Sensing-Communication-Computing-Control Closed-Loop Optimization for 6G Unmanned Robotic Systems

Cyber physical system (CPS) refers to the system that integrates communication, computing, and control to get information from the physical world, perform analysis, and change the physical world [1, 2]. In recent decades, a new kind of CPSs–unmanned robotic system–has received great attention for their human-like capabilities [3]. Driven by technologies such as smart computing and cloud-fog architectures, these systems have made significant strides in industrial automation, significantly improving production efficiency [4, 5]. Moreover, recent advancements in field robots have extended unmanned robotic systems from indoor factories to outdoor environments. These systems are now being deployed in hard-to-access areas, replacing humans in dangerous tasks such as disaster rescue [6], oil exploitation, and space exploration [7]. To fully unlock the potential of unmanned robotic systems, supporting field robots has been identified as an important use case for the sixth-generation (6G) network [nextG].

Due to the lack of ground facilities in remote areas, aerial-borne and space-borne platforms such as unmanned aerial vehicles (UAVs) and satellites, are envisioned to provide global seamless coverage in 6G [8, 9]. These platforms are agile to carry communication and computing modules, acting as edge information hubs (EIHs) to both transfer and process information [10]. A digital twin can be further integrated into EIHs to enable intelligent decision-making [11]. During the task execution, sensors collect raw data and upload them to the EIH via the sensor-EIH link. Based on the sensing data, the EIH calculates control commands. These commands are then transmitted to the actuators via the EIH-actuator link for actions. Synergistically, the sensor, the sensor-EIH link, the EIH, the EIH-actuator link, and the actuator form an integrated sensing-communication-computing-control ($\mathbf{SC}^{3}$) loop. Through effective feedback, the $\mathbf{SC}^{3}$ loop continuously learns the behavior of the physical system and guides its evolution to the desired direction. From the perspective of biology, the $\mathbf{SC}^{3}$ loop has great similarity to the reflex arc, as we compared in Fig. 1. In biology, it is widely recognized that the presence of the reflex depends on the functional integrity of the reflex arc [12]. In this sense, it is reasonable to take the $\mathbf{SC}^{3}$ loop as an integrated structure when we investigate the unmanned robotic system.

However, due to the heterogeneity of communication, computing, and control, current applications design $\mathbf{SC}^{3}$-loop components separately. For this reason, the fifth-generation (5G) network positions itself as a communication network and primarily focuses on data transmission. It applies multiple access and duplex techniques, e.g., orthogonal frequency-division multiple access (OFDMA) and time-division duplexing/frequency-division duplexing (TDD/FDD), to decompose the network into independent links. Although this link-level division brings high capacity for communication, the task-level connections of different components within the $\mathbf{SC}^{3}$ loop are disintegrated. The mismatch of the sensor-EIH link, computing, and EIH-actuator link impairs the overall functioning of the $\mathbf{SC}^{3}$ loop, ultimately degrading the task efficiency of the robotic system.

Unlike Shannon who separated communication from its served system and regarded communication engineering as “reproducing at one point either exactly or approximately a message selected at another point” [13], the creator of cybernetics, Wiener, noted that “The problems of control engineering and communication engineering were inseparable, and that they centered not around the technique of electrical engineering but around the much more fundamental notion of the message.” [14]. By understanding communication through Wiener’s lens, we can find that communication is not merely for data transmission but an integral part for task execution. In this context, we need to shift our focus from individual communication links to integrated $\mathbf{SC}^{3}$ loops. As a result, the communication efficiency transcends bit transmission: it involves cooperating with computing and control to achieve good task performance. Building on these insights, we use a structured lens to regard the sensor and actuator within the $\mathbf{SC}^{3}$ loop as a virtual user [15]. The sensor-EIH link and EIH-actuator link are thus the uplink and downlink (UL&DL) of this virtual user. On this basis, we consider the goal-oriented closed-loop design for the unmanned robotic system.

In the literature, related studies mostly focus on part of the $\mathbf{SC}^{3}$ loop. In wireless control systems (WCSs), the interplay between communication and control has been extensively studied. In mobile edge computing (MEC), joint optimization of communication and computing has been a central topic. In goal-oriented communication, related studies integrated the task efficiency into the communication design, and the learning-based approaches have become a dominant trend. In the following, we review related studies on WCS, MEC, and goal-oriented communication.

