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Minimum Rate For Partially Observable Linear System with Side Information: LQG Plant and Gaussian-Markov Source

Sijie Li, Hyeji Kim

arXiv:2608.26917v1cs.ITeess.SY

TL;DR

The paper asks how much information is needed to control or estimate partially observed linear systems when the controller or decoder also has side information. It identifies sufficient linear policies for the conditional directed information lower bound and derives convex scalar formulations for time-varying and time-invariant systems. Simulations illustrate that side information can change achievable control costs and reduce required rate.

  • Problem

    The paper addresses the minimum-rate problem for LQG plants and Gaussian-Markov sources with partial observation at the encoder and side information at the controller or decoder.

  • Method

    The paper identifies an optimal policy class with linear encoders based on conditional-mean state estimates and Gaussian noise, plus certainty-equivalence control or conditional-mean decoding.

  • Results

    The conditional directed information optimization is convex for scalar systems in both time-varying and time-invariant settings, with a single-letter form for time-invariant systems.

  • Takeaways & Limitations

    Side information can make otherwise unattainable control costs achievable and reduce required rate, even when its quality is lower than the partial observation.

Abstract

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This paper studies the minimum rate required for a partially observable linear system with side information. The Linear Quadratic Gaussian(LQG) plant and the Gaussian-Markov source are considered. We show that a class of linear policies is sufficient for optimizing the conditional directed information lower bound. We also show that the resulting optimization problem is convex for the scalar case in both time-varying and time-invariant systems. Our results generalize the past works that consider the case with full or partial observation only, and the case with full observation and side information. Numerical simulations are presented to illustrate the effect of side information for partially observable systems.

I. INTRODUCTION

The paper studies minimum-rate control and estimation for LQG plants and Gaussian-Markov sources when the encoder has partial observation and the controller or decoder has side information. It develops sufficient linear policies and convex scalar optimization forms for time-varying and time-invariant systems.

  • Problem setting: The encoder observes noisy plant measurements, while the controller receives codewords and additional side information to produce control signals.The codeword and control signal are causal functions of information available up to the current time.
  • Motivation: Prior work generally considered partial observation or side information separately, rather than both together.Earlier results covered full observation, partial observation, or full observation with side information.
  • Main contributions: A linear encoder based on the conditional-mean plant state and independent Gaussian noise, paired with a certainty-equivalence controller, is sufficient for the LQG optimization.For the Gaussian-Markov source, the corresponding decoder is a conditional-mean estimator.
  • Main contributions: The same sufficient linear-policy structure extends to the Gaussian-Markov source, replacing control-cost constraints with weighted mean-square error distortion constraints.The source is treated as an uncontrolled version of the plant model.
  • Main contributions: The scalar conditional directed information optimization is convex in both time-varying and time-invariant settings.For time-invariant systems with infinite horizon, the optimization admits a single-letter convex form.
  • Optimization objective: The models use prefix coding and characterize rate lower bounds through conditional directed information under control-cost or distortion constraints.The LQG formulation uses control cost, while the Gaussian-Markov formulation uses weighted mean-square error.

III. OBSERVABLE MARKOV CHAIN

The paper converts the partially observed system into an observable Markov chain using conditional means and Kalman-filter error decompositions. The residual estimation error is Gaussian and independent of the shared information set.

  • Observable Markov chain: The key step is identifying an observable Markov chain for the partially observable plant and source.The Gaussian-Markov source follows by viewing it as the LQG case with zero control input.
  • Kalman filtering: Kalman filtering supplies the conditional-mean updates and the associated error-covariance recursions.The filtering proposition establishes the update equations and covariance relationships.
  • Kalman filtering: The filtering residual is Gaussian and independent of the information available to the encoder and controller.The proposition states this independence after conditioning on the relevant observations.
  • State decomposition: The plant state decomposes into a conditional-mean component and a residual error, with the residual independent of the information set.The residual is the portion of the state that cannot be estimated from the encoder's and controller's observations.
  • Observable Markov chain: The resulting observable process is driven by independent Gaussian noises.This property supports treating the transformed process as an observable Markov model.

