Lower Bound of Networked Control with Multiple Sensors and One Controller And The Application to Tracking Gaussian-Markov Source

I. INTRODUCTION

• The system comprises an LQG plant, k sensors or encoders, and one decoder communicating through independent prefix-coded messages.Encoders observe plant outputs and transmit independently over a lossless channel; the decoder produces the plant’s control signal.Source: §I · p. 2
• Multiple-encoder, single-decoder systems without feedback have an unresolved tradeoff between data rates and control performance.The paper identifies this causal rate-distortion problem as a longstanding open problem.Source: §I · pp. 2–3
• The networked setting requires proving optimality for independent linear encoders because the encoders do not communicate.Arguments establishing linear optimality in point-to-point settings do not directly generalize to this networked case.Source: §I · p. 4
• The paper establishes a new directed-information lower bound for LQG plants with multiple sensors or encoders and one decoder.The authors describe it as the first lower bound derived for this networked control setting.Source: A. Contributions · p. 4
• Under full plant-state observation, linear independent encoders and linear decoders are sufficient for weighted-sum lower-bound optimization with LQG plants and quadratic cost.The original infinite-dimensional optimization is reduced to a finite-dimensional one, but the resulting optimization is not convex.Source: A. Contributions · p. 4
• For Gaussian-Markov sources with linear side information, the paper provides SDP formulations covering weighted mean-square error and singular noise covariance settings.The formulations include both time-variant and time-invariant systems, extending the stated side-information setting.Source: A. Contributions · pp. 4–5

B. Notations

• Lowercase letters denote vectors, uppercase letters denote matrices, subscripts denote time steps, and superscripts denote encoder indices.Source: B. Notations · p. 5
• Directed information and conditional directed information are defined for sequences of random variables used in the analysis.Source: B. Notations · p. 5

C. Related Work

• Prior work studies data-rate requirements for stabilizing linear plants, including bounds determined by unstable eigenvalues.Source: C. Related Work · p. 5
• Networked-control studies also consider multiple-access, broadcast, interference, relay, and Gaussian product channels with various achievable or stabilizing conditions.Source: C. Related Work · pp. 5–6
• Related studies extend stabilizability analysis to stochastic delays, fading channels, and multiple sensors, encoders, decoders, or controllers.Source: C. Related Work · pp. 5–6

B. Notations

• The paper is organized around network formulation, directed-information lower bounds, linear-policy optimality, Gaussian-Markov sources, side information, and SDP formulations.Source: D. Organization · p. 6
• The Gaussian-Markov application includes causal rate-distortion formulations with side information and singular noise covariance matrices.Source: D. Organization · p. 6

II. PROBLEM FORMULATION

• The system contains k sensors or encoders and one decoder connected through lossless prefix-coded channels.Source: §II · p. 7
• The plant has Gaussian process noise, while sensors receive linear signals with independent or potentially correlated Gaussian observation noises.Source: §II · p. 7
• The optimization characterizes the tradeoff between expected input length and expected quadratic control cost under a target distortion requirement.Source: §II · p. 7

A. Directed Information Lower Bounds

• A lower bound on the weighted sum rate is established through conditional directed information for the networked control problem.Source: §II · p. 7, §III · p. 8, A. Directed Information Lower Bounds · pp. 8–9
• Linear policies, including independent linear encoders and a linear certainty-equivalence controller, are sufficient for optimizing the weighted-sum lower bound.Source: §II · p. 7, §III · p. 8, A. Directed Information Lower Bounds · pp. 8–10
• The same framework yields a point-to-point side-information lower bound and an alternative proof that linear policies suffice there.Source: A. Directed Information Lower Bounds · p. 9
• The linear-policy result transforms the lower-bound optimization into a finite-dimensional problem, supporting convex optimization in related point-to-point settings.Source: A. Directed Information Lower Bounds · p. 9
• Theorem 1’s lower bound is suboptimal, although tighter bounds can be obtained; showing linear-policy optimality for those tighter expressions is difficult.Source: A. Directed Information Lower Bounds · p. 10

