Closed-Loop TXP Control Framework

• Closed-loop TXP is a feedback-driven method that adaptively adjusts transmission power and task execution using sensed error signals.
• It integrates control strategies like PID regulation and Lyapunov-based stability analysis to achieve robust, energy-efficient performance.
• Hybrid configurations employing cascaded loops ensure rapid recovery and throughput stability, reducing energy consumption by up to 60%.

A closed-loop TXP (Transmission Power/Task Execution Planning) control framework refers to a general class of feedback-driven methods for adaptively regulating transmission power, control input, planning actions, or sensor communication rates in cyber-physical and networked systems, based exclusively on dynamically sensed error signals and performance metrics. This paradigm contrasts with open-loop or periodic mechanisms by directly incorporating real-time system feedback, often leading to more robust stability, efficient resource utilization, and empirically validated performance gains in highly dynamic environments.

1. System Model and Theoretical Foundations

Closed-loop TXP control frameworks instantiate feedback regulation in diverse settings: wireless transmission power adjustment for communication links, event-triggered sensor transmission for networked control, and sequential execution planning in robotic manipulation. Across these domains, the system architecture invariably comprises a plant (physical device or agent), sensing channel, and feedback controller interconnected via quantifiable signals.

For the wireless transmission power (TXP) case, as in the BLE IoT scenario, communication is characterized by the received signal strength indicator (RSSI), throughput $T$, and peripheral power $P_{\rm sys}$, each modeled by empirical and physical equations:

• Path-loss: $$PL(d) = PL(d_0) + 10 n \log_{10}(d / d_0) + \chi_\sigma$$
• RSSI: $RSSI(d) \approx TXP - PL(d)$
• Throughput: $T(d)$ is piecewise linear in RSSI, bounded by $T_{\max}$ and $T_{\min}$
• Power: $P_{\rm sys} \approx P_{\rm idle} + \beta T$

In networked control, the transmission-lazy (TXP) scheme uses system state or output feedback, where the interconnection is described by continuous-time plant and controller dynamics: $\dot{x}_p = A_p x_p + B_p u ,\quad y = C_p x_p$

$\dot{x}_c = A_c x_c + B_c y_k ,\quad u = C_c x_c$

where $y_k$ is the most recent transmitted measurement and transmission occurs when the error $e(t) = y_k - y(t)$ exceeds a prescribed threshold (Forni et al., 2013).

2. Controller Design Strategies

The central strategy in closed-loop TXP control is adaptive feedback regulation via discrete-time controllers—most commonly PID (Proportional-Integral-Derivative)—operating on sensed metrics or error signals. In wireless power control (Zhou et al., 6 Jan 2026), two core strategies are established:

• RSSI-based control: PID regulation on the RSSI error $e_\mathrm{RSSI} = RSSI_{\rm tgt} - RSSI$, updating TXP at high frequency (100 Hz) for rapid responsiveness.
• Throughput-based control: PID acting on throughput error $e_T = T_{\rm tgt} - T$ with lower frequency (1 Hz), yielding more direct throughput guarantees but slower adjustment.

Event-triggered transmission control (Forni et al., 2013) employs state-based or observer-based policies, where new samples are sent only when the error $e(t)$ surpasses a state-relative threshold: $\|e(t)\| \geq \sigma \|x(t)\|\,,$ with rigorous Lyapunov-based stability analyses ensuring asymptotic (or exponential) convergence.

In closed-loop task execution planning, as implemented in CLOVER-style or TXP frameworks for robotics (Bu et al., 2024), controllers regulate the action $u_t$ based on embedding-space error between current and goal states, and initiate replanning upon significant divergence across sequential plan states.

3. Hybrid and Cascaded Control Architectures

Hybrid closed-loop TXP architectures integrate multiple feedback signals to leverage both fast responsiveness and high steady-state accuracy. In the BLE control framework, this entails a cascaded dual-loop system:

• The outer throughput-loop (1 Hz) PID adapts the RSSI target for the inner loop.
• The inner RSSI-loop (100 Hz) PID directly modulates TXP for fine-grained link stability.