B. Gaussian-Markov Source

For the Gaussian-Markov source and the LQG plant, the paper restricts optimization to a sufficient linear policy class and derives convex scalar formulations. The time-invariant case admits a single-letter form.

  • Convex optimization: For time-invariant systems, the scalar optimization reduces to a single-letter convex optimization form.The paper addresses convergence of two coupled recursions in establishing this result.
  • Linear policy: The linear-policy class uses linear encoders and a certainty-equivalence controller for the LQG plant.The encoder can be represented using the conditional-mean state and independent Gaussian noise.
  • Linear policy: Theorem 1 states that restricting the optimization to the linear policy set is sufficient.The policy structure recovers earlier results for full observation, side information, and partial observation cases.
  • Convex optimization: For scalar systems, the resulting optimization problem has a convex form in the time-varying case.The paper introduces a convex reformulation for the scalar optimization.

B. Time-Invariant Plant

For the time-invariant LQG plant, the infinite-horizon problem is reduced to a single-letter convex optimization form, with steady-state quantities obtained from Riccati recursions.

  • Assumptions: The analysis assumes time-invariant parameters, an infinite horizon, and a stabilizable pair (A, B) under an average control-cost requirement.Positive-definite Q and R ensure convergence of the control Riccati recursion and the steady-state control gain.
  • Steady-state quantities: The steady-state control gain K is computed from the unique solution of a Riccati equation and then used to define Θ.K = −(B^TSB + R)^−1B^TSA and Θ = K^T(B^TSB + R)K.
  • Steady-state quantities: The estimation error covariance converges to a discrete algebraic Riccati-equation solution, yielding steady-state covariance terms such as N̄.N̄ is the limiting difference between the predicted and filtered error covariances.
  • Objective: The resulting objective uses a scalar function f(P̄_t|t) expressed as a difference of logarithms of affine terms and −log P̄_t|t.The function is defined as log(αP̄_t|t + β) − log(γP̄_t|t + δ) − log P̄_t|t.
  • Convex formulation: The infinite-horizon optimization problem is transformed into a single-letter optimization problem.The transformation is stated in Theorem 3 for the time-invariant case.
  • Closed-form solution: The paper gives a closed-form solution for the single-letter convex problem, including a zero optimum when d exceeds the stated threshold.The threshold is ΘW̄ + SW + ΘP̄*; an intermediate regime is also specified by Corollary 1.

V. GAUSSIAN-MARKOV PROCESS

For the Gaussian-Markov source, linear encoders and a conditional-mean decoder suffice, and the scalar optimization remains convex with distortion replacing control cost. Simulations show that side information can substantially change rate-cost tradeoffs.

  • Model and policy: The Gaussian-Markov source is studied under a weighted mean-square-error distortion constraint.The source formulation parallels the LQG plant, but replaces the control-cost constraint with weighted distortion.
  • Convexity: The scalar Gaussian-Markov optimization is convex in both time-varying and time-invariant cases.The time-invariant formulation has the same single-letter structure as the LQG result, with distortion replacing control cost.
  • Policy set: The policy set Λ1 consists of linear encoders and a conditional mean decoder.The encoder includes an independent white Gaussian noise term, while the decoder outputs E[x_t|y_t,c_t].
  • Policy set: Because the decoder observes y_t, the codeword effectively carries the same information as L_t x̄_t plus independent Gaussian noise.This equivalence is stated for the conditional-mean representation of the encoder.
  • Policy optimality: It is sufficient to optimize over Λ1 for the Gaussian-Markov source.Corollary 2 transfers the linear-policy sufficiency result from the LQG setting.
  • Numerical simulations: Simulations compare full and partial observation, with and without side information, using black, green, red, and blue dots respectively.The experiments vary observation and side-information noise to represent different quality combinations.
  • Numerical simulations: Side information of comparable or higher quality can make control costs achievable that partial observation alone cannot attain.Even lower-quality side information can reduce the required rate when paired with partial observation.

