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Joint Communication and Control Beamforming: A Closed-Loop Control Perspective

Hao Jiang, Chongjun Ouyang, Yuanwei Liu, Arumugam Nallanathan, Robert Schober

arXiv:2608.26352v1eess.SP

TL;DR

The paper addresses how wireless interference affects long-term performance in closed-loop joint communication and control. It derives beamforming-linked LQG costs and designs vector- and scalar-control solutions, which match simulations and outperform zero-forcing, especially with limited spatial DoFs.

  • Problem

    Existing JCC studies often model only parts of the closed loop or use abstractions that do not directly connect beamforming and wireless impairments to plant-state evolution.

  • Method

    The paper combines LMMSE downlink control-input recovery, KF uplink state tracking, finite- and infinite-horizon LQG analysis, and SINR-constrained beamforming design.

  • Results

    The proposed methods consistently outperform the zero-forcing benchmark, including vector-case ΔJ reductions of 79.0% and 87.6% in reported settings.

  • Takeaways & Limitations

    Balancing communication-control interference improves long-term control performance, particularly when spatial degrees of freedom are limited.

Abstract

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A joint communication and control (JCC) framework is proposed, where a base station (BS) simultaneously serves multiple communication users (CUs) and controls a physical plant in a closed loop. In the downlink, BS-generated control inputs are transmitted to and recovered at the plant, with wireless actuation distortion incorporated into the plant-state evolution. In the uplink, the plant state is reported to the BS and tracked by a Kalman filter (KF) for subsequent control-input generation. To characterize long-term control performance under communication-control interference, finite- and infinite-horizon linear quadratic Gaussian (LQG) costs are derived, directly linking beamforming design to plant-state evolution. JCC beamforming problems are then formulated for vector- and scalar-valued control inputs to minimize the infinite-horizon LQG cost subject to per-user communication signal-to-interference-plus-noise ratio (SINR) requirements. For the vector case, a second-order cone programming (SOCP)-based successive convex approximation method is developed for the resulting nonconvex problem. For the scalar case, a closed-form infinite-horizon LQG cost is derived, and the communication-control Pareto boundary is optimally characterized by an SOCP-based bisection method. Its optimality follows from the strict monotonicity of the scalar control cost with respect to the control SINR. Numerical results show that the derived costs closely match Monte Carlo simulations, the KF accurately tracks the ground-truth plant-state trajectory, and the proposed methods consistently outperform the zero-forcing benchmark. This confirms the benefit of balancing communication-control interference, especially with limited spatial degrees of freedom (DoFs).

I. INTRODUCTION

The paper develops a MIMO JCC framework that models communication and control jointly over a wireless closed loop. It derives long-term control costs and designs beamformers to balance communication QoS with plant-control performance.

  • Motivation: JCC performance depends on temporally coupled plant-state evolution, so instantaneous communication or sensing metrics do not fully characterize control quality.Wireless control-input distortion affects subsequent observations, state estimates, actions, and long-term performance.
  • Framework: JCC uses a multi-antenna BS to serve multiple CUs while transmitting vector-valued control inputs to a physical plant.The architecture incorporates LMMSE downlink control-input recovery and KF-based uplink plant-state tracking.
  • Control metric: The derived finite- and infinite-horizon LQG costs capture actuation distortion, noise, and feedback-link state-estimation errors through beamforming-dependent control performance.This connects wireless transmission design directly to the closed-loop plant dynamics.
  • Beamforming design: For vector control inputs, beamforming minimizes infinite-horizon LQG cost under CU QoS and power constraints using an SOCP-based successive convex approximation algorithm.The problem is nonconvex, so the method obtains a suboptimal solution.
  • Beamforming design: For scalar control inputs, a closed-form control cost supports optimal Pareto-boundary characterization through SOCP-based bisection.The optimality result relies on strict monotonicity of scalar control cost with respect to control SINR.
  • Results: Numerical results show close agreement with Monte Carlo costs, accurate KF state tracking, and consistent improvement over the ZF benchmark.The benefit of balancing inter-function interference is especially evident when spatial DoFs are limited.

B. Control Model

The control model transmits a recovered control input to a stochastic linear plant while modeling communication interference and receiver noise as actuation disturbances. The resulting state evolution provides the basis for closed-loop control-cost analysis.

