While information theory has been introduced to characterize the fundamental limitations of control and filtering for a few decades, the existing information-theoretic methods are indirect and cumbersome for analyzing the limitations of continuous-time systems. To answer this challenge, we lift the information-theoretic analysis to continuous function spaces by the I-MMSE relationships. Continuous-time control and filtering systems are modeled as the additive Gaussian channels with and without feedback, and the total information rate is identified as a control and filtering trade-off metric and calculated from the estimation error of channel inputs. Fundamental constraints for this trade-off metric are first derived in a general setup and then used to capture the limitations of various control and filtering systems subject to linear and nonlinear plant models. For linear scenarios, we show that the total information rate quantifies the performance limits, such as the minimum entropy cost and the lowest achievable mean-square estimation error, in the time domain. For nonlinear systems, we provide a direct method to calculate and interpret the total information rate and its lower bound by the Stratonovich-Kushner equation. (c) 2026 Published by Elsevier Ltd.

Fundamental limitations or performance trade-offs/limits are important properties and constraints of control and filtering systems. Among various trade-off metrics, total information rate, which characterizes the sensitivity trade-offs and average performance of control and filtering systems, is conventionally studied by using the (differential) entropy rate and Kolmogorov-Bode formula. In this paper, by extending the famous I-MMSE (mutual information -- minimum mean-square error) relationship to the discrete-time additive white Gaussian channels with and without feedback, a new paradigm is introduced to estimate and analyze total information rate as a control and filtering trade-off metric. Under this framework, we enrich the trade-off properties of total information rate for a variety of discrete-time control and filtering systems, e.g., LTI, LTV, and nonlinear, and also provide an alternative approach to investigate total information rate via optimal estimation.

While information theory has been introduced to investigate and characterize the control and filtering limitations for a few decades, the existing information-theoretic methods are indirect and cumbersome for analyzing the fundamental limitations of continuous-time systems. To answer this challenge, we lift the information-theoretic analysis to continuous function spaces of infinite dimensions by using Duncan’s theorem or the I-MMSE relationships. Continuous-time control and filtering systems are modeled as an additive Gaussian channel with or without feedback, and total information rate is identified as a control and filtering trade-off metric and directly computed from the estimation error of channel input. Inequality constraints for the trade-off metric are derived in a general setting and then applied to capture the fundamental limitations of various control and filtering systems subject to linear and nonlinear plants. For the linear systems, we show that total information rate has similar properties as some established trade-offs, e.g., Bode-type integrals and minimum estimation error. For the nonlinear systems, we provide a direct method to compute the total information rate and its lower bound by the Stratonovich-Kushner equation.

A simplified analysis is performed on the Bode-type filtering sensitivity trade-off integrals, which capture the sensitivity characteristics of the estimate and estimation error with respect to the process input and estimated signal in continuous- and discrete-time linear time-invariant filtering systems. Compared with the previous analyses based on complex analysis and Cauchy's residue theorem, the analysis results derived from the simplified method are more explicit, thorough, and require less restrictive assumptions. For continuous-time filtering systems, our simplified analysis reveals that apart from the non-minimum phase zero sets reported in the previous literature, the value and boundedness of filtering sensitivity integrals are also determined by the leading coefficients, relative degrees, minimum phase zeros, and poles of plants and filters. By invoking the simplified method, a comprehensive analysis on the discrete-time filtering sensitivity integrals is conducted for the first time. Numerical examples are provided to verify the validity and correctness of the simplified analysis.

This paper introduces the f-divergence variational inference (f-VI) that generalizes variational inference to all f-divergences. Initiated from minimizing a crafty surrogate f-divergence that shares the statistical consistency with the f-divergence, the f-VI framework not only unifies a number of existing VI methods, e.g. Kullback-Leibler VI, Rényi's α-VI, and χ-VI, but offers a standardized toolkit for VI subject to arbitrary divergences from f-divergence family. A general f-variational bound is derived and provides a sandwich estimate of marginal likelihood (or evidence). The development of the f-VI unfolds with a stochastic optimization scheme that utilizes the reparameterization trick, importance weighting and Monte Carlo approximation; a mean-field approximation scheme that generalizes the well-known coordinate ascent variational inference (CAVI) is also proposed for f-VI. Empirical examples, including variational autoencoders and Bayesian neural networks, are provided to demonstrate the effectiveness and the wide applicability of f-VI.

A simplified analysis is performed on the Bode -type filtering sensitivity trade-off integrals, which capture the sensitivity characteristics of the estimate and estimation error with respect to the process input or estimated signal in continuous- and discrete-time linear time-invariant filtering systems. Compared with the previous analyses based on complex analysis and Cauchy's residue theorem, the analysis results derived from the simplified method are more explicit, thorough, and require less restrictive assumptions. For continuous-time filtering systems, our simplified analysis reveals that apart from the non-minimum phase zero sets reported in the previous literature, the value and boundedness of filtering sensitivity integrals are also determined by the leading coefficients, relative degrees, minimum phase zeros, and poles of plants and filters. By invoking the simplified method, a comprehensive analysis of the discrete-time filtering sensitivity integrals is conducted for the first time. Numerical examples are provided to verify the validity and correctness of the simplified analysis.