1) WCS: In WCS, related studies investigated the impact of imperfect communication, such as low data rate and latency, on control [16, 17]. From the theoretical perspective, Tatikonda et al. analyzed linear discrete-time systems and derived the lower and upper bounds on the data rate required for different control objectives [18]. Kostina et al. further generalized the work in [18] and derived the lower bound of the data rate given the control objective measured by the linear quadratic regulator (LQR) cost [19]. In terms of optimization schemes, Baumann et al. and Gatsis et al. devised control-aware transmission schemes that the sensor uploads sensing data only when the estimation error surpasses a certain threshold [20, 21]. By taking the LQR cost as the objective, Wang et al. optimized the UL [22], while Lei et al. and Fang et al. optimized the DL [23, 7]. For ultra-reliable and ultra-low latency communication (URLLC)-supported WCSs, Chang et al. jointly optimized communication bandwidth, power, and control convergence rate to maximize the spectrum efficiency (SE) [24], and Yang et al. optimized the UL&DL transmit power and block length to minimize a new metric named energy-to-control efficiency [25]. In addition to above theoretical advancements, researchers also established simulation platforms to testify the control performance under different communication conditions. Bhimavarapu et al. proposed a Unobtrusive Latency Tester solution to measure the communication latency and reliability in real-time control [26]. Lyu et al. proposed a novel hardware-in-the-loop simulation platform, which provided reliable test results by accurately stimulating the real wireless environment [27, 28]. Utilizing these platforms, the performance of 5G and Wi-Fi 6 was assessed, which provides valuable insights for selecting communication protocols for specific applications [26, 27, 28].

2) MEC: The core idea of MEC is to place computing resources close to mobile devices to support real-time applications. Most studies used joint communication and computing optimization to minimize the task latency [29, 30, 31], energy consumption [32][33], or maximize the utility [34, 35, 36]. For example, Wen et al. considered a TDD-mode MEC system in which each user is allocated a single slot to complete the data uploading, computing, and result downloading. The authors optimized bit, subchannel, and time allocation to minimize energy consumption [33]. Jian et al. focused on computing and DL transmission, proposing a joint MEC server-user association, CPU frequency, and bandwidth optimization scheme [34]. Wang et al. considered a joint sensing, communication, and computing framework, where the base station detects surrounding objects and uploads part of computing tasks to the cloud center using integrated sensing and communication (ISAC) signals. The authors optimized the beamformer and CPU frequency to maximize the computing rate [36].

3) Goal-Oriented Communication: The core idea of goal-oriented communication is to adapt communication strategies to the specific needs of supported tasks. For example, Wen et al. considered a real-time inference task performed by ISAC devices and an edge server. The authors took the discriminant gain as the objective and jointly optimized the transmit power, time allocation, and quantization bit allocation [37]. Girgis et al. proposed a semantic communication-control co-design that trains an encoder at the edge to abstract the low dimensional feature and a decoder at the control center to recover the state and calculate the command [38]. Shao et al. utilized the information bottleneck method to maximize the mutual information between the inference result and the encoded feature, while minimizing the mutual information between the raw input data and the encoded feature [39]. Mostaani et al. devised a distributed deep learning scheme for a multi-agent system, and maximized the long-term return by jointly optimizing communication and control policies [40].

In summary, these studies offer valuable insights into the design of the $\mathbf{SC}^{3}$ loop. Most of these studies have focused on part of the $\mathbf{SC}^{3}$ loop, such as UL and computing or DL and control, using indirect task metrics like latency or computing performance as their optimization objectives. However, for unmanned robotic systems, the reflex-arc-like $\mathbf{SC}^{3}$ loop is an integrated structure. The loop performance is reflected by the control actions on the physical world rather than the intermediate metrics. There is a lack of work that jointly optimizes the UL, computing, and DL within and across the $\mathbf{SC}^{3}$ loops from a goal-oriented perceptive.