  • Downlink control-input transmission: The downlink control-input transmission is affected by communication interference and additive Gaussian receiver noise at the plant.These impairments are collected into the disturbance affecting control-input recovery.
  • Downlink control-input transmission: Because the control-input vector is real-valued, the received complex signal is converted into stacked real and imaginary components before recovery.The real-valued representation accounts for the variance split between complex disturbance components.
  • Downlink control-input transmission: An LMMSE combiner recovers the normalized control input and induces an additive actuation disturbance from interference and receiver noise.The recovered original control input is obtained by rescaling the normalized estimate.
  • Plant-state evolution: The plant evolves as a discrete-time stochastic linear system driven by the recovered control input and additive Gaussian control-process noise.The state xn is governed by constant matrices A and B, with vn representing process noise.
  • Control-process factors: Transmission imperfections and control-model imperfections are distinct sources affecting the control process.The former include interference and receiver noise, while the latter arise from plant uncertainty and unmodeled dynamics.

2) Uplink Plant-State Reporting:

The JCC system reports the plant state uplink to the BS, where a Kalman filter updates the state estimate used to generate subsequent control inputs. Finite- and infinite-horizon LQG costs characterize long-term performance under wireless actuation and estimation effects.

  • Uplink Plant-State Reporting: The plant normalizes its updated state and transmits it uplink to the BS using SVD-based eigenmode precoding.The state-reporting link is modeled with plant transmit power and additive Gaussian noise; under link separation, it does not interfere with downlink transmission.
  • Uplink Plant-State Reporting: The BS applies a Kalman filter to each received state report, producing a posterior estimate for the next control-input generation.The true state is represented as the KF estimate plus an additive estimation error whose covariance quantifies residual uncertainty.
  • Finite-Horizon Control Cost: Finite-horizon LQG cost minimization uses backward Riccati recursion and feedback controls based on the KF state estimate.The terminal Riccati condition is Θ_N = S, and the optimal finite-horizon inputs are u_n,⋆ = −K_n x̂_n.
  • Infinite-Horizon Control Cost: The infinite-horizon average cost is evaluated in a stationary regime using steady-state Riccati and KF error-covariance quantities.Under stabilizability and detectability, the closed-loop state covariance converges to a finite stationary value, making the average cost finite.
  • Beamforming Dependence: Communication and control beamformers affect the control cost through communication-interference covariance and the plant’s equivalent control channel.The dependence is summarized by M̄ = M̄(W_c, W_p) and Σ_e = Σ_e(W_c, W_p).

III. BEAMFORMING DESIGN FOR JCC SYSTEMS WITH VECTOR-VALUED CONTROL INPUTS

The vector-control beamforming problem minimizes the infinite-horizon LQG cost subject to communication QoS and total-power constraints. Because the formulation is nonconvex, it is handled through SOCP reformulation and successive convex approximation with backtracking.

  • Problem Formulation: The vector JCC design minimizes infinite-horizon control cost while enforcing per-user SINR targets and a total transmit-power limit.The communication constraints specify minimum QoS rather than priority over control; the control cost remains the direct optimization objective.
  • Problem Formulation: The problem is nonconvex because the LQG objective depends nonlinearly on beamformers through LMMSE recovery, Riccati quantities, and KF error covariance.The fractional SINR constraints are also nonconvex before reformulation.
  • SOCP Reformulation: Communication SINR and power constraints are converted into convex second-order cone constraints using phase rotation and norm-affine representations.The phase choice preserves relevant squared magnitudes, covariance matrices, and power terms without loss of optimality.
  • SOCP-Based SCA: Successive convex approximation constructs a strongly convex local surrogate of the control-cost objective and solves the resulting convex problem iteratively.The optimization variable stacks real and imaginary beamformer components, while gradients are computed numerically by central finite differences.
  • SOCP-Based SCA: Backtracking increases the proximal weight until the candidate decreases the original objective, after which iterations continue until the relative objective change meets tolerance.The procedure is summarized in Algorithm 1 and uses solver, SCA-iteration, and backtracking counts to characterize complexity.

