Rate-cost tradeoffs in control

System model

• The system consists of a linear stochastic plant whose encoder observes sensor outputs and sends channel inputs to a controller that selects control actions causally.The encoder and controller both access their respective observation histories.Source: A. System model · p. 1
• The LQR cost balances state deviation from the target state with control power through weighting matrices Q, R, and S_t.The special case Q = I_n, R = 0, and S_{t+1} = I_n gives average mean-square state deviation.Source: B. The rate-cost tradeoff · pp. 1–2

Information constraint

• Directed mutual information captures causal dependence of the control sequence on the observations and is no greater than full mutual information.The system dynamics determine the causal plant kernels, while encoder, channel, and controller form the causal channel kernels.Source: B. The rate-cost tradeoff · p. 2

Rate-cost tradeoff

• The rate-cost function is the minimum directed mutual information compatible with a target LQR cost and lower-bounds the channel capacity needed to sustain that cost.This operational connection applies to both noiseless and noisy channels, while noiseless channels can approach the bound using variable-length lattice quantization of the innovation.Source: B. The rate-cost tradeoff · p. 2

Prior work

• Earlier work established data-rate conditions for stabilization, while later results linked LQG control under communication constraints to causal rate-distortion theory.Prior converse results include bounded-noise and mean-square stabilization settings, with fixed-rate quantizers unable to maintain bounded cost under unbounded noise.Source: C. Prior art · p. 2

• A variable-length lattice quantizer that transmits only the state innovation closely approaches the converse, with the gap vanishing as b approaches bmin.Source: D. Our contribution · p. 3, A. Fully observed system · pp. 4–5
• The framework extends to partially observed systems when system and observation noises are Gaussian, and applies across fixed-rate, variable-rate, noisy-channel, and non-Gaussian settings.Source: D. Our contribution · p. 3, A. Fully observed system · p. 6
• The causal Shannon lower bound supports the converse, while high-resolution quantization and lattice constructions establish near-achievability; the entropy-cost function also has direct operational meaning through variable-length quantization.Source: D. Our contribution · p. 3, E. Technical approach · p. 3, A. Fully observed system · p. 5
• The paper derives a lower bound on the rate-cost function for fully observed linear stochastic systems with system noise satisfying h(V) > −∞.Source: A. Fully observed system · p. 4
• The rate-cost bound decreases with larger target cost, approaches log |det A| as b grows, and exhibits substantial cost improvement from increasing rate from 1 to 3 nats per sample.Source: A. Fully observed system · p. 4
• The converse remains valid for unrestricted quantizers using the entire observation history, without assuming linear structure.Source: A. Fully observed system · p. 5
• The achievability scheme needs no common randomness, and nonuniform rate allocation across time is unnecessary for attaining the stated bound.Source: A. Fully observed system · p. 5
• For m < n, the bounds from Theorems 1 and 3 lose their dependence on b, while Theorem 4 remains decreasing in b.Source: A. Fully observed system · p. 5

• The controller’s minimum cost separates into contributions from system noise and observation noise under unconstrained communication.Source: B. Partially observed system · p. 6
• Theorem 5 lower-bounds the rate-cost function using the steady-state covariance of the encoder’s state-estimation innovation.Source: B. Partially observed system · p. 6
• A scheme that quantizes the innovation, recursively estimates the state, and controls from the estimate performs provably close to R(b) in the high-rate regime.Source: B. Partially observed system · p. 7
• Theorem 6 upper-bounds the entropy-cost function for Gaussian partially observed systems at any LQR cost b > bmin.Source: B. Partially observed system · p. 7
• Theorem 7 provides a lower bound on the rate-cost function for controllable and observable Gaussian partially observed systems.Source: B. Partially observed system · p. 7
• Theorem 8 extends the lower bound to systems with control dimension satisfying m ≤ k ≤ n and reduces to Theorem 5 when k = m = n.Source: B. Partially observed system · p. 7

• The rate-cost and entropy-cost functions are defined as limiting mutual-information and entropy optima subject to an LQR cost constraint.Source: C. Operational implications · p. 7