B. Proof of Theorem 1

The proof establishes lower bounds through an observable conditional-mean process and a Gaussian comparison, then shows that linear encoders with certainty-equivalence control attain the relevant structure.

  • Controller choice: Under the linear policy set, the objective is independent of the controller choice, making the certainty-equivalence controller optimal.The objective becomes a sum of log determinants determined by encoder and side-information covariances.
  • Lower bound: The rate is lower bounded by conditional mutual information between the control signal and the observable conditional-mean process.This replaces direct dependence on the hidden plant state with the observable process x̄_t = E[x_t|I_t].
  • Gaussian comparison: For any feasible policy, the conditional directed information is lower bounded by its Gaussian version.This is formalized as Lemma 3.
  • Measure construction: A policy measure Q is constructed recursively from the conditional law of the control and the observable process transition.The construction defines Q over successive control, observation, and conditional-mean variables.
  • Measure construction: The constructed Q and Gaussian measure G share the relevant joint distribution, so their second moments and induced control costs agree with those under P.The equality Q(u_t,y_t,x̄_t)=G(u_t,y_t,x̄_t) supports preservation of the control-cost calculation.
  • Linear realization: The policy constructed through Q can be represented by a linear encoder with independent Gaussian noise.The encoder is written using the coefficient of z_t in the linear least-mean-square estimate.
  • Lower-bound proof: The mutual-information chain uses Markov structure and nonnegativity of mutual information to complete the comparison with feasible encoders and controllers.These steps establish the required inequality for the theorem.

C. Proof of Theorem 2

The proof converts the scalar optimization into a convex problem by expressing its objective through covariance variables and proving convexity of the resulting logarithmic terms.

  • Objective representation: Within the linear policy set, the objective is written as a sum of differences of log determinants of covariance matrices.This representation isolates the covariance variables governing the scalar optimization.
  • Objective representation: The Woodbury matrix identity and a covariance decomposition rewrite the objective using scalar functions f_t(P̄_t|t).The decomposition uses the independence of the estimation-error terms and the covariance N̄_t.
  • Convexity argument: Each scalar function has the form log(α_tP̄_t|t + β_t) − log(γ_tP̄_t|t + δ_t) − log P̄_t|t.The coefficients are positive under the filtering relations used in the proof.
  • Convexity argument: The proof shows f_t(P̄_t|t) is convex for P̄_t|t > 0.Convexity follows by evaluating the second derivative and establishing positivity of the relevant coefficients.
  • Convexity argument: Because the full objective sums the f_t terms and −log P̄_T|T, it is jointly convex in the positive covariance variables.The initial side-information covariance is treated as a constant.
  • Constraints: The covariance feasibility constraint is reformulated using the Schur complement, while the control-cost constraint is expressed for the certainty-equivalence controller.These transformations provide the constraints used in the convex formulation.

D. Proof of Theorem 3

The proof reduces the infinite-horizon problem to a single-letter convex optimization by using certainty equivalence and establishing convergence of the coupled recursions.

  • Controller reduction: The certainty equivalence controller is optimal once linear sensors and side information are fixed, so controllers can be replaced accordingly.The objective is independent of the controller choice under the stated linear policy structure.
  • Limit construction: The time-varying quantities ¯Wt and ¯Nt converge to their time-invariant limits ¯W and ¯N, supporting the infinite-horizon reduction.These limits are used together with compactness and subsequence arguments in the proof.
  • Convex reduction: The resulting objective is bounded below through a single-letter convex form because f(·) is convex and Jensen’s inequality applies.The proof combines subsequence bounds, Lemma 5, and convexity to obtain the lower bound.
  • Recursion convergence: The proof establishes convergence of the covariance recursions to a feasible limiting covariance for any initial value P1|0 > 0.Lemma 6 states convergence for any initial positive covariance.
  • Objective and cost limits: The limiting mutual information term and average control cost converge as the covariance recursion converges.The mutual information limit is obtained directly, while the control-cost convergence follows by a Cesàro-sum argument.