Sensitivity of linear continuous-time SISO feedback systems, subject to control and measurement noise, is analyzed by deriving the lower bounds of Bode-like integrals via an information-theoretic approach. Bode integrals of four different sensitivity-like functions are employed to gauge the performance limitations of feedback systems. When the signals of the control system are stationary Gaussian, these four different Bode-like integrals can be represented as the differences between mutual information rates. These mutual information rates and hence the corresponding Bode-like integrals are proven to be bounded below by the unstable poles and zeros of the plant model, if the signals of the control system are wide-sense stationary.

Bode integrals of sensitivity and sensitivity-like functions along with complementary sensitivity and complementary sensitivity-like functions are conventionally used for describing performance limitations of a feedback control system. In this paper, we show that in the case when the disturbance is a wide sense stationary process the (complementary) sensitivity Bode integral and the (complementary) sensitivity-like Bode integral are identical. A lower bound of the continuous-time complementary sensitivity-like Bode integral is also derived and examined with the linearized flight-path angle tracking control problem of an F-16 aircraft.

SUMMARYFor a class of linear dynamical systems with constant unknown parameters, an adaptive control scheme is developed that provides stable adaptation in the presence of input magnitude constraints. Whereas for open‐loop stable systems the results are global, for open‐loop unstable systems, the problem of nonconservative estimation of the nonempty positive invariant set is cast into an LMI framework, which can be efficiently solved numerically via convex optimization. To achieve this, a standard result toward invariant set characterization is appropriately extended to accommodate bounded disturbance and model uncertainties. In addition to closed‐loop stability, performance bounds of the adaptive closed‐loop system are analyzed, and the degradation due to the possible control deficiency is quantified. Simulation examples of aerospace applications are included to illustrate the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

This paper analyzes causal closed-loop continuous-time systems in the presence of limited information. Assuming that the exogenous signals can be modeled as a stochastic process, a mutual information rate inequality is obtained that can be viewed as an extended Bode-type formula for stationary processes. The tightness of the resulting Bode's integral inequality is then analyzed for the linear time invariant closed loops. Within the developed framework we consider the control-communication interplay and analyze the underlying fundamental limitations.

In this paper, we establish a Bode sensitivity integral formula for a class of feedback closed-loop systems with stochastic switched plants and controllers. Using information theory, we study the information conservation law, based on which a log integral theorem is obtained for the closed loops of interest. Furthermore we develop several algebraic conditions to explicitly capture the performance limitations. Application of this theoretical framework to Networked Control Systems (NCS) is used as an illustrative example.

This paper addresses an optimal state estimation problem in the presence of limited communication and noiseless feedback. In this setup, the state dynamics is estimated via an additive white Gaussian channel with input power constraint. We present a new communication and estimation strategy based on Kalman-Bucy filtering theory and water filling optimization algorithm. The optimality is established with respect to the minimal mean-square estimation error. As an example, we propose an analogue amplitude modulation scheme for state-estimation of a linear planar dynamics.

We address the noise attenuation problem for linear time-invariant (LTI) systems in the presence of Gaussian communication channels. A novel Linear Matrix Inequality (LMI) approach is formulated to solve the problem. A simulation example is discussed to verify the proposed solution.

This paper derives robust stability margin of an L1 adaptive controller in the gap metric. For that purpose, first a classical result on robustness of nonlinear input-output systems is reviewed, and an appropriate extension on biased-gain stability is introduced. This extension circumvents the singularity inherent to the analysis of robustness of adaptive systems, and leads to computable robust stability margin of L1 adaptive controller. A time-delay margin of the L1 controller is computed as an illustration of the theoretical results.

This paper considers both synthesis and analysis problems for continuous-time stochastic systems in the presence of limited information. First, a novel linear matrix inequality (LMI) optimization problem is formulated to obtain a linear quadratic controller over an additive white-noise channel subject to a power constraint. We next consider a more general setting and, using tools from information theory, derive an extended Bode-type formula to analyze fundamental performance limitations.

This paper presents a proof of stability and performance bounds of the L1 adaptive control architecture in the presence of input constraints. We prove that by appropriate modification of the state predictor, which is used for the definition of the error signal in the adaptive laws, the stability and the performance bounds of the L1 adaptive controller can be quantified within an appropriately defined domain of attraction. While for open-loop stable system the results are global, for open-loop unstable systems the domain of attraction is shown to depend upon the conservative knowledge of the uncertain parameters, the choice of the filter and the amplitude constraint of the actuator. The performance bounds can be systematically improved by increasing the rate of adaptation. Simulations verify the theoretical findings.

This paper presents a convex optimization method for the feedback-loop tradeoff of L1 adaptive controller. Both problems of performance improvement and time-delay margin maximization are shown to be cast into Linear Matrix Inequality (LMI) type conditions. First, each of these conditions is studied separately towards a distinct objective, and next two similar LMI algorithms are proposed for optimization of one of the objectives with a prespecified constraint on the other.

Recent papers have introduced a new paradigm for design of adaptive controllers that enables a priori prediction of performance bounds and also analytical quantification of the time-delay margin of the closedloop nonlinear system. In this paper, we revisit the main architecture from Refs. and consider several filter design methods for maximizing the time-delay margin, while retaining the same performance bounds.