Update equations are of the form: $$\Delta TXP[k] = K_p e_R[k] + K_i \sum e_R + K_d \Delta e_R[k]$$

$$RSSI_{\rm tgt}[k] = RSSI_{\rm tgt}[k-1] + K_p^T e_T[k] + K_d^T \Delta e_T[k]$$

The hybrid strategy achieves minimal throughput variance and rapid recovery from link fades ($<150$ ms), outperforming single-signal approaches (Zhou et al., 6 Jan 2026).

4. Stability, Performance, and Robustness Guarantees

Closed-loop TXP frameworks are generally constructed to guarantee robust stability and constraint satisfaction under feedback regulation. In transmission-lazy sensor systems (Forni et al., 2013), Lyapunov functions $V(x,e)$ and dwell-time timers rigorously exclude Zeno behavior and certify global asymptotic or exponential convergence. Similarly, tube-based zonotopic predictive control (Farjadnia et al., 2024) over-approximates admissible models with matrix-zonotopes and constructs positive-invariant error tubes around nominal trajectories, ensuring recursive feasibility and robust exponential stability under bounded noise.

In wireless transmission power control, stability and power efficiency are quantified experimentally:

• The hybrid PID framework achieves up to 60% energy reduction relative to fixed TXP and maintains throughput within 5% target error across 0–50 m link distances.
• Responsiveness is maximized via the inner RSSI loop; throughput stability via the outer loop (Zhou et al., 6 Jan 2026).

5. Implementation Details and Experimental Results

Closed-loop TXP frameworks have been validated on commodity hardware (Nordic nRF54L15 DK, nRF21540 FEM), with detailed measurement of all inputs and outcomes:

• Sampling rates: RSSI at 100 Hz (direct chip register), throughput at 1 Hz.
• Hardware: Power profiling via Nordic PPK II; BLE settings include ATT MTU=498 B, Conn. Interval=400 ms, PHY=2 Mbps.
• Control parameters: Typical RSSI target $-60$ to $-65$ dBm; throughput targets 100–800 kbps, with TXP adjustments bounded to $|\Delta TXP| \leq 2$ dB per cycle.
• Performance metrics: Steady-state mean/STD for RSSI and throughput, power consumption (mW), and recovery times from link drops.

Quantitative table summarizing performance (Zhou et al., 6 Jan 2026):

Method	Mean RSSI (dBm)	T STD (kbps)	Power (mW)	Recovery time
Fixed TXP=+20 dBm	–45.3	114.5	87.1	n/a
Fixed TXP=–10 dBm	–71.4	129.7	24.8	n/a
RSSI-based	–58.5	–	34.8	150 ms
Throughput-based	–51.5	101.1	34.4	3 s
Hybrid	–50.6	82.1	46.5	150 ms

6. Practical Design Guidelines and Application Scope

Closed-loop TXP frameworks are recommended in scenarios requiring adaptive, robust, and energy-efficient control, particularly in dynamic wireless (BLE, mesh) or networked cyber-physical domains. Controller parameter selection is informed by empirically driven tradeoffs:

• RSSI-loop $K_p$ in [0.1, 0.3], $K_i$ in [0.005, 0.02]
• Throughput-loop $K_p$ in [0.005, 0.01], $K_d$ in $[10^{-4}, 10^{-3}]$
• Update rates: RSSI 100 Hz, throughput 1 Hz
• Set RSSI targets 5-10 dB above sensitivity; throughput per application SLAs

In networked control and TXP-enabled robotic planning, parameters $\sigma$ (error-to-state threshold) and $\gamma_x$ (Lyapunov decay rate) directly trade off communication frequency and closed-loop convergence speed. Observer-based variants utilize Luenberger observers and composite Lyapunov functions to guarantee stability with only output feedback (Forni et al., 2013). Hybrid and tube-based MPC architectures, as in (Farjadnia et al., 2024), further generalize the approach to settings with model uncertainty and bounded noise.

7. Scope and Adaptability

Closed-loop TXP frameworks unify feedback regulation concepts across wireless power control, cyber-physical system stabilization, networked control, and robotic task execution. They support cascaded and multi-signal controller architectures, robust Lyapunov-based or predictive optimization, adaptive error quantification, and explicit empirical tuning. These principles have been shown scalable to mesh and multi-node topologies, sensor networks, and modular robot planning regimes, with direct implications for future mission-critical IoT and autonomous system deployments.