In this paper, we consider an unmanned robotic system formed by a multi-functional UAV, sensors, and actuators. The UAV carries communication and computing modules, with a digital twin simulating the physical process in real time. During the task execution, the UAV acts as an EIH that collects data, calculates commands, and distributes commands to actuators to take actions. Together, the EIH, sensors, and actuators form reflex-arc-like $\mathbf{SC}^{3}$ loops. Different from current studies that devise $\mathbf{SC}^{3}$ loop components separately, our study takes the $\mathbf{SC}^{3}$ loop as an integrated structure and jointly configures UL, computing, and DL within and across the $\mathbf{SC}^{3}$ loops from a goal-oriented perspective. The main contributions are listed as follows.

1. 1.

   We investigate an unnamed robotic system formed by multiple reflex-arc-like $\mathbf{SC}^{3}$ loops. To effectively use limited resources, we propose a goal-oriented closed-loop optimization scheme that jointly optimizes bandwidth, time, and CPU frequency within and across $\mathbf{SC}^{3}$ loops, with the objective of minimizing the total LQR cost.

2. 2.

   In the intra-loop configuration, we derive optimal closed-form solutions for UL&DL bandwidth and time, along with the optimal LQR cost. Based on these results, we demonstrate the task-level balance between the UL&DL, as well as the interchange relationships between communication bandwidth and computing CPU frequency.

3. 3.

   In the inter-loop configuration, we propose an iterative algorithm to optimize the bandwidth and CPU-frequency allocation. Under the condition of adequate CPU frequency availability, we derive an approximate closed-form solution for the inter-loop bandwidth allocation and analyze the allocation principles regarding communication and control parameters.

4. 4.

   We conduct comprehensive simulation to validate our findings. We show the superiority of the goal-oriented closed-loop optimization by comparing it with separate schemes and communication-oriented schemes. We show that the proposed scheme achieves a two-tier task-level balance within and across the $\mathbf{SC}^{3}$ loops, which is crucial for the overall performance of the unmanned robotic system.

The rest of this paper is organized as follows. Section II introduces the model of the unmanned robotic system and the related $\mathbf{SC^{3}}$ loop. Section III presents the goal-oriented closed-loop optimization scheme and its solution. Section IV presents simulation results and discussion. Section V draws conclusions.

Throughout this paper, vectors, matrices, and sets are represented by bold lowercase letters, bold uppercase letters, and curly uppercase letters, respectively. $\mathbb{R}^{n\times n}$ represents the set of $n\times n$ real matrices, $\mathbf{I}_{n}$ is the $n\times n$ unit matrix, and $\mathbf{0}_{n}$ is the $n\times n$ zero matrix. $\lambda(\mathbf{A})$ denotes the eigenvalue of matrix $\mathbf{A}$ and $\det\mathbf{A}$ denotes the determinant of matrix $\mathbf{A}$. The complex Gaussian distribution of zero mean and $\sigma^{2}$ variance is denoted as $\mathcal{CN}(0,\sigma^{2})$. The optimal solution of $x$ is denoted as $(x)^{*}$.

In this paper, we aim to minimize the total LQR cost of the unmanned robotic system by jointly optimizing bandwidth, CPU frequency, and time within and across the $\mathbf{SC}^{3}$ loops. The optimization problem is formulated as follows:

where $B_{\max}$ and $f_{\max}$ denote the maximal bandwidth and CPU frequency, respectively. Due to the coupling relationships among different variables, (P1) is a highly complex problem with non-convex constraints (14e), (14f) and (14g). To address this challenge, we begin by investigating the intra-loop configuration within an $\mathbf{SC}^{3}$ loop and move to investigate the inter-loop optimization across multi-$\mathbf{SC}^{3}$ loops.

For simplicity, the index $k$ is omitted when we consider an $\mathbf{SC}^{3}$ loop. When $K=1$, the optimization problem becomes

As the lower bound of the LQR cost (15b) monotonically decreases with the closed-loop information, minimizing the LQR cost is equivalent to maximizing the closed-loop information for an $\mathbf{SC}^{3}$ loop. In addition, as there are no other $\mathbf{SC}^{3}$ loops competing for resources, the optimal CPU frequency is its maximum allowable value, i.e., $(f)^{*}=f_{\max}$. As a result, the critical focus in the single-$\mathbf{SC}^{3}$-loop optimization is the UL&DL configuration,

where the stable condition (15d) is omitted. We test it after solving (PA-2). If the optimization result, $(D^{\mathbf{SC}^{3}})^{*}$ satisfies (15d), we calculate the LQR cost according to (10). Otherwise, the controlled system cannot be stabilized, and the LQR cost is infinite.

