Input-to-state stabilization of linear systems under data-rate constraints

A communication and control strategy is proposed for feedback stabilization of linear systems under data-rate constraints in the presence of completely unknown disturbances, establishing input-to-state stability (ISS) with respect to the disturbance.

System Model and Information Structure

The paper considers a continuous-time linear system described by the differential equation:

x˙ = Ax + Bu + Dd, x(0) = x0, where x is the state, u is the control input, and d is an unknown disturbance assumed to be Lebesgue measurable and locally essentially bounded. The first basic assumption requires that the pair (A, B) is stabilizable. The information structure involves a sensor sampling the state at times tk = kτs with a fixed sampling period τs > 0, where each sample x(tk) is encoded as an integer ik ∈ [0, Nnx + 1], resulting in a data transmission rate of log2(Nnx + 2)/τs bits per unit time. A second basic assumption imposes the data-rate condition: Λ:=∥eAτs∥ < N, which links the sampling period τs and integer N to impose a lower bound on the admissible data rate.

Communication and Control Strategy

The proposed strategy alternates between stabilizing and searching stages based on whether the state is visible or lost within a quantization range Rk, which approximates the reachable set of the state. If in a stabilizing stage, if x(tk) ≤ Ek/N, the sensor transmits ik = 1; otherwise, it transmits an index ik ≥ 2 corresponding to a cell center ck. The controller then applies u(t) = Kxˆ(t), where xˆ evolves according to ˙x̂ = Ax̂ + Bu with the boundary condition x̂(tk) = ck. If in a searching stage, the sensor transmits ik = 0, and the control input is set to zero on [tk, tk+1). The quantization range Rk is dynamically adjusted using formulas derived from reachable-set approximations.

Stability Analysis and ISS Guarantee

The main result establishes an input-to-state stability (ISS) property with respect to the disturbance, characterized by functions γ1, γ2, and γ3 belonging to the class K∞. The closed-loop solution satisfies:

x(t) ≤ γ1(x0) + γ2(∥d∥[0,∞)) ∀ t ≥ 0 (3)

and

lim sup t→∞ x(t) ≤ γ3 lim sup t→∞ d(t) (4).

The proof relies on alternating between searching and stabilizing stages. During stabilizing stages, the function Vk:= V (x∗k, Ek) is shown to be a Lyapunov-type function with exponential decay, satisfying Vk+1 ≤ νVk where ν ∈ (0, 1). The analysis uses several technical lemmas to bound the state evolution:

(33)

x(tk) ≤ C3ν(k−l−1)/2p Vl + Φ∥d∥[tk−1,tk]

During searching stages, the strategy guarantees finite-time (re)capture of the state. Lemma 6 ensures that if the state is lost at t0 = 0, it is captured at some sampling time ti0 satisfying an inequality involving ηx and ηd functions. Lemma 8 further guarantees that if the state escapes at tj > 0, it is recaptured at some sampling time ti > tj.

Characterization of Gain Functions

The gain functions γ1, γ2, and γ3 are constructed by combining bounds derived from the alternating stages.

(42)

γ1(s):= max[γˆx0(s), χ¯1(E0, s)]

γ2(s):= max[γˆd0(s), χ¯2(E0, s), γˆ(s), γ¯(s)]

where κ∞ functions are used to ensure the required properties.

The final bound (4) is established by showing that if only finitely many searching stages occur, the system eventually remains in a stabilizing stage, leading to:

lim sup k→∞ x(tk) ≤ Φ lim sup t→∞ d(t) with γ3(s):= max[Φs, γˆ(s), γ¯(s)].

Simulation Example

A simulation example illustrates the effectiveness of the approach on a linear system with specific parameters (A, B, D, K, τs=0.1 s, N=5). The design parameters were set as E0 = 0.5, ε = 0.2, δ = 0.1, ψ = 0.5, ρ = 150, ϕ = 0.

As a diligent researcher, I have analyzed the provided scientific paper, Input-to-state stabilization of linear systems under data-rate constraints, focusing on its core contribution: establishing Input-to-State Stability (ISS) for linear systems subject to communication and quantization constraints in the presence of unknown disturbances.

The improvements derived from this work are primarily in developing robust, low-bandwidth control architectures for real-world networked AI/control systems.

Here are the specific improvements and capabilities an improved AI system could possess based on this research:

)1. Robust Low-Bandwidth Control Architecture (The Core Improvement):

The system can implement a feedback loop that maintains stability even when state information is severely limited (sampled and quantized). This means the AI control logic does not require continuous, high-bandwidth data streams.

)2. Adaptive Quantization Strategy for Unknown Dynamics:

Unlike previous schemes that relied on fixed quantization ranges or external disturbance bounds, this system can dynamically adjust its sensing resolution (zooming/moving-center quantization) based on real-time estimates of the disturbance and the current state's proximity to the equilibrium.

• Capability: The AI can intelligently decide when to zoom in (increase resolution near a suspected instability or disturbance) and when to zoom out (reduce resolution when operating stably) to maximize data rate efficiency while guaranteeing stability.

)3. Guaranteed Disturbance Rejection via State Estimation:

The system utilizes a novel disturbance-estimation framework derived directly from the quantization parameters, rather than assuming a known bound on the disturbance beforehand.

• Capability: The AI can actively estimate unknown noise or external perturbations (like sensor drift or adversarial inputs) by analyzing the error dynamics and update formulas. This allows it to proactively adjust its control effort to counteract these disturbances before they cause system instability.

)4. Hybrid Control Mode for Uncertainty Handling (Search vs. Stabilize):

The control strategy explicitly alternates between two modes:

• Capability: When the state measurement is lost or corrupted (searching stage), the AI can immediately switch to a safe, zero-control mode to prevent catastrophic failure while simultaneously using a recovery mechanism (based on the estimate of the disturbance) to ensure finite-time capture once information is reacquired. This makes the system resilient to temporary communication dropouts.

)5. Explicit Performance Guarantees (ISS):

The system provides mathematically rigorous, explicit bounds on how much an external disturbance can affect the state's trajectory (Input-to-State Stability).

• Capability: For critical applications (e.g., autonomous vehicle control, industrial robotics), this means the AI can guarantee that even under worst-case unknown environmental noise or communication latency, the system will not diverge beyond a mathematically defined threshold relative to the disturbance magnitude.

)6. Implementation in Networked/Resource-Constrained Environments:

Since the scheme is based on sampled and quantized measurements, it is inherently suitable for edge computing or IoT devices with limited communication bandwidth (low data rate constraints).

• Capability: The AI can operate reliably in environments where transmitting raw state vectors is too costly or slow, relying only on sending compact, quantized indices and symbols.