B. Linear Policies Are Sufficient

• Linear encoders and decoders suffice under the full observation condition, reducing the original problem to a finite-dimensional optimization.The condition requires that jointly collected sensor signals can recover the full plant state.Source: B. Linear Policies Are Sufficient · p. 10
• The optimal policy belongs to a class with linear, independent encoder components and a corresponding linear controller.The encoder components are separated according to the information available to each encoder.Source: B. Linear Policies Are Sufficient · pp. 10–11
• The finite-dimensional optimization searches over matrix parameters defining encoder noise and linear transformations.The policy is parameterized by matrices such as H_t, N_t, and F_t^i.Source: B. Linear Policies Are Sufficient · p. 11
• The proof constructs Gaussian and linear policies preserving the relevant covariance, weighted rate lower bound, and control cost.Independent encoder parts are then obtained by decomposing the linear function of the aggregate control signal.Source: B. Linear Policies Are Sufficient · p. 11
• Certainty-equivalence control is not optimal for the weighted-sum directed-information lower bound in this policy class.Replacing the sum-of-codewords controller with a linear function of all codewords invalidates the sufficiency result.Source: B. Linear Policies Are Sufficient · p. 11

IV. CAUSAL LOSSY COMPRESSION WITH SIDE INFORMATION

• The causal lossy compression problem uses a Gaussian-Markov source, weighted mean-square error, and linear side information at the decoder.The decoder observes y_t = C_t m_t + v_t, with independent Gaussian observation noise.Source: §IV · p. 12, A. Problem Setting · p. 12
• The paper presents SDP formulations for time-variant and time-invariant systems, including singular noise covariance matrices.The formulation covers full-rank system matrices and explicitly extends prior SDP treatment to singular N_t.Source: §IV · p. 12, B. Semidefinite Programming Forms · p. 14
• The encoder is causal and measurable, while the optimal decoder is the conditional expectation given messages and side information.The encoder depends on the source history and previous messages; the decoder estimates m_t from a[t] and y[t].Source: A. Problem Setting · pp. 12–13
• Theorem 3 lower-bounds the optimization by a conditional directed-information problem whose optimal policy lies in a restricted policy set.The stated policy includes the conditional-expectation decoder.Source: B. Semidefinite Programming Forms · pp. 13–14
• For the time-invariant case, a Riccati recursion exists when A has no eigenvalues on the unit circle and (A, W) is detectable.These conditions accompany the long-term formulation of the lower bound.Source: B. Semidefinite Programming Forms · p. 14

1) Time-Variant System:

• The time-variant formulation optimizes directed information subject to a weighted mean-square error constraint over a finite horizon.The SDP representation handles singular noise matrices and full-rank system matrices.Source: 1) Time-Variant System · pp. 14–15
• The time-variant covariance recursion incorporates linear side information through Kalman-filter updates.The conditional covariance is defined from estimation errors given the message and side-information histories.Source: 1) Time-Variant System · pp. 14–15
• The optimal time-variant message is linear Gaussian, with a_t = F_t m_t + h_t and h_t distributed as N(0, H_t).The message parameters are obtained from the SDP solution.Source: 1) Time-Variant System · p. 15
• The time-invariant formulation studies the limit as T grows and characterizes long-term average behavior.It uses constant system, side-information, and noise parameters across time.Source: 2) Time-Invariant System · p. 16
• Under detectability, singular N = LL^T, full-rank A, and no unit-circle eigenvalues, the time-invariant lower bound has an SDP formulation.The corresponding linear Gaussian message is obtained through a singular-value decomposition, with rank(H) fixed by the covariance expression.Source: 2) Time-Invariant System · p. 16