From (17), we can see that the task-level balance between UL&DL is to match the task-related information extracted from the raw data transmitted via the UL with the information transmitted via the DL. This balance can be illustrated by the analogy to water pipes. The UL and computing are considered as one pipe, while the DL is another. Just as the maximum water flow through the interconnected pipes is limited by the pipe with the smallest capacity, the optimal solution requires equalizing the capacities of both pipes, as indicated by the equilibrium in (17).

Based on the (17) and (18), we can express the closed-form information as an expression of the bandwidth,

From (19), we observe that $D^{\mathbf{SC}^{3}}$ is monotonically decreasing with $[\frac{1}{\rho B^{u}r^{u}}+\frac{\alpha}{\rho f_{\max}}+\frac{1}{B^{d}r^{d}}]$. This expression represents the time to transfer or process one-bit task-related information, where $[\frac{1}{\rho B^{u}r^{u}}]$, $[\frac{\alpha}{\rho f_{\max}}]$, and $[\frac{1}{B^{d}r^{d}}]$ represent the time for UL, computing, and DL, respectively. Therefore, we move to consider the time-minimization problem as follows:

From (24a), we can see that the closed-loop SE is jointly determined by the UL&DL SEs. It has the following approximation,

In fact, $\rho r^{u}$ and $r^{d}$ represent the task-level SE of UL&DL, respectively. From (25), we can learn that closed-loop SE is determined by the weak link. A task-level balance of UL&DL SEs is required to ensure a large closed-loop SE.

In addition, from (22), we observe an interesting trade-off between bandwidth and CPU frequency for a given closed-loop performance. This trade-off is achieved by adjusting communication and computing time. For instance, increasing the bandwidth reduces communication time, $\frac{1}{(B_{\max}+\Delta B)r^{\text{comm}}}$, which in turn allows for an increase in computing time, $\frac{1}{(f_{\max}-\Delta f)r^{\text{comp}}}$, leading to a reduction of the CPU frequency. Consequently, the relationship between $\Delta B$ and $\Delta f$ can be derived to illustrate how bandwidth and CPU frequency can be exchanged for a given closed-loop performance:

From the above expression, we observe that $\Delta B$ is influenced by the working point of the $\mathbf{SC}^{3}$ loop, characterized by the bandwidth, CPU frequency, and SE-to-CE ratio, $(B_{\max},f_{\max},\frac{r^{\text{comm}}}{r^{\text{comp}}})$. With a fixed $\Delta f$, a large bandwidth is required when $B_{\max}$ is high, $f_{\max}$ is low, and $\frac{r^{\text{comm}}}{r^{\text{comp}}}$ is high, which corresponds to the communication-saturated region.

Based on the optimal intra-loop configuration, we move to the inter-loop allocation problem. Using the optimal LQR cost expression (23), (P1) is simplified into the following inter-loop bandwidth and CPU frequency allocation problem:

where $B_{k}$ denotes the bandwidth allocated to the $k$-th $\mathbf{SC}^{3}$ loop and we denote $\mathcal{B}=\{B_{k}\}_{k=1}^{K}$. By calculating the second-order derivative of the right-hand side of (27b), it is easy to prove that it is a convex expression of $D_{k}^{\mathbf{SC}^{3}}$. Thus, the only non-convex constraint in (PB-1) is (27c). To address this, we first transform it into

Since $[\frac{1}{x}]$ is a convex expression of $x$ when $x>0$, we can see that only the right-hand side of (28) does not satisfy the rule of convex optimization. By using the successive convex approximation, we propose an iterative algorithm. The convex optimization problem in the $s$-th iteration is given by

where the right side of (29c) is the Taylor expansion of $\big{[}\frac{1}{D^{\mathbf{SC}^{3}}_{k}}\big{]}$ at the result obtained in the $(s-1)$-th iteration, $(D^{\mathbf{SC}^{3}}_{k})^{s-1}$. We summarize the proposed iterative algorithm in Algorithm 1. The convergence of the algorithm is proved in Appendix C.