IV. BEAMFORMING DESIGN FOR JCC SYSTEMS WITH SCALAR-VALUED CONTROL INPUTS

The scalar-control specialization models one-dimensional actuation with a multi-antenna BS and a single-CU, single-plant setting. It derives scalar recovery, state evolution, KF estimation, and a closed-form infinite-horizon LQG cost.

  • Scalar Specialization: The scalar case specializes the vector model to one-dimensional actuation and uses a single CU and single plant to study the communication-control trade-off.The BS still uses multiple antennas for transmission and reception, while the plant has a single-antenna architecture for control reception.
  • Control-Input Recovery: At the plant, LMMSE estimation recovers the normalized scalar control input from the effective control channel in the presence of aggregate interference and noise.The recovered input is denormalized before entering the plant state evolution.
  • State Feedback: The scalar plant state evolves under the scalar counterparts of the system matrices, while the BS uses KF estimates and estimation-error variance for state feedback.The true state equals the posterior estimate plus an additive KF estimation error, explicitly contributing to control cost.
  • Control Cost: The scalar finite-horizon LQG cost penalizes state deviation, applied control effort, and terminal state, with the infinite-horizon cost defined as the long-run average.The stationary analysis takes the control-input variance toward a limiting value.
  • Closed-Form Cost: For scalar control, the infinite-horizon average cost and optimal control input admit closed-form expressions based on the scalar Riccati solution.The scalar closed-loop coefficient must satisfy |A_cl| < 1 so the state variance and average cost remain finite.

B. Pareto Boundary Characterization for the Scalar Case

In the scalar setting, the Pareto boundary is found by maximizing feasible control SINR for each communication SINR requirement. Strict cost monotonicity converts control-cost minimization into an SOCP-based bisection search.

  • Pareto Boundary: For a fixed communication SINR requirement Γ_c, the scalar Pareto-boundary point is obtained from a beamforming problem balancing communication QoS, control SINR, and transmit power.The procedure varies Γ_c over its feasible range to trace the communication-control boundary.
  • Monotonicity: The scalar infinite-horizon control cost is strictly decreasing in control SINR whenever the effective closed-loop control coefficient is nonzero.Thus, minimizing the control cost is equivalent to maximizing control SINR.
  • SOCP-Based Bisection: The algorithm uses an outer search over candidate control SINR and an inner SOCP feasibility problem to find corresponding communication and control beamformers.Feasible candidates raise the lower bound, while infeasible candidates lower the upper bound.
  • Feasibility: The communication constraint must first satisfy P_max ≥ Γ_cσ² before bisection can proceed.If this condition fails, the communication QoS requirement cannot be met with the available power.
  • SOCP-Based Bisection: Bisection terminates when the control-SINR interval width is at most ε and returns the largest feasible lower-bound value.The number of bisection iterations scales as O(log2((ϱ_ub − ϱ_lb)/ε)).

V. NUMERICAL RESULTS

Numerical evaluations validate the theoretical control-cost expressions, KF tracking, and convergence of the proposed algorithms, while showing improved communication-control trade-offs over Joint-ZF, particularly with limited spatial DoFs.

  • Cost validation: Theoretical infinite-horizon LQG costs converge with empirical running-average costs for both vector- and scalar-valued control inputs.This validates Theorem 2 and Corollary 1 while showing the transition from transient to steady-state operation.
  • KF tracking: Most steady-state KF tracking-error realizations are below 0.05 for the vector case and 0.04 for the scalar case.The empirical CDFs use the last 100 time slots from 200 Monte Carlo runs.
  • Algorithm convergence: The proposed algorithms’ control-cost degradation decreases monotonically and converges below the Joint-ZF benchmark in both cases.The vector case converges more slowly because multiple control streams and communication beamformers are more strongly coupled.
  • Vector-case trade-off: For Mt = 10, the proposed JCC design reduces ∆J by 79.0 % for K = 4 at 7.5 dB and by 87.6 % for K = 6 at 5 dB versus Joint-ZF.The advantage is attributed to balancing inter-user and inter-function interference rather than completely nulling interference.
  • Scalar-case trade-off: In the scalar case, ∆J increases with the communication SINR threshold, while increasing Mt improves control performance and reducing Mt widens the gap versus Joint-ZF.The JCC design is globally optimal in the single-CU, single-plant scalar setting.

