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Consider that an autonomous linear time-invariant (LTI) plant is given and that each member of a network of LTI observers accesses a portion of the output of the plant. The dissemination of information within the network is dictated by a pre-specified directed graph in which each vertex represents an observer. This paper proposes a distributed estimation scheme that is a natural generalization of consensus in which each observer computes its own state estimate of the plant using only the portion of the output vector accessible to it and the state estimates of other observers that, according to the graph, are available to it. Unlike straightforward high-order solutions in which each observer broadcasts its measurements throughout the network, the average dimension of the state of each observer in the proposed scheme does not exceed the order of the plant plus one. We determine necessary and sufficient conditions for the existence of a parameter choice for which the proposed scheme attains asymptotic omniscience of the state of the plant at all observers. The conditions reduce to certain detectability requirements that imply that if omniscience is not possible under the proposed scheme then it is not viable under any other scheme-including higher-order LTI, nonlinear, and time-varying ones-subject to the same graph.

Consider a discrete-time remote estimation system formed by an encoder, a transmission policy, a channel, and a remote estimator. The encoder assesses a random process that the remote estimator seeks to estimate based on information sent to it by the encoder via the channel. The channel is affected by Bernoulli drops. The instantaneous probability of a drop is governed by a finite state machine (FSM). The state of the FSM is denoted as the channel state. At each time step, the encoder decides whether to attempt a transmission through the packet-drop link. The sequence of transmission decisions is the input to the FSM. This paper seeks to design an encoder, transmission policy and remote estimator that minimize a finite-horizon mean squared error cost. We present two structural results. The first result in which we assume that the process to be estimated is white and Gaussian, we show that there is an optimal transmission policy governed by a threshold on the estimation error. The second result characterizes optimal symmetric transmission policies for the case when the measured process is the state of a scalar linear time-invariant plant driven by white Gaussian noise. Use-dependent packet-drop channels can be used to quantify the effect of transmission on channel quality when the channel is powered by energy harvesting. In the expanded version of this paper, an additional application to a mixed initiative system in which a human operator performs visual search tasks is presented.

Consider that a linear time-invariant (LTI) plant is given and that we wish to design a stabilizing controller for it. Admissible controllers are LTI and must belong to a pre-selected subspace that may impose structural restrictions, such as sparsity constraints. The subspace is assumed to be quadratically invariant (QI) with respect to the plant, which, from prior results, guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided. This paper addresses the previously unsolved problem of extending Youla's classical parametrization so that it admits QI subspace constraints on the controller. In contrast with prior parametrizations, the one proposed here does not require initialization and it does not require the existence of a stable stabilizing controller. The main idea is to cast the stabilizability constraint as an exact model-matching problem with stability restrictions, which can be tackled using existing methods. Furthermore, we show that, when it exists, the solution of the exact model-matching problem can be used to compute an admissible stabilizing controller. Applications of the proposed parametrization on the design of norm-optimal controllers via convex methods are also explored. An illustrative example is provided, and a special case is discussed for which the exact model matching problem has a unique and easily computable solution.

Quadratic invariance is a condition which has been shown to allow for optimal decentralized control problems to be cast as convex optimization problems. The condition relates the constraints that the decentralization imposes on the controller to the structure of the plant. In this technical note, we consider the problem of finding the closest subset and superset of the decentralization constraint which are quadratically invariant when the original problem is not. We show that this can itself be cast as a convex problem for the case where the controller is subject to delay constraints between subsystems, but that this fails when we only consider sparsity constraints on the controller. For that case, we develop an algorithm that finds the closest superset in a fixed number of steps, and discuss methods of finding a close subset.

Consider a first order linear time-invariant discrete time system driven by process noise, a pre-processor that accepts causal measurements of the state of the system, and a state estimator. The pre-processor and the state estimator are not co-located, and, at every time-step, the pre-processor transmits either a real number or a free symbol to the estimator. We seek the pre-processor and the estimator that jointly minimize a cost that combines two terms; the expected squared state estimation error and a communication cost. In our formulation, the transmission of a real number from the pre-processor to the estimator incurs a positive cost while free symbols induce zero cost. This paper proves analytically that a symmetric threshold policy at the pre-processor and a Kalman-like filter at the estimator, which updates its estimate linearly in the presence of free symbols, are jointly optimal for our problem.