Next, we derive the approximate closed-form solution for the inter-loop bandwidth allocation. We assume that the CPU frequency is sufficiently adequate such that the computing time becomes negligible compared to the communication time, $\frac{1}{B_{k}r_{k}^{\text{comm}}}+\frac{1}{f_{k}r_{k}^{\text{comp}}}\approx\frac{1}{B_{k}r_{k}^{\text{comm}}}$. On this basis, the LQR cost can be approximated as follows:

where the term [$-1$] in the denominator is omitted from its original expression (23), under the assumption that the system operates in the assured-to-be-stable region, i.e., $T_{k}B_{k}r_{k}^{\text{comm}}\gg\log_{2}|\det\mathbf{A}_{k}|$. Using (30), (PB-1) is simplified into a convex bandwidth allocation problem,

where $\text{tr}\left(\mathbf{\Sigma}_{\mathbf{v}_{k}}\mathbf{S}_{k}\right)$ is omitted as it does not influence the allocation results. The stable condition is also omitted as in the assure-to-be-stable region.

We can learn from (32) that bandwidth allocation is influenced by both the communication parameter, $r_{k}^{\text{comm}}$, and the control parameter, $e_{k}$. The $\mathbf{SC}^{3}$ loop with poor communication performance (smaller $r^{\text{comm}}$) or controlling a more unstable system (larger $e_{k}$) is allocated more bandwidth. This shows that this inter-loop bandwidth allocation is fairness-minded, which balances the control processes across different $\mathbf{SC}^{3}$ loops.

In Fig. 3, we present the relationships between UL&DL SEs and the closed-loop SE, as described in (25). In the left figure, we set the information extraction ratio as $\rho=0.01$, which creates the unbalance of UL&DL SEs at the task level. In this case, the closed-loop SE is constrained by the UL. As indicated by the red and black arrows, the most effective way to enhance the closed-loop SE is by improving the UL SE, while improving the DL SE provides little improvement. In contrast, the right figure illustrates the case where the information extraction ratio is set as $\rho=1$, resulting in relatively balanced task-level UL&DL SEs. In this case, the most effective way to enhance the closed-loop SE is to simultaneously improve UL&DL SEs. It is noted that, in practice, the raw data transmitted via the UL often contain redundant and irrelevant information, leading the information extraction ratio a very small number, $\rho\ll 1$. An enhanced UL is usually required for keeping the task-level balance between UL&DL.

In Fig. 4, we present the LQR cost under different UL&DL configurations. Here, we compare the proposed scheme with two classical schemes: equal allocation and proportional allocation.

• •

  Equal allocation: The bandwidth is equally divided for UL&DL like FDD, $B^{d}=B^{u}=\frac{B_{\max}}{2}$.

• •

  Proportional allocation: The UL&DL bandwidth is proportional allocated per the information extraction ratio, $B^{u}=\frac{1}{1+\rho}B_{\max}$ and $B^{d}=\frac{\rho}{1+\rho}B_{\max}$.

In above two schemes, the DL transmission time is set as $t^{d}=\frac{T}{3}$, and the remaining time is adjusted between UL and computing by letting the UL transmitted information rightly processed by computing, i.e., $t^{u}+\frac{\alpha D^{u}}{f_{\max}}=\frac{2T}{3}$. The maximal bandwidth is set as $B_{\max}=500$ kHz, and the CPU frequency is set as $f_{\max}=0.5$ GHz. Fig. 4 is a dual-axis chart with the left side representing the task-related information and the right side representing the LQR cost. From the bar, we can see that, under the proposed scheme, the UL&DL are aligned to transmit the same amount of task-related information, while the other two schemes fail to achieve this balance. Their weak ULs constrain the control performance of the $\mathbf{SC}^{3}$ loop, leading to the high LQR cost (shown by the black curve). In addition, we can observe that the LQR cost is higher with more degree of UL&DL imbalance, indicating the importance of keeping task-level UL&DL balance within the $\mathbf{SC}^{3}$ loop.