E. Proof of Corollary 1

The corollary derives the optimal value by analyzing the scalar feasible region and separating cases according to the control-cost bound.

  • Objective monotonicity: The derivative of f(x) is negative for x > 0, so the objective is minimized at the largest feasible covariance value.The feasible maximum is determined by the constraints analyzed in the scalar problem.
  • Feasible region: The positive-semidefinite constraint yields a positive root ¯P∗ and restricts feasible covariances to ¯P ≤ ¯P∗.This follows from the quadratic polynomial g(¯P) and the condition F^2A^2 > 0.
  • Parameter regimes: When |A| ≥ 1, the corresponding scalar constraint is satisfied for every ¯P ≥ 0; when 0 < |A| < 1, it reduces to an additional constraint.The proof therefore treats the two regimes separately.
  • Corollary conclusion: The proof concludes by identifying the optimal value stated in Corollary 1.The preceding case analysis supplies the feasible optimizer used for that value.
  • Optimal value: When d > Θ ¯W + SW + Θ ¯P∗, the optimizer is ¯P = ¯P∗ and the objective evaluated there is zero.The complementary case uses ¯P = (d − SW)/Θ as described in the proof.

F. Proof of Lemma 3

Lemma 3 proves the conditional directed information lower bound by rewriting the non-Gaussian expression and comparing it with its Gaussian version through KL divergence.

  • Chain-rule decomposition: The proof rewrites the conditional directed information using repeated applications of the chain rule of probability.The factorization expands the relevant joint and conditional densities across the time horizon.
  • Gaussian comparison: The difference between the original and Gaussian mutual-information sums is expressed using a KL-divergence term.The comparison is made between the induced measure P and its Gaussian counterpart G.
  • Lower-bound conclusion: Nonnegativity of KL divergence yields that the conditional directed information under P is lower bounded by its Gaussian version.The initial conditional distribution is Gaussian, and Lemma 2 supplies the remaining equality needed in the argument.

G. Proof of Lemma 4

The proof establishes convergence of the covariance recursions by bounding the sequence and showing that its limit inferior and superior must coincide at a unique fixed point.

  • Distributional induction: The induction argument propagates equality between the relevant distributions from time t to time t + 1.The base case follows from joint Gaussianity, and the induction step uses the stated identities and Lemma 2.
  • Asymptotic terms: The limiting parameters in the recursion converge, allowing the time-varying logarithmic terms to vanish through a Cesàro-sum argument.The proof separately establishes convergence for the parameter ratios and applies the same reasoning to the remaining terms.
  • Sequence bounds: The covariance sequence is eventually bounded below by a positive constant and above by 1/c1.Monotonicity of ρ(x), convergence of ¯Wt, and the recursion bounds establish these two-sided bounds.
  • Fixed-point characterization: Both the limit inferior and limit superior are fixed points of the limiting recursion on the relevant interval.Compactness, uniform convergence, and subsequence arguments establish the fixed-point property.
  • Uniqueness and convergence: Concavity and monotonicity of the limiting composite map imply that only one fixed point exists for x ≥ ¯W, forcing convergence.The map is positive above ¯W and falls below the diagonal beyond 1/c1.

VIII. CONCLUSION

The paper characterizes rate-optimal policies for partially observed linear systems with side information and establishes convex scalar optimization forms. It also identifies extensions to multiple encoders and vector systems as future directions.

  • The optimal policy set consists of linear encoders based on plant-state estimates and Gaussian noise, paired with certainty-equivalence control or conditional-mean estimation.
  • For scalar systems, the resulting optimization problem is convex in both time-varying and time-invariant settings.
  • Time-invariant systems admit a single-letter optimization form whose tightness follows from convergence of two nested Riccati recursions.
  • Future work includes rate-distortion tradeoffs with one full-observation and one partial-observation encoder, and extending the scalar optimization form to vector systems.
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