APPENDIX A IMPLEMENTATION OF KALMAN FILTER

The appendix specifies the BS Kalman filter for tracking the plant state from an evolution model with wireless recovery distortion and an uplink observation model. The recursion treats unknown recovery and process disturbances as effective process noise, then performs prediction and posterior correction.

  • State and observation models: The KF tracks the true plant state at the BS using the plant state-transition and observation models.The control input is known to the BS, while recovery error and control-process noise are generally unknown.
  • State and observation models: The unknown recovery and process disturbances are combined into zero-mean effective process noise with covariance BΣe,n−1B^T + Σv.
  • State and observation models: The observation model is rewritten in real-valued form with Gaussian measurement noise covariance R ≜ (σ²f/2)I2Mr.
  • KF recursion: Prior prediction updates the state estimate using the known control input and propagates the covariance with the effective process-noise covariance.
  • KF recursion: Posterior correction forms the innovation and updates the state estimate and error covariance using the Kalman gain.

Q + ATSA

The derivation applies Bellman recursion to KF-based state estimates, yielding finite-horizon value functions and optimal controls that include estimation-error contributions. Under stabilizing stationary conditions, the Riccati recursion and estimation covariance converge to an infinite-horizon control law and average cost.

  • Finite-horizon solution: The optimal control input is obtained by differentiating the quadratic Bellman expression with respect to the preceding control input.
  • Finite-horizon solution: The value function retains a quadratic form in the estimated state plus a covariance trace term and an additive constant.The recursion is applied backward from the terminal time.
  • Finite-horizon solution: The optimal finite-horizon control cost is expressed through the initial KF state estimate and the recursively defined value-function terms.
  • Infinite-horizon limit: Under standard stabilizing conditions, the finite-horizon Riccati recursion converges to the stabilizing solution of the infinite-horizon DARE.
  • Infinite-horizon limit: In the stationary regime, the estimation-error covariance converges, the infinite-horizon average cost is obtained as the horizon grows, and the control becomes un,⋆ = −Kx̂n.

APPENDIX D PROOF OF COROLLARY 1

For scalar control, the matrix Riccati derivation reduces to scalar quantities and yields an explicit stabilizing control gain and closed-form infinite-horizon average cost. The scalar Riccati equation is quadratic, so its nonnegative root provides a real-valued solution.

  • Scalar specialization: The scalar specialization replaces the vector system quantities with scalar counterparts and reduces the auxiliary matrix to ϕ = m̄²(D + B²θ).
  • Scalar specialization: When m̄ ≠ 0 and D + B²θ > 0, the scalar auxiliary term is invertible.
  • Scalar specialization: The scalar optimal control gain is k = ABθ/[m̄(D + B²θ)].
  • Closed-form cost: The scalar control input admits an infinite-horizon average-cost expression derived from the scalar Riccati solution.
  • Closed-form cost: The scalar Riccati equation becomes a quadratic in θ, whose nonnegative root yields a real-valued solution because its discriminant satisfies Δ ≥ 0.

APPENDIX E PROOF OF LEMMA 1

The proof separates the scalar infinite-horizon cost's dependence on control SINR into steady-state estimation and plant-state terms. It then establishes that the resulting cost is strictly decreasing in control SINR whenever the effective closed-loop control coefficient is nonzero.

  • Cost decomposition: The proof analyzes how the infinite-horizon average control cost depends on control SINR ρp by separating its constituent terms.
  • Steady-state moments: The steady-state plant-state second moment is defined as I2(ρp) = E[x²n+1], while normalized estimation error is represented by I3.
  • Steady-state moments: The posterior estimation-error variance is obtained from the real-imaginary stacked observation model and enters the steady-state second-moment analysis.
  • SINR dependence: I3 is independent of ρp because its defining steady-state equation depends only on I1 and η, which are independent of ρp.
  • SINR dependence: The actuation-disturbance variance and plant-state moment are substituted into the cost, after which differentiation with respect to ρp proves strict decrease when m̄k ≠ 0.
  • SINR dependence: Whenever m̄k ≠ 0, dJ∞/dρp < 0, establishing strict monotonicity of the scalar infinite-horizon cost.
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