stabilization with bounded inputs is impossible; thus, the focus is on "maximally coarse" quantization for systems that are already Lyapunov stable.

Methodology
The proposed solution utilizes a $\kappa$-history-dependent policy, where the control sequence is determined by the history of quantized states. The methodology relies on the following structural assumptions and constructions:

1. System Properties: The pair $(A, B)$ is assumed to be reachable in $\kappa$ steps. Without loss of generality, a change of basis allows the authors to treat $A$ as an orthogonal matrix.
2. Quantizer Design: The quantizer is designed to be "radial." For states outside a specific radius $r$ (the boundary of a ball $B_r$), the quantizer maps the state to a point on the boundary $\partial B_r$. Inside the ball $B_r$, the quantization rule is arbitrary regarding mean-square boundedness. The quantizer ensures that the angle $\phi$ between a state $z$ and its quantized version $q(z)$ is bounded.
3. Control Policy: The control policy applies a sequence of $\kappa$ control inputs at each step $t$ based on the quantized state $q(x_{\kappa t})$. The policy is defined as:
   $$\bar{u}_{\kappa t} = -R_{\kappa}(A, B)^+ A^{\kappa} q(x_{\kappa t})$$
   where $R_{\kappa}(A, B)$ is the reachability matrix and $(\cdot)^+$ denotes the Moore-Penrose pseudoinverse. This control law attempts to drive the state toward the origin using the quantized information.
4. Stability Analysis: The proof of boundedness relies on Proposition 3, a result derived from [5], which establishes mean-square boundedness for non-negative random variables if they exhibit a negative drift when large and have bounded fourth-order increments.
   • The authors define a Lyapunov-like variable $\xi_t = \|x_{\kappa t}\|$.
   • They demonstrate that for $\|x_{\kappa t}\| > r$, the expected change in the norm is negative (drift condition), provided the radius $r$ and the maximum quantization angle $\phi$ satisfy specific inequalities involving the system's singular values and noise bounds.
   • They verify that the fourth moment of the state increments is bounded, satisfying the conditions of Proposition 3.

Key Results
The paper presents Theorem 2, which provides sufficient conditions for the mean-square boundedness of the closed-loop system:

• If the quantizer satisfies specific geometric constraints (finite alphabet, radial projection on the boundary of a ball of radius $r$, and a bounded angle $\phi$ between the state and its quantized value), and
• If the control bound $U_{max}$ is sufficiently large relative to the system parameters ($U_{max} \geq r / \sigma_{\min}(R_{\kappa}(A, B))$),
• Then the successive application of the proposed $\kappa$-step control policy ensures that $\sup_{t} \mathbb{E}[\|x_t\|^2] < \infty$.

The theorem implies that a quantizer with a finite number of bins can stabilize the second moment of the state, even in the presence of unbounded noise, provided the system is Lyapunov stable.

Significance and Claims
The authors claim that this work addresses a gap in the literature regarding stabilization with finite alphabets for Lyapunov stable systems. While previous works (cited as [2, 4, 1, 3, 7, 8]) have focused on communication constraints and systems with unstable eigenvalues, this paper targets "maximally coarse" quantization for systems that are already stable.

Key claims regarding the contribution include:

• Novelty: The authors state they are not aware of prior work dealing with stabilization with finite alphabets in the context of Lyapunov stable systems.
• Feasibility: The procedure demonstrates that finite-alphabet quantizers are sufficient to ensure mean-square boundedness without requiring the communication constraints typically associated with unstable systems.
• Recovery of Previous Results: As the density of the bins on the $r$-sphere increases (i.e., as the angle $\phi \to 0$), the proposed policy recovers the results and lower bounds on $U_{max}$ established in the authors' previous work [6].

Simulation
The paper includes a simulation of a 2D system with a rotation matrix $A$ and additive Gaussian noise. The results compare the average square norm of the state over 1000 runs using the proposed policy with different bin counts (2, 4, and 8 bins) against the policy from [6]. The simulation illustrates the performance of the quantized control strategy in maintaining bounded state variance.