In this paper, we investigate control strategies for a scalar, one-step delay system in discrete-time, i.e., the state of the system is the input delayed by one time unit. In contrast with classical approaches, here the control action must be a memoryless function of the output of the plant, which comprises the current state corrupted by measurement noise. We adopt a first order state-space representation for the delay system, where the initial state is a Gaussian random variable. In addition, we assume that the measurement noise is drawn from a white and Gaussian process with zero mean and constant variance. Performance evaluation is carried out via a finite-time quadratic cost that combines the second moment of the control signal, and the second moment of the difference between the initial state and the state at the final time. We show that if the time-horizon is one or two then the optimal control is a linear function of the plant's output, while for a sufficiently large horizon a control taking on only two values will outperform the optimal affine solution. This paper complements the well-known counterexample by Hans Witsenhausen, which showed that the solution to a linear, quadratic and Gaussian optimal control paradigm might be nonlinear. Witsenhausen's counterexample considered an optimization horizon with two time-steps (two stage control). In contrast with Witsenhausen's work, the solution to our counterexample is linear for one and two stages but it becomes nonlinear as the number of stages is increased. The fact that our paradigm leads to nonlinear solutions, in the multi-stage case, could not be predicted from prior results. In contrast to prior work, the validity of our counterexample is based on analytical proof methods. Our proof technique rests on a simple nonlinear strategy that is useful in its own right, since it outperforms any affine solution.

Consider a discrete-time networked control system, in which the controller has direct access to noisy measurements of the output of the plant. However, information flows from the controller to the actuator via a channel that features Bernoulli erasure events. If an erasure occurs, the channel outputs an erasure symbol; otherwise, it transmits a real finite-dimensional vector. We determine necessary and sufficient conditions for the stabilizability of an unstable linear time-invariant finite-dimensional plant. Given a minimal state-space representation for the plant, the necessary and sufficient conditions for stabilizability are expressed in terms of the probability of erasure at the channel and of the spectral radius of the one-step state transition matrix. There are two main results in the technical note. The first result shows that if the actuator has processing capabilities, then the necessary and sufficient conditions for stabilizability remain unchanged with or without acknowledgements from the actuator to the controller. The second result shows that the stabilizability conditions are identical for two types of actuators: (Type I) Processing at the actuator has access to the plant's model; (Type II) Processing at the actuator uses a universal algorithm that does not depend on the model of the plant. Thus, neither the knowledge of the model of the plant at the actuator, nor the presence of acknowledgements from the actuator to the controller, can be used to alter or relax the necessary and sufficient conditions for stabilizability.

Consider two identical discrete-time, finite-dimensional, linear and time-invariant systems denoted as the leader and the follower. The leader system is driven by a deterministic control input and by a zero-mean white Gaussian process noise. In this paper, we address the problem of designing a networked control scheme for the follower system, which guarantees that the state of the follower system tracks the state of the leader system optimally, according to a mean-squared cost. We adopt a networked control architecture featuring two erasure links and three design blocks: a controller acting on the follower system and two encoders, one at the output of each system. The controller has remote access to noisy measurements of the state of both systems via two erasure links that are used to connect each encoder to the controller. We consider erasure links whose erasure events occur according to a Bernoulli process. If an erasure occurs, then no information is transmitted; otherwise, a finite vector of real numbers is conveyed through the link. The purpose of the encoders is to process noisy measurements of the output of each system prior to transmission over the corresponding erasure link. While the encoder of the follower system has access to current measurements, the encoder of the leader system has access to measurements that are advanced in time by a finite time interval (also denoted as preview). This paper describes a methodology for the design of controller and encoders that are jointly optimal, with respect to optimal tracking of the leader system by the follower system. We also obtain explicit necessary and sufficient conditions for the existence of a scheme that guarantees that the tracking error has finite second moment. Copyright © 2008 John Wiley & Sons, Ltd.