C. Numerical Simulation

• The simulations evaluate the causal rate-distortion function for Gaussian-Markov sources with linear side information.They include cases with singular noise covariance matrices and use parameters matching prior side-information work.Source: C. Numerical Simulation · p. 17
• The side information is set to C = I and V = 10I, while the WMSE weight uses the control gain from the Riccati equation.Noise covariances of different ranks are generated using columns from a Cholesky decomposition.Source: C. Numerical Simulation · p. 17, A. Proof of Theorem 1 · p. 18
• Higher noise-covariance rank produces a higher rate in the simulations.The rank i denotes the rank of the noise covariance matrix.Source: C. Numerical Simulation · p. 17
• The section’s proof material establishes the inequalities and policy relations used for the multi-encoder lower-bound result.The argument includes independent encoder decompositions and comparison with the original weighted-sum directed-information lower bound.Source: 1) Proof of (31) · p. 21, B. Proof of Theorem 2 · p. 21

• A nonzero-mean policy can be replaced by a zero-mean policy with the same directed information and no larger control cost.Source: 1) Proof of (31) · p. 21
• The proof introduces a Gaussian measure with the same covariance matrix as the original measure and uses least-mean-square estimation errors.Source: 1) Proof of (31) · pp. 21–22
• The resulting policy comparison completes the proof of inequality (31).Source: 1) Proof of (31) · p. 22, 3) Proof of (33) · p. 23
• Under the full-observation condition, covariance and triangular-matrix identities preserve the relevant distributions and control costs across constructed policies.Source: 1) Proof of (31) · p. 22, 2) Proof of (32) · p. 23, 3) Proof of (33) · p. 23
• A policy in Γ2 is transformed into one in Γ1 with equal control cost and equal objective functions.Source: 2) Proof of (32) · p. 23

• The causal rate-distortion expression CS(d) is rewritten as a covariance-based optimization problem.Source: D. Proof of Theorem 5 · pp. 25, 27
• A feasible point in D0 has an objective no greater than that of the original formulation (44).Source: D. Proof of Theorem 5 · p. 27
• For every feasible P in D0, a sequence of time-indexed covariance matrices converges to P while preserving the required objective comparison.Source: D. Proof of Theorem 5 · p. 27
• Detectability and the covariance recursion provide a sequence converging to (P, P̃), with the weighted covariance cost bounded by d.Source: D. Proof of Theorem 5 · pp. 27–28
• The original objective (44) is upper bounded by the finite-dimensional objective (48).Source: D. Proof of Theorem 5 · p. 28

• The proof develops lemmas for the joint process of sensor observations and controls indexed by sensor subsets.Source: E. Proof of Lemmas · p. 28
• The relevant conditional-independence equalities rely on Markov properties and generally fail unless the complementary sensor set is empty.Source: E. Proof of Lemmas · p. 28
• The resulting information comparison is I(x[T] → a[T]∥y[T]) ≥ I(x[T] → u[T]∥y[T]).Source: E. Proof of Lemmas · p. 29, F. Proof of Proposition 1 · p. 30
• Under the full-observation condition, a matrix maps the sensor output to the plant state, allowing the plant recursion to be used for the constructed process.Source: E. Proof of Lemmas · p. 29
• Independence of the white Gaussian noises from the past supports the subsequent distributional identities.Source: E. Proof of Lemmas · p. 29

G. Proof of Lemma 1

• The proof compares measures with matching covariance matrices and uses Gaussianity, Bayes’ rule, and KL-divergence nonnegativity.Source: G. Proof of Lemma 1 · pp. 31–32
• An induction establishes the required measure identity from the Gaussian initial distribution through successive time steps.Source: H. Proof of Lemma 2 · pp. 32–33
• If the constructed pair were not detectable, an unstable eigenvector in the null space of F would contradict the covariance relation.Source: §I · p. 33

VI. Conclusion

• The paper establishes a directed-information lower bound for multiple sensors or encoders with one decoder and proves linear-policy optimality for its weighted-sum rate lower bound.Source: §VI · p. 34
• The results yield SDP formulations for Gaussian-Markov causal lossy compression with linear side information and singular noise covariance.Source: §VI · p. 34
• Showing linear-policy optimality for the tighter directed-information lower bound remains a future research direction.Source: §VI · pp. 34–35