In Fig. 5, we illustrate the interchange relationships between bandwidth and CPU frequency. This figure shows the required bandwidth in exchange for 1 MHz CPU frequency under different working points of the $\mathbf{SC}^{3}$ loop, described by the bandwidth, CPU frequency, and SE-to-CE ratio, i.e., ($B$, $f$,$\frac{r^{\text{comm}}}{r^{\text{comp}}}$). The color gradient represents the amount of required bandwidth, increasing as the color shifts from blue to yellow. The figure shows that in the communication-saturated region, more bandwidth is required compared to the communication-limited region. For example, at the working point of $(1\text{MHz},2\text{GHz},5\times 10^{4})$, the required bandwidth is $\Delta B=12.6\ \text{kHz}$, whereas at $(2\text{MHz},1\text{GHz},10^{5})$, the required bandwidth is $\Delta B=500\ \text{kHz}$, with an increase of forty-fold. This indicates the marginal effect, that the effectiveness of additional bandwidth diminishes as the $\mathbf{SC}^{3}$ loop approaches the communication-saturated region.

In Fig. 6, we provide the convergence performance of the proposed iterative algorithm. In this simulation, we randomly generate UL&DL distances within the range of $[0.5,5]$ km, the maximum bandwidth is set as $B_{\max}=1$ MHz. The results show that, across all 100 simulations, the proposed algorithm consistently requires only 4 or 5 iterations to converge.

In Fig. 7 we compare the proposed scheme with a TDD-mode optimization scheme [33] and three separate schemes, with each focused on a part of the $\mathbf{SC}^{3}$ loop. For fair comparisons, we consider these schemes to be goal-oriented by taking the LQR cost as the objective.

• •

  TDD-Mode Optimization Scheme [33]: This scheme uses TDD to access four $\mathbf{SC}^{3}$ loops. Each loop is allocated with an equal time slot, $T_{k}=\frac{T}{4}=2.5$ ms, and exclusively uses bandwidth and CPU frequency resources in this slot. It optimizes the UL&DL, and computing time for each $\mathbf{SC}^{3}$ loop.

• •

  UL&Computing Optimization Scheme: This scheme optimizes UL transmission and computing, while the DL parameters are fixed as $B^{d}_{k}=\frac{B_{\max}}{2K},\forall k$ and $t^{d}_{k}=1\ \text{ms},\forall k$.

• •

  DL&Computing Optimization Scheme [34]: This scheme optimizes DL transmission and computing, with the UL parameters fixed at $B^{u}_{k}=\frac{B_{\max}}{2K},\forall k$ and $t^{u}_{k}=4\text{ms},\forall k$.

• •

  UL&DL Optimization Scheme: This scheme optimizes UL&DL transmission, with the CPU frequency fixed at $f_{k}=\frac{f_{\max}}{K},\forall k$.

It can be observed that the proposed scheme achieves the lowest LQR cost compared to the other four schemes. In the bandwidth-limited region, the DL&Computing Optimization Scheme [34] performs the worst. This is because the fixed UL is the bottleneck, which largely determines the $\mathbf{SC}^{3}$-loop performance: the data transmitted via the UL are the data processed by computing, which in turn determines the accuracy of the command transmitted via the DL. Conversely, in the bandwidth-adequate region, the TDD-Mode Optimization Scheme and UL&DL Optimization Scheme perform poorly. This is because these two schemes constrain the inter-loop adjustment, negatively affecting the balance of the control progress across different $\mathbf{SC}^{3}$ loops. This highlights the advantage of treating the $\mathbf{SC}^{3}$ loop as an integrated structure and jointly configuring resources within and across the $\mathbf{SC}^{3}$ loops.

In Fig. 8, we compare the proposed goal-oriented scheme with two communication-oriented schemes.

• •

  Communication-Oriented Max-Sum Rate Scheme: This scheme maximizes the sum of the closed-loop information, i.e., $\max\sum\limits_{k=1}^{K}{D^{\mathbf{SC}^{3}}_{k}}$.

• •

  Communication-Oriented Max-Min Rate Scheme: This scheme maximizes the minimal closed-loop information, i.e., $\max\min\limits_{k}{D^{\mathbf{SC}^{3}}_{k}}$.