In this technical note, we investigate the design of controllers, for discrete-time Markovian jump linear systems, that achieve optimal reference tracking in the presence of preview (reference look-ahead). For a quadratic cost and given a reference sequence, we obtain the optimal solution for the full information case. The optimal control policy consists of the additive contribution of two terms: a feedforward term and a feedback term. We show that the feedback term is identical to the standard optimal linear quadratic regulator for Markovian jump linear systems. We provide explicit formulas for computing the feedforward term, including an analysis of convergence.

Consider a discrete-time, linear time-invariant process, two sensors and one controller. The process state is observed in the presence of noise by the sensors, which are connected to the controller via links that feature erasure. If a link transmits successfully then a finite-dimensional vector of real numbers is conveyed from the sensor to the controller. If an erasure event occurs, then any information conveyed over the link is lost. This paper addresses the problem of designing the maps that specify the processing at the controller and at the sensors to minimize a quadratic cost function. When the information is lost over the links either in an independent and identically distributed (i.i.d.) or in a (time-homogeneous) Markovian fashion, we derive necessary and sufficient conditions for the existence of maps such that the process is stabilized in the bounded second moment sense. We also solve the optimal design problem in the presence of delayed noiseless acknowledgment signals at the sensors from the controller for an arbitrary packet drop pattern. We provide explicit recursive schemes to implement our solution. We also indicate how our approach can be extended to situations when more than two sensors are available and when the sensors can cooperate. The analysis also carries over to the case when each point-to-point erasure link connecting the sensors and the controller is replaced by a network of erasure links.

This paper addresses fundamental limitations of feedback using information theoretic conservation laws and flux arguments. The paper has two parts. In the first part, we derive a conservation law dictating that causal feedback cannot reduce the differential entropy inserted in the loop by external sources. An interpretation of this result is that the total randomness induced by disturbances, as measured by differential entropy, cannot be reduced by causal feedback; it can only be re-allocated in time or in frequency (if well defined). Under asymptotic stationarity assumptions, this result has a spectral representation which constitutes an extension of Bode's inequality for arbitrary feedback. Our proofs make clear the role of causality, as well as how stability assumptions impact the final result. In the second part, we derive an inequality unveiling that the feedback loop must be able to convey information originating from two independent sources: 1) initial states of the physical plant; 2) exogenous disturbance signals. By using such principle, we construct a variety of information rate (information flux) inequalities. Furthermore, we derive a universal performance bound which is parameterized solely by the feedback capacity and the parameters of the plant. The latter is a new fundamental limitation, which is different from Bode's classical result, indicating that finite feedback capacity brings a new type of performance bound.

A causal feedback map, taking sequences of measurements and producing sequences of controls, is denoted as finite-set if its range is in a finite set. Bit-rate constrained or digital control are particular cases of finite-set feedback. In this paper, we show that the finite gain (FG) lp stabilization, with 1 ≤ p ≤∞ , of a discrete-time, linear and time-invariant unstable plant is impossible by finite-set feedback. In addition, we show that, under finiteset feedback, weaker (local) versions of FG lp stability are also impossible. These facts are not obvious, since recent results have shown that input to state stabilization (ISS) is viable by bit-rate constrained control. In view of such existing work, this paper leads to two conclusions: (1) in spite of ISS stability being attainable under finite-set feedback, small changes in the amplitude of the external excitation may cause, in relative terms, a large increase in the amplitude of the state (2) FG lp stabilization requires logarithmic precision around zero. Since our conclusions hold with no assumption on the feedback structure, they cannot be derived from existing results. We adopt an information theoretic viewpoint, which also brings new insights into the problem of stabilization.

We study the stabilizability of uncertain stochastic systems in the presence of finite capacity feedback. Motivated by the structure of communication networks, we consider a variable rate digital link. Such link is used to transmit state measurements between the plant and the controller. We derive necessary and sufficient conditions for internal and external stabilizability of the feedback loop. In accordance with previous publications, stabilizability of unstable plants is possible if and only if the link's average transmission rate is above a positive critical value. In addition, stability in the presence of uncertainty in the plant is analyzed using a small-gain argument. We also show that robustness can be increased at the expense of a higher transmission rate.