• If past channel outputs are unavailable at the encoder, directed mutual information reduces to ordinary mutual information.Source: 1) Control over a noisy channel · p. 7
• The converse results imply lower bounds on the channel capacity necessary to stabilize the system at a target LQR cost.Source: 1) Control over a noisy channel · p. 7
• Equality is attained for scalar Gaussian control over a scalar memoryless AWGN channel, where linear innovation transmission is optimal.Source: 1) Control over a noisy channel · p. 7
• How closely the capacity lower bound can be approached over noisy channels remains open in general.Source: 1) Control over a noisy channel · p. 7

• For a noiseless bit pipe accepting r bits per channel use, both R(b) and H(b) lower-bound the minimum quantization rate required for cost b.Source: 2) Control under fixed-rate quantization · p. 7
• The converse theorems give sharp lower bounds on the minimum size of a fixed-rate quantizer compatible with LQR cost b.Source: 2) Control under fixed-rate quantization · p. 7

Operational rate-cost function

• The minimum encoded length can fall slightly below entropy when the prefix condition is removed.This follows from lifting prefix constraints in lossless compression.Source: 3) Control under variable-length… · p. 8
• The operational rate-cost function Rvar(b) is defined by the asymptotically achievable average rate for target LQR cost b.It uses the limsup of the infimum of rates satisfying the coding constraint and cost requirement.Source: 3) Control under variable-length… · p. 8

Rate-cost characterization

• Rvar(b) ≥ ψ^-1(R(b)), where ψ(x) = x + log2(x + 1) + log2 e.Thus the converse theorems provide lower bounds for variable-length quantization, while the achievability results characterize the operational tradeoff.Source: 3) Control under variable-length… · p. 8
• The converse and achievability results characterize the operational rate-cost tradeoff for variable-length quantizers.The result applies to the variable-length setting considered in this section.Source: 3) Control under variable-length… · p. 8

Prefix-free quantization

• The same theorems characterize the operational rate-cost tradeoff for variable-length prefix-free quantizers.For prefix-free compression, the minimum average length is bounded by entropy plus one bit.Source: 3) Control under variable-length… · p. 8

Causal DPCM

• Causal compression uses the decoder’s predicted state as side information and quantizes the current innovation.The DPCM encoder sends the quantized innovation, and the decoder reconstructs the state recursively.Source: §III · p. 8
• Causal rate- and entropy-distortion functions are attained within DPCM without requiring independent innovations or encoded innovations.The result follows from Proposition 2 and permits dependence across samples.Source: §III · p. 8

Control and compression

• The causal rate-distortion tradeoffs for controlled and uncontrolled Markov processes are the same when past controls are available to encoder and decoder.The processes share the same innovation process, so the same DPCM scheme encodes both.Source: §III · pp. 8–9

Causal Shannon lower bound

• Theorem 9 provides a causal rate-distortion lower bound obtained by combining Shannon’s lower bound with dynamic programming.It is presented as an extension of Shannon’s lower bound to causal compression.Source: §III · p. 9
• For scalar Gauss-Markov sources, equality holds when the expression under the logarithm is at least 1.This recovers the known scalar Gaussian result.Source: §III · p. 9

Achievability

• A DPCM quantizer with uniform rate and distortion allocations approaches the converse in the infinite-horizon limit.Consequently, nonuniform allocations cannot provide significant performance gains in this setting.Source: §III · p. 9

Refined bounds

• When the eigenvalue dynamic range is large, projecting out smaller eigenvalues can improve the bound at medium to large distortion dimensions.Further bounds remain decreasing in distortion even when the noise covariance is singular or supported on a subspace.Source: §III · p. 9

Separated design

• The controller uses an estimate of the encoder’s state estimate to form the control action, while encoder and controller mappings minimize LQR cost.The architecture passes encoded state information through the channel before control is applied.Source: §IV · p. 10

Cost separation

• The LQR cost separates into control, estimation, and communication costs for the partially observed system.The separation is established in Theorem 13 for the system and channel scenario considered.Source: §IV · p. 10
• The minimum achievable LQR cost is expressed through an infimum over admissible control sequences.Equality in the resulting decomposition is not attained in general.Source: §IV · p. 10

Special cases

• Equality in the cost decomposition holds for fully observed systems and Gaussian partially observed systems.In the fully observed case estimation terms vanish; in the Gaussian partially observed case they are control-independent through the Kalman filter.Source: §IV · p. 10
• In fully observed and Gaussian partially observed systems, rate-cost and entropy-cost functions are independent of the control sequence.They are evaluated relative to the unconstrained minimum cost bmin and causal rate- or entropy-distortion functions.Source: §IV · p. 10