These two communication-oriented schemes can be solved using a similar iterative algorithm by replacing the objective of the LQR cost with the sum-rate and min-rate objective, i.e., $\max\sum\limits_{k=1}^{K}D^{\mathbf{SC}^{3}}_{k}$ and $\max\min\limits_{k}D^{\mathbf{SC}^{3}}_{k}$, and convert the LQR constraint (29b) into a closed-loop-information constraint,

In this simulation, we set the LQR cost requirement in (34) as $l_{k}=5,\forall k$. From the figure we can see that, the Communication-Oriented Max-Sum Rate Scheme performs the worst. This is because this scheme disproportionately allocates the most resources to the $\mathbf{SC}^{3}$ loops with the highest SE and CE, amplifying the imbalance among different $\mathbf{SC}^{3}$ loops. As the bandwidth increases, this bias intensifies, leading the LQR cost of other $\mathbf{SC}^{3}$ loops to rise sharply, resulting in the increase of the sum LQR cost. In addition, we can observe that, when $B_{\max}\geqslant 1.4$ MHz, the LQR cost reaches its maximum and remains unchanged under the Communication-Oriented Max-Sum Rate Scheme. In this region, the control performance of the $\mathbf{SC}^{3}$ loops diverges into two extremes: the LQR cost of the $\mathbf{SC}^{3}$ loops with the highest SE or CE approaches its minimal value, while the LQR cost of the remaining $\mathbf{SC}^{3}$ loops barely meets the basic requirement as given in (34). Conversely, the Communication-Oriented Max-Min Rate Scheme performs similarly to the proposed scheme. They have a small performance gap, which is because the Max-Min Rate Scheme prioritizes the rate balance while overlooking the differences of controlled systems. The good performance of the Max-Min Rate Scheme and the proposed scheme indicates the importance of keeping balance across different $\mathbf{SC}^{3}$ loops.

Fig. 9 compares the approximate closed-form solution for the inter-loop bandwidth allocation (32) with the optimal solution obtained by CVX. We can see that the LQR costs under the approximate closed-form solution and the optimal solution exhibit a small gap in the bandwidth-limited region, and this gap rapidly approaches zero as the bandwidth increases. This demonstrates the accuracy of the approximate closed-form solution, particularly when the system has sufficient communication and computing resources.

Furthermore, we show the bandwidth allocation principles by comparing the proposed goal-oriented scheme with two communication-oriented schemes in Fig. 10. The maximal bandwidth is set as $B_{\max}=1$ MHz. In the top subfigure, we set the four $\mathbf{SC}^{3}$ loops with different intrinsic entropy, $\log_{2}|\det\mathbf{A}_{k}|=[10,20,100,200]$, and equal SE, $r^{\text{comm}}_{k}=0.1$ bits/s/Hz, $\forall k$. It can be seen that the proposed scheme allocates more bandwidth to the $\mathbf{SC}^{3}$ loops with higher intrinsic entropy, whereas two communication-oriented schemes allocate bandwidth equally. This is because the communication-oriented schemes focus on the data rate, and as a result, they are unable to perceive the stability differences of the controlled systems. In the bottom subfigure, we set the four $\mathbf{SC}^{3}$ loops with different SEs, $r^{\text{comm}}_{k}=[0.08,0.10,0.12,0.14]$ bits/s/Hz, and equal intrinsic entropy, $\log_{2}|\det\mathbf{A}_{k}|=20,\ \forall k$. In this case, the Max-Sum Rate Scheme disproportionately allocates the most resources to the $\mathbf{SC}^{3}$ loop with the highest SE (loop 4), leaving other $\mathbf{SC}^{3}$ loops with minimal resources to satisfy the LQR-cost constraint (34). In contrast, both the proposed scheme and the Max-Min Rate Scheme allocate more bandwidth to the $\mathbf{SC}^{3}$ loops with lower SEs, aiming to achieve the task-level and rate-level balance, respectively. This shows that the proposed scheme is fairness-minded. Instead of over-resourcing the strong $\mathbf{SC}^{3}$ loop at the expense of others, it maintains the task-level balance across $\mathbf{SC}^{3}$ loops to ensure the control performance of the whole unmanned robotic system.