In this paper, we address the stabilizability of uncertain systems in the presence of finite data-rate feedback. In our formulation, a finite data-rate link is used to transmit state measurements between the plant and the controller. We determine sufficient conditions for internal and external stabilizability of the feedback loop. Stability in the presence of uncertainty in the plant is analyzed using a small-gain argument. We show that larger uncertainty can be tolerated at the expense of higher transmission rates. A comparison is established between the sufficient conditions for robust stabilizability and the necessary conditions for the stability of the nominal model. We also suggest how our results can be used to determine the stabilizability of a class of non-linear systems.

Consider that an autonomous LTI plant is given and that a network of local observers assesses the output vector of the plant. Each local observer in the network is LTI and measures a portion of the output vector. Each local observer computes a state estimate using its measurements and the state estimates of other local observers shared according to a preselected neighborhood structure. This paper proposes an update rule that is a natural generalization of consensus, and for which we determine necessary and sufficient conditions for the existence of parameters that lead to asymptotic omniscience of the state of the plant at all local observers.

Control of miniature mobile robots in unconstrained environments is an ongoing challenge. Miniature robots often exhibit nonlinear dynamics and obstacle avoidance introduces sigificant complexity in the control problem. Furthermore, miniature robots have strict power and size constraints, drastically reducing on-board processing power and severely limiting the capability of digital implementations of nonlinear model predictive controllers. To accommodate the demands of this application area, we describe the architecture of a mixed-signal mobile robot control system using randomized receding horizon control. We compare the proposed mixed-signal implementation with purely digital control systems in terms of power requirements and precision and find that the mixed-signal implementation offers significant reductions in power consumption at an acceptable loss of precision.

This paper studies a network of observers for a distributed estimation problem, where each observer assesses a portion of output of a given LTI system. The goal of each observer is to compute a state estimate that asymptotically converges to the state of the LTI system. We consider there is a sparsity constraint that restricts interconnections between observers. We provide a sufficient condition for the existence of parameters for the observers which achieve the convergence of the state estimates to the state of the LTI system. In particular, this condition can be written in terms of the eigenvalues of the Laplacian matrix of the underlying communication graph and the spectral radius of the dynamic matrix of the LTI system.

In this paper we deal with the problem of stabilizing linear, time-invariant plants using feedback control configurations that are subject to sparsity constraints. Recent results show that given a strongly stabilizable plant, the class of all stabilizing controllers that satisfy certain given sparsity constraints admits a convex representation via Zames's Q-parametrization. More precisely, if the pre-specified sparsity constraints imposed on the controller are quadratically invariant with respect to the plant, then such a convex representation is guaranteed to exist. The most useful feature of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Q-parameter, which makes this approach suitable for optimal controller design (in the ℋ2 sense) using numerical tools readily available from the classical, centralized optimal ℋ2 synthesis. All these procedures rely crucially on the fact that some stabilizing controller that verifies the imposed sparsity constraints is a priori known, while design procedures for such a controller to initialize the aforementioned optimization schemes are not yet available. This paper provides necessary and sufficient conditions for such a plant to be stabilizable with a controller having the given sparsity pattern. These conditions are formulated in terms of the existence of a doubly coprime factorization of the plant with additional sparsity constraints on certain factors. We show that the computation of such a factorization is equivalent to solving an exact model-matching problem. We also give the parametrization of the set of all decentralized stabilizing controllers by imposing additional constraints on the Youla parameter. These constraints are for the Youla parameter to lie in the set of all stable transfer function matrices belonging to a certain linear subspace.