Noisy channels

• The separated design also yields a converse for control over noisy channels through a tracking converse.The result follows by combining the controlled and uncontrolled rate-distortion relations.Source: §IV · pp. 10–11

Information-theoretic tools

• The analysis relies on conditional entropy power inequalities to control how side information affects causal compression.In causal compression, each quantized output becomes side information for the next source sample.Source: §V · p. 11

Proof strategy

• The entropy-power scaling argument is exact when the transformation is square or the source is Gaussian.For a general rectangular transformation, Proposition 5 supplies the corresponding inequality.Source: §V · p. 11

Information-theoretic tools

• Weighted and unweighted mean-square distortion-rate functions are equated using linear transformations and data processing.The reduction uses projection onto the image of the weighting matrix.Source: §V · p. 11

Shannon lower bound

• The conditional Shannon lower bound is tight when the source decomposes into side information plus isotropic Gaussian noise.At high rates, Shannon’s lower bound can be approached beyond the Gaussian case, supporting the converse’s achievability analysis.Source: §V · p. 12

Proof strategy

• Theorem 9 is proved by lower-bounding each stage’s distortion with Shannon’s bound, entropy-power inequalities, and a recursive information-rate relation.The resulting recursion is averaged over time to obtain the causal lower bound.Source: §V · p. 12

Lattice quantization

• The lattice covering efficiency ρC measures how closely Voronoi cells approximate balls, with smaller values indicating better covering.By definition, ρC ≥ 1, and values closer to 1 correspond to more sphere-like cells.Source: §VI · p. 14

Proof strategy

• The proof analyzes a DPCM scheme with codebook generation, encoder and decoder operations, and performance analysis.The scheme uses lattice quantization and transmits quantized innovations rather than the full state.Source: §VI · p. 14

Encoder and decoder

• At each time, the encoder recursively computes the state innovation from the observed state and decoder estimate, then transmits its lattice-cell index.The decoder recovers the identified cell and forms its state estimate from the quantized innovation.Source: §VI · p. 14

Theorem applications

• Theorem 2 follows by bounding the entropy-distortion function of the fully observed process, while Theorem 6 uses the conditional entropy-distortion function of Kalman-filter estimates.Both achievability results invoke Theorem 10 through the corresponding corollaries.Source: §VI · p. 15

VII. CONCLUSION

• Theorems 1, 3, 4, 5, 7, and 8 provide sharp rate-cost lower bounds for fully and partially observed control systems.The rate-cost function characterizes communication requirements compatible with attainable quadratic cost.Source: §VII · p. 15
• At high rate and low cost, variable-rate lattice quantization approaches the converse by transmitting only the quantized state innovation.The achievability results are given for both fully and partially observed systems.Source: §VII · p. 15
• Via the separation principle, the same converse and matching achievability extend to causal compression of Markov sources.The converse is presented as a causal counterpart of Shannon’s lower bound.Source: §VII · p. 15
• Extending the partially observed analysis to non-Gaussian noises remains open.The paper also leaves open whether the converse can be approached by fixed-rate quantization or noisy-channel control.Source: §VII · p. 15

APPENDIX

• The appendix begins with an entropy-continuity tool that bounds differential-entropy differences when two random-vector distributions are close.This tool is used alongside regularity and Wasserstein estimates in the proof machinery.Source: APPENDIX · p. 17
• The Wasserstein distance is defined by an infimum over joint distributions with fixed marginals, and lattice quantizers provide an upper bound on output entropy for regular densities.These results supply key ingredients for bounding quantization entropy and distortion.Source: APPENDIX · p. 17
• Regular densities are preserved under adding an independent bounded random vector, enabling regularity control for random vectors arising during system operation.Proposition 9 changes the regularity parameters from (c0, c1) to (c0 + c1b, c1).Source: APPENDIX · p. 17
• The leading term of the achievability bound is Shannon’s lower bound, while remaining terms become negligible as dimension grows and distortion decreases.The appendix attributes this behavior to the asymptotics of the correction terms and lattice covering efficiency.Source: APPENDIX · p. 17
• Rogers’ result supplies lattices whose covering-efficiency contribution is logarithmic in dimension, supporting the high-dimensional negligibility of correction terms.Low-dimensional constants can instead be evaluated using known thin lattice coverings such as A_n^*.Source: APPENDIX · p. 17