In this paper we deal with the problem of decentralized stabilization for linear and time-invariant plants in feedback control configurations that are subject to sparsity constraints. Recent theoretical advances in decentralized control have proved that the class of stabilizing controllers, satisfying a given sparsity constraint admits a convex representation of the Youla-type, provided that the sparsity constraints imposed on the controller are quadratically invariant with respect to the plant. The most useful feature of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the free parameter, which makes this approach suitable for optimal controller design methods based on convex optimization, numerical algorithms. All these procedures rely indispensably on the fact that some decentralized, stabilizing controller is a priori known, while design procedures for such a decentralized controller to initialize the aforementioned optimization schemes are not yet available. This paper provides necessary and sufficient conditions for such a plant to be stabilizable with a decentralized controller. These conditions are given in terms of the existence of a special type of doubly coprime factorization of the plant, which we call the input/output decoupled, doubly coprime factorization. More importantly, the set of all decentralized stabilizing controllers is characterized via the Youla parametrization. The sparsity constraints on the controller are also recast as convex constraints on the Youla parameter.

This paper deals with linear systems and the disturbance attenuation problem of minimizing a given operatorial norm of the lower linear fractional transformation between a generalized plant and a controller. We show that the optimal gain attainable by causal feedback does not depend on linear, stable, additive plant perturbations, irrespective of the chosen norm. The result proves applicable to an important class of decentralized control configurations. Applications to reducing the computational effort for the optimal controller synthesis are also discussed.

This paper addresses the design of controllers, subject to sparsity constraints, for linear and time-invariant plants. Prior results have shown that a class of stabilizing controllers, satisfying a given sparsity constraint, admits a convex representation of the Youla-type, provided that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that the plant is strongly stabilizable. Another important aspect of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Youla parameter, which makes this approach suitable for optimization using norm-based costs. In this paper, we extend these previous results to the general case of possibly non-strongly stabilizable plants. Our extension is conveyed in terms of a parametrization for the class of controllers that is very similar to the Youla parametrization. In our extension, under quadratically invariant constraints, the controller class also admits a representation where the free parameter is subject to only convex constraints. While the strong stabilizability assumption has been removed our result yields the same elegant simplicity from the strongly stabilizable case.

Consider two identical discrete-time, finite-dimensional, linear and time-invariant systems denoted as leader and follower. The leader system is driven by a deterministic control input and by a zero-mean white Gaussian process noise. In this paper, we address the problem of designing a networked control scheme for the follower system that guarantees that the state of the follower system tracks the state of the leader system optimally, according to a mean squared cost. We adopt a networked control scheme featuring two erasure links and three design blocks: a controller acting on the follower system and two encoders, one at the output of each system. The controller has remote access to noisy measurements of the state of both systems, via two erasure links that are used to connect each encoder to the controller. We consider erasure links whose erasure events occur according to a Bernoulli process. If an erasure occurs then no information is transmitted, otherwise a finite vector of real numbers is conveyed through the link. The purpose of the encoders is to process noisy measurements of the output of each system prior to transmission over the corresponding erasure link. While the encoder of the follower system has access to current measurements, the encoder of the leader system has access to measurements that are advanced in time by a finite time interval (also denoted as preview). This paper describes a methodology for the design of controller and encoders that are jointly optimal, with respect to optimal tracking of the leader system by the follower system. We also obtain explicit necessary and sufficient conditions for the existence of a scheme that guarantees that the tracking error has finite second moment.

In this paper, we investigate an optimal state estimation problem for Markovian Jump Linear Systems. We consider that the state has two components: the first component of the state is finite valued and is denoted as mode, while the second (continuous) component is in a finite dimensional Euclidean space. The continuous state is driven by a deterministic control input and a zero mean, white and Gaussian process noise. The observable output has two components: the first is the mode delayed by a fixed amount and the second is a linear combination of the continuous state observed in zero mean white Gaussian noise. Our paradigm is to design optimal estimators for the current state, given the current output observation. We provide a solution to this paradigm by giving a recursive estimator of the continuous state, in the minimum mean square sense, and a finitely parameterized recursive scheme for computing the probability mass function of the current mode conditional on the observed output. We show that the optimal estimator is nonlinear on the observed output and on the control input. In addition, we show that the computation complexity of our recursive schemes is polynomial in the number of modes and exponential in the mode observation delay.

The purpose of this paper is to develop an optimal controller design framework for a class of linear time invariant (LTI) networked control systems with stochastic intertrans-mission intervals and random scheduling protocols perturbed by stochastic noise processes. Whereas the authors' previous works examine sufficient conditions that lead to robust (Lp, ISS) notions of stability ex post facto controller design ignoring the network, this paper examines controller design taking the network into account in an appropriate sense. In particular, we assume zero-order hold between transmission instants and our optimality criteria apply to the discrete-time system induced by examining the NCS only at transmission instants. We do, however, develop performance bounds between such instants (where no claim of optimality is made).

In this paper, we investigate the design of optimal state estimators for Markovian jump linear systems. We consider that the state has two components: the first component is finite valued and is denoted as mode, while the second (continuous) component is in a finite dimensional Euclidean space. The continuous state is driven by a zero mean, white and Gaussian process noise. The observation output has two components: the first is the mode and the second is a linear combination of the continuous state observed and zero mean, white Gaussian noise. Both output components are affected by delays, not necessarily equal. Our paradigm is to design optimal estimators for the current state, given the current output observation. We provide a solution to this paradigm by giving a recursive estimator for the continuous state, in the minimum mean square sense, and a finitely parameterized recursive scheme for computing the probability mass function of the current mode conditioned on the observed output. We show that when the mode is observed with a greater delay then the continuous output component, the optimal estimator nonlinear in the observed outputs.

In this paper, we investigate control strategies for a scalar, one-step delay system in discrete time, i.e., the system?s state is the input delayed by one time unit. We allow control policies that are memoryless functions of noisy measurements of the state of the delay system. We adopt a first order state-space representation for the delay system, where the initial state of the system is a Gaussian and zero mean random variable. In addition, we assume that the measurement noise is drawn from a white and Gaussian process with zero mean and constant variance. Performance evaluation is carried out via a finite-time quadratic cost that combines the second moment of the control signal, and the second moment of the difference between the initial state and the state at the final time. We show that if the time-horizon is one or two then the optimal control is a linear function of the plant?s output, while for a sufficiently large horizon a control taking only two values will outperform the optimal linear solution. This paper complements the well known counterexample by Hans Witsenhausen, which showed that the solution to a linear, quadratic and Gaussian optimal control paradigm might be nonlinear. Witsenhausen?s counterexample considered an optimization horizon with two time-steps (two stage control). In contrast to Witsenhausen?s work, the solution to our counterexample is linear for one and two stages but it becomes nonlinear as the number of stages is increased. Existing tests for linearity of the optimal memoryless control consider only the two-stage problem. The fact that our paradigm leads to non-linear solutions, in the multi-stage case, could not be predicted from prior results. In particular, the fact that the optimal solution for the two stage problem is linear, but the multiple stage might not be, also shows that dynamic programming principles cannot be used for our paradigm. Our paper provides analytical proofs which hold for any number of stages.

Consider a discrete-time networked control scheme, in which the controller has direct access to noisy measurements of the plant?s output, but the controller and the actuator are connected via a link that features Bernoulli erasure events. We determine necessary and sufficient conditions for the stabilizability of an unstable linear and time-invariant plant. We show that these conditions are identical for two types of actuators: (1) (Type I) Processing at the actuator has access to the plant?s model; (2) (Type II) Processing at the actuator uses a universal algorithm that does not depend on the model of the plant. We also identify cases where availability of acknowledgements over the controller-actuator channel is not required for stability. We also consider decentralized networked control structures, where two or more independent controller-link-actuator assemblies have access to different measurements of the plant?s output.

Consider a discrete-time, linear time-invariant process, two sensors and one controller. The process is observed by the sensors, which are connected to the controller via links that can be modeled as erasure channels. If a link transmits successfully then a finite-dimensional vector of real numbers is conveyed from the sensor to the controller. If an erasure event occurs, then any information conveyed over the link is lost. This paper addresses the problem of designing the maps that specify the processing at the controller and at the sensors for stabilizing the process in the bounded second moment sense. When the information is lost over the links either in an independent and identically distributed (i.i.d.) or Markovian fashion over time, we derive necessary and sufficient conditions for the existence of maps such that the process is stabilized. Such conditions are expressed as inequalities involving the parameters of the plant and the probabilities of link fading, and provide the least conservative stabilization conditions. We also indicate how our approach can be used if more than two sensors are available, if the sensors can cooperate and if the acknowledgment signals are also transmitted over erasure channels. The analysis also carries over to the case when the single channels are replaced by networks of erasure channels.

Consider a discrete-time linear time-invariant process being observed by two sensors, which are connected to an observer via links that can be modeled as erasure channels. If a link transmits successfully then a finite-dimensional vector of real numbers is conveyed from the sensor to the controller. If an erasure event occurs, then any information conveyed over the link is lost. This paper addresses the problem of designing the maps that specify the processing at the sensors and the observer to calculate the minimum mean square error estimate of the state of the process. We derive necessary and sufficient conditions for the existence of maps such that the estimate error is bounded in the second moment sense.

We study the stabilizability of uncertain stochastic systems in the presence of finite capacity feedback. Motivated by the structure of communication networks, we consider a stochastic digital link that sends words whose size is governed by a random process. Such link is used to transmit state measurements between the plant and the controller. We extend previous results by deriving sufficient conditions for internal and external stabilizability of multi-state linear and time-invariant plants. In accordance with previous publications, stabilizability of unstable plants is possible if the link’s average transmission rate is above a positive critical value. In our formulation the plant and the link can be stochastic. In addition, stability in the presence of uncertainty in the plant is studied using a small-gain argument.

In this paper, we consider a two-step decentralized control problem with two agents which are not co-located. The first agent is located at the source of a disturbance, it can measure the disturbance with high accuracy and it can perform local actuation, while communicating with the second agent via a Gaussian channel with a prescribed signal to noise ratio. What remains of the disturbance, after the control action of the first agent, will hit the second agent after a fixed delay. The second agent has noisy measurements as well as the information conveyed through the communication channel to decide on the best strategy to reduce the effects of the remaining disturbance. The control design paradigm is to minimize a quadratic cost combining the control power of the first agent and the final effect of the disturbance at the second agent. Such a framework can be viewed as an extension of the well known Witsenhausen's counter-example which did not include a communication channel. In the absence of a channel, Hans Witsenhausen has shown, by means of an example, that nonlinear strategies outperform the best linear strategy. This result is not obvious because the framework is linear and Gaussian, and the cost is quadratic. In this paper, we give an example illustrating that Witsenhausen's conclusion holds for our framework where a communication channel is present. We provide numerical results showing that, for a large range of the signal to noise ratio at the channel, nonlinear strategies strongly out-perform the best linear strategies.

In this paper, we show that the finite gain (FG) lp stabilization, with 1 ≤ p ≤ ∞, of a discrete-time, linear and time-invariant unstable plant is impossible by bit rate constrained feedback. In addition, we show that, under bit rate constrained feedback, weaker (local) versions of FG lp stability are also impossible. These facts are not obvious, since recent results have shown that input to state stabilization (ISS) is viable by bit-rate constrained control. We establish a comparison with existing work, leading to two conclusions: (1) in spite of ISS stability being attainable under bit rate constrained feedback, small changes in the amplitude of the external excitation may cause, in relative terms, a large increase in the amplitude of the state (2) FG lp stabilization requires logarithmic precision around zero, implying that even without bit-rate constraints FG lp stabilization is impossible in practice. Since our conclusions hold with no assumptions on the feedback structure, they cannot be derived from existing results. We adopt an information theoretic viewpoint, which also brings new insights into the problem of stabilization.

This paper addresses a fundamental limitation of performance for feedback systems, in the presence of a communication channel. The feedback loop comprises a discrete-time, linear and time-invariant plant, a channel, an encoder and a decoder which may also embody a controller. Measurements of the plant's output must be encoded for transmission over the channel. Information, at the other end of the channel, is decoded and used to generate a control signal, which is additively disturbed by a Gaussian and stationary stochastic process. We derive an inequality of the form L/spl I.bar/ /spl ges/ /spl Sigma/ max{0, log(|/spl lambda//sub i/(A)|)} - C/sub channel/, where L/spl I.bar/ is a measure of disturbance rejection, A is the open loop dynamic matrix and C/sub channel/ is the Shannon capacity of the channel. Our measure L/spl I.bar/ is non-positive and smaller L/spl I.bar/ indicates better rejection (attenuation), while L/spl I.bar/ = 0 signifies no rejection. Previous results show that C/sub channel/ > /spl Sigma/max{0, log(|/spl lambda//sub i/(A)|)} is a necessary condition for stability and now we show that the extra rate C/sub channel/ - /spl Sigma/max{0, log(|/spl lambda//sub i/(A)|)} determines a fundamental limitation for disturbance rejection. Additionally, we prove that, under stationarity assumptions, L/spl I.bar/ admits a log-sensitivity integral representation. We contrast our condition with Rode's integral formula and the water-bed effect. The new inequality shows explicitly how the capacity of the channel limits closed loop performance.

In this paper we tackle the problem of mode estimation in switching systems. From the theoretical point of view, our contribution is twofold: creating a framework that has a clear parallel with a communication paradigm and deriving an analysis of performance. In particular, our work is restricted to the class of systems that randomly switch among a finite alphabet of discrete-time finite impulse response linear operators, therein designated as modes. In our approach, the switching system is viewed as an encoder of the mode, which is interpreted as the message, while a probing signal establishes a random code. Accordingly, the estimator, which knows the code and uses noisy measurements of the output, is constructed as a decoder whose properties can be studied by means of a modification of Shannon's theory. Using a distance function, we define an uncertainty ball where the estimates are guaranteed to lie with probability arbitrarily close to 1. The radius of the uncertainty ball is directly related to the entropy rate of the switching process. It is shown that lower rates lead to smaller uncertainty. Such distance also reflects the informativity of the probing signal (code) and as such can be used as a guide on its choice. The estimator/decoder can be implemented using a low complexity algorithm.

We study the stabilizability of uncertain systems in the presence of finite capacity feedback. Motivated by the structure of communication networks, we consider a stochastic digital link that sends words whose size is governed by a random process. Such link is used to transmit state measurements between the plant and the controller. We derive necessary and sufficient conditions for the internal and the external stabilizability of the feedback loop. In accordance with previous publications, stabilizability of unstable plants is possible if and only if the link's average transmission rate is above a positive critical value. In our formulation the plant and the link can be stochastic. In addition, stability in the presence of uncertainty in the plant is analyzed using a small-gain argument. We show that the critical average transmission rate, for stabilizability, depends on the description of uncertainty and the statistical properties of the plant as well as the link.

This paper is about the mode estimation of randomly switching linear systems. We follow up on previous results, where we posed the problem in a way which is well fitted for a worst-case analysis. The aforementioned investigation relies on a divergence measure and several other quantities, such as the entropy rate of the mode switching as well as the power ratio between the system's excitation and the measurement noise. In this article we ascribe a natural interpretation to such divergence measure, while establishing a connection to rate-distortion theory. We also explore why and in what form estimation fidelity is affected by the entropy rate of the mode switching. Additionally, this analysis suggests that such entropy rate must be regarded as a fundamental quantity in the determination of the estimation fidelity.

Presents a deduction of the Kalman-filter H/sub 2//H/sub /spl infin// constrained optimization. The optimal filter is found by means of a minimax strategy applied to a suitable cost functional. The bound on the H/sub /spl infin// norm of the estimation error, is derived from the eigenvalue structure of the Hamiltonian matrices related with the optimization procedures.

In this paper, implementation and controller design issues are considered for a class of large-scale systems. The idea of block decoupling is used as the main tool to achieve great simplification and complexity reduction. Linear input flow constraints are also considered. Two application examples illustrate the main ideas on the paper.